Processivity and Velocity for Motors Stepping on Periodic Tracks

Abstract

Processive molecular motors enable cargo transportation by assembling into dimers capable of taking several consecutive steps along a cytoskeletal filament. In the well-accepted hand-over-hand stepping mechanism, the trailing motor detaches from the track and binds the filament again in the leading position. This requires fuel consumption in the form of ATP hydrolysis and coordination of the catalytic cycles between the leading and the trailing heads. Alternate stepping pathways also exist, including inchworm-like movements, backward steps, and foot stomps. Whether all the pathways are coupled to ATP hydrolysis remains to be determined. Here, to establish the principles governing the dynamics of processive movement, we present a theoretical framework that includes all of the alternative stepping mechanisms. Our theory bridges the gap between the elemental rates describing the biochemical and structural transitions in each head and the experimentally measurable quantities such as velocity, processivity, and probability of backward stepping. Our results, obtained under the assumption that the track is periodic and infinite, provide expressions that hold regardless of the topology of the network connecting the intermediate states, and are therefore capable of describing the function of any molecular motor. We apply the theory to myosin VI, a motor that takes frequent backward steps and moves forward with a combination of hand-over-hand and inchworm-like steps. Our model quantitatively reproduces various observables of myosin VI motility reported by four experimental groups. The theory is used to predict the gating mechanism, the pathway for backward stepping, and the energy consumption as a function of ATP concentration.

Significance

Molecular motors harness the energy released by ATP hydrolysis to transport cargo along cytoskeletal filaments. The two identical heads in the motor step alternatively on the polar track by communicating with each other. Our goal is to elucidate how the coordination between the two heads emerges from the catalytic cycles. To do so, we created a theoretical framework that allows us to relate the measurable features of motility, such as motor velocity, with the biochemical rates in the leading and trailing heads, thereby connecting biochemical activity and motility. We illustrate the efficacy of the theory by analyzing experimental data for myosin VI, which takes frequent backward steps and moves forward by a combination of hand-over-hand and inchworm-like steps.

Introduction

Long-distance transport in cells is essential for the targeted delivery and recycling of subcellular machinery. Cytoskeletal filaments, actin, and microtubules supply the analogous roads and highways for the molecular motor families myosin, kinesin, and dynein that catalyze intracellular transport (1). In many examples of intracellular transport, only a few motors of a particular type are attached to the cargo (2, 3, 4, 5, 6), so they usually operate in very small groups. Dimeric processive molecular motors with two identical heads such as myosin V (7), VI (8), and X (9), kinesin-1 (10), and cytoplasmic dynein (11) take multiple consecutive steps along their track for each diffusional encounter with the filament, enabling long-distance movement. During each step, the motor completes a catalytic cycle in which it occupies a number of intermediate biochemical states. Thermodynamics imposes that each step be reversible; hence, the cycle could be completed “backward” as well, although coupling with the hydrolysis of ATP to ADP and inorganic phosphate (P_i) guarantees that the motor progresses predominantly in one direction. Run termination occurs via stochastic dissociation of both the heads from the filament. Thus, long-distance movement is achieved by a strong association between the filament and the predominantly occupied biochemical intermediate states of the reaction cycle (high duty ratio) (12, 13, 14, 15).

The movement of processive motors can be monitored using a variety of single-molecule measurements, for instance, optical tweezers (16), fluorescence of bound probes (17), or light scattering from a small refractile particle (18), which are each suited to extract specific information over limited spatial and temporal ranges. Actin and microtubules are both polarized, and the specific motors move mainly toward one of the two filament ends. The predominant isoforms of all three motor families involved in long-distance transport form dimers having two actin- or microtubule-binding domains that bind successively in positions ahead of (the leading head, LH) or behind (the trailing head, TH) the center of mass. Determining the stepping mechanism, such as canonical hand-over-hand or inchworm-like, requires significant subdiffraction spatial resolution (17,19, 20, 21, 22). Tracking experiments measure the velocity, run length (mean distance traveled before dissociation), dwell times between motions, and likelihood of stepping backward. Comparing these data over a range of conditions (for instance, varying the ATP concentration) with the predictions of kinetic models enables determination of the reaction pathway and limiting biochemical processes. Whether the experimental conditions can be broad enough to fully constrain the model and reach physiological conditions such as millimolar ATP depends on the spatial and temporal resolution of the experiment and the speed of the motor. Dwell times and velocities have straightforward saturable kinetics with ATP concentration, but the run length and probability of backward stepping differ considerably among motors depending on the specifics of their kinetic schemes. Thus, theoretical modeling is an important tool for mechanistic interpretation of experimental data and hypothesis generation for behavior under different conditions, which leads to new experimental tests.

For the motor to exhibit directionality, either a mechanical or a biochemical asymmetry between the leading and trailing heads must be present (23). Biochemical asymmetry arises, for instance, in ATP binding and/or release of posthydrolysis products, and it is believed to originate from intermolecular strain between the two heads when they are both strongly bound to the filament. Because of their elastic connection, the two heads are under stress of opposite directions—the TH is pulled forward, and the LH is strained backward (24,25). This can coordinate the ATPase cycles of the heads to cause efficient coupling between ATP utilization and stepping (24). A reduced rate of a biochemical step in the LH relative to the TH is thought to be the primary gating mechanism (26). Biochemical gating is not mandatory for kinetic schemes (27), but if it is absent, then some other directional bias must be present (for instance, preferred detachment of the TH to initiate a step or promoted rebinding in the leading position, which can be conferred by structural asymmetry of the catalytic cycle). Preferential rebinding in leading position of the stepping head can be considered a mechanical type of gating, but microscopic reversibility rules out a bias that remains in the absence of an energy source (28). The mechanism of coordination adopted by a given motor is determined by structural features of the heads, intrahead linkers, and the mechanical load, and is tailored to the physiological role of the motor. Distinguishing between the distinct gating mechanisms by comparing predictions of kinetic models with experimental data and relating them to structural details are therefore critical goals of research aimed at understanding intracellular transport.

These considerations have elicited a variety of theoretical approaches to obtain insights into the stepping mechanism (12,29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40). One such method is to model motors as performing a random walk on a one-dimensional lattice. In this model, the motor is assumed to visit a number of biochemical and mechanical states in each position before completing a step. At each site l, there are N intermediate chemical states, where N is usually determined either from biochemical kinetics or by comparing theoretical predictions with experimental data. Formally, the probability of being in a state i at a location l along the track can be derived from the solution of an appropriate Markov chain or master equation. Because cytoskeletal tracks are periodic, we can consider the directed motor motion as occurring on a periodic one-dimensional lattice (see Fig. 1).

Figure 1.

One-dimensional network for a processive dimeric motor. (a) Two heads are attached to the same cargo (in black). The two heads are in shown in two conformations, with and without tension. The track is shown as green and magenta segments. Black arrows correspond to transitions between states at a fixed location along the filament (identified by the index l). Forward and backward steps are depicted as blue and red arrows, respectively. Gray arrows indicate the detachment from the track, which causes the end of a processive run. (b) A simplified representation of the more complete scheme is given, in which each symbol indicates a matrix associated with the corresponding transitions. The matrices $\hat{\text{F}}$ and $\hat{\text{B}}$ incorporate all the transitions associated with forward and backward steps, respectively. The matrix $\hat{\text{S}}$ contains all the rates related to changes in the conformation of the system at a fixed location along the track. To see this figure in color, go online.

The solution of the model involves finding expressions that connect the microscopic rates describing biochemical and structural transitions with observable characteristics of motility such as processivity, velocity, run length, and dwell time (41, 42, 43, 44). However, the kinetic scheme quickly becomes cumbersome; for instance, with two heads that each explore four biochemical states (apo, ATP, ADP-P_i, and ADP) bound to and detached from the track with asymmetry between the LH and TH, a few dozens of distinguishable intermediates are needed to describe the reaction scheme during a run, plus the detached state (an absorbing state) corresponding to run termination. Depending on the transitions allowed between these states, under the conservative assumption that only three transitions are possible per each head (involving either nucleotide binding or release or attachment or detachment from the track), the number of kinetic rates is the order of a few hundreds. Of course, many of these rates are identical or related to each other via thermodynamic considerations, ensuring that the model does not result in a perpetual motion machine. The development of a formalism that is independent of the complexity of the network of connections between the accessible states, which is the aim of this work, facilitates a quantitative description and of the dynamical features of the model.

