Effective behavior of cooperative and nonidentical molecular motors

Abstract

Analytical formulas for effective drift, diffusivity, run times, and run lengths are derived for an intracellular transport system consisting of a cargo attached to two cooperative but not identical molecular motors (for example, kinesin-1 and kinesin-2) which can each attach and detach from a microtubule. The dynamics of the motor and cargo in each phase are governed by stochastic differential equations, and the switching rates depend on the spatial configuration of the motor and cargo. This system is analyzed in a limit where the detached motors have faster dynamics than the cargo, which in turn has faster dynamics than the attached motors. The attachment and detachment rates are also taken to be slow relative to the spatial dynamics. Through an application of iterated stochastic averaging to this system, and the use of renewal-reward theory to stitch together the progress within each switching phase, we obtain explicit analytical expressions for the effective drift, diffusivity, and processivity of the motor-cargo system. Our approach accounts in particular for jumps in motor-cargo position that occur during attachment and detachment events, as the cargo tracking variable makes a rapid adjustment due to the averaged fast scales. The asymptotic formulas are in generally good agreement with direct stochastic simulations of the detailed model based on experimental parameters for various pairings of kinesin-1 and kinesin-2 under assisting, hindering, or no load.

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Availability of data and material (data transparency)

Simulation data was generated to test the model. As the main product of this manuscript is the theoretical framework, we do not view the simulated data as being of sufficient value to make publicly available.

Notes

  1. 1.

    This latter procedure is presented in more generality in a separate publication “Renewal reward perspective on linear switching diffusion systems in models of intracellular transport” by M. V Ciocanel, J. Fricks, P. R. Kramer, and S. A. McKinley

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Acknowledgements

This work grew out of conversations between JF and PRK while both were long-term visitors at the Isaac Newton Institute for the program on “Stochastic Dynamics in Biology: Numerical Methods and Applications.” The authors also would like to thank Will Hancock for discussions informing the model development. PRK would also like to acknowledge early discussions with Leonid Bogachev, Leonid Koralov, and Yuri Makhnovskii which helped me map out the general mathematical framework and approaches, while we were all supported as long-term visitors at the Zentrum für Interdisziplinäre Forschung for the program on “Stochastic Dynamics: Mathematical Theory and Applications.” Here we’ve managed to work out one of the easier cases we considered...

Funding

The work of JF and PRK are partially supported by National Institutes of Health grant R01GM122082 and PRK was partially supported by a grant from the Simons foundation. The work of JK is partially supported by an National Science Foundation RTG grant 1344962.

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Correspondence to Joseph J. Klobusicky.

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The authors declare no conflicts of interest.

Code availability (software application or custom code)

Code was developed simply to simulate the model to check the theory. As no novel computational methodology was involved, we do not view the code to be of sufficient general interest to publish it.

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Dedicated to Andy Majda for his 70th birthday, with gratitude for his lasting inspiration starting from my undergraduate and graduate days on the creative deployment of mathematical modeling and the beautiful application of analysis techniques as a lens for exploring and understanding the dynamics of physical systems - PRK.

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Appendices

A Appendix: A note on nonlinear spring models

A linear model for representing the tether between the motor and the cargo is not particularly accurate. Better theoretical tether models can involve a model which is Hookean for extension beyond a rest length, but offers no resistance to compression [25, 32, 54, 56, 67, 68], a sigmoidal stiffness dependence on force [59], or a multiple-component model for the motor-cargo tether including separate models for the neck linker and stalk [40]. We may generalize the averaging results by considering a nonlinear spring

$$\begin{aligned} F^{(i)} (r)={\bar{F}}^{(i)}\varPhi '^{(i)}(r/L_c^{(i)}) \quad 1 \le i \le N. \end{aligned}$$
(99)

Here \(\varPhi ^{(i)}(r)\) is a nondimensionalized spring potential, \(L_c^{(i)}\) is an appropriate length scale, and \({\bar{F}}^{(i)}\) is a characteristic force magnitude for each motor index i. We define \( \kappa ^{(i)} \equiv {\bar{F}}^{(i)}/L_c^{(i)} \) as an “effective” spring constant of the nonlinear spring; it agrees with the usual spring constant when the spring force model is purely Hookean, as in the main text.

