We have used our model to ask how L. monocytogenes motility is mediated by actin-mediated forces. Building a simulation from basic, well-understood structures and interactions, we have reconstituted bacterial motility in silico. Appropriate speeds and persistence of motion emerge, reproducing experimentally observed values. Additionally, our simulation yields as an emergent behavior the nanometer-scale saltatory motion reported by experimentalists. We can analyze details of the simulated bacterial trajectories to investigate characteristics of this saltation: what is the mechanism behind the stepping, and is there a favored step-size?
Our computational experiments lead us to conclude that the tethered-ratchet model is an inherent “pauser” with several important attributes. First, there is no characteristic step-size or pause length; shorter steps and pauses are more frequent than longer ones. Second, the intensity of Brownian agitation of the bacterium influences average pause duration and frequency, but this agitation is not necessary for persistent saltatory motion. Third, pauses start when a population of filament links happen to form nearly simultaneously with low strain to balance filament collision forces. Pauses end when those links catastrophically break.
To produce nanometer-scale pauses and runs, no special function need be attributed to the bacterial bound ActA protein, beyond an elastic linkage to actin filaments and some mechanism that prevents barbed end filament capping. Specifically, the ActA protein does not need a motor-protein-like stepping ability, nor need it act as a clamped-filament elongation motor (Dickinson and Purich 2002). Given the large number of filaments near the surface of the bacterium and the wide variation in angle of those filaments, it is not clear that we would expect any step-size, even if ActA were motor-like with a discrete working stroke.
The speed of motion during a run is variable, but it is mostly confined to a narrow range of speeds that depends on the parameter set (see Figure 4C). Pauses are a significant feature in our simulated trajectories; bacteria spend large fractions of their time paused (from 33% to 65% in the simulations presented here).
Lastly, we have explored the variation of key biochemical events and mechanical interactions during a typical nanometer-scale saltation, looking at both individual pause events and averages of many such events, in an effort to uncover the causal factors. We conclude that the tethered-Brownian ratchet model is an inherent pauser; forward motion is temporarily halted whenever a population of synchronously strained filament links balances filament collision forces. Different mechanisms cause pause initiation/termination with and without simulated Brownian motion of the bacterium. With simulated Brownian motion of the bacterium, we find that pauses events are largely driven by coordinated Brownian simulation forces: a series of forces in the forward direction helps establish a set of coordinately-strained links, forces in the backward direction can help maintain a pause, and lastly forces in the forward direction help break links to terminate a pause. Without simulated Brownian motion of the bacterium, we find that a coordinately strained set of filament links balances the filament collision forces and that a pause will ensue until those links break en masse. Formation of such a set of filament links is an accidental, but frequent, occurrence, explaining the shape of the step-size and pause duration histograms (see Figure 4).
That we find no characteristic step-size in our simulated nanoscale stepping constrasts with experimental results (Kuo and McGrath 2000; McGrath et al. 2003). Restricted by experimental noise, those researchers cannot see steps smaller than about 2.5 nm, if they indeed exist. Without those steps, about 30% of our simulated steps would be between 4 nm and 6 nm. We are presently sharing trajectories with the Kuo laboratory to directly compare model and experiment.
Mogilner and Oster (2003) explore stepping behavior of their elastic tethered Brownian ratchet model and, for low filament tether numbers and particular capping rates and tether stiffness, observe step sizes similar to those reported by Kuo and McGrath (Kuo and McGrath 2000; McGrath et al. 2003). A step in that model occurs when one filament tether breaks and the remaining tethers all stretch in response to the new force balance. By constrast, steps in our model occur following a catastrophic breakage of many coordinately strained tethered filaments and are not highly dependent upon tethered filament number or capping rate. Because of the method by which we resolve collisions and strained links (see Figure 8), we do not prescribe the elastic properties of the filament–ActA links, and so our stepping is also independent of those values. Dickinson and Purich (2002) proposed a completely different mechanism for nanoscale stepping, involving a putative elongation motor that demonstrates approximately 5.4-nm stepping in simulations. In their model, most filaments are in compression while a few lagging filaments prevent the bacterium from moving forward. It is the “release and relocking” of a single lagging actin filament by the elongation motor that allows a 5.4-nm step. While the mechanisms are severely different, the basic behavior of this elongation motor model is similar to ours. Of the population of filaments interacting with the bacterium in our simulations prior to a step, a subset generates collision forces and a coordinately strained subset attached to ActA proteins balances those collision forces, resisting forward motion. The concurrent breakage of this linked subset allows a step forward, analogous to the “release and relocking” in the elongation motor model, though many more filaments are involved in our “release,” and the step distance before these filaments rebind to ActA, and thus “relock”, is not prescribed.
The computational analysis of L. monocytogenes motility described here represents a new tool that should be useful for understanding many complex subcellular systems. The construction of this computational model requires experimental measurements of the biological details in L. monocytogenes propulsion and actin dynamics in general. Only in the last several years have crucial biological details come to light, e.g., the role of Arp2/3 in filament branching, or the binding sites and functionality of the ActA protein. Additionally, the implementation of the model in silico requires significant computational power, now affordable in the form of clusters of “off-the-shelf” machines (we estimate use of 30 cpu years on a 2.8 Ghz Pentium 4 in the development and exploration of this model, 3.5 cpu years of which directly contribute to this report). Powerful object-oriented languages are also recently mature (we use Java™), making it possible to write computer code to implement such models. We believe that the confluence of detailed biological information and computational power/software heralds a new approach for understanding subcellular systems in which many thousands of simple biochemical and mechanical interactions lead to complex emergent behavior.
The biological systems in which this approach will be useful are, by definition, rich in detail. This complexity favors collaborations between modelers and the experimentalists who discover and quantify the molecular details without which this study would be specious. Creating a simulation environment that makes intuitive sense to experimentalists, i.e., one in which there is clear correspondence between biological entities and their modeled counterparts, greatly facilitates communication between modelers and biologists, and it ensures appropriate refinement of the model as new biological facts are uncovered.
There are many future refinements and research directions for this model. We can incorporate a more sophisticated representation of the actin hydrolysis cycle (Bindschadler et al. 2004) and include specifics of the interactions between ActA and proteins such as Ena/VASP. Recent work with ActA-coated beads (Cameron et al. 2004) characterizes the relationships between several biophysical parameters and motion initiation, speed, and persistence; these experimental findings can also be explored in silico. Extended with additional cellular components (e.g., a dynamic cortex, microtubules, motor proteins), the model might also be used to explore any of a number of cellular phenomena, including whole cell motility and cytokinesis. Our model, encoded in an object-oriented manner, is structured in a way that is strongly delimited by nature—so while we must still embrace approximation, we can minimize abstraction.