A way to simplify the description of this complex network is to define the N × N transition probability matrix, $\hat{\text{M}}$, at any one location and to isolate the elements associated with transitions at a fixed site $\left(\hat{\text{S}}\right)$ from the forward steps $\left(\hat{\text{F}}\right)$ and backward steps $\left(\hat{\text{B}}\right)$. Periodicity imposes that the network of states and the transition matrices be identical at every site along the track. Here, we show that compact expressions emerge for the velocity, the run length (processivity), and the probability of backward stepping for an arbitrary network of states. The expressions may be derived using matrix algebra and can also be straightforwardly implemented for fitting experimental data. Our approach shares some similarities with the strategies adopted in (27) and (45) to extract the velocity and processivity for kinetic models of myosin VI and X, respectively. However, our results have the benefit of an analytical solution that holds regardless of the underlying structure of the kinetic network.

In the next sections, we first discuss the main results of the study and illustrate them in the context of a simple one-dimensional random walk. We then consider a modification of the model that allows it to account for steps of different sizes. Finally, we illustrate the utility of the method with a case study of the processivity of dimeric myosin VI. The literature contains conflicting conclusions about biochemical and mechanical gating in myosin VI (25,27,46,47), and we show that our theoretical solution to the model can be used to reveal the gating mechanism, the pathway for backward stepping, and the energy consumption of the motor.

For clarity of presentation, we relegate the formal derivation of the analytical results, some details concerning the myosin VI model, and the details of the kinetic Monte Carlo simulations to the Supporting Materials and Methods.

Methods

The model for myosin VI was numerically implemented in a Jupyter notebook (48), version 5.4.0. Functions from NumPy (49) were used to perform the matrix operations. The kinetic Monte Carlo code was also implemented in the same notebook, using the rejection-free scheme (50). The experimental data were digitized from the figures (as described in the captions of Figs. 5 and 6) using WebPlotDigitizer (51). We fitted the dwell-time and velocity distributions obtained with Monte Carlo simulations using the function curve_fit of SciPy (52). The optimal values of the parameters were obtained using an in-house script that uses the function minimize from ScyPy. All the plots were made using matplotlib (53). The details of the fitting procedure are presented in the Supporting Materials and Methods.

Figure 5.

Comparison of theoretical results to experiments on myosin VI. The red points and blue squares (including the error bars) were digitized using WebPlotDigitizer (51) from Fig. S6, c and d of Elting et al. (27) and Fig. 3 B of Ikezaki et al. (59), and the dashed lines come from the theory after fitting the parameters (see Table 1). (a) Run length, (b) velocity, and (c) probability of taking hand-over-hand and (d) inchworm-like forward steps are shown. To see this figure in color, go online.

Figure 6.

Comparison with experimental run length (L) and velocity (v) data as a function of ADP. The red dots (and related error bars) are obtained from Sweeney et al. (46). The experiments are performed at [ATP] = 2 mM. The red dashed lines report the values from the model at this ATP concentration as [ADP] is varied. The blue squares (and related error bars) are data points extracted from Fig. 5 A of (72) using WebPlotDigitizer (51). The data refer to measurements of actin gliding on a surface sparsely coated with myosin VI dimers; the experiments are performed at [ATP] = 1.4 mM and different concentrations of ADP. The blue, dash-dotted lines refer to the results from the model at [ATP] = 1.4 mM. (a) shows run length as a function of [ADP]. (b) shows [ADP] dependence of the velocity. To see this figure in color, go online.

Theory

Processivity and probability of forward stepping

To present the major results of the study, some new nomenclature is required. First, we distinguish “transitions” from “steps.” A transition occurs whenever the biochemical state or the structure of the motor changes. A step is a transition associated with a forward or backward displacement along the track. We use the index x to identify transitions and n for the total number of steps (backward and forward), and f and b refer to the counts of the forward and backward steps, respectively. A “pathway” is a series of transitions, whereas a “trajectory” refers to a collection of steps. In the discussion of the main results and in the Supporting Materials and Methods, we introduce a number of different symbols, defined in Table S1.

Let N be the number of distinguishable biochemical states accessible to the motor. The N states are assumed to be bound to the track, which means that at least one member of the dimeric motor complex is attached to the filament. If both of the heads simultaneously detach from the track, the motor dissociates from the filament; we consider this state to be “absorbing,” implying that there are no further transitions from it and the run cannot continue.

The N-dimensional vector $\overrightarrow{P}$(x) gives the probabilities of occupying any of the N states after x transitions, regardless of the location along the track. In a Markov chain, the probability vector $\overrightarrow{P}$(x + 1) is obtained multiplying $\overrightarrow{P}$(x) by a transition probability matrix whose N² entries define the probabilities of changing location (stepping) and/or the biochemical state of the motor. We consider four matrices: $\hat{\text{S}}$ contains all the transitions that occur at a fixed location on the track; $\hat{\text{F}}$ and $\hat{\text{B}}$ include all of the forward and backward steps, respectively; and $\hat{\text{M}}=\hat{\text{S}}+\hat{\text{F}}+\hat{\text{B}}$ accounts for all possible transitions between bound states. Because of the periodicity of the track, $\hat{\text{M}}$, $\hat{\text{S}}$, $\hat{\text{F}}$, and $\hat{\text{B}}$ are identical at any site along the filament; therefore, for an infinitely long track, $\overrightarrow{P}$(x + 1) is given by (see Supporting Materials and Methods for details)

$$\overrightarrow{P}\left(x+1\right)=\hat{\text{M}}\overrightarrow{P}\left(x\right)=\left(\hat{\text{S}}+\hat{\text{F}}+\hat{\text{B}}\right)\overrightarrow{P}\left(x\right).$$ (1)

The probability of being bound to the track is given by the sum of the probabilities of being in any of the N biochemical states, which is P(x) = ${\overrightarrow{1}}^{⊺}·\overrightarrow{P}\left(x\right)={\sum }_{i=1}^N{P}_i\left(x\right)$, where ${\overrightarrow{1}}^{⊺}$ is an N-dimensional vector of ones. At x = 0, P(0) = ${\overrightarrow{1}}^{⊺}·\overrightarrow{P}$(0) = 1, but as the state of the system evolves, the motors dissociate from the filament; therefore, P(x) ≤ 1 and P(x) → 0 for x → ∞.

We are interested in obtaining the probability $\mathbb{P}$(n) that the motor takes exactly n steps (forward or backward) before detaching from the track. We show in the Supporting Materials and Methods that

$$\mathbb{P}\left(n\right)={\overrightarrow{1}}^{⊺}\left(\hat{\text{I}}-{\hat{\text{P}}}_{\text{step}}\right){\left({\hat{\text{P}}}_{\text{step}}\right)}^n\overrightarrow{P}\left(0\right),$$ (2)

where

$${\hat{\text{P}}}_{\text{step}}=\left(\hat{\text{F}}+\hat{\text{B}}\right){\left(\hat{\text{I}}-\hat{\text{S}}\right)}^{-1}.$$ (3)

The matrix ${\hat{\text{P}}}_{\text{step}}$ accounts for all the possible pathways ending with a forward or backward step ($\left(\hat{\text{F}}+\hat{\text{B}}\right)$ in Eq. 3) after an arbitrary number of transitions has occurred at a fixed location ($\left(\hat{\text{I}}-\hat{\text{S}}\right)$⁻¹ in Eq. 3; see Supporting Materials and Methods for details). The expression in Eq. 2 has a simple physical meaning. It is clear that $\left({\hat{\text{P}}}_{\text{step}}\right)$ⁿ includes all the trajectories in which exactly n steps take place, regardless of the number of actual transitions. The matrix $\hat{\text{I}}-{\hat{\text{P}}}_{\text{step}}$ accounts for the detachment from the track before the occurrence of any other step but after an arbitrary number of transitions has taken place. The product of these terms ($\left({\hat{\text{P}}}_{\text{step}}\right)$ⁿ and $\hat{\text{I}}-{\hat{\text{P}}}_{\text{step}}$) is $\mathbb{P}$(n), given in Eq. 2.

