INTRODUCTION

TOP ABSTRACT INTRODUCTION THE MODEL NUMERICAL RESULTS DISCUSSION APPENDIX REFERENCES

Molecular motors constitute a class of proteins responsible for the many transport processes within eukaryotic cells, and in the organization of the mitotic spindle (see Alberts et al., 1994 or Lodish et al., 1995 for a general introduction). Motors can be characterized as consisting of three domains: the "head" or "motor" domain, in which force is produced, the "tail" which attaches to a load, and the "body," which links the head to the tail. These motors work by moving their motor domains along relatively rigid polymers with a load attached to their tails. They are commonly divided into three families, myosins, dyneins, and kinesins (see Hirokawa, 1998 for a family tree). Myosins are associated with actin filaments, whereas kinesins and dyneins are associated with microtubules (MTs). These motor proteins have been the subject of many ingenious experiments, as attempts to characterize them have grown more ambitious. The discovery of the kinesin family in the late 1980s (Hirokawa et al., 1989; Scholey et al., 1989), along with advances in imaging technology, opened the door to a set of exciting experiments. In gliding motility assays (Howard et al., 1989; Hancock and Howard, 1998), a microscope slide is coated with kinesin, a microtubule placed on top of the slide, and the motion of the center of mass of the MT is tracked as a function of time. In the optical-tweezer assay (Block et al., 1990), a single motor is tethered to a latex bead, which is then held in an optical potential well. Attempts by the motor to drag the bead out of the well yield important quantitative information about the motor's force. These motor assays allowed the recording of the position (Howard et al., 1989; Block et al., 1990; Svoboda et al., 1993), velocity (Block et al., 1990), and force (Finer et al., 1994) applied by a single motor, all with unprecedented sensitivity.

As a more complete characterization of kinesin motors emerged, a number of questions became apparent. First, how do these motors move? Some kind of walking model is believed to be the correct way to think about this (Peskin and Oster, 1995; Derényi and Vicsek, 1996; Vicsek, 1997). Second, the "fuel" used by these motors is well known, but the question of the stochiometry between fuel consumption and steps walked has recently been answered by Hua et al., (1997), Schnitzer and Block (1997), and Coy et al. (1999), who have shown that consumption of one ATP molecule results in kinesin taking exactly one 8-nm step. Third, in gliding assays, the speed of microtubule movement is found to be independent of both the length of the microtubule, and the density of kinesin adsorbed onto the substrate (Howard et al., 1989; Hancock and Howard, 1998). It is not fully understood why this should be so. And last, a graph of speed versus kinesin density tells us very little about the behavior of a single motorhow long does it remain attached to the microtubule, how much force does it exert, how is it correlated with the behavior of the others, etc.

In this work, we focus on the last two of these questions. For this purpose, we develop a stochastic model for the dynamics of a microtubule in gliding assay experiments, which includes the viscous drag acting on the moving fiber and the interaction with the kinesins. For those motors, we construct a simple mechanical model extracting the involved constants based on experimental data. We show how our model quantitatively reproduces the results of the gliding-assay experiments of Howard et al. (1989) and Hancock and Howard (1998). Having established the validity of the model, we then examine various aspects of motor behavior to try to understand these observations. We use a combination of detailed numerical calculations with analytic analysis in different limits.

	THE MODEL

TOP ABSTRACT INTRODUCTION THE MODEL NUMERICAL RESULTS DISCUSSION APPENDIX REFERENCES

We describe the dynamics of a MT in terms of the location of its center of mass  = xî + y (where î and are units vectors along the x and y axes), and the angle  between the horizontal x-axis and the unit vector pointing along the MT.

This model is two-dimensional, because it has been shown (Hunt and Howard, 1993) that, in motility assays at least, the vertical distance between the head and tail of a kinesin attached to a microtubule is only about 20% of its total lengthin other words, the motors mostly lie in a plane parallel to the microscope slide. The motion of the microtubule is overdamped because it involves low Reynolds numbers (Hunt and Howard, 1993). This dimensionless number, defined as  vL/, where v is the relative velocity between the object and the fluid, L is its dimension,  and  are, respectively, the density and viscosity of the fluid, indicating the relative importance of inertial to viscous effects. For a fish swimming in water,   100, whereas, for a microtubule moving in an aqueous solution,   8 × 10⁶, clearly indicating that, although the fish experiences inertia, the MT does not. Its motion is Aristotelian rather than Newtonian. A nice introduction to this concept is given in Purcell (1977) and Berg (1983).

