how simulations reveal dynamics, disorder, and the energy landscapes of biomolecular function

DOI: 10.1002/ijch.201400018

How Simulations Reveal Dynamics, Disorder, and theEnergy Landscapes of Biomolecular FunctionJeffrey K. Noel[a] and Paul Charles Whitford*[b]

1 Introduction

The development of molecular simulation methods hasprovided a vast range of quantitative tools for studyingtheoretical representations of biomolecular energetics.With these techniques, exploration through simulationcan complement experimental measurements by suggest-ing modes of regulation in naturally occurring systems.Many examples of the cooperation between simulation,theory, and experimentation are found in studies of fold-ing and functionally relevant conformational rearrange-ments of proteins and RNAs, for which diverse modelshave converged on consistent descriptions of the dynam-ics, thereby implicating characteristic features of biomo-lecular energetics.[1] Of principal interest herein is howmodels that span from coarse-grained to explicit-solventrepresentations are able to provide complementary in-sights and suggest common features of biomolecular dy-namics.[2]

Some of the first computational approaches applied inthe study of biomolecules, for which the 2013 Nobel Prizein Chemistry was awarded, provided a technical founda-tion with which broad classes of models can now addressincreasingly complex questions. In the 1960s and 1970s,Nobel Prize winners Warshel, Karplus, and Levitt beganpioneering the application of computing technology tostudy the dynamics of biomolecules,[3] which they contin-ued to develop in subsequent decades.[4] At that time, theworld�s fastest computers could perform tens of megafloating-point operations per second (MFLOPS); this isabout 1/1000th of the computational power of a modernhandheld electronic device. Their groundbreaking work

towards creating computer models that mixed quantummechanics with classical mechanics were a feat of techni-cal prowess and personal dedication. While studentstoday cannot imagine relying on such limited technology,these bold efforts set a precedent for modern computa-tional biology, which itself has blossomed as hardwareand software developments have pushed the scale of com-putation, both in terms of simulated time and system size(Figure 1).

Enabled by advances in structure determination,[5]

high-performance computing (HPC),[6] and computationalmodeling techniques,[7] theoretical methods can now beused to predict physicochemical quantities for large-scalemolecular assemblies.[8] Although the dynamic nature ofmolecular machines, such as the ribosome,[9] has posedsignificant challenges in structural studies, revolutionaryadvances[10] have provided atomic models of these molec-ular behemoths at distinct points along their functionalcycles. With increasingly accessible atomic data and grow-

Abstract : Over the last 40 years, the area of computationalmolecular biophysics has grown and developed to the pointwhere simulations can now provide detailed mechanistic in-sights, suggest theoretical principles that underpin function,and provide frameworks for understanding and interpretingexperimental measurements. The success of molecular sim-ulations has been the result of the unrelenting developmentof novel theoretical models, exponential growth in computa-tional resources, and advances in structural biology tech-

niques. Through the continued refinement and applicationof diverse classes of models, general themes in biomolecu-lar dynamics are beginning to surface. In particular, molecu-lar simulations are highlighting the pervasive role thatorder�disorder events play in molecular biology. These dy-namic modes of function are transforming our perspectiveon the role that entropy has in many large-scale biologicalprocesses.

Keywords: biomolecular disorder · computational chemistry · protein folding · molecular devices · ribosomes

[a] J. K. NoelCenter for Theoretical Biological PhysicsRice University, 6500 Main St MS-654Houston, TX 77054 (USA)

[b] P. C. WhitfordDepartment of Physics, Northeastern UniversityDana Research Center, 123, 360 Huntington AveBoston, MA 02115 (USA)Phone: (+1) 617-373-2952Fax: (+1) 617-373-2943e-mail: [email protected]

Isr. J. Chem. 2013, 53, 1 – 16 � 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim &1&

These are not the final page numbers! ��

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ing computing capacity,[6a] we are now entering an erawhere molecular simulations may be systematically ap-plied to large-scale, complex assemblies (Figure 2). Theinsights provided by simulations will then, in turn, guidethe design of experiments that precisely delineate the en-ergetic contributors to cellular function.

This review covers several aspects of the developmentand application of biomolecular simulation methods.

First, advances in computing technology and recent appli-cations of HPC are discussed, with a focus on the techni-cal advances that have increased the scale of explicit-sol-vent simulations. This is followed by a discussion on thediversity of molecular simulations tools, which are fre-quently developed in the context of protein folding andthen applied in the study of assemblies. Finally, commonthemes that have emerged from simulation studies arediscussed, with a particular emphasis on the impact thatorder�disorder transitions have on the dynamics of func-tional conformational transitions.

2 Advancing the Limits of HPC

Scientific computing has benefited from the relentlesslyincreasing speed of integrated circuits. The historicaltrend that the number of transistors in an integrated cir-cuit doubles every 18–24 months, called Moore�s Law, hasallowed long-term exponential growth in computing capa-

Dr. Noel received his B. S. in appliedmath, engineering, and physics fromhis home state at the University of Wis-consin, Madison. He then pursueda Ph.D. in physics at the University ofCalifornia, San Diego, largely focusingon the folding and dynamics of topo-logically complex proteins. He is cur-rently a Postdoctoral Scholar at theCenter for Theoretical Biological Phys-ics at Rice University.

Dr. Whitford received his B. S. in phys-ics with high distinction from Worces-ter Polytechnic Institute in 2003. In2009, he received his Ph.D. in physicsfrom the University of California, SanDiego, where he focused on modelinglarge-scale conformational processes inproteins. He subsequently pursued ap-plications of high-performance comput-ing as a Director’s Postdoctoral Fellowat Los Alamos National Laboratory,and as a Senior Scientist at Rice Uni-versity. He is now an Assistant Profes-sor of Physics at Northeastern Univer-sity.