It may be easier to understand Eq. 2 by considering the case for N = 1 depicted in Fig. 2. When N = 1, the transition probability matrices $\hat{\text{S}}$ = 0, $\hat{\text{F}}$ = k₊/Σ, $\hat{\text{B}}$ = k₋/Σ, and $\hat{\text{M}}$ = (k₊ + k₋)/Σ are all scalars, with Σ = k₊ + k₋ + γ being the sum of all the rates. It is easy to show that ${\hat{\text{P}}}_{\text{step}}$ = (k₊ + k₋)/Σ = π and that $\hat{\text{I}}-{\hat{\text{P}}}_{\text{step}}$ = γ/Σ = 1 − π, where π is the probability of stepping either forward or backward and 1 − π is the probability of dissociating from the track. Plugging these expressions in Eq. 2 (in the N = 1 case, ${\overrightarrow{1}}^{⊺}=\overrightarrow{P}$(0) = 1), we find that $\mathbb{P}$(n) = (1 − π)πⁿ, a well-known and intuitive result. Therefore, Eq. 2 generalizes the N = 1 case to an arbitrary number of biochemical states.

Figure 2.

Molecular motor without intermediate biochemical states. Detachment, forward stepping, and backward stepping and stepping occur only from state (1), with rates γ, k₊, and k₋, respectively.

Once the distribution $\mathbb{P}$(n) is known, it is straightforward to derive the average number of steps completed before detachment. After some algebraic manipulations (see the details in the Supporting Materials and Methods), the result is

$$\langle n\rangle =\sum _{n=0}^{\infty }n\mathbb{P}\left(n\right)={\overrightarrow{1}}^{⊺}{\hat{\mathrm{P}}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}{\left(\hat{\mathrm{I}}-{\hat{\mathrm{P}}}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}\right)}^{-1}\overrightarrow{P}\left(0\right)={\overrightarrow{1}}^{⊺}\left(\hat{\mathrm{F}}+\hat{\mathrm{B}}\right){\left(\hat{\mathrm{I}}-\hat{\mathrm{M}}\right)}^{-1}\overrightarrow{P}\left(0\right).$$ (4)

Again, it is useful to consider the N = 1 case, in which Eq. 4 becomes $\langle n\rangle$ = π/(1 − π), another well-known result.

Analogously, we derived the probability of detaching after f forward steps have been completed, regardless of the number of backward steps or transitions at a fixed site. As shown in the Supporting Materials and Methods, we obtained the average number of forward steps before detachment as

$$\langle f\rangle ={\overrightarrow{1}}^{⊺}{\hat{\mathrm{P}}}_{\mathrm{F}}{\left(\hat{\mathrm{I}}-{\hat{\mathrm{P}}}_{\mathrm{F}}\right)}^{-1}\overrightarrow{P}\left(0\right)={\overrightarrow{1}}^{⊺}\hat{\mathrm{F}}{\left(\hat{\mathrm{I}}-\hat{\mathrm{M}}\right)}^{-1}\overrightarrow{P}\left(0\right),$$ (5)

with

$${\hat{\text{P}}}_{\text{F}}=\hat{\text{F}}{\left[\hat{\text{I}}-\left(\hat{\text{S}}+\hat{\text{B}}\right)\right]}^{-1}.$$ (6)

The physical reasoning is the same as before. The matrix ${\hat{\text{P}}}_{\text{F}}$ accounts for all pathways ending with a forward step $\left(\hat{\text{F}}\right)$ after an arbitrary number of backward steps and transitions at a fixed location have occurred ($\left(\hat{\text{I}}-\hat{\text{S}}-\hat{\text{B}}\right)$⁻¹). Similarly, the average number of backward steps completed before detachment, regardless of the number of forward steps or total transitions, is

$$\langle b\rangle ={\overrightarrow{1}}^{⊺}{\hat{\mathrm{P}}}_{\mathrm{B}}{\left(\hat{\mathrm{I}}-{\hat{\mathrm{P}}}_{\mathrm{B}}\right)}^{-1}\overrightarrow{P}\left(0\right)={\overrightarrow{1}}^{⊺}\hat{\mathrm{B}}{\left(\hat{\mathrm{I}}-\hat{\mathrm{M}}\right)}^{-1}\overrightarrow{P}\left(0\right),$$ (7)

with

$${\hat{\text{P}}}_{\text{B}}=\hat{\text{B}}{\left[\hat{\text{I}}-\left(\hat{\text{S}}+\hat{\text{F}}\right)\right]}^{-1}.$$ (8)

Note that Eq. 7 is the same as Eq. 5 after swapping $\hat{\text{F}}$ and $\hat{\text{B}}$.

As before, it helps to think about the N = 1 case, in which ${\hat{\text{P}}}_{\text{F}}$ = k₊/(k₊ + γ) and $\hat{\text{I}}-{\hat{\text{P}}}_{\text{F}}$ = γ/(k₊ + γ) , leading to $\langle f\rangle$ = k₊/γ. Similarly, we find that when N = 1, $\langle b\rangle$ = k₋/γ.

We define the probability of stepping forward as the average number of forward steps divided by the total number of steps completed before detachment,

$$P_{\text{FWD}}=\frac{\langle f\rangle }{\langle n\rangle }.$$ (9)

As we show in the Supporting Materials and Methods, $\langle n\rangle =\langle f\rangle +\langle b\rangle$.

Finally, the average run length is given by

$$\langle L\rangle =\Delta \left(\langle f\rangle -\langle b\rangle \right),$$ (10)

where Δ is the average step size of the motor. When N = 1, P_FWD = k₊/(k₋ + k₊) and $\langle L\rangle$ = Δ(k₊ − k₋)/γ, which are the expected results.

Average run time

Let $\langle \tau \rangle$ be the average run time of the motor, that is, the average duration of a processive run. Here, we briefly outline how to obtain an expression for $\langle \tau \rangle$ independent of the number of biochemical states and the connectivity between them. We begin by defining the vector $\overrightarrow{P}$(t), which incorporates the probability of being at any of the N states bound to the track at time t, anywhere along the filament. The evolution in time of $\overrightarrow{P}$(t) is obtained by solving the master equation,

$$\frac{d\overrightarrow{P}\left(t\right)}{dt}=\hat{\text{K}}\overrightarrow{P}\left(t\right),$$ (11)

where $\hat{\text{K}}$ is the N × N rate matrix. We ought to point out that $\hat{\text{K}}$ includes the transitions toward the absorbing state, although the absorbing state is not included in the N states considered. This means that the sum of a column of $\hat{\text{K}}$ is not zero: in the stationary state, the motors will be detached from the filament. They will occupy a state (the detached state) not included in the N described by $\overrightarrow{P}$ and $\hat{\text{K}}$. Using a classic result (54), the run time (or mean first passage time to detachment) is

$$\langle \tau \rangle =-{\overrightarrow{1}}^{⊺}{\hat{\text{K}}}^{-1}\overrightarrow{P}\left(0\right).$$ (12)

Velocity

To obtain an expression for the average velocity, we introduce the probability p(L, τ)dτ of detaching after a run length L and a run time τ. The most direct definition of the average velocity is $v=\sum _L\int d\tau L/\tau p\left(L,\tau \right)=\langle L/\tau \rangle$. Such an approach has been taken in the one-state (38) and two-state models (55) for kinesin; in these cases, analytical expressions for p(L, τ) can be found. Alternative definitions of v are 1) the limit lim_t→∞d$\langle l\left(t\right)\rangle$/dt (29,41), where l(t) is the location of the motor at time t and the average is performed over the ensemble of motors that are still bound to the track at time t (56); 2) the “instantaneous” velocity, taken as the average step size multiplied by the rate of stepping; and 3) the ratio between the average run length and the mean run time $\langle \tau \rangle$,

$$v=\frac{\langle L\rangle }{\langle \tau \rangle }.$$ (13)

The definitions of velocity given have different merits; however, Eq. 13 and $\langle L/\tau \rangle$ benefit from being directly related to two quantities, the run length and run time, that are readily measured in experiments. However, obtaining an analytical expression for $\langle L/\tau \rangle$ in the general case presents technical challenges, which are avoided if one uses Eq. 13.

