Myxococcus xanthus cells self-organize into periodic bands of traveling waves, termed ripples, during multicellular fruiting body development and predation on other bacteria. To investigate the mechanistic basis of rippling behavior and its physiological role during predation by this Gram-negative soil bacterium, we have used an approach that combines mathematical modeling with experimental observations. Specifically, we developed an agent-based model (ABM) to simulate rippling behavior that employs a new signaling mechanism to trigger cellular reversals. The ABM has demonstrated that three ingredients are sufficient to generate rippling behavior: (i) side-to-side signaling between two cells that causes one of the cells to reverse, (ii) a minimal refractory time period after each reversal during which cells cannot reverse again, and (iii) physical interactions that cause the cells to locally align. To explain why rippling behavior appears as a consequence of the presence of prey, we postulate that prey-associated macromolecules indirectly induce ripples by stimulating side-to-side contact-mediated signaling. In parallel to the simulations, M. xanthus predatory rippling behavior was experimentally observed and analyzed using time-lapse microscopy. A formalized relationship between the wavelength, reversal time, and cell velocity has been predicted by the simulations and confirmed by the experimental data. Furthermore, the results suggest that the physiological role of rippling behavior during M. xanthus predation is to increase the rate of spreading over prey cells due to increased side-to-side contact-mediated signaling and to allow predatory cells to remain on the prey longer as a result of more periodic cell motility.
Spatial self-organization of developing cells, which results the formation of complex dynamic structures, remains one of the most intriguing phenomena in modern biology [1]–[4]. Analogous developmental behaviors are observed as bacterial cells form biofilms, which are populations of surface-associated cells enclosed in a self-produced matrix [5], [6]. The dynamic self-organization in biofilms formed by the soil bacterium Myxococcus xanthus is dependent on the ability of the cells to move on solid surfaces [7], [8], while sensing, integrating and responding to a variety of intercellular and environmental cues [9]–[12]. M. xanthus is the preeminent model system for bacterial social development. At high density and under nutrient stress M. xanthus cells execute a complex multicellular developmental program by aggregating into multicellular mounds, termed fruiting bodies, and differentiating into dormant, environmentally resistant myxospores [11]. In addition, these bacteria exhibit complex behaviors when they cooperatively prey on other microorganisms by collectively spreading over the prey cells, producing antibiotics and lytic compounds that kill and decompose their prey [13], [14]. One of the most intriguing forms of collective dynamics exhibited by M. xanthus is their ability to self-organize into ripples – travelling bands of high-density wave crests [15]–[18]. Although the M. xanthus counter-traveling waves appear to pass through each another, they actually reflect off of one another and are termed “accordion waves” [16], [18]–[21]. These waves are distinct from the waves originating from Turing instability diffusion-reaction patterns, such as those in chemical systems or observed during development of the other well-studied model social microorganism, the amoeba Dictyostelium discoideum [22]–[24]. The initial studies of the mechanisms underlying M. xanthus rippling motility focused on this behavior during starvation-induced multicellular fruiting body development [16]–[20], [25]–[27]. The application of mathematical modeling to developmental rippling revealed that the wave properties are consistent with contact-induced reversal signaling [18]–[21], [28]. This signaling was hypothesized to originate from ‘head-to-head’ collisions of cells moving in opposite directions and to result in an exchange of C-signal that accelerates the reversal clock [16], [19], [20]. C-signal is an extracellular protein that controls aggregation and sporulation via contact-dependent pole-to-pole transmission [12]. Developmental aggregation and motility coordination are induced through the C-signal-dependent stimulation of the frz chemotaxis-like system, which includes an unconventional soluble cytoplasmic chemoreceptor homologue FrzCD [12], [29], [30]. An opportunity to reevaluate and replace the pole-to-pole collision-mediated model was prompted by a new report of FrzCD protein clusters that appear to transiently align and stimulate reversals in cells making side-to-side contact [8] and by the recent discovery that more robust rippling occurs during predation [13], [15]. In this paper we have investigated predatory rippling behavior with a combination of mathematical modeling and experimentation. We have constructed a mathematical model that faithfully reproduces the travelling wave behavior by adapting the recently proposed reversal-inducing side-to-side contact-mediated signaling model [8] and incorporating the properties of the patterns resulting from these interactions.