Therefore, neglecting inertial effects, the translational bidimensional Brownian motion is described by the Langevin equation

(1)

whereas, for the rotational Brownian motion, we have

(2)

where the dot means the time derivative, is the translational drag force, _r is the rotational friction constant, is the applied external force by the molecular motors on the MT, and  is the associated torque. The translational and rotational Gaussian fluctuating Langevin forces are respectively denoted by _t(t) and _r(t).

In what follows, we assume that the motors are located randomly in the plane, with the position of the ith motors tail fixed at _i and the position of the head at _i. (See Fig. 1 for schematic.) We model each motor as a simple spring, parameterized by a spring constant k_s and equilibrium length L₀. Each motor will exert a force on the MT equal to (see Appendix for further justification for our model)

(3)

The components of the force exerted by the spring parallel and perpendicular to the MT axis along the unitary vector are

(4)

whereas the magnitude of the torque is given by

(5)

where _i is the angle between the rod and the spring.

View larger version (14K): [in this window] [in a new window]   FIGURE 1   A schematic of a motor walking on a microtubule, indicating the position of the center of mass of the microtubule , the (fixed) tail of the ith motor at i and the moving head at i. Kinesin is a plus-end-directed motor, as indicated by the arrow marked v0. Note that only a small section of the microtubule is shown, because it is so much larger than the motor.

Combining Eqs. 1-5, we have (see Appendix for more details)

(6)

where '_i denotes a sum only over those motors that are attached to the MT, and , are unitary vectors along the x and y axes, respectively.

In Table 1, we show the parameter values that we use to model the behavior of the motors, as determined from experiment. A typical length L for an MT in the mitotic spindle is 10 microns (Gliksman et al., 1993). In the gliding-assay experiments of Howard et al., the length of the MT is 2.2 ± 1.4 µm (Howard et al., 1989) using bovine brain kinesin and 2.05 ± 0.92 µm (Hancock and Howard, 1998) using recombinant Drosophila kinesins. The spring constant k_s is taken from experimental work (Coppin et al., 1995). The value of the equilibrium length L₀ of the spring is based on the facts that the length of kinesin is about 80 nm, and the extension of the spring, which is sufficient to stall the motor, is about 25 nm. A precise value for the unloaded motor velocity v₀ is not known (see Howard et al., 1989; Svoboda et al., 1993), but it is in the range 0.5-1.0 µm s¹. The parameter w describes the minimal proximity required between a motor and the microtubule for capture to occur. No experimental values exist, but clearly it should be no larger than the equilibrium length of the motor itself (~80 nm).

	NUMERICAL RESULTS

TOP ABSTRACT INTRODUCTION THE MODEL NUMERICAL RESULTS DISCUSSION APPENDIX REFERENCES

In Fig. 2, we show a gliding assay picture, obtained from our numerical results for the MT motion at different times for low and high kinesin densities. There we see that, at low densities, the MT can rotate as it moves, whereas, at high densities, the rotation is quenched.

View larger version (24K): [in this window] [in a new window]   FIGURE 2   Illustration of results from simulation of microtubule gliding assays at (A) low kinesin density (kin = 5 µm2) and (B) high kinesin density (kin = 100 µm2). At sufficiently high densities, the Brownian angular fluctuations are damped, and the motion is in a straight line. The MT is represented by the long rod (its color changes from dark to light, to represent the passage of time). The locations of the motors are indicated by the small spheres.

We also looked at the shape of the trajectories of the center of mass of the MT, as a function of kinesin density, _kin. Figure 3 shows typical trajectories as a function of kinesin density, using the parameters in Table 1. For some densities between _kin = 1 µm² and _kin = 5 µm², there is clearly a crossover from Brownian motion to a more directed kind of motion, in which the angular fluctuations are damped out. This critical density has been predicted (Duke et al., 1995) to be **_kin ~ 0.05 µm², independent of microtubule length. However, their calculation assumes a finite persistence length for the microtubules of ~5 nm, whereas our simulations assume the rods are completely rigid (infinite persistence length). Therefore, the critical density in our analysis should be larger.

View larger version (19K): [in this window] [in a new window]   FIGURE 3   Comparison of typical trajectories for w = 40 nm, as the kinesin density is varied. Initial conditions are unchanged in each run. Panels a-f correspond to densities of kin = 0.1, 1.0, 5.0, 20.0, 60.0, 100.0 µm2, respectively. Note that not all trajectories represent equal lapses in time.