Multiple stepping mechanism

So far, we have assumed that all the different stepping pathways displayed by a motor are characterized by an identical average step size (Δ in Eq. 10 is the periodicity of the track). However, this is not always the case. For instance, dynein and myosin VI display broad step-size distributions (22,57). In particular, myosin VI (discussed in detail later) combines hand-over-hand with inchworm-like displacements (57). The theory presented here can be generalized to incorporate step-size variability. Suppose that a dimeric motor exists in two states, D (as in “distal”), with the two heads separated by a repeat of the filament, and C (as in “close”), in which the motors are bound to adjacent sites. This scenario is represented in Fig. 3 and is inspired by a model for myosin VI proposed by Yanagida and co-workers (57, 58, 59) (discussed in further detail in the next section). The number of D and C states are N_D and N_C, respectively, and N = N_D + N_C is the total number of bound states available to the motor. Let $\overrightarrow{P}$(x) be the N-dimensional vector describing the probabilities of occupying the N different states anywhere along the filament. The matrices ${\hat{\text{S}}}_{\text{D}}$ and ${\hat{\text{S}}}_{\text{C}}$ contain the transitions within states D and C, respectively, explored without changing location along the track. The forward and backward transitions associated with hand-over-hand steps are described by the matrices ${\hat{\text{F}}}_{\text{HoH}}$ and ${\hat{\text{B}}}_{\text{HoH}}$, respectively, and connect states D at different filament sites. Inchworm steps are accounted for by the matrices ${\hat{\text{F}}}_{\text{Iw}}$ and ${\hat{\text{B}}}_{\text{Iw}}$ and connect D and C states. Let $\hat{\text{S}}={\hat{\text{S}}}_{\text{D}}+{\hat{\text{S}}}_{\text{C}}$; we show in the Supporting Materials and Methods that under the assumption of periodic and infinite track, the following equation holds:

$$\overrightarrow{P}\left(x+1\right)=\left(\hat{\text{S}}+{\hat{\text{F}}}_{\text{HoH}}+{\hat{\text{B}}}_{\text{HoH}}+{\hat{\text{F}}}_{\text{Iw}}+{\hat{\text{B}}}_{\text{Iw}}\right)\overrightarrow{P}\left(x\right),$$ (14)

which becomes identical to Eq. 1 upon defining $\hat{\text{F}}={\hat{\text{F}}}_{\text{HoH}}+{\hat{\text{F}}}_{\text{Iw}}$ and $\hat{\text{B}}={\hat{\text{B}}}_{\text{HoH}}+{\hat{\text{B}}}_{\text{Iw}}$. Similarly to what we have done before, we can compute the average number of forward and backward hand-over-hand and inchworm steps, $\langle f_{\text{HoH}}\rangle$, $\langle b_{\text{HoH}}\rangle$, $\langle f_{\text{Iw}}\rangle$, and $\langle b_{\text{Iw}}\rangle$, with $\langle n\rangle$ equal to the total number of steps. It follows that we can define a probability of taking a forward hand-over-hand or inchworm step,

$$P_{\text{FWD},\text{HoH}}=\frac{\langle f_{\text{HoH}}\rangle }{\langle n\rangle },P_{\text{FWD},\text{Iw}}=\frac{\langle f_{\text{Iw}}\rangle }{\langle n\rangle },$$ (15)

and the average run length is

$$\langle L\rangle ={\Delta }_{\text{HoH}}\left(\langle f_{\text{HoH}}\rangle -\langle b_{\text{HoH}}\rangle \right)+{\Delta }_{\text{Iw}}\left(\langle f_{\text{Iw}}\rangle -\langle b_{\text{Iw}}\rangle \right),$$ (16)

where the inchworm step size is Δ_Iw = Δ_HoH/2.

Figure 3.

Combination of hand-over-hand and inchworm-like steps. The two heads forming the dimer are shown in red and blue, and the track is in green and magenta. The bottom three states (enclosed in a gray box) are of type D, with the two motors occupying sites separated by a filament repeat. In the case of C states (upper two states, within a blue box), the two heads are closer to each other. Under the pictorial representation of the dimeric motor, the letters in parenthesis refer to the location of the motor along the filament. For instance, (l, l + 1) indicates that the TH is in position l and the LH in position l + 1. Hand-over-hand steps connect D states at different sites along the filament, and the corresponding transition probabilities are contained in the matrices ${\hat{\text{F}}}_{\text{HoH}}$ and ${\hat{\text{B}}}_{\text{HoH}}$. There are two types of inchworm-like steps, both of which connect D states with C states. The motor could either step from D to C (matrices ${\hat{\text{F}}}_{\text{IwD}}$ and ${\hat{\text{B}}}_{\text{IwD}}$) or from C to D (${\hat{\text{F}}}_{\text{IwC}}$ and ${\hat{\text{B}}}_{\text{IwC}}$). Transitions that do not change the location of the motor are described by the probability matrices ${\hat{\text{S}}}_{\text{D}}$ and ${\hat{\text{S}}}_{\text{C}}$ for states D and C, respectively. Dissociation from the filament may occur from either D or C, and it is identified with the green arrows pointing toward the absorbing state, labeled “OUT.” For simplicity, we are only showing 2HB states to identify D and C conformations. A more complete representation of the kinetic scheme implemented is reported in Figs. S1–S3. To see this figure in color, go online.

Observables

From the kinetic mechanism, we can compute the average number of times that any specific event occurs. For instance, we may be interested in monitoring the energy consumption of the motor, which can be calculated by counting the average number of ATP molecules hydrolyzed per processive run. Let q be the count of the event of interest. To compute $\langle q\rangle$, we collect in the matrix $\hat{\text{Q}}$ all the transitions corresponding to the occurrence of the event of interest, and $\hat{\text{M}}-\hat{\text{Q}}$ is the probability matrix associated with all the remaining transitions. We show in the Supporting Materials and Methods that

$$\langle q\rangle ={\overrightarrow{1}}^{⊺}{\hat{\mathrm{P}}}_{\mathrm{Q}}{\left(\hat{\mathrm{I}}-{\hat{\mathrm{P}}}_{\mathrm{Q}}\right)}^{-1}\overrightarrow{P}\left(0\right)={\overrightarrow{1}}^{⊺}\hat{\mathrm{Q}}{\left(\hat{\mathrm{I}}-\hat{\mathrm{M}}\right)}^{-1}\overrightarrow{P}\left(0\right),$$ (17)

with

$${\hat{\text{P}}}_{\text{Q}}=\hat{\text{Q}}{\left[\text{I}-\left(\hat{\text{M}}-\hat{\text{Q}}\right)\right]}^{-1}.$$ (18)

There is a similarity between this expression and the average number of steps previously discussed. For instance, let $\hat{\text{Q}}=\hat{\text{F}}$, in this case $\hat{\text{M}}-\hat{\text{Q}}=\hat{\text{S}}+\hat{\text{B}}$ per definition, and ${\hat{\text{P}}}_{\text{Q}}=\hat{\text{F}}\left(\hat{\text{I}}-\hat{\text{S}}-\hat{\text{B}}\right)$⁻¹, which is the same as ${\hat{\text{P}}}_{\text{F}}$ in Eq. 6. Therefore, when $\hat{\text{Q}}=\hat{\text{F}}$, Eq. 17 is the same as Eq. 5. Similar arguments show that from Eq. 17, we recover Eq. 4 if $\hat{\text{Q}}=\hat{\text{F}}+\hat{\text{B}}$ and Eq. 7 if $\hat{\text{Q}}=\hat{\text{B}}$.

Results and Discussion

Myosin stepping pathways

Myosins are molecular motors typically composed of three structural regions with distinct functions: 1) a head domain, which hydrolyzes ATP and binds actin; 2) the oblong lever arm, which amplifies the small structural transitions occurring in the myosin head; and 3) a tail domain responsible for multimerization and cargo binding.

Processive myosins (M) are tightly bound to the actin filament (A) when they are in complex with ADP (AM·D) or in the apo state (AM) (13, 14, 15). ATP binding to the motor induces detachment from actin. After ATP hydrolysis, myosin binds actin tightly, undergoes the power stroke (a structural transition that propels the lever arm forward), and releases the P_i. The repriming stroke occurs upon ATP hydrolysis, when the motor is dissociated from the actin filament, and rotates the lever arm to the pre-stroke position.

