A stochastic model for kinesin bidirectional stepping.

A stochastic model for kinesin bidirectional stepping Xiaojun Yao and Yujun Zheng Citation: The Journal of Chemical Physics 140, 084102 (2014); doi: 10.1063/1.4865934 View online: http://dx.doi.org/10.1063/1.4865934 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stochastic dynamics of small ensembles of non-processive molecular motors: The parallel cluster model J. Chem. Phys. 139, 175104 (2013); 10.1063/1.4827497 Bidirectional transport of motor-driven cargoes in cell: A random walk with memory AIP Conf. Proc. 1512, 140 (2013); 10.1063/1.4790950 Analysis of the nucleotide-dependent conformations of kinesin-1 in the hydrolysis cycle J. Chem. Phys. 131, 015104 (2009); 10.1063/1.3157256 Effective stochastic dynamics on a protein folding energy landscape J. Chem. Phys. 125, 054910 (2006); 10.1063/1.2229206 Stochastic regulation of gene expression AIP Conf. Proc. 502, 191 (2000); 10.1063/1.1302384

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THE JOURNAL OF CHEMICAL PHYSICS 140, 084102 (2014)

A stochastic model for kinesin bidirectional stepping Xiaojun Yao ()1,2 and Yujun Zheng ()1,a) 1 2

School of Physics, Shandong University, Jinan 250100, China Taishan College, Shandong University, Jinan 250100, China

(Received 5 October 2013; accepted 3 February 2014; published online 24 February 2014) In this paper, a hand-over-hand stochastic model for the dynamics of the conventional kinesin is constructed. In the model, both forward and backward motions are taken into consideration. First passage time distributions, average velocities, dwell times, and forward/backward step ratios are investigated based on the model. A good agreement between the results of the model and experimental data is achieved under a variety of external loads. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4865934] I. INTRODUCTION

The transportation of intracellular vesicles containing proteins and other necessary nutrients plays a vital role in living cells. A pure diffusion fails to satisfy the nutrient need of metabolic activities.1 Nature solves this problem elegantly by “inventing” different kinds of molecular motors in eukaryotic cells such as dyneins, myosins, and kinesins. These nanoscale motors are driven by adenosine triphosphate (ATP) hydrolysis and can convert the chemical energy released into mechanical work. The protein motor family kinesin has drawn a lot of research interest. They move on microtubules (MT) and partake in transportations of membranous organelles, mRNAs, and protein complexes.2 Currently, there are 14 groups of kinesins and many ungrouped kinesins.3 Kinesin-1, also called the conventional kinesin, is the first member discovered in the kinesin family (abbreviated as kinesin below). Although the structures and biochemical properties of kinesin have been widely investigated (see Appendix A), its stepping mechanism is still not well explained. Many theoretical models have been proposed to explain how they move unidirectionally under thermal fluctuations: rectified diffusion models,4, 5 flashing ratchets,6 2D flashing ratchets,7 forced thermal ratchets,1 an adenosine diphosphate (ADP)dependent model,8 and an electrostatic model.9 The first two models have been ruled out from experimental data.10 Among the models, there are mainly two walking patterns: the inchworm mechanism and the hand-over-hand mechanism.11 One experiment rules out the inchworm mechanism and supports the symmetric hand-over-hand mechanism12 while other experiments indicate, however, that an asymmetric hand-overhand mechanism is more plausible.13, 14 The asymmetry originates from the switch of the neck linker between two conformations: one is being immobilized and extended towards the plus end when kinesin binds with a microtubule and an ATP molecule; the other one is reverting a more mobile conformation when the γ -phosphate is released.15 These two conformations are like “docking” into and “undocking” from the catalytic core and induce tensions between the two heads. The a) Electronic mail: [email protected]