We have performed about 300 realizations of the gliding-assay simulations, each with a different initial random arrangement of kinesins, and random initial position and orientation of the MT. For all realizations at a given density, we drew graphics like Figs. 3 and 4, ensuring that the track was straight, and then computing the average speed of the MT, as v(_kin)  (d(t)  d(0))/t, where d(t) is the distance of the center of mass displacement at time t from its initial position d(0). We also computed error bounds for each v(_kin). The results are shown in Fig. 5. Our principal finding, in agreement with experiment, is that the speed of the microtubule remains constant over approximately two orders of magnitude in the kinesin density, _kin (i.e., from _kin = 2 µm² to _kin = 200 µm²). For densities less than _kin = 1 µm², we find it difficult to gather data, because the motors are so few and far between that there is seldom any directed motion. At densities beyond _kin = 200 µm², the opposite is truethere are so many motors that the simulation time-step is driven down by orders of magnitude, causing the simulations to be impracticably slow.

View larger version (17K): [in this window] [in a new window]   FIGURE 4   Typical data for d(t), the distance moved by the MT center of mass, from its initial position, as a function of time t. The inset shows the path taken by the MT center of mass, as it moves across the (square) slide. The microtubule length is 10 µm, so, clearly, the microtubule has moved by almost its own length in this simulation. The speed of the microtubule is clearly constant for the length of the simulation. Here kin = 10 µm2. Inset shows the expected noisy nature of the same data.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   Comparison of the results obtained by simulation using the parameters of Table 1, with those experimentally obtained by Howard's group. Simulated data represent 257 individual MTs. In the experiments by Howard et al. (1989), the results came from measurements of drawings by hand on acetate-sheet overlays of taped video images, acquired at 33 ms intervals. At low kinesin density, the speed was determined as the rate at which the MT's trailing end approached the fixed point at which the kinesin molecule was located. Results were averaged over 233 individual MTs in two different experiments (hence different symbols), and error bars correspond to standard errors of the means. The experimental data is taken from Howard et al. (1989) that used bovine brain kinesin. These results are very similar to those obtained using two-headed recombinant Drosophila kinesins (Hancock and Howard, 1998). See text for a more detailed discussion of this figure.

A possible partial explanation for this: first, we consider low kinesin densities (_kin  5 µm²). With only one motor attached (on average) at a given time, it is impossible for the microtubule to move faster than the motor can walk. For medium kinesin densities (5 < _kin < 50 µm²), there may be only two or three motors attached, so there is a finite probability that they will cooperate with each other. At high kinesin densities (50 < _kin  200 µm²), there is a larger number of motors attached, and it is likely to have some averaging out, with some motors pushing and others pulling. It is interesting to examine the detailed dynamics of the initial attachment of motors to the microtubule. An example is shown in Fig. 6. At low densities, there is a brief "attachment" phase, as the motors attach themselves to the microtubule. The number of motors attached rapidly reaches a relatively constant value, and the microtubule begins to move. At higher kinesin densities, the attachment phase happens rapidly too, but motion does not begin immediately. There may be a waiting period, during which the microtubule undergoes no directed motion. It simply sits, buffeted by the Brownian noise. This is visible in Fig. 6, for t < 0.7 s. This kind of behavior can be explained by our theoretical model (see below).

View larger version (15K): [in this window] [in a new window]   FIGURE 6   Typical microtubule path at high kinesin density, from simulation, showing the attachment phase, followed by waiting, and finally directed motion. The weakness of the data shown in this graph is in the length of time simulated. The MT moves only a few tenths of a micron, whereas its own length is 10 microns. Movement on this scale would not be easily detectable in an experiment.

Analysis of the effective force

There are two questions here that are worth looking at: what is the force required to move an MT? and why is the speed such a weak function of kinesin density? Or, to use a well known analogy (Leibler and Huse, 1993), why should a boat with 8 rowers move no faster than one with 4?

A simple back-of-the-envelope calculation provides a qualitative answer to the first question. Let us take Eq. 6 and assume an MT aligned and moving only along the x-axis. Take the time average of both sides of Eq. 6 (which eliminates the noise term, by definition), and ask what is the time-averaged longitudinal force necessary to produce the observed velocity. The force is given by F = k_BT/D. Taking k_BT = 4.1 × 10²¹ J,  = 650 nm s¹ (see Fig. 5) and D = 10¹³ m² s¹, we find F ~ 0.027 pN. Analysis of the force exerted during our simulations shows a figure within a factor of two of this force (see Fig. 7). However, as we can see from a simple force-velocity curve (Fig. 8), the maximum force that a single motor can exert is larger than this by a factor of 100 and up to 5 pN. A more detailed analysis given in the Appendix leads to an answer that agrees quantitatively with the numerical calculations.

