Effect of detachment of motor protein from track on its transport

Abstract

The transportation of the cargoes in biological cells is primarily driven by the motor proteins on filamentous protein tracks. The stochastic nature of the motion of motor protein often leads to its spontaneous detachment from the track. We formulate a mathematical model to study the effect of the detachment of motor protein on its track bound transport. We calculate two quantities: the distance traveled by the motor protein in given time, and the average time taken by a single motor protein to reach a target distance. Expectedly, both of these quantities decrease with the increasing detachment rate if the motor velocity is kept fixed. However, the existing experimental data suggest that a change in the detachment rate also affects the velocity of the motor protein. This relation between motor protein speed and its detachment rate results in a non-monotonic dependence on the distance traveled in fixed time and transport rate to a fixed distance. Therefore, we demonstrate that optimal motor speeds can be identified for the time and distance controlled conditions.

Supplementary Information

The online version contains supplementary material available at 10.1007/s10867-022-09613-z.

Introduction

Motor proteins are deployed for the directional intra-cellular transportation. These proteins move along the filamentous tracks, formed of actin and tubulin proteins, at the expense of the energy released from ATP hydrolysis [1, 2]. The prominent examples of these motor proteins include kinesin, dynein and myosin which move along the microtubules and actin filaments. Kinesin and dynein motors move in the opposite directions on microtubules, whereas myosins travel towards the positive end of the F-actin [1–3]. The dynamics of the motor protein transport along the tracks has been studied extensively, and the aspect of the spontaneous motor protein detachment, that is without application of any external force, from the tracks has also been under investigation [4–6] (and references therein). The transport of the track-bound motor protein and their detachment from it has been experimentally investigated by tracking of single motors in physiological [7–14] and non-physiological conditions [15, 16] as well as of their engineered counterparts [17–19]. Similarly, modeling approaches have also studied the influence of the spontaneous motor detachment [20–23] and binding [24] with the track on its overall transport properties. In all of these works the distance covered by the motor proteins before its detachment from the track has been defined as ‘processivity’ and the influence of different motor intrinsic properties (amino acid sequence of the motor protein [17, 18], motor heads [8, 19]) and external factors (temperature [15, 16], track structure [14]) on processivity has been quantitatively studied by the measurement of motor speed and force generation.

Despite the availability of extensive quantitative measurements, the rates of spontaneous motor protein detachment from the track have not been measured explicitly under diverse conditions where motor speed and force are known to be influenced. As a result, in different modelling studies the spontaneous detachment rate is considered to be independent of the motor speed [21, 22]. Recent experimental evidence, however, demonstrates that the rate of spontaneous motor detachment from the track and its speed are correlated for Kinesin and Dynein [15, 17, 18]. In this work we have studied the effect of this experimental observation on the transport properties of a single motor protein using a one-dimensional random walk model. The effect of the motor speed and detachment rate relationship is studied in two conditions—time controlled setup and distance controlled setup. In these two conditions we obtain the distance travelled by the motor protein in a given duration and the time taken by the motor protein to cover a pre-specified distance, respectively.

Materials and methods

Mathematical model

Basic assumptions

We consider the transport of a single motor protein molecule along the corresponding filament (for example unbranched filamentous actin for myosin or microtubule for kinesin and dynein). Since we are focusing on the transport of a single motor protein, we do not take into account any interaction between motor proteins, such as cooperative [25] or competitive [26] behaviors. Further, we also assume the filament track to be stable and ignore its polymerization and depolymerization dynamics. Since we are looking at the transport kinetics on the track, once the motor protein is detached from the track we do not consider its reattachment. The model presented here, however, can be extended to include the effect of the motor protein reattachment to the track.

Drift-diffusion with detachment

The detachment of motor protein from the track limits it from reaching its target destination, the end of the filament track within a finite time. Therefore, an estimate of the distance covered by the motor protein in given duration (time controlled setup) and the time required for covering a pre-specified distance (distance controlled setup) are crucial aspects for complete understanding of the motor driven cellular transportation.

