Association-dissociation process with aging subunits: Recursive solution

The coupling of stochastic growth and shrinkage of one-dimensional structures to random aging of the constituting subunits defines the simple association-dissociation-aging process which captures the essential features of the nonequilibrium assembly of cytoskeletal filaments. Because of correlations, previously employed mean-field methods fail to correctly describe filament growth. We study an alternative formulation of the full master equation of the stochastic process. An ansatz for the steady-state solution leads to a recursion relation which allows for the calculation of all emergent quantities with increasing accuracy and in excellent agreement with stochastic simulations.

Figure 1

Definition of simple association-dissociation-aging process (SADAP). The internal state of each subunit is described by a binary variable $S_i=0,1$. State-1 subunits enter the filament at the active end with rate ${\omega }_{+}$. State-1 subunits are converted into state-0 subunits by aging with rate ${\omega }_{\text{a}}$. State-1 subunits (state-0 subunits) leave the filament at the active end with rate ${\omega }_1$ (${\omega }_0$). The state of the $N$ terminal subunits is represented by the sequence $\left(S_1,S_2,\cdots ,S_N\right)$.

Figure 2

Probability $\langle S_1\rangle$ as a function of the association rate ${\omega }_{+}$. Shown are analytical results employing the mean-field approximation [13] or the method in Ref. [20] (dashed blue line), the Eqs. (10) and (11) combined with the “mean-field cluster approximation” (cf. text, dash-dotted green line), and Eq. (12) with $g_1=0, g_3=0$, and $g_5=0$, respectively (three solid red lines). Simulation results are shown as black dots. Parameter values consistent with actin filaments are chosen, i.e., ${\omega }_1=0.16$/s, ${\omega }_0=6$/s, and ${\omega }_{\text{a}}=0.01$/s, see Ref. [11].

Figure 3

Average growth velocity $v$ as a function of the association rate ${\omega }_{+}$. The same parameter values and color code as in Fig. 2 are used. The solid red line corresponds to $g_5=0$.

Figure 4

Phase diagram featuring growth and shrinkage phases. Shown is the $({\omega }_{+},{\omega }_{\text{a}})$ plane for the (rescaled) parameter values ${\omega }_0=1$ and ${\omega }_1={10}^{-3}$. The (dashed) blue line represents the phase boundary ($v=0$) as obtained from the mean-field approximation, while the (solid) red line was calculated via Eq. (12) with $g_5=0$. At the position of the + (–) signs, simulated filaments grow (shrink). This indicates that the red (solid) and not the blue (dashed) line represents the correct phase boundary. Inset: Complete phase diagram, where $v=0$ is given by ${\omega }_{+}={\omega }_1$ for ${\omega }_{\text{a}}\to 0$, and ${\omega }_{+}={\omega }_0$ for ${\omega }_{\text{a}}\to \infty$.

Figure 5

Joint probabilities $P_{{\alpha }_1,{\alpha }_2,{\alpha }_3}\equiv \text{prob}\left(S_1={\alpha }_1,S_2={\alpha }_2,S_3={\alpha }_3\right)$ and individual probabilities $\langle S_n\rangle$ as functions of time. The steady states from simulations (colored circles) of ${10}^5$ filaments match the analytical results (colored lines) from Eq. (12) with $g_5=P_{000001}=0$. The joint probabilities $P_{{\alpha }_1,{\alpha }_2,{\alpha }_3}$ of particular configurations are very small, while $\langle S_n\rangle$ decays only with $\langle S_n\rangle =b_1^{n-1}{r}_1$, where $r_1\simeq 0.920$ and $b_1\simeq 0.826$. Parameter values as in Fig. 2 and with ${\omega }_{+}=0.2$/s.