View larger version (17K): [in this window] [in a new window]   FIGURE 7   The time-averaged total force exerted by motors on a microtubule. Black circles indicate ensemble means, error bars indicate standard deviations of the ensemble ( = (1/N) i=1,N ). Negative values indicate that the force is directed toward the minus end of the microtubule, which is consistent with motors walking toward the plus end. The force necessary to propel a microtubule at the observed averaged speed of ~650 nm s1 is 0.027 pN.

View larger version (14K): [in this window] [in a new window]   FIGURE 8   A typical experimentally determined force-velocity curve for kinesin, taken from Svoboda et al. (1993). For low values of the applied force (the load), the velocity is near its maximum value. The velocity decreases as the load increases, until the stall force Fstall is reached and the velocity is zero. Under loads greater than the stall force, the tendency is for the motor to become detached from the microtubule, rather than to move backward.

We note that the mean force exerted at all densities is extremely small, much smaller than the maximum force exerted by a single motor. This distribution of forces is important to understand the kinesin density independence of the average MT velocity. Of course, the distribution of forces exerted at a given time varies as a function of density _kin. This variation is explicitly shown in Fig. 9, which gives a Gaussian distribution of forces exerted on the MT over the length of a numerical run. The force changes as a function of the longitudinal force exerted in the MT. Here we show results for several densities: _kin = 5, 10, 20, 30, 50, and 100 µm². We point out that the results for densities below 130 µm² have error bars due to the ensemble of different initial conditions, but for higher densities the calculations are very CPU intensive, and we only show the results for one initial condition. These results, however, follow qualitatively and even quantitatively the experimental results.

View larger version (25K): [in this window] [in a new window]   FIGURE 9   Distribution over time of the total force exerted on a MT. The labels in the panels in the figure indicate the kinesin densities of kin = 5, 10, 20, 30, 50, and 100 µm2, respectively. It illustrates that the mean force exerted is always a very small fraction of the stall force, though the maximum instantaneous force exerted increases with kinesin density, and may be several times Fstall.

Attachment lifetime analysis

We have also examined the distribution in times for which each motor remains attached to the microtubule, shown in Fig. 10. The times shown may include several detachment and reattachment processes. What matters is the total length of time a motor is attached, whether continuously or not. We find that, of all the motors with which a microtubule may have contact, most remain attached only briefly. This is illustrated in Fig. 11, which shows the integrated probability distribution of total attachment times. A small number of these motors remain attached for extended periods of time, forming the long tails of these distributions, and it seems reasonable to believe that these perform most of the work involved in moving the microtubule. What is clear from this figure is that, at higher density, more motors remain attached for longer periods of time: the tail is much longer at _kin = 10 µm² than at _kin = 5 µm².

View larger version (26K): [in this window] [in a new window]   FIGURE 10   Histogram showing the length of time for which each motor is attached to the microtubule. The horizontal axis gives a unique number to every motor that has ever attached to the microtubule. The vertical axis indicates the total length of time for which this motor is attached during the course of the simulation (it may detach and then re-attach). (a) the results for kin = 10 µm2; (b) those for kin = 5 µm2. It is clear that only a small fraction of the motors remain attached for a significant length of time, indicating that a small number of them do most of the work. The total length of time for the run was 10 s.

View larger version (15K): [in this window] [in a new window]   FIGURE 11   Integrated probability of attachment lifetimes. The abscissa P(ti < t) indicates the probability that any motor i will have a total attachment time less than the ordinate, t. The total length of time for the run was 10 s. (a) kin = 10 µm2; (b) kin = 5 µm2.

Sticky motors

It has been found in experiments (J. Howard, University of Washington, private communication), that not all motors are identical: there is a natural variation in speed among motors of a given type and, in addition, some motors may, for some reason, not be fully functional. We have examined the effects of such nonuniformity in motor speed, in two different ways.

First, we allowed a motor to be either fast (v_fast = 800 nm s¹) or slow (v_slow = 200 nm s¹). We call this the delta-function velocity distribution. The fraction of slow motors, q, lies between 0 and 1. For q = 0, all the motors are fast, and for q = 1, they are all slow. Using the methods described previously, we measured the speed of the microtubule for several different values of kinesin densities, _kin, as q was increased from 0 to 1. The results are shown in Fig. 12 A. From Fig. 12 A, two things are clear: the overall behavior of the microtubule speed, as a function of q, is nonlinear; and a small admixture (say, q  0.1) of slow motors has little effect on the overall speed of the microtubule. Likewise, a small admixture of fast motors (q  0.9) has little effect.

