A large set of differential equations determine how our state variables change with time. We solve these equations numerically, but not in a standard way because discontinuities in time occur frequently as objects collide suddenly and as objects suddenly spring into existence or disappear (due to new filament nucleation and depolymerization). To solve these thousands of differential equations, we divide time into discrete steps (typically tens of microseconds) balancing the necessities of capturing the system dynamics and accomplishing the simulation in reasonable human time (typically 3–5 d). At the beginning of each time-step, the biochemical events and forces experienced during the last time-step will have changed the state of the system. New collisions and link forces may have arisen, as well as new objects. Existing links may break if they experience excessive strain for several consecutive time-steps. Each explicit player thus experiences a net force vector; in the next step, we move each explicit player in this vector direction so as to reduce or eliminate the strain energy associated with its collisions and links. To accomplish this practically, we calculate the forces required to resolve each individual collision (or strained linkage) in a single time-step. Figure 8 demonstrates this calculation for a collision between two spherical bodies; a similar approach is taken for all pair-wise collisions or links. In brief, we sum all forces, attenuating all their magnitudes by the same factor without changing their directions, so that acting during the time-step they produce just enough displacement to separate objects that collided in the prior time-step. This process avoids prescription of elastic constants and is equivalent to proceeding through a series of quasi-static equilibria, a formally valid approach if the biochemical dynamics are slow relative to the resolution of force imbalance.
Each individual computer run simulates bacterial motion for a period of up to many minutes. We run hundreds of such simulations and then statistically analyze the ensemble of runs. Fitting straight line segments to each trajectory and filtering those segments by slope (speed of motion) reveals that each simulated bacterial trajectory is composed of a sequence of pauses (of varying duration) separated by near-constant speed runs between pause locations. After we identify all the pauses, we measure the distances between adjacent pauses; these are the putative step-sizes. Histograms of pause duration and step-size, distilled from multiple simulations, then allow comparison with experimental observations and reveal whether there exists a preferred step-size or pause duration. Figure 9 shows a segment of trajectory data and the progressive stages of our line-fitting analysis.
We average many thousands of pause events into portraits characterizing system behavior preceding, during, and following the typical pause. To do this, we align pauses, time and space shifting short sections of the path projected trajectories that span a single pause event so as to superimpose their starting or stopping points (Figure 10). These pauses are of different duration, so our average response will be most meaningful near the alignment point. To improve this analysis, we can also select and average only pauses of similar duration or create ensemble portraits from start-aligned and stop-aligned analyses.
Any trend remaining after the averaging of many thousands of events will reveal significant system behavior near the alignment point. Any individual event, however, might not exhibit all the trends revealed in such an average, so that the interpretation of these average profiles should be tempered accordingly.
Not all of the capabilities of our model have been enabled in the simulations contributing to this study. Our calculations show that the local depletion of the implicit players, due to their incorporation into a larger assembly, is not significant for the concentrations, rate constants, and geometries of this system (data not shown). Thus, we do not simulate the diffusion of any of the implicit players (proteins), but rather assume that each exists at a constant concentration (see Table 1). With this assumption, we need not accurately represent the depolymerization of the bacterium's comet tail in modeling the movement of the bacterium. (This depolymerization could otherwise have had an important role in regenerating depleted stocks of some implicit players.) We therefore depolymerize F-actin in the most computationally efficient way: we assume an artificially high pointed end depolymerization and ignore cleavage by ADF/Cofilin.
In addition, F-actin interacts with cellular components in vivo that are not explicitly represented in either the dendritic nucleation model or our simulations (e.g., with other cytoskelet al.filaments). Some of these interactions have the effect of locking down actin tail in cellular space. We approximate their effect with a time- and actin length-dependent application of adhesions that eventually fix F-actin and the actin tail in our simulation space.