Discrete Kinetic Models for Molecular Motors: Asymptotic Velocity and Gaussian Fluctuations

Abstract

We consider random walks on quasi one dimensional lattices, as introduced in Faggionato and Silvestri (Random Walks on Quasi One Dimensional Lattices: Large Deviations and Fluctuation Theorems, 2014). This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.

Appendix: Random Time Change of Cumulative Processes

Consider a sequence $( w_i, \tau _i )_{i \ge 1}$ of i.i.d. 2d vectors with values in ${\mathbb {R}} \times (0,+\infty )$. For each integer $m \ge 1$ we define

$$\begin{aligned}&W_m:= w_1+w_2 + \cdots + w_m,\end{aligned}$$

(90)

$$\begin{aligned}&\mathcal {T}_m:= \tau _1+\tau _2 + \cdots + \tau _m. \end{aligned}$$

(91)

We set $W_0=\mathcal {T}_0=0$. Note that $\lim _{m \rightarrow \infty } \mathcal {T}_m= +\infty$ a.s. As a consequence, we can univocally define a.s. a random process $\left\{ \nu (t) \right\} _{t \in {\mathbb {R}} _+}$ with values in $\{0,1,2,3, \dots \}$ such that

$$\begin{aligned} \mathcal {T}_{\nu (t)}\le t < \mathcal {T}_{\nu (t)+1 }\,, \qquad t \ge 0. \end{aligned}$$

(92)

Note that $\nu (t)= \max \{ m\in {\mathbb {N}} :\, \mathcal {T}_m \le t \}$. Finally, we define the process $Z:[0,\infty ) \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} Z_t:= W_{\nu (t) }. \end{aligned}$$

(93)

Note that $Z_0=0$. The resulting process $Z= (Z_t)_{t \in {\mathbb {R}} _+}$ is therefore obtained from the cumulative process $(W_m)_{m \ge 0}$ by a random time change, and generalizes the concept of (time-homogeneous) random walk on ${\mathbb {R}}$. For example, if $w_i$ and $\tau _i$ are independent and $\tau _i$ is an exponential variable of parameter $\lambda$, then the process $Z$ is a continuous time random walk with exponential holding times of parameter $\lambda$ and with jump probability given by the law of $w_i$. If $\tau _i \equiv 1$ for all $i$, then $Z_t= W_{\lfloor t \rfloor }$ ($\lfloor \cdot \rfloor$ denoting the integer part) and $(Z_n)_{n \in {\mathbb {N}} }$ is a discrete time random walk on ${\mathbb {R}}$ with jump probability given by the law of $w_i$.

The skeleton process $X^*$ is indeed a special case of process $Z$ (recall the definition of the random time $S$ given in (2)):

Lemma 11.1

Consider a sequence $( w_i, \tau _i )_{i \ge 1}$ of i.i.d. vectors, with the same law of the random vector $\bigl (X^*_S,S\bigr )\in \{-1,1\} \times (0,+\infty )$ when the process $(X_t)_{t \in {\mathbb {R}} _+}$ starts at $0_*$. We define $(Z_t )_{t \in {\mathbb {R}} _+}$ as the stochastic process built from $( w_i, \tau _i )_{i \ge 1}$ according to (93). Then $(Z_t)_{t \in {\mathbb {R}} _+}$ has the same law of $( X_t^*)_{t \in {\mathbb {R}} _+}$ with $X^*_0= 0$.

The proof of the above lemma is very simple and therefore omitted.

We state our main results for $(Z_t)_{t \in {\mathbb {R}} _+}$:

Theorem 2

The stochastic process $(Z_t)_{t \in {\mathbb {R}} _+}$ satisfies:

• (i) [LLN] Assume that ${\mathbb {E}} ( \tau _i) < \infty$. Then almost surely $\lim _{t \rightarrow \infty } \frac{Z_t}{t} = v:= \frac{ {\mathbb {E}} (w_i) }{{\mathbb {E}} (\tau _i)}$.