Figure 1. The scale of detailed molecular simulations is rapidly in-creasing. A) The X-ray crystal structure of the potassium channelKcsA spurred simulations in the early 2000s of a highly complexmixture of proteins, lipids, and ions that totaled 40k atoms. B) Sim-ulations of the full ribosome containing over two million atomsthat span microseconds are now possible. These simulations aresufficiently long to study relaxation times of the fastest conforma-tional motions. C) The largest molecular dynamics simulation ofa biological system, to date, is a 64 million atom representation ofa HIV-1 viral capsid simulated for 100 ns. At length scales exceed-ing 100 nm, viral capsid simulations are close in size to the lightmicroscopy limit. The assemblies are shown to scale in C). PDB IDs:A) 1BL8,[28] B) 2Y0U/2Y0V,[29] C) 3J3Y.[30]

Figure 2. The elongation cycle of the ribosome involves manylarge-scale conformational rearrangements. Upon initiation of elon-gation (top, center), there is a single tRNA molecule in the riboso-mal P site. Next, the incoming aminoacyl-tRNA (yellow) is deliveredto the ribosome by EF-Tu (purple). After aa-tRNA accommodation,a peptide bond is formed (not shown). To allow for the next frameof the mRNA to be read, EF-G facilitates tRNA hybrid-state forma-tion and translocation, which are followed by tRNA dissociation. Al-though cryo-EM and X-ray crystallography provide insights into theatomic details of stable configurations, simulations bridge thesestatic representations and elucidate the structural and energeticcontent of the transition-state ensembles (TSEs) during each transi-tion. The atomic models depicted are described elsewhere.[7c,29,34]

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bilities. While this trend has a dominant effect on the im-provement of the performance of individual processors(cores), the gains in scientific computing does not directlyfollow the same law, since these applications use ad-vanced algorithms to utilize many computing cores con-currently. Parallel computation, which takes many forms,has led to a number of significant jumps in the overallcomputing speed of biomolecular simulations. In the last10–15 years, simulations have moved from being limitedto hundreds of computing cores per calculation, to tens ofthousands of computing cores connected by high-per-formance networking. In addition to highly scalable appli-cations that can utilize large numbers of central process-ing units (CPUs),[11] there has also been the graphical pro-cessor unit (GPU) revolution, in which GPUs have beenintegrated into modern supercomputers. GPUs offer tre-mendous computing power by interconnecting thousandsof smaller compute cores, each operating with the sameinstruction set. The overall flow of data on a GPU is fun-damentally different from a CPU, which has necessitatedsignificant programming efforts for molecular simulationsto efficiently utilize the hardware.[12] Finally, and most no-tably, there has recently been a leap in biomolecular sim-ulation performance enabled by computing hardware ad-vances. D. E. Shaw Research designed ANTON, which isa supercomputer that contains hardware specifically tail-ored to perform a single task: all-atom explicit-solventmolecular dynamics simulations.[1e,13] In doing so, compu-tations no longer have to be coded into atomic floating-point operations that are interpreted by a general pur-pose processor, but are instead performed completely atthe hardware level, where data communication is alsofully optimized. In all of these advances, it should bestressed that the major force has been parallelism: theability to simultaneously utilize many computing cores toincrease the speed of a single simulation.

With rapidly increasing computing capacity, it is no sur-prise that over the last ten years we have seen incredibleincreases in the scale of biomolecular simulations, both interms of accessible simulated time and simulation size.Before continuing, it is important to note that manytopics fall under the general category of molecular simu-lations, such as implicit-solvent[14] and coarse-grainedmodeling,[15] and Monte Carlo simulations.[16] However, toprovide a direct comparison of performance increases, welimit the immediate discussion to a single class of molecu-lar computation, explicit-solvent MD simulations,[17]

whereas other models are discussed in Sections 3 and 4.

2.1 Increased Simulation Duration

While early molecular simulations spanned only picosec-onds, in 1998, Duan and Kollman pushed the power ofcomputing to a point where they were able to performthe first microsecond simulation for a biopolymer in ex-plicit solvent.[18] Through the use of parallel computing,

this simulation represented a nearly 100-fold increase intimescale over the rest of the field. By pushing to longertimes, this simulation was able to detect a single partialfolding event for the villin headpiece subdomain. Thissimulation was certainly a technical accomplishment,though the system of interest was not chosen for immedi-ate scientific or medical relevance. Instead, this proteinwas chosen because it was known to be one of the fastestautonomously folding polypeptide sequences. Unfortu-nately, being limited to only fast folding proteins had thedrawback that the physical insights may not be transferra-ble to full-length naturally occurring proteins. Further-more, since all biomolecular processes are stochastic,many conformational events must be observed to drawphysically meaningful conclusions. While anecdotal evi-dence may be gleaned from individual trajectories, ob-taining sufficient statistics requires tens to hundreds oftimes the computing power needed for a single trajectory.Despite these limitations in interpretation, this technicaladvance was a significant milestone.

Ten years after the first microsecond simulation, multi-ple algorithmic advances by Schulten and co-workers[19]

enabled a relatively small simulation (30000 atoms) to ef-ficiently scale to over 300 compute cores. With this sub-stantial increase in scalability, the team was able toobtain over 100 ns of simulated time per day, and theyperformed a single simulation that spanned a total of10 ms. As a point of comparison, at that time it wascommon for entire studies to be limited to tens of nano-seconds. Consistent with the study of Duan and Kollman,this study focused on a fast-folding protein: the WWdomain. A striking finding of their study was that, whilenumerous collapsed configurations were populated, theydid not sample the native (experimentally determined)topology. While the authors may have been less than ex-cited to find that the simulation did not recover thenative configuration, the ability to obtain such long tra-jectories marked a new era of force-field refinement,where detailed aspects of the models have continued tobe improved.[20]

Only two years after the simulation by the Schultenteam, Shaw et al. shattered the record for longest simula-tion by performing a 1 ms explicit-solvent simulation.[13]

This incredible increase in speed was accomplished bybuilding a computer, from the ground up, that was specifi-cally designed for all-atom explicit-solvent simulations ofbiomolecules.[6b] With this hardware, the team fully foldedthe WW domain and Villin headpiece, in addition to ob-taining a 1 ms simulation that demonstrated the relaxa-tion processes in a single protein (even a small one) couldspan many orders of magnitude. More exciting than thisinitial study have been the team�s subsequent simulations,in which numerous small proteins have been folded,[1e] al-lowing identification of general features about the dy-namic of protein folding with particular explicit-solventmodels. In other applications, they have simulated sponta-



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neous drug-binding events,[2b,21] tested the robustness offolding results to force-field parameters,[22] and have evenobserved spontaneous large-scale functionally relevantconformational rearrangements.[2b]