View larger version (11K): [in this window] [in a new window]   FIGURE 12   Effects of distribution in motor speeds on the MT velocity, at various kinesin densities. To compensate for the slight dependence of the MT velocity on kinesin density, all velocities at a given density have been normalized to the velocity for that density, with no distribution in speed. (A) Delta-function distribution p(v) = (1  q)(v  vfast) + q(v  vslow). vfast = 800 nm s1, and vslow = 200 nm s1. Data for three different kinesin densities are shown: , , and  represent data for kin = 10, 40, and 100 µm s1, respectively. In addition,  indicate the linear relationship v = 1  0.75q, which would be achieved if the MT speed varied linearly with q. (B) Gaussian distribution p(v) = (2)1/2 exp((v  v0)2/2). Mean motor speed, v0, is 800 nm s1, and the distribution width vel is normalized to this value. Data for three different kinesin densities are shown: , , and  represent data for kin = 10, 60, and 100 µm2, respectively.

We also looked at a more realistic scenario, in which the motor speeds are distributed according to a normal or Gaussian distribution. This is characterized by specifying a mean speed (v_mean) and a standard deviation from the mean (_vel). For the Gaussian distribution in motor speeds shown in Fig. 12 B, the principal effect of allowing the motors to assume speeds over a wide distribution seems to be a slight increase in the mean speed for very wide distributions of motor speeds. It seems that, in general, the speed of the microtubule goes as the mean speed of the motors.

Theoretical analysis

One of the main results of this paper is the agreement between experiment and our numerical results, shown in Fig. 5. To provide further understanding to the agreement of these results, we have carried out approximate analytical calculations of the model Langevin Eqs. 6. We have analyzed the short and long time limits of the equations for one, two, and a very large number of motors attached to the MT. The details of these calculations are given in the Appendix. The analytic calculations differ from the numerical ones in that we do not consider the dynamic attachment and detachment of motors from the MT, but assume that the motors are attached all the time. We do find, nonetheless, that the asymptotic value of the average displacement velocity of the MT is given by the mean speed of the motors, that quantitatively agrees with the experiments and our numerical calculations.

	DISCUSSION

TOP ABSTRACT INTRODUCTION THE MODEL NUMERICAL RESULTS DISCUSSION APPENDIX REFERENCES

We have introduced a simplified stochastic model to describe the behavior of kinesin-based microtubule gliding assays. We set up a phenomenology-based theoretical description for the motion of a single microtubule moving through a viscous medium (assumed to be water), across a bed of attached kinesins. Comparing the results of our simulation and analytic analysis with those obtained experimentally (Howard et al., 1989; Hancock and Howard, 1998), we find rather good agreement. By analyzing the total force exerted by the motors on the microtubule as a function of time, we have shown how cancellations between the forces exerted by the individual motors can produce a resultant that is orders of magnitude smaller than the maximum force exerted by a single motor, but in agreement with the predicted value of the total force required to produce the experimentally observed motion. There are differences in the explicit expressions for the force used in numerical analysis, which yield results that compare with the experimental results, in contrast to the forces used in analytic calculations. These differences are, however, not important to explain the general qualitative property of having an almost constant MT displacement speed for different kinesin densities.

Our numerical analysis for the duration of each motor attachment to the microtubule reveals that, although a microtubule may come into contact with hundreds (at low kinesin densities) or thousands (at high kinesin densities) of motor molecules, only a small fraction of these remain attached for long periods of time, and therefore most of the work appears to be performed by a relatively small number of motors. We have also examined the effect of allowing a distribution in the maximum speeds of the motors, to mimic the natural nonuniformity in motor behavior. Taking a binomial distribution in motor speeds, where a fraction q of the motors have speed v_slow and the remaining motors have speed v_fast, we find that the microtubule speed generally scales with the average motor speed. Taking a Gaussian distribution, we find that the microtubule speed remains more or less unchanged. In each case, it seems that the speed of the microtubule is proportional to the mean maximum motor speed. Note that our mechanical model description does not contain biochemical elements that may also play a roll, like the energy consumption of ATP that depends on the total number of motors moving along the MT. We do find, however, a good semi-quantitative agreement between our model and experiment.

In summary, the goal of our work was first to come up with a simple mechanical model that can agree quantitatively with the experimental results (Howard et al., 1989; Svoboda et al., 1993; Hunt et al., 1994; Coppin et al., 1995) and with previous theoretical analysis (Duke et al., 1995). Second, we wanted to develop a model that we could use to study the more complicated mitotic spindle formation (J.-F. Chauwin, F. Gibbons, and J. José, manuscript in preparation). Previous theoretical analysis has also provided some understanding to this problem (Duke and Leibler, 1996). Here we show, however, that our simplified model can provide numerical results that are in good quantitative agreement with experiment. Furthermore, we also provided an approximate analytic understanding of the kinesin density independence of the MT averaged displacement velocity.