Design of a Constant Force Clamp and Estimation of MolecularMotor Motion using Modern Control Approach

Subhrajit Roychowdhury, Shreyas Bhaban, Srinivasa Salapaka, Murti Salapaka

Abstract— Since its inception, optical traps have be-come an important tool for single molecule investigationbecause of its precise ability to manipulate microparti-cles and probe systems with a force resolution of theorder of fN. Its use as a constant force clamp is ofparticular importance in the study of molecular motorsand DNA. The highly nonlinear nature (specially thepresence of hysteresis) of the force-extension relationshipsin such biomolecules is traditionally modelled as a linearHookean spring for small force perturbations. For theselinear models to hold, high disturbance rejection band-widths are required so that the perturbations from theregulated values remain small. The absence of systematicdesign and performance quantification in the currentliterature is addressed by designing an optimized PIand a H∞ controller, that significantly improve theforce regulation and its bandwidth. A major applicationof constant force clamps is in step detection of bio-molecules, where due to the presence of thermal noise,one has to extract the stepping data via postprocessing. Inthis paper, a real time step-detection scheme, currentlylacking in literature, is achieved via a mixed objectiveH2/H∞ synthesis. In the design, the H∞ norm for forceregulation and stepping estimation error is minimizedwhile keeping the H2 norm of the thermal noise on thestepping estimate is kept bounded.Keywords: Constant force clamp, isotonic clamp, ki-nesin stepping, real time step detection, mixed objectiveH2/H∞ synthesis, optical trap.

I. INTRODUCTION

Engineering concepts have an important role to playin biophysical research where there is an urgent needfor faster instruments for single-molecule studies tounderstand the mechanism of motor-protein based in-tracellular transport at native cellular condition. Motorproteins, such as kinesin and dynein, serve as carriersfor cellular cargo in intracellular transport that influenceglobal aspects of cell biology including establishmentof cell polarity, maintenance of genomic stability, reg-ulation of higher brain functions, developmental pat-terning and suppression of tumorigenesis. Malfunctionsin intracellular transport underlie a host of medical

maladies including neurodegenerative diseases such asAlzheimers [7], [12].

The challenge in gleaning information about motorprotein dynamics arises from the requirements of nmspatial resolution, second to millisecond temporal res-olution, high degree of thermal noise and the fragilityof the bio-matter (requires very small probing force).Optical tweezers, a laser based probing system, areuniquely positioned to address these challenges becauseof their capability to probe systems with fN forceresolution and detect events with nm spatial resolution[6]. Typically, the motor proteins are decorated witha polystyrene bead which is then probed by the lasertrap to interrogate various aspects of the motor proteindynamics indirectly.

In many in-vitro experimental scenarios, it is impor-tant for the biologists to keep the load force constant onthe bead despite external disturbances (such as motorstepping, unfolding of domains and thermal force). Thiswould allow them to simulate and study the in-vivocondition of constant load force on the molecule whileavoiding complications arising due to the nonlinearnature of the force-extension relationship of the bio-molecule under tension. Also, constant load eliminatesthe need for linkage corrections to infer displacementinformation of the bio-molecule attached from theposition of the bead [13].

The existing force clamps can be broadly dividedinto two categories: active clamps and passive clamps.Active clamps [13], [8] employ feedback to movethe trap location such that a preset distance betweenthe trap location and the center of the trapped bead(and hence a preset force on the bead) is maintained,whereas in passive clamps [10], [5] the trapped beadis placed in a constant force region of the trappingpotential. The main drawback of existing active clamps[13], [8] is their slow disturbance rejection bandwidthand occasional instability, which stems from neglectingthe instrument dynamics and lack of formal controllerdesign methodology. The passive clamps on the otherhand, despite their higher bandwidth, is not suitable

2013 American Control Conference (ACC)Washington, DC, USA, June 17-19, 2013

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for fast moving motor proteins due to reduced trappingforce [10] and limited extent of constant force region[5].

In this paper we circumvent the aforementionedproblems by employing modern control techniques andby precisely characterizing the instrument dynamics.A H2/H∞ mixed objective controller synthesis is alsoapplied to estimate the stepping motion in real timeas well, where H∞ synthesis alone are shown to beinsufficient.

