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Current address: Department of Biochemistry & Biophysics, University of California San Francisco, San Francisco, California, United States of America

Conceived and designed the experiments: SMR JBA GMO. Performed the experiments: SMR JBA. Analyzed the data: SMR JBA. Contributed reagents/materials/analysis tools: SMR JBA GMO. Wrote the paper: SMR JBA.

The authors have declared that no competing interests exist.

Cells tightly regulate the branched actin networks involved in motility, division, and other important cellular functions through localized activation of the Arp2,3 protein, which nucleates new actin filaments off the sides of existing ones. The pathogenic bacterium,

The actin driven motility of

To understand the dominant force mechanisms regulating bacterial speed, we combine this simulation with new experimental results on ActA distribution patterns. A population of

It may seem obvious that concentrating more ActA at the ‘business end’ of the bacterium, where the polymerization it catalyzes most effectively moves it forward, would enhance bacterial speed. This turns out to be true, but for subtle reasons. Consider the following statements about the bacterial-actin interactions:

The cylindrical geometry of the bacterium is important to its motion, with filaments nucleated at the sides contributing weakly, or not at all, to forward forces, while filaments nucleated at and/or interacting with the pole push the bacterium forward

Filaments generated at the sides grow autocatalytically, due to Arp2,3-dependent branching. Thus the sides provide many pre-polymerized filaments already integrated into the comet-tail actin network and primed to push when, due to forward displacement of the bacterium, they find themselves at the rear end.

All filaments are also able to retard the bacterium due to their interactions with the ActA protein, either directly or possibly via Arp2,3 or Ena/VASP, which create transient forces tethering the tail to the bacterium reviewed in

Filaments and the bacterium exchange kinetic friction forces proportional to the contact force of each interaction, though the importance of this force in determining motility is unknown.

Motion of the bacterium feeds back onto this system –the slower the bacterium moves, the more filaments it can accumulate to generate propulsive forces, and vice versa.

The interplay between these competing propulsive and restraining mechanism ultimately determine the effect of ActA quantity and distribution on bacterial speed.

Here we experimentally alter the degree of polarity of ActA on the surface of

We created populations of

A) Images of normal (top) and ultrapolar (bottom) bacteria expressing ActA-RFP. Left panels show ActA signal, right panels show bright-field illumination. B) ActA intensity linescan of one normal and one ultrapolar bacterium labeled in A. The average intensity over the width of the bacterium is plotted at each point along its length. Intensities are normalized to the maximum in the linescan. Ultrapolar ActA bacteria display relatively less ActA along the sides than normal ActA bacteria.

Ultrapolar and normal populations were mixed to create a continuum of ActA surface distributions that we could directly compare within the same motility assay experiment. We performed time-lapse video microscopy of steady-state movement of these mixed polarity populations. To avoid confusion, we excluded bipolar ActA bacteria

To obtain a continuous measure of the degree of polarity, we calculated, from measured ActA linescans, the 1^{st} moment of the intensities along the bacterium (normalized to bacterial length and maximum ActA intensity). The 1^{st} moment describes how asymmetric the ActA intensity linescan is around the center of the bacterium, with higher 1^{st} moments representing distributions with greater asymmetry (linescans on axis in ^{st} moment and to the total ActA linescan intensity (a measure of total ActA computed by integrating the ActA distribution over the surface of the bacterium) in this population (^{st} moment of ActA distribution). Approximations of this function as polynomials in the two variables will become arbitrarily accurate as the polynomial degrees increase. To ascertain how high the polynomial degrees should be before diminishing returns makes further increases in degree pointless, we generated fits to our measured data using all degrees less than 4. We found only slight increases in the R^{2} goodness of fit criteria above a fit linear in both 1^{st} moment and total ActA intensity (