• (ii) [Invariance Principle] Assume that ${\mathbb {E}} ( w_i^2), {\mathbb {E}} ( \tau _i^2) < \infty$. Given $n\in \{1,2, \dots \}$ define the rescaled process

  $$\begin{aligned} B^{(n)} _t := \frac{1}{\sqrt{n}} \left\{ Z_{ nt} - vnt \right\} \end{aligned}$$

  in the Skohorod space $D( {\mathbb {R}} _+; {\mathbb {R}} )$. Then as $n\rightarrow \infty$ the rescaled process $B^{(n)}$ weakly converges to a Brownian motion on ${\mathbb {R}}$ with diffusion constant

  $$\begin{aligned} \sigma ^2:=\frac{ \mathrm{Var}(w_1-v\tau _1)}{ {\mathbb {E}} (\tau _1) }. \end{aligned}$$

  (94)

1.1 Proof of the Law of Large Numbers in Theorem 2

The proof is rather standard, we give it for completeness as short. Since $\lim _{m \rightarrow \infty } \mathcal {T}_m= \infty$ a.s., we have $\lim _{t \rightarrow \infty } \nu (t)= \infty$ a.s. Hence from the LLN for $(W_n)_{n \ge 1}$ and $(\mathcal {T}_n )_{ n \ge 1}$ we deduce that $\lim _{t\rightarrow \infty } \frac{ W_{\nu (t) }}{\nu (t)} ={\mathbb {E}} (w_i)$ and $\lim _{t \rightarrow \infty } \frac{\mathcal {T}_{\nu (t)}}{ \nu (t) } = {\mathbb {E}} (\tau _i)$ a.s. From the last limit and the bounds (recall (92))

$$\begin{aligned} \frac{\mathcal {T}_{\nu (t)}}{ \nu (t) } \le \frac{t}{\nu (t)} < \frac{\mathcal {T}_{\nu (t)+1 }}{\nu (t)+1}\cdot \frac{ \nu (t)+1}{\nu (t)} \end{aligned}$$

we get $\lim _{t \rightarrow \infty } \nu (t)/t = 1/ {\mathbb {E}} (\tau _i)$ a.s. Since $Z_t/t= [W_{\nu (t) }/\nu (t)]\cdot [\nu (t)/t]$ we get the thesis.

1.2 Proof of the Invariance Principle in Theorem 2

For each $n\ge 1$ and $t\in {\mathbb {R}} _+$ let

$$\begin{aligned} A^{(n)}_t := \frac{1}{\sqrt{n} } \left\{ W_{\lfloor nt \rfloor }- {\mathbb {E}} ( w_i) nt \right\} \,,\qquad D^{(n)}_t:= \frac{1}{\sqrt{n} } \left\{ \mathcal {T}_{\lfloor nt \rfloor }- {\mathbb {E}} ( \tau _i) nt \right\} . \end{aligned}$$

Then, since $v= {\mathbb {E}} (w_i)/ {\mathbb {E}} (\tau _i)$, the following identity holds:

$$\begin{aligned} B^{(n)}_{t}&=\frac{1}{\sqrt{n}} \left\{ W_{\nu (nt)}- {\mathbb {E}} (w_1) \nu (nt)\right\} +\frac{1}{\sqrt{n}} \left\{ {\mathbb {E}} (w_1) \nu (nt) - v \mathcal {T}_{\nu (nt)}\right\} + \frac{v}{\sqrt{n}} \left\{ \mathcal {T}_{\nu (nt)}- nt\right\} \nonumber \\&=A^{(n)}_{\nu (nt)/n }- v D^{(n)} _{\nu (nt)/n}+ \frac{v}{\sqrt{n}} \left\{ \mathcal {T}_{\nu (nt)}- nt\right\} . \end{aligned}$$

(95)