Before closing the discussion on increased simulationduration, it is important to note that the preceding discus-sion was focused completely on advances that have ena-bled longer continuous simulated trajectories. It is theperspective of many that the holy grail of molecular simu-lations is to fully characterize the thermodynamics and ki-netics of biological systems through a single simulation.In the words of the Schulten group, the philosophy is touse the computer as a “computational microscope” in thestudy of biology.[23] From this perspective, the two mainobstacles are to simulate systems for physiologically rele-vant timescales and to develop sufficiently accurate force-fields. Long continuous trajectories are unarguably thegold standard, although they are not the only approachcapable of providing kinetic and thermodynamic informa-tion. Large numbers of short trajectories can be generat-ed by utilizing poorly connected, but highly redundant,cloud computing platforms such as Condor[24] or Fol-ding@home (folding.stanford.edu). Therefore, there havebeen some efforts to obtain information on large-scaleconformational events by piecing together many shorttrajectories.[25] The most difficult aspect of these ap-proaches is that the ensemble of short simulations mustbe properly initialized. That is, the initial configurationsmust represent a thermodynamically equilibrated distri-bution. With the inherent limitations of this approach, thequantitative details of the predictions are often controver-sial, although the simulated time accessible with Antonprovides the necessary benchmarks for validating thesetechniques or determining their limits. Obtaining thermo-dynamics does not necessitate long continuous trajecto-ries, and therefore, does not require specialized machines.For example, trivially parallelizable schemes, such as rep-lica exchange, have been utilized for molecular simula-tions for decades.[26] These methods accelerate thermody-namic convergence by simulating multiple copies of thesystem, each with slightly different parameters (e.g., tem-perature, energetic perturbations). The configurations ofthe replicas are then interconverted according to MonteCarlo criteria that ensure detailed balance is satisfied.Due to the instantaneous jumps in parameters, kinetic in-formation is compromised, but the probability distribu-tion for a given parameter set may be evaluated throughfree-energy, or weighted histogram, approaches.[27]

2.2 Increased Simulation Size and Complexity

In addition to pushing the timescales of explicit-solventsimulations, advances in HPC have allowed for increasesin the size and complexity of simulated systems. Ratherthan provide an exhaustive list of technical achievements,we discuss some representative examples of simulations

that have addressed increasingly complex systems, whichhave also been accompanied by increases in simulationscale.

2.2.1 Ion Channels

In the early 2000s, large-scale supercomputers were al-lowing researchers to move beyond simulations of smallpolypeptides, and begin simulating more complex molec-ular assemblies. For example, atomic-resolution crystalstructures of membrane ion channels (first available forthe K+ channel KcsA in 1998[28]) allowed for the con-struction of molecular simulation models. Pioneered byRoux and co-workers,[8d,31] simulations of this assemblyare notable not for their massive size, although they werereasonably large for the time (�40000 atoms), rather, thegreater challenge was to prepare biologically relevantstructural models that included the protein, membrane,water, and ions. We remark on simulations of ion chan-nels because, in addition to their complexity, these studieswere scientifically very successful.[31c] In particular, simu-lations provided insights into the locations of free-energyminima for K+ ions inside the channel,[32] the conduc-tance of the channel under saturating conditions,[31a] andthe mechanism of selectivity.[31b] In fact, many resultswere consistent with concurrent experimental studies andsometimes even preceded them.

2.2.2 Ribosomes

A major milestone in the use of computing applied tomolecular biology was reached in 2005 with the first ex-plicit-solvent simulation of a complete bacterial riboso-me.[8c] The solvated ribosome contained over two millionatoms, which necessitated the use of nearly 1000 computecores on one of the most powerful supercomputers. Withsuch massive dimensions, even state-of-the-art HPC re-sources could only yield trajectories of a few nanosec-onds. While this is far shorter than biological timescales,by applying targeted molecular dynamics methods[33] thatforce a conformational change to occur over a predefinedtime interval, the large-scale tRNA “accommodation” re-arrangement was visualized for the first time.

Subsequent to the first simulation of the ribosome, in2010, we reported the first explicit-solvent simulations ofa ribosome that extended to over 100 ns (seven individualtrajectories, for a cumulative 1.5 ms of sampling).[2c] Incontrast to earlier calculations, these simulations werecompletely unrestrained, allowing the ribosome to natu-rally sample its phase space. Although these calculationswere the longest simulations of their size, only local rear-rangements could be sampled on such short intervals.However, by moving to these timescales, it was possibleto characterize diffusive properties of individual mole-cules within the complex, for which relaxation times for



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short-scale displacements were in the order of about 10–20 ns.[35]

In 2013, we further extended the duration of explicit-solvent simulations of the ribosome, and reported thefirst continuous explicit-solvent simulation of a multimil-lion atom system that exceeded 1 ms.[36] As discussedlater, this three-orders of magnitude increase in simulatedtime, relative to the first ribosome simulations, has trans-formed explicit-solvent simulations of the ribosome froma method that suggests possible dynamics into a tool thatcan provide physicochemical measures that bridge theo-retical descriptions of energy landscapes and experimen-tal kinetics.

2.2.3 Viruses

Last year, the record for largest biomolecular simulationwas broken by the Schulten team, who performed thefirst all-atom explicit-solvent simulation of a completeHIV-1 viral capsid.[30] This simulation included 64 millionatoms and spanned over 100 ns. This was enabled by im-provements in the scalability of molecular dynamics simu-lation software (in this case NAMD[11a]) that allowed ef-fective utilization of one of the world�s largest supercom-puters, Blue Waters. Given the size of a viral capsid,100 ns is not sufficient to observe large-scale conforma-tional processes, although it is nonetheless an impressivetechnical achievement. Despite the current limitations intimescales, if exponential growth in computing capacitycontinues, one may expect millisecond simulations ofviral capsids to become standard in the future.[6a]

3 The Ever-Increasing Utility of MultiscaleModeling for Biomolecular Simulations

In the 1970s, the 2013 Nobel Prize winners pioneered theuse of molecular dynamics to help speed up the calcula-tion of quantities that depended on incredibly detaileddegrees of freedom. Their QM/MM approach monitoredthe detailed electronic degrees of freedom in the regionof the biomolecule where chemistry was taking place, andrelied on classical mechanics, for example, the explicitsolvent models described above, to describe the dynamicsof the rest of the system.[37] This technique of removingextraneous degrees of freedom is generally known as“multiscale modeling” or “coarse graining” and is wide-spread in computational molecular biophysics. The rapidincreases in computing power over recent decades havenot diminished the need for multiscale approaches, andadvances in multiscaling techniques have only broadenedin applicability and scope. One clear benefit of reducingthe number of degrees of freedom is the ability to reachlonger timescales and larger length scales, since there arefewer forces to calculate and fewer atoms to move. A lessclear, but arguably more important, benefit of multiscale

modeling is that by reducing the complexity of a simula-tion, the results are more easily used to obtain intuitivephysical insights about the system. In the field of MDsimulations, there are two broad classes of physics-basedcoarse-graining. The first are bottom-up approaches thatattempt to rigorously define a set of coarse-grained sitesand interactions between these sites based on a more-de-tailed Hamiltonian.[38] The second are top-down ap-proaches that define a simplified Hamiltonian based ontheoretical considerations of the global energy landscap-e.[15b,39]