A) 3-dimensional plot of the average experimentally measured bacterial speed vs. its 1^{st} moment and total ActA linescan intensity. The linescan graphs along the bottom right axis display the ActA distribution for respective 1^{st} moment values (intensity normalized to the maximum to highlight how the 1^{st} moment describes degree of polarity). The larger the 1^{st} moment, the greater the degree of polarity in ActA distribution. The total ActA intensities span a ten-fold range. Actual data points are shown in red. The best-fit plane is shown in gray. Speed increases with both ActA polarity (1^{st} moment) and ActA intensity. The bacteria that provided the prototypical normal and ultrapolar distributions used in simulations (^{2} residual values obtained when fitting the data shown in A by functions of increasing polynomial degree. The numbers on the x- and y-axes represent the highest polynomial degree of the function used for the fit. For example, the small round point at “2” on the ActA intensity axis and “1” on the 1^{st} moment axis represents the R^{2} value for the fit Z = a+bX+cX^{2}+dY where Z is the speed, a, b, c, and d are fitting constants, X is the ActA intensity and Y is the 1^{st} moment (a similar plot representing the mixed non-linearities, i.e. where the yellow star would represent the function Z = a+bXY was also considered; data not shown). The greatest increase in R^{2} occurs for a linear fit in ActA intensity and polarity (yellow star) –higher degree polynomials only marginally improve the fit. This suggests that a linear fit in both 1^{st} moment and ActA intensity sufficiently describes the trend in the data. The plane shown in A and B corresponds to this fit. D) Speed vs. ActA intensity for ultrapolar and normal ActA bacteria. Bacteria were classified into ultrapolar and normal categories based on the value of their 1^{st} moment. Bacteria with 1^{st} moments below 0.045 and above 0.075 could be easily distinguished by eye. Bacteria with intermediate 1^{st} moments (between 0.045 and 0.075) were ignored in this analysis. Normal bacteria are shown as gray circles and ultrapolar bacteria as black crosses. Linescan inset contrasts two examples each of ultrapolar (black) and normal (gray) ActA distributions.

We found we could easily distinguish, by eye, ActA distributions with 1^{st} moments below 0.045 from those with 1^{st} moments above 0.075, and thus categorized these bacteria into normal and ultrapolar classes respectively (linescans in ^{st} moments (0.045–0.075) from this analysis in order to make a more stark comparison between bacteria with normal and ultrapolar ActA distributions. The average speed of bacteria was positively correlated to the total ActA intensity in both the normal and ultrapolar populations (p values 1e-3 and 2e-5 respectively;

To explore the mechanisms by which polarity affects bacterial speed, we incorporated into the simulation examples of both normal and ultrapolar ActA distributions from our experimental dataset with comparable total ActA intensities (

A) Representative examples of a normal ActA distribution and an ultrapolar ActA distribution selected from the experimental dataset for use in the simulation. These distributions are scaled such that the total amount of ActA is equal. B) Representative set of simulation runs with normal and ultrapolar ActA distributions for simulations exhibiting a constant drag as in

On the other hand these same filaments form tethers with the bacterial surface, via ActA, and thus also generate pulling forces retarding bacterial motion. Since the population of filaments along the sides of the bacterium can only restrain the bacterium in our model (pushing by filament tips is assumed to be in a direction normal to the bacterial surface), we initially reasoned that ultrapolar bacteria might be made to move faster than normal ActA bacteria if we simply changed the nature of the actin filament-bacterial surface tethers, allowing side-filaments to retard motion more than they enhance motion in a filament number-dependent fashion and thus slowing normal ActA bacteria more than ultrapolars. Extensive parameter searches, varying tether toughness, breakage criteria, and number (by adjusting the parameters governing new and autocatalytic branch creation) failed to find parameter sets matching our

We additionally explored values of other simulation parameters that might affect the polarity-speed dependence, including parameters affecting actin growth (actin nucleation (50-fold range), depolymerization (4-fold range) and branching rate (25-fold range)), strength of the attachments between the comet tail and its surroundings (50-fold range), and the viscosity of the environment (6-fold range). These parameters did affect the overall speed of bacteria, and the nature of the trajectories (e.g. smooth vs. hoppy motion), and some produced simulation runs in which ultrapolar ActA bacteria moved almost as fast as normal bacteria. However, ultrapolar ActA bacteria consistently moved more slowly than normal ActA bacteria (

We introduced a representation of both fluid coupling between the bacterium and the actin network around it and a representation of friction forces between individual filaments and the bacterium (

With this qualitatively realistic frictional drag force by side filaments, our simulation replicated the polarity-speed dependence of two specific experimental ActA distributions. However, our experimental results indicate a dependence of speed on both ActA polarity and on total ActA intensity. To test this experimental result further in our simulation, we constructed an ad-hoc mathematical function, as the sum of two sine waves with one varying parameter (^{st} moment and the total ActA intensity (p = 8e-80, 4e-19, and 2e-80 both variables together and each separately, respectively). This suggests that our simulation, by incorporating a frictional force between actin filaments and the bacterium, captures the experimentally observed speed dependence for a continuum of polarities and intensities.

