Directional Stepping Model for Yeast Dynein: Longitudinal- and Side-Step Distributions

Abstract

Motor proteins are biological machines that convert chemical energy stored in ATP to mechanical work. Kinesin and dynein are microtubule (MT)-associated motor proteins that, among other functions, facilitate intracellular transport. Here, we focus on dynein motility. We deduce the directional step distribution of yeast dynein motor protein on the MT surface by combing intrinsic features of the dynein and MTs. These include the probability distribution of the separation vector between the two microtubule-binding domains, the angular probability distribution of a single microtubule-binding domain translation, the existence of an MT seam defect, MT-binding sites, and theoretical extension that accounts for a load force on the motor. Our predictions are in excellent accord with the measured longitudinal step size distributions at various load forces. Moreover, we predict the side-step distribution and its dependence on longitudinal load forces, which shows a few surprising features. First, the distribution is broad. Second, in the absence of load, we find a small right-handed bias. Third, the side-step bias is susceptible to the longitudinal load force; it vanishes at a load equal to the motor stalling force and changes to a left-hand bias above that value. Fourth, our results are sensitive to the ability of the motor to explore the seam several times during its walk. Although available measurements of side-way distribution are limited, our findings are amenable to experimental check and, moreover, suggest a diversity of results depending on whether the MT seam is viable to motor sampling.

Significance

The function of the microtubule-associated protein motors, kinesin and dynein, is essential for a myriad of intracellular processes. Different measurements on yeast cytoplasmic dynein stepping characteristics appear to be unrelated to each other. We provide a unified physical-statistical model that combines these seemingly independent features with a theoretical expression that accounts for the exertion of a longitudinal load force to yield the longitudinal step distribution at various load forces. The latter is in excellent accord with the measured distributions. Moreover, we deduce the side-step distribution, which, surprisingly, is susceptible to longitudinal load forces and comprises a right- or left-handed bias. This side-way bias is consistent with observations of the helical motion of a nanoparticle carried by a number of motors.

Introduction

The function of the microtubule (MT)-associated protein motors, dynein and kinesin, is essential for a myriad of intracellular processes such as cargo transfer, cell motility, and cell division. Kinesins usually perform either retrograde or anterograde movement (1,2) corresponding to the position of the motor domain (C-terminal kinesin or N-terminal kinesin, respectively)—although a bidirectional motion of kinesin has also been observed (3, 4, 5). Cytoplasmic dynein usually performs retrograde movement toward the MT minus end. Regarding intracellular transport, this implies motion of cytoplasmic dynein toward the centrosome (i.e., the MT organization center) that resides in the vicinity of the nucleus. In this work, we focus on cytoplasmic dynein. In general, dynein is a homodimer motor protein with two heavy chains; each heavy chain of dynein is composed of a tail, linker, AAA + ring (“head”), stalk, buttress, and MT-binding domain (MTBD) (6) (see Fig. 1 a). In broad strokes, the dynein superfamily can be divided into three groups: axonemal dyneins, intraflagella transport dyneins (cytoplasmic dynein-2), and cytoplasmic dyneins (cytoplasmic dynein-1). The latter group can be further divided into subcategories: metazoan (e.g., mammalian dynein) and nonmetazoan (e.g., yeast dynein) (7).

Figure 1.

(a) Illustration of dynein’s structure (not to scale). (b) An illustration of the MTBDs and dynein pivot motion with respect to the MT is given. The dynein pivot is represented by a large (green) dot and the MTBDs by small (red) dots. Sizes are not to scale. To see this figure in color, go online.

Over the years, dynein motors have whetted the interest of many research groups (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18), with a focus on the dynein mechanism of motion and regulation. In large part, the stepping mechanism of dynein (the dynein “power stroke”) has been resolved (6,8,9). However, despite numerous attempts to elucidate the peculiar and broad longitudinal-step-size distribution of dynein (10, 11, 12, 13), the reasons beyond such a broad distribution are yet to be resolved, mainly because of the high number of the elements that might affect dynein step size (such as the dynein and MT structures) and the possible interplay between them. Furthermore, recent studies provide evidence that unlike most types of kinesin, yeast cytoplasmic dynein motors are not limited to steps that are parallel to the MT long axis (x axis) and, in fact, perform steps in various directions that include a (vector) component in the direction of the MT short axis (y axis) (5, 6, 7). Consequently, dynein stepping-size distribution seems to become even more convoluted than previously reported (10,13). In this work, we focus on cytoplasmic yeast dynein motion (henceforth referred to as “dynein”). However, we will also draw some conclusions regarding mammalian cytoplasmic dynein.