In general, the motion of the motor proteins on track is constituted of forward (and in some cases backward too) steps of different sizes where each step is driven by ATP hydrolysis [1]. Additionally, the motor protein can also spontaneously get detached from the track at any cycle with a detachment probability $p_e$. Considering the motion of the motor proteins on the tracks as a one-dimensional random walk, its position on the track at any given time can be characterized by the probability density P(x, t) which follows the Fokker-Planck equation [27]

$$\begin{array}{c}\hfill \frac{\partial P(x,t)}{\partial t}=-v\frac{\partial P(x,t)}{\partial x}+D\frac{{\partial }^{2}P(x,t)}{\partial x^2}-P(x,t)r_e\end{array}$$ 1

where v and D are the motor speed and its diffusion constant on the track, respectively, whereas $r_e$ rate of motor detachment from the track. We can obtain $r_e$, the rate of motor protein detachment, from the available discrete experimental data (see Sect. 2.1.5) for detachment probability. For this, we assume the transport of the motor protein to be a Markov process [28] for which the relationship between transition rate and probability is given by [29, 30]

$$\begin{array}{c}\hfill r_e=-log(1-p_e)/\tau ,\end{array}$$ 2

where $\tau$ is the time required for the hydrolysis of single ATP. It has to be noted that for small values of $p_e$, $r_e=p_e/\tau$. In order to understand the properties of the stochastic processes, the first passage time distribution is often used [27]. In the context of the motor protein motion the first passage time distribution, f(L, t), is defined as the time distribution for the motor protein crossing a target distance L. In other words, the probability that the motor protein will cover a distance L between time t and $t+dt$ is given by f(L, t)dt. Using the approach described in [27] we can derive the first passage time distribution corresponding to the Eq. (1) as

$$\begin{array}{c}\hfill f(L,t)=\frac{L}{\sqrt{4\pi D{t}^3}}exp\left(-,r_e,t,-,\frac{{(L-vt)}^2}{4Dt}\right)\end{array}$$ 3

In the scenario where the motor does not detach from the track ($r_e=0$), the above expression reduces to the first passage time distribution for the Brownian particle with drift. In the following, we will use the first passage time distribution in the time and distance controlled setups.

Time controlled setup: distance travelled in fixed time

A measure of a motor protein transport can be obtained in terms of the distance covered by it in a pre-specified duration. The average distance covered in time T can be written as

$$\begin{array}{c}\hfill L_m(T)=\underset{0}{\overset{\infty }{\int }}L\left[\underset{0}{\overset{T}{\int }},\left(\frac{\partial f}{\partial L}\right),d,t\right]dL\end{array}$$ 4

where the integral in $[·]$ represents the probability density of the motor protein at a distance L after time T. The integral over L is performed to calculate the mean value of this distribution. In general, the analytical evaluation of the above integral is not straight forward but it can be calculated for some special cases. For purely ballistic motion, that is $D=0$, we can evaluate the above integral by substituting the first passage time distribution $f(L,t)=e^{-r_{e}t}\delta \left(t,-,L,/,v\right)$ to obtain

$$\begin{array}{c}\hfill L_m\left(T,,,D,=,0\right)=\frac{v}{r_e}\left(1,-,e^{-r_{e}T}\right).\end{array}$$ 5

On the other hand, for the purely diffusive motion ($v=0$), we obtain

$$\begin{array}{c}\hfill L_m\left(T,,,v,=,0\right)=\sqrt{\frac{D}{r_e}}\text{erf}\left(\sqrt{r_{e}T}\right)\end{array}$$ 6

where $\text{erf}(x)$ is the error function. From the relations (5) and (6), it is apparent that $L_m$ attains finite values $L_s=v/r_e$ and $L_s=\sqrt{D/r_e}$, respectively, for $T\to \infty$ and is defined as processivity.

