Theoretical Perspectives on Biological Machines

Mauro L. Mugnai∗

Department of Chemistry,University of Texas, Austin,TX 78712

Changbong Hyeon†

Korea Institute for Advanced Study, Seoul 02455,Republic of Korea

Michael Hinczewski‡

Department of Physics,Case Western Reserve University,OH 44106

D. Thirumalai§

Department of Chemistry,University of Texas, Austin,TX 78712

Many biological functions are executed by molecular machines, which like man mademotors consume energy and convert it into mechanical work. Biological machines haveevolved to transport cargo, facilitate folding of proteins and RNA, remodel chromatinand replicate DNA. A common aspect of these machines is that their functions are drivenby fuel provided by hydrolysis of ATP or GTP, thus driving them out of equilibrium.It is a challenge to provide a general framework for understanding the functions of bi-ological machines, such as molecular motors (kinesin, dynein, and myosin), molecularchaperones, and helicases. Using these machines, whose structures have little resem-blance to one another, as prototypical examples, we describe a few general theoreticalmethods that have provided insights into their functions. Although the theories rely oncoarse-graining of these complex systems they have proven useful in not only accountingfor many in vitro experiments but also address questions such as how the trade-off be-tween precision, energetic costs and optimal performances are balanced. However, manycomplexities associated with biological machines will require one to go beyond currenttheoretical methods. We point out that simple point mutations in the enzyme coulddrastically alter functions, making the motors bi-directional or result in unexpected dis-eases or dramatically restrict the capacity of molecular chaperones to help proteins fold.These examples are reminders that while the search for principles of generality in biologyis intellectually stimulating, one also ought to keep in mind that molecular details mustbe accounted for to develop a deeper understanding of processes driven by biologicalmachines. Going beyond generic descriptions of in vitro behavior to making genuineunderstanding of in vivo functions will likely remain a major challenge for some timeto come. In this context, the combination of careful experiments and the use of physicsand physical chemistry principles will be useful in elucidating the rules governing theworkings of biological machines.

CONTENTS

I. Introduction 2

II. Structures 4A. Molecular motors 4B. Catalytic Cycle 5

III. Biological machines are active systems. 6

∗ maurolm83@gmail.com† hyeoncb@kias.re.kr‡ mhincz@gmail.com§ dave.thirumalai@gmail.com

IV. Stochastic Kinetic Models 7A. The Master Equation 8B. Thermodynamics 9C. Rate of Entropy Production 10D. Fluctuation Theorems 10

V. Molecular Motors – Models without Detachment 12A. Molecular Motors are Processive Enzymes 12B. Processive Motor Velocity 13C. Periodic Lattice Model 13

1. One-state Models 142. Multi-state, Uni-cycle Models 153. Multi-cycle Models 17

D. Additional remark on non-equilibrium nature ofmotors 18

E. Applications 191. Myosin V and Kinesin-1 19

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2. Dynein – an erratic motor 20

VI. Molecular Motors – Models with Detachment 20A. Velocity Distribution 21B. Alternate Models with Detachment 22

VII. Polymer Physics-based Approaches IncorporatingStructural Features Into Kinetic Theories 23

VIII. Simulations using Coarse-Grained Models 28

IX. Cost-precision Trade-off and Efficiency of MolecularMotors 30

1. Cost-precision trade-off and its physical bound ofmolecular motors 30

2. Transport Efficiency 31

X. Molecular chaperones 33

XI. Universal characteristics of Helicases 36

XII. Discussion 38A. Sometimes Details Matter 38B. Efficiency and optimality 38C. Specificity versus Promiscuity 39D. Biological complexity 39

XIII. A Final remark 39

Acknowledgments 40

References 40

I. INTRODUCTION

The opening sentence of a perspective by Bustamante(Bustamante et al., 2011) on the workings of nucleic acidtranslocases begins with the quote, “The operative indus-try of Nature is so prolific that machines will be eventu-ally found not only unknown to us but also unimaginableby our mind”, attributed to Marcello Malpighi, who isconsidered the founder of microscopic anatomy, histol-ogy, and embryology. This statement, made over threecenturies ago, is even more relevant today. It is a re-minder that mechanical forces must play a fundamentalrole in biology. In modern times this vast subject fallsunder the growing field of mechanobiology. The perva-sive role of mechanics in living systems controls motil-ity on all length scales, from motion of a single cell ona substrate and collections of cells to dynamics at themolecular level. Cooperative interactions between var-ious modules at the molecular level is thought to con-trol functions at the mesoscale. A number of complexdynamical processes such as transcription, translation,transport of vesicles and organelles, folding of proteinsand RNA, and chromosome segregation control the sus-tenance and growth of cells. At some level all these bio-logically important processes involve molecular machines,whose ability to perform their functions, often but alwayswith high efficiency, in a noisy crowded environment istruly remarkable. It is worth remembering that the abil-ity to execute a variety of functions distinguishes living

!1FIG. 1 Illustration of the complexity of transportation ofmelanosomes, which are vesicles containing melanin. Bothdynein and kinesin-2 compete for the same binding site ondynactin mediated by p150Glued. Depending on the func-tion (aggregation of melanosomes or their dispersion through-out the cell) one or the other wins. When melanosomes aredispersed in the cell they are transported by kinesin-2 andmyosin V whereas when they aggregate dynein moves thecargo. Figure extracted from Soldati and Schliwa (2006).

and abiotic systems. Because of functional demands inliving systems, which also includes adaptation to chang-ing environmental conditions, it is virtually impossibleto fully describe biology without evolutionary considera-tions. Although not the focus of our perspective, the roleof evolutionary constraints must also be integrated withphysical models in order to discover general principlesgoverning the functions of biological machines.

What are the characteristics of biological machines?First, there are many varieties of machines, all of whichshould be thought of as enzymes, which consume energyand perform mechanical work to carry out specific tasks.A few of the machines that we consider here are kinesin,myosin, and dynein, which are collectively referred to asmolecular motors (Belyy et al., 2014a; Block, 2007; Reck-Peterson et al., 2018; Sun and Goldman, 2011a; Valeand Milligan, 2000). These cytoskeletal motors trans-port cargo by walking, almost always unidirectionally,on filamentous actin and microtubules (MT). Under invivo conditions the motors cooperate or there could be atug-of-war in the process of transport (see Fig. 1) (Grosset al., 2002; Levi et al., 2006). To illustrate the diversityof motor-like functions and point out certain emerginggeneral principles, we also provide theoretical descrip-tions for the functions of molecular chaperones, whichassist in the folding of proteins and ribozymes (RNA en-zymes) that cannot do so spontaneously, and helicaseswith multiple functions that includes separation of dou-ble stranded (ds) DNA strands. Both molecular chap-erones and helicases bear no structural resemblance orsequence similarity to molecular motors. Nevertheless,

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we are convinced that by comparing the characteristicsof these seemingly unrelated machines, integrating the-ory and experiments, unifying themes and differences be-tween them could be elucidated. Second, all these motorsand others not covered here (for example polymerases,ribosomes, and packaging motors) are all multidomainproteins, whose architectures are spectacularly different.Despite the vastly different sequences, structures, andevolutionary origins it is indeed the case that all of thesemachines utilize some form of chemical energy (gener-ated by ATP or GTP hydrolysis) in order to amplify thesmall local conformational changes through their struc-tural linkages to facilitate large conformational changesfor functional purposes. Such conformational amplifica-tions are examples of remarkable allostery or action ata distance, and could also be couched in terms of in-formation transfer between structural subunits that arespatially well separated, which in some cases (dynein forexample (Bhabha et al., 2014)) can be as large as 25nm.

A large number of experiments have unveiled many ofthe details of how these machines move by convertingchemical energy into mechanical work (see for example(Hartman et al., 2011; Spudich and Sivaramakrishnan,2010) (De La Cruz and Ostap, 2004; Holzbaur and Gold-man, 2010; Sweeney and Houdusse, 2010). A remarkablenumber of experimental methods have been developedto address various aspects of biological machines. Theseinclude, but are not restricted to, ensemble experiments(stopped flow and fluorescent labeling) that provide themuch needed data on ATP hydrolysis and ADP releaserates, single molecule experiments that yield dwell timedistributions in molecular motors, processivity and veloc-ity as a function of external loads in motors and helicases.In addition, a combination of ensemble experiments anddetermination of structures using X-ray crystallographyand cryo-EM experiments has produced a vivid picture ofthe molecular basis of chaperone function. Of particularnote are optical trap experiments, which have been usedto obtain mean motor velocity as a function of a resis-tive force and ATP concentration in kinesin, myosin, anddynein, and the dependence of velocity and processivityin a number of helicases. No point would be served inreviewing the detailed results from these experiments asthere are many articles that the interested reader mightconsult.

Our focus here is to describe theoretical approachesthat are rooted in a number of areas in physics in order tounderstand principally outcomes of in vitro experiments.The theoretical approaches in these studies were devel-oped, and continue to be the focus of current research,in order to quantitatively explain the experimental ob-servations, and shed light on new puzzles that seem toarise with ever improving advances in experimental tech-niques. Perhaps, the most detailed view of how biologicalmachines operate might be obtained from molecular dy-namics simulations. Although atomically detailed molec-

FIG. 2 Significant physical timescales associated with myosinV dynamics, from the coarse-grained polymer theory modelof Hinczewski et al. (2013).

ular dynamics simulations have been performed to get amolecular picture of certain aspects of the functions ofmotors (Hwang et al., 2008, 2017) and other machines(see for example (Elber and West, 2010; Ma et al., 2000;Stan et al., 2005), the current limitations of such ap-proaches prevent them from making direct contact withexperiments. Improvements in the development of ac-curate energy functions (also known as force fields) andenhancement in computer capacity to enable simulationsfor long times will in the future bridge the gap betweenwhat is currently possible and what is needed for realisticdescription of molecular machines.

The actions of molecular machines, like all biologicalprocesses, are complicated, involving cooperative dynam-ics on multiple time scales. This is illustrated using thetypical time scales involved in a single step of myosin V(Fig. 2). In many problems in physics it is the hoped thatsimple models, that capture the essence of the problemcan be created, which be can be solved, in order to ob-tain insights into highly non-trivial systems. This strat-egy is hard to implement for biological machines becausethe interaction energies are highly heterogeneous (like inspin glasses), thus making it hard to construct a reason-able coarse-graining strategy. Nevertheless, in order tomake progress one has to devise tractable coarse-grainedmodels, which can be either simulated or solved ana-lytically (at least approximately). The efficacy of suchapproaches can then be assessed by direct comparisonswith experimental results, and their abilities to provideinsights into the mechanisms of their functions. Remark-ably, inspired by advances in experiments in the lastdecade, several theoretical models have been proposed,which have greatly contributed to our understanding ofmolecular machines. These developments have occurredin different contexts, which belie the underlying unifyingprinciples. To bring these issues to the fore, we pro-

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vide our perspectives on the application of these models,which should be thought of as coarse-grained networkmodels or their generalizations. Such models have beencreated for explaining not only the motility of motors,but also the functions of molecular chaperones, and he-licases. We focus on the applications of these theoreticalideas in order to account for the functions of these in-trinsically non-equilibrium systems. Collectively, theseapproaches show that, by examining in detail the work-ings of many machines, universal principles, both at theconceptual and practical levels, might emerge.

The literature on the functions of biological machinesis vast. Therefore, we restricted ourselves to only a fewtopics that focus on theoretical approaches, which aresufficiently general that they can be applied to an ar-ray of problems in the field. Rather than describe manyresults in detail, we walk the the reader through a se-lection of theoretical methods, which is necessarily bi-ased, so that she or he can access the literature read-ily. Before getting to the details of this review we willbe remiss if we did not point out one prescient mono-graph (Howard, 2001) and two forward looking reviews(Julicher et al., 1997; Kolomeisky and Fisher, 2007). Themonograph by Howard, written nearly twenty years ago,covers all aspects of cytoskeletal motors and is a land-mark in this field. It provides conceptual and practi-cal guidelines needed to understand the fundamentalsof motor mechanics. Questions of generality, such asdescribing movement of a generic motor either in iso-lation or as a collection, were addressed using princi-pally the Brownian ratchet model by Julicher, Adjari,and Prost (Julicher et al., 1997), whereas Kolomeiskyand Fisher (Kolomeisky and Fisher, 2007) summarizedthe development and practical applications of stochas-tic kinetic models. Here, our focus is on the more re-cent developments and applications of physical princi-ples that have started to provide a unified perspectivefor not only molecular motors but a large of class ofmachines with vastly different functions. These new in-sights have become possible by expanding the scope oftraditional stochastic chemical kinetics models for mo-tors, helicases, and molecular chaperones, incorporationof polymer physics concepts to account for the architec-ture of motors, and use of coarse-grained models in sim-ulating the stepping kinetics of motors and the dynamicsof large scale allosteric transitions.

II. STRUCTURES

The structures of the biological machines that we con-sider in our perspective (kinesin, myosin, dynein, theE. Coli. chaperonin machinery GroEL/GroES, and he-licases) are shown in Fig. 3. Inspection of the figuresshows that there are considerable variations in the ar-chitectures, although kinesin-1 (or conventional kinesin)

8 nm 20 nm

35 nm

Kinesin Dynein

Myosin V

a b c

14 nm

23 nm

GroEL-GroESd

Helicase

10 nm

e

FIG. 3 Schematic representations of the structures of six bio-logical machines. Molecular motors, like conventional kinesin(a), dynein (b), and myosin V (c), show great variations intheir structures and sizes. The motor heads in kinesin andmyosin V, shown in dark blue, bind directly to the micro-tubule and actin, respectively (not shown) but in dynein themicrotubule binding domain (light blue) is separated from themotor domain (hexameric ring structures) by nearly twentyfive nanometers. (d) Structure of the GroEL in the symmet-ric state (Protein Data Bank (PDB) code 4PKO (Fei et al.,2014)), which is the functional state in the presence of mis-folded proteins. The blue and aqua colors represent the sevenfold symmetric GroEL rings. The red and the pink correspondto the co-chaperonin GroES. (e) Structure of the DnaB he-licase with six subunits assembled as a ring. The figure wascreated using the coordinates in the PDB code 4ESV (It-sathitphaisarn et al., 2012). Structures (a)-(c) are adaptedfrom (Vale, 2003a).

and myosin V share some structural similarities.

A. Molecular motors

Let us first consider the three motors whose structuresare schematically shown in Fig. 3 (a-c). Although thereare differences between them, the parts list is roughlythe same. These motors are dimeric. The nucleotidebinding sites are in the two motor heads, which are con-nected through a mobile linker (the lever arm in myosin)to a tail domain involved in dimerization and cargo-binding. Changes due to nucleotide binding and hydrol-ysis in the motor heads result in conformational changesin the linker that propel the motor on the cytoskele-tal filaments. For the motors to be processive, whichmeans they take multiple steps before disengaging fromthe track, one head has to be bound till the detached headrebinds to a site on the track. This involves communi-cation between the heads and is referred to as gating,the origin of which is still not fully understood at themolecular level.

Kinesin: There are at least forty five members be-longing to the kinesin superfamily in mouse and humangenomes (Hirokawa et al., 2009). They are all micro-

5

tubule (MT) bound motors, which walk unidirectionallytowards the plus end of the MT (for example kinesin-1)or to the minus end (Ncd motor). Kinesin-1 takes pre-cisely 8.2 nm steps, which corresponds to the distancebetween two adjacent α/β tubulin dimers, which are thebuilding blocks of the MT. The size of kinesin-1 is a fewnanometers whereas the length of the stalk is in the rangeof 30-40 nm. The kinesin-1 velocity depends on the con-centration of ATP, saturating at high values. The motormoves towards the plus end of the MT with maximal ve-locity of approximately 800 nm/s (Visscher et al., 1999),and is capable of resisting forces on the order of 7 pN(Carter and Cross, 2005; Visscher et al., 1999) (kBT =4.1 pN·nm where kB is the Boltzmann constant and T isthe temperature).

Myosin: The number of genes encoding for myosin mo-tors in roughly thirty five (Sellers, 2000). The superfam-ily of the actin bound myosins are divided into fifteenclasses (Hartman and Spudich, 2012; Sellers, 2000). Withthe exception of myosin VI all other members of this fam-ily walk towards the plus end of actin. The structure ofmyosin V (Fig. 3) shows that the motor heads are con-nected to the lever arms, which are made up of six IQmotifs. The lever arm is ≈ 36 nm-long stiff unit (thepersistence length exceeds 100 nm), whose size is com-mensurate with half the helical repeat length of F-actin.The maximal velocity of myosin V, whose motor head islarger than kinesin-1, is roughly 500 nm/s (Baker et al.,2004) with a stall force between 2-3 pN (Kad et al., 2008;Mehta et al., 1999; Uemura et al., 2004a; Veigel et al.,2002).

Dynein: Cytoplasmic dynein, discovered over fiftyyears ago (Gibbons and Rowe, 1965) and pictured in Fig.3b, walks somewhat erratically with a broad step-sizedistribution (DeWitt et al., 2012; Reck-Peterson et al.,2006a) on the MT towards the minus end. The struc-tural features of dynein are different when compared withother cytoskeletal motors (see Fig. 3). First, the motorhead belongs to the class of AAA+ family, which meansdynein must have evolved from a different lineage com-pared with myosin and kinesin. Second, other AAA+enzymes, such as bacterial chaperonin GroEL (Fig.3d)and protein degradation machines, are oligomeric assem-blies. In contrast, the hexameric ring that constitutesthe motor domain assembles from a single polypeptidechain! Third, the size of the dynein motor head is sig-nificantly larger than those of kinesin and myosin. Thelength of the motor head of dynein along its longest axisis about 25 nm, which is in contrast to the diameter ofkinesin, which is only about 5 nm. Finally, althoughthere are six nucleotide binding sites in dynein, hydroly-sis in only two (perhaps three) are relevant for its motil-ity. The velocity of dynein at 1 mM ATP is roughly ∼100nm/s (Reck-Peterson et al., 2006b) with a stall force thatis ≈ 7pN (Gennerich et al., 2007; Toba et al., 2006).

Chaperonins: The beautiful and unusual structurewith seven fold symmetry of the E. Coli. chaperoninmachinery, consisting of a complex between GroEL andGroES, resembles an American football (Fig. 3d). TheGroEL/GroES machine helps in the folding of recalci-trant proteins that do not fold spontaneously. The struc-ture in Fig. 3d (reported in (Fei et al., 2014)) is the func-tional state of this stochastic machine that is one of thepopulated states during the catalytic cycle of GroEL inthe presence of misfolded substrate proteins (SPs). Thereare two chambers in which the SPs could be sequestered.The volume of each of the chambers in the structure inFig. 3e is about 185,000 A3, which is about twice the vol-ume in the relaxed structure in the absence of nucleotidesand GroES, the co-chaperonin. The spectacular changein the chamber volume that occurs during the catalyticcycle, with accompanying alterations in the chemical na-ture of the cavity interior, changing from hydrophobicto polar, is the mechanism of annealing action of thismachine (Todd et al., 1996).

Helicases: Helicases, which are involved in all aspectsof nucleic acid metabolism (Lohman, 1992; Lohman andBjornson, 1996), function by coupling nucleoside triphos-phate (NTP) hydrolysis, to either translocate on singlestrand nucleic acids (ssNAs) or unwind double-stranded(ds) DNA. Depending on their sequences they are clas-sified into six super families (SFs). The structural di-versity of helicases can be appreciated by noticing thatsequences classified under the SF1 and SF2 families arenon-ring forming whereas those in SF3-SF6 form ringstructures. The hexametric structure (Itsathitphaisarnet al., 2012) of the replicative helices (DnaB) from bacte-ria, which does belong to the AAA+ family, is shown inFig. 3d. However, unlike GroEL (see Fig. 3d) DnaB hasthe expected crystallographically allowed six fold symme-try. How the NTP chemistry is coupled to translocationand unwinding remains an outstanding unsolved theoret-ical problem.

B. Catalytic Cycle

All machines undergo a catalytic cycle, not unlikeman-made motors, in which fuel, usually in the form ofATP, bound to a nucleotide binding site (or sites) is hy-drolyzed. These events trigger conformational changesthat produce motion, which in motors and helicases re-sults in stepping on the polar tracks or translocationon single stranded nucleic acids. In chaperones the nu-cleotide chemistry is linked to conformational changes,which in turn perform work on the protein or ribozymeto be folded. The link between ATPase cycle and func-tion is somewhat different in GroEL, which we describebelow. The catalytic cycle for myosin V in the simplestform, which suffices for our purposes, is reproduced inFig. 4. There are four crucial steps (see Fig. 4). In

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FIG. 4 A simple representation of the catalytic cycle ofmyosin V describing the stepping of the trailing head towardsthe plus end of actin, which is the dominant pathway in theabsence of resisting force. The main text describes the de-tails. As described later in this perspective there are fourother pathways that have to be accounted for in order toproduce a quantitative theory of the stepping kinetics of thismotor. The figure is reproduced from (Vale, 2003b)

the initial state, ATP binds to the trailing head (TH),with ADP in the (LH). The premature release of ADPfrom the LH is slowed by rearward tension, which is anexample of gating. Upon ATP binding to the TH theinteraction with F-actin is weakened, resulting in its de-tachment from F-actin. During the diffusive search bythe TH for the forward binding site, ATP is hydrolyzed,producing ADP and the inorganic phosphate, Pi. In thisstate, the TH binds to F-actin after which Pi is releasedfrom the new LH followed by ADP release from the TH,and the cycle continues till the processive run ends. Ofcourse, this simple description is incomplete because therates for nucleotide binding vary depending on the nu-cleotide concentration. Nevertheless, this simple reactioncycle, whose main features hold for all myosin motors, issufficient to nearly quantitatively characterize many ex-perimental observables (see below). The catalytic cyclefor kinesin is similar except that the interaction betweenthe MT and kinesin is weakest if the motor head con-tains ADP. All other nucleotide states (the no nucleotideapo, ATP bound state, the state with ADP and Pi) bindstrongly to the MT.

III. BIOLOGICAL MACHINES ARE ACTIVE SYSTEMS.

Biological machines are non-equilibrium systems thatare driven by non-conservative forces require constantsupply of energy. Therefore, it is not surprising thatdetailed balance (DB) relation and the fluctuation-dissipation theorem (FDT) are violated (Battle et al.,

2016; Gladrow et al., 2016). These are the key featurescharacterizing the out-of-equilibrium nature of biologi-cal machines driven by non-conservative forces. Conse-quently, molecular machines can be thought of as activesystems, and hence a note on diffusive motion is war-ranted. Chemical free energy released upon hydrolysis ofATP or GTP is the driving force for directed motion ofmolecular motors. However, if the energy source is notexplicitly modeled in the description of the associateddynamics, molecular motors could be regarded as self-propelled active particles. To our knowledge such an ap-proach has not been pursued to calculate experimentallymeasurable quantities, such as the force-velocity relationor the distribution of run lengths of motors.

The dynamics of active particles is fundamentally dif-ferent from passive particles at equilibrium or movingunder the influence of a conservative external field. Forexample, the mean drift velocity of a spherical colloidalparticle with a drag coefficient γ, and charge q subject toa constant electric field E is VD = qE/γ. This relationis readily obtained by balancing the Stokes force (γVD)and the force exerted by the field (qE). The diffusionconstant is D = kBT/γ, which is the Stokes-Einstein re-lation. Hence, the distribution of the particle position isgiven by the probability density,

P (x, t) ∼ exp

[− (x− x0 − VDt)

2

4Dt

]. (1)

Starting from an initial position x0, the particle moves onaverage at velocity VD and the distribution of positionsspreads with time as 〈(δx)2〉 = 2(kBT/γ)t. Of particularnote is that D is defined independently of the particlevelocity VD, so that the magnitude of D is not altered byincreasing the field strength. Conversely, the velocity ofthe particle does not depend on the ambient temperatureT . Thus, VD and D are mutually independent of oneanother.

In stark contrast, for an active particle, whose motionis powered by the internal fuel, the diffusivity is no longerindependent from the driving velocity. The effectivediffusion constant, Deff, defined as limt→∞〈(δx)2〉/2t,depends on a set of non-thermal parameters and vio-lates the FDT (Liu et al., 2011; Tailleur and Cates,2008). For transport motors exhibiting one-dimensionalmovement along the cytoskeletal filament, both V =limt→∞ d〈x(t)〉/dt and D = (1/2) limt→∞ d〈δx(t)2〉/dtcan be measured directly using a given time trace of amotor, which can be measured using single molecule opti-cal tweezer experiments. Indeed, such measurements onkinesin-1 (Visscher et al., 1999) were used to obtain thevelocity (V ) and diffusion coefficient (D) from the globalanalysis of the stepping trajectories. These values wereused (Visscher et al., 1999) to estimate the randomnessparameter r = 2D/d0V where d0(≈ 8.2 nm) is the stepsize of kinesin-1. When the diffusion constant D is cal-culated from the randomness parameter with the knowl-

7

edge of V and d0, and D is compared with V measuredunder the same conditions, it can be shown that D in-creases monotonically with V (Hwang and Hyeon, 2017)in contrast to passive diffusion. This result is a con-sequence of the enzyme catalytic turnover (Hwang andHyeon, 2017). As already discussed the effective diffu-sivity D for kinesin-1, which is an active particle whosedynamics is fueled by ATP hydrolysis free energy, doesnot obey the FDT. This illustration shows that these en-ergy consuming enzymes operate out of equilibrium.

IV. STOCHASTIC KINETIC MODELS

In this section we discuss the development of stochas-tic kinetic models (SKMs) as a means of describing thefunction of different types of molecular machines. Asstated above and shown explicitly in Fig. 4 for myosinV, all molecular machines go through cycles during whichthey hydrolyze ATP (or GTP) and perform a some func-tion. Bulk-kinetics, single-molecule experiments, andstructural studies have shown that the intermediates ex-plored by the molecular machines during the cycle havedistinguishable structural or kinetic features and haveprovided the rate for going from one state to the next.The temporal resolution in these studies is on the orderof milliseconds [although new techniques enable the ex-ploration of time-scales as fast as ≈ 55µs (Isojima et al.,2016)]. It follows that the dynamics of the molecularmachine can be described as a discrete-state (the dis-tinguishable intermediates should be experimentally ob-served), continuous-time Markov model – a frameworkthat falls under the rubric of the “master equation.” Inthis model, the cycle of a molecular machine is reduced toa network of states connected by edges that identify theavailable transitions between the intermediates. Thus,by solving the dynamics associated with an appropriatenetwork it is possible, in principle, to account for exper-imental observables in terms of the underlying networkparameters.

SKMs have a number of appealing features: (i) theyhave a direct connection with the biochemistry associ-ated with the molecular machines; (ii) the rates thatconstitute the model are measurable; (iii) it is some-times possible to find an analytical solution, and (iv)more complex networks may be solved numerically; (v)finally, SKMs can be directly related with thermodynam-ics, which makes the model instructive and appeals to theinterests, and intuition of a vast scientific community.

Nevertheless, SKMs also have some shortcomings: firstand foremost, they often have a large number of param-eters. For instance, a simple model for a molecular ma-chine performing mechanical work against a fixed loadand described by a single-cycle network with N interme-diate states has 4N − 2 independent parameters, whichhave to be determined by fitting to appropriate experi-

ments. However, typically the number of observables arevery few, making matching the predictions of the SKMto experiments is difficult. Furthermore, it is often dif-ficult to interpret the physical meaning of the extractedparameters in terms of the underlying motor architectureand the underlying biochemical cycle. As a consequence,it is necessary to strike a balance between the “minimal-ity” of a model, which reduces the risk of over-fittingand increases the generality of SKMs and the predic-tive power, and the “comprehensiveness” of the networkconsidered. This comes at some risk; for instance, onemay be tempted to neglect certain transitions that are“fast” compared to others, and therefore are not expectedto contribute to the overall phenomenology of the ma-chine. In addition, it may seem reasonable to ignore off-pathway, slow and rare transitions. On the other hand, inorder to understand the function of molecular machines,experimentalists often probe their response to changes inthe environment (for instance modifying the ATP con-centration), or by applying mechanical perturbations andstudying the response of the motor. As the environmentor perturbations change, the nature of the “rate-limiting”step as well as the likelihood of alternative stepping path-ways might change. Therefore, a simplified descriptionthat ignores certain intermediate states might only workfor a limited set of experimental conditions, thereby re-ducing the number of measurements that can be used totrain the parameters or to falsify the predictions. Finally,we note en passant that the determination of an appro-priate catalytic cycle is predicated on a few experimentalobservables only, and it may not be complete enough forall the states that a machine might sample during itsfunction.