To model collective cell behavior we needed a modeling formalism that would allow us to connect the motility of individual cells, intercellular interactions, and the resulting population patterns. To this end, we employed an agent-based model (ABM) approach [19], [31]–[33]. Individual cells are represented as agents that move and interact according to the rules and equations that correspond to experimental observations. Unlike continuous, cell-density-based approaches, the ABM approach allows cell variability and modular implementation of interactions to be easily incorporated. The details and equations describing our ABM are summarized in the Materials and Methods Section. Here we qualitatively describe the main model ingredients that result in predatory rippling behavior. Each agent is simulated as a self-propelled rod on a 2-D surface. The agents move continuously along their long axis and periodically reverse by switching the polarity of their two ends simultaneously. As in the previous models [19]–[21], we expected the ripples to emerge as a result of intercellular signaling, which leads to synchronized cellular reversals among the cell population. The side-to-side contact-induced signaling mechanism used here is based on the recent observations by Mauriello et al. [8], which demonstrated that when cells make transient side-to-side contact, their FrzCD clusters align causing one or both of the cells to reverse. The reversals stimulated by this intercellular signaling would be somewhat similar to the reversals induced by pole-to-pole collisions that were hypothesized to occur due to C-signal exchange during M. xanthus development [18]–[21]. Based on this and other experimental observations, our model incorporates four rules to guide the agents' interactions (see below). These rules are converted to mathematical equations that describe rippling motility (see the Materials and Methods Section). The first three rules are sufficient for the model to produce rippling behavior (Figure 1, top row; Video S1). Starting from a uniform aligned population of agents (0 hrs), the model results in their self-organization into periodic traveling bands (ripples) within about 3 hrs. As in previous models [18]–[21], [25]–[27], the ripples emerge from the synchronized reversals. However, this model, which is based on a side-to-side contact-mediated signaling mechanism, appears to be more robust than the previous models that utilized pole-to-pole collision-mediated signaling (Figure S2). Rule (iv) is not necessary for rippling, but it allows the model to reflect the cell reversal behavior exhibited at low densities when cell contacts are rare, and it does not significantly change the high-density motility patterns studies here. The mean value of the native reversal period is chosen to be about 8 min (Figure S3 A) to achieve agreement with experimental observations by us here and others [11]. Within the framework of the proposed model, each rule (i) – (iii) is necessary to generate rippling behavior. Specifically, rule (i) is necessary because eleminating intercellular reversal-generated signaling abolishes rippling motility (data not shown) and eliminating the assumptions that signaling occurs only between counter-moving agents has the same effect (Figure S4 A and B). It is noteworthy that rippling motility is robust to the minimal overlap between agents that is required for them to engage in side-to-side signaling (Figure S5). Hereafter, an arbitrary value of 50% as a minimal overlap threshold is assumed in all simulations. Moreover, Figure S4 C vs D show that rippling motility occurs regardless of whether each signaling event is bidirectional (when cell #1 signals to cell #2, cell #2 also signals to cell #1) or unidirectional (cell #1 signaling to cell #2 and cell #2 signaling to cell #1 are independent events). In our simulations we use unidirectional signaling assumptions for the reasons explained below. The refractory period (rule ii) is also required for ripples, as reducing it to a very short duration leads to the dissapearance of the waves (Figure S4 E and F). In our simulations, the refractory period is a stochastic quantity with a mean value of 2. 7 min and standard deviation of 0. 