Motor Protein Transport Along Inhomogeneous Microtubules

Following the general TASEP paradigm, we construct a discrete model from first principles which generalizes previous models for homogeneous MTs to incorporate a defective region. Briefly, the TASEP paradigm states: (i) each binding site may be occupied by a maximum of one motor; (ii) motors move unidirectionally on the lattice and (iii) motors enter the lattice on the left side and exit the lattice on the right side ParFraFre2003 . Here we also account for the attachment and detachment of motor proteins on the MT interior as in works similar in scope focused on modeling ParFraFre2003 ; ParFraFre2004 ; Fre11 and experiment Var09 . Another recent work has focused on stochastic modeling with the goal of revealing how motor protein and filament properties affect transport Dal19 .

Specifically, consider a one-dimensional lattice $\left\{0=x_{0}<x_{1}<...<x_{M}=\ell\right\}$, representing sites which a motor may occupy on a microtubule of length $\ell$. Introduce $\rho_{i}^{n}$ which is the probability of finding a motor at site $x_{i}$ at time step $n$. Probabilities at each lattice site $0\leq\rho_{i}^{n}\leq 1$ for $1\leq i<M$ change during one time step via

The first term on the right-hand side of (1) says that the probability of finding a motor at site $x_{i}$ increases due to a possible jump of a motor from site $x_{i-1}$ to $x_{i}$ provided that the following jump condition is satisfied: there is a motor at site $x_{i-1}$ and site $x_{i}$ is vacant. As it is done in previous works on one-dimensional transport of active motors along a microtubule ParFraFre2003 ; ParFraFre2004 ; Fre11 , consider the problem in the mean-field approximation, that is, correlations are negligibly small and the probability of the jump condition is simply $\rho_{i-1}^{n}(1-\rho_{i}^{n})$. Additionally, the coefficient $v_{i-\frac{1}{2}}$ accounts for inhomogeneity of the microtubule: if the jump condition is satisfied on the interval $[x_{i-1},x_{i}]$, then the jump occurs with probability (motility rate) $v_{i-\frac{1}{2}}$, and these coefficient may change from site to site. The second term on the right-hand side of (1) is similar to the first term, but accounts for the decrease in probability $\rho_{i}^{n}$ due to a possible jump from site $x_{i}$ to $x_{i+1}$. The third term on the right-hand side of (1) describes the interaction of the microtubule with the exterior environment: a motor from outside can attach to the microtubule at site $x_{i}$ as well as a motor already occupying site $x_{i}$ can detach from the microtubule. Parameters $\omega_{A}$ and $\omega_{D}$ are the corresponding attachment and detachment rates.

Stationary states of (1) solve the following system:

This system is supplemented with boundary conditions corresponding to the attachment rate $\alpha$ at the left end and detachment rate $\beta$ at the right end of the microtubule:

We note here that it is assumed that boundary attachment rates have a corresponding relationship to those inside the microtubule, that is,

and $v_{i}\equiv v$, then the solution of (2)-(3) is simply a constant: $\rho\equiv\omega_{A}/(\omega_{A}+\omega_{D})$. However, in practice, rates $\alpha$ and $\beta$ are different from (4) which leads to non-trivial stationary solutions possessing interior jumps, even in the homogeneous case $v_{i}\equiv\text{const}$ ParFraFre2003 ; ParFraFre2004 .

We end this subsection with another form of (2) which is helpful for understanding the continuous limit presented in Section 2.2. Introduce the following notation for the flux between sites $x_{i-1}$ and $x_{i}$:

Note that if one interpolates $\rho_{i}$ by a smooth function $\rho(x)$ such that $\rho(x_{i})=\rho_{i}$, then performing Taylor expansions one can verify that

where $\rho_{i-\frac{1}{2}}=\rho(x_{i-\frac{1}{2}})$, $\Delta x=x_{i}-x_{i-1}$, and $x_{i-\frac{1}{2}}=x_{i}-\frac{1}{2}\Delta x$.