Previous studies showed the existence of dynein longitudinal motion toward and away from the MT’s minus end, with and without the exertion of external longitudinal force (10,13,14). So far, most studies have focused on the longitudinal motion, yet only a few of them (11,12,15) have detected dynein side steps (presumably because of experimental difficulties). In this work, to obtain profound knowledge regarding dynein vectorial motion and provide novel, to our knowledge, insights, we deduce the vectorial (directional) step distribution of dynein. We infer such step distribution by predicting the MTBDs’ vectorial motion, which can be readily converted into the dynein pivot vectorial motion. Our prediction is based on a stochastic model that incorporates several elements, including the influence of an external longitudinal force. In the following sections, we will discuss the model basic assumptions, elements, and predictions.

Methods

Our computation is based on the following key elements: 1) prior knowledge of the binding sites of the MT to the dynein MTBD (16,17), 2) the measured probability distribution of the separation vector between the two MTBDs belonging to the same dynein motor (12), 3) the angular probability distribution of a single MTBD translation (12), and 4) theoretical extension to the existence of a load (either positive, i.e., pulling, or negative, i.e., assisting) force acting on the motor. The computer simulations algorithm and description are discussed in the Supporting Materials and Methods, “Computer Simulations” Section (in the same section in the Supporting Materials and Methods, we also provide statistical analysis of the time-autocorrelation of ${\ell }_{\text{x}}$ and ${\ell }_y$, showing that our model generates short-term memory that disappear, however, after just a few motor steps).

The computed distribution for the vectorial step ($\overrightarrow{\ell }$) of MTBDs is then split into distributions for ${\ell }_x$ (the component of $\overrightarrow{\ell }$ along the x axis), describing longitudinal steps, and ${\ell }_y$ (the component of $\overrightarrow{\ell }$ along the y axis), describing side steps. In a similar manner, the vectorial step of dynein pivot is denoted as $\overrightarrow{\text{L}}$, with its components L_x and L_y along the x and y axes, respectively. The mechanical balance between the two MTBDs implies that the dynein pivot is always located between the two MTBDs; hence $\overrightarrow{\text{L}}=\overrightarrow{\ell }/2$, allowing the calculation of the pivot step distribution (see Fig. 1 b). In what follows, we describe the above model inputs in more detail.

Dynein stepping model

Dynein MTBDs only bind to specific binding sites

The MT is a hollow cylinder with a radius of ∼12.5 nm and composed of ∼13 parallel protofilaments that are slightly displaced with respect to one another in the direction of the MT x axis, creating a 3-start pseudohelix structure (18). This pseudohelical structure is created because of a mismatch that breaks the MT helical continuity, also termed as “lattice seam” or, in short, “seam” (19). Each protofilament contains repeating and alternating α- and β-tubulin proteins (18). It was demonstrated that the MTBD attaches to the MT surface mainly at specific sites that are located between the α- and β-tubulin (16,17). Thus, we assume here that MTBDs can only attach to these sites. Under this assumption, an examination of the top projection, i.e., “bird’s-eye view,” of an MT with its seam located at the bottom (see Fig. 2 b) reveals a two-dimensional (2D) array of attachment sites that resides on transverse diagonals (see Fig. 2 c). The longitudinal distance between each neighboring pair of diagonals is 8 nm (18), and the angle between each diagonal and the y axis is ∼0.15 radians (or 7.5°) (see Fig. 3; Fig. S1 b). We further assume that, because of obvious steric effects, MTBDs of a dynein—whose pivot is located in the middle of the cylindrical “upper” half—cannot attach to sites that reside on the cylindrical “lower” half (see Fig. 2, b and c). Consequently, the width of the 2D array is 25 nm.

Figure 2.