With the same nondimensionalization as before, the equations of motion become, for 1 \(\le i \le N,\)

$$\begin{aligned}&d{\tilde{X}}^{(i)}({\tilde{t}}) = \left( \epsilon ^{(i)} g(s^{(i)} {\tilde{\kappa }}^{(i)}(\lambda ^{(i)})^{-1}\varPhi '^{(i)}(\lambda ^{(i)}(\tilde{X}^{(i)}({\tilde{t}})-\tilde{Z}({\tilde{t}}))))d{\tilde{t}}+\sqrt{{\hat{\rho }}^{(i)} \epsilon ^{(i)}}\hbox {d}W^{(i)}(\tilde{t})\right) Q^{(i)}({\tilde{t}}) \end{aligned}$$
(100)
$$\begin{aligned}&\quad + \left( -\left( \varGamma ^{(i)}\lambda ^{(i)}\right) ^{-1}\tilde{\kappa }^{(i)}\varPhi '^{(i)}\left( \lambda ^{(i)}\left( \tilde{X}^{(i)}({\tilde{t}})-{\tilde{Z}}({\tilde{t}})\right) \right) d{\tilde{t}}+(\varGamma ^{(i)})^{-1/2}\hbox {d}W^{(i)}(\tilde{t})\right) (1-Q^{(i)}({\tilde{t}})), \quad \nonumber \\&d{\tilde{Z}}({\tilde{t}})=\left( \sum _{j = 1}^N (\lambda ^{(i)})^{-1}{\tilde{\kappa }}^{(i)}\varPhi '(\lambda ^{(i)}(\tilde{X}^{(j)}(\tilde{t})-{\tilde{Z}}({\tilde{t}})))-\tilde{F}_T\right) d{\tilde{t}} + d W_{z}(\tilde{t}), \end{aligned}$$
(101)

where we have introduced the new nondimensional parameter

$$\begin{aligned} \lambda ^{(i)} = \frac{\sqrt{2k_\mathrm{B}T/{\bar{\kappa }}}}{L_c^{(i)}}. \end{aligned}$$
(102)

which describes the length-scale of thermally induced variations on the tether relative to the length scale of variation of the tether force law. Calculations for the averaged drifts \({\bar{g}}^{(i)}\), and thus \(G_+\) and \(G_-\), are similar, but now involve pairing drift functions with non-Gaussian stationary distributions for unattached motors and cargo (the forms for these equations are nearly identical to those found in Appendix A in McKinley, Athreya, et al [34]). For detachment jumps, no assumptions of distribution type are made for \(p_R(r)\), and therefore, the calculations for distributions in Sect. 4.3.2 only need to be adjusted to refer to the mean cargo position under nonlinear tethers.

The random variable \(\varDelta M^{(\mathrm {a})}_{i}\) describing the change of position of the cargo tracking variable after motor attachment may be computed similarly as in Sect. 4.3.3, but it will no longer be normally distributed or have mean zero in general. The calculations in Sect. 5 otherwise go through for a nonlinear tether model, with only the modified contribution to the moments of the cargo tracking variable changes at attachment and detachment jumps.

B Appendix: Derivation of effective diffusion for two motors

The following derivation for the effective diffusion of the cargo tracking variable during a state with both motors attached to the microtubule follows the multiscale expansion method illustrated in Pavliotis and Stuart [75], with rigorous exposition in Veretennikov and Pardoux [94] for the unbounded state space case relevant to our application. Having computed the effective drift \( V_{1,2}\) in Eq. (34) in this state, we pass to a diffusive scaling centered about this mean drift \( {\bar{t}} \rightarrow t/\epsilon ^2 \), \( M \rightarrow \epsilon (M-V_{1,2}t) \), with the internal configuration variable R unscaled (\( R \rightarrow R\)). Note in this appendix, \( \epsilon \) is just a formal small parameter used to push to long time; it is unrelated to the physically meaningful nondimensional parameters \( \epsilon ^{(i)} \) and \( {\bar{\epsilon }} \) in the main text. Moreover, for simplicity for calculations within this appendix, we use the undecorated variables tMR to describe the dynamics under this centered diffusive rescaling, which read:

$$\begin{aligned} \hbox {d} M(t)&= \frac{1}{\varepsilon }\left( G_+( R(t))-V_{1,2}\right) \hbox {d}t+\frac{\sqrt{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2}}{2}\hbox {d}W^{(1)}(t)+\frac{\sqrt{\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}}{2} \hbox {d}W^{(2)}(t) \nonumber \\\end{aligned}$$
(103)
$$\begin{aligned} \hbox {d} R(t)&= \frac{1}{\varepsilon ^2} G_-( R(t)) \hbox {d}t+\frac{1}{\varepsilon }\left( \sqrt{\rho ^{(1)}}\hbox {d}W^{(1)}(t)-\sqrt{\rho ^{(2)}} \hbox {d}W^{(2)}(t)\right) . \end{aligned}$$
(104)