Going back to definition (5), we point out that, in terms of $J_{i\pm\frac{1}{2}}$, equation (2) has the following form:

We focus on the asymptotic behavior of solutions of (2)-(3) as $M\to\infty$ in the framework of the mean-field limit. Specifically, we introduce parameter $\varepsilon:=\ell M^{-1}\ll 1$ (here $\ell$ represents the total length of microtubule) and lattice $\left\{x_{i}=i\varepsilon,\,i=0,..,M\right\}$ with the distance between lattice points $\varepsilon$. For small $\varepsilon$, we approximate the solution with the continuous function $\rho_{\varepsilon}(x)$ defined on $0<x<\ell$ and derived from a discrete set of unknowns $\rho_{i}$ associated with the lattice points $x_{i}$, i.e., $\rho_{\varepsilon}(x_{i})=\rho_{i}$. Then for $\varepsilon\ll 1$, the system of algebraic equations (2) for the unknown $\rho_{i}$’s becomes a second order ordinary differential equation for unknown function $\rho_{\varepsilon}(x)$:

where $v(x)$ is the velocity the motor proteins move with at location $x$, and $\Omega_{A/D}=M\omega_{A/D}$ are properly rescaled attachment/detachment rates. The equalities in (3) become boundary conditions for $\rho_{\varepsilon}(x)$:

One can also consider (8) in the distributional sense. Then using the definition of $J_{\varepsilon}(x)$, the integration of (8) in $x$, and the absolute continuity of integrals together imply the continuity of flux $J_{\varepsilon}(x)$:

In what follows below, we assume that for all $\varepsilon>0$, there exists unique smooth solution $0\leq\rho_{\varepsilon}(x)<1$ which solves (8) and satisfies the boundary conditions (9). Moreover, there exists a piecewise smooth function $\rho_{0}(x)$, referred to as the “outer solution”, such that

By calling $\rho_{0}(x)$ the outer solution we stick to the standard terminology of singularly perturbed ordinary differential equations Hol2013 . While the outer solution $\rho_{0}(x)$ is the pointwise limit of $\rho_{\varepsilon}(x)$ for $0\leq x\leq\ell$, it does not approximate the exact solution $\rho_{\varepsilon}(x)$ uniformly on $0\leq x\leq\ell$. Specifically, the outer solution $\rho_{0}(x)$ approximates $\rho_{\varepsilon}(x)$ poorly in the vicinity of the jumps $\left\{x_{J}\right\}$. To make the approximation uniform, one takes into account boundary layer terms of the form $\rho_{\text{corrector}}(x)=Y((x-x_{J})/\varepsilon)$ whose distinguishing feature is that its slope, derivative in $x$, is of the order of $\varepsilon^{-1}$. Observe also that, even though $\rho_{0}(0)=\alpha$ and $\rho_{0}(\ell)=1-\beta$, it is possible that the outer solution $\rho_{0}(x)$ does not satisfy the boundary condition in the following sense: either $\lim_{x\to 0^{+}}\rho_{0}(x)\neq\alpha$ or $\lim_{x\to\ell^{-}}\rho_{0}(x)\neq 1-\beta$.

Consider a constant motor protein motility rate $v(x)$, representing a homogeneous microtubule, that is, $v_{i}\equiv v_{0}$. The solution in the homogeneous case has been studied previously (e.g., ParFraFre2003 ; PopRakWilKolSch2003 ; ParFraFre2004 ). In this case, (8) reduces to

Recall that $\ell$ is the length of a microtubule.

Equation (14) is a second order nonlinear ODE. If one formally passes to the limit $\varepsilon\to 0$ in (14), then the term with the second derivative of $\rho_{\varepsilon}$ vanishes and this equation becomes first order where the solution cannot, in general, satisfy both boundary conditions in (15) as the boundary value problem is overdetermined. To describe the limiting solution, $\lim\limits_{\varepsilon\to 0}\rho_{\varepsilon}(x)$, we introduce an auxiliary function $g(x;s,a)$, which is the solution of the initial value problem of the first order obtained from the formal limit as $\varepsilon\to 0$ in (14):

In other words, function $\rho_{0}(x):=g(x;s,a)$ is the smooth outer solution subject to a single boundary condition (or, equivalently, initial condition): $\rho_{0}|_{x=s}=a$. Equation (16) can be solved in an explicit form in terms of special functions (see ParFraFre2004 ).

Note that initial value problem (16) is not well-posed for $a=0.5$. If $a=0.5$, then there is no solution for $x>s$, and for $x<s$ we define $g$ as the function solving the differential equation in (16) subject to the following condition:

In other words, for the initial condition with $a=0.5$, equation (16) admits two solutions for $x<s$: one solution is less than $0.5$, another one is above $0.5$, and to describe the outer solution $\rho_{0}(x)$ we will need restrict our consideration to the upper one (the upper solution is stable in a certain sense, see Appendix A).

