Nonequilibrium actin polymerization treated by a truncated rate-equation method

Abstract

Actin polymerization time courses can exhibit rich nonequilibrium dynamics that have not yet been accurately described by simplified rate equations. Sophisticated stochastic simulations and elaborate recursion schemes have been used to model the nonequilibrium dynamics resulting from the hydrolysis and subsequent exchange of the nucleotide bound within the actin molecules. In this work, we use a truncation approach to derive a set of readily accessible deterministic rate equations which are significantly simpler than previous attempts at such modeling. These equations may be incorporated into whole-cell motility models which otherwise quickly become computationally inaccessible if polymerization of individual actin filaments is stochastically simulated within a virtual cell. Our equations accurately predict the relative concentrations of both monomeric and polymerized actin in differing nucleotide hydrolysis states throughout entire polymerization time courses nucleated via seed filaments. We extend our model to include the effects of capping protein. We also detail how our rate-equation method may be used to extract key parameters from experimental data.

FIG. 1

Time courses of T^T/N (solid), η^T (circles) and ${\mathit{\eta }}^{P_i}$ (triangles) obtained via stochastic simulation of 3 µM ATP actin polymerized from 10 nM ATP pentamers. Here, it is seen that T^T /N acheives only 70% of the steady-state η^T value while ${\mathit{\eta }}^{P_i}\approx 30\%$ at the steady state. Inset) The time course of η^D closely follows that of T^D /N.

FIG. 2

Values of η^T obtained from Equation 11 (solid) against those obtained via stochastic simulation (circles) of 3 µM ATP actin polymerized from 10 nM ATP pentamers.

FIG. 3

Comparison between stochastically simulated data (circles) and calculated data (curves). A) The relatively slow polymerization of 1.0 µM ATP G-actin from N = 1 nM of 1 µm ADP seed filaments is modeled well by both the simple, single tip state model (dashed) and our multi-tip state model (solid). B) Rapid polymerization, where N is increased to 10 nM, is modeled well by our multi-tip state model but not by the single tip state model.

FIG. 4

Simulated (shapes) and calculated (curves) subunit hydrolysis states for the polymerization time course shown in Figure 3b. ATP: solid, triangles; ADP+P_i: dashed, circles. ADP: dotted, squares.

FIG. 5

Simulated (shapes) and calculated (curves) plus end hydrolysis states for the polymerization time course shown in Figure 3b. ATP: solid, circles; ADP + P_i: dashed, triangles. ADP: dotted, squares.

FIG. 6

The fractional error between the polymerization time course obtained via stochastic simulation and that predicted by our rate equations is below 3.0% across a broad range of experimentally accessible conditions.

FIG. 7

Stochastically simulated polymerization time course (circles) of 5 µM actin from 5 nM of 1 µm ADP seed filaments in the presence of capping protein (k_cap = 0.016 s⁻¹) compared to same as predicted by our rate equations (solid). Inset) The percentage of capped filaments obtained via stochastic simulation (circles) compared to those predicted by our rate equations (solid).

FIG. 8

Time courses of 1.0 µM ATP actin polymerized from 10.0 nM ADP seed pentamers. The non-negligible repolymerization which can occur at long times as predicted via stochastic simulation (circles) is not predicted (solid) by our rate equations when many filaments completely depolymerize.