Illustration of the 2D binding sites array: (a) MT three-dimensional structure: α- and β-tubulin are denoted as large circles (orange and yellow, respectively), and an example of an MT-binding site to the MTBD is denoted as a small (green) circle (or “dot” henceforth). (b) MT structure as a cylinder with a grid of binding sites is shown. Binding sites are denoted as (green) dots. The arrow indicates the angular position of a dynein pivot (“center of mass”). The “lower” MT cylindrical half (colored in red) indicates a forbidden area where MTBDs cannot bind because of steric effects. (c) The grid of binding sites, or 2D array, is shown as projected from above with respect to dynein pivot/center of mass. The tilted horizontal (blue) lines are the “transverse diagonals” referred to in the main text, and the vertical (black) lines denote the tubulin protofilaments. The longitudinal distance (along the MT long axis) between each pair of adjacent diagonals is 8 nm, and the width of the 2D array (along the MT short axis) is 25 nm, which is the MT diameter. To see this figure in color, go online.

Figure 3.

Illustration of the 2D array structure (in detail) and binding of dynein. Dynein motor protein pivots (pink and green large circles) and their MTBDs (pink and green small circles) binding to the MT lattice are shown. Blue lines denote the transverse diagonals. Black lines denote the protofilaments. Teal lines denote the seam that leads to a 4 nm longitudinal shift of the 2D binding array. Notably, for any perfect helix, the helix angle, α = π/2 − θ, must obey the following equation: α = tan⁻¹(2πr/(d × m)), where r is the helix radius (12.5 nm for MT), m is an integer, d is the helical pitch (8 nm for MT), and d × m is the helical rise after one encirclement (beginning at the helix edge). Because in the MT case, a noninteger value of m is required (m = 1.5, it is not a perfect helix and demands a seam defect. (a) The 2D binding array (as seen from above) with the seam at the middle is shown, along with possible configurations for an MTBD pair as they (the MTBDs) cross the seam. The red line highlights one transverse diagonal with the purpose of emphasizing the symmetric break. Notice that when MTBDs cross the seam, they break the rules and reside on the same diagonal but do not break the rules regarding the forbidden angles. (b) A different perspective of the same 2D array (unlike the case of (a), here the MT cylnder is “spread” to a sheet) is shown, with the seam located at the left side, along with configurations of MTBDs that do not cross the seam. Thus, the MTBDs obey all the rules mentioned in the main text without any exceptions. The pale gray lines indicate protofilaments that are not in the 25 nm width range. The angle θ of 0.15 radians (or 7.5°) is denoted in yellow and located between MT short axis and the transverse diagonal (both denoted in red). (c) An illustration of the spread of the MT three-dimensional cylinder to a 2D sheet is given.

The theoretical interplay between two MTBDs

Consider two MTBDs that belong to the same dynein molecule, henceforth referred to as MTBD1 and MTBD2. Although the forces that act between them are fascinating per se, the explicit description of these forces is beyond the scope of this work. Instead, we chose to use experimental results as follows.

Angular distribution. According to the 2D attachment array (see Fig. 3), if, hypothetically, MTBD1 and MTBD2 have attached to the same diagonal and have not crossed the seam, the sharp angle between the separation vector (between the two MTBDs) and the y axis would be 7.5°. However, by marking the two AAA + rings (“heads”) with different colors and measuring the angle between them (12), it has been shown that there is zero probability for 7.5° angle between two rings (of the same motor). Although each marker is on the AAA + ring and not on the MTBD itself (i.e., it is separated by a 15 nm stalk, not precisely in the cylinder radial direction (6)), those measurements do give a strong indication regarding the MTBDs location. Thus, we assume that there is also zero probability for 7.5° angle between the two MTBDs, suggesting that they cannot occupy the same diagonal, as long as they do not cross the seam. If the MTBDs do cross the seam, they can attach to the same diagonal without breaking the 7.5 angle limitation. Other than the latter angular constraint, we do not use the measured angular distribution explicitly.