Lemma 11.2

Given $\varepsilon >0$ and $s>0$ we have

$$\begin{aligned} \lim _{n\rightarrow \infty } {\mathbb {P}} \left( \sup _{0\le t \le s} \left| \frac{1}{\sqrt{n}}\left\{ \mathcal {T}_{ \nu ( n t ) } -nt\right\} \right| >\varepsilon \right) =0. \end{aligned}$$

(96)

Proof

As proven above $\lim _{t \rightarrow \infty } \nu (t)/t= \theta := 1/ {\mathbb {E}} (\tau _i)$ a.s. Hence, fixed $\delta >0$, it holds ${\mathbb {P}} ( E_n^c) \le \delta$ for $n$ large enough as we assume, where $E_n$ is the event $\{\nu ( n s ) \le 2\theta n s \}$. Trivially the event $E_n$ implies that $\nu (nt)\le 2\theta n s$ for each $t \in [0,s]$. Now we observe that, due to (92), $\mathcal {T}_{\nu (nt)}\le nt < \mathcal {T}_{\nu (nt)+1}$, hence $0\le nt -\mathcal {T}_{\nu (nt)} \le \tau _{\nu (nt)+1}$. Due to the above considerations, calling $F_n$ the event in (96) we conclude that

$$\begin{aligned} {\mathbb {P}} (F_n)\le {\mathbb {P}} (E_n^c)+ {\mathbb {P}} \left( \max _{ 1 \le i \le 2\theta n s+1 } \tau _i > \varepsilon \sqrt{n}\right) \end{aligned}$$

(97)

Since ${\mathbb {P}} (E_n^c) \le \delta$ eventually, by the arbitrariness of $\delta$ we only need to show that the last probability in (97) goes to zero as $n \rightarrow \infty$. This is a general fact. Let $(X_i)_{i \ge 1}$ be i.i.d. positive random variables with ${\mathbb {E}} (X_i^2) < \infty$ (in our case $X_i= \tau _i$). Then, given $a >0$,

$$\begin{aligned} {\mathbb {P}} (\max _{1\le i \le N} X_i \le a \sqrt{N})&= {\mathbb {P}} ( X_1\le a \sqrt{N})^N =[ 1- {\mathbb {P}} ( X_1> a \sqrt{N})]^N \\&= e^{N \ln [ 1- {\mathbb {P}} ( X_1> a \sqrt{N})] } \sim e^{- N {\mathbb {P}} ( X_1> a\sqrt{ N})}. \end{aligned}$$

Note that the last equivalence holds since $\lim _{N \rightarrow \infty }{\mathbb {P}} ( X_1> a\sqrt{ N})=0$. At this point, in order to prove that $\max _{1\le i \le N} X_i / \sqrt{N}$ weakly converges to zero we only need to show that $\lim _{N \rightarrow \infty } N {\mathbb {P}} ( X_1> a\sqrt{ N}) =0$, or equivalently that $\lim _{t \rightarrow \infty } t^2 P(X_1>t)=0$. This follows form the fact that ${\mathbb {E}} (X_i^2) < \infty$ (see Exercise 3.5, page 15 of [5]). $\square$

Due to Lemma 11.2 we can disregard the last addendum in (95) in order to prove the invariance principle for $B^{(n)}$. Let us now consider the random path $\Gamma ^{(n)}$ in $D({\mathbb {R}} _+; {\mathbb {R}} \times {\mathbb {R}} \times {\mathbb {R}} _+)$ defined as

$$\begin{aligned} \Gamma ^{(n)}: {\mathbb {R}} _+ \ni t \rightarrow \left( A^{(n)}_t ,D^{(n)}_t , \nu ( n t )/n\right) \in {\mathbb {R}} \times {\mathbb {R}} \times {\mathbb {R}} _+. \end{aligned}$$

