SMASIS A STOCHASTIC MECHANO-CHEMICAL MODEL FOR COOPERATIVE MOTOR PROTEIN DYNAMICS

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1 Proceedings of SMASIS 8 8 ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems October 8-3, 8, Ellicott City, Maryland, USA SMASIS8-585 A STOCHASTIC MECHANO-CHEMICAL MODEL FOR COOPERATIVE MOTOR PROTEIN DYNAMICS Arjun Krishnan Department of Mechanical Engineering University of Michigan Ann Arbor, Michigan arjunkc@umich.edu Bogdan I. Epureanu Department of Mechanical Engineering University of Michigan Ann Arbor, Michigan epureanu@umich.edu ABSTRACT The Peskin-Oster model for single molecule experiments on kinesin is extended to analyze the motion of two motors cooperatively pulling a bead in a kinesin motility assay. The various assumptions inherent in the model are tested for the case of two motors, and justified rigorously. We formulate a Markov process similar to the single motor case, and simulate the master equations on a restricted set { N,,N} for each motor. The model is simulated for different initial configurations, and motors with identical parameters are found to synchronize in expectation, both spatially and in phase. It is found that synchronization is faster at higher loads and for higher adenosine triphosphate (ATP) concentrations. From a physical perspective, it is shown that the motors move to a configuration where the load is shared equally between them. INTRODUCTION The cell contains many mechanisms to transport nutrients, proteins and organelles, and to perform various organizational activities during essential cellular processes like mitosis. Smaller molecules like glucose can diffuse to their target locations through the cytosol, but larger organelles like mitochondria need to be carried from one location to other. Some cells carry appendages such as flagella and cilia, which are moved like flexible oars for propulsion. These mechanical and transport activities are performed mainly through the use of molecular motors (motor proteins) like dynein, kinesin and myosin, which move Address all correspondence to this author. along track-like structures called microtubules (MTs). The MTs are hollow cylindrical structures formed by bundles of protofilaments aligned in parallel. The proto-filaments themselves are long single-strand polymers of tubulin dimers. MTs have a specific polarity, and a set of equally spaced chemical binding sites where motor protein attach. A kinesin motor protein consists of two globular domains called heads. Each of the heads is a fully functional enzyme capable of hydrolyzing adenosine triphosphate (ATP). The heads are both attached to the end of a long coiled-coil tether. The other end of the tether is attached to the cargo. The heads have radii of about 1 nm, and the tether is about 6 nm long when unstretched. Kinesin consumes ATP and uses the resulting chemical energy to walk along the microtubule by moving its heads alternately from one binding site to another. The resulting steps are 8 nm long. Kinesins have been studied since the 9s, and experimental techniques have now become sufficiently precise to obtain useful data about their behavior. Their dynamics are rich and complex. First, it is surprising that a device so small can produce directed motion in the thermal environment of the cell. Second, the details of its chemical and mechanical cycle are sparse, and many mechanisms governing their function are unknown. For example, it is not known precisely how the two ATPases synchronize their enzymatic cycles to produce directed mechanical motion [1]. Third, molecular motors are seen to organize MTs into functional remarkably regular structures during mitosis []. It is also hypothesized that kinesins work cooperatively in organized teams to move loads inside the cell. Understanding the physics 1 Copyright c 8 by ASME

