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  • We follow pairwise evolutionary competitions between strains that differ both in their ability to produce extracellular polymeric substances (EPS) and the extent to which this behavior is under quorum-sensing control. For our simulation study, we focus on three strains with the following behavior: (1) no polymer secretion and no quorum sensing (EPS−), (2) constitutive polymer secretion and no quorum sensing (EPS+), and (3) polymer secretion under negative quorum-sensing control such that EPS secretion stops at high cell density (QS+). A fourth strain for which polymer secretion is under positive quorum-sensing control is omitted from the main analysis because its behavior was found to be qualitatively identical to that of the EPS+ strain (see Discussion, Text S1, and Figure S1). Our simulations examine quorum-sensing control of a single trait (EPS) in response to the concentration of a single autoinducer. In reality, bacteria often use more than one autoinducer to regulate multiple traits, and indeed, several quorum-sensing circuits may be linked via parallel or serial signaling pathways within the cell [15,16,41]. There is a rich scope, therefore, for additional study of many potential complexities of quorum-sensing–regulated social behaviors, which we leave open here. Biofilm development involves a number of interacting physical and biological processes, including growth, neighbor-pushing, solute diffusion, and other cell–cell and cell–solute interactions, all of which occur largely at the spatial scale of single cells. We use individual-based modeling methods to explore the emergent characteristics of these processes at the level of whole biofilms [42]. Simulated cells behave independently according to user-defined kinetic rate expressions designed to represent the essential features of bacterial metabolism. Our simulations begin with one or more colonizing cells, which are attached to a uniformly flat surface and grow in a two-dimensional (2-D) space with horizontal periodic boundary conditions. The model framework used here allows the definition of any number and kind of bacterial and solute species [43]. As cells consume substrate according to their strain-specific metabolism kinetics and produce additional biomass, they grow and divide once a maximum cell radius is achieved. Movement of cells, which are modeled as rigid circles, results from forces exerted between neighbors as they grow and divide. Summed over all the cells present, these forces cause the biofilm front to advance. Solutes diffuse across a boundary layer between the biofilm and a bulk fluid in which solute concentrations are assumed to be homogeneous and constant. Inside this boundary layer, we determine the dynamics of solute spatial distributions by solving the 2-D diffusion-reaction equations. In so doing, we assume that solute concentrations reach their diffusion-reaction equilibria much faster than bacterial cells grow and divide [43,44]. The biofilm simulation framework and its associated numerical methods have previously been described in detail [42,43,45]. Following Xavier and Foster [36], we assume that bacteria consume a substrate, S, and invest it in the production of biomass and EPS (for a full list of model notation, see Table 1). This allows a simple definition of the strains based upon their biomass versus EPS investment strategies. Non-EPS producers (EPS−) devote all substrate taken up to biomass production, whereas unconditional EPS producers (EPS+) always allocate a proportion f to EPS synthesis. Table data removed from full text. Table identifier and caption: 10.1371/journal.pbio.0060014.t001 Notation Summary Our third strain, QS+, is intended to represent a hypothetical first step in the evolution of quorum sensing. We assume that QS+ cells have gained the ability to detect a waste chemical produced by conspecific bacteria. This chemical can be envisioned as a byproduct of metabolism or cellular housekeeping that has been co-opted as a primitive autoinducer for monitoring local population density. This scenario is consistent with many real-world autoinducers, especially those of Gram-negative bacteria and some unicellular yeasts, which are closely related to, or simply are, metabolic waste products [4,15,46,47]. One way that the transition from a nonresponsive to a responsive quorum-sensing phenotype could occur is through mutation in a preexisting transcription factor, which allows it to bind the accumulating autoinducer. Binding the autoinducer may then alter the transcription factor's ability to control the expression of an EPS synthase. This abstraction conforms very well with the molecular mechanism underlying LuxI/R-type quorum-sensing circuits widely observed among bacteria [4,15,48]. Bacteria grow according to Monod saturation kinetics, and we assume that all cells secrete an autoinducer without cost and at a constant rate (Table 2). Following the pattern exhibited by V. cholerae, QS+ cells synthesize EPS only when local autoinducer concentration is