Distance controlled setup: motor transport to fixed distance

Similar to time controlled setup, the motor transport can also be analyzed in the distance controled setup where the rate of motor travel to a fixed distance is calculated. In this scenario, due to its detachment from the track the motor protein is not necessary to reach the pre-specified distance. Therefore, first we need to calculate the probability that the motor protein can reach the given distance. This probability is given by

$$\begin{array}{c}\hfill P_r(L)=\underset{0}{\overset{\infty }{\int }}f(L,t)dt.\end{array}$$ 7

In the case it can cover the distance ($P_r(L)>0$), the average time taken to achieve it can be written as

$$\begin{array}{c}\hfill T_m(L)=\frac{1}{P_r(L)}\underset{0}{\overset{\infty }{\int }}tf(L,t)dt.\end{array}$$ 8

It has to be noted that for $r_e=0$, the above integral diverges but $r_e>0$ results in finite $T_m$. Further, in this distance controlled setup the ratio $\varphi (L)=P_r/T_m$ can be defined as the rate of motor transport to a fixed distance. The ratio $\varphi (L)$ can be interpreted as the fraction of the motor proteins reaching a target distance per unit time. Low rate of motor protein transport to a specified distance ($\varphi (L)\sim 0$) could either be due to fast detachment of the motor protein from the track or due to its extremely slow motion along the track. On the other hand, $\varphi (L)$ takes high values only if $P_r$ is high and $T_m$ is low.

Similar to the time controlled setup, we can evaluate the probability and average time taken for reaching the target distance in the limiting cases. For ballistic motion, we obtain

$$\begin{array}{cc}\hfill P_r\left(L,,,D,=,0\right)& =e^{-r_{e}L/v}=e^{-L/L_s},\hfill \end{array}$$ 9

$$\begin{array}{cc}\hfill \text{and}{T}_m\left(L,,,D,=,0\right)& =\frac{L}{v}.\hfill \end{array}$$ 10

where $L_s=v/r_e$ (see Eq. (5)) for ballistic motion. For diffusive motion

$$\begin{array}{cc}\hfill P_r\left(L,,,v,=,0\right)& =exp\left(-,\sqrt{\frac{L^2{r}_e}{D}}\right)=e^{-L/L_s},\hfill \end{array}$$ 11

$$\begin{array}{cc}\hfill \text{and}{T}_m\left(L,,,v,=,0\right)& =\frac{L}{2\sqrt{D{r}_e}}\hfill \end{array}$$ 12

where $L_s=\sqrt{D/r_e}$ (see Eq. (6)) for diffusive motion. Expectedly, for ballistic motion the average time $T_m$ is not dependent on the detachment probability. On the other hand for diffusive transport ($v=0$), we obtain $T_m\to {\displaystyle \frac{L}{2}}\sqrt{{\displaystyle \frac{\tau }{D{p}_e}}}\sim 1/\sqrt{p_e}$ as $p_e\to 0$.

Experimental observations

The distance based processivity, as defined previously, does not give any insight into the detachment probability, $p_e$, at any given time. To obtain the experimental estimates of the $p_e$, we looked into the literature where duration of the motor attachment to the protein tracks was measured along with its speed. We found three studies where the duration of motor attachment and its velocity were reported [15, 17, 18] (see Tables S1–S3 in the supplementary information for detailed experimental data).

In the work by Nara and Ishiwata [15] the motion of kinesin was reported at different temperatures to study the dependence of velocity and duration of attachment on temperature. On the other hand the works by Yildiz et al. [17] and Cleary et al. [18] studied the effect of the neck linker lengths on the velocities and attachment duration for kinesin and dynein motor proteins. For these three reports, when plotted on log-log scale, the average duration of the motors’ attachment to the track $T_{\mathit{avg}}$ and their speeds reveal monotonic relationships which can be approximated as power law relations, $T_{\mathit{avg}}\sim v^{-\alpha }$ with $\alpha >0$ (Fig. 1a). This correlation between the motor speed and rate of detachment was not highlighted in the respective studies [15, 17, 18]. Furthermore, a compilation of $T_{\mathit{avg}}$ and the speeds of different motors (myosins, kinesins, dynein, KIF1 motor proteins) from multiple independent experimental reports from [7–11, 14, 15, 17–19, 31, 32] also demonstrated similar relation with $\alpha =1.1$ (Fig. 1b, see Table S4 in supplementary information). The positive values of $\alpha$ indicate a tradeoff between the motor speed and its duration of attachment to the track as an increase in the speed leads to faster detachment. In the following we have utilized this speed-attachment duration relation to explore the transport of motor proteins.