II. MODELLING, CHARACTERIZATION AND DESIGN

OBJECTIVES

A. Physical ModelAn optically trapped bead carried by a molecular

motor in a fluid can be modeled as a spring massdamper system. Let xT , xb and xm denote the positionsof the trap center, bead centre and the motor headon the microtubule respectively and β represents thevisocus damping coefficient. The trap and the motorare assumed to act as linear Hookean springs withrespective stiffnesses kT and km, under the assumptionsthat the excursions of |xT − xb| and |xm − xb|, or themagnitude of corresponding forces fT and fm remainsmall at all times. The validity of this constraint isensured by imposing appropriate design constraints.Also, we ignore any geometric effects due to the finitesize of the bead.

The system is highly overdamped [2], [11] and hencethe equation of motion of the bead in presence ofthermal noise η is written as,

βxb = kT (xT − xb) + km(xm − xb) + η. (1)

After taking Laplace trasform and rearranging, we get,

xb(s) = G(s)(kTxT (s) + kmxm(s) + η(s)) (2)

where G(s) = 1βs+kT+km

.

B. Instrumentation DynamicsThe control hardware and actuation system dynamics

significantly affect the overall system [2], [11] andthus should be included for a good control design. Weintroduce the transfer function D(s) to represent theinstrumentation dynamics, i.e., change in trap positionxT (s) = D(s)u(s) for a trap movement commandu(s). Thus, in absence of any motor force fm andthermal noise, for a given command u, the measuredbead position in open loop is governed by the equation,

xb(s) = xb(s)+n(s) = kTGp(s)D(s)u(s)+n(s). (3)

where n is the measurement noise and Gp(s) = 1βs+kT

.

Fig. 1. (a) Magnitude response and (b) phase response ofan optically trapped bead including the instrumentation dy-namics. The red curve represents the experimentally obtainedresponses while the blue curve corresponds to those of a 2-zero, 3-pole fit of the experimental data. (c) Validation of theestimated transfer function by comparing the experimental(in red) and simulated step response (in blue).

C. System CharacterizationThe parameters of the physical system estimated by

power spectrum method [11] yielded β = 4.721×10−6

pNs/nm and kT = 0.05 pN/nm. To estimate D(s),we give a chirp input with frequency varying from10 Hz - 10000 Hz to the system. By fitting a 2-zero, 3-pole transfer function to the experimentallyobtained frequency response data, the transfer functionfrom control effort u to xb is estimated as H(s) =

kTGp(s)D(s) = 712.3(s−1.431×105)(s−5.123×104)(s+2852)(s2+5.317×104s+1.561×109) . The

transfer function is validated by matching experimen-tally obtained frquency response data and step re-sponses to those of the estimated system (see Fig. 1).From the estimated H(s), D(s) is estimated as D(s) =0.067255(s−1.431×105)(s−5.123×104)(s+1.059×104)

(s+2852)(s2+5.317×104s+1.561×109) .

D. Design ObjectivesA block diagram representing the optical trap with

the feedback system is shown in Fig. 2(a). The re-quirements of a good isotonic clamp with an additonalobjective of estimation of motor motion is summarizedbelow in terms of design objectives. For the sake ofsimplicity, we use H to denote any transfer functionH(s), and any transfer function from input a to outputb is denoted by Tab.• A small value of ‖TxmeF ‖H∞ would ensure that

the motor stepping motion does not significantlyalter the force on the system, which is the mainobjective of the isotonic clamp. Here eF denotes

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Fig. 2. (a) The block diagram showing different inputs,disturbances, weighted fictitous outputs, measured signalsand control signals. Here the main aim is to obtain anestimate xm of motor motion xm while maintaining theforce fm on the motor regulated at a constant value fr. Theinfluence of thermal noise η needs to be small on the estimatexm and the actuator effort xT should be limited to avoidlosing the bead from the trap. (b) The modern control designframework showing the plant P , controller K, exogenousinput vector w, weighted signal vector z, measured signalvector y and control signal vector u.

the force regulation error, the difference betweenthe desired force fr to be maintained on themolecule and the trap force fT . The desired band-width of force regulation is captured in ze by thecorresponding weighting function We.