Dimeric myosins walk predominantly by a hand-over-hand mechanism (9,17,19,20), in which the TH detaches from the filament, overtakes the bound head (BH), binds to a new actin subunit, and becomes the new LH. As discussed in the Introduction, this stepping mechanism is facilitated by two types of processes: 1) biochemical gating and 2) mechanical gating. The former ensues from the interhead tension between the two bound motors and is expected to facilitate the detachment of the TH before the LH by modifying the rates of nucleotide binding and release from the two heads. The second is structural, due to the combined effect of the power stroke in the leading head and the repriming stroke of the detached head, which favor the reattachment of the stepping head in the leading position.

However, exceptions include 1) backward steps, in which the LH detaches first and moves rearward (57); and 2) inchworm-like steps (57), leading to transitory states in which the two motors bind at adjacent actin sites. Also, 3) the interhead tension experienced by the two actin-bound motors may affect the stability of the actomyosin complex both in apo and in ADP-bound state (25); hence, it may facilitate the spontaneous detachment from the filament (i.e., occurring before ATP binding). We use “spontaneous” to indicate detachment of the motor without ATP binding, that is, AM·D → A + M·D or AM → A + M. We refer to the transition AM + T → A + M·T as the “ATP-induced” detachment from the filament. Finally, 4) a stepping head may “stomp” and rebind to the same location on actin from which it detached (60,61). Stomps may correspond to futile cycles, in which an ATP molecule is hydrolyzed without motor movement.

Ideally, a model for myosin should incorporate all of the above features, and their measured occurrence should emerge as a natural consequence of the model. The method presented in this manuscript provides a straightforward route to a comprehensive model that encompasses all of these stepping modes.

Myosin VI

We apply the theory to myosin VI, the only member of the myosin superfamily that is known to move toward the pointed end of the polarized actin filament (62) and a processive motor that is also known to take frequent backward and inchworm-like steps (57). Despite a number of insightful studies, the nature of the gating mechanism in some of the myosin motors, including myosin VI, continues to be unresolved. It has been suggested that ADP release is slower in the LH than in the TH (ADP-release gating) (27,47). Alternatively, it was proposed that ATP binding occurs more slowly in the LH (ATP-binding gating) (25,46,63). In addition, recent experiments have suggested that the probability of forward stepping (compared to backward stepping) for myosin VI depends on the concentration of ATP (59), a property anticipated for either ADP-release or ATP-binding gating. Thus, our model, which naturally includes all of the stepping modes, provides clarity to the debate by accounting fully for the published experimental data (27,59).

Modeling the biochemistry of dimeric myosin VI

Fig. 4 shows the biochemical states explored by each individual myosin head. We ignore the collision complexes—all the nucleotide binding events are considered to be pseudo-first order—and we assume that ATP binding induces a fast detachment from actin. Furthermore, the transition A + M·T → AM·D combines 1) ATP hydrolysis, 2) formation of the transitory AM·D·P_i complex, and 3) actin-activated phosphate release. We ignore the slow phosphate release from the detached motor in the one-head-bound (1HB) state. Moreover, the model does not account for ATP synthesis or dissociation of ATP from a myosin head. Finally, we assume that there is only one AM·D (ADP- and actin-bound) state for myosin VI, although experimental evidence indicates that two such states could exist for both myosin V and VI (13,14,64).

Figure 4.

States accessible to a single myosin. The scheme is a simplified representation of the myosin cycle shown in (13, 14, 15). A is actin; myosin is M; ADP and ATP are shown as D and T, respectively; and P_i is orthophosphate. All the rates are described in Table S2. ATP binding to actomyosin is assumed to lead to a fast detachment of the motor from the filament. Rebinding is a combination of ATP hydrolysis, binding to actin, and releasing the phosphate. ATP binding is assumed to be irreversible. r is a parameter describing the enhanced rate of actin binding in the ADP-bound and apo states compared to the ATP-bound state. θ relates the rate of actin binding in the apo and ADP-bound states.

The model for myosin VI dimers is sketched in Fig. 3, in which we show both states of type C (as in “close”), in which the heads are bound to adjacent actin sites, and states of type D (“distal”), with the two heads separated by 34–36 nm. This type of model has been proposed by Yanagida and co-workers in a number of remarkable experimental studies of myosin VI processivity (57, 58, 59). For the dimeric motor, N_D = 16, with four states in which both heads are actin bound (2HB states) and 12 in which only one head is attached to the track (1HB states). In the C states, experiments suggest that the two heads step with equal probability (58), and as a consequence, there are only three states with 2HB and six states with 1HB, for a total of N_C = 9. The total number of actin-bound states is N = 25 with 152 transitions, including those between bound states and those toward the absorbing, detached state (see Figs. S1–S3 for a graphical representation). The parameters and rates of the model are described in Table S2.

We constructed the model under the premise that interhead tension affects the rates for binding and releasing nucleotides and the filament. In a myosin dimer, interhead tension arises when the two heads pull on each other to attain the nucleotide-dependent favored orientation of the lever arm. For myosin VI, experiments suggest that the lever arm undergoes a rotation forward upon releasing ADP (ADP bound → apo) (62,65,66). Thus, in principle, if either of the two heads releases or binds ADP, the changes in the favored lever arm orientation of that myosin could modify the interhead tension and consequently might affect the rates for both heads. However, we assume that in the 1HB state and in C conformations, there is no interhead tension. As a consequence, the rates for one head associated with changes of conformations within this set of states ((1HB or C) → (1HB or C)) do not depend on the nucleotide bound to the companion head. In other words, as long as the dimer remains in a C conformation or in the 1HB state, the two heads go through their catalytic cycle independently. In contrast, interhead tension affects the transitions between D conformations and to D conformations. 1) Tension gives rise to gating (that is, the rates for ADP release and ATP binding in the TH and in the LH are different), and 2) it alters the affinity of both heads for the actin filament. On general grounds, we expect that 3) the rates of one head depend on the nucleotide bound to the partner myosin because changes in the chemical state of the companion head change the interhead tension. However, the reported changes to the lever arm orientation upon releasing ADP (62,65,66) 1) are small and 2) have only been shown for monomeric myosin VI. Furthermore, 3) the pliancy of the converter/lever arm region of myosin VI (8,20,40,67, 68, 69, 70, 71) might reduce the effect of these structural changes. Thus, we surmise that even in the D conformations, we may neglect the changes of interhead tension associated with ADP release, leading to a model in which in D conformations as well, the rates describing the transitions for the two myosins are independent of the biochemical state of the partner head. We refer to this assumption as the “independence hypothesis” (IH).

In the Supporting Materials and Methods, we show that the IH approximation implies that the ADP dissociation constant for a myosin head bound to actin is the same in the LH, TH, and 1HB conformations. A survey of the experimental literature on myosin VI ((14,25,46,63); see Supporting Materials and Methods for details) indicated that myosin VI affinity for ADP is not very sensitive to mechanical strain, lending support to the IH approximation. However, the independence of ADP affinity for actomyosin on the position (LH or TH) of the head in the double-headed bound state requires that if there is ADP-release gating, then both the ADP on rate and ADP dissociation rate must change together. This characteristic does not impact fitting of data in the absence of ADP but might restrict some models. Therefore, we relaxed the IH and generated a “weak dependence hypothesis” (WDH) model in which the ADP dissociation constant may differ in the LH and in the TH but is independent of the chemical state of the companion head. This also requires that the affinity for actin in the D conformations depends on the nucleotide bound to the partner head. In the Supporting Materials and Methods, we show that the results with the WDH and IH are essentially identical. In the main text, we focus on the simpler IH model.

In both the models (IH and WDH), ATP binding (AM + T → A + M·T) does not depend on the state of the companion head, although it depends on the location of the head in a D conformation (TH or LH) because of biochemical gating. Moreover, binding to actin of the posthydrolysis head (M·T → AM·D) is regulated by mechanical gating (see the later section, Rates for Actin Binding) but is independent of the chemical state of the partner myosin for both IH and WDH.