Another issue with SKMs is that structural informa-tion is incorporated in the model only in a very approxi-mate way, normally by introducing a parameter that al-lows the rate to depend on the load applied accordingto the Bell model (Bell, 1978). This reduces the capa-bility of the model to incorporate the wealth of experi-mental structural information, and decreases the predic-tive power of the proposed paradigm. However, in somecases a tractable analytical theory that incorporates leverarm structural information into a kinetic model is possi-ble (Hinczewski et al., 2013), as discussed in more detailbelow. Despite these limitations, the framework underly-ing SKMs not only provides a convenient way to analyzeexperiments, but also has been used to address concep-tual questions related to the efficiency of biological ma-chines. For these reasons, the theory underlying SKMsis an integral part of describing the function of molecularmotors.

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A. The Master Equation

We assume that the system is described by N distin-guishable intermediate states. The probability of being instate i at time t is given by pi(t), and the time evolutionof this probability is governed by the master equation,

dpi(t)

dt=∑j

pj(t)wji −∑j

pi(t)wij , (2)

where wij ≥ 0 is the rate for the transition i → j,subject to the constraint

∑i pi(t) = 1. Because of mi-

croscopic reversibility, if wij 6= 0 then wji 6= 0. Ast → ∞, the probabilities pi(t) become independent oftime. This stationary solution of the master equationdescribes an equilibrium system (peqi ) if detailed balanceholds, that is if across all the edges of the network thenet flux is zero: peqi wij − peqj wji = ∆Jij = 0. However,dpi/dt = 0 is also satisfied by the less stringent relation∑j(p

ssi wij − pssj wji) =

∑j ∆Jij = 0. Under these con-

ditions the system could be in a non-equilibrium steadystate (NESS). An isolated system is expected to reachan equilibrium state, whereas coupling with an externalenergy source enables the creation and persistence of aNESS.

From a mathematical standpoint, given a kinetic net-work described by the master equation, the stationaryprobabilities pssi (or peqi ) can be obtained using graph-theoretical arguments. The details may be found in theworks of Hill (Hill, 1966, 2005a; Hill and Chen, 1975).Briefly, one may construct the set of partial diagramssuch that all the states are visited but no cycles areformed (see Fig. 5b). The stationary probability of be-ing in state i is proportional to the sum of all the par-tial graphs oriented in such a way that the fluxes con-verge towards state i, or σi. The normalization factor isΣ =

∑i σi, so that the stationary probability is,

pssi =σiΣ. (3)

The stationary flux along one edge may be written asa sum of contributions from all the cycle fluxes of thesystem. Each cycle can be completed in two directions:one, counterclockwise, labeled as “+”; the other, clock-wise, termed “-”. The cycle fluxes in the “+” and “-”directions are given by the following relationships (Hill,2005a),

Jν± =ΣνΠν±

Σ, (4)

where Σν is a combination of rates that are specific forcycle ν, Πν± is the products of the rates of the cycle per-formed in the “+” (Πν+) or “-” (Πν−) direction, and thedenominator Σ is defined above (see Fig. 5c). It followsthat,

Jν+

Jν−=

Πν+

Πν−= eβAν , (5)

FIG. 5 Example of kinetic network. The network has 6 statesand 7 edges [see panel (a)]. Panel (b): the partial graphs ori-ented in order to extract the stationary probability of state1. Panel (c): There are three cycle fluxes, F (upper threefigures), B (bottom three), and D (central figure). From thefigure we extract the values of Σν in Eq. 4 for the three cy-cles. These are given by the product of the rates flowing to-wards the cycle [see T. L. Hill (Hill, 2005a) for details]: ΣF =w32w45 +w43w32 +w34w45; ΣB = w12w65 +w61w12 +w16w65;ΣD = 1.

where Aν (or βAν) has been termed the action func-tional (Lebowitz and Spohn, 1999) or affinity (Schnaken-berg, 1976) of the cycle ν. Note that the net direc-tion of completion of cycle ν is given by the sign of∆Jν = Jν+ − Jν− . It is easy to show that (Hill andSimmons, 1976),

∆JνAν ≥ 0. (6)

The above equality holds only when Jν,+ = Jν,−. Inother words, ∆Jν and Aν have the same sign, whichmeans that the value of the affinity dictates the direc-tion of the cycle (Hill and Simmons, 1976).

In order to identify the physical interpretation ofthese mathematical identities we need to connect therates with thermodynamic quantities, such as energy,entropy, and the chemical potential. Following threedifferent approaches we show that the affinity (Eq. 5)is related to the entropy produced during a cycle. Thefirst two strategies involve the Shannon entropy (Liepeltand Lipowsky, 2007b; Lipowsky and Liepelt, 2008;Schnakenberg, 1976) and fluctuation theorems (Crooks,1998; Seifert, 2005a,b, 2012), and they will be dis-cussed without specifically referring to the experimentalhallmarks for motor velocity described in the previoussections. The last method, which is based on estab-lishing a connection between pseudo-first-order rateconstants and the chemical potential (Hill, 2005a), willbe developed in the context of molecular motors.

9

B. Thermodynamics

Molecular machines are enzymes (E) that catalyze thechemical transformation of substrate molecules (S → P ),which in general can be described by the Michaelis-Menten kinetics (E + S ES → E + P ). Of course,the process is reversible, and the enzyme also catalyzesthe reverse reaction – the transformation of P into S.In the case of molecular machines, the substrate is ATP,and the products of ATP hydrolysis are ADP and or-thophosphate (Pi). (Some molecular machines catalyzethe hydrolysis of GTP; mutatis mutandis our considera-tions do not change.)

We imagine the following experimental setup: a molec-ular machine operates in a solution containing ATP,ADP, and Pi. The chemical potentials of these threespecies are given by (Hill, 2005b),

µATP = µ0ATP + kBT ln [ATP]

µADP = µ0ADP + kBT ln [ADP]

µPi = µ0Pi

+ kBT ln [Pi]

X = −∆µhyd = µATP − µADP − µPi =

= kBT lnKeq[ATP]

[ADP][Pi],

(7)

where µc (µ0c) is the (standard) chemical potential of

species c, [c] is the concentration in solution, and Keq ≈4.9 · 105 M (Howard, 2001) is the equilibrium constantfor ATP hydrolysis. (Note that we should use activitiesac in Eq. 7 instead of concentrations [c]; throughout thereview we will assume that ac ≈ [c].) At equilibrium,([ADP][Pi]/[ATP])Eq = Keq, and ∆µhyd = 0. On theother hand, if [ATP]Keq � [ADP][Pi], then ∆µhyd < 0,and the hydrolysis of ATP is a spontaneous reaction.We exclude un-catalyzed hydrolysis/synthesis of ATP, asit occurs over time-scales beyond our interest (Hulett,1970); as a consequence, in the absence of molecular ma-chines the concentrations [ATP], [ADP], and [Pi] are con-stant. The presence of the molecular machine does notalter the equilibrium features of ATP hydrolysis (namely,Keq), however it accelerates the rate of ATP synthe-sis/hydrolysis. Let the initial concentrations of ATP,ADP, and Pi make ATP hydrolysis a spontaneous re-action. After each catalytic cycle the enzyme attainsthe same conformation that it had at the start. How-ever, the solution conditions have changed as a substrate(ATP) has been consumed and products (ADP and Pi)have been created. Thus, in the presence of the molecularmachine the solution approaches the equilibrium ratio ofconcentrations of ATP, ADP, and Pi, and monitoring thedynamics under these conditions corresponds to study-ing the time-dependent relaxation towards equilibrium.

At equilibrium, the rate of hydrolyzing and synthesizingATP is the same.

However, this is not what happens in a cell, wherethe concentration of ATP is maintained under homeo-static control (Wang et al., 2017) far away from equilib-rium; typical concentrations are [ATP] ≈ 1mM, [ADP] ≈10µM, [Pi] ≈ 1mM (Howard, 2001), resulting in a chem-ical potential X = −∆µhyd of ≈ 25 (Howard, 2001),which makes ATP hydrolysis a spontaneous reaction(∆µhyd < 0). This means that the enzymatic cycle of themolecular machine will be driven in the direction thatconsumes ATP, and ATP hydrolysis provides the driv-ing force that enables the molecular machine to performwork. Without accounting explicitly for the whole cellu-lar apparatus involved in dictating and maintaining theset-point ATP level, in the simplest theoretical model thesystem is assumed to be in contact with some devices re-ferred to as “chemostats” capable of maintaining the ini-tial concentrations of ATP, ADP, and Pi (Lipowsky andLiepelt, 2008; Lipowsky et al., 2009; Qian and Beard,2005; Seifert, 2011b). After each cycle the chemostatremoves the products from solution and replenishes thesubstrates, thereby ensuring that after each enzymaticcycle the initial condition is reset, which enables the cre-ation of a NESS for t→∞.

The system is also in contact with a thermal reser-voir with which it exchanges heat (Lipowsky and Liepelt,2008; Lipowsky et al., 2009; Seifert, 2011b). We assumethat the thermal reservoir operates “quasi-statically” onthe time-scale of the heat exchange, and the combinationof the system and the thermal reservoir is energeticallyisolated. Because after a cycle the motor and the solu-tion have not changed, the entropy produced is equal tothe ratio between the heat absorbed by the thermal reser-voir, Q, and the constant temperature T . We exclude pVwork from our formulation, but we include the possibilitythat the molecular machine performs work against a fixedload, so that for a forward (backward) step Wmech = fd0

(Wmech = −fd0), where the choice of the sign impliesthat a positive force opposes forward movement. Notethat the load f is assumed to be clamped, so at everystep, regardless of the position of the motor, the cycle isrepeated under identical conditions.

The energy X = −∆µhyd > 0 consumed during a cycleis partitioned into work performed (−Wmech) and heatreleased (−Q), that is,

∆µhyd +Q+Wmech = 0, (8)

implying that the total change of entropy of the systemplus environment over one cycle is given by,

∆S =Q

T=−∆µhyd − fd0

T≥ 0. (9)

(For backward steps, the sign in front of fd0 is the oppo-site). The inequality is due to the second law of thermo-dynamics, which states that we should to find an increase

10

in entropy during the cycle because the combination ofsystem and environment is isolated. Eq. (9) leads to theobservation that the maximum amount of force that themotor can resist is given by,

fmax =−∆µhyd

d0. (10)

A number of authors have discussed variations ofthis thermodynamic framework. In particular, Seifertelucidated the contribution of the thermodynamiccontribution of chemostats (Seifert, 2011b); Qian andBeard discussed a network of reactions in which somemetabolites are clamped, while other are introduced ata constant rate (Qian and Beard, 2005).

C. Rate of Entropy Production

Let the entropy be,

S = −kB

∑i

pi ln pi, (11)

where the sum extends over all the available conforma-tions of the system. Let us take a time derivative ofS, and after imposing the conservation of probability(∑i dpi/dt = 0) condition, and plugging in the master

equation we get,

dS

dt= kB

∑i

∑j

(piwij − pjwji) ln pi. (12)

Note that in a NESS,∑j(piwij − pjwji) =

∑j Jij =

0, and as a consequence dS/dt = 0, as expected. Wefirst symmetrize the result with respect to the indexes iand j, and then add and subtract 1/2kB

∑i

∑j(piwij −

pjwji) ln(wij/wji), which leads to (Lipowsky and Liepelt,2008; Schnakenberg, 1976),

dS

dt= kB

1

2

∑i

∑j

Jij lnpiwijpjwji

− kB1

2

∑i

∑j

Jij lnwijwji

.

(13)The first of the two terms on the r.h.s. of Eq. 13 is alwaysnon-negative, and it has been identified as the rate of en-tropy production, diS/dt (Lipowsky and Liepelt, 2008;Schnakenberg, 1976). We note parenthetically that someauthors (Hill and Simmons, 1976) do not include the fac-tor 1/2 as they interpret the summation to be carried outover the edges of the network, and not over the states.Because in a NESS dS/dt = 0, it follows that the secondterm of the r.h.s of Eq. 13 is the rate of entropy out-flux,and,

diS

dt= kB

1

2

∑i

∑j

Jij lnpiwijpjwji

=

= kB1

2

∑i

∑j

Jij lnwijwji≥ 0.

(14)

Consider now a cyclic network of N states; the onlytransitions allowed from state i are i i + 1 and i i− 1. (The cyclic nature of the network implies periodicboundary conditions, and so state i = 0 is the same asstate i = N). In a NESS, the condition

∑j Jij = 0

becomes ∆Ji,i+1 = ∆Ji−1,i, so that the net flux acrossall the edges is the same and equal to the net cycle flux,∆J = J+ − J−. It follows that,

diS

dt= kB∆J ln

N−1∏i=0

wi,i+1

wi+1,i=

∆JA

T(15)

where wi,i+1 and wi+1,i are the forward and backward

rates, respectively. The term A = kBT ln∏N−1i=0

wi,i+1

wi+1,iis

the affinity (see Eq. 5), which Hill refers to as a ther-modynamic force driving the system out of equilibriumand imposing a NESS (Hill, 2005a). Liepelt and Lipowkyidentified the thermodynamic force with the entropy pro-duction (Liepelt and Lipowsky, 2007b); if the rate ofcompleting a cycle is ∆J > 0 (which implies that the cy-cle is preferentially completed in the forward, or “+”, di-rection), one can identify the entropy ∆S produced overone cycle as,

T∆S =T

∆J

diS

dt= kBT ln

N−1∏i=0

wi,i+1

wi+1,i= A ≥ 0. (16)

The argument can be extended to the case in which thekinetic network is characterized by multiple cycles (Lie-pelt and Lipowsky, 2007b); if ∆J−1

ν is the average timefor completing a cycle ν, and ∆Sν is the entropy pro-duced with that cycle, we can write,

diS

dt=∑ν

Jν∆Sν ≥ 0. (17)

Equation 6 indicates that each term in the summation isnon-negative (Hill and Simmons, 1976).

To summarize, in this section we showed how theaction functional (or affinity) can be identified with theentropy produced. We now present a different argumentbased upon fluctuation theorems leading to the sameconclusions.

D. Fluctuation Theorems

Let the probability density of taking a forward path ofn steps be PF (i0, t0; i1, t1; · · · ; in, τ), in which the systemis in state i0 at t0, transitions to state i1 at t1 and so onuntil it reaches state in at time τ (see Fig. 6a). We usethe subscript “F” to indicate that the transitions arecompleted forward in time. Using the Markov property,we can rewrite this joint probability as (Crooks, 1998),

PF (i0, t0; i1, t1; · · · ; in, τ) =

= pi0(t0)P (i1, t1|i0, t0) · · ·P (in, τ |in−1, tn−1).(18)

11

FIG. 6 Path in state space executed forward (a) and back-ward (b) in time. The starting point is the dot, waiting timesat a fixed state are in black, transitions are in blue.

Assuming that the system is in a steady state at t =t0, we replace pi0 = pssi0 . The conditional probabili-

ties are given by P (j, tj |i, ti) = wije−Wi(tj−ti), where

Wi =∑j wij . The conditional probability P (j, tj |i, ti)

is the product of two probabilities. The first is wij/Wi,the probability of making a transition to state j amongall the other possibilities when starting in state i. Thesecond is Wie

−Wi(tj−ti), the probability that this transi-tion happens after time (tj − ti). The reason for the Wi

in the exponent is that the mean waiting time to transi-tion from state i to any other state is W−1

i , independentof state j. This independence is a well-known propertyof escape times in Markov networks, which may be com-puted through well-known methods (Van Kampen, 2007).Therefore, the joint probability density becomes (Seifert,2012),

PF (i0, t0; i1, t1; · · · ; in, τ) =

= N−1pssi0

n−1∏α=0

wiα,iα+1e−Wiα (tα+1−tα),

(19)

where tn = τ and N (t0, τ) is a normalization factor toensure that

∑i0,i1,·,in

∫ τt0dt1∫ τt1dt2 · · ·

∫ τtn−1

dtnPF = 1.

The probability of completing the time-reversed path(Fig. 6b) is,

PB(in, τ , in−1, tn−1; · · · ; i0, t0) =

= N−1pssin

N−1∏α=0

wiα+1,iαe−Wiα (tα−tα+1),

(20)

in which we started in state in at time tn = τ − tn andretrace the forward path until we reach state i0 at timet0 = τ − t0. Note that the initial probability is now

pssin and because each path could be performed forward

or backward in time, the normalization factor N = N .Following Seifert (Seifert, 2005b), we define R to be thelogarithm of the ratio between the probability for for-ward and backward paths, and for the sake of simplicitywe consider only cyclical pathways, in which i0 = in.Therefore,

R = lnPFPB

= ln

n−1∏α=0

wiα,iα+1

wiα+1,iα

. (21)

Note that the time-dependent terms in the joint proba-bilities cancel out, so we neglect them in the following.(In alternative, we may consider the time at which thesejumps occur to be fixed.) Here, R has a structure anal-ogous to that of the affinities introduced in the previoussections. More precisely, given a set of states visited wecan rewrite,

R =∑ν

nνβAν , (22)

where nν is the net number of times the cycle ν hasbeen completed during the n-step path associated withR. In order to justify this expression, note that everytime that during the path a cycle is not completed, thepath re-traces it self and those branch-like excursions donot contribute to R. If we average over all paths thatcomplete a cycle of length n, we find the following se-quence of identities (Seifert, 2005b),

〈e−R〉 =∑

paths

PF e−R =

∑paths

PB = 1. (23)

Using Jensen’s inequality, 〈e−R〉 ≥ e−〈R〉, leads to,

〈R〉 ≥ 0. (24)

From Eq. 22 we obtain,

〈R〉 =∑ν

〈nν〉βAν ≥ 0. (25)

We now introduce an Arrhenius-type relationship be-tween the rates, wij/wji = e−β∆Fij , where ∆Fij =Fj − Fi is the free energy difference between state i andstate j. The free energy difference accounts for threecontributions: (i) the intrinsic free energy of a state maychange; (ii) the motor may bind/release ATP, ADP, andPi; (iii) the motor may perform work against an exter-nal load. After a cycle, the motor returns to the initialstate, and as a consequence the intrinsic free energy doesnot carry any contribution to the cycle. The hydrolysisof ATP contributed with the release of energy equal to−∆µhyd, and the work performed per each displacementof size d0 against a load f is equal to −fd0. It followsthat (Lipowsky and Liepelt, 2008),∏

|i,j〉∈ν

kijkji

= e−nν,ATPβ∆µh−βflν = e∆SνkB , (26)

12

where, following the notation of Lipowsky and Lie-pelt (Lipowsky and Liepelt, 2008), |i, j〉 corresponds to adirected edge from state i to j, and the sum is extendedover all the directed edges belonging to cycle ν. Here,nν,ATP is the net number of ATP hydrolyzed during cy-cle ν, and lν is the net displacement along the track (amultiple of d0). The last identity follows from Eq. 9, andfrom Eq. 25 gives,

kB〈R〉 =∑ν

〈nν〉∆Sν ≥ 0. (27)

V. MOLECULAR MOTORS – MODELS WITHOUTDETACHMENT

In this section we develop models of molecular motorswith increasing complexity, from one-state models,to uni-cyclic models, ending with multi-cycle kineticnetworks. We show that experimental evidence andthermodynamic insights suggest that the introductionof multiple cycles creates models that are physicallymore sensible. We discuss more in detail the efficiencyof the motors, and we conclude the section by describingmodels that account for the detachment of the molecularmotor from the track.

A. Molecular Motors are Processive Enzymes

The catalytic cycle of kinesin, dynein, and myosin pro-ceeds via multiple steps, in which ATP binding, hydroly-sis, and release of ADP and Pi lead to substantial changesin the motor conformation. The presence of the cy-toskeletal filament, CF (F-actin in the case of myosin,microtubules for dynein and kinesin), increases the rateof ATP hydrolysis (De la Cruz et al., 2001, 1999; Hack-ney, 1988; Homma and Ikebe, 2005; Ori-McKenney et al.,2010). When the structural changes are properly recti-fied under the specially designed interactions with theappropriate CFs, the conformational fluctuations in Mare transduced to a predominantly uni-directional motionalong the track, and generates mechanical forces againstan external load. The free energy source necessary toperform this movement (or work) is provided by the hy-drolysis of ATP.

Enzymes that work in conjunction with substrates (asis the case for Ms and CFs) are said to be “processive”if they perform multiple catalytic cycles without fullydisengaging from the CFs (Schnitzer and Block, 1995).Processive motors move long distances along the CFs byhydrolyzing one ATP molecule per step without dissoci-ating from the CF. In general (although with some fasci-nating exceptions (Inoue et al., 2002; Post et al., 2002))processive movement requires the cooperation of mul-tiple enzymes: some members of the ensemble proceed

forward while the others hold tight onto the CF. Fromhere on, we focus our discussion exclusively on dimericprocessive motors made of two identical enzymes of thesame family, which we refer to as “heads.” With thiswe may identify the leading head (LH) as the one thatis in front of the dimeric complex, while the other isthe trailing head (TH). We reserve the word “motor”for the motile construct, which is a dimer in the case ofmyosin V (Mehta et al., 1999), VI (Rock et al., 2001),and X (Sun et al., 2010), conventional kinesin (kinesin-1) (Howard et al., 1989), and cytoplasmic dynein (Reck-Peterson et al., 2006b).

In the well-accepted hand-over-hand stepping mecha-nism (Okten et al., 2004; Sun et al., 2010; Toba et al.,2006; Yildiz et al., 2003, 2004a,b), the TH of the motordetaches from the filament, overcomes the bound head,and advances the motor by reaching a forward targetbinding site (TBS). After the step is completed, the roleof the two heads is reversed, and the process is repeatedidentically, until the motor detaches from the CF. A fewkey ingredients are necessary for processive, hand-over-hand movement to occur. First, in order to prevent pre-mature end of the processive run, the head that remainsbound to the track during a step should not dissociatefrom the CF at least until the detached head reaches theTBS. Second, the two heads should go through their cat-alytic cycles out-of-phase (Block, 2007), so that the THis more likely to detach from CF than the LH. This re-quirement necessarily implies that the two heads oughtto communicate with one another through action at adistance, which may be viewed as a complex form of al-lostery (Thirumalai et al., 2019). Third, during a stepa combination of conformational transitions in the twoheads biases the diffusive motion of the free head towardsthe TBS unless a large resistive force is applied.

Meeting the first two conditions requires specific fea-tures of the hydrolytic cycles that the two heads mustundergo. For instance, if the rate-limiting step occursin a CF-bound conformation, the chances of prematurelyending the processive run are diminished. In addition,the interaction between the two heads is believed to bekey in ensuring that the TH steps first, and that theLH is unlikely to detach during a step. This is possi-ble if the interplay between the two heads slows down(or “gates”) some of the steps of the LH catalytic cycle,or accelerates them in the trailing head (also referred toas “gating”) (Block, 2007; Hancock, 2016; Sweeney andHoudusse, 2010). The structural bias towards the TBS isprovided by a conformational transition of the CF-boundmotor that projects forward the stepping head. Forthe three classes of motors mentioned before, the leverarm swing in myosin, the neck-linker docking in kinesin,and the bent→straight transition of the linker domain indynein provide the requisite forward bias. It is remark-able that these structural transitions, induced by ATPbinding and hydrolysis, which occur on relatively small

13

length scales are amplified by responses in other struc-tural elements. The terms “power stroke” (Dominguezet al., 1998) and “Brownian ratchet” (Rice et al., 1999)have been used to schematize this forward bias, depend-ing on whether the emphasis is placed on the mechanistic(order→order) or stochastic (disorder→order) nature ofthe conformational transitions.

In the simplest cases, the step-size of a motor iscommensurate with the filament repeat – for motorsmoving along the MT, d0 ≈ 8.2 nm, whereas d0 ≈ 36 nmfor actin-based motors (myosins). However, not allmotors walk precisely by following this rule: myosinVI and dynein, for instance, display a broad step-sizedistribution, in which hand-over-hand steps are inter-twined with inchworm-like movements and frequentbackward steps (Nishikawa et al., 2010; Reck-Petersonet al., 2006a). Of course, the variability in the stepsizes is also a consequence of the architecture of themotor. Thus, both the structural design of the motorand the coupling to catalytic cycles determine not onlythe stepping kinetics but also the precision of the stepsizes.

B. Processive Motor Velocity

A variety of single-molecule techniques are able toinform on the motile properties of molecular motors.The motor or the track may be labeled with a fluores-cent (Nishikawa et al., 2010; Yildiz et al., 2003) or re-fractive (Isojima et al., 2016; Mickolajczyk et al., 2015)probe. Monitoring the time-dependent changes in thelocation of the such probes enables the determinationof the velocity of the motor as a function of nucleotideconcentration. Alternatively, optical trapping techniquesmay be used to follow the displacement of the motor orthe filament, and to investigate the motor response tothe external load (Block et al., 1990; Finer et al., 1994;Howard et al., 1989). Much of our current knowledgeabout kinesin comes from single molecule experiments(Block et al., 1990; Howard et al., 1989), which were de-veloped approximately at the time of the discovery of ki-nesin (Brady, 1985; Vale et al., 1985). It has been shownthat each step of kinesin motors along MTs is tightlycoupled with hydrolysis of one ATP molecule, that is,a single event of ATP hydrolysis by kinesin leads to ∼8nm step along MTs (Schnitzer and Block, 1997). Thus,within the Michaelis Menten (MM) scheme, the velocityof kinesin is associated with enzyme turnover rate is,

V = d0kcat[ATP]

KM + [ATP], (28)

where d0 = 8 nm is the step size of the kinesin, kcat

and KM are the rate of catalysis and the Michaelis con-stant, respectively. For kinesin-1, the typical experimen-

tal value for the Michaelis constant is KM & 50 µM,and for [ATP] & 1 mM the velocity of kinesin is satu-rated to its maximum value Vmax = d0kcat ≈ 8 nm/10ms = 0.8 µm/s (Kolomeisky and Fisher, 2007; Visscheret al., 1999). Similar single-molecule studies have beenconducted on other motors, showing that the Michealis-Menten scheme provides a good paradigm to rationalizethe ATP concentration dependence of velocity. However,the parameters of the fit depend on the motor: for myosinV it was found that Vmax ≈ 0.55µm/s, with KM ≈ 163µM (Baker et al., 2004), whereas the slower myosin VI hasVmax ≈ 0.16µm/s and KM ≈ 274µM [fit of Eq. 28 todata from Elting et al. (2011)].

Experiments employing optical tweezers have shownthat external loads affect the ATP chemistry in the cat-alytic site as well as the motility of the motor (Blocket al., 2003; Oguchi et al., 2008; Veigel et al., 2005; Viss-cher et al., 1999). Thus, the resulting force-velocity-ATPrelationship has been a landmark measurement, which isused as a constraint to decipher the mechanism under-lying kinesin motility, and to construct the appropriateSKMs. Incorporation of the effect of load into MM modelwas suggested by making the parameters kcat and KM

force-dependent (Schnitzer et al., 2000),

V (F, [ATP]) =d0kcat(F )[ATP]

kcat(F )/kb(F ) + [ATP](29)

where kcat(F ) = kocat/(pcat + (1 − pcat)eFδcat/kBT ) and

kb(F ) = kob/(pb + (1 − pb)eFδb/kBT ). The fit of the

F -dependent velocity data of motor using Eq.29 isreasonable as long as the magnitude of load (F ) is small(Schnitzer et al., 2000). However, the conventional MMmodel has an intrinsic drawback when the external loadapproaches the stall force value (≈ 7 pN for kinesin)and becomes greater than the stall force. An increaseof load ought to induce backward stepping (V < 0); yetno modification of Eq.28 produces a negative velocity!One possible fix for the failure of the MM model atlarge load is to permit in the model a reverse reactioncurrent, which is realized by rendering every elementaryreaction step within the kinesin cycle “reversible.” Wediscuss later how such models have been constructed inthe context of Markov jump processes.

