NECK-LINKER TENSION AND THE LOCOMOTION OF KINESIN ALONG MICROTUBULES

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 18, Number 3, Fall 2010 NECK-LINKER TENSION AND THE LOCOMOTION OF KINESIN ALONG MICROTUBULES PETER W. BATES AND ZHIYUAN JIA ABSTRACT. This paper concerns the detailed modeling of kinesin locomotion along microtubules. Experimentalists have inserted polymers into the neck-linkers of kinesin, extending their lengths and changing their flexibility. Observations of the progression of these mutants along microtubules provide some insight into the mechanisms of locomotion. We formulate a mathematical model of this locomotion that takes into account the rates of various biochemical processes, orientationdependent binding affinities, and the entropic forces involved in the constrained random walk of a tethered head. Our analysis and simulation of the model with neck-linkers of varying lengths indicate that the processivity of kinesin is obtained through the coordination of the chemical states of two heads, which is regulated by the tension in neck-linkers. We find that both frontgating and rear-gating mechanisms are required for processivity consistent with experimental evidence. 1 Introduction Kinesin-1 (for brevity, we always write kinesin for kinesin-1) is a member of kinesin motor family that walks on microtubules by utilizing chemical energy, hydrolyzed from ATP (Adenosine triphosphate) molecules to take mechanical movements in a locomotion process. Kinesin is composed of two identical heavy chains, each of which includes a motor domain (the head), in which there is an ATP binding site. The neck-linker is the segment in each heavy chain connecting the head to the coiled-coil stalk. Growing from the other end of the coiled-coil stalk are the two light chains (arms), which can hold cargo (e.g. mrnas or protein complexes). Carefully performed experiments have shown that kinesin walks along microtubules in a hand-over-hand manner [2, 27]. The two heads of the kinesin molecule alternately bind to and unbind from the microtubule with mechanisms that provide a Copyright c Applied Mathematics Institute, University of Alberta. 229

2 230 P. W. BATES AND Z. JIA bias to the Brownian motion expected, as explained below. Experiments show that the center of mass of the kinesin molecule moves 8 nm with each step, which is exactly the length of one αβ tubulin dimer. Kinesin typically hydrolyzes one ATP molecule each step, meaning that kinesin tightly couples a chemical reaction to a mechanical movement [5, 22, 23]. The binding of ATP causes part of the neck-linker to dock towards the front of the microtubule-bound head (called zipping). This provides a significant part of the bias towards forward binding (rather than backward binding or neutral rebinding) of the tethered head. The heads have different microtubule binding affinities when they are in different nucleotide states [25], i.e., the core containing either ATP, ADP, or being empty (nucleotide-free). The binding affinity is either strong, weak, or of intermediate strength, when it is in the ATP-bound, ADP-bound, or the nucleotide-free state, respectively. While it is possible for both heads to be in the ADP-bound state simultaneously, such a state could lead to kinesin detaching completely from the microtubule and either terminating the transport of a cargo or else re-attaching after some brief time and continuing. The latter scenario is probably rare and in any case may be indistinguishable from futile cycles in the experiments performed so far. It has been observed that kinesin walks processively on a microtubule continuously for well over 100 steps without falling off and terminating the transport. This processivity requires that there must be at least one head bound to the microtubule at all times in the walking process. So there must be a mechanism to prevent both heads from being in the ADP-bound state when both are attached to the microtubule. In other words, there should be mechanisms to ensure that the chemical states of the two heads are out of phase. The tension of the neck-linkers is such a candidate in the regulation of the chemical states of the two heads [8, 9, 19]. There are two hypotheses concerning the tension of the neck-linkers: the frontgated-head model [8, 19] and the rear-gated-head model [9]. The frontgated-head model postulates that in the two-head-bound state with the leading head in its empty nucleotide state and the trailing head in its ADP-bound state, the tension in the leading neck-linker prevents ATP molecules binding to the leading head until the trailing head detaches from the microtubule. The rear-gated-head model [9] postulates that the tension in the trailing neck-linker favors the dissociation of the trailing head from the microtubule. To test how the tensions in the neck-linkers help coordinate the chemical states of the two heads, Amet Yildiz et al. [28] performed experiments in which they inserted polymers with different lengths and prop-

3 TENSION OF NECK-LINKERS OF KINESIN 231 erties into the native neck-linker. As is shown in Figure 1, they inserted three amino acids, represented by K(lysine)KG(glycine) in Figure 1, at the junction of the neck-linker and the coiled-coil stalk so that the joint between the two neck-linker parts is more flexible. FIGURE 1: Illustration of the various mutants. This is part A of Figure 1 in [28]. (All the experimental results are shown by permission from Cell Press.)