Fig. 1.

a Relationship between experimentally measured motor speed, v, and average duration of its attachment to the track, $T_{\mathit{avg}}$ for Kinesin and Dynein motors under different conditions as reported in [15, 18] and [17]. b The speed and the average attachment duration for different types of Myosins, Kinesins, Dynein, KIF1 motor proteins to their respective tracks as per experimental reports in [7–11, 14, 15, 17–19, 31, 32]. The lines in (a) and (b) show the power law fit $T_{\mathit{avg}}\sim v^{-\alpha }$

Assuming the average duration of the motor attachment to the track $T_{\mathit{avg}}$ to be inversely proportional to detachment probability $p_e$, we get $p_e\sim v^{\alpha }$. Assuming a reference motor protein velocity $v_0$ corresponding to reference detachment probability $p_e^0$, this gives

$$\begin{array}{c}\hfill \frac{p_e}{p_e^0}={\left(\frac{v}{v_0}\right)}^{\alpha }.\end{array}$$ 13

By substituting $p_e=1-e^{-r_e\tau }$ from Eq. (2) we obtain

$$\begin{array}{c}\hfill \frac{1-e^{-r_e(v)\tau }}{1-e^{-r_e^0\tau }}={\left(\frac{v}{v_0}\right)}^{\alpha }\end{array}$$ 14

where $r_e^0$ is the reference rate of detachment. We substitute the expression for $r_e$ from the above relation to obtain the average distance covered in finite time and the transport rate to a finite distance as a function of the exponent $\alpha$.

It has to be pointed out here that here we are only considering a relation between the motor protein velocity and its rate of detachment. For simplicity, the diffusivity of the track-bound motor protein is assumed to remain unchanged. This simplification can be justified as following. Since for long distances (with respect to the motor protein step size on the track) the motor protein transport is dominated by its ballistic motion the effect of its diffusion is small.

Results and discussion

Parameter values

In this paper we have taken the transport of kinesin motor on microtubule as an example to understand the effects of the speed dependent detachment rate. The values of the model parameters for kinesin motor are shown in Table 1.

Table 1.

Values of the model parameters for kinesin motor used in the current study

Parameter	Value	Reference
$v_0$	640 nm/s	[2]
$p_e^0$, $r_e^0$	0.01, 0.35/s	[15]
$\tau$	1/80 s	[2]
D	$2500n{m}^2/s$	[33]

The similar analysis can also be performed for dynein and myosin motor proteins.

Influence of motor detachment from the track

The detachment of the motor protein from the track limits the distance it can cover in finite time and the rate of its transport to a specific distance. In order to study the effect of the change in velocity with the detachment probability we first calculated the processivity of the motor protein when its velocity does not change with $p_e$. This is shown for kinesin motor on microtubule track for fixed velocity and diffusivity in Fig. 2a. If the motor velocity remains fixed, for small values of $p_e$ its processivity decreases as $1/p_e$ and $1/\sqrt{p_e}$ for velocity and diffusivity dominant transports, respectively. This decrease in processivity, however, is contrary to the experimental observations where it shows an increase with $p_e$ [15, 18]. This shows that the assumption of fixed rate of detachment [21, 22] independent of motor speed may not be correct.

Fig. 2.