Length distribution. Demanding that the two MTBDs do not occupy the same diagonal suggests that the shortest possible separation distance between them—the distance between two (parallel) neighboring diagonals—is 7.99 or ∼4.62 nm if they cross the seam (see Fig. 3, a and b, and Fig. S1 c and S3). In addition, to account for the forces that act between MTBD1 and MTBD2, we use the probability histogram as presented in Qiu et al.’s work, Fig. 5 b (we would like to underline the fact that numerical values of the inter-MTBD distance probabilities were extracted from the figures in (7) with estimated error less than 10% because we do not have the actual numerical data).

Figure 5.

Dynein pivot displacements under ATP-affluent conditions and under exertion of a pulling longitudinal force for a pseudohelical MT structure with adequate sampling of the seam. Plots are computed from 2.5 × 10⁷ computer steps each. (a) Displacements along the MT x axis (longitudinal steps—see Fig. 1b for a spatial orientation) are shown. (b) Displacements along the MT y axis (side steps—see Fig. 1b for a spatial orientation) are shown.

A single theoretical probability distribution of the MTBD translation vector

The same experiment mentioned above (12) measures the angular distribution of the “heads” motion and shows that the “heads” never move in the 7.5° direction. Using the same reasoning as in The Theoretical Interplay between Two MTBDs, it follows that MTBD1 or MTBD2 never moves in the 7.5° direction. This implies that unless the MTBD crosses the seam during its motion, MTBD cannot make a step along the same diagonal on which it already resides. As a result, considering the 25 nm array width (along the y axis), $\left|{\ell }_x\right|$ cannot be shorter than 4.25 nm (Document S1. Supporting Materials and Methods and Figs. S1–S6, Document S2. Article plus Supporting Material e) if the MTBD does not cross the seam, whereas it is ∼0.37 nm (Fig. S3) if the MTBD does cross the seam; see Fig. 3, a and b. This implies that if the MTBD does not cross the seam, the smallest value of |L_x| in a single MTBD stepping event is 2.125 nm (i.e., L_x values between −2.125 and +2.125 nm are forbidden).

Stepping rules deduced from subsections 1–3

The discussion in subsections 1–3 leads to the following MTBD stepping rules. First and foremost, 1) each MTBD can only attach to specific sites on the MT surface. In addition, if the MTBDs do not cross the seam, 2) the two MTBDs cannot reside on the same diagonal, and 3) each one of them must change diagonal as it moves to another attachment site. As a consequence of rules 1–3, as long as the MTBDs do not cross the seam, each MTBD cannot step onto a diagonal on which the other MTBD already resides. According to the above rules, the distribution for $\overrightarrow{\text{L}}$ varies according to the temporal configuration of the two MTBDs.

Inclusion of a longitudinal load force

An MTBD stepping probability function $\text{P}\left(\overrightarrow{\ell }\right)$ is built as a product of two functions, $\text{H}\left(\overrightarrow{\ell }\right)$ and G$\left({\ell }_{\text{x}}\right)$, and a renormalization of the resulting probability. The former, $\text{H}\left(\overrightarrow{\ell }\right)$, is a general function that accounts for the model rules that were mentioned above. The latter, G$\left({\ell }_{\text{x}}\right)$, is a force exertion function, which was previously shown to be in accord with experimental longitudinal step distributions of dynein under force (20). This function was derived based on detailed balance between forward and backward stepping rates, under a mechanical potential composed of the motor-generated force and the pulling force. $\text{G}\left({\ell }_{\text{x}}\right)$ thus accounts for the forward bias in terms of the motor internal force F_s (working forward), which derives from ATP hydrolysis, and an external longitudinal pulling force F and reads

$$\text{G}\left({\ell }_{\text{x}}\right)=\frac{1}{1+\text{exp}\left[-\frac{{\ell }_{\text{x}}\left({\text{F}}_{\text{s}}-\text{F}\right)}{2{\text{k}}_{\text{B}}{\text{T}}_{\text{eff}}}\right]}=\frac{1}{1+\text{exp}\left[-\frac{{\ell }_{\text{x}}}{2}\text{Z}\left(1-\frac{\text{F}}{{\text{F}}_{\text{s}}}\right)\right]},$$

where Z = F_s/k_BT_eff, k_B is the Boltzmann constant, and T_eff is an effective temperature that will be taken as a fit parameter (see discussion at the end of the next section). Equivalently, we can use Z as a fit parameter, which is advantageous, as described later. By construction, the internal motor force is exactly equal in magnitude to the stall force, F_s = 4 pN for yeast dynein (as described in (21,22)).