C. Periodic Lattice Model

In a molecular motor (i) binding, release, and chemi-cal transformations of ATP, ADP and Pi from the mo-tor head domain, and (ii) the advancement along thetrack occur over much slower times (ms - s) comparedto the fluctuations of molecular conformations (ns - µs).The time-scale separation enables the construction of aMarkov jump processes in which the state of the system

14

is defined by two coordinates: (i) one of the N intermedi-ates defining the chemical state of the substrate, and (ii)the location l along the track. The probability densityof the i-th state (i = 1, 2, . . . , N) at filament site l obeysthe master equation,

dPi(l, t)

dt=

l+1∑l′=l−1

∑j

[wj,l→i,l′Pj(l′, t)− wi,l→j,l′Pi(l, t)]

(30)where

∑l

∑i Pi(l, t) = 1, and wj,l→i,l′ is the rate for go-

ing from state j in location l to state i on F site l′ = l, l±1.Under the assumption that the track is infinite and pe-riodic, that is that the rates do not depend on l, onemay sum the r.h.s and the l.h.s of the master equationand eliminate l. As a result, we obtain the master equa-tion in Eq. 2, and we can use the theoretical frameworkpresented before.

Periodic one-dimensional models, which were orig-inally suggested by Derrida (Derrida, 1983) in orderto study the mean velocity and diffusion constant ofrandom systems, have been widely adopted in describingthe dynamics of molecular motors. In particular,Fisher and Kolomeisky (Fisher and Kolomeisky, 1999a,2001; Kolomeisky and Fisher, 2007) have pioneeredthis approach and specifically used the reversible ki-netic model to describe the motility of kinesin-1 andmyosin V. Numerous other studies have used vari-ants of these models to discuss the thermodynamicfeatures of stochastic motors. We review the mainsuccesses and shortcomings of these models in the con-text of molecular motors as their complexity is increased.

1. One-state Models

In the simplest possible model, the motor is defined bya single (N = 1) chemical state at each site along thetrack (Fig. 7a). The rate of forward stepping, u, dependson the concentration of ATP. Microscopic reversibilitydemands that the backward transition is possible as well,which is then nominally associated with ATP synthesis.The rates of forward and backward stepping are modeledas pseudo-first order processes,

u = u∗[ATP], w = w∗[ADP][Pi], (31)

leading to u/w = u∗/w∗ · [ATP]/([ADP][Pi]). In equi-librium, the probability of going forward is identical tothe probability of backward stepping, which implies that(u/w)Eq = 1. From Eq. 7, at equilibrium ∆µhyd = 0and ([ADP][Pi]/[ATP])Eq = Keq. As a consequence, us-ing Eq. 7 we obtain that the ratio between forward and

12

⋯ 34

N − 1

N⇌u1 = u*1 [ATP]

w2

⇌ u2w3

⇌u3w4

⇌wNuN−1

uN

w1 = w*1 [ADP][Pi]⇌

1l 1l+1⋯⋯1l−1 ⇌⇌w = w*[ADP][Pi]

u = u*[ATP]

(a)

(b)FIG. 7 Example of simple kinetic networks. Panel (a): thenetwork has only one statte, N = 1. The forward rate is givenby u = u∗[ATP], the backward rate is w = w∗[ADP][Pi]. Bothforward and backward rates are pseudo-first-order, with unitsmM−1s−1 and mM−2s−1, respectively. Panel (b): multi-statemodel. Here, the rates u1 and w1 are arbitrarily chosen todepend on [ATP] and [ADP][Pi], respectively. Panel (c): acycle corresponding to the multi-state model.

backward rates is given by,

(a)u

w=u∗

w∗[ATP]

[ADP][Pi]= eβX

(b)u∗

w∗= Keq,

(32)

where we recall that X = −∆µhyd. The average, station-ary velocity is given by the step-size times the differencebetween the forward and backward rates,

v = d0(u− w) = d0w(eβX − 1

). (33)

The crucial insight from this expression pertains tothe interplay between the non-equilibrium steady-state(NESS) homeostatically maintained by the cell andmolecular motor function: in the cytosol X > 0, thusv > 0; at equilibrium, X = ∆µhyd = 0, implying thatv = 0.

In the absence of applied load, the motor moves with-out performing any work. In this case, the entire free en-ergy extracted from the hydrolysis of ATP is dissipatedinto heat, Q = −∆µhyd. From Eq. 15, the rate of heatdissipation per step is given by,

TdiS

dt= Q = kBT (u− v) ln

u

v= (u− v)(−∆µhyd) ≥ 0.

(34)Clearly, Q ≥ 0, and the equality holds in equilibrium,where u = v. If instead the motor is subjected to aforce, f , opposing its movement, at each step the motor

15

performs mechanical workW = fd0. In this case, the freeenergy released by ATP hydrolysis is partitioned betweenwork and heat, Q = −∆µhyd−W . In order to account forthe presence of applied load, the rates may be modifiedusing the Bell model (Fisher and Kolomeisky, 1999b),

(a) u(F ) = u∗[ATP]e−βθfd0 ,

(b) w(F ) = w∗[ADP][Pi]eβ(1−θ)fd0 .

(c)u

w= eβ(X−fd0)

(35)

Here, the coefficient θ indicates how the load is parti-tioned between the forward and backward steps, and itidentifies the position of the transition state along thedirection of motion (Fisher and Kolomeisky, 2001). Forθ = 0, the forward step is unaffected by force, whereasbackward steps are insensitive to load if θ = 1. These twoextreme cases have been referred to as “power stroke”and “ratchet”, respectively (Howard, 2006). In a “powerstroke”, the chemical step (crossing the transition state)occurs before the movement forward of the motor. Incontrast, in the case of a “ratchet” the motor fluctuatesbetween the sites l and l + 1 until it is captured in theforward site by ATP hydrolysis. In the presence of back-ward load the velocity becomes,

v = d0w(0)eβ(1−θ)fd0

(eβX−βfd0 − 1

). (36)

It is clear that the motor stalls under a resistive load of,

fstall =X

d0= −∆µhyd

d0, (37)

at which the mean motor velocity becomes v = 0. It isclear that fstall = fmax for this model. Using Eq. 37 to-gether with the value of ∆µhyd in the cell and the averagestep-size of a motor, one can predict the value of fstall: forkinesin fstall ≈ 12.5pN, for myosin V fstall ≈ 2.8nm. Inthe case of myosin V, fstall approximates the value of thestall force measured experimentally; for kinesin, the fmax

is about twice as large as the stall force. Note also thataccording to Eq. 37, Fs depends on [ATP]; a few experi-ments have suggested that this is not the case for a vari-ety of motors (Carter and Cross, 2005; Gennerich et al.,2007; Nishiyama et al., 2002; Toba et al., 2006; Uemuraet al., 2004a), although other studies have observed suchadependence (Visscher et al., 1999). The independenceof Fs on [ATP] might be reasonable because each headhas only one binding pocket for ATP, and hence [ATP]should not determine Fs. The relationship v(f,∆µ) al-lows one also to compute the power output of the motor,defined as P = vF , and the motor efficiency,

η(f,X) =fd0

X, (38)

that is the ratio of the work produced over the energyreleased by the hydrolysis of ATP. Note that the power

is zero when f = 0 and at the stall force (where v = 0),thus for some force 0 < f∗ < Fs it reaches a maximumP ∗. The force-velocity curve, power output, efficiency,and efficiency at maximum power (η∗) have been used tocompare ratchet-like motors (θ = 1) with power-stroke-driven (θ = 0). Given a value of the driving force X,power strokes result in larger velocity at fixed F , anddisplay higher efficiencies and exert more power at a givenvelocity (Wagoner and Dill, 2016). Furthermore, bothη∗ and P ∗ at fixed X increase as θ → 0 (Schmiedl andSeifert, 2008). Finally, from Eq. 15 we write,

TdiS

dt= Q = (u− w) ln

u

w= (u− w)(−∆µhyd − fd0),

(39)where we used Eq. 35c and Eq. 36. Note that, again,Q ≥ 0, and the equality holds at equilibrium (u = w).

One-state models are appealing because of theirsimplicity; for instance, they have only two fitting pa-rameters with straightforward physical interpretations:w∗ sets the time-scale (see Eq. 33), and θ establishes thelocation of the transition state during a step. However,these models fail in reproducing the dependence of thevelocity on ATP concentration observed in experiments(Eq. 28). According to Eq. 33 v grows without saturat-ing as [ATP] increases, which is not physical. In orderto solve this problem, one must use kinetic models withN > 1.

2. Multi-state, Uni-cycle Models

Figure 7b shows a kinetic model for molecular motorswith N > 1 intermediates. Assuming that the filament isperiodic, we can drop the label l and construct an equiv-alent uni-cycle in Fig. 7c, where the rate uN is associatedwith a forward step, and w1 with a backward displace-ment. From Eq. 4 we obtain the following relationships,

(a) J+ =

∏Ni=1 ui

Σ({u,w}) , J− =

∏Ni=1 wi

Σ({u,w}) ,

(b)

∏Ni=1 ui∏Nj=1 wj

=J+

J−= eβX ,

(c)u∗1∏Ni>1 ui

w∗1∏Nj>1 wj

= Keq,

(40)

where J+ and J− are the fluxes to complete a cyclein the clockwise and counterclockwise direction, respec-tively, and Σ({u,w}) is a function of all the rates (seeEq. 3). The connection with thermodynamics is providedby Eq. 40b, which is equivalent to Eq. 32a; Eq. 40c holdsbecause in equilibrium X = 0 (see Eq. 32b for the N = 1case). For the case of uni-cyclic network models, the net

16

steady state flux (∆J = J+ − J−) along the cycle is ob-tained by calculating ∆J = wi,i+1p

ssi −wi+1,ip

ssi+1 at any

edge between the two neighboring chemical states alongthe cycle. Therefore, the velocity, which is obtained as

the flux across edge NuN�w1

1, is given by,

v = d0∆J = d0J−(eβX − 1) = d0

∏Ni=1 wi

Σ({u,w}) (eβX − 1),

(41)The similarity between Eq. 33 and Eq. 41 is clear: bothof the expressions relate the velocity to the exponentialof the free energy released upon the hydrolysis of ATP,and in both cases at equilibrium the motor does notmove. However, the expression in Eq. 41 saturates atlarge [ATP], in agreement with experiments (see Eq. 28).

Compared to the case of a single chemical state, newload distribution factors are necessary for each transitionrate, i.e.

ui(f) = ui(0)e−βθ+i fd, wi(f) = wi(0)eβθ

−i fd. (42)

The distribution factors obey the relationship∑Ni=1(θ+

i +θ−i ) = 1 (Fisher and Kolomeisky, 1999b). It follows that,∏N−1

i=0 ui(f)∏N−1i=0 wi(f)

=J+(f)

J−(f)= eβX−βfd0 , (43)

and given ∆J(f) = J+(f) − J−(f), the force-velocityrelation is,

v(f) = d0∆J(f) = d0

∏Ni=1 wi(f)

Σ(f)

(eβX−βfd0 − 1

). (44)

Although v(F ) is a complicated function of all the pa-rameters (Fisher and Kolomeisky, 1999b; Wagoner andDill, 2016), the expression for a theoretical estimation ofthe stall force is the same as the one found for N = 1,given by Eq. 37, and it is again equal to fmax in Eq. 10.

In principle, one could design kinetic networks with anarbitrary number of intermediates. However, because thenumber of free parameters becomes 4N − 2, it is desir-able to establish the minimal N that enables an accu-rate description of the experimental data as accuratelyas possible. Let D = limt→∞

ddt (〈x(t)2〉 − 〈x(t)〉2) be

the dispersion, and v = limt→∞ddt 〈x〉 is the velocity; the

randomness parameter for a quantity x is defined as,

r =2D

d0v. (45)

It can be shown that N ≥ 1/r (Kolomeisky and Fisher,2007; Koza, 2002), and therefore r−1 sets a lower boundto N that is needed to account for measurements. Notethat r is a function of ATP concentration and externalload.

Following Eq. 15, the rate of heat released during acycle (Hill, 2005a; Qian, 2007; Qian and Beard, 2005;Toyabe et al., 2010; Zimmermann and Seifert, 2012) is,

Q = ∆JkBT lnJ+

J−= ∆J(−∆µhydr−fd0) = E+W ≥ 0,

(46)where E is the rate of free energy expended per cy-cle. Note that in uni-cyclic network models, W =−∆J(f)Fd0 = 0 either at f = 0 or if the motor is subjectto an opposing stall force (f = Fs), where ∆J(Fs) = 0;thus the work production (W ) is a non-monotonic func-tion of f , whereas E and Q decrease monotonically withF .

Although the conventional N-state unicyclic mod-els (Fisher and Kolomeisky, 1999a, 2001; Kolomeisky andFisher, 2003) appear successful in describing the motil-ity and thermodynamics of molecular motors, unicyclicmodels encounter two serious problems, especially whenthe molecular motor is stalled or starts taking backwardsteps at large hindering loads (Astumian and Bier, 1996;Hyeon et al., 2009; Liepelt and Lipowsky, 2007a). First,the backward step in the unicyclic network, by construc-tion, is produced by a reversal of the forward cycle, whichimplies that the backward step is always realized via thesynthesis of ATP from ADP and Pi. This is the casefor rotary ATP-synthases, which function as transducersof electrochemical potential into the synthesis of ATP inthe mitochondria (Alberts et al., 2008), and dependingon the sign of ∆µhyd and on environmental conditions(e.g. the presence and strength of a proton gradientand applied load), these spectacular motors can reversetheir function (Itoh et al., 2004; Turina et al., 2003). Al-though ATP hydrolysis is reversible for linear molecularmotors (Bagshaw and Trentham, 1973; Hackney, 2005),back-stepping has not been associated with ATP synthe-sis. Rather they are associated with ATP-independent“slippage” or coupled to fuel consumption just as for-ward stepping (Carter and Cross, 2005; Clancy et al.,2011; Gebhardt et al., 2006; Ikezaki et al., 2013). Thisrequires an update of the kinetic network.

In addition, the unicyclic network models lead toQ = 0 under stall conditions (Eq. 46), which contradictsphysical reality. For example, an idling car still burnsfuel and dissipates heat (Q 6= 0)! There certainly existfundamental differences between molecular world andmacroscopic counterpart in that the former is subject toa large degree of fluctuations, which permit the reverseprocess of ATP hydrolysis (i.e., synthesis) or negativeheat dissipation for a sub-ensemble of the entire realiza-tions. Yet, the mean of entropy production Q/T is stillbound to be positive as demanded by thermodynamics,and the probability of a local violation of this principlebecomes vanishingly small as the system size grows. Toameliorate the aforementioned physically problematicinterpretation, associated with the backward step

17

DRAFTFig. 1. Analysis of kinesin-1 based on the 6-state double-cycle kinetic model. A. Schematics of the 6-state double-cycle kinetic network for hand-over-hand dynamics ofkinesin-1, where T , D, and „ denote ATP-, ADP-bound, and apo state, respectively. Through binding [(1)æ(2)] and hydrolysis of ATP [(2)æ(5)], and release of ADP [(5)æ(6)],kinesin moves forward in the F -cycle [(1) æ (2) æ (5) æ (6) æ (1)], where it takes a backstep in the B-cycle [(4) æ (5) æ (2) æ (3) æ (4)]. The arrows in thefigure depict the direction of reaction currents. In both cycles, each chemical step is reversible and the transition rate from the state (i) to (j) is given by kij . B. Reactioncurrent JF , JB , and J as a function of load. The three cartoons illustrate the amount of current along the F and B cycles at each value of load. f < 0 and f > 0 correspondto the assisting and hindering load, respectively. C. The ratio between the forward and backward fluxes JF/JB as function of f at fixed [ATP]. The stall forces, determined atJF/JB = 1 (dashed line), are narrowly distributed between f = 6 ≠ 8 pN. (D–H) V , D, Q, W , E as a function of f and [ATP]. The white dashed lines depict the locus of[ATP]-dependent stall force, and the dashed lines in magenta indicate the condition of [ATP] = 2 mM. I. Dependences of E, Q, W on f at [ATP] = 2 mM. The star symbol in Dindicates the cellular condition of f ¥ 1 pN and [ATP] ¥ 1 mM, which will be discussed later.

be obtained using a proper kinetic network model that candelineate the dynamical characteristics of the system (14). Ouranalyses on motors show that Q is a complex function of f and[ATP] and is sensitive to a subtle variation in motor structure.Transport and rotory motors studied here are semi-optimizedin terms of Q under cellular condition.

Results and Discussion

Chemical driving force, steady state current, and heat dis-sipation of double-cycle kinetic network for kinesin-1. Thechemomechanical dynamics of a molecular motor can bemapped on an appropriate kinetic network, which allows usto express the transport properties of the motor, V and D, interms of a set of rate constants {kij}, where kij is associatedwith the transition from the i-th to j-th state (7, 15, 16) (seeSI text). The formal expressions of V and D in terms of{kij(f, [ATP])} are used to fit simultaneously the experimen-tal data of V and D obtained under varying conditions of ATPand f (14). The set of rate constants {kij} decided from thefit further allows us to calculate the a�nity (A, net drivingforce), and reaction current (J) of the network, and hence theheat dissipation from the motor, Q (see below).

For kinesin-1, we considered the 6-state network (17), con-sisting of two cycles, F and B (Fig. 1A). This minimal ki-netic scheme accommodates 4 di�erent kinetic paths: ATP-hydrolysis induced forward/backward step and ATP-synthesisinduced forward/backward step. Although the conventional(N=4)-state unicyclic kinetic model confers a similar resultwith the 6-state double-cycle network model at small f (com-pare Figs.1 and Fig.S3), it is led to a physically problematicinterpretation when the molecular motor is stalled and startstaking backsteps at large hindering load. The backstep inunicyclic network, by construction, is produced by a reversalof the forward cycle (18), which implies that the backstep is

always realized via the synthesis of ATP from ADP and Pi.More importantly, in calculating Q from kinetic network, theunicyclic network results in Q = 0 under the stall condition,which is not compatible with the physical reality that an idlingcar still burns fuel and dissipates heat (Q ”= 0). To build amore physically sensible model that considers the possibility ofATP-induced (fuel-burning) backstep, we extend the unicyclicnetwork into a multi-cyclic one which takes into account anATP-consuming stall, i.e., a futile cycle (17–20).

In the double-cycle model, kinesin-1 predominantly movesforward through F-cycle under small hindering (f > 0) orassisting load (f < 0), whereas it takes a backstep throughthe B-cycle under large hindering load. In principle, thecurrent within the F-cycle, JF , itself is decomposed into theforward (J+

F ) and backward current (J≠F ), and JF = J+

F ≠J≠F > 0 is satisfied in NESS. Although a backstep could be

realized through an ATP synthesis (21), corresponding to J≠F ,

experimental data (18, 22, 23) suggest that such backstepcurrent (ATP synthesis induced backstep, J≠

F ) is negligible incomparison with J+

B (ATP hydrolysis induced backstep).The dynamics realized in the double-cycle network can be

illuminated in terms of variation of JF and JB with increasingf (Fig.1B). Without load (f = 0), kinesin-1 predominantlymoves forward (JF ∫ JB). This imbalance diminishes withincreasing f . At stall conditions, the two reaction currentsare balanced (JF = JB), so that the net current J associatedwith the mechanical stepping defined between the states (2)and (5) vanishes (J = JF ≠ JB = 0), but nonvanishing currentdue to chemistry still remains along the cycle of æ (2) æ(3) æ (4) æ (5) æ (6) æ (1) æ (2) æ. A further increaseof f beyond the stall force renders JF < JB, augmenting thelikelihood of backstep.

For a given set of rate constants, it is straightforward tocalculate the rates of heat dissipation (Q), work production(W ), and total energy supply (E). The total heat generated

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXX

125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186

187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248

Hwang et al.

f < fstall f > fstall

DRAFTFig. 1. Analysis of kinesin-1 based on the 6-state double cycle kinetic model. A. Schematics of the 6-state double cycle kinetic network for hand-over-hand dynamicsof kinesin-1, where T , D, and „ denote ATP-, ADP-bound, and apo state, respectively. Through binding ((1)æ(2)) and hydrolysis of ATP ((2)æ(5)), and release of ADP((5)æ(6)), kinesin moves forward in the F -cycle ((1) æ (2) æ (5) æ (6) æ (1)), where it takes a backstep in the B-cycle ((4) æ (5) æ (2) æ (3) æ (4)). Thearrows in the figure depict the direction of reaction currents. In both cycles, each chemical step is reversible and the transition rate from the state (i) to (j) is given by kij . B.Reaction current JF , JB , and J as a function of load. The three cartoons illustrate the amount of current established in F and B cycles at each value of load. C. The ratiobetween the forward and backward fluxes JF/JB as function of f at fixed [ATP]. The stall forces, determined at JF/JB = 1 (dashed line), are narrowly distributed betweenf = 6 ≠ 8 pN over the range of [ATP] = 10 µM ≠ 10 mM. (D–H) V , D, Q, W , E as a function of f and [ATP]. The white dashed line in each panel depicts the locus of[ATP]-dependent stall force, and the dashed lines in magenta indicate the condition of [ATP] = 2 mM. I. Dependences of E, Q, W on f at [ATP] = 2 mM.

network model that can capture the dynamical characteristicsof the system (11). Our analysis on motors will show that Qis a complex function of f and [ATP], and that Q(f, [ATP])is sensitive to subtle variations in motor structure.

Results and Discussion

Chemical driving force, steady state current, and heat dis-sipation of double cycle kinetic network for kinesin-1. Thechemomechanical dynamics of a molecular motor can bemapped on an appropriate kinetic network where nonzerosteady state current created by ATP hydrolysis free energy(≠�µhyd > 0) acts as the driving force. Given a proper net-work model, the transport properties of the motor, V andD, can be expressed in terms of a set of rate constants {kij},where kij is associated with the transition from the i-th to j-thstate (4, 12, 13) (see SI text). The formal expressions of Vand D in terms of {kij(f, [ATP])} are used to fit experimentaldata of V and D obtained under varying conditions of ATPand f simultaneously (11). The set of rate constants {kij}decided from the fit further allows us to calculate the a�nity(A, chemical driving force) and reaction current (J) of thenetwork, and hence the heat dissipation from the motor, Q.

For kinesin-1, we considered a 6-state network (14), con-sisting of two cycles, F and B (Fig. 1A). This minimal ki-netic scheme allows us to accommodate 4 di�erent kineticpaths: ATP-hydrolysis induced forward/backward step andATP-synthesis induced forward/backward step. Although theconventional (N=4)-state unicyclic kinetic scheme confers asimilar result with the 6-state double cycle network schemewhen f is small, it is led to a physically problematic interpre-tation when the molecular motor is stalled and starts takingbacksteps at high load. The backstep in unicyclic reactionschemes, by construction, is produced by a reversal of theforward cycle (15), which implies that the backstep is alwaysrealized via the synthesis of ATP from ADP and Pi. More

importantly, in calculating Q from kinetic network, the uni-cyclic reaction scheme is led to Q = 0 under the stall condition,which is not compatible with the physical reality that an idlingcar still burns fuel and dissipates heat (Q ”= 0). To build amore physically sensible model that considers the possibilityof ATP-induced (fuel-burning) backstep, it is indispensableto extend the unicyclic reaction scheme into a multi-cyclicscheme which takes into account an ATP-consuming stall, i.e.,a futile cycle (14–17).

In the 6-state double cycle model, kinesin-1 predominantlymoves forward through F-cycle under small hindering or as-sisting load, whereas it takes a backstep through the B-cycle.In principle, the current within the F-cycle, JF , itself is de-composed into the forward (J+

F ) and backward current (J≠F ),

and JF = J+F ≠ J≠

F > 0 is satisfied at NESS. Although abackstep could be realized through an ATP synthesis (18)and a reversal of the normal current direction of F cycle, i.e.,J≠F , experimental data (15, 19, 20) suggest that such backstep

current (ATP synthesis induced backstep, J≠F ) is smaller than

J+B (ATP hydrolysis induced backstep).

The dynamics realized in the double cycle kinetic networkcan be illuminated in terms of variation of JF and JB withincreasing f (Fig.1B). Without load (f = 0), kinesin-1 predom-inantly moves forward (JF ∫ JB). This imbalance diminisheswith increasing f . At stall conditions, the two reaction cur-rents are balanced (JF = JB), so that the net current Jassociated with the mechanical stepping defined between thestates (2) and (5) vanishes (J = JF ≠ JB = 0), but nonvan-ishing current due to chemistry still remains along the cycleof æ (2) æ (3) æ (4) æ (5) æ (6) æ (1) æ (2) æ. Afurther increase of f beyond the stall force renders JF < JB,augmenting the likelihood of backsteps.

For a given set of rate constants, it is straightforward tocalculate the rates of heat dissipation (Q), work production(W ), and total energy supply (E) for varying conditions of

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXX

125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186

187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248

Hwang et al.

[T]

[T]

hyd

hyd

D

D

DT

T

DT

T

DRAFTFig. 1. Analysis of kinesin-1 based on the 6-state double cycle kinetic model. A. Schematics of the 6-state double cycle kinetic network for hand-over-hand dynamics ofkinesin-1, where T , D, and „ denote ATP-, ADP-bound, and apo state, respectively. Through binding [(1)æ(2)] and hydrolysis of ATP [(2)æ(5)], and release of ADP [(5)æ(6)],kinesin moves forward in the F -cycle [(1) æ (2) æ (5) æ (6) æ (1)], where it takes a backstep in the B-cycle [(4) æ (5) æ (2) æ (3) æ (4)]. The arrows in thefigure depict the direction of reaction currents. In both cycles, each chemical step is reversible and the transition rate from the state (i) to (j) is given by kij . B. Reactioncurrent JF , JB , and J as a function of load. The three cartoons illustrate the amount of current along the F and B cycles at each value of load. f < 0 and f > 0 correspondto the assisting and hindering load, respectively. C. The ratio between the forward and backward fluxes JF/JB as function of f at fixed [ATP]. The stall forces, determined atJF/JB = 1 (dashed line), are narrowly distributed between f = 6 ≠ 8 pN. (D–H) V , D, Q, W , E as a function of f and [ATP]. The white dashed lines depict the locus of[ATP]-dependent stall force, and the dashed lines in magenta indicate the condition of [ATP] = 2 mM. I. Dependences of E, Q, W on f at [ATP] = 2 mM. The star symbol in Dindicates the cellular condition of f ¥ 1 pN and [ATP] ¥ 1 mM, which will be discussed later.

motor e�ciency, if any, that Q is minimized. To evaluate Q,one should know Q, D, and V of the system (see Eq.2), whichcan be gained by building a proper kinetic network model thatcan delineate the dynamical characteristics of the system (11).Our analysis on motors will show that Q is a complex functionof f and [ATP], and that Q(f, [ATP]) is sensitive to a subtlevariation in motor structure.

Results and Discussion

Chemical driving force, steady state current, and heat dis-sipation of double cycle kinetic network for kinesin-1. Thechemomechanical dynamics of a molecular motor can bemapped on an appropriate kinetic network where nonzerosteady state current created by ATP hydrolysis free energy(≠�µhyd > 0) acts as the driving force. Given a proper net-work model, the transport properties of the motor, V andD, can be expressed in terms of a set of rate constants {kij},where kij is associated with the transition from the i-th to j-thstate (4, 12, 13) (see SI text). The formal expressions of V andD in terms of {kij(f, [ATP])} are used to fit simultaneouslythe experimental data of V and D obtained under varyingconditions of ATP and f (11). The set of rate constants {kij}decided from the fit further allows us to calculate the a�nity(A, net driving force), and reaction current (J) of the network,and hence the heat dissipation from the motor, Q.

For kinesin-1, we considered a 6-state network (14), con-sisting of two cycles, F and B (Fig. 1A). This minimal ki-netic scheme accommodates 4 di�erent kinetic paths: ATP-hydrolysis induced forward/backward step and ATP-synthesisinduced forward/backward step. Although the conventional(N=4)-state unicyclic kinetic scheme confers a similar resultwith the 6-state double cycle network scheme at small f , itis led to a physically problematic interpretation when themolecular motor is stalled and starts taking backsteps at large

hindering load. The backstep in unicyclic reaction schemes,by construction, is produced by a reversal of the forward cycle(15), which implies that the backstep is always realized viathe synthesis of ATP from ADP and Pi. More importantly,in calculating Q from kinetic network, the unicyclic reactionscheme is led to Q = 0 under the stall condition, which isnot compatible with the physical reality that an idling carstill burns fuel and dissipates heat (Q ”= 0). To build amore physically sensible model that considers the possibility ofATP-induced (fuel-burning) backstep, we extend the unicyclicreaction scheme into a multi-cyclic scheme which takes intoaccount an ATP-consuming stall, i.e., a futile cycle (14–17).