0021-9606/2014/140(8)/084102/8/$30.00

tension may detach the ADP-bound, trailing head from the microtubule, and throw the tethered head forward. A simple neck linker docking model cannot explain the movement of kinesin completely by energy arguments.16, 17 It is found that the intramolecular strain helps to keep the two heads out of phase, favours the dissociation of the rear head from the microtubule, and increases the ADP unbinding rate when the ADP-bound head is in the front.18, 19 In such sense, the motion is more likely gated by the rear head.20 It is still undecided what fraction of the 16 nm displacement is covered by the conformational change and what part is associated with a Brownian motion. The neck linker docking process is believed to play an important role in the processive movement and more studies are needed.10, 18, 20 In this manuscript, using the key experimental facts about kinesin (they are summarized in Appendix A), we develop a stochastic model based on the hand-over-hand mechanism to describe both the forward and backward motions of kinesin in a unified framework. We then investigate average velocities, dwell times, and forward/backward step ratios under a variety of external loads. A good agreement between numerical results of our model and experimental data is reached. The manuscript is organized as follows. In Sec. II, we develop our hand-over-hand stochastic model. In Sec. III, we show our numerical calculation results and compare them with experimental data. Discussions and a concise conclusion are also given in this section.

II. HAND-OVER-HAND STOCHASTIC MODEL

In this section, we present our hand-over-hand stochastic model, which has five steps in the forward and backward motions, respectively. All the model parameters are summarized in Table I. The chemical reaction rates ki (i = 1, 3, 4, 5) and ki (i = 1, 2, 4, 5) are taken from experimental results. The corresponding references are also indicated in Table I. The other parameters in our model are chosen by fitting experimental data, and the details of parameter estimates are explained in Appendix B. Varying external loads Fext are applied to the motor. It is assumed that the external load is sustained by the two heads with an equal weight when both of them bind to a

140, 084102-1

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TABLE I. Parameters used in the model. Parameter

Forward (μM−1 s−1 )21

ATP binding ATP hydrolysis (s−1 )14, 21 ADP dissociation (s−1 )22 P release (s−1 )14, 21 VS (pN nm) ai (nm) F0 (pN) c0 (pN) MM constant (μM) k (pN/nm) x0 (nm)

Backward

k1 = 4 k1 = 4 k3 = 600 k4 = 600 k4 = 140 k2 ∼ 1 k5 = 600 k5 = 600 VD = 11 VT = 23 ai = 4 (x > xi ), ai = 3 (x < xi ) −6 13.5 4 1.8 KM = 24 633 0.72 1.96

MT. After one head detaching from the MT, the tethered head sustains the whole load. A. Forward motion

In this subsection, we describe the stochastic model for the forward motion, which is schematically shown in Fig. 1. In the model, the chemical cycle for the forward motion is k1 k2 − D·M+T− − − D · M + T −−→ k−1 k3 k4 T · M · D −−→ (D + P) · M · D −−→ k5 (D + P) · M + D −−→ D · M + P + D, where P represents the γ -phosphate and D · M · T means both the ADP-bound rear head and the ATP-bound front head bind with a MT. Before an ATP molecule arrives, a harmonic potential energy is stored in the neck linker. The energy is gained from the last step, no matter whether it is forward or backward. Its final source is the ATP hydrolysis of the last step. The motion is triggered by the ATP molecule’s binding to the front head, which changes the charge configuration (the ATP-bound head has −2e charge and the empty head has +2e charge9 where e = 1.6 × 10−19 C). The charge distribution rearrangement may induce the experimentally observed neck

FIG. 1. Model for the forward motion of kinesin.

FIG. 2. Free energy A(x) along the reaction coordinate x on MT. There is no binding site at x = 8 nm for the tethered ADP-bound head since it has already been occupied by the ATP-bound head.

linker docking, which favors the detachment of the weakly bound rear head containing an ADP. Then the ADP-bound head diffuses randomly in both directions. The part of the free energy A(x) that excludes the change caused by the ATP hydrolysis along the reaction coordinate (i.e., the MT track since the mechanical motion and the chemical reaction are tightly coupled) is A(x) = V (x) + Fext x,