A) 3-dimensional plot of the average bacterial speed vs. its 1^{st} moment and total ActA linescan intensity for 253 simulated bacteria. A range of simulated ActA polarities was chosen to correspond to the experimentally observed range (^{st} moment values (intensity normalized to the maximum). Actual data points are shown in red. The best-fit plane is shown in gray. B) An alternate view of the dataset shown in A, highlighting the quality of the fit and scatter around the best-fit plane. C) 3-dimensional plot of the R^{2} value obtained when fitting the data shown in A by functions of increasing polynomial degree. As in ^{2} occurs for a linear fit in ActA intensity and polarity (yellow star) –higher degree polynomials only marginally improve the fit.

The nanoscale details that lead to microscale

But it is a fair criticism to point out that the simulation, due to its very complexity, doesn't build intuition. We have thus encapsulated the major mechanisms from our complex model into a vastly simpler one-dimensional continuum model (

A) The bacterium is spatially discretized into 0.1 µm long mesh elements along the long-axis (the x-direction). In each of the mesh elements, two state variables

In the solution of this model the bacterium is spatially discretized into a one-dimensional set of elements, each of size 0.1 µm, spanning the bacterium at the optical resolution of our experimental images of ActA distribution and bacterial motility (

The predictions of this model are coarsely consistent with our results from the agent-based model, but less specific about the nature of the restraining forces. The differential-equation model predicts that ultrapolars will move faster than normals with any restraining mechanism that is filament number dependent, whether linear or cooperative (

Our experimental observation of how the speed of

We explored force-based mechanisms that lead to ultrapolars moving faster
than normals, but there are reasonable biochemical explanations as well. The
most obvious of these is actin monomer depletion at the rear of the bacterium.
If the denser actin growth from normal ActA distributions leads to substantially
greater actin monomer depletion, relative to the ultrapolar bacteria, then
slower speeds for normal bacteria could result simply from decreased
polymerization. Estimates (

Additionally, neither the agent-based or continuum models we present explicitly consider gel effects, and they thus ignore the mechanisms proposed in elastic propulsion models

Our experimental dataset spans a large range of different ActA amounts and polarities (1^{st} moments). We explored ^{st} and potentially higher moments). From our results with the agent-based model, we explain the ActA polarity-speed dependence of

Of the restraining forces shown in

The only filament-number-dependent restraining force considered in other

The slower, normal ActA distribution is, indeed, the “normal” (i.e. typical) distribution for

The qualitative differences between a simple continuum model and the complex agent-based model motivate a discussion of biological modeling methodologies in general. These two models were constructed under different design principles – the partial-differential model was constructed with a single narrow question in mind, while the agent-based model, designed to incorporate a large degree of low-level “realism”, can address many different questions. The failure of our particular continuum model to exhibit qualitative differences in outcome corresponding to qualitative differences in restraining mechanism does not denigrate analytical models in general –a modification of Gerbal et al.

Why do the agent-based and continuum models reach different conclusions? We assert that the average relationships of the continuum model mask a smaller time-scale process critical to the correct emergent behavior. Thus, we find that the parameters chosen as constants in the continuum model are, in reality, functions of ActA distribution, our experimental variable.

To construct our continuum model, which tracks the dynamics of three state variables (barbed-ends, filamentous actin, bacterial speed) through time and in one-dimensional space, we have to assume some average relationships. We assume: an average propulsive force for each filament on the rear hemispherical cap of the bacterium, an average restraining force per unit actin (for fluid coupling) or per barbed-end (constant for ActA-filament tethers or filament number dependent for kinetic friction), an average autocatalytic and de novo barbed-end creation rate, and an average f-actin growth rate. We found that the continuum model is insensitive to the nature of the three restraining mechanisms we have modeled: fluid coupling between filaments and the bacterium, ActA-filament tethers, and kinetic friction. We interpret this insensitivity to mean that behaviors critical to the correct emergent ActA distribution response occur on a time-scale for which the assumed average relationships of the continuum model are not valid. Consider a filament born 1 µm from the rear tip of the 1.7 µm long bacterium, a bit forward of the centroid. If the bacterium moves at 100 nm/s (a typical speed) then it might interact with the bacterium for as long as 10 seconds before becoming part of the trailing actin tail. During that interval this filament may undergo many polymerization, collision, ActA tethering, and branching events. In other words, filaments stochastically sample a large number of possible states during their lifetime, and this is especially so for the filaments that interact the most with the bacterium. For instance, the ability of a filament (or branch structure, if it is part of one) to impart large forces on the bacterium generally increases until the bacterium has long passed (see actin distribution

