The gross behavior of our simulated bacterium is life-like; model bacteria move in a qualitatively similar way to wild-type L. monocytogenes. Average speeds of motion varied from ten to hundreds of nanometers per second, as do real observed bacterial speeds in different experiments (40 nm/s in purified proteins in Loisel et al. 1999; 140 nm/s in cytoplasmic extracts in Cameron et al. 2004; 1.4 μm/s in vivo in Dabiri et al. 1990). Videos at any scale may be rendered from our simulations (Videos S1 and S2; other full-length simulations at www.celldynamics.org). Figure 3 merges several frames from one of those movies, showing the large-scale formation, hydrolysis, and depolymerization of the actin comet-tail.
The microrheology experiments of Kuo and McGrath (2000) and McGrath et al. (2003) present an opportunity to illustrate the utility of our stochastic model founded on small-scale details. They reported that L. monocytogenes motility is multiphasic; motion of the bacterium that appears smooth on the micrometer length-scale actually consists of pauses that last many milliseconds, discrete nanometer-scale steps, as well as uninterrupted runs. No current model of this bacterial motility has fully explained these discrete steps, though numerical simulations with the tethered ratchet (Mogilner and Oster 2003) can exhibit saltatory motion. Possible explanations involving strained links between ActA and actin filaments and nucleotide dependent filament templates are discussed in McGrath et al. (2003).
Figure 4 shows the distribution of step sizes and pause durations that our model produces (using the values of Tables 1 and 2 and one set of mechanistic hypotheses) with three different assumptions for the Brownian motion of the bacterium (see under The Model). In all cases, we find that there is no characteristic step-size, but rather a continuum of steps with the smaller steps being more probable than larger ones. The qualitative shapes of these histograms are insensitive to changes in all parameters we might reasonably vary, barring values that disrupt persistent bacterial motion. The parameters we have varied include link characteristics (e.g., link length, link force, and maximum link strain), Arp2/3 branching rate, and temperature. In fact, even doubling the size of each actin monomer (easily done in a simulation) does not change these histograms significantly (data not shown).
These results suggest that our simulated bacterium does not move with steps of any prefered size, and specifically not with a step-size related to the actin monomer dimensions. The pause in forward progress might equally be considered the defining event in the bacterium's motion; a “step” in this case is just a run made along the path between adjacent pauses. But what physical process stalls the actin polymerization ram to initiate pauses, and what physical process breaks the bacterium out of each paused state into a run? To answer these questions, we need to examine how key descriptive system features vary before, during, and after pauses in our simulations. We do this by looking both at individual pauses and at the average of these system outcomes for many thousands of pauses (see Materials and Methods).
In Figure 5, we follow actin polymerization, link formation and breakage, link number, and path-directed forces for several sequential pause events during a single simulation. Owing to the stochasticity introduced by the Brownian simulation forces in this simulation, it is difficult to find trends in such single simulation profiles. We can learn more by turning off the Brownian simulation forces on the bacterium as has been done in figure 6. Now frequent long pauses are observed that clearly reveal the force relationships during pause initiation and termination. Both filament link force and collision force increase in magnitude synchronously during a pause, indicating resistance to forward progress by a population of ActA–actin filament links. The bacterium moves rapidly forward only when the total filament link force suddenly plummets. This sudden decrease in link force can only be attributed to a cascade of link breakage. This result indicates that the highly strained filament links that had balanced the filament collision force during a pause are rapidly exchanged for unstrained links when a pause ends.
The average of these system outcomes further clarifies pause causality and reveals differences between the two cases illustrated in Figures 5 and 6. From Figure 7 we see that pauses can occur with or without Brownian motion of the bacterium. But when we simulate the Brownian motion of the bacterium, we observe that pauses are correlated with an accidental sequence of similarly directed Brownian simulation forces (forward to initiate a pause, backward to sustain a pause, and again forward to break out of a pause). Note that any individual pause event in our average ensemble may experience only a subset of the correlated Brownian simulation force profile presented in Figure 7. On that figure we have partitioned this correlated Brownian simulation force into three temporal regions, labelled A, B, and C. An individual pause event might correlate with the Brownian simulation force sequence of A, or only A and C, or only B and C, or any other combination. Additionally, some pause events might be entirely uncorrelated with any Brownian simulation force trend. In other words, this averaging method reveals system trends that occur frequently, but need not be present in every contributing event.
We contrast the Brownian/no Brownian motion cases to better understand pause initiation, maintenance, and termination. Our most realistic simulation will incorporate effects from each extreme case. With simulated Brownian motion of the bacterium, we suggest the following causal temporal sequence for pause initiation, maintenance, and termination (with the caveat that most individual pause events will experience only a subset of this sequence):
A particularly large Brownian simulation force (or accidentally correlated sequence of forces) in the forward path direction causes an unusually rapid but small forward displacement of the bacterium (region A).
The steady-state rate of filament link turnover increases slightly as highly strained links break and are replaced by an ensemble of new links that all form nearly simultaneously in an unstrained state, thus creating a larger than steady-state population of coordinately unstrained links.
A particularly large Brownian simulation force, or correlated sequence of forces, opposite to the path direction forces the bacterium backward against the population of linked barbed end actin filaments; filament collision force increases, filament link force falls, and actin polymerization near the surface decreases. A pause ensues.
During the pause new filaments form and existing but distant barbed ends “catch up” with the bacterium, thus increasing the filament collision force forward which the links restrain. The pause terminates when a particularly large Brownian simulation force (or an accidentally correlated sequence of forces) in the forward path direction is sufficient to break a few of the strain-synchronized filament links. As these links break, the force stretching each remaining link increases, setting in motion an avalanche of cooperative link breakage and initiating a run.
We are justified in interpreting these correlations of Brownian simulation forces as causal because those forces are generated in our simulations so as to be random in direction and magnitude (representing a Gaussian distribution). Nothing in our model can cause such Brownian simulation force “accidents.” Their correlation with pause initiation or termination must therefore be causal.
Absent Brownian simulation forces on the bacterium, the system response throughout the course of a pause is very different. In this case, a pause occurs only when a population of synchronized filament links is able to balance the filament collision forces, which on average increase only slightly during a pause, until a cascade of breaking links allows the bacterium to run again. Judging from the shape of the step-size histograms in Figure 4, the generation of a set of strain-synchronized links that initiate a pause is likely a random event. That figure reveals a Poisson process-like distribution of step-sizes with weak or no Brownian simulation forces; moreover, the step termination (and therefore pause initiation) appears to occur with a constant probability through time. This should be contrasted with the case of simulated Brownian motion appropriate for an unconstrained bacterium, in which step-termination (pause initiation) is correlated with forward path-directed Brownian simulation forces.
The small amplitude of experimentally measured fluctuations of bacteria in vivo (Kuo and McGrath 2000) suggest that the simulations absent Brownian motion of the bacterium come closest to representing the biological reality; the coincidence of similarly directed Brownian movements is probably less important than the balance between filament–bacterium collision and link forces.