Stochastic models for plant microtubule self-organization and structure

Abstract

One of the key enablers of shape and growth in plant cells is the cortical microtubule (CMT) system, which is a polymer array that forms an appropriately-structured scaffolding in each cell. Plant biologists have shown that stochastic dynamics and simple rules of interactions between CMTs can lead to a coaligned CMT array structure. However, the mechanisms and conditions that cause CMT arrays to become organized are not well understood. It is prohibitively time-consuming to use actual plants to study the effect of various genetic mutations and environmental conditions on CMT self-organization. In fact, even computer simulations with multiple replications are not fast enough due to the spatio-temporal complexity of the system. To redress this shortcoming, we develop analytical models and methods for expeditiously computing CMT system metrics that are related to self-organization and array structure. In particular, we formulate a mean-field model to derive sufficient conditions for the organization to occur. We show that growth-prone dynamics itself is sufficient to lead to organization in presence of interactions in the system. In addition, for such systems, we develop predictive methods for estimation of system metrics such as expected average length and number of CMTs over time, using a stochastic fluid-flow model, transient analysis, and approximation algorithms tailored to our problem. We illustrate the effectiveness of our approach through numerical test instances and discuss biological insights.

Appendices

Appendix 1: Input parameter sets (dynamics)

$$\begin{aligned} Q&= [q_{m,n}], m,n \in \{GS,GP,SS,SP,PS,PP\} \\ V&= diag(v^m), m \in \{GS,GP,SS,SP,PS,PP\} \end{aligned}$$

Parameter Set I:

$$\begin{aligned}&Q = \begin{bmatrix} -8.925&\quad 6.72&\quad 1.485&\quad 0&\quad 0.72&\quad 0 \\ 2.427&\quad -4.632&\quad 0&\quad 1.485&\quad 0&\quad 0.72 \\ 3.537&\quad 0&\quad -11.192&\quad 6.72&\quad 0.935&\quad 0 \\ 0&\quad 3.537&\quad 2.427&\quad -6.898&\quad 0&\quad 0.935 \\ 5.05&\quad 0&\quad 2.376&\quad 0&\quad -14.1&\quad 6.72 \\ 0&\quad 5.05&\quad 0&\quad 2.376&\quad 2.427&\quad -9.85 \end{bmatrix}. \\&V = diag(0.91,3.69,-8.66,-5.88,-2.78,0) \end{aligned}$$

Parameter Set II:

$$\begin{aligned}&Q=\begin{bmatrix} -7.234&\quad 6.72&\quad 0.236&\quad 0&\quad 0.278&\quad 0 \\ 2.427&\quad -2.941&\quad 0&\quad 0.236&\quad 0&\quad 0.278 \\ 5&\quad 0&\quad -13.27&\quad 6.72&\quad 1.55&\quad 0 \\ 0&\quad 5&\quad 2.427&\quad -8.977&\quad 0&\quad 1.55 \\ 25.125&\quad 0&\quad 12.75&\quad 0&\quad -44.595&\quad 6.72 \\ 0&\quad 25.125&\quad 0&\quad 12.75&\quad 2.427&\quad -40.302 \end{bmatrix} \\&V=diag(0.72,3.5,-11.78,-9,-2.78,0) \end{aligned}$$

Parameter Set III:

$$\begin{aligned}&Q=\begin{bmatrix} -9.617&\quad 6.72&\quad 2.338&\quad 0&\quad 0.559&\quad 0 \\ 2.427&\quad -5.324&\quad 0&\quad 2.338&\quad 0&\quad 0.559 \\ 12.438&\quad 0&\quad -21.908&\quad 6.72&\quad 2.75&\quad 0 \\ 0&\quad 12.438&\quad 2.427&\quad -17.614&\quad 0&\quad 2.75 \\ 8.75&\quad 0&\quad 4.375&\quad 0&\quad -19.845&\quad 6.72 \\ 0&\quad 8.75&\quad 0&\quad 4.375&\quad 2.427&\quad -15.552 \end{bmatrix} \\&V=diag(3.72,6.5,-15.18,-12.4,-2.78,0) \end{aligned}$$