Rates for nucleotide binding/release

A consequence of the assumption that the two heads are equivalent in 1HB states and C conformations is that biochemical gating is not active in these states. Instead, biochemical gating tunes the relative rate of ADP release from and ATP binding to the LH of a 2HB state in conformation D: $k_{\text{D},\text{off}}^{\text{LH}}$ = $k_{\text{D},\text{off}}^{\text{TH}}$/g_ADP and $k_{\text{T},\text{on}}^{\text{LH}}$ = $k_{\text{T},\text{on}}^{\text{TH}}$/g_ATP, where g_ADP and g_ATP are factors identifying the strength of ADP-release and ATP-binding gating. For the TH rates of ADP release and ATP binding, we adopt previous estimates for the TH of a dimer (27) (in close agreement with single-head values (14), as required by the velocity at ATP saturation) or from experiments monitoring the single-head rate (14). Because we use these values to describe the rates of the bound head (BH) of a 1HB myosin, we refer to these rates as $k_{\text{D},\text{off}}^{\text{BH}}$ and $k_{\text{T},\text{on}}^{\text{BH}}$ (see Table 1). For the myosin head detached from actin (free head (FH), thus $k_{\text{D},\text{off}}^{\text{FH}}$ and ${\text{k}}_{\text{T},\text{on}}^{\text{FH}}$), we also used rates obtained from experiments (14) (see Table 1). When the heads are bound to adjacent subunits (states C) the rates $k_{\text{D},\text{off}}^{\text{BH}}$ and $k_{\text{T},\text{on}}^{\text{BH}}$ apply to both heads. In the IH, the ADP dissociation constant is the same for all BHs $\left(K_{\text{D}}^{\text{BH}}\right)$ but can be different in the detached head $\left(K_{\text{D}}^{\text{FH}}\right)$. We assume that the ADP dissociation constant is known from experiment (14,63) (see Table 1), and we construct the ADP-binding rates accordingly (see Eq. S43).

Table 1.

Parameters Used in the Myosin VI Model

Scheme	Rate/Diss. Cnst.	Applied to	Value
AMD → AM + D	${\text{k}}_{\text{D},\text{off}}^{\text{TH}}={\text{k}}_{\text{D},\text{off}}^{\text{BH}}$	2HB (D, TH) or 2HB (C) or 1HB	5.1/sa
AMT → A + MT	${\text{k}}_{\text{T},\text{on}}^{\text{TH}}={\text{k}}_{\text{T},\text{on}}^{\text{BH}}$	2HB (D, TH) or 2HB (C) or 1HB	0.016/(s · μM)a
AMD → A + MD	kMD,off	1HB	0.07/sb
AM → A + M	kM,off	1HB	0.004/sb
A+MT → AMD	kreb	1HB	57/sa
MD → M + D	${\text{k}}_{\text{D},\text{off}}^{\text{FH}}$	1HB or 0HB	5.6/sb
M + T → MT	${\text{k}}_{\text{T},\text{on}}^{\text{FH}}$	1HB or 0HB	0.14/(s · μM)b
MDPi → MD + Pi	${\text{k}}_{{\text{P}}_{\text{i}},\text{off}}^{\text{FH}}$	0HB	0.04/sb
AMD $\rightleftharpoons$ AM + D	$K_{\text{D}}^{\text{BH}}$	2HB or 1HB	34 μMc
MD $\rightleftharpoons$ M + D	$K_{\text{D}}^{\text{FH}}$	1HB	21.5 μMb

Parameter	Applied To	Value
gADP	2HB (D, LH)	3.9d
gATP	2HB (D, LH)	0.69d
gMEC	1HB or 0HB	34d
s	2HB (D, TH) or 2HB (D, LH)	9.0d
r	1HB or 0HB	90d
θ	1HB	0.090

The left column shows the name of the parameter or the corresponding transition. The central column indicates whether the specific parameter is used for a 2HB state, D conformation, and TH (2HB(D, TH)) or LH (2HB(D, LH)); for a C conformation (2HB(C)); for a 1HB (1HB); or to determine the initial state (zero head bound, or 0HB). The last column gives the numerical value of the parameter. θ was obtained using the listed ADP-release rates and spontaneous detachment rates and the ADP-binding rates from (14). After the minimization, we obtained χ²/n_data ≈ 11.08, where n_data is the number of data points considered. As described in the main text, ${\text{k}}_{\text{D},\text{off}}^{\text{LH}}$ = k_D,off/g_ADP and ${\text{k}}_{\text{T},\text{on}}^{\text{LH}}$ = k_T,on/g_ATP. In addition, ${\text{k}}_{\text{MD},\text{off}}^{\text{LH}}={\text{k}}_{\text{MD},\text{off}}^{\text{TH}}$ = sk_MD,off, and ${\text{k}}_{\text{M},\text{off}}^{\text{LH}}={\text{k}}_{\text{M},\text{off}}^{\text{TH}}$ = sk_M,off. Note that although the binding of ATP induces the detachment from the actin filament, we do not multiply $k_{\text{T},\text{on}}^{\text{LH}}$ or $k_{\text{T},\text{on}}^{\text{TH}}$ by the factor s. This is because the rate-limiting transition associated with this rate is binding ATP, not releasing the filament.

^aValues are taken from (27).

^bThese refer to the mutant T406A of (14).

^cThe ADP dissociation constant labeled was taken from the T406A mutant in (63).

^dValues are fitted to the data (see Methods).

Rates for spontaneous detachment from actin

We obtain k_M,off and k_M·D,off from experiments (14). These are the rates of spontaneously detaching from actin in the apo and ADP-bound state for single-head myosin. We adopt these rates for the 1HB state and the C conformations. In the presence of interhead tension, we expect that the interaction of a myosin head with actin is destabilized (25), and therefore, we introduce a parameter s to account for the increased rate of detachment in D conformations compared with 1HB and C conformations,

$$k_{\text{M}·\text{D},\text{off}}^{\text{TH}}=k_{\text{M}·\text{D},\text{off}}^{\text{LH}}=s·k_{\text{M}·\text{D},\text{off}},$$ (19)

$$k_{\text{M},\text{off}}^{\text{TH}}=k_{\text{M},\text{off}}^{\text{LH}}=s·k_{\text{M},\text{off}}$$

Rates for actin binding

The rate for the A + M·T → A·M·D transition is k_reb = 57 s⁻¹ (27), and it is the sum of the rate for binding in the leading position, in the trailing position, or on a site adjacent to the BH irrespective of the chemical state of the BH. (We refer to state of a myosin head dissociated from actin as either M·T or M·D·P_i; because we do not explicitly account for ATP hydrolysis, there is no practical difference between these two states.) The binding rate of a dissociated ADP-bound myosin head is rk_reb, where the factor r is an adjustable parameter. For an apo head, we use rθk_reb, with θ set via thermodynamic considerations (see later and Supporting Materials and Methods). We expect r > 1 because of the high duty ratio of myosin VI (14).

The motor can stomp or complete a hand-over-hand step (from state D to D; see continuous lines in Document S1. Supporting Materials and Methods, Figs. S1–S18, and Tables S1 and S2, Document S2. Article plus Supporting Material a) of size Δ_HoH = 34 nm (27,47) or an inchworm-like step (D → C or C → D; see dashed lines in Fig. S3) with Δ_Iw = 17 nm. The detached head in a 1HB state may bind to three possible sites along actin: in the leading position, in the trailing position, or adjacent to the BH. We made the following crucial hypothesis: the relative likelihood of these three 2HB geometries depends on the orientation of the lever arm of the two heads when the free motor binds actin. Let g_MEC be the strength of the mechanical gating due to either the repriming stroke in the detached head or the power stroke of the attached head. The probability of attaining a 2HB state when both heads are in their preferred orientation is enhanced by a factor $g_{\text{MEC}}^2$. If only one of the lever arms must rotate away from the preferred angle, the factor is g_MEC, whereas there is no enhancement factor if the dimer geometry at the time of binding actin is frustrated in both the heads simultaneously. The lever arm of an actin-bound head is always in the post-power-stroke orientation; that is, it points forward (we make no distinction between the lever arm orientation in the ADP-bound and apo states (62,65,66) because their difference is small compared to the pre-power-stroke conformation of the lever arm). In contrast, for the lever arm of the FH, we distinguish two scenarios, depending on whether the detached head of a 1HB state is or is not bound to ATP. The scenarios are portrayed in Fig. S6.