Bottom-up coarse-graining uses statistical mechanicalprinciples as a theoretical foundation for coarse-graining.The free energy F is expressed in terms of a reduced setof coordinates RCG,[38a] instead of in terms of all the 3Natomic coordinates r [Eq. (1)]:

expð�bFÞ /Z

dr exp½�bVðrÞ� !Z

dRCG exp½�bVCGðRCGÞ�

ð1Þ

The challenge then becomes to define a set of RCG suf-ficient to capture the process of interest and a set of inter-actions between these sites that converges as closely aspossible to the underlying distribution. While coarse-graining reduces the computational demand when evalu-ating thermodynamic properties, the accuracy of thecoarse-grained potential, VCG, is dependent on the degreeto which the more-detailed Hamiltonian is initially simu-lated, since it is the benchmark against which the coarse-grained model is calibrated. Voth and co-workers usedtheir version of multiscale coarse-graining (MS-CG)[40] toexplore the assembly of biomolecular complexes. Theyfound that the long length-scale dynamics of actin fila-ments could be affected by subtle differences in actin mo-nomer packing.[41] Of interest to the previously discussedHIV-1 viral capsid, Grime and Voth used experimentaldata to parameterize a MS-CG model and explored time-dependent assembly of the capsid,[42] which occurred ona timescale completely out of reach to the 64 millionatom explicit-solvent simulation.

Applications of top-down coarse-graining have beenvery effective at revealing the intricate role of configura-tional entropy during function, although direct interpreta-tion of the energetic contributions is often less clear. Theenergy landscape theory for protein folding[39,43] has beenextensively used as a theoretical basis for creating biomo-lecular models with Hamiltonians that are explicitly fun-neled.[15b,44] This theory states that evolution has achievedrobustly folding biomolecules according to the principleof minimal frustration, by selecting for sequences inwhich the interactions present in the native structure(native contacts) are, on average, more attractive thannon-native interactions. The simplest realizations of fun-neled energy landscapes are called “structure-basedmodels” (SBMs, or SMOG models)[45] because they



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define a biomolecular Hamiltonian with an explicitenergy minimum at a native configuration, where nativeinteractions are attractive and non-native interactionspreserve excluded volume. With their relative simplicity,interpretation of general physicochemical relationshipscan be established, offering great potential to uncover or-ganizing principles of biomolecular dynamics. Becausethe interactions are soft (i.e., contacts can break), as op-posed to elastic network models,[46] SBMs can explore dy-namics across a range of length scales, from detectingsubtle entropic allostery,[47] elucidating TSEs of globularproteins,[44b,48] and exploring conformational transitions inmolecular machines.[2c,49] Through continued growth inthe scale of HPC facilities, and accessibility of the modelsthrough web-based tools,[50] SBMs can now be used to cal-culate thermodynamic relationships in large biomolecularassemblies, even the ribosome.

4 Recent Physical Insights from Simulations

The true power of molecular simulations is realized whenthey are used to provide physical insights that are not di-rectly measureable. In this section, we discuss two distincttypes of advances that are unique contributions of molec-ular simulations. First, there have been many instances inwhich molecular simulations suggest order�disorder dy-namics that are functionally relevant. Oftentimes, the pre-diction of disorder persists in multiple theoretical models,in which the principal difference in the prediction is thedegree to which disorder occurs. By establishing the theo-retical viability of these events, existing experimental datais reinterpreted, and new experiments are devised to dis-tinguish the subtle details of each process. A secondmajor contribution of molecular simulations is that theyprovide a means to explore the relationship betweenenergy landscapes and biological dynamics. These struc-tural�energetic frameworks provide a theoretical founda-tion with which deeper insights may be drawn from ex-perimental measurements. We review progress in the con-text of large-scale, functionally relevant, conformationaltransitions, and their relation to folding and disorder.

4.1 The Role of Order�Disorder Transitions at the MolecularLevel

A key theme that has emerged from computational ap-proaches is that biomolecules often undergo partial (orcomplete) unfolding events during function. These largechanges in configurational entropy can directly governthe biological dynamics, and can manifest in qualitativelydistinct ways. One example is found in the Rop dimer, forwhich it was predicted that interconversion betweenactive and inactive conformations required complete un-folding of the monomers,[51] as later verified in single-mol-ecule experiments.[52] In other systems, the molecules

become only partially disordered, as predicted for adeny-late kinase (Adk),[53] epidermal growth factor receptor(EGFR) kinase,[2b] and the ribosome.[2c] We survey somerepresentative examples of cases where simulations havepredicted order�disorder transitions that are related tofunctional dynamics. A more comprehensive discussionon these events, and the associated computational meth-odologies, may be found elsewhere.[15b] It should be em-phasized that the general theme of order�disorder transi-tions during function is not model specific because theyare frequently predicted in all types of models, includingcoarse-grained, highly detailed and explicit-solvent simu-lations, as well as by non-simulation-based computationalapproaches.

4.1.1 Cracking

Cracking is the process by which biomolecules transientlyunfold isolated sets of residues during functional confor-mational rearrangements. This was first suggested bynormal mode analysis (NMA) of Adk,[53a,54] which pre-dicted that during conformational transitions large levelsof internal “strain” energy accumulated in specific resi-dues. Furthermore, the scale of the predicted strainenergy was greater than the overall stability of a protein(>20 kcal mol�1). Such high degrees of strain energywould implicate large free energy barriers, such that inter-conversion rates would be far slower than biological time-scales. Based on these observations, Miyashita et al. pro-posed that residues might have relieved strain by partiallyunfolding during the conformational transition.[53a] Thispartial unfolding releases stabilizing interactions, and theincreased configurational entropy of the TSE further re-duces the free-energy barrier of interconversion(Figure 3). While the cracking paradigm is an elegant res-olution to the apparent paradox of strain accumulationand biological kinetics, its prediction was based ona simple elastic-network model. The simplicity of the ap-proach led to doubts, particularly whether the barrierheights were an artifact of the model,[55] which called forfurther investigation with more highly detailed ap-proaches.