Effect of kinesin detachment from microtubule on its transport for fixed speed and diffusivity. a Dependence of kinesin processivity $L_s$ under diffusive transport ($v=0$ and $D=2500n{m}^2/s$, dashed line), ballistic transport ($v=640nm/s$ and $D=0$, dash-dotted line), and mixed transport ($v=640nm/s$, $D=2500n{m}^2/s$, solid line) on detachment probability $p_e$. Inset shows the enlarged version of the same plot for large $p_e$. b Probability, $P_r$, of covering a distance of 100nm by kinesin on a microtubule track before detachment under diffusive (dashed), ballistic (dash-dotted) and mixed (solid) modes of transport. c Rate of kinesin transport on microtubule to a distance of 100nm for aforementioned three modes of transport

In the distance controlled setup, for fixed velocity and diffusivity the probability of motor protein reaching a target distance decreases with $p_e$ (Fig. 2b) in both of the diffusion and drift dominated modes of transport. The transport rate to the target distance $\varphi (L)$, however, shows different behaviors in diffusion and drift dominated motions (Fig. 2c). For purely drift driven motor protein transport, the transport rate decreases with $p_e$. But in the diffusion driven transport, the transport rate follows a non-monotonic relationship with $p_e$. This is because for large $p_e$, the probability of reaching the target distance is negligible whereas for very small $p_e$ the the time taken for complete transport is very large (see relations (11) and (12)). The disagreement between the experimental observations of increased processivity with the detachment rate and model predictions with constant motor speed suggests that the dependence of motor speed on the detachment rate has significant effect on its transport properties. In the following we will study the effect of motor speed-detachment rate relationships on its transport under time and distance controlled conditions.

Effect of speed dependent detachment rate

Motor transport in fixed time

The experimental data showed (Fig. 1) that the motor speed and its detachment rate can be characterized as power law with exponent $\alpha$. With this power-law relationship, the distance covered by the motor protein in finite time (Fig. 3a) shows a non-monotonic dependence on $p_e$ which is unlike monotonically decreasing dependence when the motor speed does not change (Fig. 2a). Since very small values of $p_e$ correspond to negligible motor speed (thanks to the relation between $T_{\mathit{avg}}$ and v, Fig. 1), the distance covered in finite time takes a constant value ($L_m(T)=2\sqrt{DT/\pi }$) which is determined by the diffusivity of the motor protein alone. On the other hand, for drift dominated scenario ($D=0$) the distance covered in finite time increases as $p_e^{1/\alpha }$ for small values of $p_e$. For large $p_e$, the motor protein gets detached from the track very fast and does not traverse any distance. Figure 3b shows the distance covered by the kinesin on microtubule in finite time for different $\alpha$. Apart from the maximum value attained by $L_m$, the nature of the dependence of $L_m$ on detachment probability does not change with the power law exponent $\alpha$. As $T\to \infty$ the distance covered by the motor on track converges to its processivity, $L_s$ and it is shown in Fig. 3c. As opposed to the distance covered in finite time, the qualitative nature of processivity depends significantly on $\alpha$. It can be seen that for the motor proteins with $\alpha <1$, such as kinesin, the processivity increases with the detachment probability, whereas for motor proteins like dynein, for which $\alpha >1$, it decreases. In order to understand this dependence of processivity on $\alpha$ we can substitute the value of v from relation (13) in (5) to obtain

$$\begin{array}{c}\hfill L_s=\underset{T\to \infty }{lim}{L}_m=\frac{v_0}{r_e}{\left(\frac{p_e}{p_e^0}\right)}^{1/\alpha }.\end{array}$$ 15

Fig. 3.

Distance travelled by kinesin on microtubule in $100\mu s$ with velocity dependent detachment rate. a Distance covered under diffusive (dashed), ballistic (dash-dotted) and mixed (solid line) modes of transport with $\alpha =1$. b Distance covered by kinesin for $\alpha =0.8,1.2,1.6,2.0$. c Dependence of motor protein processivity on $p_e$ for different $\alpha$

For small values of $p_e$ this simplifies to

$$\begin{array}{c}\hfill L_s=\frac{v_0\tau }{{\left(p_e^0\right)}^{1/\alpha }}{p}_e^{(1-\alpha )/\alpha }.\end{array}$$ 16

for drift dominated transport. This relation shows that for $\alpha <1$ an increase in detachment probability leads to much larger increase in the motor velocity and hence increased processivity and vice versa.