Lemma 11.3

The random path $\Gamma ^{(n)}$ weakly converges to the random path

$$\begin{aligned} \left( \,\left( B_1(t), B_2(t) , \theta t \right) \,\right) _{t \in {\mathbb {R}} _+} \end{aligned}$$

where $\left( \,\left( B_1(t), B_2(t)\right) \,\right) _{t \in {\mathbb {R}} _+}$ is a zero mean bidimensional Brownian motion such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{Var}\bigl (B_1(1) \bigr ) =\mathrm{Var} (w_1) ,\\ \mathrm{Var} \bigl ( B_2(1) \bigr )=\mathrm{Var}(\tau _1) ,\\ \mathrm{Cov}\bigl ( B_1(1), B_2(1) \bigr )=\mathrm{Cov}(w_1,\tau _1). \end{array}\right. } \end{aligned}$$

Proof

We first show that the random path $\bigl (\nu (n t )/n\bigr )_{t \in {\mathbb {R}} _+}$ weakly converges to the deterministic path $(\theta t )_{t \in {\mathbb {R}} _+}$. To this aim it is enough to show the convergence in probability w.r.t. the uniform distance on finite intervals (this implies the convergence in the Skohorod topology). In particular, we claim that for any $s , \delta >0$ it holds

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathbb {P}} \left( \sup _{0 \le t \le s} \Big | \frac{ \nu (nt)}{n} - \theta t \Big | > \delta \right) =0. \end{aligned}$$

(98)

By monotonicity $\nu (n u)\le \nu (n t) \le \nu (n v)$ if $u \le t \le v$, thus implying that

$$\begin{aligned} \Big | \frac{ \nu (nt)}{n} - \theta t \Big |\le \max \left\{ \Big | \frac{ \nu (nu )}{n} - \theta u \Big |+\theta |u-t|, \,\Big | \frac{ \nu (nv )}{n} - \theta v\Big |+ \theta |v-t|\right\} . \end{aligned}$$

At this point the convergence (98) follows easily from the convergence in probability for fixed times, i.e. from the fact that $\nu (n r)/n \rightarrow \theta r$ a.s. for each fixed $r$.

Since by the invariance principle for sums of independent vectors it holds

$$\begin{aligned} \Big ( A^{(n)}_t ,D^{(n)}_t \Big ) _{t \in {\mathbb {R}} _+}\Longrightarrow \left( \, \bigl ( B_1(t), B_2(t)\bigr ) \,\right) _{t \in {\mathbb {R}} _+} \end{aligned}$$

and since we have just proved that $\bigl (\nu (nt)/n\bigr )_{t \in {\mathbb {R}} _+} \Longrightarrow \bigl ( \theta t \bigr )_{t \in {\mathbb {R}} _+}$ (where the r.h.s. is a deterministic path), the thesis follows from Theorem 3.9 of [1]. $\square$

We can now conclude the proof of the invariance principle. Since the map ${\mathbb {R}} _+ \ni t \rightarrow A^{(n)}_{ \nu (nt)/n } \in {\mathbb {R}}$ if simply the composition $f \circ g (t)$ where $f(u)= A^{(n)}_{u }$ and $g(t)= \nu (nt)/n$, and similarly for ${\mathbb {R}} _+ \ni t \rightarrow D^{(n)}_{ \nu (nt)/n } \in {\mathbb {R}}$, we can simply combine the above lemma with the Lemma in [1][page 151] to derive the weak convergence

$$\begin{aligned} \Big ( \,A^{(n)}_{ \nu (nt)/n } - v D^{(n)}_{ \nu (nt)/n } \,\Big ) _{t \in {\mathbb {R}} _+} \; \Longrightarrow \; \Big ( \,B_1(\theta t) -v B_2(\theta t) \, \Big )_{t \in {\mathbb {R}} _+}. \end{aligned}$$

(99)

Since the last process is a Brownian motion with diffusion constant (94), combining the above convergence with (95) and Lemma 11.2 we get the invariance principle for $B^{(n)}$.