4 232 P. W. BATES AND Z. JIA This mutant is labeled 0P. Into the mutant 0P, they inserted polyproline (P) helices with different lengths at the position between KK and G to form the mutants 2P, 4P, 6P, 13P, 19P and 26P. Finally, 14GS is the mutant formed by inserting seven more flexible glycine-serine (GS) repeats instead of polyproline, resulting in neck-linkers having lengths close to those of the 19P mutant but with much shorter persistence lengths. Their experimental results showed that all of the mutants still walk over 100 steps and have more or less the same run length before falling off as the wild type. However, the speed of these mutants decreases as the mutants neck-linker becomes longer and the inserted material is soft (has a shorter persistence length). The measured ATPase rates of the mutants are the same as the wild type but more futile cycles (where ATP is used but no step is taken) were found in the mutants. The coupling ratio of ATP, defined as the quotient of the number of steps over the number of hydrolyzed ATP molecules, decreases in a way similar to the speed. The coupling ratio for the wild type is about 80% but the 14GS has a ratio of only 10%. The motility of kinesin is tested at different nucleotide conditions. If there is no ATP present to fuel the motor, under external forces, kinesin can walk toward either the plus or the minus end of a microtubule depending on the direction and the magnitude of the external forces. An external force of 3 pn is required for the motor to walk toward the plus end and a force of 6 pn is required for the motor to move towards the minus end. In this situation, there is no neck-linker docking (zipping) and the direction of the movement is decided only by the direction of the external force. The authors of [28] also tested the motor with only ADP or AMPPNP (Adenylyl-imidodiphosphate, a nonhydrolyzable ATP analog) present and found that the amplitude of the external forces for the motor to walk toward the plus end is 1 pn in an ADP solution and 9 pn in an AMPPNP solution. A force of 2 pn in ADP solution and 12 pn in AMPPNP solution are required for the motor to walk towards the minus end. In particular, the amplitude of the external forces for the kinesin to walk toward the plus end is different in the different nucleotide solutions, 1 pn in ADP solution, 3 pn in no nucleotide condition, and 9 pn in AMPPNP solution. This confirms the dependence of binding affinities on the nucleotide states. On the other hand, the amplitude of the external forces for the kinesin to walk toward the minus end is also different in the different nucleotide solutions, 2 pn in ADP solution, 6 pn in no nucleotide condition, and 12 pn in AMPPNP solution. These

5 TENSION OF NECK-LINKERS OF KINESIN 233 differences in the external forces needed for movement under the same nucleotide conditions clearly indicate that there exists asymmetric steric binding affinities for kinesin, in each of the nucleotide states. The step size histogram for each of the mutants indicates that kinesin still continues to walk forward with the extended neck-linkers. There are only about 5% more backward steps for mutants compared with the wild type. With the long neck-linker and soft GS inserted, 14GS has much more lateral binding, meaning that the motor binds to the sites on the protofilaments adjacent to the one on which the bound head lies. There are several works on modeling and theoretical analysis of the walking mechanism of kinesin (see [14] and reference therein). To the best of our knowledge, there is no detailed simulation of the walking process of kinesin that faithfully follows the experimentally established biochemical and mechanical processes. In this paper, we have developed a set of algorithms which tightly follow the mechanochemical transition process of kinesin motors. The movement of the unbound head from release to reattachment is described by a three-dimensional Langevin equation, solved numerically by the Euler scheme for stochastic differential equations. The chemical reaction process is simulated by a Monte Carlo method. We performed detailed simulations of the walking of the wild type and its mutants with extended neck-linkers and, by adjusting unknown parameters, obtained results consistent with the experimental results of [28]. We discuss different approaches to estimate the tension in the neck-linkers by using models from polymer science. We explore the binding mechanism by devising and testing different binding probability formulas for the tethered head. In our analysis of the processivity of kinesin, we also clarified the role of the front-gated-head and the reargated-head hypotheses in the regulation of the processivity. Our conclusion is that both of them must be included to produce qualitatively accurate simulations. 2 Regulation of tension in the neck-linker 2.1 Processivity of kinesin walking In the experiments reported in [28], the mutants with longer neck-linkers walk over 100 steps as does the wild type. The mutants having long and (or) soft neck-linkers have small tensions in their neck-linkers. If tension is the only factor regulating the processivity, then small tensions in the mutants should lead to the loss of processivity. Therefore, the processivity of the mutants indicates there are other mechanisms involved in coordinating the chemical states. We will distinguish the two heads by the labels H1 and H2. In Fig-