• For good command tracking, we need ‖TfreF ‖H∞

to be small.• For good estimation of motor motion, ‖Txmxm

‖H∞

should be small, where xm = xm−xm denotes theerror in estimation of motor motion. zm capturesthe desired bandwidth of estimation through Wm.

• Small ‖Tηxm‖H2

ensures reduced effect of thermalnoise on the estimate of the motor motion. zηcaptures the desired range of filtering via Wη.

• To avoid losing the bead from the trap dueto sudden spikes in trap position, we require‖TxmxT

‖H∞ ≤ 1, as that would ensure that theinput disturbance is not magnified to the trapposition. Also, for the linearity assumptions tohold good, trap movement should be small. Thisconstraint takes care of the actuator saturation aswell, which has a much higher limit.

• Any controller should be robust enough to endurecertain variation of the motor spring constant km.

• The measurement noise in similar systems [2] isobserved to be substantially below the thermalnoise response and thus is not considered underthe design objectives.

It can be seen that the requirements for different ob-jectives are conflicting and thus all of them can not besatisfied simultaneously for all frequencies. In the fol-

Fig. 3. Comparison of performances from frequency re-sponses between the PI controller (shown in blue) and theH∞ controller (shown in red) for different performanceobjectives : (a) ‖TxmeF ‖ and ‖W−1e ‖ (b) ‖TfreF ‖ and‖W−1e ‖ (c) ‖TxmxT

‖ and ‖W−1T ‖. The inverse of weightingfunction is shown in gray.

lowing sections we present LMI-based control designmethodology where the resulting controller optimizescertain objective transfer functions while guaranteeingpractical bounds on values of other transfer functions(where these objectives can be characterized by H2

or H∞ norms on the transfer functions). We presentand solve a few such optimal control problems eachof which reflects a specific experimental scenario andobjective.

III. DESIGN OF A CONSTANT FORCE CLAMP

In this section, we design controllers for constantforce clamps that would have good error regulation(small ‖TxmeF ‖H∞) and good command tracking (small‖TfreF ‖H∞) in closed loop. We first design a PI con-troller and then a H∞ controller to show improvmentin performance over the former. For the H∞ design,we incorporate an additional objective of avoiding am-plification of input disturbances at the actuator output(small ‖TxmxT

‖H∞) to avoid losing the bead duringexperiment.

A. PI controller

By using loop shaping techniques, we obtain a PIcontroller as KPI(s) = kp(1 + 1

Tis), where Kp = 58.5

and Ti = 4.1667 × 10−5 s. The frequency plots andsimulation results in Fig. 3 and Fig. 4 shows theclosed loop system with PI controller to have goodcommand tracking and error regulation for a reasonablebandwidth.

B. H∞ controller

Design: To meet the specified performance objec-tives,we choose z = [ze zT ]T , w = [xm fr]

T , u = uand y = eF (see Fig. 2(a)), making our generalized

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Fig. 4. Simulation results: (a) Change in bead positionxb (shown in red) and the corresponding control effort xT(shown in blue) in response to input step disturbances xm ofamplitude 8 nm (shown in black) with the H∞ controller inpresence of input noise η with σnoise = 5 pN. (b) Regulationerror eF with the PI controller (shown in blue) and thatwith the H∞ controller (shown in red) in presence of theaforementioned input disturbance and noise for fr = 5 pN.

open loop plant from [w | u]T to [z | y]T to be

P =

WekTkmG We −WekTD(1− kTG)−WTkmG 0 WTD(1− kTG)

kTkmG 1 −kTD(1− kTG)

(4)

Our objective is to design a stabilizing controller Kthat would minimize ‖Ψ‖H∞ , where Ψ is the closedloop transfer function from w to z (see Fig. 2(b)). Weuse the performance of the PI controller as a guidelinefor choosing We. The choice of WT is dictated by thefact that we want to avoid amplification of disturbancexm at xT .