Motor transport to fixed distance

Similar to the track-bound transport of motor protein in finite time we can also study the effect of the $v-p_e$ relations (Fig. 1 and Eq. (14)) on its transport to a fixed distance. The probability of reaching a target distance by the kinesin motor on microtubule is shown in Fig. 4a which shows that for drift dominated scenario ($D=0$) the probability $P_r$ attains a maximum. This is opposed to the scenario when the motor speed does not change with the detachment probability (Fig. 2b). On the other hand, for the rate of motor transport $\varphi (L)$ both of the drift and diffusion dominated regimes attain local maxima (Fig. 4b). Further, for small values of $p_e$ the rate of transport $\varphi$ increases as $\sqrt{p_e}$ and $p_e^{1/\alpha }$ in the drift and diffusion dominated motions of the motor protein on the track, respectively. Figure 4c and d show the dependence of $P_r$ and $\varphi$ on $p_e$ for difference values of $\alpha$. It has to be noted that the maxima attained by the $P_r$ vanishes for $\alpha >1$. However, the rate of transport $\varphi$ shows a maximal transport for all values of $\alpha$. This demonstrates that for track-bound transport of motor proteins to a fixed distance there exists an optimal velocity which maximizes its rate of transport to the target location.

Fig. 4.

Transport of kinesin to a fixed distance with velocity dependent detachment rate. a Probability $P_r$ of reaching and b rate of transport of kinesin motor to distance of $100\mu m$ before detachment under three modes of transport. c The probability $P_r$ and d the transport rate to the target distance of $100\mu m$ for $\alpha =0.8,1.2,1.6,2.0$

Optimum motor velocity

As shown in Figs. 3b and 4d the track-bound transport of the motor proteins has non-monotonic dependence on the motor speed (or detachment rate). Therefore, faster motor speed on a track does not necessarily mean its faster transport. In fact, these two non-monotonic relationships show that one can estimate two optimal motor velocities—firstly, the velocity for its transport to farthest distance in a given time (from Fig. 3b); secondly, the velocity for which the rate of motor protein transport is fastest to a pre-specified distance (from Fig. 4d). A comparison of these two optimal velocities with the physiological ones can shed light on the potential mechanisms of motor protein driven transport.

For kinesin, its velocity in physiological conditions $v=640nm/s$ turns out to be optimal for its farthest transport in time $T\sim 10s$. For the cellular functions, this duration is too large. On the other hand, the same physiological velocity of kinesin is optimal for the fastest transport to a distance of $L>5\mu m$. This distance is roughly of the same order of magnitude as that of the size of a single cell. This demonstrates that kinesin motor is more likely to move at speeds to maximize the transport rate to a given distance instead of maximizing the distance in given time. This observation is reasonable as in physiological conditions the transport of cellular cargo is to fixed distances as the cell size remains more or less unchanged.

It has to be noted that in physiological conditions motor protein does not constantly move on a single track. For instance, in case of kinesin transport on microtubule tracks it keeps hopping among different microtubule filaments [34] and also undergoes diffusive motion in cytosol. Furthermore, consumption of ATP is another factor governing the motor transport in biological systems. From the economy viewpoint, the optimal motor velocity can also be determined by the one which requires minimal ATP consumption. Therefore the estimation of optimal speed of the motor protein has to take these factors into account which is not in the scope of present study.

Biological implications and future directions

In this paper, using existing experimental data, we have demonstrated a novel monotonic relationship between the speed of the motor protein and the duration of its attachment to the track as a tradeoff between the two. Although an insight into the exact nature of this relationship will require a detailed experimental and modelling endeavour, a physical understanding of its monotonic nature for bipedal motors can be obtained by the following argument. The kinesin and other bipedal motors move on the tracks by the sequential attachment and detachment of the two heads [3]. The speed and the detachment probability depends on the duration of attachment of each motor head in each cycle. In case of high (low) motor speed, its both heads undergo fast (slow) attachment-detachment transitions which also leads to high (low) detachment probability. For the range of the motor speeds presented here, as shown in Fig. 3c and relation (16), a power-law approximation leads to the prediction that increase in the speeds of the kinesin ($\alpha <1$) and dynein ($\alpha >1$) motors would result in higher and lower processivity, respectively. For kinesin, it has already been observed experimentally by varying motor speed by temperature changes [16]. However, the same is yet to be experimentally verified for dynein.