(1)

where V (x) is potential energy given by Eq. (10) and (11), and Fext is positive for the backward direction. A(x) is plotted in Fig. 2 for the case of Fext = 4 pN. The changes in the energy landscape are obvious by comparing Fig. 2 and Fig. 3 due to the neck linker harmonic potential and the external force. The ADP-bound head is less probable to move backward since the harmonic potential increases rapidly when the neck linker is overstretched. But this gives no directional motion. It is the ATP hydrolysis (whose rate is increased significantly by the kinesin-MT complex) that fixes the motion direction. If the ADP-bound head diffuses to the binding site at x = 16 nm, the rapid ATP hydrolysis followed provides a big drop in the free energy (see the red dashed line in Fig. 2 where the free energy released is depicted as a sharp Gaussian function). Then the process becomes irreversible and the kinesin makes a net forward movement. If the ADP-bound head wanders around the original binding site and after some time the ATP is hydrolyzed, the motor keeps its original

FIG. 3. Affinity potential Vaff . The parameters in Table I are used.


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position and waits for the next try. If the ADP-bound head happens to diffuse to the binding site at x = −8 nm, the ATPhydrolysis rate is extremely slow since no experimental evidence indicates that the kinesin-MT complex still has the enzyme activity when the kinesin is overstretched. So, it is still possible that the ADP-bound head at x = −8 nm can diffuse to x = 16 nm before the ATP hydrolysis. During the process the neck linker “spring” is relaxed first and then is extended. After the stepping motion a potential energy in the neck linker is stored for the next cycle which comes from the free energy released by the ATP hydrolysis. When the front ADP and the produced γ -phosphate are released from the two heads, the chemical state returns to its original one with a net forward 8 nm displacement in the center of mass. 1. ATP binding trigger

The ATP binding in the first step is assumed to be a Poisson process with a parameter λ = k1 [ATP], where [ATP] denotes the ATP concentration. The average event number during a time period τ is λτ . Letting the average number be 1 leads to the time consumed in the first step, namely, t1 =

1 1 = . λ k1 [ATP]

(2)

t1 is also the time needed for the substrate (ATP) to reach the chemical equilibrium with the enzyme-substrate complex (kinesin+ATP+MT). 2. Michaelis Menten rate

The first two steps have a character of Michaelis Menten kinetics, thus the rate of T · M · D production is v=

[ATP] d[T · M · D] = Vmax , dt KM + [ATP]

KM + [ATP] . [ATP]

(3)

3. Drifting diffusion

The drifting-diffusion process is described by the Langevin equation ∂V (x) + (t) − Fext , ∂x where a backward external load is defined with a positive sign and γ = 6π ηR is the drag coefficient for a spherical object with a radius R. The kinesin head is approximated as a sphere with a diameter 5 nm. η is the dynamic viscosity, mx¨ = −γ x˙ −

∂V (x) + (t) − Fext , ∂x where (t) is white noise satisfying γ x˙ = −

(4)

(t) = 0, (t)(t ) = 2γ kB T δ(t − t ).

(5)

The equation of motion is a stochastic differential equation and its solution at time t is given by a distribution density function w(x, t) which satisfies the Fokker-Planck equation (in this case also called Smoluchowski equation)23 ∂w(x, t) = LF P (x, t)w(x, t), (6) ∂t where LFP (x, t) is the Fokker-Planck operator. It is defined as LF P (x, t) = −

∂ ∂2 D1 (x, t) + 2 D2 (x, t), ∂x ∂x

(7)

with the drift term D1 (x, t) = − ∂V∂x(x) − Fext and the diffusion term D2 (x, t) = 2kγB T . The initial condition is w0 = w(x, 0) = δ(x − xi ) (xi is the initial position of the moving head). For a finite period time T, the formal solution of Eq. (6) can be written as T LF P (x, t)w0 dt w(x, t) = w0 + 0

+

T t

t

LF P (x, t )LF P (x, t)w0 dtdt + · · · .