The infinitesimal generator \({\mathcal {L}}\) for (103)–(104) is defined by its action on a test function \(v = v(m,r)\), with

$$\begin{aligned} \mathcal {L}v(m,r) = {\mathbf {h}}\cdot \nabla v+\frac{1}{2} \varGamma : \nabla \nabla v. \end{aligned}$$
(105)

Here \({\mathbf {h}}(m,r) = ((G_+(r)-V_{1,2})/\varepsilon , G_-(r)/\varepsilon ^2)\) is the drift vector, and \(\frac{1}{2} \varGamma \) is the diffusion tensor, where

$$\begin{aligned} \varGamma&= \begin{bmatrix} \frac{\sqrt{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2}}{2}&{} \frac{\sqrt{\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}}{2} \\ \frac{\sqrt{\rho ^{(1)}}}{\varepsilon }&{} -\frac{\sqrt{\rho ^{(2)}}}{\varepsilon }\\ \end{bmatrix} \begin{bmatrix} \frac{\sqrt{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2}}{2}&{} \frac{\sqrt{\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}}{2} \\ \frac{\sqrt{\rho ^{(1)}}}{\varepsilon }&{} -\frac{\sqrt{\rho ^{(2)}}}{\varepsilon }\\ \end{bmatrix}^T \end{aligned}$$
(106)
$$\begin{aligned}&= \begin{bmatrix} \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{4}&{} \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{2\varepsilon } \\ \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{2\varepsilon } &{}\frac{\rho ^{(1)}+\rho ^{(2)}}{\varepsilon ^2}\\ \end{bmatrix}. \end{aligned}$$
(107)

We have also used notation for the Frobenius inner product for matrices, where for matrices \(A = (a_{i,j})_{nxm}\) and \(B = (b_{i,j})_{nxm}\), we define \(A:B = \sum _{i,j} a_{i,j}b_{i,j}\).

We may write out (105) explicitly as

$$\begin{aligned} \mathcal {L}v(m,r)&= {\mathbf {h}}\cdot \nabla v+\frac{1}{2} \varGamma : \nabla \nabla v \end{aligned}$$
(108)
$$\begin{aligned}&= \frac{1}{\varepsilon }(G_+(r)-V_{1,2})v_m+\frac{1}{\varepsilon ^2} G_-(r)v_r \end{aligned}$$
(109)
$$\begin{aligned}&\quad +\frac{1}{2} \left[ \left( \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{4}\right) v_{mm}+ \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{\varepsilon }v_{mr} + \left( \frac{\rho ^{(1)}+\rho ^{(2)}}{\varepsilon ^2}\right) v_{rr}\right] . \end{aligned}$$
(110)

The generator may be decomposed to match powers of \(\varepsilon \) as

$$\begin{aligned} {\mathcal {L}} = \frac{1}{\varepsilon ^2}{\mathcal {L}}_0+ \frac{1}{\varepsilon }\mathcal {L}_1+ {\mathcal {L}}_2, \end{aligned}$$
(111)

with

$$\begin{aligned} {\mathcal {L}}_0&= G_-(r)\partial _r +\frac{\rho ^{(1)}+\rho ^{(2)}}{2}\partial _{rr}, \end{aligned}$$
(112)
$$\begin{aligned} {\mathcal {L}}_1&= \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{2}\partial _{mr}+\left( G_+(r)-V_{1,2}\right) \partial _m, \end{aligned}$$
(113)
$$\begin{aligned} {\mathcal {L}}_2&= \left( \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{8}\right) \partial _{mm}. \end{aligned}$$
(114)

Assuming a multiscale solution \(v = v_0+\varepsilon v_1+\varepsilon ^2 v_2 +\dots \) for the backward Kolmogorov equation

$$\begin{aligned} \frac{\partial v}{\partial t} = \mathcal {L}v, \end{aligned}$$
(115)

we match powers of orders \(1/\varepsilon ^2, 1/\varepsilon ,\) and 1 to obtain

$$\begin{aligned} {\mathcal {L}}_0 v_0&= 0, \end{aligned}$$
(116)
$$\begin{aligned} -{\mathcal {L}}_0 v_1&= {\mathcal {L}}_1 v_0, \end{aligned}$$
(117)
$$\begin{aligned} -{\mathcal {L}}_0 v_2&= -\frac{\partial v_0}{\partial t}+ {\mathcal {L}}_1 v_1+ \mathcal {L}_2 v_0. \end{aligned}$$
(118)