Inspired by the prediction of cracking, numerous theo-retical models have been applied to identify the degree ofcracking in Adk.[55,56] Using a coarse-grained model, forwhich only interactions in the endpoint configurations arestabilizing, we found that backbone distortions were mostpronounced in the same residues that were predicted tobe under high strain by NMA.[56b] For a similar coarse-grained model developed by Daily et al. , traces of crack-ing were detected, although to a lesser extent than in ourmodel.[57] As discussed in detail elsewhere,[15b] the attenu-ated levels of cracking may be rationalized by consideringsubtle differences in the protocols used to construct thesimplified models. Regardless of these differences, sincethe protein was described at a coarse-grained level, meas-



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ures of cracking were limited to large-scale backbonefluctuations. In simulations of protein folding with all-atom models, both using simplified energetic schem-es[44e,58] and explicit-solvent,[13] it has been observed thatthere is differential timing of backbone and side-chain or-dering. This heterogeneous distribution of disorder sug-gests that cracking dynamics may be isolated to individualresidues, or side chains, and not necessarily groups of resi-dues, as originally envisioned. In support of this notion,explicit-solvent simulations of Adk later predicted thatcracking was primarily associated with changes in disor-

der of side chains, and less significant changes were ob-served in the backbone atoms.[56c]

An alternate (and commonly employed) conceptualframework for describing rearrangements is the “hinge-rod” model, where specific residues are described as ex-tremely flexible and tertiary structure is approximated asbeing rigid.[59] In the case of Adk, evidence is continuingto mount in support of aspects of both the hinge-rod de-scription and the cracking paradigm. Most likely, theextent to which each type of motion is applicable will becontext dependent.[53b] With the continued application ofdiverse models, and increasingly convergent statistics ena-bled by computing advances, the precise balance betweeneach is sure to be illuminated in the coming years.

Cracking is not isolated to the dynamics of Adk, and ithas been implicated in kinesin motor proteins, as well.These molecules transport cargo throughout the cell byundergoing cyclic “walking” motions along microtu-bules.[60] The stepping motion, which is driven by hydroly-sis of adenosine triphosphate (ATP), involves domain dis-placements of about 80 �.[61] Through the application ofcoarse-grained models that included a time-dependent in-teraction to mimic the role of hydrolysis, Hyeon andOnuchic predicted that cracking was also likely to occurduring the power stroke of the kinesin head domain.[44d,49]

Through simulation, they demonstrated that the time-de-pendent effect of hydrolysis could lead to transient inter-nal competition between stabilizing interactions, whichdistorted the linker region connecting the domains. Theyfurther demonstrated that a delicate balance between ri-gidity and disorder was likely to control the kinetics of ki-nesin walking. That is, when the linker is too flexible,strain cannot be transferred between the domains, and in-effective stepping attempts follow. In contrast, when thelinkers are too rigid, there is not sufficient flexibility toallow for global rearrangements of the domains; this isconsistent with predictions from analytical theories forother motor proteins.[62] This balance has also been de-tected in Ncd, which is a motor protein similar to kinesin-1 that processes along the microtubule in the opposite di-rection.[63]

It is becoming clear that there is a propensity to crackin many protein families, and prediction of its presence isnot an artifact of a specific model. Using a statistical me-chanical model, Itoh and Sasai found evidence of crack-ing in the C-terminal tail of calmodulin.[64] In a similarstudy, Tripathi and Portman developed a variationalmodel that implicated cracking in the N-terminal domainof calmodulin.[65] Hyeon et al. also applied their coarse-grained models to protein kinase A and found a propensi-ty to crack in the catalytic domain.[66] While direct obser-vation of cracking is difficult experimentally, current datais now being found to be in agreement with the crackingframework for Adk.[67] Using an instantaneous normalmode (INM) calculation to enhance sampling of domainrearrangements in Adk,[68] it was observed that the INMs

Figure 3. A) When proteins undergo large-scale, functionally rele-vant, conformational transitions, there is the possibility that largelevels of strain energy will accumulate. By transiently partially dis-ordering (i.e. , cracking) strained regions, the free energy barriers ofinterconversion can be reduced. This novel mode of function isone way that enzymes can accelerate conformational kinetics, evenwhen the endpoint configurations are fully ordered. Two proteinsfor which cracking has been predicted are B) Adk and C) EGFRKinase. The endpoint configurations of each transition are shown,with each colored by domain.



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immediately preceding domain rearrangements were dis-tinct from the endpoint values. Since low-frequencymodes describe low-energy, rigid-body-like rearrange-ments, their inability to account for Adk�s rearrange-ments suggests that more localized fluctuations, such ascracking, may be required. In a survey of proteins, Oka-zaki et al. found that cracking was a common feature intheir dynamics.[69] Most recently, using the powerfulAnton supercomputer, the Shaw team used all-atom ex-plicit-solvent simulations and observed that, during thetransition between active and inactive forms of EGFRkinase, the protein adopted a metastable basin, in whichspecific residues underwent increased fluctuations;[2b] thisis again consistent with the proposed role of crackingduring conformational interconversion steps.

4.1.2 Fly Casting

Inside of the cell there are many kinetic processes thatmust maintain a proper balance to ensure survival.[70]

Thus, kinetic accessibility of interactions is required forproper decision making at the molecular level. If bimolec-ular association kinetics were governed completely by dif-fusion of the center of mass, then the ability to modulaterates would be limited to a very small range of values. Toexplain the apparent ability of biology to tune associationrates, the fly-casting mechanism was proposed,[71] whereproteins partially unfold, thereby extending disorderedchains to search for a substrate. Upon substrate recogni-tion, the chain reorders and signaling occurs. Using free-energy functional techniques, Shoemaker et al. showedthat this form of rearrangement could exponentially in-crease the accessibility of substrate capture, leading tosignificantly increased bimolecular association rates.[71]

Subsequent studies using simulations of coarse-grainedmodels explicitly demonstrated the balance of order/dis-order and association kinetics.[44c,72] In further support ofthe fly-casting mechanism, bioinformatic analysis suggeststhat the electrostatic composition of disordered tails canbe conducive to fly casting.[73]

In addition to disordered regions enabling fly casting,studies that employ coarse-grained models have demon-strated that DNA-binding proteins may utilize disorder toincrease the rate at which proteins search for the targetsite. Specifically, disordered regions allow for “sliding”motions,[74] which reduce the search process to a 1D diffu-sive motion.[75] As proteins search DNA, they slide andalso “hop” between sites. Each hop requires dissociationand reassociation with the DNA, where the fly-castingmechanism assists each reassociation event. These simula-tions have demonstrated that the degree to which eachmechanism is utilized is determined by a combination offlexibility, disorder, and energetics.