2 of such synchronized behavior in a fundamentally stochastic system is a challenging problem. Modeling kinesin is complicated by its discrete and continuous (hybrid) dynamics and the inherent randomness due to thermal noise [3]. There are many models that treat the stepping and kinetics as a Markov chain [] or as a Brownian particle diffusing over energy wells [5], [6]. The first physically consistent model that combined mechanical, kinetic and stochastic aspects of kinesin s cycle was the Peskin-Oster model [7]. Other notable models that try to use detailed mechanics are Aztberger and Peskin [8], Hendricks, Epureanu and Meyhofer [9], and Derenyi and Vicsek [1]. Current models for cooperative multiple motor behavior are abstract, and formulated in terms of thermal ratchet models [11]. Our analysis is based on Peskin-Oster model, which is more suited to our goal of obtaining greater physical insight into the dynamics. The Peskin-Oster Model The Peskin-Oster (PO) model and other variants based on it [, 7, 8, 1] have been fairly successful in predicting various aspects of kinesin s mechanism. The PO model is one dimensional, tracking only coordinates along the MT. The chemistry is assumed to have only two steps, ATP binding and subsequent hydrolysis. The bead is attached to the elastic tether, which is assumed to obey Hooke s law. This is an assumption that is well-justified by the analysis of Atzberger and Peskin [8], who obtained tether energies from experimental data. The other end of the tether is attached to a hinge-point to which both motors are attached. Crystallographic studies show that the heads are attached to a common point by their neck-linkers [13]. The neck-linkers may be modelled as a elastic or rigid elements. Let us define two binding states for each head as S (strongly bound to the MT) and W (weakly bound to the MT). The bead, hinge, bound head and free head locations are denoted by x b, x h, x bnd and x. The hinge point is defined to be between the two heads at all times. The various steps in the mechanochemical cycle of this model can be summarized as follows: 1. The motor starts in the state S S. In this state, the motor is rigid and the bead undergoes a Brownian motion in some potential well established by various elastic components of the system. Then, x bnd1 =, x bnd = 8, x h =.. Next, ATP binds to one of the two heads, and the system undergoes a transition S S S W, i.e., one of the heads is now weakly bound and is free to diffuse in a potential biased towards the next forward binding site. This biasing potential is another modeling input. One simple choice is to assume a quadratic potential. Suppose ATP binds to the forward head. Then, x bnd = 8 nm, x h = 8+x, x = 8+x. 3. While the ATP hydrolysis is taking place, the free head and the bead diffuse together, governed by an equation of the form: p p D b x + D h b x + D h h k B T ( V ) p + x h x h D b k B T ( V p x b x b ) p t =, (1) where p is the joint density of the bead and diffusing head locations x b and x h, D b and D h are the diffusion coefficients, k B T is thermal energy, and V is the potential energy of the system. The subscripts b and h stand for bead and head respectively.. As soon as ATP hydrolysis completes, the free head regains its affinity for the MT and quickly finds a free binding site to attach to. The potential in which the head diffuses is biased in the forward direction thus creating a power stroke, i.e., the minimum of the biasing potential is located close to the forward bind site. 5. Once this binding takes place, ATP hydrolysis is complete and the by-products of the hydrolysis are ejected from the nucleotide binding pocket. This brings the system back to its original state S S with both heads free of nucleotide. One of the heads has travelled a distance of 16 nm and the bead has moved 8 nm, the spacing on the MT. There are two distinct states in which we must define the system potential energy: (1) when both heads are bound to the MT, and () when one head is free and diffuses along with the bead. The coupled diffusion of the bead and the head is difficult to solve analytically. A key simplification made here is to separate time-scales by assuming that the head s diffusion is much faster than the bead s. This is because the approximate radius of the head is much smaller than that of the bead - a ratio of 1/1 - implying that the diffusion coefficient is much larger. That is, at any instant, the head is at relative equilibrium with the bead. Then, we can write an effective potential for the bead, and solve for its probability density. Furthermore, if the tether stiffness is large, we can consider the time-scale for the bead to reach its equilibrium density to be smaller than the chemical time scale (see Section A.1). This means that relative to the chemical processes, the bead can also be considered to be at its equilibrium position at every instant. Then defining a single state variable k that is allowed to take integer (k I) and half-integer values (k H), we can designate whether the motor has both heads bound or just one, respectively. Constructing a Markov process over the states k, we can derive expressions for the rate of change of the mean and variance of the state (and hence the bead position). Copyright c 8 by ASME