Results

Effective temperature of dynein

We applied the model rules 1–4 in Monte Carlo computer simulations and computed the resulting distribution of $\overrightarrow{\text{L}}$ in ATP-affluent environment, with (or without) the exertion of external longitudinal pulling forces. When we use the room temperature (298 K) as T_eff to calculate the relative fractions of forward and backward steps in the case of an ATP-affluent system and without an external force (F = 0), we find almost 100% forward steps, which contradicts the experimental results reported in (12), 77% and 23%, of forward and backward steps, respectively. This apparent contradiction is resolved by replacing the actual temperature by the effective temperature that is used as a fit parameter or, alternatively, a fit of Z. This leads to Z $\cong$ 0.1997 nm⁻¹, implying that for F_s = 4 pN, the effective temperature that recovers the experimental values mentioned above (77% and 23%, for forward and backward steps, respectively) is T_eff = 1451 K. The success of the use of a single effective temperature is further established by the excellent agreement (we assume that the seam is hardly sampled by the motor, so it does not affect the motor motion (see Fig. 6)) between our model results and the experimental results reported in (10), in which dynein motion under different values of external pulling force was measured. Moreover, the mean longitudinal step of dynein that stems from T_eff = 1451 K is roughly 9 nm, whereas the use of room temperature leads to 12 nm. This value, 9 nm, is in agreement with several experiments that report a longitudinal step size distribution with a mode of 8 nm and a tail of long steps (> 8 nm) (12,13).

Figure 6.

Dynein pivot displacements under ATP-affluent conditions and under exertion of a pulling longitudinal force for a pseudohelical MT structure without the sampling of the seam defect. Plots are computed from 2.5 × 10⁷ computer steps each. (a) Displacements along the MT x axis (longitudinal steps—see Fig. 1b for a spatial orientation) are shown. (b) Displacements along the MT y axis (side steps—see Fig. 1b for a spatial orientation) are shown.

More generally, because F_s measurements are controversial (e.g., reported values are 7 pN (10,13) and 4 pN (21,22)), we can report on the fit value of Z because our fit to the experimental data—regarding the ratio between forward and backward steps under native condition (F = 0)—remains the same. Thus, for F_s = 4 pN, the fit value Z corresponds to an effective temperature of T_eff = 1451 K (whereas, for F_s = 7 pN, T_eff = 2540 K). The fact that the fit of Z is independent of the measured value of F_s allows to use our results even if the value of F_s is modified.

Dynein pivot mean stepping vector

The results for the angular distribution of the dynein pivot movement under different longitudinal force values are shown in Fig. 4. Note that although the angular distribution provides information regarding the relation between longitudinal and side steps, it does not represent the vector size of the dynein pivot. Therefore, for further analysis, we deduce the L_x and L_y distributions for two cases: in the first case, the dynein encounters the seam (Fig. 5), whereas in the second case, the dynein never encounters the seam (Fig. 6).

Figure 4.

Angular histograms of MTBDs displacements under ATP-affluent conditions and with an external force. The MT long axis is parallel to the 90–270° axis (90° is the MT minus-end direction). The MT short axis is parallel to the 0–180° axis; left and right (as referred to in the text) correspond to 0 and 180°, respectively (see Fig. 1b for a spatial orientation). F is the longitudinal pulling force, meaning that it pulls against the motor direction toward the MT plus end. (a–d) Angular histograms for F/F_s = 0, 0.43, 0.86, and 1.43 are given, with mean angles of 139 ± 0.0171, 154 ± 0.0171, 180 ± 0.0171 (or 0), and 205 ± 0.0171, respectively (SEM, n = 25,000,000 each). To see this figure in color, go online.