Based on our analysis with the agent-based model, we postulate that it might be possible to rectify the discord between models by introducing a third independent variable, filament age, into the PDE description. This modified continuum model would then track barbed-ends and filamentous actin in time, 1D-space, and age, thus allowing a filament age-dependent formulation of propelling and restraining forces, and perhaps of branching and elongation rates. We could use the agent-based model to establish the average age-dependence (as in

As a last word, we believe that the historical narrative of our work can serve as a parable for others. We described extensive parameter searches with the agent-based model, when it lacked a cooperative restraining mechanism, that all failed to match our experimentally observed polarity-speed relation. This iterative process was very time-consuming, as simulation of a single bacterium (representing one parameter set) requires several days of computer time. Ultimately, we found that this model produced robustly wrong behaviors, and this pointed to its lacking a critical mechanism. With the inclusion of kinetic friction, the model robustly predicts that ultrapolar bacteria move faster than normals. We might have saved ourselves many months searching through parameter space to find values that might make a qualitatively wrong model yield predictions that agreed with our

JAT-396 bacteria (constitutively expressing ActA-RFP) _{600} of approximately 9.0 then used continuously in multiple independent motility assays for 5 hours (maintained on ice;

Microscopy was performed on an Olympus IX70 equipped with an x-y-z automated stage (Applied Precision, Issaquah, WA) and a cooled CCD camera (CoolSNAP HQ; Photometrics, Tucson, AZ). Timelapse images were taken using a 60×, 1.4NA PlanApo lens and collected every 5 s for 2 minutes using Softworx software (Applied Precision, Issaquah, WA).

Bacteria were tracked at their centroid using the semi-automated threshold dependent “track objects” function in Metamorph (Universal Imaging, Downington, PA). Tracked bacteria were imaged between 1.5 and 2.5 hours after mixing with extract to ensure this analysis included only bacteria moving at steady-state. Bacteria displaying bipolar ActA distributions

To generate a continuous measure of ActA polarity (from least to most polarized) the ActA linescan was used to calculate the 1^{st}, 2^{nd}, etc. moments by calculating:_{n}_{t}_{j}_{j}^{st} moment was the most significant of the first 5 moments (0^{th} to 4^{th} moment) for describing the experimental polarity-speed dependence. The 2^{nd} moment is useful to differentiate between unipolar and bipolar ActA bacteria. However, since all bacteria with bipolar distributions were removed from the dataset, this measure was not critical to our analysis. The total amount of ActA on a bacterium was determined as the sum of the intensities at each camera pixel in the linescan (also the same as the 0^{th} moment).

No correlation between speed and bacterial length was found (data not shown), but to ensure no effect from extremely long or short bacteria, the final analyzed datasets were limited to bacteria between 10 and 25 pixels (1.06 and 2.26 µm) in length (>90% of tracked population). The final filtered dataset shown in

3-dimensional plots showing average speed per bacterium as a function of 1^{st} moment (polarity) and total ActA were created in Mathematica (Wolfram Research, Champaign, IL). To determine which polynomial functions best fit the data, the R^{2} values from different polynomial-order fits were plotted; the improvement in fit beyond linear was minimal. The statistical significance of the linear fit was calculated with the “regression” analysis tool in Microsoft Excel. The values within each category (i.e. average speed etc.) were randomized and recombined. A regression analysis of these randomized data confirmed no significance in the linear fits (p≫0.1).

We made two different computer models. One is a nanoscale-detail-oriented stochastic model of the biochemical and force-based interactions between a rigid

The stochastic model is computationally intensive, with typical run-times (for 6 minutes of simulated time) of three to five days, depending on parameters, on modern Linux servers running Java 1.6. These simulations additionally require up to 2 GB of main memory per server, again depending on the particular parameter set.

