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A computational model of Listeria monocytogenes propulsion that explicitly simulates a large …


Biology Articles » Biophysics » In Silico Reconstitution of Listeria Propulsion Exhibits Nano-Saltation » Figures

Figures
- In Silico Reconstitution of Listeria Propulsion Exhibits Nano-Saltation

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Figure 1.Model Schematic

(A) shows a simple cartoon of the bacterium and some actin filaments (explicit players) against a backdrop of a moving diffusion–reaction grid attached to the bacterium. We use this spherical coordinate grid, whose origin moves with the bacterium, to keep track of the diffusion and biochemical interactions of the scalar protein concentrations fields that characterize implicit players. Protein size greatly impacts diffusion of proteins in a cellular environment (Luby-Phelps 1994, 2000), so we modify the nominal diffusion coefficients of implicit players accordingly. The dotted line is the path trajectory in 3D space of the bacterium.

(B) illustrates the origin of forces that act on a bacterium in our simulation. Actin filament α is bound to an ActA protein on the bacterial surface, generating a link force that acts to hold the two objects together. Filament β is shown colliding with the bacterium, generating a collision force that acts to push the two objects apart. The barbed end of filament γ is nominally too close to bacterial surface to allow addition of an actin monomer, but Brownian motion might bend the filament into the dotted configuration, thus allowing polymerization and creating a collision. We model our filaments as rigid rods, but simulate this filament flexibility with a polymerization probability function that approaches zero with the filament–bacterium gap; i.e., it is possible for a filament to add a monomer even if the nominal filament–bacterium gap is less than 2.7 nm (the filament length increase per actin monomer). The bacterium, actin filaments, and structures of actin filaments are all subject to Brownian simulation forces and torques, represented by δ.

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Figure 2.A Simulation Video Frame Showing Actin Filament Branching near the Bacterial Surface

Arp2/3 seeds branches off of existing filaments at a characteristic 70° angle. Different ATP/ADP forms of actin have differing affinities for proteins such as ADF/Cofilin. Filament barbed tips can be capped unless they are protected through interaction with an ActA molecule; as indicated, surface-bound ActA molecules elastically link the bacterium to an actin filament near the barbed end.

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Figure 3.A Simulation Time Series

Several video frames from one simulation show, on a gross scale, the de novo ActA filament nucleation and Arp2/3 branching from existing filaments that form a comet-tail, the hydrolysis of actin filaments in the comet-tail, and the subsequent pointed end depolymerization of ADP-actin (filament severing by ADF/Cofilin is not active in this simulation).

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Figure 4.Step-Sizes, Pause Durations, and Speeds of Motion with Different Brownian Simulation Force Attenuation

The legend shows the multiplier by which the Brownian simulation force on the bacterium is attenuated, such that a multiplier of 1 corresponds to a Brownian simulation force appropriate to an unconstrained bacterium and a multiplier of 0 signifies no simulated Brownian motion of the bacterium (see discussion of the relationship between applied Brownian simulation force and numerical time-step in the model description). We line-fit and filter trajectories by slope (speed) to identify pauses in the bacterial motion. Any line with slope less than the pause threshold might indicate a pause. The distance between adjacent pause locations is the step-size, assembled into a histogram in (A). No characteristic step-size is evident; smaller steps are simply more probable than larger ones. We exclude steps shorter than 0.2 nm. (B) shows a pause duration histogram; here we exclude pauses 50 ms in duration. For a Brownian multiplier of 1, we used 30 simulations with 56,239 pause events for (A) and (B) and 6,098 runs for (C) (1,929 s total simulation time, 644 s total paused time, 116 nm/s average speed, pause threshold 40 nm/s). For a multiplier of 0.5, we used 13 simulations with 49,207 pause events and 2,287 runs (1,534 s total simulation time, 700 s total paused time, 93 nm/s average speed, pause threshold 30 nm/s). For a multiplier of 0, we used 18 simulations with 9,748 pause events and 1,179 runs (940 s total simulation time, 612 s total paused time, 56 nm/s average speed, pause threshold 20 nm/s).

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Figure 5.System Outcome Profiles for Several Adjacent Pauses

Pauses are shown by red horizontal lines in a single simulation run in which the Brownian force multiplier was 1 (unconstrained bacterium). Listed for each pause are the pause duration (δt) and distance to the following pause (δs) as reported by our line-fitted analysis. The vertical line segments in the lower half of the figure show discrete events. From the bottom, these are the number of polymerization events on filaments very close to the bacterium, the number of new links formed between the bacterium and filaments, and the number of these bacterium–filament links broken. Above these are plotted the total number of bacterium–filament links. At the top, the net filament link force and the net filament collision force are given in picoNewtons (see Figure 1B), along with the total force. Some general trends aligned with pauses are apparent, such as decreased actin and link dynamics during a pause, but any characteristic biochemical or force response is obscured by the Brownian agitation of the bacterium.