In the double cycle model, kinesin-1 predominantly movesforward through F-cycle under small hindering (f > 0) orassisting load (f < 0), whereas it takes a backstep throughthe B-cycle under large hindering load. In principle, thecurrent within the F-cycle, JF , itself is decomposed into theforward (J+

F ) and backward current (J≠F ), and JF = J+

F ≠J≠F > 0 is satisfied in NESS. Although a backstep could be

realized through an ATP synthesis (18), corresponding to J≠F ,

experimental data (15, 19, 20) suggest that such backstepcurrent (ATP synthesis induced backstep, J≠

F ) is negligible incomparison with J+

B (ATP hydrolysis induced backstep).The dynamics realized in the double cycle network can be

illuminated in terms of variation of JF and JB with increasingf (Fig.1B). Without load (f = 0), kinesin-1 predominantlymoves forward (JF ∫ JB). This imbalance diminishes withincreasing f . At stall conditions, the two reaction currentsare balanced (JF = JB), so that the net current J associatedwith the mechanical stepping defined between the states (2)and (5) vanishes (J = JF ≠ JB = 0), but nonvanishing currentdue to chemistry still remains along the cycle of æ (2) æ(3) æ (4) æ (5) æ (6) æ (1) æ (2) æ. A further increaseof f beyond the stall force renders JF < JB, augmenting thelikelihood of backstep.

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXX

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Hwang et al.

hyd

[T]

D

DT

T

DT

a

b

c

FIG. 8 b. Schematics of the network model representing thedynamics of double-headed kinesin-1. T , D, and φ denoteATP-, ADP-bound, and apo state, respectively. ThroughATP binding [(1)→(2)], mechanical step [(2)→(5)], releaseof ADP [(5)→(6)], and hydrolysis of ATP [(6)→(1)], kinesinmoves forward in the F-cycle [(1)→ (2)→ (5)→ (6)→ (1)],backward in the B-cycle [(4) → (5) → (2) → (3) → (4)].The arrows in the figure depict the direction of major coun-terclockwise reaction current in each cycle. In both cycles,each chemical step is reversible and kij defines the transitionrate from the i-th to j-th state. c. Reaction current JF , JB,and J as a function of load. The three cartoons illustrate theamount of current flowing along the F and B cycles as a func-tion of f . f < 0 and f > 0 are the assisting and hinderingload, respectively.

mechanism, it has been proposed to extend the unicyclicnetwork into a network with multiple cycles (Clancyet al., 2011; Hyeon et al., 2009; Liepelt and Lipowsky,2007a; Yildiz et al., 2008), so that the kinetic pathwayof ATP-induced (fuel-burning) backward step or ATP-consuming stall can naturally be considered in the model.

3. Multi-cycle Models

The (N = 6)-state double cycle network in Fig. 8b isdiscussed here as a minimal kinetic model to account forsubtleties in the physics of kinesin under external load.In the model, the rate constants at the edges between the

adjacent chemical states are given as

k25(f) = ko25e−θfd0/kBT

k52(f) = ko52e(1−θ)fd0/kBT

kij(f) = 2koij(1 + eχijfd0/kBT )−1 for ij 6= 25 or 52 (47)

For the sake of simplicity, it is assumed that (2) (5),the step associated with the switching of the trailing andleading head positions (the yellow arrow in Fig.8b), obeysthe Bell-like force dependence. When f > 0, the forceexerted on the motor hinders the forward step and en-hances the backward step. Other steps (ij 6= 25 or 52)are abolished (kij → 0) at large f(> 0), which corre-sponds to the physical situation where the large externalforce impedes the chemical steps such as ATP binding,hydrolysis, and ADP release by deforming the confor-mation of molecular motor. The model consists of twosub-cycles F ((1) (2) (5) (6) (1)) and B((2) (3) (4) (5) (2)). At small (f ≈ 0) or as-sisting force (f < 0), which renders k25 � k52, the reac-tion current is mainly formed along the counterclockwisedirection of F cycle (F+); in contrast, at large hinderingforce (f > 0) above the stall condition (f = fs), the cur-rent flow along the counterclockwise direction of B cycle(B+), which corresponds to the ATP hydrolysis-inducedbackstep.

Owing to the microscopic reversibility, each cycle canin principle be performed in both counterclockwise (+sign, F+,B+) and clockwise (− sign, F−,B−) direc-tions. Forward (backward) steps occur via the (2)→ (5)((5) → (2)) transition; the edges (1) → (2) ((2) → (1))and (4)→ (5) ((5)→ (4)) are associated with ATP bind-ing (dissociation); the edges (6) → (1) ((1) → (6)) and(3)→ (4) ((4)→ (3)) are associated with ATP hydrolysis(synthesis) (The cyan arrows in F and B cycles highlightATP hydrolysis). The model in principle accommodates4 different pathways: (i) ATP-hydrolysis-induced for-ward step (F+); (ii) ATP-hydrolysis-induced backwardstep (B+); (iii) forward step that synthesizes ATP (B−);(iv) backward step that synthesizes ATP (F−). The re-action currents JF flowing through (6) (1) and JBthrough (3) � (4) can be calculated by the generatingfunction technique by Koza (Hwang and Hyeon, 2017;Koza, 1999), or utilizing the approach based on the largedeviation theory (Lebowitz and Spohn, 1999). The ex-pressions of JF and JB in terms of {kij} for the double-cycle network are generally lengthy and too complicatedto be shown here (see Eq. S25 in (Hwang and Hyeon,2017)).

In the double-cyclic network model the total heat gen-erated from the kinetic cycle depicted in Fig. 8b is de-composed into two contributions from the subcycles, QFand QB, each of which is the product of reaction cur-rent and affinity (Barato and Seifert, 2015; Ge and Qian,2010; Liepelt and Lipowsky, 2007a; Qian, 2004; Qian and

18

βQ = β (JFAF + JBAB)

= JF log

(k12k25k56k61k21k52k65k16

)+ JB log

(k23k34k45k52k32k43k54k25

)= JF log

(k12k56k61k21k65k16

)JB log

(k23k34k45k32k43k54

)︸ ︷︷ ︸

chemical

+ (50)

+ (JF − JB) log

(k25k52

)︸ ︷︷ ︸

mechanical

(51)

Beard, 2005; Seifert, 2012)

Q = JFAF + JBAB, (48)

where the two driving forces (affinities) A are given by,

(a) AF = kBT log

(k12k25k56k61

k21k52k65k16

)= −∆µhyd − fd0

(b) AB = kBT log

(k23k34k45k52

k32k43k54k25

)= −∆µhyd + fd0

(49)Note that at f = 0, the chemical driving forces for Fand B cycles are identical to be −∆µhyd. The above de-composition of affinity associated with each cycle intothe chemical driving force and the work done by themotor follows naturally from the f -dependent expressionof {kij} given in Eq.47 (Fisher and Kolomeisky, 1999a,2001; Liepelt and Lipowsky, 2007a). The entropy pro-duction from the double cycle model is expressed as thesum of entropy produced from the two sub-cycles F andB. Or simply from Eq. 48 and Eq. 49, Q can be re-cast into the differnece between the total free energyinput [E = (JF + JB)(−∆µhyd)] and work production

[W = (JF − JB)fd0]:

Q = JFAF + JBAB= (JF + JB)(−∆µhyd)− (JF − JB)fd0

= E − W . (52)

Notice that in order to reiterate our earlier remark thatthe mechanical equilibrium at the stall condition does notcorrespond to the thermodynamic equilibrium, the lastline of βQ (Eq.51) is decomposed into the contributionsby chemical and mechanical processes. In the proposeddouble-cycle kinetic model, the average velocity of themotor is given by V = d0(JF − JB). Thus, if the num-bers of forward and backward steps taken over time arebalanced to each other, satisfying JF = JB (Carter andCross, 2005), we expect no net directional movement ofthe motor (〈x(t)〉 = 0), which is equivalent to say V = 0.It is important to recognize that displacement (or traveldistance) x(t) is the most direct observable to an exter-nal observer if one were to use optical tweezers or fluores-cence dyes. In the stall condition (JF = JB ≡ Js 6= 0),

however, because the chemical processes associated with“two” ATP hydrolysis events, one along the forward andthe other along the backward step, are still at work, theheat production is finite (Q > 0)

βQ = Js log

(k12k23k34k45k56k61

k21k32k43k54k65k16

)= 2Js(−∆µhyd).

(53)

This is a point of great importance, which cannot becapture by the uni-cyclic network model (Fig.8a).

Figure 8c depicts the currents flowing through the twosub-cycles JF and JB calculated from the set of rate con-stants ({kij(f)}) determined against the motility dataof kinesin-1. Under small hindering (f & 0) or assist-ing load (f < 0), kinesin-1 predominantly moves forwardthrough the F-cycle, whereas it starts taking more num-ber of backsteps through the pathway B+ as the loadincreases further. At stall conditions, the net current as-sociated with the mechanical stepping defined betweenthe states (2) and (5) vanishes (J = JF − JB = 0);however, non-vanishing current along the futile cycle hy-drolyzing ATP still persists along the reaction path of(1) (2) (3) (4) (5) (6) (1) (see Figure8b). A further increase of f beyond fs leads to JF < JB,increasing the chance of backsteps via ATP hydrolysis.Although a backward step satisfying JF < 0 could berealized through an ATP synthesis, a theoretical analy-sis (Hyeon et al., 2009) of the experimental data (Carterand Cross, 2005; Nishiyama et al., 2002) suggest thatsuch event (JF < 0, ATP synthesis induced backstep) ispractically negligible compared with the one associatedwith ATP hydrolysis-induced backstep (JB > 0), so that|JB| � |JF |. This is in agreement with energetic analysissuggesting that backward and forward steps are not thereverse of each other (Hackney, 2005).

D. Additional remark on non-equilibrium nature of motors

It is worth mentioning a couple of recent studies thatfurther highlight the non-equilibrium aspect of molecularmotors. First, the Harada-Sasa equality (Harada andSasa, 2006) quantifies the heat generated from Langevinprocesses out of equilibrium as follows,

Q =

N∑i=1

γi

{v2i + γ

∫ ∞−∞

[Cii(ω)− 2kBTR′ii(ω)]

dω

2π

}.

(54)

It equates the total heat dissipation rate from the sys-tem Q with the expression in terms of friction coefficientγi of an i-th variable xi, the Fourier component of auto-correlation of the velocity fluctuation Cii(t) ≡ 〈(xi(t) −vi)(x(0) − vi)〉0, and the real part of the response func-tion R′ii(ω). At equilibrium, when the detailed balancecondition is satisfied, vi = 0, which leads to the standard

19

fluctuation dissipation theorem, Cii(ω) = 2kBTR′ii(ω).

Thus, the heat dissipation is zero. Therefore, the equalityrelates the extent of violation of the fluctuation-responserelation in NESS with the rate of energy dissipation fromthe system into the bath. At least two studies have sofar used this equality to experimentally assess the heatdissipation from molecular motors, one for kinesin andthe other for F1-ATPase. Ariga (Ariga et al., 2018) haveadapted this equality in the form,

Qx = γx

{v2x +

∫ ∞−∞

[Cvv(ω)− 2kBTR′vv(ω)]

dω

2π

}(55)

where the displacement of kinesin motor x(t) was experi-mentally monitored, and the Fourier component of auto-correlation function and response function of velocityfluctuation were directly calculated using the time tracesobtained from single molecule measurements. Theyshowed that the total heat dissipation assessed usingEq.55 and the work done by the kinesin do not add up tothe total chemical free energy input to the motor. Theauthors suggested that there are other elements of heatdissipation that do not involve the dynamics of the motoralong the direction of motion x(t) that can be monitoredby their experiment. Although the proposed double cyclemodel in Fig.8b has already accommodated other possi-bilities, such as ATP-hydrolysis involved futile cycle with-out stepping, other scenarios such as slippage induced bymechanical force without chemical process (Yildiz et al.,2008) are not included. Further elements of heat dissi-pation could be included in kinetic models in order todescribe the function of the motors.

The finding by Ariga et al. should be contrastedwith another experimental study using the Harada-Sasaequality on a single F1-ATPase (Toyabe et al., 2010).These authors showed that the total free energy inputto the enzyme was partitioned to the heat and workproduction with little loss. This indicates that the cycleof ATP synthesis corresponds to the reversed cycle ofthe hydrolysis-driven motor rotation. However, a recentcareful theoretical analysis by Sumi and Klumpp onF1-ATPase (Sumi and Klumpp, 2019) suggested thatthe reversibility (100% efficiency) of the rotary motoris only attained under certain conditions. Mechanicalslip can occur in the presence of high external torquewithout chemomechanical coupling, the effect of whichis amplified at low ATP and ADP concentrations. Thisreduces the previously estimated 100 % free-energytransduction efficiency. It was argued that in additionto the viscous dissipation of the probe, heat dissipa-tion could occur from the rotary motor itself as thetorque applied to the biological nanomachine inducedmechanical slip. This occurs as a result of deformationof molecular conformation, which is best optimized tofunction in the absence of torque.

E. Applications

1. Myosin V and Kinesin-1

SKMs have been successfully used to model the func-tion of molecular motors. Sometime ago Leibler andHuse (Leibler and Huse, 1993) devised a model to investi-gate the differences between motors that act as “porters”and those that function as “rowers.” The former, likethe processive molecular motors described so far, workin small groups in order to transport cargoes; myosinII and axonemal dynein are examples of the latter, andwork in large teams in order to slide or bend filaments.

In two landmark papers Fisher and Kolomeisky an-alyzed the data for kinesin-1 (Fisher and Kolomeisky,2001) and myosin V (Kolomeisky and Fisher, 2003) inorder to train the parameters in their SKMs. A 2-stateand a 4-state model for kinesin-1 described accuratelythe motor velocity as a function of ATP and resistiveload (see Fig. 9a-b); although the 2-state model was alsoused to describe the run-length of kinesin (see the sec-tion VI for details about the run length), the analysis ofthe randomness parameter as a function of [ATP] andf highlighted the need for a N = 4 model. In the caseof myosin V, Kolomeisky and Fisher considered the datafor ATP and load-dependent dwell time before a forwardstep (Mehta et al., 1999); they fit the parameters of their2-state model (Fig. 9c) and predicted the value of the ran-domness parameter as a function of [ATP] and f . Theanalysis of the parameters trained with the experimen-tal data revealed in both circumstances the existence ofa substep, and that most of the load dependence on therates was carried by the reverse processes, indicating thatthe transition states are closer to the initial state than tothe final state.

Liepelt and Lipowsky (Liepelt and Lipowsky, 2007a)devised a multi-cycle model (see Fig. 8b), which allowedthem to incorporate backward steps fueled by ATP hy-drolysis. The experimental results of Carter and Cross(2005) and Visscher et al. (1999) were recovered with adi-cyclic, N = 6 model (see Fig. 10), although Liepeltand Lipowsky showed that another state (and a new cy-cle) was necessary in order to account for the velocity asa function of ADP concentration (Schief et al., 2004).

A further development was put forth by Hyeon et al.(2009), where a more sophisticated network was proposedin order to consider an alternative pathway for backwardstepping based upon known structural features of thekinesin-1 dimer. Clancy et al. (2011) proposed yet an-other cycle for kinesin, in this case to fit the data for amutant with an extended neck linker; the model success-fully recovered the velocity, randomness parameter, andratio between number of forward and backward steps as afunction of ATP and resistive load, even in the super-stallregime.

20

FIG. 9 Unicycle models for kinesin-1 and myosin V. Panel(a-b): comparison for kinesin-1 of the results of a uni-cyclemodel with experiments from Visscher et al. (1999). Panel(c): myosin V, comparison between a N = 2 model formyosin V and the velocity from experiments, obtained fromthe mean dwell times τ (Mehta et al., 1999) via the rela-tionship v = 36nm/τ . Panels (a-b) and panel (c) are adaptedfrom Fisher and Kolomeisky (2001) and from Kolomeisky andFisher (2003), respectively.

2. Dynein – an erratic motor

In part due to its complex architecture and the paucityof details of the nucleotide chemistry much less is knownabout the kinematics of dynein stepping kinetics. Weshould note that both from the perspective of structuredetermination (Schmidt et al., 2015) and single moleculeexperiments (Belyy et al., 2014b; DeWitt et al., 2012;Reck-Peterson et al., 2006a) there have been spectacularadvances in recent years, which could be used to createnew theories. A theoretical model different from thosedescribed in this perspective, which by necessity treatedthe structural and transition kinetics between pre- andpost-power stroke states approximately, was introduced(Tsygankov et al., 2011) to calculate the distribution ofstep size. Despite several untested approximation, exper-iments (Reck-Peterson et al., 2006a) and kinetic MonteCarlo simulations of the model were in fair agreement (see

FIG. 10 Fit of experimental data for kinesin-1 with a di-cycle,N = 6 model. The data in panels (a) and (b) is from Carterand Cross (2005), whereas the experiments considered in pan-els (c) and (d) are from Visscher et al. (1999). The figure isreproduced from Liepelt and Lipowsky (2007a).

Fig. 3A in (Tsygankov et al., 2011)). A much more elabo-rate model that couples structural aspects of dynein witha model for the catalytic cycle, which is similar in spiritto the theory for myosin V (Hinczewski et al., 2013),was proposed more recently (Sarlah and Vilfan, 2014).The predictions of the model, which has a large numberof parameters, were successful in obtaining the step-sizedistribution as well as an estimate of the stall force. How-ever, simple theories that can account for the unusuallybroad step size distributions reflecting the erratic natureof this motor, force-velocity curves as a function of ATPconcentration, as has been done for kinesin and myosinV, are lacking.

VI. MOLECULAR MOTORS – MODELS WITHDETACHMENT

All molecular motors take only a finite number of stepsalong their tracks before detaching. Therefore, promi-nent features of motor motility are the run-length, L,and the run-time, T , which correspond to the distancecovered during a processive run, and the amount of timespent bound to the filament before detaching, respec-tively. In addition, the ratio between L and T consti-tutes an alternative and intuitive definition of velocity,v = L/T , which is simply the ratio between the spa-tial displacement of a motor and the amount of timetaken to complete the movement. Processive motors takemany steps along the filament before dissociating fromthe track. The number of steps depends on the motor,

21

and on a number of parameters such as the concentrationof nucleotides, and the value of the applied load. Thisunderscores the importance of accounting for the end ofthe processive run in a way that is consistent with the en-zymatic cycle of a specific motor and the experimentallydetermined run-length, run-time etc.

The models described in the previous sections assumethat the motor is permanently attached to the track,and ignore the end of the processive run. We discusshere a number of strategies that have been used inorder to account for the detachment from the filament.(i) The most direct way is to increase the numberof states by Nd, the number of detached states, andto introduce the rates of releasing and attaching tothe filament. This method, which was used to modelthe function of kinesin-1 with Nd = 1 (Liepelt andLipowsky, 2007a), enables a steady-state treatment ofthe corresponding Markov jump process, but requires anincrease in the number of states, the number of edges,and the number of cycles. (ii) Alternatively, it is possibleto enforce a new stationary state in the Markov jumpprocess by diverting the flux into the detached statetowards the filament-bound conformations. Althoughthis approach does not require any other state (Nd = 0),it is necessary to increase the number of edges, whichresults in the creation of new cycles. A theoreticaldescription of this steady-state approach can be foundin the remarkable work by Hill (Hill, 2005a). It isunlikely that the methods described by Hill could beused to calculate velocity and run-length distribution.However, one could calculate their averages as a functionof nucleotide concentration or external load. (iii) Itis also possible to account for the detached trajecto-ries by re-normalizing quantities such as the averagevelocity over the decreasing number of bound motors.This strategy, put forth by Kolomeisky and Fisher, isdescribed in (Kolomeisky and Fisher, 2000), and hasbeen used to test an approximate equation for kinesinrun length (Fisher and Kolomeisky, 2001). (iv) Onecould define the average run length as the ratio betweenthe average velocity v and the rate of detachment γ(L ≈ v/γ). In order to do so, γ could be estimatedas the product between the stationary probabilities ofoccupying the “vulnerable” states and newly introduceddetachment rates (γ =

∑i∈vulnerable p

Si ki,det) (Fisher

and Kolomeisky, 2001; Maes and Van Wieren, 2003).(v) Finally, the detached state could be treated as anabsorbing state. A stationary solution is not possible:after a sufficient time (t >> γ−1) all the motors willbe absorbed to the detached state. However, one mayaccount for all of the possible trajectories leading toabsorption, and perform averages over the ensemble ofthese paths. In the remainder of this section we focuson this last method.

200 100 0 100 2000.0000

0.0003

0.0006

v (nm/s) v (nm/s)- 400 - 200 0 200 400

0.0000

0.0010

0.0020

0.0030

0 2 4 6 80.0

0.2

0.4

0.6

F (pN)

P(v

=0)

5pN7.6pN

4pN

8.5pN

3pN A B

P(v

≠0)

P(v

)

i i+1 i-1

k+ 𝛾

k-

��

𝐸 𝐸

|𝑑 | |𝑑 |

𝑘 𝑘

Activ

atio

n En

ergy

A B C

D E

FIG. 11 One-state model for kinesin motility with detach-ment. (A) Kinesin walks along the MT, subject to an externalload F . (B) The MT is represented as a linear discrete set ofsites separated by d0 = 8.2nm; the motor advances towardsthe + end with rate k+, backsteps with rate k−, and at ev-ery site γ establishes the detachment rate. (C) Load affectsthe rates according to Bell model, which in turn correspondsto a modification of the energy profile: k+ diminishes as loadincreases, whereas the resistive force favors back-stepping (in-creases k−). The transition state location with respect to the

starting state, θ = |d−|(|d−|+|d+|) . (D) The velocity distribution

displays a markedly bimodal shape at a variety of resistiveforces, with the negative peak that increases as the load ap-proaches stall. (E) Simulations performed assuming an exper-imentally motivated error on the determination of d0 maintainthe bimodal structure of P (v). Figure adapted from Vu et al.(2016).

A. Velocity Distribution

The simplest model of molecular motor with detach-ment is shown in Fig. 11, in which forward (backward)steps occur with rate k+ (k−), and the detachment rateis γ. For this model it is possible to determine analyti-cally the probability distribution p(v), where the veloc-ity v is the ratio between the net number of steps taken(n = m− l, where m is the number of forward steps andl is the number of backward steps) divided by the run-time (v = n/T = v/d0). Vu et al. showed that (Vu et al.,2016),

p(v ≷ 0) =

=γ

|v|∞∑n=0

( n|v|)n+1 1

n!(k±e−kT/|v|)n0F1(;n+ 1;

n2k+k−

|v|2 ),

(56)

where 0F1(;n+1; n2

|v|2 k+k−) is a hypergeometric function,

and kT = k+ +k−+γ. For an alternative derivation, seea more recent study (Zhang and Kolomeisky, 2018). Thismodel was adopted to describe the stepping mechanism

22

of kinesin-1; the parameters k+ and k− were establishedfrom experiments and the run length and velocity distri-bution (Walter et al., 2012) were fitted using only one pa-rameter, γ. The result in Eq. 56 hold also in the presenceof force, which increases the probability of back-steppingand detaching from the microtubule. Using this model anumber of noteworthy results were obtained (Vu et al.,2016): (i) even at zero load, the velocity distribution isnot Gaussian, although it approaches a normal distribu-tion if the run length is large (that is, k+/γ >> 1 andk+/k− >> 1); (ii) the probability distribution of the in-stantaneous velocity (the size of the step divided by thedwell time) differs from Eq. 56, and does not match theexperimental distribution (Walter et al., 2012), suggest-ing that it is not the correct way to compute velocity; (iii)most surprisingly, the velocity distribution is bimodal,with distinct peaks for v > 0 and v < 0. The predictionthat p(v) is bimodal, becoming most prominent at stallforce, was unexpected. At f = fstall, the naıve expecta-tion would be that p(v) would have a peak at v = 0, andthe area under p(v) with v > 0 and v < 0 would be thesame. The bimodal distribution is a consequence of thediscrete nature of kinesin step size, and it is exaggeratedat large f , when the probability of taking backward stepsand the probability of detaching are larger.

The model in Fig. 11 was recently extended in orderto include an intermediate step (Fig. 12), which was usedto describe the kinetics of a molecular motor as a com-bination of an ATP-dependent and an ATP-independenttransition (Takaki et al., 2019). The resulting velocitydistribution is,

p(v ≷ 0) =

γ

v

∑m,l

m≷l

m− lv

√π

m!l!

kn+1(k+)m(k−)l

|k − (k+ + k− + γ)|n+1/2

(m− lv

)n+1/2

e−k+k++k−+γ

2m−lv In+1/2

( |k − (k+ + k− + γ)|2

m− lv

),

(57)where I is a modified Bessel function of the first kind. In-terestingly, the bimodal structure of the velocity distribu-tion is robust to changes of the nucleotide concentration,and it is independent of which step (1 → 2 or 2 → 1)is ATP-dependent. Therefore, an experiment aimed attesting the predicted multi-modality of p(v) may be con-ducted under arbitrary ATP concentration and resistingload.

The model in Fig. 12 was used to tackle a vexingconundrum concerning the stepping mechanism ofkinesin (Takaki et al., 2019). Recently, two groups haveperformed similar experiments in which they monitoredthe movement of one kinesin head conjugated witha gold-nanoparticle (AuNP) by tracking the positionof AuNP (Isojima et al., 2016; Mickolajczyk et al.,2015). From the analysis of the trajectories, one groupproposed that ATP binds to kinesin when the trailing

1l 1l+1⋯⋯1l−1

→k

detached state

2l

→γ

→k+→k−

FIG. 12 Two-state model with detachment. The forward andbackward stepping rates are k+ and k−, respectively, and theyconnect state 2 to state 1. The 1→ 2 transition occurs withrate k. Detachment occurs only from state 2, with rate γ.

head is still bound to the MT (or at least in the vicinityof the pre-stepping site) (Mickolajczyk et al., 2015),the other group suggested that the dissociation fromthe MT and forward movement of the trailing headdetachment precede ATP binding (Isojima et al., 2016).Using the model in Fig. 12, Takaki et al. (Takaki et al.,2019) predicted that the ATP-dependent profile ofthe randomness parameter may be used in order todetermine whether ATP binds to the two-head-bound orone-head-bound conformation of the kinesin-1 dimer.

B. Alternate Models with Detachment

More complicated kinetic schemes that include mo-tor detachment have been used in order to compute theprocessivity and velocity of molecular motors. Elting etal. (Elting et al., 2011) found an analytical solution fora 8-state model of myosin VI which they proposed in or-der to investigate the nature of the gating mechanism.This model did not account for the possibility of back-ward stepping, which was instead included by Caporizzoet al. (Caporizzo et al., 2018) in a kinetic model that wasdevised in order to investigate the motility of myosin Xas a function of the structure of the filament (individ-ual versus bundled actin) and the geometry of the tail ofthe dimer (parallel versus anti-parallel). More recently, anew theoretical framework capable of extracting motilitycharacteristics such as average velocity, average and dis-tribution of number of steps, and probability of backwardstepping for a kinetic network of arbitrary geometry hasbeen proposed (Mugnai et al., 2019). The dynamics ofthe motors is described in terms of the following Markovchain,

~Px+1 = (S + F + B)~Px = M~Px, (58)

23

l l + 1l − 1

FB

detachment

S

FIG. 13 Model for processive motor moving on a periodictrack. From each site l of a filament the motor can changeits conformation (e.g. the chemical state) within the same

location (transitions included in the matrix S, in black) move

forward (the matrix F , in red, accounts for these steps) or

backward (the matrix B, in blue, refers to these displace-ments) along the track, or detach (magenta).

where the N -dimensional vector ~Px contains the proba-bility of occupying any of the N filament-bound statesof the motor after x transitions have occurred, and S,F, and B are the N ×N transition probability matricesfor undergoing a transition at a fixed location, steppingforward, or backward, respectively (see Fig. 13). Eq. 58holds under the assumption that the track is periodic,which makes S, F, and B independent on the location ofthe motor along the filament. By accounting for all of thepossible stepping pathways, it was found that the prob-ability of taking n steps (forward or backward) beforedetaching is,

P(n) = ~1ᵀ(I− Pstep)(Pstep)n ~P0, (59)

where ~1ᵀ is the transpose of an N -dimensional vector ofones, ~P0 is the initial probability (appropriately normal-

ized, ~1ᵀ · ~P0 = 1), and Pstep = (F + B)(I − S)−1. Thisexpression is a generalization of a well-known result forN = 1, in which P(n) = (1 − π)πn, where π < 1 isthe probability of stepping and 1 − π is the probabilityof detaching. The average number of steps 〈n〉 may becomputed using simple linear algebra, as well as the dis-tribution and average number of forward and backwardsteps, 〈m〉 and 〈l〉, respectively, with 〈m〉 + 〈l〉 = 〈n〉.The average run length is then 〈L〉 = d0(〈m〉 − 〈l〉),and by using a classical result for the mean-first-passage-time to absorption it is possible to compute 〈τ〉 (Op-penheim et al., 1977), and define an average velocityv = 〈L〉/〈τ〉. With Kinetic Monte Carlo simulations it ispossible to show that 〈L〉/〈τ〉 differs from 〈L/τ〉, whichwas adopted in (Takaki et al., 2019; Vu et al., 2016). Onthe other hand, it was shown empirically that the dif-ferences become smaller as the average number of stepstaken by the motor increases (Mugnai et al., 2019). Theadvantages of this theoretical framework are its flexibil-ity (it works with any network) and ease of implemen-

tation (it only requires matrix algebra). Therefore, itcan be used to fit complicated models against experi-mental data. The model was used to solve a compli-cated model for myosin VI motility inspired by modelof Yanagida and coworkers (Ikezaki et al., 2012, 2013;Nishikawa et al., 2010), which accounted for a motor tak-ing both hand-over-hand and inchworm-like steps (Mug-nai et al., 2019), and predicted the gating mechanism [inagreement with (Dunn et al., 2010; Elting et al., 2011)],size of the backward steps [matching experimental ob-servations (Altman et al., 2004; Nishikawa et al., 2010)within discrepancies likely related to fluctuations and thesite of probe attachment], pathway for backward stepping[as suggested by (Ikezaki et al., 2013)], and importanceof foot-stomping in breaking the tight-coupling of myosinVI stepping.