6 234 P. W. BATES AND Z. JIA FIGURE 2: The figure on the left shows a chemomechanical cycle of kinesin. The letters represent the nucleotide states of a kinesin catalytic core; E represents the empty state, T is for the ATP-bound state, D is for the ADP-bound state, and DP is for the intermediate state after the ATP molecule is hydrolyzed. The dark solid oval represents H2 and the light solid oval represents H1. The time intervals defined for H1 and H2 in Section 2.1 are illustrated here. The right figure is an illustration of the available binding sites for wild type kinesin. The three vertical stripes represent three protofilaments of the microtubule. We assume that kinesin can only bind to the sites on these three neighboring protofilaments. The oval with X inside denotes the bound head, i.e., head H2. The empty oval in the figure on the right is H1, the light head in the left panel. The five forward binding sites for H1 are represented by squares. The number of possible binding sites for the mutants will increase depending on the lengths of their neck-linkers. Notice that the binding sites are arranged to reflect the helical structure of the microtubule. ure 2, H1 is represented by the light head and H2 is represented by the dark head. We now examine how the chemical states of the two heads change during one step in the walking process. Starting from the moment, denoted by t H1 0 = t H2 0 = 0, when kinesin is in the two-headbound state with the trailing head in the ADP-bound state and the leading head in the empty state, we will decompose the time period for a forward step into several subdivisions according to the temporal order as follows:

7 TENSION OF NECK-LINKERS OF KINESIN 235 T H1 = T dmt + T diffusion + T dadp T H2 = T AT P + T AT P hydro + T dpi T dmt = t H1 dmt th1 0, where t H1 dmt is the moment when head H1 unbinds from the microtubule. If H1 temporarily binds to a backward binding site in this chemomechanical cycle, then T dmt also includes the time taken for H1 to unbind from the site again. T diffusion = t H1 fbinding th1 dmt, where t H1 fbinding is the moment when H1 binds to a forward binding site. T dadp = t H1 dadp th1 fbinding, where th1 dadp is the moment when the ADP is released from H1, which is believed to take place after binding in the forward position. Note that the torsion on H1 created by the tension in the neck-linker distinguishes between H1 in its forward or trailing bound state. This torsion may also be partly due to the binding orientation with respect to the microtubule [13]. The difference in orientation may be due to one of the changes in configuration due to zipping, ADP release, ATP capture, or hydrolyzation. T AT P = t H2 AT P th2 0, where t H2 AT P is the moment when an ATP molecule is captured by H2. We assume that zipping happens quickly after ATP capture but its rate also depends on tension. T AT P hydro = t H2 ADP P th2 AT P, th2 ADP P is the moment when the ATP is hydrolyzed into ADP P. Finally, T dpi = t H2 ADP th2 ADP.P, where t H2 ADP is the moment when H2 releases the phosphate after hydrolysis. If T H1 > T H2, then both heads will be in the ADP-bound state simultaneously, which has the weakest binding affinity to the microtubule, and so the kinesin is likely to fall off. Therefore, processivity requires T H1 T H2. Given the inequality T H1 T H2, we next determine how T H1 and T H2 change with the length and the tension of the neck-linkers. The front-gated-head and rear-gated-head hypotheses support the following conclusions, respectively: T AT P decreases as the tension decreases. T dmt increases as the tension decreases. On the other hand, it is obvious that T diffusion increases as the neck-linker length increases. But also, as the length of the neck-linker increases, the average tension in the neck-linker decreases. Therefore, we have T H1 increases and T H2 decreases as the tension decreases and the neck-linker length increases.

8 236 P. W. BATES AND Z. JIA These changes with respect to the tension and the neck-linker length may break the inequality T H1 T H2 and further induce the loss of processivity. To restore the balance, we could either decrease T H1 or increase T H2. To decrease T H1, T dadp ought to decrease as the tension is reduced. This conclusion so far has not been supported by experimental results. To increase T H2, we can increase either T AT P hydro or T dpi or both. Under the condition of saturated ATP, the rate-limiting step in the ATP hydrolysis process turns out to be the P i release. Therefore it is reasonable to postulate that T dpi is regulated by tensions. This is the other part in the updated rear-gated-head model [24]: Increased tension in the neck-linker enhances the release of P i after the ATP molecule is hydrolyzed. T dpi depends on tensions and T dpi increases as tension decreases. Although the front- and rear-gated hypotheses are not mutually exclusive, the front-gated-head model has obtained more support, especially because it is more consistent with new data [8]. Here our analysis strongly suggests that the rear-head-gated mechanism is also required to guarantee the processivity of kinesin in the manner observed. 2.2 Biochemical reaction cycle of kinesin The biochemical reaction pathway of kinesin can be described as follows. E represents kinesin and M is the microtubule. M E means that the kinesin is bound to the microtubule. (1) M E + ATP M E ATP M E ATP M E ADP P M E ADP + P i M E + ADP Here an ATP binding process is divided into two steps. First an ATP molecule arrives and binds to the catalytic core weakly and is easy to dissociate. If this weak binding induces a conformational change of the catalytic core, denoted by E, then the ATP molecule is trapped into a tight binding state. After the ATP is trapped, it will go through the hydrolysis process. Theoretically every biochemical reaction is reversible. Because some reverse reaction rates of the hydrolysis process are very small, we ignore them and use the following biochemical reaction process for our model. (2) M E + ATP k+ AT P M E ATP kzip M E ATP k AT P hydro k AT P M E ADP P k dp i k M E ADP + P dadp i M E + ADP