0

(8)

where KM is the ATP concentration when the rate reaches half the maximum. Here KM is a model parameter and determined by experimental data. The first passage time in the diffusion step is employed in our model to approximate Vmax . t1 is excluded in Vmax since one assumption in the Michaelis Menten kinetics is the instantaneous chemical equilibrium between the substrate and the enzyme-substrate complex. Thus the time in the second step is given by t2 ≈ tdiff

and η = 0.001 Pa s for water at 293 K, which is close to the cellular environment. From the kinesin head mass 110 kDa, = 3.89 × 10−12 s, which is the Langevin relaxation time is m γ so small that the inertial term is negligible. The overdamped Langevin equation is then written as

For the forward motion, the boundary condition w(xahead , t) = 0 is exposed (xahead is the position of the front binding site, 16 nm apart from the initial position). The travelling time, from xi to xahead in the diffusion step, is the first passage time tdiff . The first passage time distribution W (T ) can be calculated as follows:23 xahead ∂ W (T ) = − w(x, T )dx. (9) ∂T −∞ 4. Potential

The potential term in the Fokker-Planck equation consists of two parts: the affinity between kinesin heads and binding sites on a MT and the neck linker harmonic potential. To describe the binding affinity, we use an exponential function with a parabolic exponent. The potential, depending on the chemical state of the head: ADP-bound or ATP-bound, is given as follows: − x−ri 2 e ai , (10) Vaff (x) = −VS i

where ri ’s are the binding positions and ai ’s control the force range. VS has two values: VD for the ADP-bound head and


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TABLE II. Detaching forces of kinesin heads from MT measured in Ref. 24 and calculated in the model.a State and direction

Measured force (pN)

Calculated force (pN)

ADP, plus ADP, minus ATP/empty, plus ATP/empty, minus

3.3–3.4 3.6–3.9 6.1–6.9 9.1–10

3.34 4.45 6.98 9.30

(12)

6. ADP and P release

They depend on the directions of the detaching pulls, especially for the ATP-bound head.

VT for the ATP-bound head. For simplicity, we only consider the contributions from neighboring sites, namely, we choose i = 1, 2 in Eq. (10). In this case, r1 denotes the starting point and r2 is the next binding site ahead, separated by 16 nm. The term when ri = 8 nm is excluded since that site is already taken by the fixed ATP-bound head. The parameters in the affinity potential are estimated by equating the maximum (x) to the force detaching a head from affinity force −∂V ∂x max its binding MT (see Appendix B). Both the experimental24 and theoretical values of detaching forces are listed in Table II which depend on the pull directions. The dependence of a1 and a2 on the direction of motion introduces the necessary asymmetry into the model. The affinity potential for the ADP-bound head is plotted in Fig. 3. The neck linker harmonic potential, which is used to model the neck linker’s function, is given by 1 k(x1 − x2 − x0 )2 , 2

|F −F | |F −F |d 1 − extc 0 − ext 0 0 = k 3 e kB T ≡ k 3 e , t3

where c0 is a model constant and controls the function shape. Fitting details are provided in Appendix B.

a

Vnl =

an exponential function

(11)

where x1 and x2 are the positions of the two heads and x0 determines the equilibrium point. In the forward case, x1 = x is the variable and x2 is fixed. The force in the neck linker at the moment of the ATP binding is −k(−8 − x0 ). The parameters k and x0 are estimated from the stall force 7.2 pN and the energy consideration (see Appendix B).

5. Load-dependent ATP hydrolysis

Experiments have shown that some processes concerned with ATP are external-load-dependent.25 In our model, the ATP hydrolysis is assumed to be load-dependent, which can be justified by electrostatic considerations. It has been shown that the empty, ADP-bound and ATP-bound heads have charges of +2e, −e, and −2e, respectively.9 In the forward motion of the model, the ATP hydrolysis happens when the ATP-bound head is in the rear while the ADP-bound head is in the front. Therefore, a backward electrostatic repulsion acts on the ATP-bound head. The electrostatic force has an 2e2 approximate value 4π 2 = 7.2 pN for r = 8 nm, which is 0r of the same order as the external forces. The electrical force leads to a mismatch between the ATP-bound head and the MT, resulting in a smaller hydrolysis rate. Now if a forward external force with a proper magnitude is applied to balance the electrostatic force, a perfect match is achieved, resulting in a maximum hydrolysis rate. If the external force is too large or too small, the net force is still not zero and the mismatch still exits, lowering the hydrolysis rate. The dependence is through