The method by which we simulate the interactions of filaments/branches with each other, and with other cellular entities not explicitly simulated, has changed greatly. In Alberts and Odell

The simulation detects collisions between actin filaments and the bacterium at each time-step, and these collisions are resolved through application of equal and opposite forces. . Let

The bacterium was given a constant shape-based drag in Alberts and Odell

The probability that an actin filament encountering the bacterial surface will interact with an ActA protein is now dependent on the concentration of ActA proteins at that location on the bacterium (the concentration depends only on the arc length measured from the posterior pole – not on the azimuthal angle). We previously assumed that all filaments colliding with the bacterium would immediately interact with an ActA protein – a good approximation in regions of high ActA density, which is distribution dependent – but no longer relevant when ActA concentration tapers off gradually towards the front end of the bacterium.

Random number generation is too important to be left to chance. Instead of using the built-in Java pseudo-random number generator (period of 2^{48}−1), we now employ the scientific standard Mersienne Twister method (period of 2^{19937}−1). This is implemented from the free and open-source Java package provided by

Description of the agent-based model.

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Derivation and solution of the continuum model.

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Mathematica notebook solution of discretized continuum model.

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Snapshots of

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Of the restraining forces considered in the agent-based model, only the frictional force between filaments and the bacterial surface is cooperative. A) The top bacterium has three transient ActA-filament tethers along the side of the bacterium, while the bottom one has six. A two-fold increase in tethers will, on average, restrain the bacterium with only twice the force, i.e. restraint by ActA-filament tethers is approximately linear in the number of interacting filaments. B) At low Reynolds number the induced fluid velocity field can extend a large distance from an object's surface. The top figure shows a hypothetical fluid flow profile -vf will equal the bacterial velocity v at the surface and decrease parabolically from there. Any nearby filaments will interact with this fluid and induce a drag on the bacterium in a complicated way, dependent on their own position, orientation and velocity. We approximate this interaction by simply counting the number of filaments within a ‘shell’ about the bacterium and increase the drag coefficients of the bacterium linearly with that number. The induced drag will increase linearly, at most, as demonstrated in the bottom figure. Filaments i and ii will independently interact with the bacterium's fluid velocity field; identically positioned, oriented, and moving filaments will contribute equally to the drag on the bacterium, i.e. the drag will increase approximately linearly with the number of filaments. A filament pair as in iii, however, will effectively appear as a single filament in this fluid coupling -at high filament densities we might expect the drag to increase even less than linearly with the number of filaments. C) By Amonton's law, kinetic friction is proportional to the normal contact force, as shown in the top figure. The bottom figure illustrates how many filaments can cooperate to increase the average normal force, i.e. N∝β where n is the number of contributing filaments and β is an unknown exponent of dependence. The total friction force is just a summation of the contribution from each of the n filaments, and thus is cooperative in n by the factor β, i.e. F_{Total}∝n^{1+β}. In

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Examination and determination of continuum model parameters through analysis of average agent-based model relationships. A) Averaging of data for three

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We devised a simple function, the sum of two sine waves (i. e. the first two terms in a Fourier series approximation), to distribute ActA on our

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Comparison of

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Concentrations used in the agent-based model (reproduced from Alberts and Odell 2004). These values are not for any specific cell type, but are typical biological concentrations and similar to those used for

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Rates used in the agent-based model (reproduced from Alberts and Odell 2004). A hydrolysis rate is given for a vectorial ATP hydrolysis model; experimental evidence currently supports the random hydrolysis model but we have, for simplicity, implemented a vectorial scheme for this analysis. That is, we assume that there is a distinct border within each filament between the ATP actin, ADP-Pi actin, and ADP actin regions; only monomers adjacent to these borders can transition from ATP actin to ADP-Pi actin or from ADP-Pi actin to ADP actin. We can readily switch to a random hydrolysis model in future studies. The values in angle brackets, for the interactions between ActA, Arp2/3, and actin monomers, are calculated considering the diffusive flux onto the bacterium's surface (see Alberts and Odell, 2004:Dataset S2). These values are thus dependent upon ActA density, the concentrations of Arp2/3 and actin, and a heuristic adjustment of these rates to balance new filament nucleation and side-branching in order to achieve realistic tail morphologies. The on-rates in brackets listed here apply to the concentrations in

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Example trajectories of normal (on the left) and ultrapolar (on the right)

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We are grateful to the Theriot lab for providing labeled actin and Peter Jackson for providing and housing