9 TENSION OF NECK-LINKERS OF KINESIN 237 Notation Value Remark kdmt 0 60s 1 Reaction rate k + AT P 3µM 1 s 1 Reaction rate k AT P 150s 1 Reaction rate kzip 0 700s 1 Reaction rate k AT P hydro 100s 1 Reaction rate kdp 0 i 120s 1 Reaction rate k dadp 300s 1 Reaction rate D nm 2 /s Diffusion coefficient dt s Step size of time TABLE 1: The parameter values used in the simulation. The neck-linker zipping takes place in the step where the ATP molecule is trapped, changing from the weak binding state to the strong. The reaction rates used in the simulation are taken from the experimental results in [6, 26]. Considering the regulation of the reaction rates by tension [10, 16], we adjust the reaction rates according to k dmt = kdmt 0 ef δ1 c /KBT, k zip = kzip 0 e F δ3 c /KBT, k dpi = kdp 0 i e F δ2 c /KBT, where the force F is computed as a scalar by (5) and δc i, i = 1, 2, 3, are the characteristic distances along the chemical reaction coordinates. In the simulation δc 1 = 0.7 nm, δc 2 = 2.0 nm, and δc 3 = 1.0 nm. 2.3 Tension estimate of the neck-linkers The neck-linker is considered as a worm-like chain (WLC) [10, 16] considering both the elastic and entropic effects of the polymer molecule in the force-extension formula. The force exerted at the end of the WLC is given by (3) F W LC (x) = K BT l p ( 1 4(1 x l c ) x l c where x is the extension of the polymer, here computed as the end-toend distance, the total length of the polymer is denoted by l c, K B is Boltzmann s constant, and T is the absolute temperature. The persistence length of the polymer is denoted by l p. The persistence length is related to the material property and the shape of the cross section of the polymer and describes how rigid the polymer is. The persistence lengths ),

10 238 P. W. BATES AND Z. JIA of the native neck-linker, the proline insertion and the GS insertion in the experiment are l wt p = 1.4 nm, l proline p = 4.4 nm, and l GS p = 0.8 nm, respectively [28]. When there is no external force applied to a worm-like chain, its mean square end-to-end length is given by [10, 16] (4) R 2 = 2l 2 p ( e lc/lp 1 + l ) c. l p In the supplemental documents of [28], the authors used the length of each amino acid to be 0.38 nm and the length of a coil of the polyproline helix to be 0.31 nm in their computation but they did not include the lengths of the three amino acids, KKG, in their length estimate. Here we include KKG in the total length computation. Therefore the l c values in Table 2 are neither the same as the values in Figure 1 nor the same as those in the supplemental documents of [28]. In the supplemental material of [28], there is an estimate of the tension in the neck-linkers when the kinesin is in its two-head-bound state. They used l c = 11.4 nm, l p = 1.4 nm, l n the natural length of the polymer as 2.05 nm, x = 8 l n = = nm and K B T = 4.1 pn.nm in (3), and found 3.9 pn as the tension in two head bound state with the neck-linker undocked, i.e., unzipped. We made several attempts to obtain the value of 2.05 nm for l n. A mean square endto-end formula for a freely-jointed (rather than worm-like) chain taking into account the 15 native amino acids, the seven GS pairs, and ignoring the KKG inserts, gives us the best result nl 2 a = ( ) = Here, l a = 0.38 nm is the length of an amino acid. The same procedure is followed to estimate the tension when the neck-linker is zipped. Molecular dynamics simulations [11] were used to estimate the internal tension in the neck-linker and found it to be about pn when both heads are bound, with the neck-linker of the trailing head zipped. Calculation using (3) shows that the tension is 6.9 pn when the motor is in its two-head-bound state with the trailing neck-linker zipped. In that calculation, since zipping reduces the free neck-linker length by 2nm, x = 8 2 = 6 nm and l c = = 9.4 nm. We are assuming there are 15 amino acids comprising each neck-linker (there is some debate about this) with each amino acid having length 0.38 nm. We take K B T = 4.2 pn nm and l p = 1.4 nm. The experiments in [28] suggest that kinesin will easily dissociate from the microtubule when the applied external force is larger than around 10 pn and so our calculation gives a magnitude that allows forward binding to the microtubule.