The remaining two chemical steps are the ADP dissociation from the new leading head and the release of the γ phosphate from the trailing head. The reaction time in each step is the reciprocal of the corresponding rate, namely, t4 = t5

1 , k4

(13)

1 = . k5

The total time in one step is T = t1 + t2 + t3 + t4 + t5 . For a n-step forward motion, the average velocity is defined as the total displacement nL divided by the total n-step time,26 nL v+ = n

i=1 Ti

=

L . T

(14)

In this work L = 8 nm. The average time is obtained by taking a weighted average over the first passage time distribution W (tdiff ) in the diffusion step where tdiff is related to t2 via Eq. (3) v+ = ∞ 0

L . (15) (t1 + t2 + t3 + t4 + t5 )W (tdiff )dtdiff

B. Backward motion

In this subsection, we give a concise description of the backward motion in the stochastic model. The backward motion in the stochastic model is schematically shown in Fig. 4. The triggering mechanism is the same as the forward motion, so t1 = t1 . The binding of an ATP molecule induces charge rearrangements and favors the detachment of the rear ADP-bound head. If the external load is larger than 7.2 pN, the ADP-bound head struggles to diffuse to the next binding site in the plus direction (see the rapid increase in the

FIG. 4. Model for the backward motion of kinesin.


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first passage time as the backward external force increases in Sec. III). It is also hard for it to diffuse 8 nm backward since the overstretched neck linker “spring” exerts a large force on it. Therefore, most of the time it rambles around the original site with little net motion till its ADP dissociates. Then the empty head binds to the MT more strongly and the front ATPbound head is dragged out of its binding site by the spring tension and the external force and starts to diffuse randomly in two directions. The ATP hydrolysis fixes the direction of motion by releasing a large amount of free energy. It only happens when the ATP-bound head diffuses to the backward neighboring binding site, because in the forward neighboring binding site the kinesin is overstretched and the kinesin-MT complex has no enzyme activity. If the ATP is hydrolyzed in the original site, no net motion occurs and the motor waits for the next try. The ATP hydrolysis at the backward neighboring binding site makes the process irreversible with a net backward motion. The chemical cycle for the backward motion is as follows: k1 k3 k2 −− D·M+T − − D · M · T −−→ D + M · T −−→ k−1 k5 k4 T · M −−→ (D + P) · M −−→ D · M + P. Since the ADP dissociation happens when it is located in the rear head, its rate decreases a 100 fold. Furthermore, the k assumption k 2 1 in the Michaelis Menten kinetics is no −1 longer satisfied. Thus we abandon Michaelis Menten reaction rate when considering the backward motion and simply . The diffusion process is almost use t2 = k1 and t3 = tdiff 2 the same as in the forward motion except for the boundary condition and the potential parameters. The affinity potential

J. Chem. Phys. 140, 084102 (2014)

parameter VS is replaced with VT and the new binding site xback is located on the minus direction of the microtubule. A new boundary condition w(xback , t) = 0 is utilized and Eq. (9) now becomes ∞ ∂ W (T ) = − w(x, T )dx. (16) ∂T xback The load dependence of the ATP hydrolysis still holds true in the backward motion by the same argument as in the forward case. However, in the backward case a backward force favors the ATP hydrolysis because it occurs when the ATP-bound head is in the rear and the front head is empty. The empty head has positive charge and thus attracts the ATPbound head. The mismatch needs to be balanced by a proper backward force. Finally, the γ -phosphate release step has the same rate as that in forward case, t5 = t5 . Taking the minus sign of the displacement into account and using t3 = tdiff converts Eq. (15) to L v− = − ∞ . t1 + t2 + t3 + t4 + t5 W (t3 )dt3 0

(17)

III. NUMERICAL RESULTS AND DISCUSSIONS

We apply the Monte Carlo method to solve the FokkerPlanck equation for the first passage time distribution. The time variable is discretized on an order of magnitude γ . A Gaussian random number generator is applied to simulate the diffusion by generating a sequence of positions. Once the sequence hits the boundary, the stimulating time is taken as the first passage time. The simulated first passage time distributions under −10 pN, 1.5 pN, 4 pN, and 9 pN loads are shown

FIG. 5. First passage time distributions under −10 pN (a), 1.5 pN (b), 4 pN (c), and 9 pN (d) external loads.