below the quorum-sensing threshold concentration. Once this threshold level is exceeded, QS+ cells terminate EPS synthesis and invest only in biomass production [9]. The timing and density dependence with which QS+ bacteria reach a quorum depends upon three key factors: (1) how quickly the autoinducer is produced, (2) how quickly the autoinducer diffuses away from the biofilm, and (3) the critical quorum-sensing autoinducer concentration. For example, fast autoinducer production, slow autoinducer diffusion, and a low critical quorum-sensing autoinducer concentration will all lead to a quorum being reached more quickly and at lower cell density. To account for the dependence of quorum-sensing behavior on all of these factors, we group them into a single parameter, , where σ is the autoinducer production rate per unit bacterial biomass, DAI is the autoinducer diffusion coefficient, and φ is the quorum-sensing threshold autoinducer concentration. ρX, the bacterial biomass density, and L, the length of the biofilm simulation space, are included in α to form a dimensionless group. Using a dimensionless group to describe the quorum-sensing process allows us to make qualitative predictions that are independent of the specific values of the parameters contained in α, albeit within the bounds of systems that have these physical properties. Table data removed from full text. Table identifier and caption: 10.1371/journal.pbio.0060014.t002 Stoichiometry of Bioprocesses Included in Simulations Strains with the same α value will reach their respective quorums at the same time after the initiation of biofilm growth, irrespective of the different potential combinations of σ, DAI, φ, ρX, and L that can produce a particular α value. Although α accounts for multiple factors that simultaneously contribute to quorum-sensing dynamics, to aid intuition, one may hold all parameters other than φ constant and think of α as the critical quorum-sensing autoinducer concentration. α simply measures how readily QS+ cells switch from low to high cell-density state: for higher α, QS+ bacteria will reach a quorum at higher cell density and relatively later on in the course of biofilm growth. In order to determine whether simple quorum-sensing behavior (QS+) provides a fitness advantage over the unconditional behavioral strategies EPS+ and EPS−, we first consider competition in mixed biofilms initialized with the same number of either (1) QS+ and EPS+ or (2) QS+ and EPS−. We replicate these simulations over a range of α values for the QS+ strain in order to examine how the timing and density dependence of quorum sensing influence the outcome of competition. Simple Competition: QS+ versus EPS+, and QS+ versus EPS−: Simulations were parameterized with empirically estimated values (Table 3), initialized with 50 cells of each strain placed randomly on the solid substratum, and allowed to run for 14 simulated days (Figure 1), which is close to the maximum duration of a V. cholerae infection [49]. The proportion of energy invested in EPS secretion (f) will determine the extent to which EPS production allows one strain to displace others from a biofilm. As Xavier and Foster have discussed [36], for a given set of simulation parameters, there exists some evolutionarily stable strategy for EPS production, f*, which will out-compete any strain that invests either more or less in EPS. To find this optimum strategy, we performed an evolutionary stability analysis in which EPS+ strains with incrementally larger or smaller f values were competed against each other (see Text S1 and Figure S2). We found that, for our model conditions, the evolutionarily stable strategy for EPS investment independent of quorum sensing is approximately f* = 0.5, which was used for both the EPS+ and the QS+ strains (when below its quorum) in all subsequent simulations. Table data removed from full text. Table identifier and caption: 10.1371/journal.pbio.0060014.t003 Parameters Used in Biofilm Simulations Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pbio.0060014.g001 Direct Competition between QS+ and EPS+ Bacteria Initialized with Equal Numbers of Both StrainsAutoinducer (AI) concentration is shown in the background, where isoconcentration lines represent 0.1-mg/l steps. Both strains behave identically, producing both EPS and biomass, until the autoinducer quorum-sensing threshold is reached. QS+ cells then turn off polymer secretion, devote all resources to biomass production, and achieve a growth burst at locations on the upper surface of the biofilm where substrate availability is highest. A movie for this simulation is provided as Video S1. Competitions between the QS+ and EPS+ strains and between the QS+ and EPS− strains were repeated for a range of α values. We included two controls, one (α = ∞) in which the QS+ strain never reaches its quorum and behaves identically to the EPS+ strain, and another (α = 0.001) in which the QS+ strain reaches a quorum immediately after simulations begin, and behaves identically to the EPS− strain thereafter. The frequency of QS+ cells within the biofilm was calculated at each time step and averaged over all replicate simulations to generate a mean QS+ frequency plot for each α value used in both sets of competitions (Figure 2A and 2B). Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pbio.0060014.g002 Summary of Simple Competitions(A) A quorum-sensing strain that down-regulates polymer secretion at high cell-density (QS+) is competed against a constitutive polymer-secreting strain (EPS+). (B) QS+ versus non-polymer producer (EPS−). Each competition (QS+ vs. EPS+, and QS+ vs. EPS−) was replicated 50 times for each of the α values: ∞, 0.01, 0.008, 0.005, and 0.001, where α captures how quickly the QS+ strain will switch from low to high cell-density state (see main text). For higher α, QS+ bacteria will reach their quorum at higher cell density, relatively later on during biofilm growth. Plotted lines represent mean QS+ frequency time series from each set of 50 simulations and are shown with shaded 95% confidence intervals. Note that in (A) and (B), the plotted lines corresponding to α = ∞ are control treatments in which QS+ behaves identically to EPS+ throughout simulations because autoinducer concentrations never reach the QS+ quorum-sensing threshold. Similarly, in (A) and (B), the plotted lines corresponding to α = 0.001 are control treatments in which QS+ behaves identically to EPS− throughout simulations because autoinducer concentrations always exceed the QS+ quorum-sensing threshold. Competition between QS+ and EPS+. In a mixed competition between the quorum-sensing strain and a constitutive EPS producer, all cells are initially phenotypically identical; that is, they all secrete EPS. However, as cells grow and population density increases, the autoinducer accumulates, and at a time point dependent upon their α value, quorum-sensing (QS+) cells turn off polymer secretion and invest all their resources in growth. Near the upper surface of the biofilm, where substrate availability is highest, QS+ cells achieve a burst of cell division (Figure 1, days 9–13). In the short term, the QS+ strain increases in frequency over and above that of the constitutive EPS producer. The advantage is temporary, however, because the EPS+ strain continues to secrete polymer and eventually produces towers that suffocate neighboring QS+ cells (see for example Figure 2A, α = 0.005), analogous to the case of competing EPS+ and EPS− cells [36]. Quorum-sensing control of EPS production, therefore, provides a competitive advantage over constitutive EPS production, but only for a limited time window. Moreover, the location of this window within the period of biofilm growth is determined by how quickly the QS+ strain reaches a quorum. Strains with higher α attain growth bursts later in the course of biofilm formation (Figure 2A). Competition between QS+ and EPS−. Without having to pay the cost of EPS production, EPS− cells rapidly divide at the beginning of simulations and achieve a higher initial frequency than QS+ cells. By secreting EPS, however, the QS+ strain rises up and over the top of neighboring cells, suffocating those that do not secrete polymer. After its initial disadvantage due to lower growth rate, the QS+ strain rapidly ascends to a majority in the biofilm and remains there indefinitely. Unlike the EPS+ strain, QS+ cells switch to pure biomass production after they have suffocated their EPS− neighbors; at this point, investment into EPS is no longer advantageous. As a result, the QS+ strain will out-compete non-EPS producers by even larger margins than the constitutive EPS producer (Figure 2B). The simple competition simulations described above suggest that bacteria for which EPS production is under quorum-sensing control have a time-dependent advantage over strains that are not capable of responding to changes in population density. However, a within-group competitive advantage need not translate into evolutionary success when the advantage comes at a strong cost to overall productivity [50]. More concretely, if successfully suppressing another strain in a biofilm causes the entire biofilm to grow poorly, the net effect on fitness may be deleterious [36]. We investigated this possibility through evolutionary invasion analyses to determine whether rare-mutant QS+ cells can increase in frequency in populations of either EPS+ or EPS− cells, and whether a successful QS+ strain, once in the majority, can subsequently resist invasion by rare EPS+ and EPS− mutants. To do this, we simply compare the number of cell divisions of the invading strain in a focal biofilm to the mean number of cell divisions by the majority strain taken across all biofilms in the population. More formally, we first define the fitness of a bacterial strain as the average number of cell divisions that it achieves on a defined time interval [0, tend]: where NS,t is the number of cells of strain S present within the biofilm at time t. Letting S1 be a rare mutant, we define its ability to invade a majority strain, S2, as follows: where wS1 is the fitness of the potential