11 TENSION OF NECK-LINKERS OF KINESIN 239 Material types l c (nm) l p (nm) l n (nm) Wild type P P P P P P P GS TABLE 2: The total lengths, effective persistence lengths and natural lengths of the neck-linkers. The difference between the theoretical estimate in [11] and the experimental measurement in [28] might be due to the lack of data for the structures of the two heads in the microtubule bound state so that the authors of [11] had to modify the data of the monomers to create a structure for a both heads microtubule-bound state from which their simulation could be carried out. From (3) we see that F W LC (l c ) =. This is consistent with the entropic nature of the WLC model because there is only one configuration when the extension equals the total length, corresponding to the minimal entropy state. Thus, for worm-like chains, the entropic effect becomes dominant when the extension approaches the total length. To avoid computational problems and in consideration of the fact that kinesin detaches at a force of around 10 pn, we modify (3) when the extension is large. A reasonable compromise is to cut off the force at 25 pn. We also modified (3) by introducing the natural length l n so that no compressive forces may be influential. The resulting modified force is (5) 0, 0 x l n ( ) K B T 1 F (x) = l p 4(1 x ln 2l c l n ) x l n, l n x l 25 2l c l n 25, x l 25

12 240 P. W. BATES AND Z. JIA where l 25 is the length such that F (l 25 ) = 25 and l p is the effective persistence length of the neck-linker of a mutant. This is computed by (6) l p = l wt p where l mutant p and l mutant c l wt c l c + l mutant p l mutant c l c, are the persistence length and the actual lengths of the polyproline inserts, respectively. If the two neck-linkers of a mutant are viewed as one worm-like chain, then the natural length is calculated as the square root of R 2 : (7) l n = R 2 = 2l 2 p ( e 2lc/lp 1 + 2l ) c, l p In Table 2, the total length of one neck-linker is given for the wild type and for each of the mutants. The effective persistence lengths and natural lengths are also given. In our simulation, we model the motion of the tethered head in threedimensional space. Formula (5) only gives us the magnitude of the force. We use X H1 to denote the position vector of the tethered head and X H2 to denote the position vector of the bound head. The extension x in (5) is equal to Euclidean norm of the vector X H2 X H1, i.e., x = X H2 X H1. The orientation is ˆn = vector acting on the tethered head is (8) F(X H1 ) = F (x)ˆn XH2 XH1 X H2 X H1. Therefore the force 3 Algorithm In this section, ζ is used to denote a random number with uniform distribution between 0 and 1. The movement of the tethered head is modeled by a Langevin equation. Let X H1 be the position vector of the tethered head relative to the center of the bound head. Newton s second law gives the equation for the motion of X H1 : (9) mẍh1 = γẋh1 + F(X H1 ) + 2K B T γ dw dt, where m is the mass of the head and γ is the drag coefficient. Though not a true derivative, dw /dt represents white noise and W (t) is Brownian motion. The force F(X H1 ) is the sum of the entropic force and any external force acting on the head. In our simulations, we have no

13 TENSION OF NECK-LINKERS OF KINESIN 241 external force and the entropic force is computed by (5) (8). The order of magnitude of the mass is and the inertial time scale, defined as m/γ is of order s, which is so small that the inertial term can be ignored. The above equation then becomes an overdamped Langevin equation (10) γẋh1 = F(X H1 ) + 2K B T γ dw dt. Solving this stochastic differential equation by using the Euler scheme, we have the following iteration formula for (10) from t n to t n+1 = t n +dt, (11) X n+1 H1 = X n H1 + 1 γ F(Xn H1 ) dt + 2D dt (W n+1 W n ) where we have used the Einstein relation D = K B T /γ. We consider a binding probability P binding dependent on the distance between the head and the binding site on the microtubule and the orientation of the head with respect to the axis of microtubule, i.e., (12) P binding = P d P o. When the distance between the tethered head and some binding site is less than a given threshold, r cutoff = 2.5, a binding event may happen if ζ < P binding. The following was used in our simulations to mimic the binding through the electrostatic attraction. 1, d 0.1 (13) P d = 1 c 1 e c 2 K B T d, 0.1 d 2.5, and P o = e c3 sin(α/2) l, where α = θ + c c s 4 l p ω and ω is a random variable with a normal distribution of mean zero and standard deviation one. Here d is the distance between the tethered head and the binding site and θ is the angle between the positive axis of the microtubule and the vector from the bound head to the available binding site near the tethered head. s is defined as the distance between the tethered head and the bound head. We took c 3 = 8 in the simulation. c 4 = 0.4 l p π/l c is used for the wild type and 0P to 13P and c 4 = 0.6 l p π/l c is used for 19P, 26P, and 14GS. c 1 and c 2 are chosen to make P d (0.1 + ) be close to 1 (within 10 4 ). We used c 1 = 1.3 and c 2 = r cutoff K B T ln(c 1 ).