VII. POLYMER PHYSICS-BASED APPROACHESINCORPORATING STRUCTURAL FEATURES INTOKINETIC THEORIES

Even with coarse-grained numerical models like the oneproposed by Craig and Linke (2009),which is describedbelow, collecting statistics from multiple simulations cov-ering entire motor trajectories (i.e. for myosin V each runaveraging tens of steps until detachment) can be compu-tationally expensive. To more comprehensively explorehow motor architecture affects stepping dynamics, par-ticularly in light of experiments that perturb structuralfeatures like lever arm length (Oke et al., 2010; Sakamotoet al., 2005), alternative approaches are needed. We fo-cus on one particular example, an analytical theory formyosin V dynamics (Hinczewski et al., 2013), that alsohighlights several aspects discussed earlier: the startingpoint of the theory is a six-state stochastic kinetic net-work model consisting of multiple cycles that explicitlyincludes detachment of the myosin V dimer from theactin filament. Incorporating detachment allows us tomodel finite run lengths of the motor on actin, whilethe multi-cycle network topology captures the dominantpathways of myosin V dynamics under load forces belowor near stall (. 2 − 3 pN): these include both forwardand backward steps as well as so-called “stomps”, whereeither the trailing or leading head detaches and then reat-taches near its original binding location. Stomps arechallenging to detect with single-molecule fluorescencetechniques, but have been observed experimentally us-ing high-speed atomic force microscopy (Kodera et al.,2010).

However unlike the kinetic models treated so far,the force-sensitive transition rates will not be describedthrough a phenomenological Bell-like exponential depen-dence as in Eq. (35). Instead, the goal of the approachby Hinczewski et al. (2013) is to model the structuralmechanics underlying this force dependence through a

24

coarse-grained polymer theory for the myosin V dimer.This replaces the discrete semiflexible polymer descrip-tion of Craig and Linke (Craig and Linke, 2009) (inter-acting monomers representing the motor head and leverarm IQ domains) with an even simpler construct: a sin-gle continuous semiflexible polymer that represents thecombined motor head and lever arm domains. Thus, thedimer becomes two polymer “legs” attached at a flexiblejunction. The load force transmitted through the cargodomain is modeled by an effective force F applied at thejunction. Because the load force changes the ensembleof conformations for the two polymer legs, and hencethe three-dimensional distribution of positions exploredby the unbound motor head (see Fig. 14), it affects therates at which the unbound head reaches potential bind-ing sites.

The most direct advantage of this simpler coarse-grained description is that the effects of force on steppingdynamics become (to an excellent approximation) ana-lytically tractable. It is also readily generalized to morecomplex contexts. For example, while the model we fo-cus on here considers only a flexible junction, backwardsload forces parallel to the actin axis, and restricts steps toactin binding sites separated at actin helical half lengths(the most probable step separation for myosin V), all ofthese assumptions can be relaxed. A recent extension ofthe polymer theory approach (Hathcock et al., 2019) ex-plores the effects of a possible structural constraint at thejunction (inspired by experimental evidence (Andreckaet al., 2015)), considers off-axis forces, and incorporatesthe full distribution of steps at all possible actin bindingsites. This allows us to understand previously observedstep distributions of mutant myosins with various leverarm lengths (Oke et al., 2010; Sakamoto et al., 2005), aswell as the robustness of myosin V dynamics in experi-ments where various off-axis load forces are applied to themotor via optically trapped cargo (Oguchi et al., 2010).Moreover the polymer approach is not limited to myosinV: an analogous treatment of dynein, with the two motordomains approximated as rigid rods (the large stiffnesslimit of semiflexible polymers), can successfully repro-duce the complex details of dynein step distributions onmicrotubules (Goldtzvik et al., 2019). For both myosin Vand dynein each half of the dimer is modeled as a singlepolymer / rod, and this description is likely to be appli-cable to other dimeric processive motors with fairly stifflever arms, like myosin XI (Tominaga and Nakano, 2012).However there are cases, like myosin VI or X, wherethe lever arm structure is more heterogeneous, mixingstiff and flexible domains (Sun and Goldman, 2011b).Here any future attempt to apply coarse-grained poly-mer modeling would need to represent each lever arm asa series of polymer chains with different bending rigidi-ties.

To understand the basic details of the polymer the-ory for myosin V in its simplest form, we first summa-

rize the underlying six-state kinetic model, illustrated inFig. 15A. State 1 is the waiting state, where both mo-tor heads have bound ADP and are strongly attachedto actin. The lever arms for both heads are in thepost-powerstroke conformation: in other words in theabsence of other constraints the lever arms would ori-ent in the forward direction (toward the plus end ofactin), but because the junction pulls the lever arm ofthe leading head backwards the entire structure is in astrained state known as the telemark or reverse arrow-head stance (Kodera et al., 2010; Walker et al., 2000).State 1′ is also a waiting state, but with the entire mo-tor displaced forward by one half helical length of actin(∆ = 36 nm). The possible kinetic pathways are as fol-lows:

1. Forward step (1→ 2→ 3→ 4→ 1′): ADP releasefrom the trailing head is followed by ATP binding,which leads to detachment of the head from actin(1 → 2). We assume saturating ATP conditions,so ATP binding is fast compared to ADP release,and hence the entire process is subsumed into a sin-gle transition described by a rate t−1

d1 . Here td1 isthe mean detachment time scale, dominated by thewaiting time for ADP release, td1 ∼ 12 s−1 (De laCruz et al., 1999). The next transition (2→ 3) in-volves ATP hydrolysis into ADP+Pi, along with arecovery stroke where the head / lever-arm orien-tation changes from post- to pre-powerstroke. Thistransition occurs at a rate t−1

h = 750 s−1 (De laCruz et al., 1999). While there are scenarios wherethe reverse hydrolysis rate is significant (for exam-ple in motors with modified light chain composi-tion (Dunn and Spudich, 2007)), for this discussionwe make the typical assumption that the forwardhydrolysis rate dominates. In general, while ev-ery transition arrow in Fig. 15 has an associatedreverse transition in principle, here we make thesimplifying assumption that the reverse rates arenegligible relative to the forward rates. This as-sumption captures the dominant kinetic pathwaysof the motor (our focus here), but we would need toexplicitly consider reverse rates to look at thermo-dynamic features of the system (like entropy pro-duction). Once ATP is hydrolyzed, the motor headhas the ability to associate with actin again. Thethree-dimensional diffusive search can result in ei-ther binding to the forward site (36 nm ahead of thebound leg), or the original binding site. The rate atwhich it binds to the forward site is (t+fp)

−1, which

depends on the mean first passage time t+fp to the

forward site. The dependence of t+fp on the loadforce is related to how the force biases the diffusivesearch, and the underlying physics is discussed inmore detail below. A binding to the forward site(3→ 4) is quickly followed by Pi release and a pow-

25

FIG. 14 Examples of the diffusive search trajectory of the detached myosin V head for two different values of backwards loadforce: A. F = 0; B F = 2 pN. The actin filament lies along the z axis, and the trajectories are projected onto the radialdistance

√x2 + y2 away from the actin axis. The colors represent time, with lighter colors (yellow) occurring earlier than

darker colors (red). In panel A the trajectory ends in a forward step (the detached head going from the backward bindingsite at z = −36 nm to the forward binding site at z = 36 nm. In contrast, panel B shows a backwards step, which becomesincreasingly favored as a kinetic pathway under larger load forces. Superimposed on the trajectories are cartoon snapshots ofthe two polymer legs (each leg representing a lever arm plus head) of the coarse-grained model for myosin V, at a time shortlyafter detachment. The white dot is the junction of the two legs, and the cyan dot is the location of the detached head at thattime step. The contours correspond to the equilibrium probability distribution P(r) of the detached head, calculated from theanalytical polymer theory (darker contours correspond to higher probabilities). As load force is increased, the attached legand junction are pulled backwards, biasing the entire distribution away from the forward binding site, and thus increasing thechance of backwards versus forward steps (see also the corresponding experimental data of Fig. 16A). Adapted from Hinczewskiet al. (2013).

ADP PiADP

ADP

ADP Pi

ADP Pi ADP

ADP

ADP

ADPADPADPADP

ADP

ATP

0.0 0.5 1.0 1.5 2.010−4

10−3

10−2

10−1

100

Path

way

pro

bab

ility

Force F [pN]

forward step

leading stomp

termination

backward step

trailing stomp

1 1'

2 3

45

6

A Btermination

termination

forward steptrailing stomp

leading stompbackward step

FIG. 15 A. Six-state kinetic model for myosin V that underlies the coarse-grained polymer analytical theory of Hinczewskiet al. (2013). States 1 and 1′ are identical waiting states, with both motor heads bound to ADP and actin, except that 1′ isdisplaced 36 nm (a half helical length of actin) toward the plus end of the filament. Arrows are marked by inverse timescalesthat denote the transition rates, the details of which are described in the text. Colored arrows are transitions that belongto specific kinetic pathways: forward step (dark blue); trailing stomp (green); leading stomp (purple); and backward step(red); termination (light blue). Gray arrows are transitions that are shared between multiple pathways. B. Predicted pathwayprobabilities from the model, using the kinetic network together with mean first passage-times for the diffusive search step t±fpfrom polymer theory. Adapted from Hinczewski et al. (2013).

26

erstroke (4→ 1′), which occurs with a rate t−1ps . We

assume this step is rapid relative to the others inthe network, such that tps � td1, t+fp, th, and hencetps will not enter explicitly into our estimates formotor properties below.

2. Trailing stomp (1 → 2 → 3 → 5 → 1): This path-way starts in the same way as the forward step,with unbinding of the trailing head (1 → 2) fol-lowed by hydrolysis and the recovery stroke (2 →3), but the diffusive search now ends at the orig-inal binding site from where the head detached(3 → 5). The mean first passage time to this siteis t−fp (which also depends on force as describedbelow). However the head no longer binds in thesame configuration as before, because the lever armhas undergone a recovery stroke and is now in thepre-powerstroke orientation. Thus binding requiresnot only first passage to the site but overcoming theenergetic barrier of an unfavorable geometric orien-tation. We effectively model this through a factor0 < b < 1 that reduces the rate, so that the overallrate from 3 → 5 is given by b(t−fp)

−1. From fittingto experimental data, described below, the valueof b ≈ 0.065. As in the forward step, binding isfollowed by Pi release and a power stroke (5 → 1)occurring at a fast rate t−1

ps .

3. Leading stomp (1′ → 6→ 1′): This pathway is ini-tiated when the leading head detaches with ADPstill bound (1′ → 6). The reason for this alternativehead detachment mechanism is an asymmetry thatarises from the strain in the myosin V dimer in thewaiting state. The backward tension on the leadinglever arm in the telemark stance (estimated to beon around 2.7 pN (Hinczewski et al., 2013)) inhibitsADP release by a factor of 50-70 (Kodera et al.,2010; Rosenfeld and Lee Sweeney, 2004). Ratherthan releasing ADP and binding ATP to detachfrom actin, the dominant pathway for a head un-der backward load is direct detachment retainingthe ADP. This has been observed experimentally insingle-headed myosin V, where backwards loads ofaround 2 pN lead to an detachment rate t−1

d2 = 1.5s−1 (Purcell et al., 2005) independent of ambientATP and ADP concentrations. We thus take t−1

d2

as the transition rate from 1′ → 6. The differencebetween the overall detachment rates for the trail-ing and leading heads, with t−1

d1 eight-fold largerthan t−1

d2 , underlies the gating (Purcell et al., 2005;Veigel et al., 2005, 2002) mechanism: the trailinghead is much more likely to detach first. Since de-tachment of the leading head occurs with the ADPstill bound, the head can directly reattach back tothe forward binding site, with one caveat: the post-powerstroke configuration creates an energy barrierto reattachment due to geometry (since the lever

arm has to adopt a strained stance) and so as be-fore we introduce a factor b to scale the rate to theforward site. Thus the overall reattachment ratefrom 6→ 1′ is b(t+fp)

−1.

4. Backward step (1′ → 6 → 1): This pathway startswith leading head detachment (1′ → 6) like theleading stomp, but the detached head after diffu-sion finds the backward binding site. The post-powerstroke conformation with ADP bound is ge-ometrically favorable for binding to this site (be-cause the lever arm does not have to be bent) andhence the rate from 6→ 1 is (t−fp)

−1. Note that thisdescription of backward stepping is approximatelyvalid for load forces near or below stall, but doesnot include additional features that may becomeimportant for larger superstall forces (F > 3 pN),such as power stroke reversal (Sellers and Veigel,2010).

5. Termination: For the kinetic pathways describedabove, if the motor is in one of the three stateswhere only one head is bound to actin (2, 3, or6), then detachment of the second head (with ratet−1d1 ) will lead to dissociation of the entire motor

from the actin filament, terminating the run.

In the kinetic network described above, mechanicalforces enter in two ways: the first is through the inter-nal strain that leads to the gating mechanism, and thesecond is through the external load on the junction thataffects the diffusive search and hence the first passagetimes t±fp. We have already taken into account the asym-metry due to internal strain by using experimentally es-timated values for td1 and td2, but the force dependenceof t±fp has not been specified. This is precisely what thecoarse-grained polymer description allows us to do. Wetake advantage of a separation of time scales: when ei-ther head detaches, the equilibration time tr over whichthe polymer legs relax to an approximately equilibriumdistribution of conformations is fast compared to the firstpassage times, tr � t±fp. Brownian dynamics simulationsand analytical arguments (Hinczewski et al., 2013) showthat tr . 5 µs, while the fastest first passage times areat least t±fp ∼ O(0.1 ms). A summary of all the timescales in the problem is shown in Fig. 2. Thus to an ex-cellent approximation the detached head fully exploressome equilibrium distribution of positions, P(r), beforereaching either of the binding sites. In this case the firstpassage time to reach the forward (+) or backward (-)binding site at position r± is inversely proportional toP(r±), the probability density of finding the detachedhead at that position (Hinczewski et al., 2013):

t±fp ≈1

4πDhaP(r±), (60)

Here Dh is the diffusion coefficient of the detached head(which can be estimated as Dh ≈ 5.7× 10−7 cm2/s from

27

the crystal structure using HYDROPRO (Ortega et al.,2011)). The capture radius a ≈ 1 nm is the distance be-tween the head and binding site for which the interactionsbecome strong enough that binding occurs, comparableto the Debye screening length under physiological con-ditions. Eq. (60), which can be derived from standardfirst passage time analytical approaches like the renewalmethod (Van Kampen, 2007), is extremely useful: it con-verts the dynamical problem of finding mean first pas-sage times of a complex diffusion process into the moretractable problem of calculating the equilibrium distri-bution P(r) of the end-point of a two-legged semiflexiblepolymer system. The latter can be found analyticallyby extending an earlier mean-field theory for semiflexiblepolymers (Thirumalai and Ha, 1998), incorporating theorientational constraint of the bound leg due to the post-powerstroke conformation of the lever arm with respectto the motor head. Only a handful of structural parame-ters enter into the theory, determining P(r): the contourlength L = 35 nm and persistence length lp ≈ 310 nm ofthe polymer leg; the angle of the orientational constraintwith respect to the actin filament, θc ≈ 60◦; and a pa-rameter νc describing the strength of the orientationalconstraint. The first three are all known from earlier ex-perimental estimates (Craig and Linke, 2009; Dunn andSpudich, 2007; Moore et al., 2004). νc is along with bone of the two free parameters in the entire theoreticaldescription, and can be estimated based on fitting to ex-perimental data.

The full expression for P(r) and its derivation can befound in Hinczewski et al. (2013). The contours in Fig. 14illustrate P(r) for two different values of backwards loadforce: A) F = 0 pN; B) F = 2 pN. Superimposed aresample trajectories of the detached endpoint, correspond-ing to a forward and backward step respectively, alongwith snapshots of the polymer conformation at a timeshortly after detachment. The actin filament lies alongthe z-axis, with z = 0 corresponding to the location of at-tached head, and the forward/backward binding sites forthe detached head at z± = ±∆. At zero force the peak ofthe distribution is at z > 0 due to the post-powerstrokeorientational constraint on the bound leg. Thus the end-point probabilities are biased toward the forward bindingsite and P(r+)� P(r−). When F = 2 pN the situationis reversed: the load force pulls the junction backwards,counteracting the post-powerstroke constraint, and thedistribution is shifted such that P(r−) � P(r+). Thedependence of P(r) on load force translates into corre-sponding changes in t±fp through Eq. (60).

Once t±fp as a function of F is known, a variety of phys-ical quantities can be calculated directly from the kineticnetwork model (Hinczewski et al., 2013). For exampleboth forward steps and trailing stomps have the samemean duration: the average time tTb from the detach-ment of the trailing head to its subsequent reattachmentat either the forward or backward site. Similarly both

backward steps and leading stomps have a mean dura-tion tLb associated with how long the leading head takesto reattach. These two timescales are given by:

tTb = th +t+fp

1 + bα, tLb =

t+fpb+ α

(61)

where α ≡ t+fp/t−fp. Note that tTb is bounded from be-

low by th, because hydrolysis is a necessary intermediatestep after trailing head detachment, whereas it is not in-volved after leading head detachment. The probabilitiesthat the motor takes a forward step (Pf ), trailing stomp(PTs), leading stomp (PLs) and backward step (Pb) are:

Pf =g

1 + g

t2d1

(1 + bα)(td1 + th)(td1 + tTb − th),

PTs = bαPf , PLs =1

1 + g

btd1

(b+ α)(td1 + tLb),

Pb = b−1αPLs,

(62)

where g ≡ td2/td1 = 8 quantifies the strength of gat-ing. These are plotted as a function of load force for thesubstall regime in Fig. 15B, along with the terminationprobability Pt = 1−Pf −PTs−PLs−Pb. Forward stepspredominate at small forces, but are overtaken by trailingstomps as F approaches the stall value Fstall ≈ 1.9 pN,defined as when Pf = Pb. The force dependence of stompprobabilities has not yet been measured, and so consti-tutes a prediction of the theory, but there is experimen-tal data on the backward-to-forward ratio Pb/Pf (Kadet al., 2008). This is shown in comparison to the theo-retical curve in Fig. 16A, and exhibits excellent agree-ment. Other experimentally observable quantities arecompared in the remaining panels of Fig. 16: B) themean run length zrun along the actin before termination,from (Baker et al., 2004; Clemen et al., 2005; Pierobonet al., 2009; Sakamoto et al., 2000); C) the mean velocityvrun = zrun/trun, where trun is the mean run duration,from (Clemen et al., 2005; Gebhardt et al., 2006; Kadet al., 2008; Mehta et al., 1999; Uemura et al., 2004b).The theoretical expressions for these are:

zrun = vruntrun, vrun ≈∆

td1

(1

1 + bα− α

g(b+ α)

),

trun ≈gt2d1

tLb + gtTb.

(63)

The theory curves in Fig. 16 are simultaneous best-fitswith only two free parameters, νc and b, and overall showthat the theory quantitatively captures the main featuresof the motor dynamics (within experimental uncertain-ties evident in the scatter of data points collected underdifferent buffer and ATP condtions). Equally important,the theory provides insights into how motor propertiesdepend on both kinetic parameters (the gating ratio g,

28

A

B

C

FIG. 16 Best-fit theoretical results (solid curves) from thecoarse-grained polymer theory for myosin V (Hinczewskiet al., 2013) compared to experimental results (symbols) forthe following dynamical quantities: A. the ratio of backwardto forward steps, Pb/Pf ; B. the mean run length zrun onactin before detachment; C. mean velocity vrun. The sourcesof the experimental data are listed in the legends (Baker etal., 2004; Clemen et al., 2005; Gebhardt et al., 2006; Kad etal., 2008; Mehta et al., 1999; Pierobon et al., 2009; Sakamotoet al., 2000; Uemura et al., 2004) with buffer conditions inparentheses (first value is ATP concentration, second value isKCl concentration). Where the KCl value is not listed, theconcentration is 25 mM. All experiments / theory are for satu-rating ATP conditions except (Gebhardt et al., 2006) in panelC, where the theory has a modified t−1

d1 detachment rate toaccount for low ATP. Adapted from Hinczewski et al. (2013).

the reduction in binding rates described by b due to un-favorable lever arm conformations), and structural pa-rameters (L, lp, θc, and νc that control the distributionduring the diffusive search). This allows clear connec-tions to mutation experiments that perturb the latter,for example by extending or shortening lever arm lengthto change L (Oke et al., 2010; Sakamoto et al., 2005)

(see (Hathcock et al., 2019) for a fuller discussion).

VIII. SIMULATIONS USING COARSE-GRAINEDMODELS

Several models that can be simulated readily (Alhad-eff and Warshel, 2017; Goldtzvik et al., 2018; Hyeonand Onuchic, 2007a,b; Mugnai and Thirumalai, 2017;Mukherjee et al., 2017; Mukherjee and Warshel, 2013;Nam and Epureanu, 2016; Tehver and Thirumalai, 2010;Zhang et al., 2017; Zhang and Thirumalai, 2012) havebeen introduced in order address a variety of issues re-lated to stepping kinetics. These include but are notrestricted to the mechanism of stepping of conventionalkinesis on microtubule, gating mechanism, and allosterictransitions by which the motor heads communicate witheach other and the cytoskeletal filaments. The simula-tions have to be based on coarse-grained models (Hyeonand Thirumalai, 2011) because of the long time scalesand interplay of multiple length scales involved in themotor motility.

Rather than survey the findings in all these studies,which vary greatly in both details and foci, we describea particularly illuminating minimal CG mechanochemi-cal model (Craig and Linke, 2009) for myosin V, whichbeautifully illustrates strain-mediated gating mechanism(required for maintaining processivity) as well as char-acteristics such as speed and stall force. The CG model(see Fig. 17) , which takes the architecture of myosin Vin account, was constructed using the following assump-tions. First, the level arm is treated as a semiflexiblepolymer by representing the six IQ motifs by three inter-acting moieties. In this sense the model is similar to thesubsequent analytical polymer model (Hinczewski et al.,2013), described above. Second, the junction betweenthe head and the adjacent IQ motifs (points 2 and 8 inFig. 17) was treated as a semiflexible joint with equilib-rium angles that depend on the nucleotide state of themotor. Such an assumption is justified by comparisonto EM images. The forward rotation of the lever arm,which changes the angle from ΘA to ΘB , is taken to bedependent on phosphate release, a crucial step in the re-action cycle of myosin V. Third, the joint between the twolever arms is assumed to fully flexible, which would im-ply that the tethered head diffuses freely about this joint(see also (Hinczewski et al., 2013)). It should be notedthat recent experiments suggest that this may not be thecase. It has been pointed out that the angle between thelever arms is constrained (Andrecka et al., 2015), whichhas to be taken into account in describing the diffusivesearch (Hathcock et al., 2019). Fourth, the filamentousactin is treated as a passive one dimensional track withbinding sites that are space ∆ = 36 nm apart, as wasalso assumed in the analytical theory (Hinczewski et al.,2013). More importantly, if the tethered head, which has

29

FIG. 17 Illustrating a general strategy to construct and sim-ulate a coarse-grained model for myosin V (Craig and Linke,2009). The strategy involves making a coarse-grained modelbased on the structure (see (A) and (B)), which is coupledto the enzyme chemistry given in (C). (A) The lever arm isrepresented using three rigid segments that are connected toeach other. The two lever arms meet at point 5, which rotatesfreely during the stepping process. (B) The angle between thelever arm and the head (points 2 and 8) is assumed to changefrom ΘA to ΘB upon phosphate release. (C) The reactioncycle with various rates indicated in the figure is coupled tothe mechanical model. Brownian dynamics simulations wereused to calculate the observable quantities. The figure wasreproduced from Craig and Linke (2009).

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b·W = χ1

T21T22

η(ηC − η)

FIG. 18 a. Cost-error tradeoff relation and its physicalbound, Q = Qε2X > 2kBT . Accessible region for Q is in cyan.b. Power-efficiency tradeoff. Accessible region for power (W )is demarcated in cyan.

undergone hydrolysis, diffuses close to a binding site itinteracts with the binding site with an attractive electro-static interaction in order to complete a step.

A potential energy function based on the mechanicalmodel, which can be simulated using Brownian dynam-ics, is coupled to the catalytic cycle in one of the motorheads. In order to produce realistic dynamics variousrates in the cycle were taken from experiments (see Ta-ble 1 in (Craig and Linke, 2009)). The simulations weresuccessful in reproducing the run length distribution andthe value of the stall force (fS ≈ 2 − 3 pN). One of theadvantages of the CG simulations is that the stiffness ofthe head-neck and neck-neck joints, encoded by the termsVHN and VNN could be changed to assess the effect onthe motor properties. For example, they discovered thatthe value of fS depends on the stiffness of the lever arm.It has to be stiff but not overly so in order to reproducethe experimental data, which was later confirmed theo-retically (Hinczewski et al., 2013).

The strategies used in the models described in this sec-tion and the previous one are the following. First, the do-mains that execute mechanical movements are modeledusing available structural data. Second, the mechanicalmodel is coupled to the catalytic cycle, which allows oneto predict the dependence of measurable quantities asa function of control parameters such as ATP concen-tration and external load. The level of coarse-graining inthe first step is largely guided by intuition. In Hinczewskiet al. (2013) the use of polymer representation affordedanalytic solution whereas by discretizing the level armusing discrete connected links Craig and Linke (Craigand Linke, 2009) had to resort to numerical simulations.It is this general strategy that is likely to be successfulin tackling the nuances of dynein stepping and perhapsmotor functions in vivo.

30

IX. COST-PRECISION TRADE-OFF AND EFFICIENCYOF MOLECULAR MOTORS

1. Cost-precision trade-off and its physical bound of molecularmotors

From the perspective of thermodynamics, biologicalsystems are clearly in non-equilibrium, which means en-ergy is constantly injected and dissipated as heat. Be-cause they are subject to incessant thermal and non-thermal fluctuations, cellular processes are inherentlystochastic and error-prone. Thus, a plethora of energy-consuming machineries have evolved to fix any errorthat may be deleterious to biological functions. In thepresence of large fluctuations inherent to cellular pro-cesses, harnessing energy into precise motion and sup-pressing the uncertainty are critical for accuracy in cel-lular computation. Trade-off relations between the ener-getic cost and information processing have been a recur-ring theme for many decades in biology (Alberts et al.,2008; Banerjee et al., 2017; Bennett, 1982; Ehrenberg andBlomberg, 1980; Hopfield, 1974; Lan et al., 2012; Mehtaand Schwab, 2012). Recently, a concise and fundamen-tal relationship relating cost-precision trade-off and itsphysical bound, which is called the thermodynamic un-certainty relation (TUR), was first conjectured by Baratoand Seifert (Barato and Seifert, 2015) and extensivelystudied in the statistical physics community.