The first equation implies that \(v_0\) is only a function of m and t. From here, the second equation may be simplified to

$$\begin{aligned} -{\mathcal {L}}_0 v_1 =\left( G_+(r)-V_{1,2}\right) \partial _{m}v_0(m,t). \end{aligned}$$
(119)

As the operator \({\mathcal {L}}_0\) only depends on r, we may express \(v_1\) as

$$\begin{aligned} v_1(m,r,t) = \chi (r)\partial _{m}v_0(m,t). \end{aligned}$$
(120)

Proceeding to the third equation of the asymptotic expansion, the Fredholm alternative states that for (118) to have a solution, its right-hand side must be orthogonal to \(p_R(r)\), or

$$\begin{aligned} \frac{\partial v_0}{\partial t}&= \int _{\mathbb {R}} p_R(r){\mathcal {L}}_2 v_0(m,t)\hbox {d}r+\int _\mathbb R p_R(r){\mathcal {L}}_1(\chi (r)\partial _m v_0(m,t))\hbox {d}r \end{aligned}$$
(121)
$$\begin{aligned}&: = I_1+I_2. \end{aligned}$$
(122)

We look at each integral in turn. First,

$$\begin{aligned} I_1&= \int _{\mathbb {R}} p_R(r) {\mathcal {L}}_2v_0(m,t) \hbox {d}r \end{aligned}$$
(123)
$$\begin{aligned}&= \int _\mathbb R p_R(r)\left[ \left( G_+(r)-V_{1,2}\right) \partial _m v_0 (m,t)+\left( \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{8}\right) \partial _{mm}v_0(m,t)\right] \hbox {d}r \end{aligned}$$
(124)
$$\begin{aligned}&= \left( \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{8}\right) \partial _{mm}v_0(m,t). \end{aligned}$$
(125)

The second integral may be broken up further, as

$$\begin{aligned} I_2&= \int _{\mathbb {R}} p_R(r){\mathcal {L}}_1(\chi (r)\partial _m v_0(m,t))\hbox {d}r \end{aligned}$$
(126)
$$\begin{aligned}&= \int _{\mathbb {R}} p_R(r)\Big [\left( \frac{\rho ^{(1)}\kappa ^{(1)}-\rho ^{(2)}\kappa ^{(2)}}{2}\right) \partial _{mr}(\chi (r)\partial _m v_0(m,t)) \end{aligned}$$
(127)
$$\begin{aligned}&\quad +\left( G_+(r)-V_{1,2}\right) \partial _m(\chi (r)\partial _m v_0(m,t))\Big ]\hbox {d}r \end{aligned}$$
(128)
$$\begin{aligned}&:= I_3+I_4. \end{aligned}$$
(129)

The first part satisfies

$$\begin{aligned} I_3&= \int _{\mathbb {R}} p_R(r)\left( \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{2}\right) \partial _{mr}(\chi (r)\partial _m v_0(m,t))\hbox {d}r \end{aligned}$$
(130)
$$\begin{aligned}&= \left( \left( \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{2}\right) \int _\mathbb {R} p_R(r)\partial _{r}\chi (r)\hbox {d}r\right) \partial _{mm} v_0(m,t) \end{aligned}$$
(131)
$$\begin{aligned}&:= A_1\partial _{mm} v_0(m,t). \end{aligned}$$
(132)

Finally, we have

$$\begin{aligned} I_4&= \int _{\mathbb {R}} p_R(r)\left[ \left( G_+(r)-V_{1,2}\right) \partial _m(\chi (r)\partial _m v_0(m,t))\right] \hbox {d}r \end{aligned}$$
(133)
$$\begin{aligned}&= \int _{\mathbb {R}} p_R(r)\left[ \left( G_+(r)-V_{1,2}\right) \chi (r)\right] \hbox {d}r\partial _{mm} v_0(m,t) \end{aligned}$$
(134)
$$\begin{aligned}&:= A_2\partial _{mm} v_0(m,t). \end{aligned}$$
(135)

The closed form equation for \(v_0(m,t)\) is thus given by

$$\begin{aligned} \frac{\partial v_0}{\partial t} = \frac{1}{2}\left( \frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{4}+2A_1+2A_2\right) \partial _{mm}v_0(m,t), \end{aligned}$$
(136)

and is the backward Kolmogorov equation for the SDE

$$\begin{aligned} \hbox {d} M(t)&=\sqrt{\frac{\rho ^{(1)}({\tilde{\kappa }}^{(1)})^2+\rho ^{(2)}({\tilde{\kappa }}^{(2)})^2}{4}+2A_1+2A_2}\hbox {d}W(t) \end{aligned}$$
(137)
$$\begin{aligned}&\equiv \sqrt{2D^{(1,2)} }\hbox {d}W(t), \end{aligned}$$
(138)

where W(t) is a standard Brownian motion.