14 242 P. W. BATES AND Z. JIA We also tested the following binding probability formula (results not shown). (14) P binding = { e βd2 cosθ, e βd2 (cosθ) 2, wild type mutants. We used β = 1.5 in the simulation. The idea to multiply by (cos θ) 2 or cos θ is to make the orientation a more significant factor in binding and including a factor to represent the greater flexibility of the mutants. The whole simulation process can be described as follows. Head1 dissociation Start from a two-head-bound state on the microtubule, with the leading head (H2, the head closer to the positive end of the microtubule) in its nucleotide free state and the trailing head (H1) in its ADP-bound state (see Figure 2). Test for the random dissociation of H1 from the microtubule and ATP binding in H2. The neck-linker zipping induced by ATP binding in H2 may or may not imply the detachment of H1 from the microtubule depending on the length and extension of the necklinkers. If H1 remains bound we advance the time by one step and repeat. Tethered diffusion. H1 experiences a 3D diffusion process during which it might bind to an available binding site, including those to the rear. If it binds to a rearward site, this binding is weak and H1 will detach again. If H1 diffuses to a forward binding position and binds to it, then it releases ADP rapidly, binds to the microtubule strongly and completes a step, with the hydrolysis of ATP in H2 and the release of a phosphate, as described below. The chemical state of H2 is stochastically updated. If it has not bound an ATP, then we continue to test for random ATP binding and neck-linker zipping in H2 as we advance time. The random test of ATP binding is as follows. If ζ < k + AT P [AT P ]dt, then an ATP molecule binds to the catalytic core of H2, where [AT P ] represents the concentration of ATP. The test for zipping is then performed similarly. If ATP has bound and been trapped, then we test for random ATP hydrolysis and Pi release until H2 arrives at the ADPbound state.

15 TENSION OF NECK-LINKERS OF KINESIN 243 When H2 is in its ADP state after ATP hydrolysis, kinesin may fall off or just release the bound ADP molecule. If H1 happens to be in a rearward bound state, then H2 will either release the ADP molecule and not unbind from the microtubule or stay in its ADPbound state at least for the next update. If H1 is in the tethered diffusion state, H2 will take one of three choices: Release the ADP, stay in its ADP-bound state attached to the microtubule for the next time step, or possibly unbind from the microtubule. The latter choice depends to some extent on the relative position of H1 to H2. Let the positive direction of y in (x, y, z) coordinates point to the positive end of the microtubule, (see Figure 2) and let X y H1 (t n) and X y H2 (t n) be the y-coordinate of H1 and H2 at time t n. If X y H1 (t n) X y H2 (t n) + C gating l c, then ADP release in H2 is considered, i.e., that random event is tested. We use C gating = 0.2 in the simulation. Otherwise unbinding of H2 will compete with ADP release in H2. If unbinding of H2 takes place, then the kinesin falls off. If the ADP is released in H2, then a futile ATP hydrolysis cycle is recorded. Forward binding. P binding is tested against a random number in [0, 1] and if it is greater, then H1 binds to the binding site immediately. After H1 binds to a forward binding site, it releases the ADP quickly and then is in a strong binding state. H2 continues the process from step B and eventually arrives in the ADP state. Next we stipulate that ATP binding does not occur in H1 until ATP hydrolysis is complete in H2 and H2 is in a weak binding state. At this point, H1 and H2 have exchanged their trailing and leading roles and are ready for a new step. To compute the average speed and run length of kinesin and its mutants, we run 5000 chemomechanical cycles for each of them. All the continuous paths (i.e., those without kinesin falling off the microtubule) are selected from those 5000 cycles. Assume that there are N path of them. The run number, N rn, of a path is defined as the total number of steps in this continuous path and the run length, L run, is defined as the total distance traveled (note that steps may have different sizes, especially those of mutants). Three different approaches may reasonably be used to compute the average speed. The first formula is given by (15) V 1 = The total distance traveled in N path paths The total time spent in N path paths.