Parameter Set IV:

$$\begin{aligned}&Q=\begin{bmatrix} -7.537&\quad 6.72&\quad 0.535&\quad 0&\quad 0.282&\quad 0\\ 2.427&\quad -3.244&\quad 0&\quad 0.53521&\quad 0&\quad 0.282\\ 6.211&\quad 0&\quad -16.036&\quad 6.72&\quad 3.105&\quad 0\\ 0&\quad 6.211&\quad 2.427&\quad -11.742&\quad 0&\quad 3.105\\ 15.6&\quad 0&\quad 5.6&\quad 0&\quad -27.92&\quad 6.72\\ 0&\quad 15.6&\quad 0&\quad 5.6&\quad 2.427&\quad -23.627 \end{bmatrix} \\&V=diag(-0.28,2.5,-8.98,-6.2,-2.78,0) \end{aligned}$$

Parameter Set V:

$$\begin{aligned}&Q=\begin{bmatrix} -11.806&\quad 6.72&\quad 2.343&\quad 0&\quad 2.743&\quad 0\\ 2.427&\quad -7.512&\quad 0&\quad 2.343&\quad 0&\quad 2.743\\ 3.05&\quad 0&\quad -15.82&\quad 6.72&\quad 6.05&\quad 0\\ 0&\quad 3.05&\quad 2.427&\quad -11.527&\quad 0&\quad 6.05\\ 1.556&\quad 0&\quad 1.378&\quad 0&\quad -9.653&\quad 6.72\\ 0&\quad 1.556&\quad 0&\quad 1.378&\quad 2.427&\quad -5.36 \end{bmatrix} \\&V=diag(-0.78,2,-6.58,-3.8,-2.78,0) \end{aligned}$$

Parameter Set VI:

$$\begin{aligned}&Q=\begin{bmatrix} -8.925&\quad 6.72&\quad 1.485&\quad 0&\quad 0.72&\quad 0 \\ 2.427&\quad -4.632&\quad 0&\quad 1.485&\quad 0&\quad 0.72 \\ 3.537&\quad 0&\quad -11.192&\quad 6.72&\quad 0.935&\quad 0 \\ 0&\quad 3.537&\quad 2.427&\quad -6.898&\quad 0&\quad 0.935 \\ 5.05&\quad 0&\quad 2.376&\quad 0&\quad -14.1&\quad 6.72 \\ 0&\quad 5.05&\quad 0&\quad 2.376&\quad 2.427&\quad -9.85 \end{bmatrix}. \\&V=diag(3.69,3.69,-5.88,-5.88,0,0) \end{aligned}$$

Appendix 2: Proofs

Lemma 1

Let us define $z_i=(y_i-x_i)$ $i=1,\ldots ,N$. We group $z_i$ values in three sets as follows:

$$\begin{aligned} \begin{aligned} I_+&=\{i \in \{1,N\}: z_i>0\}\\ I_0&=\{i \in \{1,N\}: z_i=0\}\\ I_-&=\{i \in \{1,N\}: z_i<0\}. \end{aligned} \end{aligned}$$

As $\sum _{i=1}^N y_i-\sum _{i=1}^N x_i=0$, it follows that

$$\begin{aligned} \sum _{i=1}^N z_i=\sum _{i\in I_+} z_i + \sum _{i\in I_-} z_i=0. \end{aligned}$$

(23)

Let us divide $z_i$ values into infinitesimal pieces of the same size, denoted by $\varDelta _z>0$, such that for each $i\in I_+$, $z_i=w_i \varDelta _z$ and for each $i\in I_-$, $z_i=-w_i \varDelta _z$, $i=1,\ldots ,N$, where $\varDelta _z>0$ and $w_i$ are positive real numbers. Equation (23) can be rewritten as

$$\begin{aligned} \sum _{i\in I_+} w_i - \sum _{i\in I_-} w_i=0. \end{aligned}$$

(24)

Hence, we have an equal number ($\sum _{i\in I_+} w_i =\sum _{i\in I_-} w_i :=W$) of $\varDelta _z$ pieces that belong to sets $I_+$ and $I_-$. Note that $\forall$ $z_i$ with $i \in I_+$ and $z_j$ with $j \in I_-$, it follows from the definition of $z_i$ values that $z_i>z_j$, and consequently $x_i>x_j$ from the given ordering relation between the sequences $x_i$, and $z_i$; and finally $f(x_i)<f(x_j)$ due to the decreasing property of $f(\cdot )$. Let us redefine the sequence of $x_i$, $i=1,\ldots ,N$ such that its values are copied $w_i$ times for each $x_i$ value so that every $\varDelta _z$ has its corresponding $x'_{i'}$ and $f(x'_{i'})$ values, where $i'=1,\ldots ,W$. Adjusting $I_+$ and $I_-$ accordingly as $I'_+$, $I'_-$, for each $\varDelta _z$ value that belongs to set $I_+$, there is a $\varDelta _z$ which is multiplied with a larger value in $I_-$ in the following equation:

$$\begin{aligned} \sum _{i'\in I'_+} f(x'_{i'}) \varDelta _z - \sum _{i'\in I'_-} f(x'_{i'}) \varDelta _z<0, \end{aligned}$$

which gives the desired result.