7 min (Figure S3 B). Side-to-side signaling and rippling motility can only occur in a locally aligned cell population, and thus, physical interaction aligning cells, rule (iii), is necesary to maintain the cells' long axes approximately parallel. Since in our simulations the rules (i) – (iii) induce rippling motility, we addressed the question of which rule is modulated to ensure that rippling motility is observed only when prey cells or the macromolecules associated with their lysis are present. The initiation and maintance of ripples seems to depend on the probability of reversal-inducing signaling events (Figure S6), which must exceed a threshhold value of ∼5–10%. If the probability is below 5%, then the ripples will not form and the agents will remain uniformly distributed on the 2-D surface. When the signaling probability exceeds the threshold value, the uniform distribution becomes unstable and the agents self-organize into ripples. Therefore, we hypothesize that the presence of prey-associated macromolecules indirectly stimulates rippling by increasing the probability that side-to-side contact generates successive signaling events (reversals). Although the biochemical mechanism of this induction is unknown, various macromolecular substrates, such as peptidoglycan, bovine serum albumin, and salmon testes chromosomal DNA, have been shown to induce rippling motility [13], [15]. Thus, we predict that the presence of these substrates should increase the probability of reversal-inducing signaling. Although our experimental arrangement does not allow direct testing of this prediction, we can quantitatively compare the emergent properities of the rippling patterns in the model and in the experiments. It should be noted that the experiments demonstrating side-to-side signaling were preformed in the absence of prey cells or prey-associated macromolecules [8]. However, the results reported by Mauriello et al. [8] are consistent with a low probability of side-to-side signaling and the assumption that signaling is unidirectional. This is because in their observations only one of the cells engaged in side-to-side contact signaling reverses its gliding direction [8] (see also Figure S7). If the probability of signaling is low, it is unlikely that two signaling events will occur simultaneously. Furthermore, once one of the cell reverses, both cells will then be moving in the same direction and as a result, they will no longer be capable of signaling one another. To test the modeling predictions experimentally, we observed cell motility on a solid nutrient agar surface in the presence of prey cells. The ripples were observed with fluorescence and differential interference contrast (DIC) time-lapse microscopy, allowing us to track cell density changes and the motility of a small percentage (0. 5%) of GFP expressing cells in a wild-type population (see Materials and Methods section and Video S2). These images allowed us to calculate the global properties of the ripples: wavelength (distance from one wave crest to the next) and wave-crest width, and at the same time to measure the behavioral properties of individual cells: coordinates, velocity, reversal period, and the time/position of cellular reversals. These data provided crucial input into the model and allowed us to test our modeling predictions. It is clear that the experimental ripple patterns appear very similar to those produced in the simulation (Figure 1). To compare the timing of wave initiation between the mathematical model and the experimental results, the time point when M. xanthus cells fully cover the prey in the field of view was chosen as the starting time (0 hrs in Figure 1; Video S3). Snapshot images at 0,1, 3 and 5 hrs were selected to show the process of ripple formation in both systems. The experimental process of wave initiation appears to follow the same dynamics as the simulations. Initially, the cells homogeneously cover the field of view and the cells align as they cover the prey. During the first 3 hrs the reversals of individual cells become synchronous and result in the formation of ripples. By 5 hrs the ripples are pronounced and are easily discernible. These results indicate that the ABM is capable of qualitatively reproducing the dynamics of rippling motility observed under our experimental conditions. Interestingly, waves generated with the ABM appear somewhat more pronounced than experimentally observed ripples, which have a smaller cell density gradient from crest to trough. This observation suggests that not all the cells in the biofilm participate in rippling behavior. To compare the rippling patterns produced by the ABM to those of the experiments, we quantitatively characterized the ripples and related their patterns to the behavior of individual cells. Previous models of rippling motility [20], [21] proposed a simple equation, which relates wavelength (λ), individual agent speed (v), and agent reversal period (τ): (1) This equation indicates that cells in two colliding crests (relative speed) reverse their directions every time the crests are superimposed. This prediction was confirmed by both the ABM and experimental results of developing cells [19]. However, our analysis of the measurements by Berleman et al. [13], showed that wavelengths of their predatory ripples were ∼50% larger than those predicted by Eq. (1). Using their experimental values of v = 3 µm/min and τ = 8 minutes, the calculated λ should be 48 µm, however their observed λ was ∼70 µm. To determine if the wavelength relationship, Eq. (1), works for our new ABM of rippling motility, two sets of simulations were conducted. First, the agent speed was fixed at 6 µm/min, while the spontaneous reversal period was varied between 5 min and 30 min (corresponding to the variation between 3 and 12 min of an actual average reversal period, which is smaller due to early reversals triggered by side-to-side contact signaling; Figure 2A, solid line). Second, the spontaneous reversal period was fixed at a value corresponding to an average reversal period of approximately 6. 