16 244 P. W. BATES AND Z. JIA To compute the second, for each path, we find the corresponding run number and compute the median value of all the run numbers, N median rn. (16) (17) V i = The length of the ith path with N rn Nrn median The corresponding time for the i th path V 2 = V i, over all the paths with N rn N median rn., To compare the run lengths for the simulation and the experiment, we follow the method used in supplemental material in [28] where the mean run length is defined as (18) L mean run = Σ Lrun 500L run The total number of the paths with L rn 500. Then a third way to compute the speed is (19) V 3 = The length of the i th path with L run 500 The corresponding time for the i th. path The average speeds calculated from these three different definitions turn out to be very similar (we do not show the results here) and so we are confident that we may compare our results with those reported from experiments, even if the approaches used by some experimentalists are not stated explicitly. 4 Simulation results First of all, we tested our algorithm by showing the Michaelis-Menton chemical kinetics relation between the speed of the kinesin and the ATP concentration is satisfied (we do not show the results here). Mutants with extended neck-linkers have smaller tension when taking an 8 nm step and so they tend to take more time to detach from the microtubule, according to the rear-gated head hypothesis. Also the long neck-linker mutants can reach more backward binding sites so that they may have more backward temporary binding. More backward binding then requires more time to dissociate from the microtubule. Our results support this scenario, with Figure 3 showing clearly the average dwelling time to be an increasing function of the length of the neck-linker. Here the dwelling time is defined as the average time when H1 is bound to the microtubule in the course of one chemomechanical cycle. Our simulation of the stepping process of kinesin and its mutants with extended neck-linkers reproduces the experimental results qualitatively.

17 TENSION OF NECK-LINKERS OF KINESIN 245 FIGURE 3: The average dwelling time of the wild type, 0P-26P and 14GS mutants. There are two notable differences between our simulation results and the experimental outcomes. First, the coupling ratios of mutants are not consistently decreasing as speeds are (see the right panel of Figure 4). In the simulation, 14GS has a larger coupling ratio than that of 26P but its speed is smaller than that of 26P. This may be due to the large dwelling time and small step size for 14GS. The second difference is that the magnitude of the speed from the simulations is larger than it should be and the speeds of the mutants in our simulations do not decrease so dramatically with increased neck-linker length or with flexibility as in the experiments. On the other hand our results clearly show the reduced speed for the mutants with the longer neck-linkers. The discrepancy is very probably due to the choice of parameter values we used, shown in Table 1. As we pointed out, these parameter values are not given in [28] but are instead from the review paper [6, 26]. Experimental results indicate different speeds for kinesins from different animals. The wild type kinesin used in [28] was expressed from a truncated human kinesin gene, which is also used as a template for mutagenesis. Its speed shown in [28] is about 400 nm/sec at a saturated ATP concentration of 1 mm (we also used this value in our simulation).

18 246 P. W. BATES AND Z. JIA FIGURE 4: The left column is the experimental outcome from Figure 1 in [28]. The right column shows the simulation results corresponding to the left column. On the horizontal axes of the figures in the right column, 1 represents the wild type. The mutants 0P to 26P are represented by 2 to 8, respectively, while 9 denotes the 14GS mutant. However, the speed of kinesin coming from bovine brain at saturated ATP concentration is reported to be about 700 nm/sec in [10], which is closer to the value we obtain. The authors in [6, 26] did not identify the cell types for the kinesin parameter values listed in their articles. Trajectory samples from our simulations are shown in Figure 5(b). Each sample represents the trajectory of a ten-step run of one head of kinesin. Because samples are taken randomly, the slopes of these

19 TENSION OF NECK-LINKERS OF KINESIN 247 FIGURE 5: (a) Trajectory samples of wt, 6P, 13P, 19P, 26P, and 14GS from the experimental results. These trajectories have more or less the same slope because they are obtained using increasing ATP concentrations. (b) Simulation results for the trajectories of wt, 6P, 13P, 19P, 26P, and 14GS. A fixed ATP concentration of 1 mm is used in all these simulations. trajectories do not necessarily represent the average speeds of each type of kinesin, which is shown in the simulation result in Figure 4 but they

20 248 P. W. BATES AND Z. JIA do show the qualitative behavior of the motion. From the simulation results in Figure 5 it is easy to see that the speed of the wild type is greater than that of the mutants. The histograms of step sizes of wild type, 6P, 13P, 19P, 26P, and 14GS motors in the experiments of [28] and the corresponding simulation results are shown in the Figure 6. FIGURE 6: The left column shows histograms of step sizes of the wild type and mutant kinesin from Figure 3 in [28]. The right column shows the histograms of step sizes from the simulation results for the wild type, 6P, 13, 19P, 26P, and 14GS mutants. The simulation results are consistent with the experimental results with the exception that no backward steps are produced in the simulation.