Proposition 1

According to Lyapunov’s stability theory, a sufficient condition for the global asymptotic stability of an equilibrium point $P^*$ is existence of a Lyapunov function $\mathcal {L}(\cdot )$ such that

• $\mathcal {L}(t)\mid _P>0$, $\forall P\ne P^*$ and $\mathcal {L}(t)\mid _P=0$ only for $P=P^*$,

• $\frac{\partial \mathcal {L}(t)}{\partial t}\mid _P<0$, $\forall P\ne P^*$ and $\frac{\partial \mathcal {L}(t)}{\partial t}\mid _{P=P^*}=0$.

We set our Lyapunov function as the entropy metric, $H(\cdot )$ in Eq. (2). We already know that the first condition holds for $H(\cdot )$, as $H(t)\mid _{P=P^*}=0$ and $H(t)\mid _P>0$ $\forall P\ne P^*$. What is left to check is the sign of the derivative of the Lyapunov function with respect to $t$, which is given by

$$\begin{aligned} \frac{\partial H(t)}{\partial t}=-\sum _{\theta =0}^{179} k'(\theta ,t) ln (k(\theta ,t))+k'(\theta ,t), \end{aligned}$$

(25)

where

$$\begin{aligned} k'(\theta ,t)=\frac{\partial \,\, k(\theta ,t)}{\partial t}. \end{aligned}$$

(26)

Let us define the total density of CMTs with length $l$ and angle $\theta$ at time $t$ by

$$\begin{aligned} \bar{p}(l,\theta ,t):=\bar{p}_G(l,\theta ,t)+\bar{p}_S(l,\theta ,t)+\bar{p}_P(l,\theta ,t). \end{aligned}$$

Using Eq. (10), we can rewrite Eq. (26) as

$$\begin{aligned} k'(\theta ,t)=\frac{\left( \int _0^\infty l \frac{d \bar{p}(l,\theta ,t)}{dt}dl\right) \sum _{\theta =0}^{179}\int _0^\infty l \bar{p}(l,\theta ,t)dl\!-\!\left( \int _0^\infty l \bar{p}(l,\theta ,t)dl\right) \sum _{\theta =0}^{179}\int _0^\infty l \frac{d \bar{p}(l,\theta ,t)}{dt}dl}{ \left( \sum _{\theta =0}^{179}\int _0^\infty l \bar{p}(l,\theta ,t)dl\right) ^2}. \end{aligned}$$

Summing equations in (5) side by side, multiplying both sides with $l$ and integrating with respect to $l$ over $(0,\infty )$, we obtain the derivative of total length of all CMTs with angle $\theta$ at time $t$ as

$$\begin{aligned} L'_\theta (t) :&= \int _0^\infty l \frac{d \bar{p}(l,\theta ,t)}{dt}dl\\&= (v_G^+-v_S^-) \tilde{p}_{GS}(\theta ,t)+v_G^+ \tilde{p}_{GP}(\theta ,t)\\&\quad -(v_S^++v_S^-) \tilde{p}_{SS}(\theta ,t)-v_S^+\tilde{p}_{SP}(\theta ,t)-v_S^- \tilde{p}_{PS}(\theta ,t)\\&\quad -v_G^+\int _0^\infty \bar{p}_G(l,\theta ,t) l dl\sum _{\theta ' \in \varTheta ^*} p_b|sin(\theta -\theta ')|\int _{l'} dl' l' \bar{p}(l',\theta ',t)\\&\quad +v_G^+ \int _{l'} dl' l' \bar{p}(l',\theta ,t)\sum _{\theta ' \in \varTheta ^*} p_b|sin(\theta -\theta ')\int _0^\infty \bar{p}_G(l,\theta ',t) l dl, \end{aligned}$$