4.1.3 Entropically Directed Machine Dynamics

The previous examples illustrate how simulations havepredicted roles for disorder transitions in rather smallbiomolecules. As computing capacity continues to grow,large-scale machines are now being simulated, and moreelaborate roles of disorder are being detected. The largestmolecular machine for which extensive computationalstudies have been performed is the ribosome. Usinga combination of all-atom models that employ simplifiedenergetics (i.e., SBMs), in addition to highly detailed ex-plicit-solvent simulations, we found strong signatures ofconfigurational entropy changes during the elongationcycle of the ribosome (Figure 2). During the translationof mRNA sequences into protein sequences, the ribosomerecruits transfer RNA (tRNA) molecules one at a time.After this initial selection process, each tRNA moleculemoves nearly 100 �, placing the new amino acid withinthe catalytic center of the ribosome (i.e., the peptidyl-transferase center (PTC)). Based on enthalpic considera-tions of the tRNA�mRNA interactions, thermodynamicdifferences between correct (i.e., cognate tRNA�mRNApairs) and incorrect tRNA species can explain error ratesof around 1 : 100. However, the error rate under physio-logical conditions is about 1 :103–104. Thus, there must beadditional mechanisms by which the tRNA is “proofread”after initial association.[76]

An energetic mechanism for proofreading, which fitswithin the context of ribosome function, was proposed byHopfield, and is known as “kinetic proofreading”.[77] Ac-cording to this framework, molecular systems may be“loaded” into a marginally stable intermediate state byan energy releasing step, such as GTP hydrolysis. By driv-ing the system to a metastable state, there may bea larger kinetic difference in the forward and rejectionprocesses than can be explained by thermodynamic differ-ences alone. On the ribosome, elongation factor-Tu (EF-Tu) provides the loading step for each tRNA, whereenergy released by guanosine triphosphate (GTP) hydrol-ysis[29,78] is used to position the tRNA in an energeticallystained “A/T” configuration.[79] After hydrolysis, thetRNA may either fully enter the ribosome (i.e., adopt theA/A configuration) or be rejected and disassociate.Within the kinetic proofreading framework, the A/T-to-A/A transition (“accommodation”) provides the necessa-ry proofreading step/s during translation. Consistent withthis notion, theoretical kinetic models have shown that ki-netics and fidelity must be properly balanced; this impos-es significant limitations on possible rates for ribosomalprocesses.[80] When accommodation is too fast, insur-mountably large energetic contributions would be re-quired for incorrect tRNA molecules to be rejected. Ad-ditionally, when accommodation is slow, all tRNA mole-cules would have an elevated propensity to dissociate,thereby leading to many nonproductive tRNA associationevents, even for correct tRNA.



Review


Many investigations have implicated an interplay be-tween energetics, kinetics, and fidelity during tRNA ac-commodation,[82] which is ultimately governed by detailedatomic interactions found in the endpoints,[9a,29,83] as wellas the structural content of the TSEs.[2c,8c] As discussedearlier, Sanbonmatsu et al. were the first to perform a sim-ulation of the full ribosome, using explicit-solvent. [8c] Inthis calculation, which was for a duration of only a fewnanoseconds, targeted molecular dynamics protocols wereused to induce the accommodation process. While the si-mulated timescale was roughly four to six orders of mag-nitude faster than that in the cell, there were some struc-tural signatures detected. Specifically, the calculation il-lustrated the presence of a sterically accessible corridorthrough which the tail of the tRNA may navigate as it ap-proaches the PTC. Later, we extended a class of all-atomstructure-based models, which had primarily been used tostudy protein folding,[44e] and applied them to this sameconformational transition. With these software advances,we were able to simulate the accommodation processwithout using biasing techniques.[2c] While the earlier ex-plicit-solvent simulation implicated a single route for thetRNA, by using a model in which full-scale accommoda-tion events were spontaneous, we found that the immenseflexibility of the tRNA tail (i.e. , the 3’-CCA end) enableda variety of pathways and also resulted in large changesin the configurational entropy. This entropic contributionalso leads to a multistep accommodation process,[2c]

where a specific sequence of events is followed. After thetRNA is delivered to the ribosome, it adopts a disperseensemble of configurations, which is associated with anincrease in the configurational entropy of the tRNA tail.As the tRNA moves into the PTC, movement of the tailbecomes restricted (Figure 4), suggesting that a decreasein configurational entropy resists entry of the tail into thePTC. At the end of accommodation, the tRNA is highlyordered, which is necessary for peptide-bond formationto occur.[84]

At first glance, the order�disorder-order transition inthe tRNA is reminiscent of protein folding, where a bal-ance between enthalpy and entropy determines the mech-anism and kinetics. In addition to being a physically entic-ing description, this interpretation carries significant bio-logical implications. Specifically, the tail of the tRNA isuniversally conserved across all tRNA species, and thetail is covalently bonded to the incoming amino acid.Thus, this isolated entropic change may provide resistancethat ensures amino acid entry into the catalytic center isthe last step of accommodation. This late entry of theamino acid may then allow the use of any proofreading“machinery” before incorporating an incorrect aminoacid into the growing protein chain.

The central role that entropy has on ribosome dynam-ics is also found in our recent simulation study of tRNAhybrid-state formation. In contrast to the results for ac-commodation, simulations of hybrid-state formation sug-

gest that the tRNA has a constant (or possibly increasing)degree of disorder as it moves through the ribosome[81]

(Figure 4). Again, this can be interpreted from physicaland biological perspectives. Physically, this indicates thatthe forward hybrid-state motion is not hindered by CCAflexibility. Biologically, the entropic resistance observedduring accommodation is not necessary for hybrid-stateformation, since hybrid-state formation occurs after thepeptide bond has been formed. That is, there is no appar-ent need for movement of the tRNA tail to be delayed,since this step is not associated with fidelity. Followingthat logic, it is not clear why hybrid-state formationshould have any factors that disfavor forward motion,since the more rapidly the tRNA moves away from theA/A configuration, the more quickly the ribosome maybegin reading the next frame of the mRNA.