3 Figure 1. Schematic of likely typical experimental setup 1 Modeling Two Motors Here, we extend basic PO model to characterize the motion of two kinesin motors pulling the bead and make predictions regarding their cooperative behavior. The objective is to compare this with single-motor experiments, explore the cooperative dynamics, and to serve as a more severe test of the model. The PO model requires very little modification. All we have to do is to keep track of an extra coordinate, that of the additional motor as shown schematically in Fig. 1. The state of the system is represented by ( j,k), where j and k, as before, can both be either integers or half-integers. If j or k is a half-integer, one head of the corresponding motor is unbound from the MT and is diffusing, and hence has another coordinate associated with it, the positions x j and x k respectively. The bead position is denoted by x b. There are three possible combinations of states - depending on whether each motor has a head diffusing or not - and three associated potentials. The potential V is a function of the head locations (if applicable) x j, x k, bead position x b and the separation between motors k j. The definitions of x bnd and x h j (k) are as before, and are uniquely determined by x j(k). One obtains V(x j,x k,x b ) = f x b + K th + K th ( x h j x bnd ) + ( x hk x b ) +W(x j,x k ), where f is the force on the bead, W(x) is the biasing potential and K th is the tether stiffness. The biasing potential W(x) is modelled as in [8], and depends on the state { j,k}, as: j I,k I K bias ( xhk (x bnd j + x ) ) j I,k H W(x j,x k ) = K bias ( xh j (x bnd j + x ) ). (3) + ( xhk (x bnd j + x ) ) j H,k H + K bias We continue to make the assumptions made in the original () PO model for a single motor, plus a few more as follows: 1. Each head equilibriates instantaneously with the bead when diffusing. This is still valid because of the ratio of diffusion coefficients of the head and the bead, D h /D b.. The time scale of diffusion of the bead is still negligible. The tethers are in parallel, making the effective potential nearly twice as stiff, which makes the time-scale smaller than that for a single motor (see Section A.1). 3. A head, when diffusing, almost instantaneously binds to a binding site once hydrolysis completes. That is, the probability that the hydrolysis processes of both motors complete simultaneously in the small time window required for binding is negligible. It might turn out that when the motors are both perfectly synchronized, the hydrolysis processes might be coupled together possibly through load dependence. Including the possibility that the hydrolysis processes complete together is not very hard. Calculating the probabilities of binding forward and backward for each reduces to a first-passage problem in two dimensions. In our particular case, the problem is simple since the potentials that each head experiences are decoupled as V(x j,x k,x b ) = V 1 (x j,x b )+ V (x k,x b ), clearly, from Eq.. In general the equations are elliptic (but linear) and yield to simple finite-differencing schemes. Simulation As for a single motor, we construct a Markov process, with the states and transitions as indicated in Fig.. The only complication here is that the state-space is infinite, and the transition probabilities of the Markov process depend on state. We have to calculate each probability numerically, and there are limited analytical options. To circumvent the problem of infinite number of states, we resort to a truncation. We choose a sufficiently large number of binding sites { N,, N}, and make the boundaries absorbing, by stating that once either motor reaches ±N, they are absorbed and do not undergo any further transitions. We then simulate the governing differential equations for the probabilities of being in each site, given different initial configurations. Let C j,k be the probability of being in state ( j,k). Let P j,k represent the probability of binding forward for a motor - whether it is for the motor at j or k depends on whether they are integers or half-integers. When the motors are both diffusing - j and k are both half-integers - and hydrolysis completes, either motor binds instantaneously and we must calculate two probabilities of binding forward for each motor P1 j,k, P j,k associated with state { j,k}. The other chemical parameters are α, β b, β f and γ, the same as in the PO model. We need to write four different types of differential equations for the four different kinds of 3 Copyright c 8 by ASME