where $\tilde{p}_{m}(\theta ,t):= \int _0^{\infty } p_m(l,\theta ,t) dl$, $m$ $\in$ $\{GS,GP,SS,SP,PS,PP\}$ stands for the total density of CMTs with angle $\theta$ at state $m$ at time $t$. We denote the sum of $L'_\theta (t)$ over all $\theta$ by $\sum L'(t):= \sum _{\theta =0}^{179} L'_\theta (t)$, which gives the derivative of total length of all CMTs in the system at time $t$. Defining total length of CMTs with angle $\theta$ at time $t$ as $L_\theta (t):=\int _0^\infty l p(l,\theta ,t)dl$, and the total length of all CMTs at time $t$ as $\sum L(t):=\sum _{\theta =1}^{360} L_\theta (t)$, and plugging these into Eq. (25) results in

$$\begin{aligned} \frac{dH(t)}{dt}=-\sum _{\theta =0}^{179} \frac{L'_\theta (t)\ln \left( \frac{L_\theta (t)}{\sum L(t)}\right) \sum L(t)-L_\theta (t)\ln \left( \frac{L_\theta (t)}{\sum L(t)}\right) \sum L'(t)}{(\sum L(t))^2}. \end{aligned}$$

Rearranging terms, we obtain

$$\begin{aligned} \frac{dH(t)}{dt}=-\sum _{\theta =0}^{179} \frac{\frac{L'_\theta (t)}{\sum L'(t)}\ln \left( \frac{L_\theta (t)}{\sum L(t)}\right) -\frac{L_\theta (t)}{\sum L(t)}\ln \left( \frac{L_\theta (t)}{\sum L(t)}\right) }{\sum L(t) \sum L'(t)}. \end{aligned}$$

(27)

Let us denote $a_{\theta }:=\frac{L'_\theta (t)}{\sum L'(t)}$ and $b_{\theta }:=\frac{L_\theta (t)}{\sum L(t)}$. By definition, it follows that $\sum _{\theta =0}^{179} a_{\theta }=1$, $\sum _{\theta =0}^{179} b_{\theta }=1$, and $b_i>0$. Assuming $\sum L'(t) >0$, i.e. the net total CMT length change in time is positive, we are interested in the sign of

$$\begin{aligned} \sum -\ln (b_{\theta }) (a_{\theta } -b_{\theta }). \end{aligned}$$

(28)

If the sign for Expression (28) is negative, then Eq. (27) is negative, i.e. $\frac{dH}{dt} <0$, and the stability condition is satisfied. A sufficient condition to ensure this follows from Lemma 1 as $-ln(b)$ is a decreasing function of $b$ for $0<b<1$. Accordingly, we require the two sequences $a_\theta$ and $b_\theta$, and their difference $a_\theta -b_\theta$ to increase and decrease in the same order. This roughly means that if CMTs with an angle $\theta$ have a larger total length compared to the total length of CMTs with angle $\theta '\ne \theta$, they also grow larger in ratio in total length on average, and vice versa. This property follows by careful observation of model equations and the property that $p_b(l)$ is decreasing in $l$.

Finally, in order to fulfill $\sum L'(t) >0$, it is required that the problem parameters satisfy

$$\begin{aligned}&\sum _\theta (v_G^+-v_S^-) p_{GS}(\theta ,t)+v_G^+ p_{GP}(\theta ,t)-(v_S^++v_S^-) p_{SS}(\theta ,t)-v_S^+p_{SP}(\theta ,t)\nonumber \\&\qquad -v_S^- p_{PS}(\theta ,t)>0, \end{aligned}$$

(29)

$\forall t$, which can be stated as

$$\begin{aligned} \sum _\theta p^+_G(\theta ,t) v_{G^+}-p^+_S(\theta ,t) v_{S^+}- p^-_S(\theta ,t) v_{S^-}>0, \end{aligned}$$

(30)

by rearranging terms.

