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Simulating an Actomyosin in Vitro Motility Assay: Toward the Rational Design of Actomyosin-Based Microtransporters Yuki Ishigure and Takahiro Nitta , Member, IEEE

Abstract—We present a simulation study of an actomyosin in vitro motility assay. In vitro motility assays have served as an essential element facilitating the application of actomyosin in nanotechnology; such applications include biosensors and biocomputation. Although actomyosin in vitro motility assays have been extensively investigated, some ambiguities remain, as a result of the limited spatio-temporal resolution and unavoidable uncertainties associated with the experimental process. These ambiguities hamper the rational design of nanodevices for practical applications. Here, with the aim of moving toward a rational design process, we developed a 3D computer simulation method of an actomyosin in vitro motility assay, based on a Brownian dynamics simulation. The simulation explicitly included the ATP hydrolysis cycle of myosin. The simulation was validated by the reproduction of previous experimental results. More importantly, the simulation provided new insights that are difficult to obtain experimentally, including data on the number of myosin motors actually binding to actin filaments, the mechanism responsible for the guiding of actin filaments by chemical edges, and the effect of the processivity of motor proteins on the guiding probabilities. The simulations presented here will be useful in interpreting experimental results, and also in designing future nanodevices integrated with myosin motors. Index Terms—Actin filament, biomedical engineering, bionanotechnology, biophysics, biosensors, microelectromechanical systems, myosin, nanobioscience, nanobiotechnology.

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I. INTRODUCTION

ATURE HAS developed various biomolecular machines to maintain cellular functions. These molecular machines, especially motor proteins, are appealing for nanotechnology, and have inspired nanotechnological developments. One strategy that is applied in nanotechnology is to determine the design principles of natural molecular machines and develop artificial counterparts. This strategy is exemplified by the synthesis of DNA nanomachines [1], and the development of catalytic nanomotors [2]–[5]. Another strategy is to use the motor proteins as a component in synthetic environments, such as microelectromechanical systems (MEMSs) and lab-on-a-chip [6]–[17]. Motor proteins have been used in biosensors [18]–[21], surface imaging [22], [23], molecular reactors [24], computation [25], and molecular communications [26], [27]. In vitro motility Manuscript received February 10, 2015; revised May 16, 2015, June 05, 2015; accepted June 05, 2015. Date of publication June 17, 2015; date of current version August 28, 2015. This work was supported by JSPS Grant-in-Aid for Young Scientists (B) under Grant 24760203. Asterisk indicates corresponding author. Y. Ishigure is with the Department of Mathematical and Design Engineering, Gifu University, Gifu, 501-1193 Japan (e-mail: [email protected]). *T. Nitta is with the Department of Mathematical and Design Engineering, and Applied Physics Course, Gifu University, Gifu, 501-1193 Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TNB.2015.2443373

assays have served as an essential element in facilitating the implementation of motor proteins in synthetic environments. The in vitro gliding assay is a standard technique for study of motor proteins, in which cytoskeletal filaments are translated by their associated motor proteins, which are adhered on substrates. This configuration is preferred in applications, because the cytoskeletal filaments glide over the substrate almost without dissociation, and the gliding movements can be guided using microfabricated substrates [28]–[30]. By designing the microfabricated guiding tracks, various functional modules can be achieved [31]. Although motility assays have been extensively investigated and fairly well characterized, some ambiguities remain, and these ambiguities hamper the rational design of devices for practical applications. For example, the following questions should be answered prior to the fabrication of such devices: How many motors are needed to support the continuous motility of filaments? How many motors actually bind to the filaments, which fluctuate in their heights from substrates, as well as in their directions along the substrates? How does the guiding occur, and how does it fail on guiding tracks? Although experiments have provided insights into these issues [32]–[36], some ambiguities remain, because of the limited spatio-temporal resolution and unavoidable uncertainties associated with experiments, such as uncertainty on the number of motor proteins actually binding to the associated cytoskeletal filaments. Computer simulations are useful for the investigation of the above mentioned issues [37], [38]. Simulations of microtubules (MTs) driven by kinesin or dynein motors were performed to elucidate the guiding of microtubules by microfabricated tracks [39], [40], and to design patterns of kinesin and hypothetical minus-end-directed motors [41]. Although the motility of actin filaments (AFs) has been investigated using computer simulations in the context of biophysics, and valuable insights have been achieved [42]–[45], 3D simulations have not yet been performed, and the mechanism responsible for the guiding of AFs by microfabricated tracks has not been investigated using computer simulations. In this study, we focused on the gliding of actin filaments over myosin. The gliding of AFs over myosin is markedly different from the gliding of MTs gliding over kinesin, in terms of the translation speed and persistence length. AFs glide on myosin approximately 10 times faster than MTs glide on kinesin. This fast translation is beneficial, because it allows the rapid operation of devices using motor proteins. AFs gliding over myosin [46] have a shorter path persistence length, compared with MTs gliding over kinesin [47]. This short path persistence length is