Function $g(x;s,a)$ is not necessarily defined globally, for all $x$. For example, consider $x\geq s$, then the solution exists on the interval $(s,s+x_{a})$ for some $x_{a}>0$ and at $x=s+x_{a}$ the slope of $g$ becomes unbounded. For example, if $a<0.5$, then the value of $x_{a}$ can be found from the condition $g(s+x_{a};s,a)=0.5$ which can be written as

where function $\boldsymbol{v}(g)$ is introduced in such a way that the differential equation from (16) is equivalent to $\partial_{x}g=\boldsymbol{v}(g)$. One can compute the integral on the right-hand side of (18) to obtain an analytic formula for $x_{a}$:

The following theorem gives an explicit formula for the limiting solution of the (14)-(15) as $\varepsilon\to 0$.

The result of this theorem is consistent with previous works where the system (14)-(15) was studied (e.g., see Fre11 ; Fre07 ; Fre18 ). We relegate the proof of this theorem and examples of the application of the representation formula (20) to Appendix A.

The following corollary is important for the study of inhomogeneous microtubules in Section 2.4.

Condition (25) excludes the case of low density solutions, that is, we exclude outer solutions of the form: $\rho_{0}(x)<0.5$ for all $x\in[0,\ell)$. This regime is not consistent with jamming, which is the focus of this work. In other words, condition (25) imposes that the “left” part of solution, $g(x;0,\alpha)$, cannot be extended to entire interval $[0,\ell)$. The reason we would like to impose this condition below in the inhomogeneous case is because we then focus on cases when regions with high densities emerge, and thus traffic jams in motor transport are possible. By direct integration, condition (25) can be written as

In this section, we consider a non-constant motility rate $v(x)$. Biologically, it corresponds to a inhomogeneous microtubule with different motor protein mobilities in different regions of the microtubule. Without loss of generality, we take $\ell=1$. We restrict ourselves to the case

This case may be considered as two coupled homogeneous microtubules meeting at interface $x=x_{0}$ with coupling through the continuity of densities and fluxes. Specifically, we have the following system of equations:

and two coupling conditions:

1. (1)

   continuity of $\rho_{\varepsilon}(x)$ at $x_{0}$:

   $$\rho_{\varepsilon}(x_{0}^{-})=\rho_{\varepsilon}(x_{0}^{+})=:\rho_{\varepsilon}.$$

   (30)

2. (2)

   continuity of flux $J_{\varepsilon}(x)$ at $x_{0}$:

   $$v_{L}\left(-\dfrac{\varepsilon}{2}\rho_{\varepsilon}^{\prime}(x_{0}^{-})+\rho_{\varepsilon}(1-\rho_{\varepsilon})\right)=v_{R}\left(-\dfrac{\varepsilon}{2}\rho_{\varepsilon}^{\prime}(x_{0}^{+})+\rho_{\varepsilon}(1-\rho_{\varepsilon})\right).$$

   (31)

The outer solution $\rho_{0}(x)=\lim\limits_{\varepsilon\to 0}\rho_{\varepsilon}(x)$ is not necessarily continuous, nevertheless due to (12) it satisfies the flux continuity condition:

By analogy with function $g$ from (16) in the case of homogeneous microtubules, we introduce $g_{L}(x;s_{L},a_{L})$ and $g_{R}(x;s_{R},a_{R})$ as solutions of the following initial value problems:

In what follows we consider separately two cases:

• (i)

  fast - slow microtubule: $v_{L}>v_{R}$,

• (ii)

  slow - fast microtubule: $v_{R}>v_{L}$.

Before we formulate our main result for these two cases, we note that the difficulty in the determination of the outer solution $\rho_{0}(x)$ is finding the value of $\rho_{0}$ at the interface, $A:=\rho_{0}(x_{0})$. Once $A$ is found, one can use Theorem 2.1 to restore $\rho_{0}(x)$ in both intervals $[0,x_{0}]$ and $[x_{0},1]$.

Next, we illustrate formula (36) of Theorem 2.2 by the following two examples with $x_{0}=0.5$, $v_{L}=1.0$, $v_{R}=0.5$, $\Omega_{A}=0.8$ and $\Omega_{D}=0.2$.

Solutions from Examples 1 and 2 are depicted in Fig. 1. The main difference between Examples 1 and 2 is that in Example 2 there is an interior jump in the left sub-interval $(0,0.5)$, while the solution in Example 1 does not possesses a jump in neither of the two sub-intervals, left $(0,0.5)$ and right $(0.5,1)$ (or, equivalently, $x_{J}=0$ in (36) for Example 1).