We model the molecular mechanics of the growth/disassembly of an actin network as it interacts with a moving rod-shaped bacterium to whose surface many ActA proteins are bound at specific locations. We distinguish molecular mechanics (Howard 2001) from molecular dynamics: we are not concerned with van der Waals forces and hydrogen bonds or with conformational changes during protein–protein interactions. Our model is different from a purely continuum model, in which state variables (those dependent variables which, together, fully describe the state of a system) would characterize only the bulk properties of an actin dense tail, using average compliance and polymerization values. We instead separate the cellular world into two basic classes of entities, those that are relatively large and present in small numbers (e.g., actin filaments, a bacterium) and those that are very numerous and small (e.g., actin monomers and other diffusible proteins). We simulate the former entities, which we call “explicit players,” as individuals; our state variables keep track of the position, orientation, and biochemical state of each individual and its change with time according to appropriate physical laws (e.g., Newtonian force balance laws). Those entities that are more numerous we will call “implicit players”: we represent them with continuum field state variables, i.e., molar concentrations that vary with time and place. We use standard partial differential reaction–diffusion conservation equations to express how these continuum fields change with time as the implicit players interact with each other and with the explicit (individual) players (Figure 1A). Dataset S1 is a simplified psuedo-code of our simulation software.
Various different forces impinge on the simulated bacterium. Forces move two objects apart if they happen to collide at the end of a time-step. Likewise, elastic bonds linking two objects (e.g., an actin filament–ActA bond) exert equal and opposite forces that hold those objects together; these links break under sufficient strain. Forces of random orientation act on every explicit player to simulate Brownian motion (i.e., the sum of all the many collisions with small molecules that, in biological reality, contribute to the Brownian motion is represented in our model by a single vector force and a single vector torque). This system never approaches an equilibrium; Brownian motion and biochemical events ceaselessly create collisions and perturb protein–protein links. Thus, we must compute new forces, exchanged between new neighbors, in each time-step. Figure 1B illustrates the set of mechanisms that combine to generate the net force on the bacterium in our simulation.
At the heart of this simulation is the dendritic nucleation model of actin dynamics (Mullins et al. 1998; Pollard 2003). Asynchronously, each individual actin filament can grow or shrink at either end by actin monomer polymerization/depolymerization; hydrolyze the ATP bound to one or more of its actin monomers to ADP-Pi; dissociate the Pi from one or more such monomers; be severed by ADF/cofilin; bind Arp2/3 to an ATP-actin subunit in the filament; be capped or uncapped at either end; and nucleate new filaments through Arp2/3 initiated side-branches. Repeated nucleation of new branches from existing filaments leads to a dense meshwork of actin in the comet-tail. Besides Arp2/3 mediated branching, all other cross-linking and adhesions involving actin filaments are implicit in the age and length dependent anchoring of f-actin in our simulation space. All actin filaments accumulate adhesions that gradually increase drag coefficients and eventually lock each filament into a fixed position. Figure 2, a video frame from a typical simulation, introduces some of the explicit and implicit players.
The simulation time-step has a subtle effect on the simulation of Brownian motion for constrained objects (that is, objects linked to other objects). Applying the same forces and torques that are appropriate for free objects exaggerates the simulated Brownian motion of constrained objects since the motion restriction that results from those constraints can only be resolved over several time-steps, and those time-steps are large relative to the intrinsic time-scale of the constraints. Experimental measurements (Kuo and McGrath 2000) show very little Brownian motion (relative to similarly sized nearby vesicles) of L. monocytogenes associating with their actin tails; to match the biological reality, we need to modify our simulation of Brownian motion, since we cannot yet use much smaller time-steps. We compensate for this technical problem by carrying out simulations both for the two extreme cases (with Brownian motion appropriate for a free bacterium and with no Brownian motion of the bacterium at all) and for an intermediate degree of Brownian motion. Advances in computer processing speeds will, most likely, make such attenuation unnecessary in the near future. We will henceforth use the term “Brownian simulation force” to refer to the forces and torques that we apply to the bacterium to simulate its Brownian motion.
For our model, we use typical physiological concentrations for each of the proteins involved; these are listed in Table 1. Table 2 summarizes the reaction rate constants we used. Some crucial parameters and protein functions are as yet incompletely known. These include the exact pathways and rate constants associated with the stimulation of local actin filament polymerization by the bacterial surface-bound ActA protein. This protein has binding sites for a host of proteins, including G-actin, F-actin, Arp2/3, and Ena/VASP. In addition, while the affinity between free ActA and Arp2/3 has been measured (Zalevsky et al. 2001), that value (KD = 0.6 μM) does not sufficiently characterize that interaction since the interaction rates may be limited by the flux of Arp2/3 or G-actin onto the surface. We have calculated the flux of Arp2/3 and G-actin onto the bacterium's surface to determine the expected equilibrium number of ActA–Arp2/3 and ActA–actin complexes there, as explained in Dataset S2. We tune these approximate rate constants to create actin networks with biologically representative side-branch separation and filament numbers. Because the rate constants that we have obtained in this way will depend on the concentrations of ActA, Arp2/3, and actin monomers in the model, the rates given for ActA–Arp2/3 and ActA–actin interactions in Table 2 apply to the concentrations given in Table 1.
The ActA protein is distributed asymmetrically on the bacterial surface, with more ActA near the rearward tail-forming end. The unipolar shape of our distribution is based on measurements of the fluorescence signal from RFP-labeled ActA along the bacterial length for newly divided bacteria (S. Rafelski and J. A. Theriot, unpublished data). New filaments are produced by two pathways. By activating Arp2/3, ActA is thought to catalyze the creation of new actin filaments and side-branches. We simulate the co-binding of ActA, Arp2/3, and an existing actin filament, allowing binding in any order. This complex leads to a new side-branch on the existing filament. Binding of ActA to the actin filament can occur only at specific ATP or ADP-Pi actin sites and is binding site limited, meaning that each bound ActA occludes a linear region of five monomers on the filament from further binding. Creation of a new filament de novo in our model involves the co-binding by ActA of G-actin and Arp2/3, in any order (Boujemaa-Paterski 2001).
In conjunction with other proteins (e.g., Ena/VASP), ActA may also regulate actin dynamics in other important ways (Goldberg 2001). In this version of our model, we do not explicitly simulate Ena/VASP molecules, which can regulate actin networks by binding profilin, competing with capping protein, and regulating Arp2/3 spacing (Krause et al. 2003). Instead we assume that ActA itself can uncap any actin filament barbed end to which it binds closely (within one ActA length) and ignore the other possible Ena/VASP functions. We find that this uncapping function is necessary to obtain persistent motion with a low nucleation rate of new filaments. For the values in Tables 1 and 2, our simulated bacterium do move without this uncapping, but more slowly. Loisel et al. (1999) have reported a similar dependence in their experiments, finding that Ena/VASP is not required for bacterial motion, but that it improves the efficiency of the motion observed.
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