Barato and Seifert (Barato and Seifert, 2015) formu-lated the TUR, such that a product (Q) between the heatdissipation (Q(t)) and the square of relative error asso-ciated with a time-intergrated output observable X(t) ofthe process, ε2X(t) = 〈δX2〉/〈X〉2, is independent of mea-surement time. Based on numerical results that exten-sively sampled the rate constants kij defining the diversekinetic networks and the linear response theory. Theyfurther conjectured that Q cannot be smaller than 2kBTfor any chemical kinetic network described by Markovjump processes, which is succinctly written as,

Q = Q(t)× ε2X(t) ≥ 2kBT. (64)

The trade-off parameter Q quantifies the energetic costfor a given error and is bounded below by 2kBT . Thetime-integrated output observable X(t) can be selectedsuch that it can best represent the dynamic process ofinterest. For enzyme reactions that catalyze substrateto product, X(t) could be the product concentration,c(t). For molecular motors moving along one-dimensionaltrack, displacement (travel distance) x(t) is a natural out-put observable to represent their dynamic processes.

For motors, the relative error associated withthe motor displacement decreases with time as√

(δx(t))2/〈x(t)〉 ∝ 1/√t. Thus, if one were to decide

the displacement of a motor precisely, a longer time traceshould be generated, which demands more free energyinjection (ATP hydrolysis) and heat dissipation. The

greater the heat dissipated from the process, the smalleris the error. For molecular motors with output observablex(t), Eq.64 can be written in terms of three quantities(heat dissipation rate Q, diffusivity D, velocity V ) thatdepend on control parameters such ATP concentrationand external load.

Q = Q(t)× 〈δx(t)2〉〈x(t)〉2 = Q

2D

V 2≥ 2kBT. (65)

Notice that Q depends on a specific type of motor as wellas the conditions of [ATP] and f .

Since the Barato and Seifert’s original conjecture, therehas been impressive progress in the field. TUR hasbeen reinterpreted as the inequality relation betweengeneralized current and total entropy production rate(σtot = dS/dt), which can be written as

σtotVar(j)

〈j〉2 ≥ 2. (66)

General and elegant proofs for TUR have not only beengiven for the case of Markov jump processes on kineticnetworks by employing the large deviation theory (Gin-grich et al., 2016), but also can be deduced from theequality relation for the Fano factor of entropy produc-tion for over-damped Langevin processes (Pigolotti et al.,2017).

More recently, Dechant and Sasa (Dechant and Sasa,2018b) have generalized Eq.66 to underdamped processesin the following form

σtot ≥ B〈j〉2. (67)

where B(> 0) is a model dependent parameter, whichcan be reduced to 2/Var(j) for over-damped Langevinsystems or Markov jump processes on networks. In fact,right hand side of the inequality is always greater thanzero, namely, it recovers the second law of thermody-namics σtot ≥ 0. Therefore, one interpretation of Eq.67is that it provides tighter bound to the entropy produc-tion in terms of the square of generalized current. Al-though B is not specific but model dependent, Dechantet al.(Dechant and Sasa, 2018b) employed the above re-lation to derive a power-efficiency trade-off for engineoperating between two heat sources with temperatureT1 > T2.

W ≤ χ1T 2

1

T 22

η(ηC − η). (68)

where η = W/Q1 and ηC = 1 − T2/T1 are the thermo-dynamic efficiency and Carnot efficiency. The inequalitygives the upper bound to the power generated using thetwo heat sources. But, when η approaches to ηC , W ap-proaches to 0 as well if the model-dependent parameterχ1 is finite.

31

For pedagogical purpose, it is worthwhile to considersimple examples that can demonstrate the significance ofthe physical bound of TUR.

(i) In the context of the foregoing one-state hoppingmodel to which dynamics of molecular motors can bemapped, the rate of heat dissipation from the process isbounded as,

Q

kBT= (u− w) log

u

w≥ 2(u− w)2

(u+ w)(69)

where the inequality was discussed in Ref. (Shiraishiet al., 2016). Given a single step displacement d0, thevelocity and diffusivity of the particle moving alongthe reaction coordinate is V = d0(u − w) and D =(d2

0/2)(u + w), respectively. Therefore, inserting the ex-pressions of Q, V , and D to Eq.65 it is easy to see thatQ ≥ 2kBT . The lower bound of Q in this model is at-tained when u = w, which corresponds to the detailedbalance (equilibrium) condition. Provided that u = u∗[S]and v = v∗[P ], namely, the forward and backward rateconstants vary with the substrate (S) and product (P )concentrations, the detailed balance condition is attainedwhen [S]/[P ] = [S]eq/[P ]eq with u∗/v∗ = [P ]eq/[S]eq.

(ii) Hyeon et al. (Hyeon and Hwang, 2017) studied theover-damped Lagevin motion on tilted washboard poten-tial, which obeys

γx(t) = f − U ′(x) + ξ(t) (70)

with U(x + L) = U(x) and 〈ξ(t)〉 = 0, 〈ξ(t)ξ(t′)〉 =2γkBTδ(t−t′), where the driving of quasi-particle on thepotential is controlled by the non-conservative force f .They showed that TUR parameter Q(f) for this problemis a non-monotonic function of f , attaining its physicalbound 2kBT at both f � |U ′(x)| (near equilibrium) andf � |U ′(x)| (far from equilibrium). For f � |U ′(x)|, thequasi-particle undergoes diffusion on a rough surface. Onthe other hand, for f � |U ′(x)| the particle slides alonga smooth gradient, without feeling the effect of confin-ing potential, dissipating energy with a rate Q = γV 2

against friction. Since the diffusivity of particle in thiscase follows the Stokes-Einstein D = kBT/γ, togetherwith the velocity V one can obtain Q = 2kBT . Q(f)reaches its maximum value near the critical point wherethe potential barrier confining the particle is about tovanish.

(iii) As far as the specific models are concerned as in(i) and (ii), the minimal uncertainty condition (Qmin = 2kBT ) is attained not only under the detailed balancecondition but when there is no confining potential. Ithas been suggested that Qmin is attained when the heatdissipated from the process is normally distributed asP (Q) ∼ e−(Q−〈Q〉)2/(2〈δQ2〉) and that TUR measures thedeviation of heat distribution from Gaussianity (Hyeonand Hwang, 2017).

2. Transport Efficiency

There are a number of ways to assess the “efficiency”of engines or machines (Brown and Sivak, 2017). Histor-ically, the efficiency of heat engines has been discussed interms of the thermodynamic efficiency, the aim of whichis to maximize the amount of work extracted from twoheat sources with different temperatures (Callen, 1985).For non-equilibrium machines driven by chemical forcesthat are constantly regulated without shortage in the livecell, the power production could be a more pertinentquantity to maximize. Meanwhile, for transport motorsin the cell, the TUR parameter Q can be used to assessthe efficiency of suppressing the uncertainty in dynami-cal process by means of energy consumption, and thus isquite pertinent for evaluating the transport efficiency of amotor (or motors) (Dechant and Sasa, 2018a). The con-nection betweenQ and the transport efficiency is clear. Ifa motor transports cargos at a high speed (V ∼ 〈x(t)〉/t)with small fluctuations (D ∼ 〈δx(t)2〉/t) (which leads topunctual delivery to a target site) but consuming only asmall amount of energy (Q), such a motor would be con-sidered efficient for cargo transport. A motor efficient inthe cargo transport would be characterized by a small Qwith its minimal bound 2kBT , or as originally suggestedby Dechant and Sasa (Dechant and Sasa, 2018a), onecan consider using the definition ηT = 2kBT/Q which isbounded between 0 and 1.

Q can be used to assess the “transport efficiency” ofbiological nanomachines and to study how it changeswith varying conditions of f and [ATP]. To evaluateQ, measurement should be first carried out for Q, D,and V (see Eq.65). While V and D are straightfor-ward to calculate (V = limt→∞ dx(t)/dt and D =limt→∞(1/2)d(δx(t))2/dt), experimental measurement ofQ may be nontrivial. Although there are some reportson direct measurements of heat dissipation rate at singlecell level (Rodenfels et al., 2019; Song et al., 2019), di-rect measurement of heat dissipation at single moleculelevel is not yet known. Nevertheless, Q be estimated byconsidering a physically suitable minimal kinetic networkmodel. For a given cyclic kinetic network defined by mul-tiple chemical states (i = 1, 2, . . . N) and transition rates({kij}) connecting them, there are straightforward meth-ods (Hwang and Hyeon, 2017; Koza, 1999; Lebowitz andSpohn, 1999) to associate the measured V and D with{kij}. As long as all the rate constants {kij} defining thekinetic network are known, it is then straightforward tocalculate Q value as discussed in details in the early partof this review.

For conventional kinesin, whose chemomechanicalproperties were extensively studied by several groups,motility data for varying [ATP] and load conditions areavailable in the literature. The double-cycle networkmodel, depicted in Fig.8b, with the form of rate con-stant for each edge can be used to analyze those data,

32

FIG. 19 a. Analysis of motility data to determine the pa-rameters for rate constants ({kij} in Eq.47) for double cyclenetwork model. b. Diagram of Q as a function of [ATP] andf .

which allows one to determine the dependence of {kij} on[ATP] and f and finally to build a diagram ofQ(f, [ATP])as shown in Fig.19. The TUR diagram, Q([ATP ], f)(Fig.19), exhibit several features that are worthwhile toexplore:

(i) In terms of the load direction, assisting (f < 0)and hindering (f > 0), Q([ATP ], f) is asymmetric. Thisresult differs from that of the one-state hopping examplefor TUR calculated in Fig.20a. The fundamental differ-ence arises from the asymmetric effect of the force on thekinetic rate, particularly on k25 and k52 with θ 6= 1/2.

(ii) Q is locally minimized first at a condition of smallload (f & 0) and low [ATP] (indicated by cyan ellipse),and second at f ≈ 3 pN and [ATP] ≈ 300 µM (indicatedby the yellow arrow). The first minimum is closer to 2kBT bound; yet, this minimum was attained near thedetailed balance condition where [ATP] concentration issmall and balanced with [ADP] and [Pi]. It does nothave much of biological relevance given that molecularmotor works out of equilibrium. In fact, the second localminimum is of particular interest given that the conditionis not so far from the cellular condtion [ATP] ≈ 1 mMand f ≈ 1 pN, indicated by yellow arrow (Here, note thatf ≈ 1 pN is a rough estimate of cellular environment

FIG. 20 Two case studies for TUR. a. One-state hoppingmodel with forward and backward rate constants u and w. b.Brownian motion in tilted wash-board potential. Q(f) wasevaluated as a function f using a specific periodic function,U(x) = U0 sin 2πx/L with L = 6 nm for U0 = 5 and 10 kBT .

replete with obstacles such as cytoskeletal filaments androad blocks). The second local minimum is the very pointat which the transport efficiency defined in terms of Q isoptimized.

(iii) The high Q region (Q > 100 kBT ) at f ≈ 3 − 7pN is due to the stall condition. As explained in detailin rationalizing the double-cycle network model, energyshould be still consumed and heat should be dissipated(Q > 0) at stall condition (V ≈ 0). This particularcondition renders Q divergent at stall conditions.

Finally, the structure of Q(f, [ATP]) is sensitive tothe design of motor structure as well as the motor type.First, the diagram of Q(f, [ATP]) for a kinesin construct,Kin6AA, whose necklinker is engineered longer than thatof wild-type kinesin-1 via insertion of six amino-acids(AEQKLT) (Clancy et al., 2011) exhibits great devia-tion from that of the WT (Fig.21a). In all, the values ofQ were increased, the stall forces were reduced, and thelocal minimum observed in the WT is missing. KIF17and KIF3AB (Milic et al., 2017) show qualitatively sim-ilar structure of Q(f, [ATP]); however there are differentin terms of quantitative details from that of WT. Next,myosin V, dynein, and F1-ATPase were analyzed to cal-culate Q(f, [ATP]). Bierbaum and Lipowsky (Bierbaumand Lipowsky, 2011) employed a tri-cyclic network modelto describe the chemomechanics of myosin V in whichcycles for forward steps, energy-consuming futile steps,and force-induced mechanical slippage steps were consid-ered. The resulting Q(f, [ATP]) (Fig.21d) shows ATP-insensitive stall condition where Q is divergent with nolocal minimum as in WT kinesin-1. For dynein (Fig.21e),the uni-cyclic chemomechanical network model adoptedfor construction of Q(f, [ATP]) is in principle not satis-factory, since it would be able to account for the phys-ically correct behavior of dynein dynamics at stall andsuperstall conditions. In particular, Q(f, [ATP]) diagramshows minimization to 2kBT at the stall. Nevertheless,a suboptimal local minimum is found at f ≈ 3 pN and

33

FIG. 21 Q(f, [ATP]) for a. Kin6AA (kinesin-1 mutant); b.KIF17; c. KIF3AB; d. myosin V; e. Dynein; f. F1ATPase.The diagrams were calculated by directly analyzing the motil-ity data available in (Clancy et al., 2011) for a, (Milic et al.,2017) for b, c, and using the same double-cycle network modelas that of kinesin-1. For other motors in d, e, and f, the ki-netic network and rate constants were used as in the Ref.(Bierbaum and Lipowsky, 2011) for myosin V, Ref. (Sarlahand Vilfan, 2014) for dynesin, and Ref.(Gerritsma and Gas-pard, 2010) for F1-ATPase.

[ATP] ≈ 300 µM . Finally, Q(τ, [ATP]) (torque τ insteadof load f) for F1ATPase is shown in Fig.21f calculatedbased on uni-cyclic kinetic network model (Gerritsmaand Gaspard, 2010). For F1-ATPase, which shows near-reversibility and hence characterized with high efficiency(Toyabe et al., 2010), the use of uni-cyclic network modelwould be reasonable although this conclusion should bereached with care (see (Sumi and Klumpp, 2019)).

Some of the biological motors, kinesin family anddynein, studied here are found to be semi-optimized interms of Q under the cellular condition, which alludesto the role of evolutionary pressure that has shaped thepresent forms of molecular motors in the cell. In addi-tion, the efficiency quantified in terms of Q for variousmolecular machines presented here are ranged between 7and 20 kBT (Fig.22). Given that all these machines func-tion out-of-equilibrium, it is of great interest to discoverthat the value ofQ calculated at the working cellular con-dition is not significantly far from its physical bound 2kBT . This may arise from the fact that the molecular ma-chine analyzed here are tightly coupled machine, mean-ing that ATP hydrolysis is almost always transduced to amachanical step even though energy-wasting futile stepsstill remain as a possibility.

As long as the underlying mechanism, which offersa clear model for chemomechanical kinetic network, isknown, TUR can be studied for any time trace generatedfrom cyclic process at NESS. Other energy-intensiveprocesses, for example, error-correction processes, circa-dian cycle, and chaperonin action, would exhibit high Q.

2

q(k

BT

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physical bound (2kT)

7.27.79.19.91319

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Q/kBT<latexit sha1_base64="XkuMbV+twAjVDzhjov3bMpAfjF8=">AAAB73icbVDLSgNBEOz1GeNr1aOXwSB4irsi6DHEi8cE8oIkLLOT2WTI7MOZ3kBY8h1eRLwo+Cf+gn/jbJJLEgsGiqoauqv9RAqNjvNrbW3v7O7tFw6Kh0fHJ6f22XlLx6livMliGauOTzWXIuJNFCh5J1Gchr7kbX/8lPvtCVdaxFEDpwnvh3QYiUAwikbybLsXUhwxKrP67HbsVRueXXLKzhxkk7hLUoIlap790xvELA15hExSrbuuk2A/owoFk3xW7KWaJ5SN6ZBn831n5NpIAxLEyrwIyVxdydFQ62nom2S+nV73cvE/r5ti8NjPRJSkyCO2GBSkkmBM8vJkIBRnKKeGUKaE2ZCwEVWUoTlR0VR314tuktZd2TW8fl+qVJdHKMAlXMENuPAAFXiGGjSBwQTe4BO+rBfr1Xq3PhbRLWv55wJWYH3/AeskjwQ=</latexit><latexit sha1_base64="XkuMbV+twAjVDzhjov3bMpAfjF8=">AAAB73icbVDLSgNBEOz1GeNr1aOXwSB4irsi6DHEi8cE8oIkLLOT2WTI7MOZ3kBY8h1eRLwo+Cf+gn/jbJJLEgsGiqoauqv9RAqNjvNrbW3v7O7tFw6Kh0fHJ6f22XlLx6livMliGauOTzWXIuJNFCh5J1Gchr7kbX/8lPvtCVdaxFEDpwnvh3QYiUAwikbybLsXUhwxKrP67HbsVRueXXLKzhxkk7hLUoIlap790xvELA15hExSrbuuk2A/owoFk3xW7KWaJ5SN6ZBn831n5NpIAxLEyrwIyVxdydFQ62nom2S+nV73cvE/r5ti8NjPRJSkyCO2GBSkkmBM8vJkIBRnKKeGUKaE2ZCwEVWUoTlR0VR314tuktZd2TW8fl+qVJdHKMAlXMENuPAAFXiGGjSBwQTe4BO+rBfr1Xq3PhbRLWv55wJWYH3/AeskjwQ=</latexit><latexit sha1_base64="XkuMbV+twAjVDzhjov3bMpAfjF8=">AAAB73icbVDLSgNBEOz1GeNr1aOXwSB4irsi6DHEi8cE8oIkLLOT2WTI7MOZ3kBY8h1eRLwo+Cf+gn/jbJJLEgsGiqoauqv9RAqNjvNrbW3v7O7tFw6Kh0fHJ6f22XlLx6livMliGauOTzWXIuJNFCh5J1Gchr7kbX/8lPvtCVdaxFEDpwnvh3QYiUAwikbybLsXUhwxKrP67HbsVRueXXLKzhxkk7hLUoIlap790xvELA15hExSrbuuk2A/owoFk3xW7KWaJ5SN6ZBn831n5NpIAxLEyrwIyVxdydFQ62nom2S+nV73cvE/r5ti8NjPRJSkyCO2GBSkkmBM8vJkIBRnKKeGUKaE2ZCwEVWUoTlR0VR314tuktZd2TW8fl+qVJdHKMAlXMENuPAAFXiGGjSBwQTe4BO+rBfr1Xq3PhbRLWv55wJWYH3/AeskjwQ=</latexit><latexit sha1_base64="XkuMbV+twAjVDzhjov3bMpAfjF8=">AAAB73icbVDLSgNBEOz1GeNr1aOXwSB4irsi6DHEi8cE8oIkLLOT2WTI7MOZ3kBY8h1eRLwo+Cf+gn/jbJJLEgsGiqoauqv9RAqNjvNrbW3v7O7tFw6Kh0fHJ6f22XlLx6livMliGauOTzWXIuJNFCh5J1Gchr7kbX/8lPvtCVdaxFEDpwnvh3QYiUAwikbybLsXUhwxKrP67HbsVRueXXLKzhxkk7hLUoIlap790xvELA15hExSrbuuk2A/owoFk3xW7KWaJ5SN6ZBn831n5NpIAxLEyrwIyVxdydFQ62nom2S+nV73cvE/r5ti8NjPRJSkyCO2GBSkkmBM8vJkIBRnKKeGUKaE2ZCwEVWUoTlR0VR314tuktZd2TW8fl+qVJdHKMAlXMENuPAAFXiGGjSBwQTe4BO+rBfr1Xq3PhbRLWv55wJWYH3/AeskjwQ=</latexit>

Q(k

BT

)<latexit sha1_base64="Oat/vLb5bWROR/YK7tnr+xIx7Qo=">AAAB+XicbVDLSsNAFJ34rPUVdaebwSLUTUmqoMtSNy5b6AuaEibTaTt08mDmRiwh4K+4EXGj4E/4C/6Nkzabth4YOJxzhnvv8SLBFVjWr7GxubW9s1vYK+4fHB4dmyenHRXGkrI2DUUoex5RTPCAtYGDYL1IMuJ7gnW96UPmd5+YVDwMWjCL2MAn44CPOCWgJdc8d3wCE0pE0kyxA+wZEpyWp269de2aJatizYHXiZ2TEsrRcM0fZxjS2GcBUEGU6ttWBIOESOBUsLToxIpFhE7JmCXzzVN8paUhHoVSvwDwXF3KEV+pme/pZLanWvUy8T+vH8PofpDwIIqBBXQxaBQLDCHOasBDLhkFMdOEUMn1hphOiCQUdFlFfbq9eug66VQr9k2l2rwt1ep5CQV0gS5RGdnoDtXQI2qgNqLoBb2hT/RlJMar8W58LKIbRv7nDC3B+P4D0raS5w==</latexit>

FIG. 22 Q values for various transport motors calculated at[ATP] = 1 mM and f = 1 pN (except for F1-ATPase calcu-lated at [ATP] = 1 mM and τ = 0 pN·nm)

X. MOLECULAR CHAPERONES

Chaperones are another class of cellular molecular ma-chines that expend free energy change associated withATP binding and catalysis to facilitate the folding of cer-tain proteins and RNA that cannot fold spontaneouslyunder cellular conditions. Because they serve a key func-tion in maintaining protein homeostasis they are deemedas essential for the survival of the organisms. Among alarge class of molecular chaperones, the function of theheat-shock GroEL-GroES machinery found in E.Coli, re-ferred to as cheperonin, is now quantitatively understoodthanks to experimental and theoretical advances. The invivo function of GroES-GroEL machinery in E. Coli. isto rescue substrate proteins that are otherwise destinedfor aggregation. Just like molecular motors, GroEL un-dergoes a catalytic cycle involving a series of large scalestructural changes in response to ATP binding, hydroly-sis, and release of ADP and phosphate. GroEL/GroESmachine anneals the population of misfolded proteinsdriving them to the folded state by repeatedly goingthrough rounds of the catalytic cycle, which has beenreferred to as the Iterative Annealing Mechanism (IAM)(Tehver and Thirumalai, 2008; Todd et al., 1996).

A minimal kinetic network model of chaperonin-assisted protein folding, can be constructed, based onconsiderable experimental evidence, by assuming that aprotein exists either in an intermediate (I), misfolded(M), or folded (native) (N) state as illustrated in Fig.23.The unfolded state, a transient state right after the pro-tein is synthesized, collapses to the I state, and it furtherundergoes a spontaneous folding process via the kineticpathways with rates denoted by kIM and kIN . Only afraction (Φ) of the entire population folds correctly to theN state and the remaining fraction (1 − Φ) is misfoldedto the M state. The process of the initial populationof molecules with Φ (1 − Φ) reaching the folded (mis-

34

FIG. 23 Chaperone-assisted folding of substrate. a.Schematic of the iterative annealing mechanism (IAM) illus-trated for the hemicycle of GroEL. The T R transitionstarts when ATP and the substrate protein (SP) bind. GroESbinding and ATP hydrolysis engenders the R′ → R′′ transi-tion with a fraction of the SP partitioning to the folded struc-ture. Subsequently ADP and Pi and the SP (folded or not)are released and the R′′ → T transition completes the cycle.In the presence of the SP the machine turns over in abouta second and ADP release is accelerated by about a hundredfold. The rapid turnover comports well with the predictions ofIAM (Todd et al., 1996). b. A minimal kinetic network modelfor chaperone-assisted folding of a substrate molecule. Tran-sitions between three manifolds of collapsed intermediate (I),misfolded states (M), and native state (N) are representedin terms of rate constants. Folding, misfolding, and chaper-one assisted unfolding pathways are depicted in blue, red, andmagenta, respectively. c. Schematic of folding landscape ofsubstrate molecules. Upon spontaneous folding, the ensembleof intermediate states collapsed from unfolded ensemble reachthe native and misfolded basins of attractions with the pro-portion of Φ and 1−Φ, respectively. d. Schematic of the gen-eralized iterative annealing mechanism of chaperone-assistedsubstrate folding, from which the recursion relation for thenative yield from n-th annealing process was derived. Ni andMi denote the proportion of native and misfolded states fromthe i-th annealing process. The blue and red arrows repre-sent the pathways leading to the native and misfolded states,respectively.

folded) state is termed the Kinetic Partitioning Mech-anism (KPM). The KPM, which was first theoreticallyproposed (Guo and Thirumalai, 1995), has been used toexplain the folding of of proteins (Kiefhaber, 1995) aswell as RNA (Pan et al., 1997; Thirumalai and Hyeon,2005). Typically, substrate proteins (SPs or ribozymes)that require assistance from chaperonin action are char-acterized by extremely small values of Φ (� 1), whichimplies that the majority of population without chap-erones are trapped in the misfolded states, which couldpotentially aggregate unless rescued by the chaperones.

The function of the GroEL machine is quantitativelyexplained by the Iterative Annealing Mechanism (IAM)according to which the chaperone recognizes and actson misfolded substrate preferentially. Typically, the SPshave exposed hydrophobic residues, which make themprone to aggregation unless recognized by the GroEL-GoES machine. When the SPs bind to GroEL theybecome disordered as a result of domain movements inGroEL, which imparts a mechanical force (≈ 10pN) thatis sufficiently large to unfold (at least partially) the SPs(Thirumalai and Lorimer, 2001). As the catalytic cy-cle proceeds the SPs are encapsulated for a brief period(roughly about two seconds) during which a small frac-tion folds rapidly during the time they are in the cavity.It is worth remarking that if they fold (the probability be-ing Φ), they do so while being encapsulated in the cavityof GroEL. When the catalytic cycle is complete, the SP isejected from the cavity regardless of whether it is foldedor not. If the SP is misfolded then it once again recog-nized by GroEL and the cycle is iterated until sufficientyield of the folded state is obtained. A key requirement ofGroEL-SP interaction is that hydrophobic residues of theSPs must be exposed, which does not typically occur infolded states. It this ability of chaperone not to recognizenative proteins that enables the GroEL-GroES machin-ery to drive the misfolded states to the native state overrepeated iterations of the catalytic cycle. More specifi-cally, when the annealing process, corresponding to onecatalytic cycle, is iterated n times, the total amount ofpopulation that reaches the native state (or native yield)grows as,

Nn = 1− (1− Φ)n (71)

with n ≥ 1. As n→∞, Nn → 1.

While it is known that GroEL only recognizes mis-folded SPs, the IAM concept has to be generalized toRNA chaperones, which act on the both N and M states,but more favorably on M (Bhaskaran and Russell, 2007;Chakrabarti et al., 2017). If the proportion of N iden-tified by chaperones in comparison with the M state isdefined as κ (0 ≤ κ ≤ 1), the total yield of native stateafter n rounds of folding (annealing) in the presence ofchaperone can be calculated using the following mathe-matical formulation:

35

(i) LetNn andMn be the yields of native and misfoldedstate, respectively, after the n-th round of the annealingprocess. Note that the total amount of substrate proteinsis conserved at all times, which implies that Nn+Mn = 1for all n. In the first round of annealing, a fraction N1(=Φ) folds to the N state and M1(= 1 − Φ) partitions tothe M state;

(ii) In the n-th round of annealing, chaperones rec-ognize Nn−1 and Mn−1 differentially by the factor κ.Whereas (1 − κ)Nn−1 is left unrecognized by chaper-ones, κNn−1 is unfolded and Φ of them refold to yieldκΦNn−1 native states and κ(1−Φ)Nn−1 misfolded states.On the other hand, the entire population of Mn−1 is un-folded and Φ of them refold to yield ΦMn−1 native statesand (1 − Φ)Mn−1 misfolded states. Therefore, after the(n−1)-th round of chaperone action, the native yield Nnis determined as Nn = 1−Mn = 1−(1−Φ)Mn−1−κ(1−Φ)Nn−1;

(iii) From the resulting recursion relation of Nn = (1−κ)(1−Φ)Nn−1 +Φ with N1 = Φ, we obtain the followingexpression,

Nn = Φ1− (1− κ)n(1− Φ)n

κ+ (1− κ)Φ, (72)

native yield after n iterations. After a sufficient numberof iterations (n → ∞), the system reaches steady state,N∞ → Φ/(κ + (1 − κ)Φ). In the case of GroEL onlythe misfolded state is recognized by chaperones (κ = 0).Therefore, the GroEL-GroES machinery drives the en-tire population of substrate proteins to the native stateN∞ → 1 (Eq.71).