As expected, the longitudinal (L_x) distributions in Figs. 5 a and 6 a show that when F = 0 pN, forward steps (L_x > 0) prevail; when F = F_s, the distribution becomes symmetric; and when F > F_s, the distribution is again asymmetric, with a predominance of backward steps (L_x < 0). Fig. 6 a shows a “hole” in the L_x distribution around the origin, between −3.065 nm < L_x < +3.065 nm, which is related to stepping rule 3 and the previous discussion in A Single Theoretical Probability Distribution of the MTBD Translation Vector. On the other hand, in Fig. 5 a, in which the dynein encounters the seam, the “hole” is absent.

The transverse (L_y) distribution, depicted in Figs. 5 b and 6 b, indicates that the motion of single dynein in its native state (F/F_s = 0, F = 0 pN) is slightly biased toward the right-hand side of the y axis (L_y > 0). Calculating the mean transverse component from the distribution leads to <L_y> $\cong$ 0.04 ± 0.0028 nm SEM (n = 2.5 × 10⁷). As the pulling force increases from zero, the distribution becomes less biased and achieves complete symmetry for F = F_s, as for the distribution for L_x. When F > F_s, the bias shifts toward the left-hand side of the y axis (L_y < 0). Moreover, as the pulling force decreases from zero and turns into an assisting force, the mean transverse component value significantly increases (see Fig. 7) until it reaches a plateau for large enough forces (see Fig. S4).

Figure 7.

Force-dependence of longitudinal and transverse mean step size. The mean longitudinal $\langle {\text{L}}_{\text{x}}\rangle$ and transverse $\langle {\text{L}}_{\text{y}}\rangle$ step sizes against longitudinal pulling force F are shown; a negative force indicates an assisting force (i.e., a force that is in the MT minus-end direction), and positive indicates a resisting force (i.e., a force that is in the MT plus-end direction). Note, if one assumes equal dwell time τ for all steps, independent of the exerted force, the motor velocity is given by = $\langle \text{L}\rangle$/τ. Therefore, these graphs also depict the velocity trends. (a) Transverse mean step size against force (SEM = 0.0021 nm, 2.5 × 10⁷ computer steps for each force value) is shown. We note that regarding the results with the seam being sampled (red dots), the statistical error is likely larger than the above numerically estimated value because of an insufficient number of seam sampling events. (b) Longitudinal mean step size against force (SEM = 8.98 × 10⁻⁴ nm, 2.5 × 10⁷ computer steps for each force value) is shown. To see this figure in color, go online.

Discussion

In this work, we presented a model that aims to predict yeast dynein motion on the 2D microtubule surface. Our model assumptions are based on several distinct and independent experimental results. First, we considered data that imply the existence of specific MTBD attachment sites on the MT surface. Second, we incorporated the interplay between a pair of MTBDs, i.e., the distance and angle probability distributions between two MTBDs.

This study can be regarded as a first-principle calculation of the (phenomenological) bare dynein motor step-size distribution under stall force conditions (denoted as f₀$\left(\ell \right)$ in (20)). Moreover, our model predictions for a system under ATP-affluent conditions and exertion of external longitudinal pulling force are consistent with a variety of experimental results. Notice that we did not examine the case of ATP-depleted system because experimental results indicate that dynein remains stagnant under such conditions (10).

Let us discuss first the L_x distribution (as depicted in Figs. 5 and 6). Under external force, our model predictions are as per those described in (10). Also, our model suggests a plausible explanation for the “hole” (i.e., vanishing probability) around the origin in the L_x distribution, as measured in (23). Yet, according to our model, the effective size of the “hole” depends on whether the seam has been encountered or not. Therefore, we believe that we have acquired a reasonable explanation for the seeming discrimination between different experiments regarding the magnitude of the “hole,” e.g., a deep and wide “hole” or a smeared one.