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TABLE I PARAMETER VALUES USED IN THE SIMULATION

Fig. 1. (a) Schematic illustration of the simulation method. The thick black arrow indicates the direction of the AF movement. (b) Schematic diagram of the state conversion in the ATP hydrolysis of a myosin motor. The thick black arrows indicate the directions of the AF movement. The thin black arrow indicates myosin displacement representing its power-stroke.

beneficial for efficiently sweeping an area; it does, however, require the use of more sophisticated microfabrication techniques (such as e-beam lithography) to make guiding tracks to confine the movements of the AFs. With the aim of facilitating the rational design of nanodevices powered by myosin motors, we developed a 3D computer simulation that reproduced the motility of actin filaments gliding over myosin-coated tracks, and investigated the mechanism of guiding. To this end, we extended our previous simulations of microtubules gliding over kinesin [40] by including the ATP hydrolysis cycle of myosins. The simulation method was validated by comparing the results with results from previous experiments. Using the simulations, we achieved new insights into the motility of actin filaments gliding over a myosin-coated surface, and the mechanism responsible for the guiding. The remainder of this paper is organized as follows: In Section II, we describe the simulation method. In Section III, we describe the results of simulations of the motility of actin filaments gliding over myosin-coated planar surfaces. As described in this section, the simulation reproduced results from previous experiments, and provided additional insights into the movement. Then, because guiding is a prerequisite for high-performance biosensors integrated with motor proteins [48], in Section IV, we reproduced the guiding of AFs by chemical edges, and investigated the mechanism. Finally, a short summary of this paper and its possible impact is given in Section V. II. SIMULATION METHOD Fig. 1 shows schematics illustrating our simulation method. The simulation reproduced the 3D movements of AFs propelled by myosin motors adhered on substrates. Table I shows the parameter values that were used. The following sections give the details of the simulation method. A. Myosin Motor The myosin motor was modeled as a linear spring with a spring constant of 300 pN/ m [45].

To simulate the transformations between the various nucleotide binding states of myosin motors, we adopted the four-state model of myosin motors developed by Walcott et al. [45] (Fig. 1(b)), and used their values for the kinetic parameters. Briefly, in the absence of AFs, a myosin motor may be in an ATP-bound state or an ADP P-bound state. The myosin can be reversibly transformed between the two states, with a rate constant of or . When a myosin motor is binding to ATP, it cannot bind to an AF. Once the ATP is hydrolyzed, the myosin can then bind to an AF. After the hydrolysis of ATP, if an AF approaches within 20 nm of the myosin [34], the myosin binds to the AF at a rate of (40 s ). The point on the AF where the myosin head binds is the nearest point on the AF from the tail of the myosin. After the binding, the myosin is assumed to make a power-stroke of 10 nm. The myosin releases ADP with a rate of , where is force-dependent: (1) where is the rate constant in the absence of an applied force F, is the distance between the binding state and the peak of the energy barrier in reaction coordinates, is the Boltzmann constant, and T is temperature. F is the force applied to the myosin. The applied force originates from both the power stroke of the myosin and displacement of the actin filament by other myosins. Following the release of ADP, upon binding of ATP, the myosin head was assumed to immediately dissociate from the AF. We determined which biochemical transitions to occur in the next step according to the probabilities of the transitions to occur in a given time step . The probabilities were calculated from the reaction rates, such as . Regardless of the myosin state, when a force exceeding 9.2 pN is applied to the bound myosin, the myosin will be detached from the AF [49]. Myosin motors were randomly distributed on track regions with densities described below. The distributed myosin motors were assigned to be in either an M T or an M D P state. Until binding to an AF occurred, the myosin motors transformed between the M T and M D P states with the assigned rate constants. We neglected the ATP hydrolysis of myosin motors without AFs. A justification of this neglect is given in Supporting Information.