4 # " # " " ),./*+3./ " # &' ( )*+,-./ $ % 1 # " % :@<=>?:;<= A :;<=>?:@<= A # 1 # &' ( 1 Figure. Possible State Transitions, where j, k represents the state of a motor. Here, both j and k and assumed to be integers. Only a part of the possible states is shown, since there are an infinite number of possible states. states. Then, for N + 1 < j,k < N 1 one obtains, dc j,k dt αp1 j 1,k C j 1/,k+ αp j,k 1C j,k 1 + +α(1 P1 j+ 1,k )C j+ 1,k+ +α(1 P j,k+ 1)C j,k+ 1 + (β b + β f )C j,k = αp j 1,k C j 1,k+ +α(1 P j+ 1,k )C j+ 1,k+ β b C j,k 1 + β f C j,k+ 1 +(β b + β f + α)c j,k j I, k I j I, k H β b C j 1,k + β fc j+ 1,k + β b C j,k 1 + β f C j,k+ 1 αc j,k j H, k H. () In Eq., note that for each differential equation, there are source terms from the neighboring states from which the system can enter that particular state and one decay term for the rate at which the system leaves that particular state. So, for the boundary equations - if j, k > N 1 in Eq. - we drop the appropriate decay term and source terms if j, k = ±N. The next step is to calculate the total probabilities for each relevant state. The procedure is essentially that described in [7] for a single motor. If there is only one diffusing head (suppose it is k) determine the conditional (given x b ) head density ρ(x k x b ) and probability of binding forward p t (k,x b ). The total probability P j,k ( f) follows immediately by using the equilibrium bead density ρ bead (x b ). If both motors are diffusing, the conditional densities and probabilities continue to be independent, ρ(x j x b ), ρ(x k x b ), p t ( j,x b ) and p t (k,x b ) because the potential for the two motors decouples as mentioned in Section 1. Then, the total probabilities of binding forward can be written in terms of the the effective potential V e f f (x b ) and bead density ρ bead (x b ) as, ( (k+1/)l V e f f (x b ) = k B T log P1 j,k ( f) = P j,k ( f) = (k 3/)L ( j+1/)l ( j 3/)L exp V 1(x k,x b ) dx k k B T exp V (x j,x b ) dx j ), k B T (5) p t (x j,x b )ρ bead (x b )dx b, (6) p t (x k,x b )ρ bead (x b )dx b. (7) There are a few homogeneities in the problem we can exploit to ease computational effort: any state ( j, k) is identical to (,k j) (or (1/,k j) if j is a half-integer) but for an origin shift of jl. Since the motors are identical, parameters like Copyright c 8 by ASME

5 the probability of binding forward and the potential energy are independent of the identity of the motor. When the motors are both diffusing, if state ( j,k) has probabilities P1 j,k, P j,k, state ( j 1,k 1 ) with j 1 = k, k 1 = j, has probabilities P j,k, P1 j,k because of the obvious homogeneity. These observations significantly reduce the number of calculations. 3 Results The two motors were simulated on a grid of binding sites with { 5,,5} for two ATP concentrations - the same as the single-molecule data - for a range of forces (.5,9.5) pn. The ODEs in Eq. were simulated for a time such that edge-effects, i.e., the effect of the absorbing binding sites at 5 and at 5 on the motor dynamics was negligible. Results were obtained for different initial configurations. The multiple motor data is plotted against the single-motor predictions for the two different ATP concentrations in Fig. 3. Notice that at strongly assisting loads, the mean velocity of the bead is lower than the single-motor, owing to the fact that at least two ATP hydrolyses need to occur to move 8 nm. Because of the randomness, the hydrolysis can never be perfectly synchronous. However, because the load is shared between the two motors, the stall force is extended to nearly twice the stall-force of a single motor. We explore different aspects of the motor and bead dynamics in the following sections. 