Performance: The frequency plots in Fig. 3 showimprovements in terms of command tracking and errorregulation when compared to those of the PI controller.The simulation results in Fig. 4(b) and in the followingtable also corroborate the same.

From the frequency plots (Fig. 3) and the simu-lation results (Fig. 4(b)), the H∞ controller clearlyoutperforms the PI controller for an isotonic clamp inevery aspect. Frequency plots with the PI controllershows peaks at high frequencies (see Fig. 3) which canbe explained from second waterbed theorem and thepresence of RHP zeros in the system [11], [2]. Thisemergence of peaks poses a bottleneck on the resolu-tion and bandwidth that can be achieved, and moderncontrol design is seen to push that limit much furtherwhen compared to traditional design by eliminatingthose peaks at operating frequencies. The eliminationof the huge peak in the ‖TxmxT

‖ via modern controldesign (see Fig. 3(c)) worths special mentioning, asit eliminates the possibility of getting an experimenthampered due to the loss of the bead from the trap dueto the amplification of a sudden disturbance spike atthe input.

Fig. 5. Comparison of performances from frequency re-sponses between the H2/H∞ mixed objective controller(shown in blue) and the H∞ controller (shown in red)fordifferent performance objectives : (a) ‖TxmeF ‖ and ‖W−1e ‖(b) ‖Txmxm

‖ (c) ‖TxmxT‖ and ‖W−1T ‖. The inverse of

weighting function is shown in gray.

Another major advantage obtained via H∞ synthesisis that of robustness. For the PI controller, the systemperformance degrades as the value of the uncertainparameter km is increased from its nominal value of0.3 pN/nm, making it unstable for km = 0.375. TheH∞ controller, however, shows little degradation inperformance till km is increased to a value of 0.5pN/nm, much above of what is to be expected in anisotonic clamp operation.

IV. ESTIMATION OF MOLECULAR MOTOR MOTION

As already mentioned, an important application ofisotonic clamp is in the detection and estimation ofmolecular motor motion. The bead motion is assumedto follow the motor motion at steady state if constantforce is maintained. As an alternative to the existingscheme of xm detection, we design our controller todirectly give an estimate of xm, which would enableus to detect steps in real time. We first design aH∞ controller and point out its defficiencies for thepurpose. We then circumvent them by designing amixed objective H2/H∞ controller.

A. H∞ controller

The H∞ controller is designed to have good forceregulation (small ‖TxmeF ‖H∞), good estimation of mo-tor motion (small ‖Txmxm

‖H∞) and avoiding amplifica-tion of input disturbances at the actuator output (small‖TxmxT

‖H∞). Although the frequency plots show gooderror regulation and good bandwidth for xm estimation(Fig. 5(b) shows the frequency plot for estimationerror), the results(Fig. 6(c) and Fig. 7(c)) reveal that xmis highly corrupted with noise (specially for high noiselevel with σnoise = 10 pN), rendering the estimateineffective for real time step detection.

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Fig. 6. Comparison of performances from simulation resultsbetween the H2/H∞ mixed objective controller (shown inblue) and the H∞ controller (shown in red) for low thermalnoise (σnoise = 5 pN) and fr = 5 pN: a) Error in forceregulation. (b) Error in step estimation. (c) Estimate of steps.B. Mixed objective H2/H∞ controller

Design: We argue that to reduce the effect of inputnoise on xm, ‖Tηxm

‖H2should be small. As η and

xm comes at the same input (see Fig. 2(a)), making‖Txmxm

‖H2small serves the same purpose while mak-

ing the design process much simpler. To incorporatethis additional objective in the mixed objective designframework, we define z∞ = [ze zm zT ]T , z2 = zη,z = [z∞ z2]

T , w = xm, u = [u xm]T and y = eF (seeFig. 2(a)), making our generalized open loop plant from[w | u]T to [z | y]T as

P =

WekTkmG −WekTD(1− kTG) 0−Wm 0 Wm

−WTkmG WTD(1− kTG) 00 0 Wη

kTkmG −kTD(1− kTG) 0

(5)

Our objective is to design a stabilizing controller Kthat would minimize ‖Ψ∞‖H∞ , subject to ‖Ψ2‖H2

< ν.Here Ψ∞ and Ψ2 are the transfer functions of the closedloop channels from w to z∞ and z2 respectively. Ψ∞and Ψ2 are in turn obtained from Ψ, the closed looptransfer function from w to z. Choice of Wm is dictatedby the fact that as xm is needed at steady state, we donot need high bandwidth of estimation, as long as theerror regulation bandwidth is high enough to validatelinearity assumptions, which in turn dictates the choiceof We.