As described for molecular motors, the thermodynamicaspects of chaperonin-assisted folding can be succinctlycaptured by mapping the folding process of substratemolecule onto a kinetic network model. A uni-cyclic re-versible kinetic network model consisting of the above-mentioned three states (I, M , and N) suffices to capturethe non-equilibrium nature of chaperone-assisted proteinfolding. The relevant master Equation describing thenetwork is,

∂tP(t) =WP(t) (73)

where P(t) = (PI(t), PN (t), PM (t))T and,

W =

−(kIN + kIM ) kNI kMI

kIN −(kNI + kNM ) kMN

kIM kNM −(kMI + kMN

The probability Pi(t) of each state i evolves with timeas,

P(t) = Pss + c1~u1e−λ1t + c2~u2e

−λ2t (74)

with λ2 > λ1 > 0, and it reaches the steady statevalue P ssi at t → ∞. The steady state populationP ssi and steady state current along the reaction cycle

(J = kijPssi − kjiP ssj ) can be expressed solely using the

rate constants, as already discussed in depth in the fore-going section describing the general aspects of the vari-ous cycles. Because the function of the chaperones is topromote the formation of the folded state our primary in-terest is the steady state yield of the native state, whichis given by,

P ssN =kMI([C], [T ])kIN + kMN (kIM + kIN )

Σ([C], [T ]), (75)

where

Σ([C], [T ]) =

kMI([C], [T ])kNI([C], [T ])

+ kNI([C], [T ])(kMN + kIM ) + kMI([C], [T ])(kIN + kNM )

+ (kIM + kIN )(kNM + kMN ).(76)

The partition factor Φ can be expressed in terms of rateconstants as Φ = kIN/(kIN + kIM). Because of chaper-one action, kIN and kMI can be significant (in particularkIN � kNI), whereas kNM and kMN are negligible. Underthis condition, the native yield simplifies to:

P ssN 'kMIkIN

kMIkNI + kNIkIM + kMIkIN

' kIN

kNI

kMIkIM + kIN

=Φ

κ+ (1− κ)Φ= N∞ (77)

In the absence of either of chaperone or ATP whichredistributes the population of proteins into NESS, [C],[T ]-dependent rate constants vanish (kij([C], [T ]) = 0).In this case, the steady state population of N state be-comes

P ssN ([C] = 0 or [T ] = 0) =1

1 + kNM/kMN

=1

1 + e−∆GNM/kBT

= P eqN (78)

Replacing the two [C], [T ]-dependent rate constants(kIN and kIM ) to zero is tantamount to blocking thechaperone action and placing the M and N states inisolation. As long as they are isolated for long enoughtime greater than k−1

NM and k−1MN , the system would

finally reach the equilibrium native yield (P eqN ), as dic-tated in Eq.78 assuming that aggregation reaction canbe neglected. However, equilibrating M and N statesvia the transition paths of M N is impractical in thelight of the time scale of cellular processes because k−1

NM

and k−1MN would far exceed the biologically meaningful

time scale. These arguments suggest that chaperonesdrive the substrates out of equilibrium. In the process,they optimize the yield of the folded SPs or RNAs perunit time, which we discuss further below.

36

0 2 4 6 8 10 120

50

100

150

200

250

Force (pN)

Velocity

(bp/s)

0 2 4 6 8 100

1000

2000

3000

4000

Force (pN)

Processivity

(bp)

A B

C

D

E

FIG. 24 Helicase model. Panels (A-D): Schematic of a modelshowing the unwinding of nucleic acids by helicases. (A)Translocation process of a helicase, with k+, k−, and γ beingthe rates of forward, backward, detachment, respectively; n isthe position on the nucleic acid. (B) Opening (α) and closing(β) rates of a base pair. (C) Interactions between helicase andnuceic acids modify the stepping kinetics of the helicase andand the opening and closing rates of the base pair. The basepair location is m and j = m− n. (D) A model for the inter-action with U0 being the strength. For a passive helicase U0 iszero. Panel (E): Simultaneous fits of the force-dependence ofthe velocity and processivity (given in the inset) of T7 helicaseas a function of an external load. The blue and red lines arethe results using the theory described elsewhere (Chakrabartiet al., 2019) and the data points in circles are taken fromexperiments (Johnson et al., 2007).

XI. UNIVERSAL CHARACTERISTICS OF HELICASES

Helicases, which are molecular motors found in all or-ganisms, unwind double stranded (ds) DNA and dsRNAwhen they encounter a junction between single strand(ss)-ds nucleic acids (Delagoutte and Von Hippel, 2002,2003; Lohman, 1992; Lohman and Bjornson, 1996). Sep-aration of dsDNA strands is required for DNA replicationas well as DNA repair. Malfunction of helicases causesgenomic instability and are also implicated in cancer.In addition, certain RNA chaperones are also deemedto have helicase activity, which means they are able tounwind helices in RNA in order to facilitate its folding(Mohr et al., 2002; Russell et al., 2013). We refer thereader to a number of articles that describe a variety ofcellular functions associated with helicases (Bianco andKowalczykowski, 2000; Bustamante et al., 2011; Donget al., 1995; Lohman et al., 2008; Marians, 2000; Panget al., 2002; Pyle, 2008; Rocak and Linder, 2004; Ve-lankar et al., 1999; Venkatesan et al., 1982). Helicases,classified into six super families based on their sequences(Gorbalenya and Koonin, 1993; Iyer et al., 2004), are de-scribed as active or passive (Lohman, 1992). Active he-

licases destabilize the base pairs of the dsDNA, perhapsby exerting a force, thus separating the two strands. Ifthe helicase is passive then it binds to the ssDNA when-ever thermal fluctuations transiently open the base pairs.As the strands of the dsDNA are separated the helicasetranslocates along the ssDNA. This strand separationand translocation are intimately related. Several ensem-ble experiments have provided glimpses into the steppingmechanism (Ali and Lohman, 1997; Jeong et al., 2004;Levin and Patel, 2002; Lucius et al., 2003; Wong andLohman, 1992) of helicases. In addition, these experi-ments have also been insightful in deciphering how theyinteract with single strand–double strand (ss-ds) junc-tions, and how often they dissociate from their track.The most detailed picture of the functions of a numberof helicases have come from single molecule laser opti-cal tweezers (LOT) and Magnetic Tweezers (MT) exper-iments. These experiments provided the kinetics of step-ping and nucleic acid unwinding (Dessinges et al., 2004;Lionnet et al., 2006; Patel and Picha, 2000; Perkins et al.,2004). Such measurements are instrumental in not onlyformulating theories and simulations but also in refiningthem as additional high precision experiments becomeavailable.

In a remarkable and influential paper, Betterton andJulicher (BJ) (Betterton and Julicher, 2003, 2005a,b)presented a theory, which describes quantitatively thecoupling between translocation and strand separation.The framework used in this theory, which is another il-lustration of the SKM, has been most instrumental inunderstanding the differences between active and passivehelicases. The BJ model assumes that the helicase movesforward (backward) at a rate k+ (k−) when it is very farfrom the ss-ds junction. This aspect of the motor move-ment is similar to Fig. 5 with γ=0. When the motoris very far from the ss-ds junction the helicase merelytranslocates along the ss nucleic acids. Similarly, in iso-lation a nucleic acid base opens at a rate α and closes ata rate β. Depending on whether it is a AT or a GC basepair the rates are different but the inequality β � α holdsin both cases. Since isolated base pair opening rates aredue to thermal fluctuations they satisfy α

β = e−∆Gbp/kBT

where ∆Gbp is stability of the base pair. However, inter-actions with base pairs modify these rates. Let n be theposition of the helicase on the track and m be the loca-tion of the ss-ds junction (Fig. 24). Upon approachingthe ds-ss junction the helicase interacts with NA, whichwas modeled using a variety of potentials - all based onsome combination square well like potentials. Passivehelicases, characterized by U0 → ∞, opportunisticallystep when the base is open. For active helicases, whichforcibly rupture the base pair interactions, U0 is finite butdepends on j = m− n. The rates k+ and k− and α andβ are modified when the helicase interacts with the NA(Betterton and Julicher, 2005b). To describe the actionof the helicase one has to keep track of its position on the

37

track as well the ss-ds location. Helicase-NA interactionsmodify all the relevant rates making them position (j)dependent. Using the model, with detachment rate setto zero, the velocity of the helicase assuming that it stepsone base pair at a time, is given by,

V =1

2

∑j

(k+j + αj − k−j − βj

)Pj , (79)

where Pj is the probability of observing the helicase andjunction that are separated separated by j, αj is the rateat which the junction opens when the helicase and junc-tion are at separation j. The corresponding rates for basepair closing, forward, and backward stepping rates of thehelicase are βj , k

+j , and k−j , respectively.

Although experiments (see for example (Manosaset al., 2010)) have been analyzed using the theorysketched above the BJ model has to be generalized inorder to make precise comparison with experiments. (1)To account for the average helicase processivity (〈δ〉m),which was not addressed in the original formulation, theconsequences of detachment of the motor have to be con-sidered. As pointed out earlier, almost all of the chemi-cal kinetics models ignore detachment, which of course isunrealistic because the run-length in motors or numberof base pairs that are disrupted if finite. (2) A theorythat allows for arbitrary step size (s) is needed insteadof assuming that s is unity. Helicases, such as PcrA andNS3 interact with and possibly destabilize several basepairs that are down stream of the ss-ds junction (Chenget al., 2007; Velankar et al., 1999). In other words, stepsize s exceeds unity. (3) Experiments also apply externalforce and measure the changes in the processivity andvelocity as a function of force, f . A viable theory shouldproduce tractable expressions for both the mean velocityand 〈δm〉 as a function of of quantities. (4) Finally, howthe sequence of the NA affects 〈δm〉 and V needs to beconsidered in order to draw general conclusions.

An analytically solvable model that accounts for thefirst three effects stated above has been proposed recently(Chakrabarti et al., 2019), which was preceded by a lessgeneral theory (s was set to unity) (Pincus et al., 2015)that investigate sequence effects ob the mean velocity and〈δ〉m. These studies produced a number of unexpectedpredictions for the helicase velocity and processivity as afunction of external force and DNA sequence. (i) It waspredicted that, regardless of the underlying architectureand unwinding kinetics of the helicase or the precise DNAsequence, processivity exhibits a universal increase withapplied external force. This finding, which has subse-quently has been validated experimentally (Bagchi et al.,2018; Li et al., 2016), was used to suggest that helicasesmay have evolved to maximize processivity rather thanvelocity. (ii) The theory, which quantitatively accountsfor the experiments for force-dependent V and 〈δm〉 forT7 replisome (Fig. 6 from BJ 2019), shows that s = 2

base pairs. (iii) Normally, in analyzing experimental databack NTP-dependent stepping rates are neglected. Thisis justified while the helicase translocates along ss NA.For instance, in T7 helicase the ratio of the forward toback stepping rate is ∼ 270, a value that is not thatdifferent from kinesin at zero resistive force. The proba-bility of back-stepping is ≈ 0.3%. However, when T7 un-winds dsDNA the probability of back-stepping increasesto 26%. This estimate is not that dissimilar to obser-vations in XPD helicases belonging to a different (SF2)family, where it was shown that at 1mM ATP concentra-tion the back-stepping probability is ∼ 10%. (iii) Inter-estingly, many helicases do not function in vitro unless anexternal force is applied. Under in vivo conditions part-ner proteins that bind to single strands and impart a forceat the ss-ds junction to destabilize the base pairs (Pincuset al., 2015) are needed. That this is the case has beenshown for UvrD helicase (Comstock et al., 2015), whichbehaves as a processive motor only when 2pN force isapplied. It was noted previously that even though theseassociated proteins may or may not increase the unwind-ing velocity of a helicase they should universally increasethe processivity of the helicase.

The theories for helicase stepping are not complete be-cause they do not resolve many of the challenging prob-lems. First, there is no quantitative explanation for thevery broad velocity distribution (Johnson et al., 2007)measured during unwinding of the weakly active ring-shaped T7 helicase. It is challenging to calculate P (v)for the recent model (Chakrabarti et al., 2019), whichis the minimal that accounts for the F -dependent meanvelocity and δm accurately. Second, there is very lit-tle understanding of the structural basis of the universalincrease in 〈δm〉 as a function of F and the underly-ing dramatic variations in sequence and architectures ofthe motor. Perhaps, carefully designed simulations mightshed light on this issue (Ma et al., 2018; Yu et al., 2006).Third, a recurring theme in many aspects of motors isthat the dynamics is heterogeneous, exhibiting charac-teristics reminiscent of glasses. It was discovered thatthere are substantial molecule-to-molecule variations inthe unwinding speed of E. Coli. RecBCD helicase even ifall the enzymes are prepared under the same condition.To account for this observation it has been suggestedthat the functional landscape is likely partitioned into anumber of metastable states and the initial preparationquenches the enzyme into a specific substate (Kirkpatrickand Thirumalai, 2015) The helicase ergodically exploresall the conformations within a single metastable state butthe transitions to other states could only be achieved byresetting the ATP concentration (Liu et al., 2013). Theemergence and relevance of glass-like heterogeneous be-havior, under ambient conditions, is not understood the-oretically, and remains a challenge not only in the contextof helicases but also in other biological systems as well(Altschuler and Wu, 2010; Hyeon et al., 2012; Solomatin

38

et al., 2010).

XII. DISCUSSION

The combination of excellent experiments and few the-oretical approaches touched on here have greatly ad-vanced our understanding of how biological machineswork. However, we still do not have a complete under-standing of how these machines work even in in vitroconditions. This should not be a surprise because somebelieve that the link between allosteric communication inhemoglobin (Hb) and oxygen transport is not fully un-derstood despite over fifty years of intensive study. Itis unclear if at present there is an analogue of Hb, con-sidered the hydrogen molecule for allostery, in molecularmachines. The investigation of these machines, one at atime in vitro, seems to raise unsolved problems in eachindividual case. We outline a few of the challenging prob-lems. Clearly, the list is far from being exhaustive.

A. Sometimes Details Matter

Here, we have only described coarse-grained theoreticalmethods, which are impervious to the molecular details.There are several examples (we mention two) in whichspectacular changes occur by a single or few amino acidsubstitutions. (i) About twenty years ago, Endow andHiguchi (Endow and Higuchi, 2000) made a single aminoacid substitution in the neck linker (NL) region of a Ncd,a motor that is related to kinesin. The wild type Ncdwalks towards the minus end of the microtubule in con-trast to conventional kinesin. Upon substituting an as-parigine (a polar amino acid residue) to lysine (positivelycharged) in the NL, an element that is responsible for themotor to walk predominantly on a single protofilamentof the MT, it was found that Ncd moves in both the plusand minus direction on the MT. (ii) Myosin VI, unlikemyosin V, walks towards the filamentous actin minus endand is the only known member belonging to the myosinsuper family with this property. In humans there are re-ports of three mutations that cause deafness. A pointmissense mutation (replacement of aspartic acid by tyro-sine (D179Y) in the so-called U50 domain of the motor)leads to deafness in mouse (Hertzano et al., 2008). Ithas been suggested (Pylypenko et al., 2015), using a va-riety of experimental methods, that D179Y mutant leadsto a premature release of the phosphate Pi, a productof ATP hydrolysis, from the detached head, thus pre-venting it from rapidly binding to F-actin. In the mean-while, ATP does bind to the leading head, which de-taches the motor from actin, thus preventing processivemotion. Besides these examples there are many others,such as the link between mutations in β-cardiac myosinand hypertrophic cardiomyopathy. None of these obser-

vations are amenable to theoretical treatments exposedhere, in which molecular details are ignored. Detailedsimulations, if possible, could provide biophysical insightsbut linking such studies to functions is a daunting task.These anecdotal examples should remind us that in thesearch of principles of generality in biology one shouldnot forget that molecular details matter and could dra-matically influence functions.

B. Efficiency and optimality

A naive assessment of efficiency may be made by es-timating the theoretical stall force, based on the avail-able free energy due to ATP hydrolysis, and comparingit to the measured stall force (Kolomeisky and Fisher,2007). Consider F1-ATPase for example. This is partof the F0F1-ATP synthase responsible for synthesizingATP. This rotary motor undergoes precise 120◦ rotationsin the absence of an external applied torque (τtor) at lowATP concentrations. By controlling τtor using the electrorotation method and the chemical potential by choosingappropriate ATP, ADP, and Pi concentrations, it is pos-sible to measure the probability (ps) of rotation in thesynthetic direction (ADP and Pi are consumed to gener-ate ATP and the probability ph in the reverse hydrolyticdirection could be measured (see for example (Toyabeet al., 2011)). From the linear dependence of kBT ln

psph

on it was found that the output energy at stall is roughlyequal to the chemical potential. This implies that F1-ATPase operates at near 100% efficiency.

For myosin motors, which take roughly a d=36 nmstep, the maximum force that can be exerted is fmax ≈∆GATP

d , which is approximately 2.5 pN assuming that∆GATP ≈ 22kBT . The measured stall force (fstall) isroughly in this ball park, which suggests that myosinmotors operate efficiently (η = fstall

fmaxis very high). A

similar argument for kinesin yields (d=8.1nm) fmax ≈12pN whereas measurements report values close to 8 pN.Thus, there is about a 30-35% decrease in η for kinesinmotors. A precise computation of efficiency should beundertaken by considering the network for a given motorthat captures many aspects of motor motility. The ar-guments given above hold roughly if the motility can bedescribed in a periodic one dimensional tilted potentialwith two equivalent sites with a transition state, whichis close to the initial site. Treatments (Golubeva et al.,2012; Schmiedl and Seifert, 2008; Seifert, 2011a; Wagonerand Dill, 2019) using more elaborate models indicate thatthe the motor efficiency would be much less and woulddepend on the details of the network dynamics.

A question that is related to efficiency is optimal per-formance. In the context of the biological machines dis-cussed here, performance should be measured by velocityof movement, processivity, and for molecular chaperones

39

the time-dependent production of folded state. As men-tioned above, studies in the last decade have addressedhow biological machines might optimize speed by con-sidering models that are used to analyze force-velocitycurves in motors. One of the lessons is that speed mightbe optimized if the motor takes many sub-steps insteadof a single step (Wagoner and Dill, 2016). These studieshave not considered processivity (run length) as a func-tion of ATP and external force. For helicases, it appearsthat maximization of velocity is not as relevant as opti-mization of processivity. In addition, for GroEL it ap-pears that the rate of production of the folded state perunit time is maximized even at the consumption of lav-ish amount of energy, which would render this machinehighly inefficient. Whether questions pertaining to opti-mal performce, given available free energy, must considersimultaneously many functional requirements remain anopen problem. It is possible that optimality could dependon specific function carried out by a class of machines.

C. Specificity versus Promiscuity

The E. coli chaperonin has evolved to be a promiscu-ous machine in that it facilitates the folding of a varietyof misfolded substrate proteins (even those that are notin the E. coli proteome) that are unrelated by sequence,size, or the structure of the folded state. Using directedevolution methods a mutated GroEL/GroES, referred toas GroEL3−1 was constructed (Wang et al., 2002). Thealtered GroEL3−1 contained a single mutation in GroES(tyrosine was replaced by histidine) and two mutations(valine was substituted for alanine and glycine for aspar-tic acid) in GroEL. It was found that GroEL3−1 had en-hanced ATPase activity compared to the wild type. Moreimportantly, it was found that GroEL3−1, with a highlypolar environment in the cavity compared to the wild-type, dramatically increased the folding of green fluores-cent protein (GFP). However, the enhanced specificity offolding GFP with ease came at the expense of substan-tial reduction in the capacity of GroEL3−1 to facilitatethe folding of several other proteins. Thus, in this in-stance nature has solved the tension between specificityand promiscuity by evolving an all purpose E. coli chap-eronin that can process the folding of a large class ofproteins, albeit not as efficiently. In eukaryotes therehas been a great expansion in the number of chaperoneclasses possibly to enable the larger and more complexproteome. This example suggests that evolutionary con-straints might have to a part of addressing issues relatedto optimality. This might imply that the simple networkused to explain the out of equilibrium performance ofGroEL/GroES has to expanded to describe optimality inthe functions of chaperone networks in eukaryotes.

D. Biological complexity

We circle back to Fig. 1, which is a schematic illus-tration of transport of melanosomes (vesicles containingthe light absorbing pigment melanin found in amphib-ians). The vesicles are either dispersed throughout thecytosol or aggregate near the cell center. The transporta-tion of melanosomes is clearly complex and is controlledby the interplay of multiple motors involving kinesin-2, aplus end directed MT motor, and dynein, that walks to-wards the minus end of the MT. In addition, actin boundmyosin V is also involved in the transport. It is suspectedthat a low number of motors (about 1-2 kinesin-2 motorsand roughly 1-3 dyneins) move melanosomes during ag-gregation (Levi et al., 2006). In contrast, pigment dis-persion is mediated by kinesin-2 as well as assistance bymyosin V. Because both these motors compete for thesame attachment site (p150Glue on dynactin, see Fig. 1)it follows (Levi et al., 2006) that myosin V is releasedduring aggregation and vesicle transport is dominated bydynein. The need to change movement of melanosomes,powered by multiple motors, is thus determined by func-tion, which in this case is related to their dispersion oraggregation.

Although individual motors predominantly move uni-directionally on cytosketal filamentous tracks there arereports that motors could change directions as well. Avery impressive in vitro illustration of bidirectional motil-ity of the complex of dynein with dynactin (a complexthat is attached to the cargo and activates dynein (Fig. 1)was reported sometime ago (Ross et al., 2006). It wasfound that the complex moves processively in both di-rections on MT with ATP-dependent velocities that arenot significantly different in either direction (Ross et al.,2006). Although the authors provide a qualitative pictureof the mechanism of bidirectional transport a theory forsuch unexpected behavior is lacking. It is not even clear,at least to us, the level of coarse-graining needed to con-struct such a theory, which is clearly needed to unveil thecomplexity of vesicle transport.

XIII. A FINAL REMARK

The eventual goals of understanding biology throughthe lenses of physics are to create theoretical toolsthat are capable of describing biological functions undercrowded and noisy cellular conditions, and in the processdiscover general physical principles that control life pro-cesses. It is likely that as the scale at which living systemsare examined increases it may be possible to describe cel-lular processes using functional modules (Hartwell et al.,1999), which is a coarse grained view of biology. How-ever, it would be hard to anticipate the functions of suchmodules from their components, which are the moleculesof life. Furthermore, interactions between modules could

40

lead to new functions not encoded in isolated modules, asillustrated by many examples described in our perspec-tive. After all “more is different” (Anderson, 1972). Be-cause “Nature is an excellent tinkerer, not an engineer”(a quote attributed to Francois Jacob), and tinkering inbiology involves stochastically altering existing modulesto evolve new functions without time constraints or anyultimate design as a goal (Jacob, 1977), it is likely thatconcepts in many fields of science would have to be usedto develop an integrated view of biology.

ACKNOWLEDGMENTS

We are grateful to: Matthew Caporizzo, ShaonChakrabarty, Yale Goldman, Yonathan Goldtzvik,Sabeeha Hasnain, William O. Hancock, David Hath-cock, Chris Jarzynski, Anatoly Kolomeisky, George H.Lorimer, Micheal Ostap, George Stan, Ryota Takaki, Ri-ina Tehver, Huong Vu, Ahmet Yildiz, and Zhechun Zhangfor several useful discussions and collaborations involvingthe topics covered in this review. DT is especially grate-ful to Prof. Michael E. Fisher for innumerable inspiringdiscussions for over twenty years not only on molecularmotors but also on the potential role of physics in biol-ogy. This work was supported in part by a grant fromNational Science Foundation (CHE 19-00093) and theCollie-Welch Chair (F-0019) administered through theWelch Foundation.

REFERENCES

Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, andP. Walter (2008), Molecular Biology of the Cell, 5th ed.(Garland Science).

Alhadeff, R., and A. Warshel (2017), Proc. Natl. Acad. Sci.114 (39), 10426.

Ali, J. A., and T. M. Lohman (1997), Science 275 (5298),377.

Altman, D., H. L. Sweeney, and J. A. Spudich (2004), Cell116 (5), 737.

Altschuler, S. J., and L. F. Wu (2010), Cell 141, 559.Anderson, P. W. (1972), Science 177 (4047), 393.Andrecka, J., J. O. Arroyo, Y. Takagi, G. de Wit, A. Fineberg,

L. MacKinnon, G. Young, J. R. Sellers, and P. Kukura(2015), Elife 4, e05413.

Ariga, T., M. Tomishige, and D. Mizuno (2018), Physicalreview letters 121 (21), 218101.

Astumian, R. D., and M. Bier (1996), Biophysical journal70 (2), 637.

Bagchi, D., M. Manosas, W. Zhang, K. A. Manthei,S. Hodeib, B. Ducos, J. L. Keck, and V. Croquette (2018),Nucleic acids research 46 (16), 8500.

Bagshaw, C., and D. Trentham (1973), Biochemical Journal133 (2), 323.

Baker, J. E., E. B. Krementsova, G. G. Kennedy, A. Arm-strong, K. M. Trybus, and D. M. Warshaw (2004), Pro-ceedings of the National Academy of Sciences 101 (15),5542.

Banerjee, K., A. B. Kolomeisky, and O. A. Igoshin (2017),Proceedings of the National Academy of Sciences 114 (20),5183.

Barato, A. C., and U. Seifert (2015), Physical review letters114 (15), 158101.

Battle, C., C. P. Broedersz, N. Fakhri, V. F. Geyer,J. Howard, C. F. Schmidt, and F. C. MacKintosh (2016),Science 352 (6285), 604.

Bell, G. I. (1978), Science 200 (4342), 618.Belyy, V., N. L. Hendel, A. Chien, and A. Yildiz (2014a),

Nat. Comm. 5, 5544.Belyy, V., N. L. Hendel, A. Chien, and A. Yildiz (2014b),

Nat. Comm. 5, 5544.Bennett, C. H. (1982), International Journal of Theoretical

Physics 21 (12), 905.Betterton, M. D., and F. Julicher (2003), Phys. Rev. Lett.

91 (25), 258103.Betterton, M. D., and F. Julicher (2005a), Phys Rev E 71 (1

Pt 1), 011904.Betterton, M. D., and F. Julicher (2005b), J. Phys.: Condens.

Matter 17 (47), S3851.Bhabha, G., H.-C. Cheng, N. Zhang, A. Moeller, M. Liao,

J. A. Speir, Y. Cheng, and R. D. Vale (2014), CELL159 (4), 857.

Bhaskaran, H., and R. Russell (2007), Nature 449 (7165),1014.

Bianco, P. R., and S. C. Kowalczykowski (2000), Nature405 (6784), 368.

Bierbaum, V., and R. Lipowsky (2011), Biophysical journal100 (7), 1747.

Block, S. M. (2007), Biophysical journal 92 (9), 2986.Block, S. M., C. L. Asbury, J. W. Shaevitz, and M. J. Lang

(2003), Proceedings of the National Academy of Sciences100 (5), 2351.

Block, S. M., L. S. Goldstein, and B. J. Schnapp (1990),Nature 348 (6299), 348.

Brady, S. T. (1985), Nature 317 (6032), 73.Brown, A. I., and D. A. Sivak (2017), Proceedings of the

National Academy of Sciences 114 (42), 11057.Bustamante, C., W. Cheng, and Y. X. Mejia (2011), Cell

144 (4), 480.Callen, H. B. (1985), Thermodynamics and an Introduction

to Thermostatistics (Wiley).Caporizzo, M. A., C. E. Fishman, O. Sato, R. M. Jami-

olkowski, M. Ikebe, and Y. E. Goldman (2018), Biophysicaljournal 114 (6), 1400.

Carter, N. J., and R. Cross (2005), Nature 435 (7040), 308.Chakrabarti, S., C. Hyeon, X. Ye, G. H. Lorimer,

and D. Thirumalai (2017), Proceedings of the NationalAcademy of Sciences 114 (51), E10919.

Chakrabarti, S., C. Jarzynski, and D. Thirumalai (2019),Biophysical Journal 10.1016/j.bpj.2019.07.021.

Cheng, W., S. Dumont, I. Tinoco, and C. Bustamante (2007),Proc. Natl. Acad. Sci. 104 (35).

Clancy, B. E., W. M. Behnke-Parks, J. O. Andreasson, S. S.Rosenfeld, and S. M. Block (2011), Nature structural &molecular biology 18 (9), 1020.

Clemen, A. E. M., M. Vilfan, J. Jaud, J. S. Zhang, M. Bar-mann, and M. Rief (2005), Biophys. J. 88 (6), 4402.

Comstock, M. J., K. D. Whitley, H. Jia, J. Sokoloski, T. M.Lohman, T. Ha, and Y. R. Chemla (2015), Science348 (6232), 352.

Craig, E. M., and H. Linke (2009), Proceedings of the Na-tional Academy of Sciences 106 (43), 18261.

41

Crooks, G. E. (1998), Journal of Statistical Physics 90 (5-6),1481.

De la Cruz, E. M., E. M. Ostap, and H. L. Sweeney (2001),Journal of Biological Chemistry 276 (34), 32373.