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B. Actin Filaments Actin filaments were modeled as infinitely thin and inextensible semiflexible bead-rod polymers with a flexural rigidity of pN m [51], corresponding to the filament persistence length of 17.8 m, calculated from . No breakage of AFs was assumed. The simulated AFs consisted of segments, each with a length of 0.25 m (Supporting Information). The movements of the AFs were simulated using a Brownian dynamic simulation, as described elsewhere [40]. The time step was set to be s (Supporting Information). C. Visualization The simulation results were visualized using ParaView. D. Probability of Guiding We considered an AF arriving at a track boundary to be guided if it maintained contact with at least one myosin motor on the bottom track and continued to travel on the bottom surface and departed from the boundary by more than 0.5 m. If the AF lost contact with the bottom track surface, the AF was not considered to be guided. The probability of guiding was defined as the ratio of the number of guiding events to the total number of collision events, as a function of the approach angle. Following the method of Clemmens et al. [35], standard errors were determined for the probabilities using the formula: (2) where p is the probability of guiding, and N is the total number of AFs arriving at the boundary with an approach angle in that bin. III. MOVEMENT OF ACTIN FILAMENTS BY MYOSIN MOTORS ON PLANAR SURFACES We first investigated the movement of AFs on planar surfaces that were uniformly covered with myosin motors, and compared them with observations that were reported previously. Fig. 2 shows a representative trajectory of AFs gliding over myosins on a planar surface. Compared with microtubules gliding over kinesin (Supplementary Fig. 6), the actin filaments showed substantial lateral fluctuations during their translations. The width of the lateral fluctuation was 200 nm, which is comparable to a pixel size of microscope images. Hence, the lateral fluctuation would be difficult to observe with conventional microscopes. This lateral fluctuation was induced by the rapid binding and unbinding of myosin motors within a region where myosin motors can reach the actin filament. The region extended along the actin filament on both sides of the filament with the width of , where z is the height of the segment of the actin filament [34]. During the translation, the average tip length of actin filaments (defined as the length between the leading end and the foremost myosin) was slightly longer than the average of separations between myosin heads along the filaments (Supporting Information). Such information would be complementary with that obtained from an experiment with an in vitro motility assay modified to restrict the number of interacting myosins [52]. Since the path persistence length is an important factor determining performance of microdevices based on the in vitro motility assay [48], we investigated the path persistence length

Fig. 2. Movement of an actin filament gliding over a myosin-coated planar surface. The conformations of the actin filament (red curves) were determined every 0.1 s and overlaid. The white arrow indicates the direction of the gliding movement. The length of the actin filament was 5 m. The density of myosin motors was 5000 m . Scale bar: 2 m. Inset, close-up view of the superimposed image of the actin configurations. Scale bar: 500 nm.

of actin filaments continuously gliding over myosin-coated substrates. We found that the path persistence length was considerably shorter than the filament persistence length (Fig. 3), while a theory by Duke et al. predicts that the path and filament persistence lengths should be the same [50]. On the other hand, experiments by Vikhorev et al. showed that for phalloidin-stabilized AFs, their path persistence length was considerably shorter than their filament persistence length in solution; while for phalloidin-free AFs, their path and filament persistence length were the same [53]. Our simulation reproduced their experimental observation on phalloidin-stabilized AFs. The path persistence length calculated from the simulation was comparable with those previously reported [46], [53], and was not significantly dependent on myosin density (Fig. 3). The low value for the path persistence length, hence larger directional fluctuations, arose from sideways bindings and subsequent pulling of myosin motors within the region where myosin motors can reach the actin filament. Although interaction with myosin activates fluctuations of actin filaments in solution [54], this simulation result showed that only the sideways bindings of myosin can account for the lower value for the path persistence length of actin filaments propelled by myosin motors, than that for the filament persistence length in solution. The average elevation of the gliding actin filaments from the substrate was 1.4 nm, in the case of the myosin density of 6000 m . The distribution of the elevation was shown in Supporting Information. This average elevation was consistent under our assumption that myosin motors can be modelled as linear springs with zero equilibrium length, which gives the amplitude of thermal fluctuations of 3.7 nm that was calculated using . However, this value was considerably smaller than that reported by Persson et al. [55]. This discrepancy may arise for two reasons. Firstly, in the simulation, myosin was modeled as a linear spring with zero equilibrium length, neglecting detailed structures of myosin, such as globular catalytic domains and coiled-coils in S1 and S2. Although this assumption is widely used in previous simulations, the modelling