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800 Average Velocity (nm/s)

700

10

[ATP]=1 mM theory

Dwell Time (s)

600 [ATP]=1 mM experiment

500 400 300 200

[ATP]=1 mM theory

[ATP]=1 mM experiment

1 10-1 10-2

100 0 -15

-10

15

10-3 -20

20

-15

-10

-5 0 5 10 Applied Load (pN)

15

20

15

20

10

400 350

[ATP]=10 µM theory

[ATP]=10 µM theory

300 250 200 150 100 50 0 -50

-100 -20


[ATP]=10 µM experiment

-15

-10


15

Dwell Time (s)

Average Velocity (nm/s)

-100 -20

20

FIG. 6. Average velocities vs applied loads. Numerical results of the stochastic model are compared with the experimental data.27

in Fig. 5. The diffusion times of the order of μs are negligible compared to the times consumed in other processes. For large backward loads, the order is ms, giving a good evidence that as the backward external force increases the first passage time becomes longer. When the load approaches the stall force, the diffusion takes a time of the order of s, resulting in almost no net displacement: v+ ∼ 0 nm/s. Numerical results from the Monte Carlo method on the average velocities under varying external loads are shown in Fig. 6 for two ATP concentrations: [ATP] = 1 mM (top) and [ATP] = 10 μM (bottom). For comparison, the experimental data27 in the same condition are also plotted. The agreement achieved between our theoretical calculations and experiments is good. The average velocity has a clear dependence on the ATP concentration when kinesin steps forward while it shows no obvious difference in the two ATP concentrations for the backward motion. The reason is the slow dissociation rate of the ADP when bound to the rear head. The ADP dissociation is the rate-limiting step in the backward motion so that the different ATP binding rates caused by the two concentrations have little effect on the whole chemical cycle period. In the forward motion, the ADP dissociates fast since it is bound to the front head. When the ATP concentration is 1 mM, the diffusion time is of the order of μs for forward and small backward forces. The trend of the average velocity as load changes is captured by the ATP hydrolysis time which is load-dependent. As the forward external force increases, the average velocity drops faster than the exponential extrapolation since much more time is involved in the diffusion. For the low ATP concentra-


1

10-1

10-2 -20

-15

-10


FIG. 7. Dwell times vs applied loads. Numerical results of the stochastic model are compared with the experimental data.27

tion case, the results of our stochastic model and experiments agree well for backward and slow forward motions. However, under forward loads, there is a bizarre fluctuation around a central curve in the experimental data. The smooth simulated result fails to match the jumping behavior. We attribute this to the uncertainties and fluctuations in experiments. In Fig. 7, the dwell times under varying loads are also calculated and compared with experimental data for two different cases: [ATP] = 1 mM (top) and [ATP] = 10 μM (bottom).27 Errorbars in the experimental data are also included. In our model, the dwell time is defined as the kinesin’s staying time on one binding site plus the time consumed to escape the binding potential well ( x ∼ 8 nm). As shown in the figure, the dwell times are small under forward and tiny backward loads, corresponding to the large stepping velocities. As backward load increases, the dwell time increases dramatically. At the vicinity of the stall force, the dwell time is of the order of s. This singular behavior is justified by the fact that no net motion occurs at the stall force. Finally, the forward and backward step ratio is calculated and shown in Fig. 8 with experimental data.27 In both forward and backward motions of our model, the diffusion is governed mainly by thermal fluctuations. The ATP hydrolysis in both cases fixes the direction of motion by providing a free energy drop in the reaction coordinate. Thus the forward and backward step ratio is closely related to the relative ATP hydrolysis rate. Here, the ratio is defined as the ATP hydrolysis rates ratio between the forward and backward motions under the same external load. The assumption of the load-dependent ATP hydrolysis is justified by the good matching. Around the


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Forward/backward Step Ratio

103 theory [ATP]=1 mM experiment

102


10 1 10-1 10-2 0

2

4 6 8 Applied Load (pN)