1) If, during a step, ATP is hydrolyzed and the FH is in the primed state, then the lever arm points backward upon actin binding; that is, it preferentially binds actin in the leading position (see Document S1. Supporting Materials and Methods, Figs. S1–S18, and Tables S1 and S2, Document S2. Article plus Supporting Material a). Therefore, we adopt the following sets of rates for binding actin as the LH (l_MT), in the trailing position (t_MT), or in the vicinity of the BH (c_MT, in state C):

$$\begin{array}{l}{l}_{\text{MT}}={\text{k}}_{\text{reb}}{\Lambda }_{\text{MT}}{g}_{\text{MEC}}^2,\\ c_{\text{MT}}={\text{k}}_{\text{reb}}{\Lambda }_{\text{MT}}{g}_{\text{MEC}},\\ t_{\text{MT}}={\text{k}}_{\text{reb}}{\Lambda }_{\text{MT}},\end{array}$$ (20)

where Λ_MT = $\left(g_{\text{MEC}}^2+g_{\text{MEC}}+1\right)$⁻¹.

2) If the detached motor is in the apo or ADP-bound state, the lever arms of both heads (the one bound to the track and the free one) are in the poststroke position, and thus, the C state allows both heads to maintain their favored lever arm orientation upon rebinding, resulting in a landing rate proportional to rk_rebg_MEC/(g_MEC + 2) (see Fig. S6 b). A nucleotide-free or ADP-bound dissociated head binds actin in state D (either as the LH or as the TH) with a rate proportional to rk_reb/(g_MEC + 2) because one of the two heads must rotate the lever arm away from the preferred orientation to accommodate the 2HB geometry (see Document S1. Supporting Materials and Methods, Figs. S1–S18, and Tables S1 and S2, Document S2. Article plus Supporting Material c). From these considerations, we obtain the following rates for binding actin in the ADP-bound and apo states:

$$\begin{array}{l}{l}_{\text{M}·\text{D}}=t_{\text{M}·\text{D}}={\theta }^{-1}{l}_{\text{M}}={\theta }^{-1}{t}_{\text{M}}=r{\text{k}}_{\text{reb}}\frac{1}{g_{\text{MEC}}+2}\\ c_{\text{M}·\text{D}}={\theta }^{-1}{c}_{\text{M}}=r{\text{k}}_{\text{reb}}\frac{g_{\text{MEC}}}{g_{\text{MEC}}+2}\\ \theta =\frac{{\text{k}}_{\text{M},\text{off}}}{{\text{k}}_{\text{MD},\text{off}}}\frac{K_{\text{D}}^{\text{BH}}}{K_{\text{D}}^{\text{FH}}}.\end{array}$$ (21)

The constant θ is established from experiments and is a consequence of thermodynamic constraints applied to the kinetic scheme of myosin VI. Although the details are in the Supporting Materials and Methods, the definition of θ ensues from the following cyclic set of transition for the dimeric myosin:

$$\begin{array}{ccc}\text{AM}·\text{D.AM.D}& \rightleftharpoons & \text{M}.\text{D.AM.D}\\ \upharpoonleft \downharpoonright & & \upharpoonleft \downharpoonright \\ \text{AM.AM.D}& \rightleftharpoons & \text{M.AM.D}\end{array}$$

It is easy to verify that the constant θ ensures that the free energy change upon completing this cycle is 0.

Initial conditions

To complete the model, we need the initial condition vector, $\overrightarrow{P}$(0). We assume that the two detached myosin VI heads behave independently and that they exist in three possible states: apo, ATP bound (which we assume equivalent to ADP·P_i), and ADP bound. Movement along actin begins when both heads bind to the filament. The equations used to establish the initial condition are given in Eqs. S46 and S47.

Fitting the experimental data

The parameters adopted are reported in Table 1: 1) g_ADP, 2) g_ATP, 3) g_MEC, 4) s, and 5) r are obtained by fitting the model to experimental data, and the remaining parameters are either taken from the analysis of Elting et al. (27) or from the T406A mutant of myosin VI (14,63). To obtain the adjustable parameters, we compared the 1) run length, 2) velocity, and probability of taking forward 3) hand-over-hand and 4) inchworm steps as a function of ATP concentration with the 1) run length and 2) velocity data from Elting et al. (27) (Fig. 5, a and b) and with the probability for 3) long and 4) short forward steps measured by Ikezaki et al. (59) (Fig. 5, c and d). The model recovers quantitatively the results from the two different groups (Fig. 5). The main difference is the average run length, which can at least in part be explained because we neglected the finite size of the actin filament (27).

Goodness of fits

As discussed in the Supporting Materials and Methods, we minimized the square of the residual function, χ² (defined as the sum of the squared differences between theory and experimental data weighted by the inverse of the square of the experimental uncertainty; see Eq. S48), starting from a large number of different initial conditions. In this way, we explored more thoroughly the space of parameters, allowing us to compare different local minima. The details of the analysis are discussed in Figs. S8–S10. We found that among the parameters, only g_ADP and s are tightly constrained. The strength of ATP-binding and mechanical gating for the optimal results of the fitting procedure are constrained, but we find parameter sets with low χ² for which g_ATP and g_MEC are significantly different. The value of r changes significantly among different sets of parameters resulting in similar values of χ², indicating that more experimental data are needed to fully constrain the parameters in the model, specifically the run length at [ATP] < 1 μM and the probability of stepping backward at high ATP concentrations. If we focus on the cluster of parameter sets yielding the lowest χ², we note that ADP-release gating is more prominent than ATP-binding gating and that g_ADP and g_ATP are positively correlated, like g_MEC and s, whereas g_ADP is anticorrelated with $g_{\text{MEC}}$ (see Fig. S10).

The effect of ADP in solution

Experimental data on motility in the presence of ADP were not used in the fitting. To explore whether the model can capture the dependence of myosin VI motility on [ADP], though, we considered the run length and velocity of myosin VI reported by Sweeney et al. (46) for [ATP] = 2mM and [ADP] = 0 μM or [ADP] = 100 μM. Furthermore, Altman et al. (72) measured the gliding velocity of actin filaments over a surface sparsely coated with myosin VI at varying [ADP] and 1.4 mM [ATP]. Using the ADP dissociation constant from experiments, we achieved the agreement shown in Fig. 6 without adjusting the parameters. As [ADP] increases, the run length becomes longer, in agreement with the data from Sweeney et al. (46) (see Fig. 6 a). They also reported that velocity is nearly independent of [ADP]; although our results are within the large experimental error bars, the model predicts that myosin VI slows down as [ADP] is increased (see Fig. 6 b). On the other hand, our results are in excellent agreement with the data from Altman et al. (72), as shown in Fig. 6 b. Because at [ADP] = 0, Altman et al. reported a velocity that is smaller than the values at [ATP] = 1 mM and [ATP] = 2 mM from Elting et al. (27), which we used to train the model parameters, it is not surprising that at low [ADP], our model does not recover the velocity data from Altman et al. at low [ADP].

Predictions

In Fig. 7, we show the theoretical predictions. First, the majority of backward steps are inchworm like (Fig. 7 a, red dotted line), which is in agreement with myosin VI step-size distribution (57), as well as with earlier measurements reporting a load-independent average backward step size of ∼11 nm (72). (Note that the measured step size may depend on the geometry of the motor and the location of the probe.) Second, the majority of backward steps are spontaneous (Fig. 7 b, red dotted line), which agrees with the analysis performed by Ikezaki et al. (59). As expected, the forward steps are predominantly fueled by ATP (Fig. 7 b, solid blue line). Stomps are spontaneous at low ATP concentrations (Fig. 7 b, gray dots), but for [ATP] > 2 μM, these futile cycles waste a molecule of ATP (Fig. 7 b, gray squares). Third, we investigate whether myosin VI uses ATP parsimoniously. We make the following considerations. Backward steps effectively waste ATP because after a rearward movement, it takes an ATP-driven forward step to restore myosin to its initial position. Thus, ATP-fueled backward steps $\left({\langle b\rangle }_{\text{ATP}}\right)$ dissipate the energy from two ATP molecules per backward step, and the imbalance between spontaneous backward and forward steps $\left({\langle b-f\rangle }_{\text{spo}}\right)$ accounts for the net expenditure of one ATP per net spontaneous backward motion to restore the position. In addition, ATP is consumed also when the motor stomps after hydrolysis $\left({\langle s\rangle }_{\text{ATP}}\right)$. These considerations allow us to define the following function, W, which monitors the proportion of futile ATPase cycles:

$$W=\frac{2{\langle b\rangle }_{\text{ATP}}+{\langle b-f\rangle }_{\text{spo}}+{\langle s\rangle }_{\text{ATP}}}{\langle n_{\text{ATP}}\rangle },$$ (22)

where $\langle n_{\text{ATP}}\rangle$ is the average of the total number of ATP molecules hydrolyzed during the processive run. In Fig. 7 c, the function W has a peak around 10 μM, at which both the imbalance of forward and backward spontaneous steps and ATP-driven stomps combined to waste approximately half of the ATP hydrolyzed. At high ATP concentrations (∼1 mM), approximately 16% of the hydrolysis cycles are unproductive, predominantly because of ATP-induced stomps (Fig. 7 c, gray dashed curve). This result constitutes a novel, to our knowledge, prediction, underscores the importance of including stomps to estimate the energetics of motors, and sets an upper bound on the proportion of productive ATP hydrolysis cycles.