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Fig. 3. Path persistence length of actin filaments gliding over myosin coated . The red broken line surface as a function of the motor density shows the preset filament persistence length in solution.

Fig. 4. Translation speed as a function of ATP concentration . The solid squares are our simulation data; the open squares are data taken from Debold et al., [57]. The gray broken curve represents the 1D simulation performed by Walcott et al., [45]. The red curve is a least-squares fit of our simulation data to the Michaelis-Menten kinetics (3). The density of myosin motors was 5000 m . The length of the actin filaments was 5 m.

should be replaced with more realistic ones in future studies. Secondly, in the experiments by Persson et al., conducted with high HMM density of 6800 m [55] (similar to our simulation with 6000 m ), neighboring myosins may interact with each other and each myosin may take stretched conformations perpendicular to the substrate to reduce excluded volume interactions. Recently kinesin motors at high surface densities were found to take such stretched conformations [56]. Taking into account the estimate on a nearest neighbor distance of nm between myosin motors by Persson et al., excluded volume interactions between neighboring myosins are likely to work and myosins may take the stretched conformation. On the other hand, in the present simulation, such excluded volume interactions between neighboring myosins were not taken into account. The speed of the translation of the simulated AFs depended on the concentration of ATP (Fig. 4). In accordance with the simulations performed by Walcott et al., in which 50 active myosin heads interacted with an actin filament, the surface density of myosin was set to be 5000 m . The dependence of the speed on the ATP concentration was in agreement with the Michaelis-Menten kinetics: (3) where v is the speed of translation, is the maximum transis an apparent Michaelis constant, and [T] is lation speed, the concentration of ATP. By fitting our simulation data using (3), and were calculated to be m/s and M , respectively. These values were in agreement with results from the experiments conducted by Debold [57], and a 1D simulation performed by Walcott et al. [45], from which we adopted the kinetic parameters, as well as qualitative agreement with data from other group [58]; these agreements validated our simulation. Next, we investigated the motility of AFs on substrates with various surface densities of myosin. Above a surface density of 2000 m , the AFs glided continuously over the myosincoated surfaces. That is, all of the AFs glided over the surface for distances longer than their own length without dissociation (Fig. 5(a)). The speed of the continuous translation was approximately 6 m/s, and was almost independent of the surface density of myosins (Fig. 5(b)). In contrast, below a surface density of 500 m , no AFs glided over the myosin-coated surface for distances longer than their own length (Fig. 5(a)), and

the AFs dissociated from the surface. These simulation results were consistent with results from a previous experiment conducted by Toyoshima et al. [32], who found that in the absence of any methylcellulose in the assay solutions there was a critical surface density required to support the continuous gliding movement of the AFs. This critical density was estimated to be approximately 500 m , which was consistent with our simulation results. It should be noticed our simulation condition corresponds to the experiment by Toyoshima et al. [32], rather than that by Uyeda et al. [34], where with methylcellulose the gradual change of gliding speed against the motor density was observed, since we did not suppress lateral movements of AFs. This consistency further validated our simulation method. The simulation not only reproduced results from previous experiments, but also provided insights that are difficult to obtain using experiments. Fig. 5(c) shows the number of myosins binding to actin filaments gliding continuously over the myosins. The three plots of different length mapped onto a single curve when the number density was plotted as a function of the motor density (Fig. 5(c), inset), and below a number density of 5 m , the AFs could not move continuously. This indicated that the dissociation of the AFs from the substrate did not depend on the total number of myosin motors bound to the AFs, but on the density of binding motors. This was plausible, because upon dissociation, the unbinding of myosins from an AF occurred from either the leading or trailing edge, rather than simultaneously along the AF (Supplementary movie 2). IV. GUIDING OF ACTIN FILAMENTS DRIVEN BY MYOSIN MOTORS ON CHEMICAL TRACKS Since guiding is a prerequisite for efficient transport [48], the elucidation of the guiding mechanism is important for the design of nanodevices integrated with myosin motors. To achieve such elucidation, we simulated the guiding of AFs by chemical edges [59], [36]. The length of each filament was 2 m, in accordance with a previous experiment conducted by Sundberg et al. [36]. Fig. 6(a) shows a typical movement of AFs guided by a chemical edge. Upon crossing the boundary, the protruding part of the AF showed substantial fluctuation. If the protruding part became bound to the myosin motors on the track, the AF continued to move on the track, and was hence guided (Supplementary movie 3). Otherwise, the AF lost contact with the myosin motors and

