Cellular processes generally involve interactions among 101 to 105 gene products. These interactions can be both biochemical, as in the activation of one protein by another, and mechanical, as in the application of force between bodies. Even when each individual interaction is simple and understood in detail, neither intuition nor qualitative description can forecast the emergent behavior of the whole system. We describe a methodology to characterize such emergent behavior using a detailed computer simulation of both biochemical kinetics and mechanical dynamics. In this paper, we apply the technique to the motility of the bacteria Listeria monocytogenes, a well-studied system in which actin network growth produces a force that moves the bacterium inside of cells. We discuss the model design, compare behaviors of the computational and biological systems, use the model to explain observed features of the bacterial motion, and identify observable experimental correlates of our hypotheses through which our interpretations may be confirmed or rejected.
L. monocytogenes is a pathogenic rod-shaped bacterium that invades cells, reproduces, and spreads to neighboring cells, never exposing itself to the extracellular environment, thus avoiding a humoral immune response (Tilney and Portnoy 1989). By expressing the protein ActA (Domann et al. 1992; Kocks et al. 1992, 1995), L. monocytogenes bypasses the host cell's normal controls on actin network growth to produce a dense “comet tail” of actin. This actin tail generates a ram force, by rectifying thermal motion, to both propel the bacterium within a cell and push the bacterium into neighboring cells through distension of the cell plasma membranes.
Among experimental advances thus far made to understand this motile system are identification of the purified proteins required to reconstitute motion in vitro (Loisel et al. 1999), an ability to mimic this motion using polystyrene beads coated with the bacterial ActA protein (Cameron et al. 1999, 2001, 2004), and experiments that have revealed a discrete step-like motion on the nanometer scale (Kuo and McGrath 2000; McGrath et al. 2003). A series of complementary theoretical models have been proposed to account for some observed features of bacterial and bead motion (Peskin et al. 1993; Mogilner and Oster 1996, 2003; Gerbal et al. 2000a, 2000b; van Oudenaarden and Theriot 2000). These studies, taken together, show that L. monocytogenes' actin structures, first described by Tilney and Portnoy (1989), are created from the same protein components and perform a function similar to the actin machinery in the lamellipodia of motile cells. The dendritic nucleation model for actin network growth (Mullins et al. 1998; Pollard et al. 2000, 2001; Pollard 2003) offers a qualitative description of this biochemical network. In the absence of the bacterium, specific signals activate WASP/Scar proteins, and these in turn activate the Arp2/3 protein complex to provide new filamentous actin (F-actin) nucleation sites at or near the barbed (plus) end of existing filaments (Higgs and Pollard 2001). These new filaments form at a characteristic 70° angle to the parent filament, creating dense, highly branched networks (Mullins et al. 1998). Filament barbed ends are rapidly capped with high affinity by capping protein, making the creation/maintenance of free barbed ends critical for continued network growth. Hydrolysis of the ATP that was bound to each actin monomer favors filament disassembly, returning actin monomers to the pool of polymerization-ready G-actin. Cofilin aids in this disassembly by fragmenting F-actin, binding with much higher affinity to ADP actin than to ATP or ADP-Pi actin. The motion of L. monocytogenes exploits all of these actin network features, except that the bacteria's ActA replaces the host cell's WASP/Scar proteins and all the associated upstream signaling mechanisms that normally activate WASP/Scar to control actin polymerization (Welch et al. 1998; Zalevsky et al. 2001).
Our model differs in several ways from previous attempts to generate mathematical or physical models for L. monocytogenes motility (though see Carlsson 2001, 2003). We simulate explicitly a large number of detailed interactions of both a biochemical and mechanical nature, representing all protein–protein binding interactions with on-rate and off-rate kinetic equations. The simulation of actin filament polymerization, for example, depends on the local concentration of actin monomers and the association and disassociation rate constants (which have been experimentally determined), modulated by the steric accessibility of free barbed ends. Together these factors determine the binding/dissociation probabilities for each filament at each simulation time-step. Bulk properities of our actin “gel” arise from the contributions of the many individual parts of the actin network. Our model can thus accomodate arbitrary geometries, explicit stochastic input, and specific small-scale events. Mechanical interactions, which resolve collisions and accommodate the stretching of protein–protein linkages, follow Newton's laws.
We can represent any particular interaction in as coarse or detailed a fashion as desired, subject to the availability of computer resources, and each of these can be based either on experimental information or on simple postulates. We can determine the emergent behavior of the system, which is the dynamical outcome of all the particular interactions, only by running the computer program for many hours or days. In such a model it is neither possible nor desirable to include all details. If our model fails to characterize experimentally observed behavior, then something is missing. If our model does capture an emergent behavior, however, then we can study how quantitative changes in the underlying details (e.g., protein concentrations or specific rate constants) affect this larger scale behavior. The exploration of putative mechanisms is also straightforward, as it is easy to add, remove, or modify each individual interaction.
With our approach, we formalize experimentally based models of specific protein–protein interactions and biochemical kinetics in a direct and flexible way, but there are drawbacks. The theoretical approaches used to analyze the Brownian ratchet model and its refinements (Peskin et al. 1993; Mogilner and Oster 1996, 2003) facilitate the derivation of equations that describe important system characteristics, such as force–velocity curves. No such equations are available in our stochastic, individual molecule-based model; instead, we must distill parametric relationships from ensembles of many repeated simulations. Completing these parametric studies in reasonable human time requires considerable computer resources.
The biochemical and mechanical interactions near the bacterial surface are stochastic processes involving hundreds of filaments. We model dynamic processes on a per filament basis, rather than through bulk network properties and average filament growth. The growth of any particular filament depends upon that filament's precise location, orientation, and biochemical state, all of which change through time. There is no better way to simulate such a system than with a model that tracks each of these variables for each individual filament. In the future, this type of detail will be essential to capture (and thus explain) many observed biological phenomena.
The trajectories generated by this model of L. monocytogenes motility display repeated runs and pauses that closely resemble the actual nanoscale measurements of bacterial motion (Kuo and McGrath 2000; McGrath et al. 2003). Further analysis of the simulation state at the beginning and ends of simulated pauses suggest a new interpretation of the experimental results. We show that there is no characteristic step-size or pause duration in these simulated trajectories and that pauses can be caused by both correlated Brownian motion and by synchronously-strained sets of ActA–actin filament mechanical links.
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