3.1 Motor Dynamics A general observation from the detailed numerical study performed is that the motor synchronize for all different initial spatial separations for high and low ATP concentrations. The reason lies in the dependence on the states ( j,k) of the probability of binding forward. As mentioned before, the probability depends only on the relative separation j k. Also, note that there are two probabilities when j and k are both half-integers. The probabilities are plotted in Fig. (a) for a low and high force - clearly these are dependent solely on mechanical parameters - with j = in Fig. (a) and j = 1/ in Fig. (b). It is seen that the lagging motor has a much larger tendency to bind forward in both cases (when the leading motor is bound firmly to the MT, and when it has bound ATP and one of its heads is diffusing). A more physical picture is shown in Fig. 5. Here, the mean load felt by each motor - i.e., the extension of its tether - is plotted as a function of time. Figure 5 shows that while the total load, as it must be, is constant and equal to the external force, the load on each individual motor varies until it is shared equally on the two motors. Hence, it is unsurprising that the expected positions of the motor locations synchronize. A plot of the individual motor positions (Fig. 6) shows a dynamic picture of the motor synchronization process. We obtained plots for low and high ATP concentrations and loads to Velocitynms Velocitynms Figure Motors Init Motors Init 1 Motor 6 8 ForcepN (a) Force vs. Velocity, ATP = 16 µm 3 1 Motors Init Motors Init 1 Motor 6 8 ForcepN (b) Force vs. Velocity, ATP =. µm Two motors pulling a bead: force velocity data for different ATP concentrations compared with single motor predictions using the PO model. The multiple motor data is obtained for different initial motor configurations, and since the motors synchronize, the velocity predictions are nearly identical. Init = n represents the initial configuration j =, k = n. indicate that this synchronization happens for a large parameter range. Note that at low ATP, it takes a longer time for the load to be equally shared and for the motors to synchronize - this is a consequence of the lower rate of stepping. At higher loads, synchronization happens faster than at lower loads. The reason this happens is indicated by the probabilities of binding forward in Fig. (a), which shows that the probability of binding forward for the leading motor falls to zero quicker at higher loads than at lower loads. Another observation is that the leading motor waits for the lagging motor to catch up. Again, this fact is easily reconciled with the data from Fig.. 3. Bead Variance and Randomness The bead position and its standard deviation is easily obtained from the relative motor positions. To calculate the randomness parameter, let x b (t) be the bead position so that, r = lim t Var[x b (t)] E[x b (t)]l = lim t Var[x b (t)] t t 1 E[x b (t)] L. (8) 5 Copyright c 8 by ASME

6 35 motor 1() = motor () = 1 8 motor 1() = motor () = Motors Location site # Motors Location site # (a) Motor position (site index) vs. time for both motors, different initial separation; f =.5 pn, ATP = 16 µm (b) Motor position (site index) vs. time for both motors, different initial separation; f = 7.5 pn, ATP = 16 µm motor 1() = 3 motor () = 3 motor 1() = 3 motor () = Motors Location site # Motors Location site # (c) Motor position (site index) vs. time for both motors, different initial separation; f =.5 pn, ATP =. µm (d) Motor position (site index) vs. time for both motors, different initial separation; f = 7.5 pn, ATP =. µm Figure 6. Motor position vs. time for different initial separations, loads and ATP. Note that the motors synchronize under all conditions. The most direct way to calculate the randomness parameter is to solve Eq. for a long time interval, find the variance and expected value, and then take their ratio. It can be seen in Fig. 7(a) that the randomness parameter has not quite reached its steadystate value. However, it is reasonable to believe that it will indeed reach a limiting value if computed for a larger time interval. We know from renewal theory [1] ( when applied to a single kinesin motor), that the ratios E[x b (t)]/t and Var[x b (t)]/t approach constant limits as t. Since, essentially, this is also a renewal process - but with a much more complicated reward function - a similar limit is to be expected. Hence, a better estimate of the randomness parameter can be obtained by computing the slopes of the mean and variance of x b (t) for a large time t. This is shown in Fig. 7(b). The randomness parameter computed by the two methods is shown in Fig. 3. and is compared with the single motor predictions. Note that the randomness parameter is lower for all forces. A not-very-convincing way to physically justify this is to argue that for each renewal - one motor completes a