Figure 4. As tRNA molecules move through the ribosome, changesin configurational entropy can determine the overall conformation-al events. A) Probability distribution of the tRNA tail angle and po-sition averaged over 312 accommodation simulations with an all-atom SMOG model.[44e,50a] As the tRNA moves into the ribosome(left to right), a continual decrease in the range of accessible con-figurations was observed.[2c] B) After the peptide bond is formed,tRNA molecules move through the ribosome and adopt hybridconfigurations. Simulations of hybrid-state formation (left to right)with SMOG models suggest only modest changes in accessibleconfigurations for this transition, indicating a smaller role of entro-py.[81]



Review


While early explicit-solvent simulations of the ribosomesuggested that a minimal degree of tRNA flexibility wasrequired for accommodation to occur,[8c] our simulationsusing SBMs were the first to demonstrate that configura-tional entropy of the tRNA directly contributed to theoverall order of events during tRNA movement. Previ-ously, the flexibility of tRNA was frequently noted, al-though specific biological implications were not clear. Inmany crystallographic models, the tail of the tRNA mole-cules is not resolved,[85] or the reported Debye�Wallerfactors (i.e. , uncertainty in the coordinates) are high.[86]

Similarly, there is often only low density observed for thetRNA tail in cryogenic electron microscopy studies,[87] im-plicating conformational heterogeneity. One may alsodeduce that the tRNA tail is flexible through its ability tointeract with many binding partners,[88] such as synthetas-es,[89] EF-Tu,[90] and multiple ribosomal binding sites,[91]

each requiring that distinct tRNA configurations be ac-cessible.

4.2 Theoretical Frameworks Enabled by Simulation

In the previous sections, we highlighted structural/mecha-nistic insights that originated from theoretical considera-tions and molecular simulations. These phenomenologicaldescriptions are invaluable guides that aid the develop-ment of new experiments and the interpretation of exist-ing data. In addition to this significant contribution ofcomputation, it can also provide more precise quantita-tive descriptions. As an example, we discuss how simula-tions help in understanding general properties of theenergy landscapes of biomolecular dynamics, includingmultidimensional diffusion and the identification of ap-propriate reaction coordinates for describing the underly-ing barriers. At the most fundamental level, biomoleculardynamics may be understood as being a consequence ofan underlying energy landscape. From this perspective,conformational rearrangements are described as occur-ring through Brownian movement across the landscape.Since this diffusive picture is so pervasive in our under-standing of biomolecular dynamics, we review some ad-vances and questions related to the characterization ofthe diffusive properties of biomolecules, using proteinand RNA folding as an example. This discussion also in-cludes efforts to identify collective reaction coordinatesfor which the diffusive picture is applicable. We concludeby discussing recent studies that have aimed to adopt thestrategies developed for protein folding and apply themto study the ribosome.

4.2.1 How to Describe Energy Landscapes: Proper ReactionCoordinates and Diffusion

In the energy landscape perspective of biomolecular dy-namics, as originally developed for protein folding,[39,43-

b,44a,92] the dynamics is described as diffusive movement

along appropriately chosen collective reaction coordi-nates, where interconversion between endpoints occurswhen the system crosses the rate-limiting free-energy bar-rier. To rigorously use this theoretical approach, a �propercoordinate�, 1, cannot be chosen arbitrarily. Rather, ap-propriate coordinates must be able to distinguish the es-sential aspect of the dynamics, as defined by specific crite-ria. The simplest property is that the endpoints of theconformational process are associated with distinct valuesof the coordinate. This criterion is typically trivial to satis-fy, and inspection of structural models (from X-ray crys-tallography, or other methods) can suffice. In the contextof single-molecule methods applied to the ribosome, thisis often the primary criterion that has been used to selectcoordinates (i.e. , labeling sites).[93] When the difference inthe values of the coordinate are significantly larger thanuncertainties in the observable, and the coordinate isroughly along the direction of the underlying barrier,[79b]

the relative free energies of the endpoints can be inferredwith minimal uncertainty. Such insights are extremely val-uable to our understanding of biomolecular dynamics, al-though the full power of single-molecule methods mayonly be fully realized when they are able to probe theproperties of the TSEs. Below, we outline additional cri-teria that can be applied to the analysis of reaction coor-dinates. We draw examples from protein folding, wherethe close collaboration of theoretical and experimental in-vestigations have led to fundamental changes in our levelunderstanding.

The first kinetic criterion that an appropriate coordi-nate should satisfy is that the projected dynamics be Mar-kovian. Furthermore, the projection of the coordinate-de-pendent diffusion coefficient, D(1), and free energy pro-file, F(1), should be consistent with the kinetics of inter-conversion between the states of interest. That is, if oneprojects the motion onto a proper coordinate, the ob-served mean first passage time will be consistent with themean first passage time calculated by numerically inte-grating a Kramer�s-style double integral.[92b,94] While thenecessary properties of a proper coordinate are clear,there is no analytical expression that can be used to di-rectly calculate coordinate-dependent diffusion coeffi-cients from a trajectory. With this lack of an exact analyti-cal expression, a range of computational methods havebeen developed to extract estimates of D(1). Best andHummer have developed statistical analysis techniques,grounded in Bayesian inference,[95] that take a trajectoryalong 1 and construct a short-time propagator of the dy-namics. From the short-time propagator, D(1) and F(1)are estimated, under the assumption of diffusive dynam-ics. An alternate method for estimating D(1) and F(1)was proposed by Yang et al. ,[96] where simulated data arefit to the short-time solution of the Fokker�Plank equa-tion; thus yielding estimates of the drift and diffusion.The reliability of these projections is then evident bywhether the character of the short-time dynamics is con-



Review


sistent with that of a diffusive system. When using eitherof these methods, the projected D(1) and F(1) (as esti-mated by the short-time dynamics) may be comparedagainst the global dynamics, where agreement betweenthe observed rates would support the use of a putative co-ordinate.