Now we compute explicit expressions for constants \(A_1\) and \(A_2\). This involves solving the cell problem for \(\chi \), given by

$$\begin{aligned} -G_-(r)\chi '(r)-\left( \frac{\rho ^{(1)}+\rho ^{(2)}}{2}\right) \chi ''(r)&= {\tilde{g}}_+ (r), \end{aligned}$$
(139)
$$\begin{aligned} \quad \int _{{\mathbb {R}}}\chi (r)p_R(r)\hbox {d}r&= 0, \end{aligned}$$
(140)

where we define \({\tilde{g}}_+(r) =G_+(r) -V_{1,2}\). If we rewrite (139), using an integration factor \(\mu (r)\), as

$$\begin{aligned}{}[\mu (r)\chi '(r)]' = -\mu (r){\tilde{g}}_+(r)\left( \frac{2}{\rho ^{(1)}+\rho ^{(2)}}\right) , \end{aligned}$$
(141)

then it is straightforward to show that \(\mu (r) \) is in fact equal to the stationary distribution \(p_R(r)\).

Integrating out (141) leaves us with

$$\begin{aligned} \chi '(r) =- \int _{-\infty }^r {\tilde{g}}_+(r^{\prime })\left( \frac{2}{\rho ^{(1)}+\rho ^{(2)}}\right) p_R(r^{\prime })\hbox {d}r^{\prime }/p_R(r)+C/p_R(r) \end{aligned}$$
(142)

for some unknown integration constant C. By the subexponential growth requirement on \(\chi \) and \(\chi '\) [94], it follows that \(C= 0\). From (131),

$$\begin{aligned} A_1&= \left( \frac{\rho ^{(2)}{\tilde{\kappa }}^{(2)}-\rho ^{(1)}{\tilde{\kappa }}^{(1)}}{\rho ^{(1)}+\rho ^{(2)}}\right) \int _\mathbb {R} \int _{-\infty }^r {\tilde{g}}_+(r^{\prime })p_R(r^{\prime })\hbox {d}r^{\prime } \hbox {d}r \end{aligned}$$
(143)
$$\begin{aligned}&= \left( \frac{\rho ^{(1)}{\tilde{\kappa }}^{(1)}-\rho ^{(2)}{\tilde{\kappa }}^{(2)}}{\rho ^{(1)}+\rho ^{(2)}}\right) \int _\mathbb {R} {\tilde{g}}_+(r)p_R(r)\cdot r\hbox {d}r. \end{aligned}$$
(144)

The last equality used integration by parts, in which the boundary term vanishes under the assumption that \(p_R(r)= o(1/r^2)\) as \(r \rightarrow \pm \infty \). The calculation for \(A_2\) (135) follows from integration by parts, with

$$\begin{aligned} A_2&= \int _{-\infty }^\infty \chi (r){\tilde{g}}_+ (r)p_R(r)\hbox {d}r =- \int _{-\infty }^\infty \chi (r) ({\mathcal {L}}_0 \chi (r)) p_R(r)\hbox {d}r \end{aligned}$$
(145)
$$\begin{aligned}&=- \int _{-\infty }^\infty \chi (r) \left( G_-(r)\chi ' (r)+\left( \frac{\rho ^{(1)}+\rho ^{(2)}}{2}\right) \chi '' (r)\right) p_R(r)\hbox {d}r \end{aligned}$$
(146)
$$\begin{aligned}&= \int _{-\infty }^\infty \left( \frac{\rho ^{(1)}+\rho ^{(2)}}{2}\right) \chi '(r)^2p_R(r)\hbox {d}r \end{aligned}$$
(147)
$$\begin{aligned}&= \int _{-\infty }^\infty \left( \frac{2}{\rho ^{(1)}+\rho ^{(2)}}\right) \left( \int _{-\infty }^r {\tilde{g}}_+ (r^{\prime }) p_R(r^{\prime } )\hbox {d}r^{\prime } \right) ^2\frac{1}{p_R(r)}\hbox {d}r. \end{aligned}$$
(148)

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Klobusicky, J.J., Fricks, J. & Kramer, P.R. Effective behavior of cooperative and nonidentical molecular motors. Res Math Sci 7, 29 (2020). https://doi.org/10.1007/s40687-020-00230-7

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Keywords

  • Molecular motors
  • Stochastic averaging
  • Switched diffusion
  • Renewal-reward theory

Mathematics Subject Classification

  • 74Q15
  • 92C40
  • 65C30