Further, the observation of the tradeoff is based on different experimental studies performed under diverse experimental conditions. Therefore, a systematic experimental study is warranted to test the conditions under which the speed-detachment tradeoff holds. To achieve this the model predictions presented in this paper can be utilized.

It has to be noted that although the exponent $\alpha$ is the characteristic of the motor protein, it also depends on its microenvironment. This can be seen from the experimental data shown in Fig. 1a where the average duration of attachment of the kinesin motor to the microtubule track is obtained by two groups ([15] and [17]) differs significantly if not by an order of magnitude. Similarly, in Fig. 1B also we observe multiple values of the attachment duration and the speed of the kinesin motors. The reason for this discrepancy is the differing experimental conditions used in each setup and hence this aspect has to be kept in consideration during classification of motor proteins based on $\alpha$. Similar speed-accuracy tradeoffs have also been observed in the context of DNA replication [35], road-safety [36], human movements [37, 38], decision-making [39, 40] and perceptual learning [41]. Although the underlying mechanisms of the speed-accuracy tradeoffs might be different in different systems, its influence on the overall system dynamics would still hold.

It is important here to mention that in the model we have only considered the relation between motor protein detachment and its velocity while assuming its diffusivity to be fixed. This indeed is a simplification, especially in the context of the temperature dependent experimental conditions reported in [15] where diffusivity of motor protein on the track is also expected to change with temperature. Despite this simplification, the predictions of the model are not expected to change significantly for most physiological conditions where the ballistic transport dominates the diffusion. In order to keep the model tractable, we have also kept the duration of ATP hydrolysis $\tau$ fixed here which might not be always true. Experimental investigations are warranted to shed light on the nature of change in the diffusivity and the ATP hydrolysis kinetics to get the complete picture of the system. Further, even though we have focused only on the track-bound transport of the motor protein in reality the unbound protein also gets transported via diffusion in the cytoplasm. For a complete picture both of the states of the motor protein (track-bound and unbound) have to be taken into account.

Furthermore, it is known that motor proteins often undergo collective motions on the tracks. Therefore, it would be interesting to see the nature of the speed-detachment tradeoff during the collective motion of the motor proteins. It is not necessary that the speed-detachment tradeoff as seen in the single motor protein experiment will be present during their collective motion. Therefore, this also warrants for experimental investigations to look at the nature of motor detachment from a crowded track and its relation with its translational velocity. Once the nature of speed-detachment tradeoff for motor protein collectives is known a similar analysis can be applied to study its effect on collective transport by motor proteins. Additionally, in this work we have not considered the reattachment of the motor protein to the track after its detachment which is relatively more realistic scenario in cells. We can extend the model presented here to a two-state description (attached state and detached state) to study this aspect of cellular transport.

Conclusion

This work utilizes the available experimental observations to propose a novel relationship between the speed of motor proteins and the rate of their detachment from the track. This tradeoff is further utilized to study the characteristics of the motor transport along the track in fixed time and to a fixed distance. It is shown that faster motor speed does not necessarily mean faster or farthest transport. Using these properties optimal velocity of the motor protein is also estimated under different scenarios. The results shown here can be used for systematic experimental investigations on the role of motor protein detachment on its transport and its dependence on microenvironment.

Supplementary Information

Below is the link to the electronic supplementary material.

Author Contributions

MSR designed the study, formulated the model and simulations and wrote the manuscript.

Funding

This work was partially supported by the Institute Seed Grant from IIT Hyderabad.

Declarations

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Conflict of interest

The author declares no competing interests.