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Figure 6.System Outcome Profiles for Several Adjacent Pauses—No Brownian Motion of the Bacterium

Long pauses during this run without simulated Brownian motion of the bacterium (Brownian force multiplier of 0) reveal the force balance/imbalance conditions that cause pause–run behavior. System outcome profiles are shown for several adjacent pauses (red horizontal lines) with their duration (δt) and distance to the following pause (δs) displayed as reported by our line-fitted analysis. For a description of each of the entities plotted here, see the legend to Figure 5. Note that the bacterium pauses in its forward progress when the filament collision and link forces cancel each other; then, the collision forces tend to rise during a pause as filaments in the tail catch-up with the bacterium and generate new filament collisions. Runs are coincident with the breakage of many highly strained filament links which are quickly replaced by unstrained links; note that the filament link number does not change greatly during such an avalanche of link breakages and that it is steadily climbing over this time range in this particular simulation.

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Figure 7.Anatomy of a Pause

An ensemble of average system outcome analyses, with (solid lines) and without (dotted lines) simulated Brownian motion of the bacterium (see discussion on the relationship between applied Brownian simulation force and numerical time-step in the model description). These averages were compiled from pauses with duration greater than 10 ms, using 10,365 pauses with, and 4,358 pauses without, simulated Brownian motion (there are fewer, longer pauses without Brownian motion of the bacterium). Only sufficiently time-separated pauses contributed to these averages, so that the 10 ms preceding the pause start and the 10 ms following the pause stop are guaranteed not to include the effects of any adjacent pause. The Brownian simulation force trends can be read from the total force curve, which is only slightly offset by the link and collision forces in the case where Brownian movement of the bacterium is simulated (solid lines). Here, initiation of a pause follows a large forward-directed Brownian simulation force on the bacterium (segment A), which increases link turnover and produces a large population of synchronously strained links. Backward path-directed Brownian simulation forces (segment B) maintains the pause until, aided again by forward-directed Brownian simulation forces (segment C), the bacterium transitions back into a run. The Brownian simulation force trends can be read from the total force curve, which is only slightly offset by the link and collision forces. Without simulated Brownian motion of the bacterium (dotted lines), a pause is initiated and maintained when a population of ActA–actin filament links can resist the essentially constant total filament collision force. A pause terminates in this case when these links break en masse. Any individual pause in the averaged set of pauses might not demonstrate all of these response features.

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Figure 8.A Simple Collision Rule

We calculate the magnitude, F, of equal and opposite forces applied to colliding objects such that they no longer collide after a time-step of δt. This force is calculated by considering the maximal distance, δ, of object intersection and the shape-based viscous drag, γ, for each object. In this example, we use Stokes' law for the viscous drag on spheres. We are assuming that the time-scale for a collision to resolve itself is much shorter than the discrete time-step used in the computation. We make similar calculations for more complex shapes and collisions.

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Figure 9.Slope-Based Analysis of Bacterial Trajectories

A brief sample of a bacterial trajectory from one simulation run (black), a smoothed approximation to that trajectory (blue), line segments of near zero velocity fit to the smoothed trajectory (green), and the summary of a pause event assembled from those lines (red). Our analysis software seeks nearly horizontal segments of the trajectory (i.e., pauses), of maximal possible duration, in which excursions away from pause location lie with a jitter tolerance that we specify. Trajectories are curves in 3D space with curvature and torsion. To simplify the analysis, we use a path position variable (on the vertical axis), projecting each displacement onto the path tangent vector, instantaneously defined by the bacterial orientation. Then, we identify pauses in the resulting time series to specify displacement along the smoothed trajectory. Labels on each pause report pause durations, δt, and the displacement to the next pause (step-size), δs. Random thermal agitation forces act to buffet the bacterium, and every individual part, in our simulations; this is what makes the trajectories jagged.

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Figure 10.Calculating an Average System Response

(A) shows a segment of simulated bacterial trajectory with five identified pauses shown in red; δt is the pause duration, and δs is the distance along the path to the next pause. To obtain the average response of any system outcome, we align the pauses at their stopping (or starting) points as shown in (B).

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