step - the bead steps only by L/. Next, one might argue (based on the general properties of the randomness parameter for Poisson processes) that now both motors have to complete a step for the bead to move 8 nm, lowering the randomness parameter. Conclusions We extended the Peskin-Oster model to two motors pulling a bead to analyze cooperative behavior between multiple motors. The model was simulated on a truncated grid for a range of loads, ATP concentrations and other parameters. The graphs of force vs. mean velocity of the bead show that the velocity is lower than a single motor pulling the bead at assisting loads. This is because the both motors have to finish a hydrolysis before the bead steps 8 nm and the motors cannot be perfectly in sync in a random environment. Stall forces were seen to be extended to nearly twice the single motor range. The motors were observed to synchronize in expected value for the whole range of parameters simulated. The synchronization was faster at higher ATP concentrations because of the increased rate of stepping. Synchronization was also observed to 6 Copyright c 8 by ASME

7 Probability of Binding Forward Probability of Binding Forward Site Index f.5 pn f 7.5 pn (a) Probability of binding forward vs. site index k; j = is fixed f.5 pn, lagging f.5 pn, leading f 7.5 pn, lagging f 7.5 pn, leading Site Index (b) The two probabilities of binding forwards when both diffusing vs. site index k; j = 1/ is fixed Figure. Probabilities of motors binding to the forward site, and their dependence on the relative states - spatial separation and phase - of the two motors Load [pn] 8 6 f,m1,m 1() = f,m,m () = f = 7.5pN f 1,m1,m 1() = f 1,m,m () = f 1 =.5 pn (a) Load on each motor vs. time; motor 1 (t = ) =,, motor (t = ) =, ATP = 16 µm Load [pn] 8 6 f,m1,m 1() = f,m,m () = 3 f = 7.5pN f,m1,m 1() = f,m,m () = 3 f = 7.5pN (b) Load on each motor vs. time; motor 1 (t = ) = 3, motor (t = ) =, ATP =. µm be faster at higher loads. This was explained in terms of relative magnitudes of the probabilities of binding forward. The leading motor was observed to wait for the lagging motor to catch up. Again, the probabilities provide a good explanation for this. The reason lagging motor waits can also be understood from the graphs showing the loads experienced by each motor individually. The leading motor was shown to experience a much larger load than the lagging motor. The randomness parameter was also calculated, and as expected, was lower than the single motor prediction. This is related to the fact that both motors must take a step for the bead to traverse 8 nm. Arguing based on simple results for the Poisson process, we can conclude that including this extra substep in the multiple motor case decreases the randomness parameter. Figure 5. Load sharing between the two motors for high and low ATP and load. The dashed lines show the sum of the individual loads. They are constants, as expected. A Appendix A.1 Diffusion One result that is constantly used for overdamped Brownian motion is related to the Ornstein-Uhlenbeck process [15]. It was originally developed to show how the velocity of a fluid particle relaxes to the Maxwell-Boltzmann distribution. In terms of the velocity U, we can write Newton s Law as a stochastic differential equation in Ito form as: du(t) = βu(t)dt + σz(t) dt, (9) 7 Copyright c 8 by ASME

8 Randomness Var[xb]/(E[xb] L) f =.5 pn f = 7.5 pn Randomness Motor Motors method1 Motors method. 6 8 ForcepN (a) Randomness Parameter vs. t Figure 8. Force vs. Randomness for two motors computed by two different methods and one motor from the PO model Bead Variance Var[xb] [nm ] f =.5 pn f = 7.5 pn linear fits (b) Bead Variance Var(x b ) vs. t and linear fits Figure 7. Evolution of the bead variance and randomness parameter with time for ATP = 16 µm where β = γ/m with γ being the Stokes Law friction coefficient. Then, we may solve the Fokker-Planck equation [16] to find that U(t) is normally distributed with mean and variance given by: E[U(t)] = u e βt, (1) Var[U(t)] = σ (1 e βt ). (11) β The constant σ can be related to the thermal energy k B T by comparing it with the Maxwell-Boltzmann distribution, and all the quantities of interest can be most conveniently written in terms of a diffusion coefficient, D = k B T/γ. For an overdamped particle in an quadratic potential (corresponding to a linear spring with of stiffness K sp ), it follows that the particle relaxes to its equilibrium position