6 min and the cell speed was varied between 2 µm/min and 12 µm/min (Figure 2B, solid line). These fixed values correspond to the experimental cell motility parameters. As shown in Figure 2A and 2B, the wavelength (λ) scales linearly with agent speed (v) and average reversal period (τ). However, when no-intercept linear regression was used, regression coefficients of 15. 2 µm/min for Figure 2A and 16. 1 min for Figure 2B were obtained. Both values are slightly larger than the predicted coefficients of 2v (12 µm/min) and 2τ (13. 2 min), respectively. When we tracked the reversal points of individual agents, we observed that the agent reversals were initiated as soon as the leading edge of each crest came into contact (Test S1; Figure S6). This indicates that as the agents at the front of each crest reverse, they signal to the other cells in their crests, leading to a “chain-reaction” of signaling and reversals. Given the wave crest width Δ, the cells in each crest only move an average distance of λ−2Δ before reversing again, which results in the average reversal period τ = (λ−2Δ) /2v. Thus, we modified our wavelength equation to be: (2) To test the modified expression in our simulations, we automatically computed the average wave-crest width Δ from the simulation results (see Text S1) and used it to compute the wavelength with Eq. (2). The results demonstrate good agreement between the simulated and predicted wavelengths (Figure 2 A and B, solid vs. dashed line). To test the Eq. (2) prediction experimentally for predatory rippling motility, we tracked 37 GFP-labeled individual cells within ripples for about 2 hr (or until the cells left the field of view). Continuous 1-D wavelet transform of the microscopy images (see Text S1) was used to compute the wavelength and wave-crest width by fitting a Gaussian function to the wave crest calculations. The distributions of average speed and reversal period are shown in Figures 2 C and D; and the ABM-predicted wavelengths are in agreement with the experimentally observed wavelength (denoted by the stars in Figures 2A and B). The prediction of Eq. (2) is also in good agreement with the data from Berleman et al. [13]. Using their experimentally derived values of v = 3 µm/min, τ = 8 min, Δ∼10–15 µm, the wavelength, λ, is calculated at ∼70–80 µm, which matches their published values. Rippling motility simulated with these parameters is shown in Video S4. To further test modeling predictions, we attempted to alter rippling wavelengths with changes in agar density and initial prey-cell concentration. We have selected two plates displaying reduced wavelength for detailed analysis and cell tracking. The results show that predictions of Eq. (2) also hold for these data: a reduced wavelength resulted from a reduction in the cell speed in both movies (∼3 µm/min) and a reduction of the reversal frequency (∼4. 5 min) in one of the movies. Table S2 summarizes our experimental tests of Eq. (2). According to our ABM assumptions and predictions, most of the rippling cells should travel with the wave crest and reverse, essentially as a group, when the leading edges of the two opposing wave crests collide. To test this prediction, we observed reversals of individual cells in the context of wave-crest movement by plotting cell trajectories on the space-time florescence intensity of ripples (Figure 2E). The space-time image illustrates the timing and location of the wave crests (see the dark gray ridges in Figure 2E). By examining trajectories of GFP-labeled cells (colored lines), we observe that the tracked cells travel with the high-density crests and reverse when and where two crests collide. Statistical analysis of the position and timing of cell reversals (dots) show that 75. 0% (±2. 6%) of all tracked reversals occur during wave crests collisions, matching ABM prediction (Figure 2F). Interestingly, some cells move through a counter-propagating wave crest without reversing and subsequently reverse with the next crest. This “wave-hopping” pattern explains the small peak at ∼12 min (twice the average reversal time) in Figure 2D and the more pronounced second peak in the distribution of the average distance travelled per reversal (Figure S8 E).