In the preceding discussion, we focused on extractingdiffusion coefficients and free energies that were consis-tent with the global dynamics, although this could be anoverly stringent requirement. For example, the projecteddynamics about the endpoints may not be perfectly diffu-sive, though the overall kinetics can be described suffi-ciently well when movement about the transition regionis diffusive. In these cases, the TSE, as identified by thepeak in the projected free energy, will correspond to con-figurations that are equally likely to move towards eitherof the endpoints. In other words, the coordinate will bea monotonic function of the splitting probability[97] (fre-quently referred to as Pfold in the context of protein-fold-ing studies). Additionally, the splitting probability of con-figurations in the TSE would be 0.5. For protein folding,it was shown that intuitive reaction coordinates couldnearly fulfill this criterion,[98] even when movement alongthe full range of values was not perfectly Markovian.[99] Ifmotion is truly Markovian across the full range of values,and the coordinate is a monotonic function of the split-ting probability f(1), then one may construct the cut freeenergy as a function of the splitting probabilityFC,1(f(1))[99,100] for different sampling intervals, Dt. If thedynamics along 1 is Markovian, movement along thesplitting probability will also be Markovian, FC,1(f(1))willbe constant in f(1) and it will not depend on the value ofDt. In the case of using the number of native contacts(commonly denoted by Q) as a reaction coordinate forfolding, it was found that the motion was not perfectlyMarkovian across the full range of values, although it didaccurately reflect the splitting probability.[98] A relatedmetric is P(TPj1): the conditional probability of being ona transition path (TP) as a function of the coordinate,which is related to the splitting probability.[101] For an ap-propriate coordinate applied to a two-state transition,P(TPj1) will have a single sharp peak that reachesa value of 0.5. This property has been utilized to aid inthe design of coordinates that most accurately capturethe essential characteristics of the TSE of protein folding,both with simplified models[2a,101] and explicit solventmodels.[1e,13] In most cases, it was found that the numberof native contacts was an accurate probe for diffusivemotion along a reaction coordinate.

In the study of the ribosome, our understanding of dif-fusion and reaction coordinates is far less developed thanthat for protein folding. This large gap has been the resultof many factors, including the historical lack (until 2001)of atomic-resolution structural models for the intact com-plex, the need for large-scale computing resources toobtain long simulated trajectories, and persistent contro-

versy about the relevant functional states during ribo-some function. Over the last 15 years, the first two factorshave been largely resolved. Nonetheless, there are onlytwo simulation studies, to date, that have aimed to char-acterize the diffusive properties of the ribosome�s func-tional motions.[35,36] For those studies, we performedlarge-scale (2–3 million atoms) explicit-solvent simula-tions of the ribosome for relatively long timescales(200 ns to 1.3 ms). These are some of the largest simula-tions of biomolecules ever performed, although they arestill very short timescales from a physical perspective.That is, the short-time fluctuations of tRNA moleculesinside of the ribosome occur on timescales of 10–20 ns.[35]

Therefore, it is immediately clear that any quantitativemeasures from simulations of this duration must be inter-preted with caution. For large-scale rotary motions, whichhave been implicated in translocation dynamics of the ri-bosome, the short-scale fluctuations along the rotation co-ordinates take place over longer timescales (�50 ns),[36]

making it even more difficult to estimate kinetic quanti-ties. It is also clear that thermodynamic quantities cannotbe directly extracted from simulations of this duration.While these timescales have clear limitations, kinetic sig-natures may still be extracted, which can be used to esti-mate diffusion coefficients along various reaction coordi-nates. In our first study, we performed seven simulationsof the ribosome and obtained the effective diffusion coef-ficients along tRNA coordinates.[35] We found that thedisplacement squared increased linearly with lag time(Figure 5); this is characteristic of diffusive behavior. Theindividual estimates of D varied by roughly a factor oftwo between runs. In terms of thermodynamics, an uncer-tainty of two in D translates to an uncertainty in the bar-rier height of less than 1kBT, since the barrier is logarith-mically related to D. From these estimates of D, it is alsopossible to infer the scale of the energetic roughness.That is, the observed diffusion coefficient describes theeffective diffusion after averaging over the short length-scale energetic roughness.[92b,94,102] In the reported explicit-solvent simulations, the effective diffusion coefficientabout the A/T configuration is nearly identical to that ex-pected for free diffusion in solution; further indicatingthat the energetic roughness associated with tRNA move-ment through the ribosome is small (�1kBT). For subunitrotation, the suggested diffusion coefficient was about1 degree 2/ms.[36] Given that the mass of the small subunitis about 1 MDa, free diffusion arguments for tumblingtime in solution would suggest a diffusion coefficient of1 rad2 ms�1. Assuming random short-scale energetic rough-ness,[92b,94] this deviation from the free-diffusion limit sug-gests that the roughness is approximately 2.8 kBT(1.7 kcal mol�1). These energetic inferences are importantsteps towards revealing the full character of the energylandscapes that govern ribosome function. However, it isimportant to note that this initial progress has been predi-cated on the assumption that the coordinates (in this



Review


case, the distances that are probed in smFRET experi-ments[2c,78,103]) accurately capture the rate-limiting freeenergy barrier. To date, this question has not been ad-dressed, neither qualitatively nor quantitatively, for the ri-bosome. This lingering uncertainty clearly needs to be re-solved before we will be able to use the energy landscapeframework to unambiguously describe ribosome dynam-

ics. With continued progress in simulations and single-molecule methods applied to this system, the outlook ispromising that our physicochemical understanding of theribosome will soon mature to the level of other molecularprocesses, such as folding.

5 Summary and Outlook

What originated as highly detailed quantum-mechanicalapproaches, the field of computational biophysics hasnow matured into a broad area of scientific inquiry thatutilizes a wide range of techniques. Each method is suitedto address distinct questions about the dynamics of bio-molecules. Through the integrated use of coarse-grainedand highly detailed models, our understanding of thephysical principles that underpin biomolecular functionwill continue to be refined. Along with growth in comput-ing capacity, theoretical methods hold great promise tobecome indispensible tools for revealing the structuraland energetic contributors to biomolecular functionacross many length and time scales.

Acknowledgements

This work was supported in part by an NSF CAREERAward (Grant MCB-1350312) and the Center for Theo-retical Biological Physics sponsored by the NSF (GrantPHY-1308264) and by NSF-MCB-1214457.

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Received: January 18, 2014Accepted: February 23, 2014

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