10

12

FIG. 8. The forward and backward step ratio as a function of the applied load. Numerical results of the stochastic model are compared with the experimental data.27

stall force, the ratio is almost one, corresponding to no net motion. The stall behavior is independent of the ATP concentration, which is consistent with experiments. In summary, we have suggested the hand-over-hand stochastic model to investigate the stepping mechanism of kinesin. To this end, average velocities, dwell times, and forward/backward step ratios have been calculated and compared with experimental data. We have demonstrated how the handover-hand stochastic model can be employed to describe the walking pattern of kinesin. Also, our stochastic model manages to describe both the forward and backward steppings of kinesin and captures key features in the experiment data. ACKNOWLEDGMENTS

This work was supported by the National Science Foundation of China (Grant Nos. 11374191 and J1103212).

binding site for the kinesin head and adjacent β-tubulins are separated by 8 nm. Kinesin has two homodimeric motor domains called heads that contain catalytic cores. Each head has 350 amino acid residues with an average mass 110 kDa and a size of 7.5 nm × 4.5 nm × 4.5 nm.30 The two heads can bind with ATP, ADP, and MT. They are connected through heavy chains and neck linkers. The neck linker is made up of 11–13 amino acids and determines the direction of movement.15 The coils form the stalk which is dimerised into two heavy chains on one end and two light chains on the other to carry cargo. Experiments have shown that kinesin is a processive motor and advances 8 nm to the plus end of the microtubule each step.31 The stepping is tightly coupled to the ATP turnover under low loads: exactly one ATP is hydrolyzed in each hopping step.32, 33 It is also shown that kinesin has no ATP enzymatic activity in the absence of microtubules, and the rate of hydrolysis increases when kinesin binds to a microtubule.21, 34 The rate of ADP release in the absence of microtubules is limited ( xi ) and 3 nm for backward motions (x < xi ) where subscripts indicate the binding sites. The max2VS is treated as the detaching force. imum affinity force − √ ea VD and VT are then fitted by a modified least-square method. Since experimental values lie in certain ranges, we treat the difference between the middle value and the upper (lower) bound value as the experimental uncertainty. The best fitted values for VS are obtained by minimizing the expression ftheo − fexp,middle 2 . (B6) δexp Using Mathematica we find VD = 11 pN nm and VT = 23 pN nm. Next we determine the parameters k and x0 in the neck linker spring potential. These two parameters are estimated from the stall force 7.2 pN and the energy provided by the ATP hydrolysis. They are related by −k(−8 − x0 ) = 7.2,

Solving the two equations leads to k = 0.72 pN/nm and x0 = 1.96 nm. The other three parameters c0 , F0 , and KM are fitted by using the experimentally measured velocities and the least-square method for the forward case. In the backward case there is no Michaelis Menten kinetics, thus only c0 and F0 are fitted by using the measured step ratios. Dwell time data and the associated velocities are not used in the backward case since the uncertainties in the experiment are large. All experimental data come from Ref. 27. The fitted values are listed in Table I. Also, the ADP dissociation rate in the backward case is of the order of s−1 and is chosen to be 3 s−1 for the higher ATP concentration and 1 s−1 for the lower ATP concentration.

(B7)

because if the tension at the ATP binding moment fails to compete the load, the rear ADP-bound head struggles to make net motion. After a relatively long time the bound ADP dissociates and then backward motion starts. The second equation comes from energetics. One ATP molecule provides about 84 pN nm free energy via hydrolysis. Suppose at the stall force, all the free energy from the ATP hydrolysis is used to fuel the kinesin. The net displacement of the motion is 8 nm per step. The motion not only does work to drag the cargoes (applied external forces) but also extends the neck linker spring. Associated with the spring potential energy is the kinetic energy, which has the same value as the potential energy on average according to the virial theorem. The kinetic energy is dissipated quickly in the overdamped environment but needs to be taken into account when using the energy conservation theorem. Therefore, the conservation of energy at the stall force gives:

1 2 (B8) k(8 − x0 ) = 84. 7.2 × 8 + 2 · 2

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