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Fig. 6. The movement of actin filaments driven by myosins on chemical tracks. The configurations of the actin filaments during their gliding movements were superimposed. The red lines show the actin filaments. The white dots show the myosin motors. The motor density was 4000 m . The blue region indicates the track surface. (a) A 2D view of an actin filament guided by a chemical edge. The conformations were captured every 0.4 s. Scale bar: 1 m. (b) A 3D view of an actin filament that dissociated upon encountering a chemical edge. The conformations were captured every 0.5 s.

Fig. 5. Effect of changes in the myosin motor density on the motility of actin filaments. The solid squares, red open squares, and green circles are data for actin filaments with a length of 2, 4 and 6 m, respectively. The ATP concentration was 1000 M. (a) The fraction of actin filaments that traveled distances longer than their own filament length, as a function of the surface density of myosin. Each fraction was computed from 10 actin filaments. (b) Gliding speed of actin . filaments as a function of the surface density of myosin motors (c) The number of myosins interacting with gliding actin filaments, as a function . Inset: the number of myosins of the surface density of myosin interacting with gliding actin filaments per m.

dissociated into the solution (Fig. 6(b), Supplementary movie 4). Fig. 7 shows a plot of the probability of guiding as a function of the approach angle, as reproduced by the simulation. The probability of guiding obtained using our simulation showed reasonable agreement with that obtained by Sundberg et al. [36] (Supporting Information). The simulation reproduced the probability of guiding better than the analytical model [36]. The slightly higher probability could have been due to the presence of inactive (or dead) motors in the experimental system, where the dead motors were able to bind the filaments, but were not able to translate them. As noted by Sundberg et al. [36], upon binding with inactive motors, actin filaments are deflected from

Fig. 7. Probability of guiding of actin filaments by chemical edges. The solid squares show the simulated probability of guiding by a chemical edge with a myosin density of 4000 m . The probability of guiding was determined from 526 encounters of simulated actin filaments with the chemical edge. For comparison, the corresponding experimental and theoretical results obtained by Sundberg et al., [36] are shown as open squares and open triangles, respectively.

their gliding direction. Such deflections likely reduce the probability of guiding. The probability of guiding was not significantly different for myosin densities of 4000 and 6000 m (Fig. 8). This was also consistent with the observations of Sundberg et al. [36]. Having confirmed that our simulations successfully reproduced the guiding behavior, we investigated the mechanism of guiding using simulations. As in the case of MTs propelled by kinesin, when guiding occurred, a series of binding events occurred, starting from the pivotal point toward the leading end (Fig. 9).

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Fig. 8. Plots illustrating the effect of changes in the myosin density on the probability of guiding of actin filaments by chemical edges. The red triangles represent the simulated probability of guiding by a chemical edge with a myosin density of 6000 m . The probability of guiding was determined from 388 encounters of simulated actin filaments with the chemical edge. The simulated probability of guiding by the chemical edge with a myosin density of 4000 m (shown in Fig. 7) is replotted here with black squares.