Myxococcus xanthus cells self-organize into periodic bands of traveling waves, termed ripples, during multicellular fruiting body development and predation on other bacteria. Here we have used an approach that combines mathematical modeling with experimental observations to investigate the mechanistic basis of rippling behavior and its physiological role during predation. The resulting new mathematical model, which is more robust than previous models, is based on the recent observation of Mauriello et al. [8], that when counter-moving cells come into side-to-side contact, clusters of chemotaxis-like FrzCD receptors within the cells transiently align and thereafter one of the cells reverses. Our model shows that this side-to-side contact-mediated signaling is sufficient to induce rippling self-organization in a locally aligned cell population, assuming that there is a minimal refractory period during which the cells can not reverse again regardless of their signaling state. The existence of the refractory period has also been assumed in our previous model [19], [20] and this assumption is plausible as reversals are anticipated to require a significant reorganization of the cell-motility machinery [7], [34]. The existence of a refractory period also naturally follows from the dynamic properties of a negative-feedback oscillator (Frzilator), which was previously hypothesized to regulate cell reversals [35]. Altogether our modeling results suggest that the self-organization of cells into ripples during predation can be explained by the increased efficiency or higher probability of side-to-side signaling induced by the presence of prey macromolecules. This prediction is not tested directly in our experiments, but the emergent properties of simulated waves quantitatively match those in our predation experimental approach. Our model builds on the detailed characterization of M. xanthus predatory rippling behavior by Berleman et al. [15], which showed that rippling motility occurs during predation on the variety of microorganisms and is induced by the presence of macromolecular substances. However, our model differs from the concept promoted by Berleman et al. [15] that ripples originate solely as an interaction of individual cells with macromolecules without any self-organizing intercellular interactions. In contrast, we propose that ripples result from the self-organization of cells into traveling wave patterns, which result from the intercellular signaling that is stimulated or facilitated by the presence of macromolecules. Indeed, in our experimental approach the macromolecules are likely to be distributed uniformly and their concentration is expected to vary very little during the typical wave period (∼10 min). Moreover, even if macromolecules induce the periodicity of M. xanthus cell motility as suggested by Berleman et al. [15], this would not be sufficient to induce ripples because their formation requires temporal and spatial synchronization of cellular behavior that is unattainable without cell-to-cell signaling. Based on the previous modeling of M. xanthus developmental rippling behavior, one is prompted to ask: does the same mechanism control predatory and developmental rippling motility? Certainly this new model is similar to the previous mathematical models of developmental rippling, as they each consider that self-organization occurs when counter-moving cells interact to induce reversals [19], [20]. As expected, the new model is in good general agreement with the experimental patterns that were previously observed for developmental rippling motility [16], [18]. However, our tests reveal that this new side-to-side contact-mediated signaling model is much more robust, in that it can withstand realistic levels of variability in cell speed and reversal times (Figure S5 left panels). Specifically, when the level of randomness in cell motility consistent with the single-cell tracking experiments (fluctuations of velocity and reversal period over 30% of the mean value) is used in the pole-to-pole collision-mediated signaling model, the cells do not form ripples (Figure S5, bottom right panel). It is noteworthy that pole-to-pole signaling can result in more robust waves, if the cells are able to accumulate signals from multiple collisions and if signaling during the refractory period leads to a reduced reversal rate as the Frzilator model predicts [19]. However, for the new side-to-side contact-mediated signaling model, realistic rippling can be observed assuming only that single successful signaling events result in cellular reversals. Furthermore, the experiments of Berleman et al. [13], [15] provided evidence indicating that developmental rippling occurs as a side effect of cell lysis during aggregation, which suggests that rippling motility is likely to be a response to the released macromolecules. Thus, we propose that our new side-to-side contact-mediated signaling model of rippling describes both predatory and developmental rippling. The new model therefore explains ripples without requiring the pole-to-pole exchange of the starvation-induced C-signal. This may be biologically justified for a number of reasons. First, to date no C-signaling receptor has been identified. Second, localization of CsgA to the cell poles has not been demonstrated directly. Third, the robustness of pole-to-pole signaling-mediated mechanism is questionable as the probability of this type of collision is low. However, as C-signaling mutants fail to display rippling motility [16], it would be interesting to investigate in future studies how C-signaling affects the FrzCD cluster alignment and whether C-signaling plays