Figure 6

Expressions of $g_N$ given by Eq. (12) as an explicit function of $r_1\equiv \langle S_1\rangle$. Shown are $g_1$ in black, $g_2$ in blue, $g_3$ in green, and $g_4$ in red. The first-order approximation $g_1=0$ provides a unique physical solution $0\le r_1^{\left(1\right)-}\le 1$, see Eq. (D9), whereas higher-order approximations $g_N=0$ exhibit more than one solution with $0\le r_1^{\left(N\right)}\le 1$. As the physical solution of order $N+1$ must resemble the physical solution of order $N$, the physical $r_1^{(N+1)}$ is uniquely determined as the one with the minimal distance to the physical $r_1^{\left(N\right)}$. For this graph, we have chosen parameter values which are consistent with actin filaments, i.e., ${\omega }_{+}=0.2$/s, ${\omega }_1=0.16$/s, ${\omega }_0=6$/s, ${\omega }_{\text{a}}=0.01$/s, see Appendix pp6 and Ref. [11]. We found $r_1^{\left(1\right)}\simeq 0.861, r_1^{\left(2\right)}\simeq 0.912, r_1^{\left(3\right)}\simeq 0.918$, and $r_1^{\left(4\right)}\simeq 0.920$, while stochastic simulations led to $\langle S_1\rangle =r_1^{\left(\text{sim}\right)}\simeq 0.920$ and the mean-field approximation results in $r_1^{\left(\text{mf}\right)}\simeq 0.990$, cf. Fig. 2.

Figure 7

Probability $\langle S_1\rangle$ as function of the association rate ${\omega }_{+}$ for parameter values consistent with microtubules, i.e., ${\omega }_1=24$/s, ${\omega }_0=290$/s, and ${\omega }_{\text{a}}=0.2$/s. The mean-field approximation [13] is shown as blue (dashed) line, the Eqs. (10) and (11) combined with the “mean-field cluster approximation” as the green (dash-dotted) line, and Eq. (12) with $g_5=0$ is displayed as the red (solid) line. Simulation results are shown as black dots.

Figure 8

Probability $\langle S_1\rangle$ as function of the association rate ${\omega }_{+}$ for parameter values consistent with actin filaments in the presence of actin depolymerization factors (ADFs)/cofilins. The rates are rescaled to dimensionless form, see Appendix pp6 for details, and given by ${\omega }_0=1, {\omega }_1={10}^{-3}$, and ${\omega }_{\text{a}}={10}^{-4}$, respectively. The same color code as in Fig. 7 is used. Because ${\omega }_1/{\omega }_0={10}^3\gg 1$, correlations are very significant and the mean-field approach totally fails.

Figure 9

Average growth-shrinkage velocity $v$ as functions of the association rate ${\omega }_{+}$ for parameter values identical to Fig. 7 and consistent with microtubules. Same color code as in Fig. 7.

Figure 10

Average growth-shrinkage velocity $v$ as functions of the association rate ${\omega }_{+}$ for dimensionless parameter values identical to Fig. 8 and consistent with actin filaments in the presence of actin depolymerization factors (ADFs)/cofilins. Same color code as in Fig. 7. Because ${\omega }_1/{\omega }_0={10}^3\gg 1$, correlations are very significant and the mean-field approach totally fails.

Figure 11

Dependence of correlation coefficient $\text{corr}\left(S_1,S_n\right)\equiv \text{cov}\left(S_1,S_n\right)/\sqrt{\text{var}\left(S_1\right)\text{var}\left(S_n\right)}$ on association rate ${\omega }_{+}$ for $n=2$ and $n=9$. The black dots are obtained from simulations and the red lines are calculated from Eq. (E1) via Eq. (12) with $g_5=0$. Parameter values are consistent with actin filaments, i.e., ${\omega }_1=0.16$/s, ${\omega }_0=6$/s, and ${\omega }_{\text{a}}=0.01$/s [11].

Figure 12

Decay of correlation coefficient for parameter values consistent with actin filaments, i.e., ${\omega }_1=0.16$/s, ${\omega }_0=6$/s, and ${\omega }_{\text{a}}=0.01$/s [11] and association rate ${\omega }_{+}=0.2$/s. The black dots are from simulations while the two lines are obtained via Eq. (12) with $g_5=0$. The red (solid) line is calculated from Eq. (E1) while the blue (dashed) line displays the asymptotic behavior, given by Eq. (E2).

Figure 13

Distribution ${\mathrm{\Pi }}_L\equiv r_L-r_{L+1}$ of the cap length $L$ of state-1 subunits at the terminus, i.e., probability that the first $L$ subunits are in state 1 while the subunit at position $L+1$ attains state 0. The black dots are from simulations, while the blue squares are mean-field results and the red crosses are obtained via Eq. (12) with $g_5=0$. Apparently, correlations are significant for calculating the correct distribution. Parameter values as in Fig. 12.