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Fig. 10. Plots illustrating the effect of changes in the filament stiffness and motor processivity on the probability of guiding by chemical edges. The red triangles represent the simulated probability of guiding of actin filaments by a chemical edge with hypothetical processive motors (the surface density was 200 m ). The probability of guiding was determined from 410 encounters of simulated actin filaments with the chemical edge. The green circles represent the simulated probability of guiding of a hypothetical cytoskeletal filament with a flexural rigidity of 22.0 pN m , driven by myosins by a chemical edge (the myosin density was 4000 m ). The probability of guiding was determined from 236 encounters of the simulated hypothetical cytoskeleton with the chemical edge. For comparison, the simulated probability of guiding of actin filaments by a chemical edge with a myosin density of 4000 m (shown in Fig. 7) is replotted here as solid squares.

of guiding for the AFs driven by the hypothetical processive motors was lower than for the AFs driven by myosin motors. This indicated that the probability of guiding was not solely determined by the flexural rigidity of the cytoskeletal filaments, but was also influenced by the processivity of the motors. The visualization revealed that the pivotal points of the AFs on the myosin motors migrated more pronouncedly than those on the hypothetical processive motors (Fig. 11(b)). The migration of the pivotal points led to a higher probability for the AFs to find motors, resulting in the higher probability of guiding. Fig. 9. Mechanism of guiding of an actin filament at a chemical edge. The red line represents the actin filament. The white dots represent myosin motors. The motor density was 4000 m . The yellow dots represent myosins bound to the actin filament. The blue region indicates the track surface. The length of the actin filament is 2 m. Scale bar: 1 m.

Motivated by this insight, we investigated why AFs gliding over myosins were guided by a chemical edge over a wider range of approach angles, compared with MTs gliding over kinesin. This higher probability of guiding is typically attributed to the lower flexural rigidity of AFs [36]. Although increasing the stiffness of the filaments to 22.0 pN m (which corresponded to the stiffness of microtubules [51]) indeed lowered the probability of guiding (Fig. 10), we tested this generally accepted idea. To achieve this, we ran the simulation using a hypothetical processive motor protein having a maximum translation speed of 6 m/s, a stall force of 5 pN, and a detachment force of 9.2 pN. The value of the maximum translation speed was chosen to ensure that the gliding speed of the AFs was identical to that on myosin motors. The surface density was set to 200 m for the hypothetical motor, to ensure that the number of hypothetical motor proteins bound to an AF was similar to the number for the myosins. The probability

V. SUMMARY We extended our previous work on the simulation of microtubules gliding over kinesin motors on microfabricated tracks by performing simulations of the 3D movements of actin filaments gliding over myosin-coated surfaces. This simulation reproduced results from previous experiments investigating the motility of actin filaments gliding over myosin-coated planar surfaces, as well as results for guiding by chemical edges. More notably, the simulation provided additional insights, including information on the number of myosin motors actually binding to actin filaments, the zipping of myosin motors upon guiding, and the effect of changes in the processivity of motor proteins on the probability of guiding. The simulation also solved the contradiction between the theoretical prediction and the experimental observation on the path and filament persistence lengths. These insights are difficult to obtain using experiments; the use of the simulations here was clearly beneficial. These simulations could provide insights into AF movements on/in hollow nanowires [60], [61], and on myosin patterns with submicrometer feature sizes [62]. In these experiments, because of the small size of the hollow nanowires and the patterns, the

ISHIGURE AND NITTA: SIMULATING AN ACTOMYOSIN IN VITRO MOTILITY ASSAY

Fig. 11. Fluctuations of the protruding parts of actin filaments on (a) myosin motors (the motor density was 4000 m ), and (b) hypothetical processive motors (the motor density was 200 m ). A time series of the conformations of the actin filaments (red lines) was obtained by taking data every 0.01 s and superimposed. The white dots represent the motor proteins. Scale bar: 1 m.

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Yuki Ishigure received his B.E. and M.E. in mathematical and design engineering from Gifu University, Japan, in 2013 and 2015, respectively. He is currently working for JTEKT Corporation, Kariya, Japan.

Takahiro Nitta received his B.S. and Ph.D. degrees in the field of physics from Hokkaido University, Japan, in 1998 and 2003, respectively. He is currently an Assistant Professor of Applied Physics Course, Faculty of Engineering, Gifu University, Japan. Prof. Nitta is a member of IEEE, the Biophysical Society, the Biophysical Society of Japan, the Surface Science Society of Japan, and the Physical Society of Japan. He is serving as an Associate Editor of the IEEE TRANSACTIONS ON NANOBIOSCIENCE.