De la Cruz, Enrique M, M., A. L. Wells, S. S. Rosenfeld,E. M. Ostap, and H. L. Sweeney (1999), Proceedings ofthe National Academy of Sciences 96 (24), 13726.

De La Cruz, E. M., and E. M. Ostap (2004), Current opinionin cell biology 16 (1), 61.

Dechant, A., and S.-i. Sasa (2018a), Journal of StatisticalMechanics: Theory and Experiment 2018 (6), 063209.

Dechant, A., and S.-i. Sasa (2018b), Physical Review E97 (6), 062101.

Delagoutte, E., and P. H. Von Hippel (2002), Quarterly re-views of biophysics 35 (4), 431.

Delagoutte, E., and P. H. Von Hippel (2003), Quarterly re-views of biophysics 36 (1), 1.

Derrida, B. (1983), Journal of statistical physics 31 (3), 433.Dessinges, M.-N., T. Lionnet, X. G. Xi, D. Bensimon, and

V. Croquette (2004), Proceedings of the National Academyof Sciences 101 (17), 6439.

DeWitt, M. A., A. Y. Chang, P. A. Combs, and A. Yildiz(2012), Science 335 (6065), 221.

Dominguez, R., Y. Freyzon, K. M. Trybus, and C. Cohen(1998), Cell 94 (5), 559.

Dong, F., E. P. Gogol, and P. H. von Hippel (1995), Journalof Biological Chemistry 270 (13), 7462.

Dunn, A. R., P. Chuan, Z. Bryant, and J. A. Spudich (2010),Proceedings of the National Academy of Sciences 107 (17),7746.

Dunn, A. R., and J. A. Spudich (2007), Nature Structuraland Molecular Biology 14 (3), 246.

Ehrenberg, M., and C. Blomberg (1980), Biophysical journal31 (3), 333.

Elber, R., and A. West (2010), Proc. Natl. Acad. Sci.107 (11), 5001.

Elting, M. W., Z. Bryant, J.-C. Liao, and J. A. Spudich(2011), Biophysical journal 100 (2), 430.

Endow, S., and H. Higuchi (2000), Nature 406 (6798), 913.Fei, X., X. Ye, N. A. LaRonde, and G. H. Lorimer (2014),

Proceedings of the National Academy of Sciences 111 (35),12775.

Finer, J. T., R. M. Simmons, and J. A. Spudich (1994),Nature 368 (6467), 113.

Fisher, M. E., and A. B. Kolomeisky (1999a), Proceedings ofthe National Academy of Sciences 96 (12), 6597.

Fisher, M. E., and A. B. Kolomeisky (1999b), Physica A:Statistical Mechanics and its Applications 274 (1), 241.

Fisher, M. E., and A. B. Kolomeisky (2001), Proceedings ofthe National Academy of Sciences 98 (14), 7748.

Ge, H., and H. Qian (2010), Physical Review E 81 (5),051133.

Gebhardt, J. C. M., A. E.-M. Clemen, J. Jaud, and M. Rief(2006), Proceedings of the National Academy of Sciences103 (23), 8680.

Gennerich, A., A. P. Carter, S. L. Reck-Peterson, and R. D.Vale (2007), Cell 131 (5), 952.

Gerritsma, E., and P. Gaspard (2010), Biophysical Reviewsand Letters 5 (04), 163.

Gibbons, I., and A. Rowe (1965), Science 149 (3682), 424.Gingrich, T. R., J. M. Horowitz, N. Perunov, and J. L. Eng-

land (2016), Physical review letters 116 (12), 120601.Gladrow, J., N. Fakhri, F. MacKintosh, C. Schmidt, and

C. Broedersz (2016), Physical review letters 116 (24),

248301.Goldtzvik, Y., M. Hinczewski, and D. Thirumalai (2019),

preprint.Goldtzvik, Y., M. L. Mugnai, and D. Thirumalai (2018),

Structure 26 (12), 1664.Golubeva, N., A. Imparato, and L. Peliti (2012), Euro Phys.

Lett. 97 (6), 60005.Gorbalenya, A. E., and E. V. Koonin (1993), Current opinion

in structural biology 3 (3), 419.Gross, S. P., M. C. Tuma, S. W. Deacon, A. S. Serpinskaya,

A. R. Reilein, and V. I. Gelfand (2002), The Journal ofcell biology 156 (5), 855.

Guo, Z., and D. Thirumalai (1995), Biopolymers: OriginalResearch on Biomolecules 36 (1), 83.

Hackney, D. D. (1988), Proceedings of the National Academyof Sciences 85 (17), 6314.

Hackney, D. D. (2005), Proceedings of the National Academyof Sciences 102 (51), 18338.

Hancock, W. O. (2016), Biophysical journal 110 (6), 1216.Harada, T., and S.-i. Sasa (2006), Physical Review E 73 (2),

026131.Hartman, M. A., D. Finan, S. Sivaramakrishnan, and J. A.

Spudich (2011), Ann. Rev. Cell DEv. Biol. 27, 133.Hartman, M. A., and J. A. Spudich (2012), J. Cell. Sci.

125 (7), 1627.Hartwell, L. H., J. J. Hopfield, S. Leibler, and A. W. Murray

(1999), Nature 402 (6761supp), C47.Hathcock, D., R. Tehver, M. Hinczewski, and D. Thirumalai

(2019), .Hertzano, R., E. Shalit, A. K. Rzadzinska, A. A. Dror,

L. Song, U. Ron, J. T. Tan, A. S. Shitrit, H. Fuchs, T. Has-son, et al. (2008), PLoS genetics 4 (10), e1000207.

Hill, T. L. (1966), Journal of theoretical biology 10 (3), 442.Hill, T. L. (2005a), Free Energy Transduction and Biochemical

Cycle Kinetics (Dover).Hill, T. L. (2005b), Free energy transduction and biochemical

cycle kinetics (Dover Publications, Inc.).Hill, T. L., and Y.-D. Chen (1975), Proceedings of the Na-

tional Academy of Sciences 72 (4), 1291.Hill, T. L., and R. Simmons (1976), Proceedings of the Na-

tional Academy of Sciences 73 (1), 95.Hinczewski, M., R. Tehver, and D. Thirumalai (2013), Pro-

ceedings of the National Academy of Sciences 110 (43),E4059.

Hirokawa, N., Y. Noda, Y. Tanaka, and S. Niwa (2009), Nat.Rev. Mol. Cell. Biol. 10 (10), 682.

Holzbaur, E. L., and Y. E. Goldman (2010), Current opinionin cell biology 22 (1), 4.

Homma, K., and M. Ikebe (2005), Journal of BiologicalChemistry 280 (32), 29381.

Hopfield, J. J. (1974), Proceedings of the National Academyof Sciences 71 (10), 4135.

Howard, J. (2001), Mechanics of motor proteins and the cy-toskeleton (Sinauer associates Sunderland, MA).

Howard, J. (2006), Current Biology 16 (14), R517.Howard, J., A. Hudspeth, and R. Vale (1989), Nature

342 (6246), 154.Hulett, H. (1970), Nature 225 (5239), 1248.Hwang, W., and C. Hyeon (2017), The journal of physical

chemistry letters 8 (1), 250.Hwang, W., M. J. Lang, and M. Karplus (2008), Structure

16 (1), 62.Hwang, W., M. J. Lang, and M. Karplus (2017), eLife 6,

e28948.

42

Hyeon, C., and W. Hwang (2017), Physical Review E 96 (1),012156.

Hyeon, C., S. Klumpp, and J. N. Onuchic (2009), PhysicalChemistry Chemical Physics 11 (24), 4899.

Hyeon, C., J. Lee, J. Yoon, S. Hohng, and D. Thirumalai(2012), Nat. Chem. 4, 907.

Hyeon, C., and J. N. Onuchic (2007a), Proceedings of theNational Academy of Sciences 104 (7), 2175.

Hyeon, C., and J. N. Onuchic (2007b), Proceedings of theNational Academy of Sciences 104 (44), 17382.

Hyeon, C., and D. Thirumalai (2011), Nature communica-tions 2, 487.

Ikezaki, K., T. Komori, M. Sugawa, Y. Arai, S. Nishikawa,A. H. Iwane, and T. Yanagida (2012), Small 8 (19), 3035.

Ikezaki, K., T. Komori, and T. Yanagida (2013), PloS one8 (3), e58912.

Inoue, A., J. Saito, R. Ikebe, and M. Ikebe (2002), Naturecell biology 4 (4), 302.

Isojima, H., R. Iino, Y. Niitani, H. Noji, and M. Tomishige(2016), Nature chemical biology 12 (4), 290.

Itoh, H., A. Takahashi, K. Adachi, H. Noji, R. Yasuda,M. Yoshida, and K. Kinosita Jr (2004), Nature 427 (6973),465.

Itsathitphaisarn, O., R. A. Wing, W. K. Eliason, J. Wang,and T. A. Steitz (2012), Cell 151 (2), 267.

Iyer, L. M., D. D. Leipe, E. V. Koonin, and L. Aravind(2004), Journal of structural biology 146 (1-2), 11.

Jacob, F. (1977), Science 196 (4295), 1161.Jeong, Y.-J., M. K. Levin, and S. S. Patel (2004), Proceedings

of the National Academy of Sciences 101 (19), 7264.Johnson, D. S., L. Bai, B. Y. Smith, S. S. Patel, and M. D.

Wang (2007), Cell 129 (7), 1299.Julicher, F., A. Ajdari, and J. Prost (1997), Reviews of Mod-

ern Physics 69 (4), 1269.Kad, N. M., K. M. Trybus, and D. M. Warshaw

(2008), Journal of Biological Chemistry 283 (25), 17477,http://www.jbc.org/content/283/25/17477.full.pdf+html.

Kiefhaber, T. (1995), Proceedings of the National Academyof Sciences 92 (20), 9029.

Kirkpatrick, T., and D. Thirumalai (2015), Reviews of Mod-ern Physics 87 (1), 183.

Kodera, N., D. Yamamoto, R. Ishikawa, and T. Ando (2010),Nature 468 (7320), 72.

Kolomeisky, A. B., and M. E. Fisher (2000), Physica A: Sta-tistical Mechanics and its Applications 279 (1), 1.

Kolomeisky, A. B., and M. E. Fisher (2003), Biophysicaljournal 84 (3), 1642.

Kolomeisky, A. B., and M. E. Fisher (2007), Annu. Rev.Phys. Chem. 58, 675.

Koza, Z. (1999), Journal of Physics A: Mathematical andGeneral 32 (44), 7637.

Koza, Z. (2002), arXiv preprint cond-mat/0207203.Lan, G., P. Sartori, S. Neumann, V. Sourjik, and Y. Tu

(2012), Nature physics 8 (5), 422.Lebowitz, J. L., and H. Spohn (1999), Journal of Statistical

Physics 95 (1-2), 333.Leibler, S., and D. A. Huse (1993), The Journal of cell biology

121 (6), 1357.Levi, V., A. Serpinskaya, E. Gratton, and V. Gelfand (2006),

Biophys. J. 90 (1), 318.Levin, M. K., and S. S. Patel (2002), Journal of Biological

Chemistry 277 (33), 29377.Li, J.-H., W.-X. Lin, B. Zhang, D.-G. Nong, H.-P. Ju, J.-B.

Ma, C.-H. Xu, F.-F. Ye, X. G. Xi, M. Li, et al. (2016),

Nucleic acids research 44 (9), 4330.Liepelt, S., and R. Lipowsky (2007a), Physical review letters

98 (25), 258102.Liepelt, S., and R. Lipowsky (2007b), EPL (Europhysics Let-

ters) 77 (5), 50002.Lionnet, T., A. Dawid, S. Bigot, F.-X. Barre, O. A. Saleh,

F. Heslot, J.-F. Allemand, D. Bensimon, and V. Croquette(2006), Nucleic acids research 34 (15), 4232.

Lipowsky, R., and S. Liepelt (2008), Journal of StatisticalPhysics 130 (1), 39.

Lipowsky, R., S. Liepelt, and A. Valleriani (2009), Journalof Statistical Physics 135 (5), 951.

Liu, B., R. J. Baskin, and S. C. Kowalczykowski (2013),Nature 500, 482.

Liu, C., X. Fu, L. Liu, X. Ren, C. K. Chau, S. Li, L. Xi-ang, H. Zeng, G. Chen, L.-H. Tang, et al. (2011), Science334 (6053), 238.

Lohman, T. (1992), Molecular microbiology 6 (1), 5.Lohman, T. M., and K. P. Bjornson (1996), Annual review

of biochemistry 65 (1), 169.Lohman, T. M., E. J. Tomko, and C. G. Wu (2008), Nature

Reviews Molecular Cell Biology 9 (5), 391.Lucius, A. L., N. K. Maluf, C. J. Fischer, and T. M. Lohman

(2003), Biophysical journal 85 (4), 2224.Ma, J., P. Sigler, Z. Xu, and M. Karplus (2000), J. Mol. Biol.

302 (2), 303.Ma, W., K. D. Whitley, Y. R. Chemla, Z. Luthey-Schulten,

and K. Schulten (2018), eLife 7, e34186.Maes, C., and M. H. Van Wieren (2003), Journal of Statistical

Physics 112 (1-2), 329.Manosas, M., X. G. Xi, D. Bensimon, and V. Croquette

(2010), Nuc. Acids Res. 38 (16), 5518.Marians, K. J. (2000), Structure 8 (12), R227.Mehta, A. D., R. S. Rock, M. Rief, J. A. Spudich, M. S.

Mooseker, and R. E. Cheney (1999), Nature 400 (6744),590.

Mehta, P., and D. J. Schwab (2012), Proceedings of the Na-tional Academy of Sciences 109 (44), 17978.

Mickolajczyk, K. J., N. C. Deffenbaugh, J. O. Arroyo, J. An-drecka, P. Kukura, and W. O. Hancock (2015), Proceed-ings of the National Academy of Sciences 112 (52), E7186.

Milic, B., J. O. Andreasson, D. W. Hogan, and S. M. Block(2017), Proceedings of the National Academy of Sciences114 (33), E6830.

Mohr, S., J. M. Stryker, and A. M. Lambowitz (2002), Cell109, 769.

Moore, J. R., E. B. Krementsova, K. M. Trybus, and D. M.Warshaw (2004), Journal of Muscle Research & Cell Motil-ity 25 (1), 29.

Mugnai, M., M. Caporizzo, Y. Goldman, and D. Thirumalai(2019), bioRxiv , 684696.

Mugnai, M. L., and D. Thirumalai (2017), Proceedings ofthe National Academy of Sciences 114 (22), E4389.

Mukherjee, S., R. Alhadeff, and A. Warshel (2017), Proc.Natl. Acad. Sci. 114 (9), 2259.

Mukherjee, S., and A. Warshel (2013), Proc. Natl. Acad. Sci.110 (43), 17326.

Nam, W., and B. I. Epureanu (2016), PloS one 11 (1),e0147676.

Nishikawa, S., I. Arimoto, K. Ikezaki, M. Sugawa, H. Ueno,T. Komori, A. H. Iwane, and T. Yanagida (2010), Cell142 (6), 879.

Nishiyama, M., H. Higuchi, and T. Yanagida (2002), NatureCell Biology 4 (10), 790.

43

Oguchi, Y., S. V. Mikhailenko, T. Ohki, A. O. Olivares, E. M.De la Cruz, and S. Ishiwata (2008), Proceedings of theNational Academy of Sciences 105 (22), 7714.

Oguchi, Y., S. V. Mikhailenko, T. Ohki, A. O. Olivares, E. M.De La Cruz, and S. Ishiwata (2010), Nature chemical bi-ology 6 (4), 300.

Oke, O. A., S. A. Burgess, E. Forgacs, P. J. Knight,T. Sakamoto, J. R. Sellers, H. White, and J. Trinick (2010),Proceedings of the National Academy of Sciences 107 (6),2509.

Okten, Z., L. S. Churchman, R. S. Rock, and J. A. Spudich(2004), Nature Structural and Molecular Biology 11 (9),884.

Oppenheim, I., K. E. Shuler, G. H. Weiss, et al. (1977),Stochastic processes in chemical physics (Mit Press).

Ori-McKenney, K. M., J. Xu, S. P. Gross, and R. B. Vallee(2010), Nature cell biology 12 (12), 1228.

Ortega, A., D. Amoros, and J. G. de La Torre (2011), Bio-physical Journal 101 (4), 892.

Pan, J., D. Thirumalai, and S. A. Woodson (1997), Journalof molecular biology 273 (1), 7.

Pang, P. S., E. Jankowsky, P. J. Planet, and A. M. Pyle(2002), The EMBO journal 21 (5), 1168.

Patel, S. S., and K. M. Picha (2000), Annual review of bio-chemistry 69 (1), 651.

Perkins, T. T., H.-W. Li, R. V. Dalal, J. Gelles, and S. M.Block (2004), Biophysical journal 86 (3), 1640.

Pierobon, P., S. Achouri, S. Courty, A. R. Dunn, J. A. Spu-dich, M. Dahan, and G. Cappello (2009), Biophys. J.96 (10), 4268.

Pigolotti, S., I. Neri, E. Roldan, and F. Julicher (2017), Phys-ical review letters 119 (14), 140604.

Pincus, D. L., S. Chakrabarti, and D. Thirumalai (2015),Biophysical Journal 109 (2), 220.

Post, P. L., M. J. Tyska, C. B. O’Connell, K. Johung, A. Hay-ward, and M. S. Mooseker (2002), Journal of BiologicalChemistry 277 (14), 11679.

Purcell, T. J., H. L. Sweeney, and J. A. Spudich (2005),Proceedings of the National Academy of Sciences of theUnited States of America 102 (39), 13873.

Pyle, A. M. (2008), Annu. Rev. Biophys. 37, 317.Pylypenko, O., L. Song, A. Shima, Z. Yang, A. M. Houdusse,

and H. L. Sweeney (2015), Proc. Natl. Acad. Sci. 112 (11),E1201.

Qian, H. (2004), Physical Review E 69 (1), 012901.Qian, H. (2007), Annu. Rev. Phys. Chem. 58, 113.Qian, H., and D. A. Beard (2005), Biophysical chemistry

114 (2-3), 213.Reck-Peterson, S. L., W. B. Redwine, R. D. Vale, and A. P.

Carter (2018), Nat. Rev. Mol. Cell. Biol. 19 (6), 382.Reck-Peterson, S. L., A. Yildiz, A. P. Carter, A. Gennerich,

N. Zhang, and R. D. Vale (2006a), Cell 126 (2), 335.Reck-Peterson, S. L., A. Yildiz, A. P. Carter, A. Gennerich,

N. Zhang, and R. D. Vale (2006b), Cell 126 (2), 335.Rice, S., A. W. Lin, D. Safer, C. L. Hart, N. Naber, B. O.

Carragher, S. M. Cain, E. Pechatnikova, E. M. Wilson-Kubalek, M. Whittaker, et al. (1999), Nature 402 (6763),778.

Rocak, S., and P. Linder (2004), Nature reviews Molecularcell biology 5 (3), 232.

Rock, R. S., S. E. Rice, A. L. Wells, T. J. Purcell, J. A.Spudich, and H. L. Sweeney (2001), Proceedings of theNational Academy of Sciences 98 (24), 13655.

Rodenfels, J., K. M. Neugebauer, and J. Howard (2019),Developmental cell 48 (5), 646.

Rosenfeld, S. S., and H. Lee Sweeney (2004), Jour-nal of Biological Chemistry 279 (38), 40100,http://www.jbc.org/content/279/38/40100.full.pdf+html.

Ross, J. L., K. Wallace, H. Shuman, Y. E. Goldman, andE. L. F. Holzbaur (2006), Nat. Cell. Biol. 8 (6), 562.

Russell, R., I. Jarmoskaite, and A. M. Lambowitz (2013),RNA Biology 10 (1), 44.

Sakamoto, T., I. Amitani, E. Yokota, and T. Ando (2000),Biochem. Biophys. Res. Commun. 272 (2), 586.

Sakamoto, T., A. Yildiz, P. R. Selvin, and J. R. Sell-ers (2005), Biochemistry 44 (49), 16203, pMID: 16331980,https://doi.org/10.1021/bi0512086.

Sarlah, A., and A. Vilfan (2014), Biophys. J. 107 (3), 662.Sarlah, A., and A. Vilfan (2014), Biophysical journal 107 (3),

662.Schief, W. R., R. H. Clark, A. H. Crevenna, and J. Howard

(2004), Proceedings of the National Academy of Sciences101 (5), 1183.

Schmidt, H., R. Zalyte, L. Urnavicius, and A. P. Carter(2015), Nature 518 (7539), 435.

Schmiedl, T., and U. Seifert (2008), EPL (Europhysics Let-ters) 83 (3), 30005.

Schnakenberg, J. (1976), Reviews of Modern physics 48 (4),571.

Schnitzer, M. J., and S. Block (1995), in Cold spring har-bor symposia on quantitative biology, Vol. 60 (Cold SpringHarbor Laboratory Press) pp. 793–802.

Schnitzer, M. J., and S. M. Block (1997), Nature 388 (6640),386.

Schnitzer, M. J., K. Visscher, and S. M. Block (2000), Naturecell biology 2 (10), 718.

Seifert, U. (2005a), Physical review letters 95 (4), 040602.Seifert, U. (2005b), EPL (Europhysics Letters) 70 (1), 36.Seifert, U. (2011a), Physical review letters 106 (2), 020601.Seifert, U. (2011b), The European Physical Journal E 34 (3),

26.Seifert, U. (2012), Reports on progress in physics 75 (12),

126001.Sellers, J. (2000), Biochem. Biophys. ACTA - Mol. Cell Res.

1496 (1), 3.Sellers, J. R., and C. Veigel (2010), Nature Structural and

Molecular Biology 17 (5), 590.Shiraishi, N., K. Saito, and H. Tasaki (2016), Physical review

letters 117 (19), 190601.Soldati, T., and M. Schliwa (2006), Nature Reviews Molecu-

lar Cell Biology 7 (12), 897.Solomatin, S. V., M. Greenfeld, S. Chu, and D. Herschlag

(2010), Nature 463, 681.Song, Y., J. O. Park, L. Tanner, Y. Nagano, J. D. Rabinowitz,

and S. Y. Shvartsman (2019), Current Biology 29 (12),R566.

Spudich, J. A., and S. Sivaramakrishnan (2010), Nat. Rev.Mol. Cell. Biol. 11 (2), 128.

Stan, G., B. Brooks, and D. Thirumalai (2005), J. Mol. Biol.350 (4), 817.

Sumi, T., and S. Klumpp (2019), Nano letters.Sun, Y., and Y. E. Goldman (2011a), Biophys. J. 101 (1), 1.Sun, Y., and Y. E. Goldman (2011b), Biophysical Journal

101 (1), 1.Sun, Y., O. Sato, F. Ruhnow, M. E. Arsenault, M. Ikebe,

and Y. E. Goldman (2010), Nature structural & molecularbiology 17 (4), 485.

44

Sweeney, H. L., and A. Houdusse (2010), Annual review ofbiophysics 39, 539.

Tailleur, J., and M. Cates (2008), Physical review letters100 (21), 218103.

Takaki, R., M. L. Mugnai, Y. Goldtzvik, and D. Thirumalai(2019), in preparation.

Tehver, R., and D. Thirumalai (2008), J. Mol. Biol. 377,1279.

Tehver, R., and D. Thirumalai (2010), Structure 18 (4), 471.Thirumalai, D., and B. Ha (1998), Theoretical and mathemat-

ical models in polymer research, edited by A. Y. Grosberg(Academic Press, Boston).

Thirumalai, D., and C. Hyeon (2005), Biochemistry 44 (13),4957.

Thirumalai, D., C. Hyeon, P. I. Zhuravlev, and G. H. Lorimer(2019), Chemical reviews 119 (12), 6788.

Thirumalai, D., and G. H. Lorimer (2001), Annual review ofbiophysics and biomolecular structure 30 (1), 245.

Toba, S., T. M. Watanabe, L. Yamaguchi-Okimoto, Y. Y.Toyoshima, and H. Higuchi (2006), Proceedings of the Na-tional Academy of Sciences 103 (15), 5741.

Todd, M. J., G. H. Lorimer, and D. Thirumalai (1996), Proc.Natl. Acad. Sci. U. S. A. 93, 4030.

Tominaga, M., and A. Nakano (2012), Frontiers in plant sci-ence 3, 211.

Toyabe, S., T. Okamoto, T. Watanabe-Nakayama, H. Take-tani, S. Kudo, and E. Muneyuki (2010), Physical reviewletters 104 (19), 198103.

Toyabe, S., T. Watanabe-Nakayama, T. Okamoto, S. Kudo,and E. Muneyuki (2011), Proceedings of the NationalAcademy of Sciences 108 (44), 17951.

Tsygankov, D., A. W. R. Serohijos, N. V. Dokholyan, andT. C. Elston (2011), Biophys. J. 101 (1), 144.

Turina, P., D. Samoray, and P. Graber (2003), The EMBOJournal 22 (3), 418.

Uemura, S., H. Higuchi, A. O. Olivares, E. M. De La Cruz,and S. Ishiwata (2004a), Nat. Struct. Mol. Biol. 11, 877.

Uemura, S., H. Higuchi, A. O. Olivares, E. M. De La Cruz,and S. Ishiwata (2004b), Nat. Struct. Mol. Biol. 11 (9),877.

Vale, R. D. (2003a), Cell 112 (4), 467.Vale, R. D. (2003b), J Cell Biol 163 (3), 445.Vale, R. D., and R. A. Milligan (2000), Science 288 (5463),

88.Vale, R. D., T. S. Reese, and M. P. Sheetz (1985), Cell 42 (1),

39.Van Kampen, N. (2007), Stochastic processes in physics and

chemistry (Elsiver).

Veigel, C., S. Schmitz, F. Wang, and J. R. Sellers (2005),Nature cell biology 7 (9), 861.

Veigel, C., F. Wang, M. L. Bartoo, J. R. Sellers, and J. E.Molloy (2002), Nature cell biology 4 (1), 59.

Velankar, S. S., P. Soultanas, M. S. Dillingham, H. S. Subra-manya, and D. B. Wigley (1999), Cell 97 (1), 75.

Venkatesan, M., L. Silver, and N. Nossal (1982), Journal ofBiological Chemistry 257 (20), 12426.

Visscher, K., M. J. Schnitzer, and S. M. Block (1999), Nature400, 184.

Vu, H. T., S. Chakrabarti, M. Hinczewski, and D. Thirumalai(2016), Physical review letters 117 (7), 078101.

Wagoner, J. A., and K. A. Dill (2016), The Journal of Phys-ical Chemistry B 120 (26), 6327.

Wagoner, J. A., and K. A. Dill (2019), Proceedings of theNational Academy of Sciences 116 (13), 5902.

Walker, M. L., S. A. Burgess, J. R. Sellers, F. Wang, J. A.Hammer III, J. Trinick, and P. J. Knight (2000), Nature405 (6788), 804.

Walter, W. J., V. Beranek, E. Fischermeier, and S. Diez(2012), PLoS One 7 (8), e42218.

Wang, J., C. Herman, K. Tipton, C. Gross, and J. Weissman(2002), Cell 111 (7), 1027.

Wang, X., X. Zhang, D. Wu, Z. Huang, T. Hou, C. Jian, P. Yu,F. Lu, R. Zhang, T. Sun, et al. (2017), Elife 6, e23908.

Wong, I., and T. M. Lohman (1992), Science 256 (5055),350.

Yildiz, A., J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Gold-man, and P. R. Selvin (2003), Science 300 (5628), 2061.

Yildiz, A., H. Park, D. Safer, Z. Yang, L.-Q. Chen, P. R.Selvin, and H. L. Sweeney (2004a), Journal of BiologicalChemistry 279 (36), 37223.

Yildiz, A., M. Tomishige, A. Gennerich, and R. D. Vale(2008), Cell 134 (6), 1030.

Yildiz, A., M. Tomishige, R. D. Vale, and P. R. Selvin(2004b), Science 303 (5658), 676.

Yu, J., T. Ha, and K. Schulten (2006), Biophys. J. 91 (6),2097.

Zhang, Y., and A. B. Kolomeisky (2018), J. Phys. Chem. B122 (13, SI), 3272.

Zhang, Z., Y. Goldtzvik, and D. Thirumalai (2017), Proceed-ings of the National Academy of Sciences 114 (46), E9838.

Zhang, Z., and D. Thirumalai (2012), Structure 20 (4), 628.Zimmermann, E., and U. Seifert (2012), New Journal of

Physics 14 (10), 103023.