invader (S1) in direct competition with S2, as described in Equation 1, and is the mean fitness of S2 cells in a pure S2 biofilm, which approximates mean fitness in the population. We assume that the bacterial population as a whole contains many more biofilms than the focal simulated biofilm in which the potential invading strain (S1) has arisen. All biofilms other than the focal simulated biofilm are composed purely of the resident strain, S2, and contribute vastly more to mean population fitness. Therefore, effectively measures the fitness of S2 cells when competing solely with other S2 cells. For a rare-mutant S1 to invade a majority strain S2, must be greater than unity; that is, S1 must fare better against S2 than S2 fares against itself [51]. Length of biofilm tenure: A key variable in this analysis is the time interval [0, tend] on which wS1 and are measured. When choosing tend, we are asking: at what point during biofilm growth is it critical for long-term evolutionary success to be in the majority? We take the answer to be the time at which dispersal or disturbance occurs, and we assume that all cells within a biofilm have an equal probability of entering the propagule pool from which subsequent biofilms are seeded. This approach takes into consideration both local competition within biofilms and global competition between biofilms to determine the long-term evolutionary success of an invading bacterial strain [50]. Importantly, our method of analyzing invasiveness also assumes that dispersal or disturbance occurs in one large burst at a discrete point in time, rather than continuously throughout biofilm growth (see Discussion). Genetic relatedness at biofilm initiation: We performed reciprocal invasion analyses using simulated competitions between QS+ and EPS+ or QS+ and EPS− with a range of initial QS+ frequencies. This captures the effect of a rare mutant entering a population of another strategy, where the starting frequency of the rare strain reflects the number of strains randomly inoculated, and therefore the initial average relatedness, within the biofilm. For example, if 10 strains are present at the initiation of each biofilm, then a rare mutant will begin at a local frequency of 0.1, and average relatedness within the biofilm where the rare mutant resides will start at 0.1 [2,36]. Invasion analysis: QS+ and EPS+. We investigated whether a quorum-sensing strain that obtains an advantage in single biofilms (Figures 1 and 2) can invade a population of constitutive EPS producers and resist their reinvasion. We therefore focus on parameter values under which the QS+ strain has an advantage in the simple competition simulations. Specifically, we examine invasiveness for a disturbance interval of 9 d (tend = 9), with a QS+ strain α value (QS sensitivity) of 0.008, and we find that the QS+ strain can readily invade populations composed mostly of EPS+ cells, but not vice versa (Figure 3A and 3B). Additionally, biofilms composed entirely of QS+ cells have a higher average fitness than biofilms composed entirely of EPS+ cells. Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pbio.0060014.g003 The Quorum-Sensing Strain Can Invade Non-Quorum-Sensing Strains, but Not Vice VersaInvasiveness of a rare mutant was analyzed for different degrees of mixing among strains in biofilms, reflected in the different initial frequencies of the rare strain in the biofilm. For example, if 10 strains are randomly sampled, then the initial frequency of the rare mutant in its own biofilm will be 0.1; initial relatedness will also be 0.1 (see main text). Each box and whisker plot summarizes the results of 20 replicate simulations, and plus signs (+) denote outliers. All simulations were carried out at α = 0.008 for the QS+ strain. (A) Invasion of a rare quorum-sensing strain (QS+) into a population of unconditional EPS producers (EPS+), and (B) failure of a rare EPS+ strain to invade a population of QS+ bacteria. Biofilms composed entirely of QS+ cells attain higher average fitness than biofilms composed entirely of EPS+ cells. (C) Invasion of a rare QS+ strain into a population of non-EPS producers (EPS−), and (D) failure of a rare EPS− strain to invade a population of QS+ bacteria. Again, the QS+ strain can invade EPS−, whereas EPS− cannot invade QS+. Notably, however, biofilms composed entirely of QS+ cells have a lower average fitness than biofilms composed entirely of EPS− cells. Therefore, if all biofilms contained only a single genotype (no within-biofilm evolutionary competition), the EPS− would invade and resist invasion. Invasion analysis: QS+ and EPS.−: Again using tend = 9 d, we find that the QS+ strain invades a resident population of EPS− cells, whereas the reverse is not true (Figure 3C and 3D). It is notable, however, that biofilms composed entirely of QS+ cells have a lower mean fitness than biofilms composed entirely of EPS− cells, which reflects the fact that investment into EPS reduces total biomass production and therefore average growth rate.
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