Theorem 1

Consider $F_{ab}(l,t+h)$, where $h$ is a small positive real number. It can be written as

$$\begin{aligned} F_{ab}(l,t+h)=P\{\tau \le t+h,M(\tau )=b|L(0)=l,M(0)=a\}. \end{aligned}$$

Conditioning on the first transition from the initial state, we obtain

$$\begin{aligned} F_{ab}(l,t+h)=&P\{\tau \le t+h,M(\tau )=b|L(0)=l,M(0)=a\}\\ =&\sum _{c \ne a} P\{\tau \le t+h,M(\tau )=b|L(0)=l,M(0)=a,M(h)=c\}\\&\qquad P\{M(h)=c|M(0)=a,L(0)=l\}\\&+P\{\tau \le t+h,M(\tau )=b|L(0)=l,M(0)=a,M(h)=a\}\\&\qquad P\{M(h)=a|M(0)=a,L(0)=l\}. \end{aligned}$$

As $M(t)$ process is independent of $L(0)$, and the length would change by $v^a h$ by time $h$, when the CMT is in state $a$ at time 0,

$$\begin{aligned}&F_{ab}(l,t+h)\\&\quad = \sum _{c \ne a} P\{\tau \le t+h,M(\tau )=b|L(h)=l+v^a h,M(h)\!=\!c\}P\{M(h)\!=\!c|M(0)\!=\!a\}\\&\qquad +P\{\tau \le t+h,M(\tau )=b|L(h)=l+v^a h, M(h)=a\}P\{M(h)=a|M(0)=a\}\\&\quad = \sum _{c \ne a} P\{\tau \le t,M(\tau )=b|L(0)=l+v^a h,M(0)=c\}P\{M(h)=c|M(0)=a\}\\&\qquad +P\{\tau \le t,M(\tau )=b|L(0)=l+v^a h, M(0)=a\}P\{M(h)=a|M(0)=a\}. \end{aligned}$$

As the transition probability from state $a$ to $c$ in time $h$ is given by $q_{ac} h+o(h)$ if $c \ne a$ and $1+q_{aa} h+o(h)$ if $c = a$, where $o(h)$ is a collection of terms of higher order than $h$ such that $o(h)/h \rightarrow 0$ as $h \rightarrow 0$, it follows

$$\begin{aligned} F_{ab}(l,t+h)=\sum _{c \ne a} F_{cb}(l+v^a h,t) q_{ac} h +F_{ab}(l+v^a h,t) (q_{aa} h+1)+o(h). \end{aligned}$$

Subtracting $F_{ab}(l,t)$ from each side of the equation, dividing by $h$ and rearranging terms results in

$$\begin{aligned}&\frac{F_{ab}(l,t+h)-F_{ab}(l,t)}{h}\\&\quad =\frac{F_{ab}(l+v^a h,t)-F_{ab}(l,t)}{h}+\sum _c q_{ac}F_{cb}(l+v^a h,t)+o(h)/h. \end{aligned}$$

Letting $h\rightarrow 0$ yields Eq. (15), and rewriting in the matrix form gives Eq. (16). Next, we describe the boundary conditions for all $a,b$ and $t$. As the lifetime would be zero if CMT appeared with zero length at state $b$ such that $v^b<0$, it follows

$$\begin{aligned} F_{bb}(0,t) = 1 \qquad \hbox {for} \qquad v^b<0. \end{aligned}$$

The second boundary condition,

$$\begin{aligned} F_{ab}(0,t) = 0 \qquad \hbox {for} \qquad a\ne b, v^a<0, \end{aligned}$$

follows from the fact that although the lifetime is zero, the probability that the state is $b$ when the lifetime is reached is zero (since at time $t=0$ the state is $a$ with $v^a<0$). Finally, the last two conditions follow from the fact that lifetime cannot be reached at state $b$ at time $t=0$ if the initial state is $a \ne b$ for any initial length; or if the initial state is $b$ for a positive initial length.

Theorem 2

Taking the LT of Eq. (16) with respect to $t$ gives

$$\begin{aligned} (w I-Q) F_b^*(l,w)= V \frac{\delta F_b^*(l,w)}{\delta l}. \end{aligned}$$

(31)

Taking the LT of Eq. (31) with respect to $l$ results in

$$\begin{aligned} (w I-Q) F_b^{**}(s,w)= V [sF_b^{**}(s,w)-F^*_b(0,w)]. \end{aligned}$$

(32)

Define $e_j$ as the $j\mathrm{th}$ unit vector. Plugging in the boundary condition

$$\begin{aligned} F_b^*(0,w)=w^{-1}e_j \qquad if \qquad v^j<0, \end{aligned}$$

we get

$$\begin{aligned} (V s- w I+Q) \tilde{F}_b^*(s,w)= w^{-1}(V e_j). \end{aligned}$$

Rearranging terms yields Eq. (17).

Appendix 3: Algorithms for estimation of system metrics