a role in predatory rippling. The main hypothesis of this new computational model is that rippling behavior is initiated by side-to-side contact-mediated signaling in the presence of prey cells. This hypothesis cannot be directly tested at this time, since we do not have a complete understanding of the specific biochemical mechanisms involved. However, we can rigorously test the model by comparing the model predictions with experimental data collected by us and others. An important prediction of the model is that M. xanthus cells will reverse more frequently when prey is present. This agrees with our experimental observations (Table S3) and that of Berleman et al. [13]. Moreover, the resulting self-organization of cells into ripples provides various ways to quantitatively and qualitatively compare in silico-generated rippling motility with experimental observations. A second prediction is that if the presence of prey stimulates this side-to-side contact-mediated signaling, then the rippling would only be observed in the regions where signaling is sufficiently probable, i. e. only in the regions covering prey. This is in good agreement with our observations (Video S6) and those of Berleman et al. [13], [15]. Indeed, our simulations show that the signaling probability can serve as a bifurcation parameter that induces a transition between the homogeneous cell distribution and the formation of ripples (Figure S6). A third prediction is based on the timescale of rippling self-organization, which can be defined as the time it takes to generate ripples that consist of well-focused wave patterns, starting from an initially homogeneous cell population. Our model predicts that time to be of the order of 3 hrs, which is remarkably consistent with our experimental observations (Figure 1). The qualitative comparison of the time-lapse dynamics (Videos S1 vs. S2, S3 and S6) is also in good agreement. Interestingly, the time-scale of rippling origination in the experiments of Berleman et al. [13] is significantly longer (∼12 hrs). Although it is hard to pinpoint the source of this discrepancy, our model indicates that the cell density and the amount of noise in cell orientation can significantly affect the wave synchronization time. A fourth prediction of the model is based on measuring the rippling wavelengths and correlating them to the parameters of individual cell motility. Our new model predicts a slightly modified relationship (Eq. (2) between wavelength, wave-crest width, individual cell speed, and reversal time as compared to the previously established [19], [20]. This new relationship is confirmed by our simulations and is in excellent agreement with the experimental measurements of wavelength (Figure 2 A and B, Table S2). The wavelength prediction is also compatible with previously reported measurements [13] and with the observations of Sliusarenko et al. [15], which show that cells moving in opposite directions tend to inter-penetrate one cell length before a reversal is triggered. Figure S7 shows the sequence of events that occur during two-crest collisions. This cartoon model indicates that once the cells at the front of each crest reverse, they signal to the cells following them, which results in a chain-reaction of signaling and reversal events. This cartoon also illustrates the importance of the refractory period, because once the cells at the front of the crest reverse, it is essential for them to keep signaling to other cells to reverse without reversing themselves. A fifth prediction of the new model is based on tracking the cell reversals and locations of wave-crest collisions in time and space. Just as the model predicts (Figure 2F), the experimental results (Figure 2E) indicate that most reversals occur when and where two wave crests collide. Our previous model of developmental rippling motility suggested [28] that periodic travelling waves can ensure a more regular distribution of fruiting-body aggregates at the colony edge, as seen in the submerged culture system of Welch et al. [18]. However, the physiological implications of this observation are unclear as the developmental aggregate distribution can be well organized even without rippling [36]. Furthermore, if rippling motility is predominantly a response to predation, what is its role in these situations? Berleman et al. [37] proposed two hypotheses. The first, termed the “grinder model” speculates that the movement of the waves of M. xanthus cells during rippling motility causes a physical disruption of the prey colony. The second, termed the “population control model” suggests that waves maximize the prey-predator contact area and push excess predator cells to the edges of the rippling area. Neither of these hypotheses is likely to be correct, based on the biophysics of this environment in which the very-low Reynolds number hydrodynamics will not allow temporary periodic perturbations to affect mixing or transport [38]. Nevertheless, our mathematical model suggests several alternatives for physiological benefits of rippling to predatory cells. These predictions are consistent with the experimental observations reported here and previously. First, the model is in agreement with the observations of Berleman et al. [13] that during the expansion over prey, the presence of side-to-side contact-mediated signaling significantly facilitates the rate of M. xanthus cell spreading (Figure 3). As a result these cells cover their prey faster. This has obvious physiological benefits in the competitive soil environment. The observation is also consistent with our own experiments. Notably, this result does not require ripples per se, but only