Performance: Frequency plots for the mixed ob-jective controller show good enough error regulation

Fig. 7. Comparison of performances from simulation resultsbetween the H2/H∞ mixed objective controller (shown inblue) and the H∞ controller (shown in red) for high thermalnoise (σnoise = 10 pN) and fr = 5 pN: (a) Error in forceregulation. (b) Error in step estimation. (c) Estimate of steps.Here a tigher bound on ‖Ψ2‖H2

is imposed by sacrificing‖Ψ∞‖H∞ to some extent to counter the increased noise level.

bandwidth for the linearity assumptions to hold good(Fig. 5(a)). The xm estimation bandwidth however de-creases considerably than the previous case (Fig. 5(b)),which is to be expected as we are decreasing the H2

norm over the same channel. A trade-off is achievedthrough the choice of different weighting functions.Simulation results show (Fig. 6(c) and Fig. 7(c))thatthis bandwidth is good enough to get a clean estimateof xm even for high noise levels. Although the forceregulation deteriorate, it is still better (< 2%) thanthat reported in [14] (10%) for similar noise level.Even with higher noise level, we achieve better forceregulation(< 5%), and thus the linearity assumptionscan be assumed to hold good for all practical purposeswith the mixed objective controller.

V. CONCLUSION

The paper presents real life applications of H∞and mixed objective H2/H∞ controller synthesis anddiscusses various design trade-offs among them. Thecurrently lacking systematic design for constant forceclamp design [14], [9] is addressed via a modern con-trol design methodology. Significant improvement inforce regulation and operational bandwidth is achievedby the understanding of fundamental limitations onachievable performance due to presence of RHP zeros.Modern control design framework is successfully em-ployed to overcome those limitations within operatingrange of frequencies. This improvement in bandwidth

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and resolution1 may have a greater impact on biomolec-ular studies, as this will enable the scientists to discoverevents not previously possible, thereby refining theexisting models. Better force regulation over a higher abandwidth also validates the linearity assumptions withhigher confidence. Modern control design makes thesystem robust against possible variations of uncertainexperimental parameters as well.

The paper also develops a real time step detectionscheme for molecular motors, which to the best ofthe author’s knowledge has not been addressed sofar in the current literature. This can be of immenseimportance to experimental biologists, as they will beable to detect motility in the sample at a very earlystage of the experiment when compared to the currentlyexisting postprocessing schemes of step detection [3],[1]. The real time step detection scheme is attained bymixed objective H2/H∞ synthesis, where the effect ofbrownian noise on the estimate is decreased by keepingthe H2 norm of the channel bounded, while we mini-mize the H∞ norm over the estimation error and errorregulation channels. This scheme avoids influencing thesystem being probed and thus preserving normal systemdynamics, unlike certain other schemes of reduction ofinfluence of thermal noise by stiffeing the system itself[4].

Note that the bandwidth of estimation for the realtime scheme is limited to a few Hz (see Fig. 3(b)),which is a result of the H2/H∞ trade-off over thesame channel. From simulations, this bandwidth is seento be good enough for low motor velocities (typicallyaround 30 − 40 nm/s for kinesin) for typical noiselevels (σnoise = 5pN), which corresponds to lowATP concentration and high load force scenarios[15].This can possibly be improved by exploiting the noiseproperties to select the weighting function Wη moreeffectively and minimizing only ‖Txmxm

‖H∞ subject toappropriate bounds on other H2/H∞ norms of otherchannels, which would allow for better allocation ofcontroller efforts.

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