Next, consider dynein transverse motion (L_y) and the effect of external longitudinal pulling forces over such motion. Under native conditions and without external force (F/F_s = 0, F = 0 pN), regardless of whether the seam is adequately sampled or not sampled at all, the L_y distribution shows a meager bias toward right side steps. For the case in which the seam is not sampled adequately, a situation we assume to prevail in most cases, <L_y> $\cong$ 0.04 ± 0.0028 (SE; n = 2.5 × 10⁷). Thus, combining this result with the prediction for the mean longitudinal step size in the absence of external force, <L_x> $\cong$ 9 ± 0.0066 nm (SE; n = 2.5 × 10⁷), a single motor is predicted to move in a right-handed helical pattern with a helical pitch size of 17.7 μm. Obviously, such a motion cannot be experimentally observed because the required longitudinal run length of at least 17.7 μm exceeds the mean run length of 1.9 μm (13) by more than eightfold. Thus, we believe that, in the absence of external force, the transverse motion might usually seem unbiased. Surprisingly, we also found that when the longitudinal pulling force (F) is moderately larger than the stall force (F_s), the L_y distribution shifts to give a small bias toward left-handed side steps (L_y < 0), describing a motor that is walking backward and making a right-handed helical motion. Although a small or moderate longitudinal force does not result in significant bias, the exertion of an adequate longitudinal force (whether it is pulling or assisting) enhances the transverse bias significantly (see Fig. 7 b), e.g., for (an assisting) force F/F_s = −1.43, <L_y> $\cong$ 0.09 ± 0.0028 nm (SE; n = 2.5 × 10⁷), corresponding to a right-handed helix with a pitch size of 7.8 μm. Note that the exerted force is purely longitudinal; nevertheless, it still affects the transverse motion, implying that the cross Onsager coefficients of linear response of the motor to force, coupling the x and y axes, are nonzero (24). Measuring the side-step distribution under different values of longitudinal pulling force would thus be highly valuable to confirm these nontrivial predictions of our model. Although the exertion of a transverse force is also likely to enhance the L_y bias, we did not include such forces because of the lack of experimental results that could be used as validation.

Importantly, measurements of the helical motion of a bead, possibly being carried by more than one motor, have been reported in experiments in which the MT is lifted (well above the flow chamber surface), thus allowing free motion of the motors on the MT surface (11). The agreement is, however, only partial. First, the observed pitch helical size is ∼500–600 nm, much shorter than our prediction. Second, Can et al. also observed left-handed helical motion. We believe that these differences stem from the fact that very likely (as also mentioned by the authors of (11)), several motors are carrying the particle simultaneously. Therefore, intermotor coupling forces, which include both longitudinal and transverse components, may strongly affect the motion. This ought to motivate further investigations.

What is the physics behind the fitted effective temperature? We believe that it is caused by the partial efficiency in converting the chemical energy, released in ATP hydrolysis, to mechanical work. This implies that part of the stored energy renders into an active force noise, unable to instantaneously dissipate into the surrounding medium. Active noise has been shown in a variety of active systems to be equivalent to an effective temperature that can be several times higher than room temperature (25, 26, 27). Indeed, because a single ATP hydrolysis releases about 8.3 × 10⁻²⁰ J = 20 k_BT (28), it follows that, if the motor efficiency is η, the energy that generates an active force noise is (1 − η)20 k_BT. Thus, for T_eff = 1451 K, the efficiency, estimated from the equality k_BT_eff = k_BT + (1 − η)20 k_BT, is about η = 81%, and for T_eff = 2540 K, the efficiency is about η = 62%; both values are well within the range of reasonable values for an efficient molecular machine. Therefore, although we do not expect the concept of effective temperature and its fitted values to be entirely accurate, we can argue that these values are within a plausible range.

What can our results imply regarding the motion of mammalian cytoplasmic dynein? Although cytoplasmic mammalian and yeast dynein differ in their properties, our model can be extended to describe the motion of mammalian cytoplasmic dynein by several modifications and assumptions. First—unlike yeast dynein—to become processive, mammalian dynein requires additional cofactors: dynactin and BICD2 (21,29, 30, 31, 32). Therefore, we must assume that these cofactor proteins are present in optimal concentrations (consequently, the involved kinetics do not impair the motor processivity). Also, we need to assume similar MT-MTBD interactions. In addition, two modifications are required. First, the value of F_s must be changed to 4.3 pN, the mammalian dynein stall force (21), and, second, the MTBDs’ separation vector distribution of mammalian dynein must be acquired. We hope that investigations concerning the latter will be carried out in the future.

Author Contributions

I.F. and R.G. designed the model, analyzed data, and wrote the manuscript.

Editor: Anatoly Kolomeisky.