Figure 7.

Theoretical predictions. In all figures, blue lines refer to forward steps, backward steps are in red, gray lines are stomps. (a) Forward and backward stepping pathways are shown. Continuous and dashed lines refer to forward hand-over-hand and inchworm-like steps, respectively. Backward hand-over-hand and inchworm-like steps are shown as dash-dotted and dotted lines, respectively. (b) Conditional probabilities of taking a forward step, backward step, or stomp fueled by ATP (continuous line for forward steps, dash-dotted line for backward steps, and squares for stomps) versus spontaneous (dashed lines for forward steps, dotted line for backward steps, and circles for stomps) are shown, given that a forward step, backward step, or foot stomp occurs, respectively. (c) Waste probability W (see Eq. 22) is shown as a function of [ATP]. The black line indicates the value of W, which is the cumulative result of imbalance of spontaneous steps (magenta, dash-dotted line) and ATP-consuming backward steps (red dotted line) and stomps (gray dashed line). To see this figure in color, go online.

Conclusions

Theoretical framework

We presented a theoretical framework to predict the number of steps taken by a processive motor on a periodic lattice before the runs are terminated. Our approach involves only matrix algebra, which can be straightforwardly implemented numerically. The key assumption is that the track is periodic; that is, the set of states available to the system and the rates that connect them do not depend on the macroscopic motor location. This is physically a reasonable assumption for processive motors of the kinesin, dynein, or myosin families, which step along cytoplasmic tracks. We have also assumed that the track is infinitely long, which allows us to avoid introducing boundary effects. In comparison with previous theories, our results hold for an arbitrary network of states; no further derivations are required if an extra state or further transitions are added, only updates of the matrices $\hat{\text{S}}$, $\hat{\text{F}}$, and $\hat{\text{B}}$.

Myosin VI

As an illustration, we applied the theory to study myosin VI processivity, and we fitted the parameters to recover the experimental data (27,59). We started from the experimental evidence on myosin VI motility provided by Yanagida and co-workers and constructed a kinetic model in which both hand-over-hand and inchworm-like steps are considered. Despite the simplifications used for the single-head myosin VI catalytic cycle (see Results and Supporting Materials and Methods), the model for the dimer including ADP binding and release, ATP-induced and spontaneous detachment from the actin filament, forward steps, backward steps, and foot stomps accounted for ATP dependence of velocity, run length, and proportion of backward stepping.

Gating

On the basis of kinetic models that did not include backward stepping, Elting et al. (27) and Dunn et al. (47) argued that the differential rate of ADP release between the LH and TH underlies the myosin VI gating mechanism. In contrast, Sweeney et al. (46) and Oguchi et al. (25) suggested that myosin VI is ATP-binding gated. We identified a prominent role for ADP-release gating, but we cannot rule out an important role for ATP-binding gating in the absence of additional experiments to further constrain the model and test our prediction. In addition, in this model (like many other motor models), ATP binding does not saturate, whereas Sweeney et al. (46) proposed, on the basis of their experiments, that ATP binding to the LH is constant at about 2–3 s⁻¹. The general theoretical framework presented here is well suited to incorporate ATP binding saturation by including a plausible intermediate collision complex to elucidate the effect of saturation on processivity.

Backward stepping and waste

Our model also reproduces the finding that backward steps are shorter than forward steps (57,72) and agrees with a recent analysis suggesting that backward steps are mainly spontaneous (59). In addition, we make the important prediction that at high ATP concentrations, stomps break the tight mechanochemical coupling of myosin VI by wasting approximately 16% of the hydrolyzed ATP molecules.

Future developments

The theoretical framework proposed could be further improved in a number of ways.

• 1)The mean physical velocity is better defined as $\langle L/\tau \rangle$ instead of $\langle L\rangle /\langle \tau \rangle$. Although both $\langle L\rangle$ and $\langle \tau \rangle$ are straightforwardly measured, thereby making $\langle L\rangle /\langle \tau \rangle$ amenable to experimental testing, it would be informative to obtain L/τ for each stepping trajectory to measure the velocity distribution. It was predicted earlier using simple models that P(L/τ) is not Gaussian (38,55), and this also holds for our model (see Figs. S14 d and S15 d).
• 2)In our model, we overestimate the number of states. For instance, when one head detaches from the D or C state, the resulting 1HB states are identical to each other. However, to keep track of the stepping mechanism, we “split” these states into two: one arising from the D state, another from the C state (see Figs. S1–S3). This means that, as an example, a forward inchworm-like step may connect a 1HB C state with a 2HB D state, but the reverse transition is not possible because the 1HB state accessed from the 2HB D state would be listed as D, not C, although they are the same physical state. Therefore, model states differ somewhat from physical states. This distinction is relevant in the context of monitoring the equilibrium state of a kinetic network lacking the absorbing state and with [ATP] = 0 (see Supporting Materials and Methods). The network of model states presents lingering fluxes. However, if we collect these states into physical states, such nonphysical fluxes are eliminated because it must be at equilibrium. Nevertheless, it would be instructive to develop the theoretical framework further to produce a solution to this subtle issue to ensure that nonphysical duplicate states are absent from the model.
• 3)As mentioned before, it would be instructive to introduce the possibility of saturation for the rate for ATP binding, which would allow a more thorough comparison with the gating mechanism supported by Sweeney et al. (46). This requires introduction of an intermediate collision complex in ATP binding.
• 4)We ignored ATP and phosphate binding, ATP release before hydrolysis, and ATP synthesis. A more complete description of the biochemistry of the dimeric motor could be achieved by introducing these other transitions and by deriving the ensuing thermodynamic constraints. The theoretical framework presented here can accommodate these changes as simple modifications of the matrices $\hat{\text{M}}$, $\hat{\text{S}}$, $\hat{\text{F}}$, and $\hat{\text{B}}$. However, constraining additional parameters will require further experimental data.
• 5)There is an inference, based on the observation that the thermodynamic cycle does not close, that there may be two different actin-bound, ADP-bound states (14). These are not accounted for in our modeling but could be relevant because they might alter the thermodynamic constraints.
• 6)It would be instructive to construct a model that explicitly accounts for the states explored by the motor while detached from actin and for binding to actin from these states. In this way, the detached state would not be considered “absorbing.” The advantage of this formulation is that the initial state of the motor can be chosen appropriately. This affects the model at limiting ATP concentration, at which an arbitrary initial condition could inject energy in the system, and at saturating ATP concentration, at which the number of steps taken by the motor might be small enough that the choice of the initial state affects measurable parameters.

Finally, we mention that it is possible to obtain the results here using an alternative approach. Following (28,73,74), in the matrix $\hat{\text{K}}$ of Eq. 11, we may replace all the transitions toward the absorbing state with transitions back to the states that populate the initial probability distribution. This procedure replenishes the system with all of the trajectories absorbed in the detached state by restarting them with the appropriate probability distributions. The trajectories queued serially generate a stationary probability distribution of populating the N states available to the system. From these stationary probabilities, it is possible to obtain the fluxes for forward and backward stepping and the flux to detachment, from which the average run length can be calculated (28). We verified numerically that the averages obtained with our approach and with this steady-state method are in agreement.

Author Contributions

M.L.M. and D.T. designed the research. M.L.M. performed the research. M.L.M., M.A.C., Y.E.G., and D.T. analyzed the data and wrote the manuscript.

Editor: Anatoly Kolomeisky.

Supporting Citations

References (75, 76, 77, 78, 79) appear in the Supporting Material.