with a time constant given by γ/k sp, and a variance of k B T/K sp. A. Effective Potential The head is restricted artificially between the sites in the PO model for convenience. Hence, the effective potential that the bead diffuses in is no longer quadratic. To make the argument that the diffusion time-scale is still smaller than the chemical time-scales, we must show that the effective potential for the bead is still nearly quadratic and its stiffness is of an order of magnitude such that the time scale estimate (from Section A.1) is still much smaller than the chemical time-scales. If there are two motors, then there are two tethers in parallel, and the bead diffuses in a potential twice as stiff, making its diffusion faster than in the single motor case. Hence, it is enough if we show the following for the single motor case. It is easy to see that the only case which we have to be concerned about is when the head and the bead are diffusing. Then, we can find derivatives of the effective potential at the equilibrium point x b, and with a single application of the mean value theorem, we arrive at the following bounds: d V(x b ) dx > K th (K thl) b xb k B T, (1) d 3 V(x b ) dx 3 < (K thl) 3 xb (k b B T). (13) There are more terms in higher derivatives, but the key is that they are of the form (K th ) m (K th L) n /(k B T) n 1, where m,n are positive integers. Thus, all we require is that (K th L) /(k B T) is small. We use K th =.5 pn/nm following [7] - analysis 8 Copyright c 8 by ASME

9 of the experimental data in [8] showed that it s actually about.15 pn/nm - the thermal energy is around.1 pn nm and L = 8 nm, which gives us a value of about.9 for the ratio in question and a time-scale of about 1 µsec. Although it is rather large, it a worst case estimate in any case. The effective potential for a range of forces is shown in Fig. 9, along with quadratic fits to them. It is seen that the quadratic fits are very good, and the stiffness varies by at most.3 pn/nm at the highest load which is much less than the worst-case bound established. Energy pnnm Figure 9. 6 quadratic fits to them 5 pn, K sp. x b pn, K sp. 5 pn, K sp. Effective potentials on the bead for different loads and namics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether. Bulletin of Mathematical Biology, 68 (1), pp [9] Hendricks, A., Epureanu, B.I., and Meyhofer, E., 6. Synchronization of Motor Proteins Coupled Through a Shared Load. Proceedings of IMECE, 57, IMECE6-1575, p. 89. [1] Derenyi, I., and Vicsek, T., The Kinesin Walk: A Dynamic Model with Elastically Coupled Heads. Proceedings of the National Academy of Sciences, 93 (13), pp [11] Badoual, M., Jülicher, F., and Prost, J.,. Bi- Directional Cooperative Motion of Molecular Motors. Proceedings of the National Academy of Sciences, 99 (1), p [1] Mogilner, A., Fisher, A., and Baskin, R., 1. Structural Changes in the Neck Linker of Kinesin Explain the Load Dependence of the Motor s Mechanical Cycle. Journal of Theoretical Biology, 11 (), pp. 13. [13] Rice, S., Lin, A., Safer, D., Hart, C., Naber, N., Carragher, B., Cain, S., Pechatnikova, E., Wilson-Kubalek, E., Whittaker, M., et al., A structural change in the kinesin motor protein that drives motility. Nature,, pp [1] Karlin, S., and Taylor, H., A First Course in Stochastic Processes. Academic Press. [15] Uhlenbeck, G., and Ornstein, L., 193. On the Theory of the Brownian Motion. Physical Review, 36 (5), pp. 83. [16] Cox, D., and Miller, H., The Theory of Stochastic Processes. Chapman & Hall/CRC. REFERENCES [1] Block, S., 7. Kinesin Motor Mechanics: Binding, Stepping, Tracking, Gating, and Limping. Biophysical Journal, 9 (9), pp [] Nedelec, F., Surrey, T., Maggs, A., and Leibler, S., Self-Organization of Microtubules and Motors. Nature, 389 (668), pp. 35. [3] Astumian, R., and Haenggi, P.,. Brownian Motors. Physics Today, 55 (11), pp. 33. [] Fisher, M., and Kolomeisky, A., 1. Simple Mechanochemistry Describes the Dynamics of Kinesin Molecules. Proceedings of the National Academy of Sciences, p [5] Bier, M., Brownian Ratchets in Physics and Biology. Contemporary Physics, 38 (6), pp [6] Reimann, P., and Hänggi, P.,. Introduction to the physics of Brownian motors. Applied Physics A: Materials Science & Processing, 75 (), pp [7] Peskin, C., and Oster, G., Coordinated Hydrolysis Explains the Mechanical Behavior of Kinesin. Biophysical Journal, 68 ( Suppl). [8] Atzberger, P., and Peskin, C., 6. A Brownian Dy- 9 Copyright c 8 by ASME