reversal-inducing signaling. However, our model indicates that side-to-side contact-mediated signaling is key for rippling self-organization and the other model ingredients can easily be justified by what is known about the biophysics of M. xanthus motility [7]. Furthermore, the increase in spreading also takes advantage of the cell-density gradient of M. xanthus cells that is generated by spreading at the leading edge. It is important to note that the rippling behavior does not require a density gradient of prey cells, as the alternative chemotaxis-based explanation would predict. Second, the model predicts that cells that ripple in the absence of a cell-density gradient (i. e. when they are behind the leading edge of the swarm or once the prey is fully covered), would engage in less noisy and more periodic motion and as a result will have less of a random drift (Figure 4A). This effect would help the predatory cells to remain in the prey area for a longer time and to reduce random movement away from the prey. This prediction was confirmed by the cell-tracking assays (Figure 4B). Notably, this effect requires ripple formation, as the collective interaction of cells in the ripples leads to their synchronization. This effect is analogous to the well-known mathematical phenomena in which a collection of coupled noisy oscillators is less noisy than each oscillator on its own [39]. Third, it is likely that the formation of the ripples increases the cell alignment due to an increase in steric interactions in the denser crests. This prediction agrees with our observations and those of Berleman et al. [13]. However, it is worth noting that the causal relationship between rippling and alignment is not obvious, as ripples also require cell alignment. Therefore, it is likely that there is a positive self-reinforcing feedback loop between the formation of ripples and cell alignment: as cells align, ripples become more pronounced and their crests become more dense leading to further cell alignment. Although the physiological benefit of better alignment is not obvious, it may further enhance the rate of spreading, which contributes to the effects discussed above. Uncovering the mechanistic basis of spatial and temporal multicellular self-organization is a daunting task and a full understanding has not been achieved for even the best-studied model systems. Here, agent-based modeling, time-lapse fluorescence microscopy, and image quantification have been used synergistically to provide new insights into the mechanisms of M. xanthus self-organization into ripples. Our modeling demonstrates that a simple set of ingredients based on experimental observations is sufficient to produce rippling patterns. The subsequent experiments have tested a number of predictions based on the model and have allowed us to refine the model to achieve quantitative agreement with the experimental data. This type of combined approach is essential to further our understanding of self-organization in more complex systems such as development of multicellular organisms.
ABM are widely used to computationally simulate emerging patterns formed by multiple agents. The ABM of M. xanthus rippling presented here is kept simple yet sufficiently flexible to accurately describe the experimentally observed behaviors of M. xanthus cells. The model is an extension of the earlier ABM [19] of M. xanthus self-organization that now incorporates a side-to-side contact-mediated signaling mechanism. In this ABM, each agent represents a cell – a self-propelled rod on a 2-D surface with length of L, width of w, with a center position of (x (t), y (t) ), and orientation 0≤θ (t) ≤2π. Specifically, the agent length and width are constant throughout all simulations, whereas the center position and direction of movement are changed at each time step as the cells move and align. For each simulation, the time is updated by constant increments δt. The simulations are conducted on a fixed 2-D area in which all simulated moving agents are bounded. For most simulations periodic boundary conditions are imposed.
Myxococcus xanthus cells collectively move on solid surfaces and reorganize their colonies in response to environmental cues. Under some conditions, cells exhibit an intriguing form of collective motility by self-organizing into bands of travelling alternating-density waves termed ripples. These waves are distinct from the waves originating from Turing instability in diffusion-reaction systems, as these counter-traveling waves do not annihilate but appear to pass through each other. Here we developed a new mathematical model of rippling behavior based on a recently observed contact signaling mechanism – cells that make side-to-side contacts can signal one another to reverse. We hypothesize that this signaling is enhanced by the presence of prey-associated macromolecules and compare modeling predictions with experimentally observed waves generated on E. coli prey cells. The model predicts a modified relationship between the wavelength and individual predatory cell motility parameters and provides a physiological role for rippling during predation. We show that ripples allow predatory cells to increase the rate of their spreading to quickly envelope the prey, and subsequently to decrease their random drift to remain in the prey region for longer. These and other predictions are confirmed by the experimental observations.
complex systems
cell motility
mathematics
applied mathematics
biology
computational biology
microbiology
biophysics
microbial growth and development
The Mechanistic Basis of Myxococcus xanthus Rippling Behavior and Its Physiological Role during Predation