21 TENSION OF NECK-LINKERS OF KINESIN Discussion In the front-gated head hypothesis, it is suggested that ATP does not bind to the empty front head until the rear head dissociates from the microtubule [19]. If we assume that the movement of the tethered head is purely diffusional, then this front-gated head assumption may lead to a forward step of kinesin without consuming an ATP molecule. Indeed, a kinesin head can be seen as a sphere of radius around 3 nm. According to Stokes law [10], the diffusion constant of such a sphere is around D = nm 2 /s. Wild type kinesin tends to walk on the axis of a protofilament and so this can be modeled as onedimensional diffusion process. For a 1-D diffusion, if the particle is not subject to external forces, the first passage time [10] is equal to t fp = d 2 step /D, which gives the average time for the tethered head to diffuse through the distance d step and bind to a front binding site, completing one step. One may calculate the probability for the motor to bind an ATP molecule during the period t fp. In consideration of the diffusion constant, the time step size used in the simulation is dt = 10 8 second. Correspondingly, the first passage time t fp needs N step = [t fp /dt] = 1143 time steps. On the other hand, on average, the probability of an ATP molecule to bind is q = dt[atp]k + AT P = = when [ATP] is 1000µM and k + AT P takes the value in Table 1. Given N step = 1143, the probability for the kinesin to bind one ATP molecule is 1 (1 q) Nstep+1 = 1 ( ) 1144 = This result indicates that the diffusion is so fast that the kinesin should have finished one forward step even without an ATP binding, not to mention neck-linker docking. Block in [3] pointed out that it takes less than 100µs for a kinesin to finish a diffusive search process. A time of 100µs corresponds to iteration steps. Given N step = 10000, the probability for the kinesin to bind one ATP molecule is 1 (1 q) = 1 ( ) = 0.26, which is much larger than but still not large enough. Therefore this simple calculation suggests it is unlikely for the empty front bound head not to bind ATP molecules until the rear head unbinds from the microtubule and starts a diffusive process. To resolve this dilemma, we may assume that the forward bound head of kinesin begins ATP binding before rear head detachment. This point in time may reasonably be assumed to be at the moment when the P i is released and the rear head is in its weak binding state. This assumption is compatible with the results in [15]. There kinesin spends most of the time of a cycle in the two-head-bound state and quickly moves to the next front binding position in conjunction with ATP binding, at least when there is a high ATP concentration. Another scenario is that the rear head remains parked somewhere after it unbinds from the micro-

22 250 P. W. BATES AND Z. JIA tubule and then rapidly swings to a forward site with a force provided by the neck-linker docking that is induced by ATP binding. This is the polymer gating mechanism in [1, 7] where the tethered head parks in front of the microtubule bound head and does not itself bind to the tubulin until an ATP molecule is captured by the bound head. On the other hand, the data in [15] suggest that the tethered head parks behind the bound head instead of in front of it. More experiments may be needed to elucidate the details of this polymer gating. Another detail of the mechanism, not considered here but possibly significant, is a powered swing process induced by neck-linker zipping. The experiments [15, 28] suggest that kinesin spends most of the time in the two-head-bound state and it quickly swings to the next binding site when the neck linker is zipped, which is induced by the ATP binding. To model this process, we suggest using an overdamped beam equation to describe this swing movement, where the pair of neck-linkers is considered as an elastic beam. The potential energy is stored when the kinesin is in the two-head-bound and zipped state. We postulate that the neck-linkers are somehow twisted, and this is where the stored potential energy may reside. When the tethered head unbinds from the microtubule, the released potential, coming from the release of the strain, immediately changes the orientation of the tethered head such that it cannot easily bind back to the initial site and impels the tethered head to the next front binding position. This aspect of the walking mechanism may be considered in future work. Acknowledgements We gratefully acknowledge support from the following grants: NSF DMS , NSF UBM , NIH R01 GM , DARPA HR REFERENCES 1. M. C. Alonso, D. R. Drummond, S. Kain, J. Hoeng, L. Amos and R. A. Cross, An ATP gate controls tubulin binding by the tethered head of kinesin-1, Science 316 (2007), C. L. Asbury, A. N. Fehr and S. M. Block, Kinesin moves by an asymmetric hand-over-hand mechanism, Science 302 (2003), S. M. Block, Kinesin motor mechanics: binding, stepping, tracking, gating, and limping, Biophys. J. 92 (2007), N. J. Carter and R. A. Cross, Mechanics of the kinesin step, Nature 435 (2005),

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24 252 P. W. BATES AND Z. JIA 28. A. Yildiz, M. Tomishige, A. Gennerich and R. D. Vale, Intramolecular strain coordinates kinesin stepping behavior along microtubules, Cell 134 (2008), Corresponding author: P. W. Bates Department of Mathematics, Michigan State University, East Lansing, MI address: Department of Physics, University of California at Irvine, Irvine, CA address: