A biophysical threshold for biofilm formation

Bacteria are ubiquitous in our daily lives, either as motile planktonic cells or as immobilized surface-attached biofilms. These different phenotypic states play key roles in agriculture, environment, industry, and medicine; hence, it is critically important to be able to predict the conditions under which bacteria transition from one state to the other. Unfortunately, these transitions depend on a dizzyingly complex array of factors that are determined by the intrinsic properties of the individual cells as well as those of their surrounding environments, and are thus challenging to describe. To address this issue, here, we develop a generally-applicable biophysical model of the interplay between motility-mediated dispersal and biofilm formation under positive quorum sensing control. Using this model, we establish a universal rule predicting how the onset and extent of biofilm formation depend collectively on cell concentration and motility, nutrient diffusion and consumption, chemotactic sensing, and autoinducer production. Our work thus provides a key step toward quantitatively predicting and controlling biofilm formation in diverse and complex settings.

Similarly, studies of biofilms under defined laboratory conditions have also provided key insightssuch as by revealing the pivotal role of intercellular chemical signaling in biofilm formation (Nadell et al., 2008;Bassler and Losick, 2006;Davey and O'toole, 2000;Hall-Stoodley et al., 2004). In this process, termed quorum sensing, individual cells produce, secrete, and sense freely diffusible autoinducer molecules, thereby enabling different bacteria to coordinate their behavior (Davies et al., 1998;Sakuragi and Kolter, 2007;Bassler and Losick, 2006;Miller and Bassler, 2001;Herzberg et al., 2006;Laganenka et al., 2018;McLean et al., 1997;Paul et al., 2009). For example, in many cases, quorum sensing positively controls biofilm formation (Herzberg et al., 2006;Laganenka et al., 2018;Davies et al., 1998;Sakuragi and Kolter, 2007;McLean et al., 1997;González Barrios et al., 2006;Yarwood et al., 2004;Koutsoudis et al., 2006;Waters and Bassler, 2005;Parsek and Greenberg, 2005;Jayaraman and Wood, 2008;Hentzer et al., 2003;Kirisits and Parsek, 2006): autoinducer accumulation above a threshold concentration upregulates the expression of genes involved in biofilm formation, ultimately driving a transition from the planktonic to the biofilm state (Nadell et al., 2008). Again, however, the cellular factors that control this transition, such as the autoinducer production rate, diffusivity, and threshold concentration, can strongly vary between different species and environmental conditions. Because planktonic dispersal and biofilm formation both depend on a dizzyingly complex array of factors, these distinct processes are typically studied in isolation. Thus, while each is well understood on its own, quantitative prediction of the conditions under which a population of planktonic bacteria transitions to the biofilm state-or instead, continues to disperse away and remains in the planktonic state-remains challenging. Here, we address this challenge by developing a mathematical model that describes essential features of motility-mediated dispersal of planktonic cells and autoinducermediated biofilm formation together. Using numerical simulations of this model, we systematically examine the influence of cellular concentration, motility, and chemotactic sensing; nutrient availability, diffusion, and consumption; and autoinducer production, diffusion, and accumulation on biofilm formation. Guided by these results, we establish a potentially-universal biophysical threshold that unifies the influence of all these factors in predicting the onset and extent of biofilm formation across different species and environmental conditions. Our work therefore provides a theoretical foundation for the prediction and control of biofilm formation in diverse and complex settings, and yields new quantitative predictions to guide future experiments.

Development of the governing equations
As an illustrative example, and to connect our model to recent experiments of bacterial dispersal (Bhattacharjee et al., 2021), we consider a rectilinear geometry with a starting inoculum of planktonic cells at a maximal concentration b 1,0 and of width x 0 . In general, the continuum variable b (x, t) describes the number concentration of bacteria, where x is the position coordinate and t is time, and the subscripts {1, 2} represent planktonic or biofilm-associated cells, respectively. Following previous work (Lauffenburger, 1991;Keller and Segel, 1971;Adler, 1966a;Croze et al., 2011;Fu et al., 2018;Bhattacharjee et al., 2021), we consider a sole diffusible nutrient that also acts as 0 0  Figure 1. Competition between motility-mediated dispersal and autoinducer-mediated biofilm formation. (A) Schematic of chemotactic dispersal: planktonic bacteria (green) consume nutrient (purple) and establish a local gradient that they, in turn, direct their motion in response to. (B) Schematic of positive quorum sensing-controlled biofilm formation: accumulation of produced autoinducer (red) above a threshold concentration causes cells to transition to the biofilm state (blue). (C) Results of an example simulation of Equations 1-4 showing the dynamics of the nutrient, planktonic cells, autoinducer, and biofilm cells from top to bottom, quantified by the normalized concentrations c/c 0 , b 1 /b 1,0 , a/a * , b 2 /b 1,0 , respectively; c 0 , b 1,0 , and a * represent the initial nutrient concentration, initial bacterial concentration, and autoinducer threshold for biofilm formation, respectively. The position coordinate is represented by the normalized position x/x 0 , where x 0 is the width of the initial cellular inoculum. Different shades indicate different time points as listed. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. At t ≈ 13 h , sufficient autoinducer has been produced to trigger biofilm formation at the origin; at even longer times ( t ≳ 16 h ), nutrient depletion limits autoinducer production at this position.
However, accumulation of autoinducer by the dispersing planktonic cells triggers partial biofilm formation at x/x 0 ≈ 6 as well. This competition between dispersal and biofilm formation leads to a final biofilm fraction of f = 21% at the final time of t = 20 h . An animated form of this figure is shown in Video 1. The values of the simulation parameters are given in Supplementary file 2.
The online version of this article includes the following source data and figure supplement(s) for figure 1: Source data 1.    the chemoattractant, with a number concentration represented by the continuum variable c(x, t) with diffusivity Dc . Initially, nutrient is replete throughout the system at a constant concentration c 0 . The bacteria then consume the nutrient at a rate b 1 κ 1 g(c) , where κ 1 is the maximum consumption rate per cell and the Michaelis-Menten function g(c) ≡ c c+cchar quantifies the nutrient dependence of consumption relative to the characteristic concentration c char (Croze et al., 2011;Monod, 1949;Cremer et al., 2019;Woodward et al., 1995;Shehata and Marr, 1971;Schellenberg and Furlong, 1977;Cremer et al., 2016).
As time progresses, the bacteria thereby establish a local nutrient gradient that they respond to via chemotaxis ( Figure 1A). In particular, planktonic cells disperse through two processes: undirected active diffusion with a constant diffusivity D 1 (Berg, 2018), and directed chemotaxis with a that quantifies the ability of the bacteria to sense and respond to the local nutrient gradient (Keller and Segel, 1970;Keller and Segel, 1971;Odell and Keller, 1976;Keller and Odell, 1975) with characteristic bounds c − and c+ (Cremer et al., 2019;Sourjik and Wingreen, 2012;Shimizu et al., 2010;Tu et al., 2008;Kalinin et al., 2009;Shoval et al., 2010;Lazova et al., 2011;Celani et al., 2011;Fu et al., 2018;Dufour et al., 2014;Yang et al., 2015;Cai et al., 2016;Chen and Jin, 2011) and a chemotactic coefficient χ 1 . The planktonic cells also proliferate at a rate bγ 1 g(c) , where γ 1 is the maximal proliferation rate per cell. Finally, as the planktonic bacteria consume nutrients, they produce and secrete a diffusible autoinducer, with a number concentration represented by a(x, t) and with diffusivity Da , at a maximal rate k 1 per cell. Motivated by some previous work (Hense et al., 2012;Hense and Schuster, 2015;Kirisits et al., 2007;Bollinger et al., 2001;Duan and Surette, 2007;Mellbye and Schuster, 2014;De Kievit et al., 2001;Pérez-Osorio et al., 2010), we take this process (hereafter referred to as 'production' for brevity) to also be nutrientdependent via the same Michaelis-Menten function g(c) for the results presented in the main text, but we also consider the alternate case of 'protected' nutrient-independent production in the supplementary materials. Following previous work (Koerber et al., 2002;Ward et al., 2001;Ward et al., 2003), we also model natural degradation of autoinducer as a first-order process with a rate constant λ.
As autoinducer is produced, it binds to receptors on the surfaces of the planktonic cells with a second-order rate constant α , as established previously (Koerber et al., 2002;Ward et al., 2001;Ward et al., 2003). Motivated by experiments on diverse bacteria, including the prominent and wellstudied species Escherichia coli, Pseudomonas putida, and Pseudomonas aeruginosa (Davies et al., 1998;Sakuragi and Kolter, 2007;Bassler and Losick, 2006;Miller and Bassler, 2001;Herzberg et al., 2006;Laganenka et al., 2018;McLean et al., 1997;Paul et al., 2009;González Barrios et al., 2006;Yarwood et al., 2004;Koutsoudis et al., 2006;Waters and Bassler, 2005;Parsek and Greenberg, 2005;Jayaraman and Wood, 2008;Hentzer et al., 2003;Kirisits and Parsek, 2006), we assume that planktonic cells transition to the biofilm state at a rate τ −1 when the local autoinducer concentration exceeds a threshold value a * ( Figure 1B). Because our focus is on this transition, we assume that it is irreversible, and that cells in the biofilm lose motility. However, they still continue to consume nutrient, proliferate, and produce autoinducer with maximal rates κ 2 , γ 2 , and k 2 per cell, respectively; additional behaviors such as subsequent production of extracellular polymeric substances or transitioning back to the planktonic state can be incorporated as future extensions to this model.
Hence, while planktonic cells can disperse via active diffusion and chemotaxis, their dispersal is hindered-and biofilm formation is instead promoted-when autoinducer accumulates sufficiently, as schematized in Figure 1A-B. The central goal of this paper is to examine the processes underlying this competition between dispersal and biofilm formation. Our model is thus summarized as: where H is the Heaviside step function describing the transition from the planktonic to biofilm state.
To explore the competition between motility-mediated dispersal and autoinducer-mediated biofilm formation, we then numerically

Representative numerical simulations
The results of a prototypical example are shown in Figure 1C and Video 1. Consumption by the planktonic cells (green curves) rapidly establishes a steep nutrient gradient (purple) at the leading edge of the inoculum. This gradient forces the planktonic cells to then move outward via chemotaxis.
Video 1. Animated form of Figure 1C: Results of an example simulation of Equations 1-4 showing the dynamics of the nutrient, planktonic cells, autoinducer, and biofilm cells from top to bottom, quantified by the normalized concentrations c/c 0 , b 1 /b 1,0 , a/a * , and b 2 /b 1,0 , respectively; c 0 , b 1,0 , and a * represent the initial nutrient concentration, initial bacterial concentration, and autoinducer threshold for biofilm formation, respectively. The position coordinate is represented by the normalized position x/x 0 , where x 0 is the width of the initial cellular inoculum. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. At t ≈ 13 h , sufficient autoinducer has been produced to trigger biofilm formation at the origin; at even longer times ( t ≳ 16 h ), nutrient depletion limits autoinducer production at this position. However, accumulation of autoinducer by the dispersing planktonic cells triggers partial biofilm formation at x/x 0 ≈ 4 as well. This competition between dispersal and biofilm formation leads to a final biofilm fraction of f = 21% at the final time of t = 20 h . The values of the simulation parameters are given in Supplementary file 2. The video displays the profiles every 30 min , to retain a manageable file size; however, the temporal step size in the actual simulations is 0.1 s .
In particular, they self-organize into a coherent front that expands from the initial inoculum and continually propagates, sustained by continued consumption of the surrounding nutrient-consistent with the findings of previous studies of planktonic bacteria (Bhattacharjee et al., 2021). In this case, however, the cells also concomitantly produce autoinducer that accumulates into a growing plume (red). In some locations, the autoinducer eventually exceeds the threshold a * , thus driving the formation of an immobilized biofilm (blue). Hence, at long times, f = 21% of the overall population is biofilmassociated, while the remaining 1 − f = 79% continues to disperse in the planktonic state. Because the processes underlying motility-mediated dispersal and autoinducer-mediated biofilm formation are highly species-and environment-dependent, the values of the parameters in Equations 1-4 can span broad ranges-giving rise to different emergent behaviors under different conditions. Our simulations provide a way to examine how these behaviors depend on cellular concentration  Figure 1C, but for planktonic cells with faster nutrient consumption (larger κ 1 ). Panels and colors show the same quantities as in Figure 1C. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. However, nutrient is depleted at this position more rapidly, limiting autoinducer production; as a result, the population continues to disperse in the planktonic state and the final biofilm fraction is f = 0% . An animated form of this figure is shown in Video 2. The values of the simulation parameters are given in Supplementary file 2.
The online version of this article includes the following source data and figure supplement(s) for figure 2: Source data 1.
Figure supplement 1. Slower nutrient consumption allows greater autoinducer production, leading to more biofilm formation. and motility, quantified by {b 1,0 , D 1 , χ 1 , c − , c+} , nutrient availability and consumption, quantified by {Dc, c 0 , κ 1 , κ 2 , c char } , cellular proliferation, quantified by {γ 1 , γ 2 } , and autoinducer production, availability, and sensing, quantified by {Da, k 1 , k 2 , λ, α, τ , a * } . For example, implementing the same simulation as in Figure 1C, but for cells with faster nutrient consumption, yields a population that completely disperses in the planktonic state (the fraction of the population in the biofilm state at the final time of t = 20 h is f = 0% , as shown in Figure 2 and Video 2). Conversely, when cells consume nutrient slower, a larger fraction of the population forms an immobilized biofilm ( f = 52% , Figure 2-figure supplement 1 and Video 3). Given that the competition between motility-mediated dispersal and autoinducer-mediated biofilm formation depends sensitively on such a bewildering array of cellular and environmental factors, we ask whether these dependencies can be captured by simple, generalizable, biophysical rules. Nondimensionalization of Equations 1-4 yields characteristic quantities and dimensionless groups that can parameterize these dependencies, as detailed in Appendix 1; however, given the large number of such groups, we seek an even simpler representation of the underlying processes that could unify the influence of all these different factors. To do so, we examine the fundamental processes underlying biofilm formation in our model.

Availability of nutrient for autoinducer production
When autoinducer production is nutrient-dependent, we expect that a necessary condition for biofilm formation is that enough nutrient is available for sufficient autoinducer to be produced to eventually exceed the threshold a * . To quantify this condition, we estimate two time scales: τ d , the time taken by the population of planktonic cells to deplete all the available nutrient locally, and τa , the time at which produced autoinducer reaches the threshold for biofilm formation. While τ d and τa can be directly obtained in each simulation, we seek a more generally-applicable analytical expression for both, solely using parameters that act as inputs to the model. In particular, for simplicity, we consider Video 2. Animated form of Figure 2: Results of the same simulation as in Video 1, but for planktonic cells with faster nutrient consumption (larger κ 1 ). Panels and colors show the same quantities as in Video 1. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. However, nutrient is depleted at this position more rapidly, limiting autoinducer production; as a result, the population continues to disperse in the planktonic state and the final biofilm fraction is f = 0% .
The values of the simulation parameters are given in Supplementary file 2. The video displays the profiles every 30 min , to retain a manageable file size; however, the temporal step size in the actual simulations is 0.1 s . https://elifesciences.org/articles/76380/figures#video2 Video 3. Animated form of Figure 2-figure supplement 1: Results of the same simulation as in Video 1, but for planktonic cells with slower nutrient consumption (smaller κ 1 ). Panels and colors show the same quantities as in Video 1. The inoculum initially centered about the origin slowly consumes nutrient (purple), establishing a slight gradient that allows partial planktonic dispersal (green curves moving outward); the cells also produce autoinducer (red) concomitantly. Because nutrient is consumed slowly, autoinducer production is not limited, resulting in partial biofilm formation (blue). Autoinducer has sufficiently accumulated above the threshold after t ≈ 14 h , which causes a population of biofilm cells to form at the origin ( x/x 0 ≈ 0 ). After 20 h , the biofilm population continues to grow, and additionally, autoinducer concentration exceeds the threshold concentration at x/x 0 ≈ 10 . Thus, we see a second population of biofilm cells form, centered at x/x 0 ≈ 10 .
The slower nutrient consumption results in a greater final biofilm fraction than in Video 1-here, f = 52% .
The values of the simulation parameters are given in Supplementary file 2. The video displays the profiles every 30 min , to retain a manageable file size; however, the temporal step size in the actual simulations is 0.1 s . https://elifesciences.org/articles/76380/figures#video3 nutrient consumption and autoinducer production, both occurring at their maximal rates κ 1 and k 1 , respectively, by an exponentially-growing population of planktonic cells that are uniformly distributed in a well-mixed and fixed domain. Integrating Equations 3 and 4 then yields (Appendix 2) Three key dimensionless quantities, denoted by the tilde (˜) notation, emerge from this calculation.
, describes the yield of new cells produced as the population consumes nutrient-quantified by the rates of cellular proliferation and nutrient consumption, γ 1 and b 1,0 κ 1 /c 0 , respectively . The second, η ≡ αa * /k 1 , describes the competition between autoinducer loss and production, quantified by their respective rates αa * and k 1 , at the single-cell scale. The third, ζ 1,0 ≡ αb 0 /γ 1 , describes the loss of autoinducer due to cell-surface binding as the population continues to grow, quantified by the population-scale rates of autoinducer loss and cellular proliferation, αb 0 and γ 1 , respectively; for simplicity, this quantity neglects natural degradation of autoinducer, given that the degradation rate is relatively small, with λ ≪ αb 0 .
The ratio between Equations 5 and 6 then defines a nutrient availability parameter, D ≡ τ d /τa . When D is large, produced autoinducer rapidly reaches the threshold for biofilm formation before the available nutrient is depleted; by contrast, when D is small, nutrient depletion limits autoinducer production. Hence, we hypothesize that D ≳D * specifies a necessary condition for biofilm formation, where D * is a threshold value of order unity. The simulations shown in Figures 1C and 2 and Figure 2-figure supplement 1 enable us to directly test this hypothesis. Consistent with our expectation, the simulation in Figure 1C is characterized by D = 0.33 , near the expected threshold for biofilm formation; as a result, f = 21% . When consumption is faster as in Figure 2 ( D = 0.033 ), the available nutrient is rapidly depleted; thus, cells disperse away before sufficient autoinducer is produced to initiate biofilm formation, and f = 0% . Conversely, when nutrient consumption is slow as in Figure 2-figure supplement 1 ( D = 3.1 ), nutrient continues to be available for autoinducer production, eventually driving biofilm formation, with a larger fraction f = 52% .
Taken together, these results support our hypothesis that D ≳D * ∼ 1 is a necessary condition for biofilm formation. It is not, however, a sufficient condition: repeating the simulation of Figure 1C but for faster-moving cells yields a population that rapidly disperses without forming a biofilm at all ( f = 0% , Figure 3A and Video 4)-despite having the same value of D = 0.33 . Thus, our mathematical description of the conditions that determine biofilm formation is, as yet, incomplete.

Competition between motility-mediated dispersal and autoinducer accumulation
The results shown in Figure 3 indicate that the ability of planktonic bacteria to move, which is not incorporated in the nutrient consumption parameter D , also plays a key role in regulating whether a biofilm forms. Indeed, close inspection of Figure 3A hints at another necessary condition for biofilm formation: as shown by the magnified view in Figure 3B (e.g., at t = 4 h ), the leading edge of the dispersing planktonic cells extends beyond the plume of produced autoinducer. Therefore, we expect that even when sufficient nutrient is available for autoinducer production ( D ≳D * ∼ 1 ), autoinducer production must be rapid enough to reach the threshold for biofilm formation before cells have dispersed away. To quantify this condition, we estimate the the time τc at which the motile planktonic cells begin to 'outrun' the growing autoinducer plume. Specifically, we quantify the dynamics of the leading edge positions of the chemotactic front of planktonic cells and the autoinducer plume, x 1,edge (t) and x a,edge (t) , respectively. The front position x 1,edge (t) is known to depend on cellular motility, nutrient diffusion, and nutrient consumption in a non-trivial manner (Berg, 2004;Cremer et al., 2019;Fu et al., 2018;Amchin et al., 2022), and we are not aware of a way to compute this quantity a priori from input parameters; instead, we extract this sole quantity from each simulation by identifying the largest value of x at which b 1 ≥ 10 −4 b 1,0 . While the plume position x a,edge (t) can also be directly obtained in each simulation, we again develop a more generally applicable analytical expression by assuming that the autoinducer continually diffuses from the initial inoculum: x a,edge (t) = x 0 + √ 2Dat . Then, τc can be directly determined as the time at which x 1,edge (t) begins to exceed x a,edge (t) .  Figure 3. Enhanced motility enables cells to disperse before sufficient autoinducer accumulates, leading to complete dispersal. (A) Results of the same simulation as in Figure 1C, but for faster-moving planktonic cells (larger D 1 and χ 1 ). Panels and colors show the same quantities as in Figure 1C. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. More rapid dispersal enables the planktonic cells to 'outrun' the growing autoinducer plume, as shown by the extended and magnified view in (B). As a result, the population continues to disperse in the planktonic state and the final biofilm fraction is The ratio between τc thereby determined and τa , the time required for produced autoinducer to reach the threshold for biofilm formation (Equation 6), then defines a cellular dispersal parameter, J ≡ τc/τa . When J is large, autoinducer accumulation is sufficiently rapid to drive biofilm formation; by contrast, when J is small, the planktonic cells rapidly disperse without forming a biofilm. Hence, we hypothesize that J ≳J * specifies another necessary condition for biofilm formation, where J * is, again, a threshold value of order unity. The simulations shown in Figures 1C and 3A enable us to directly test this hypothesis. Consistent with our expectation, the simulations in Figure 1C and Figure 2-figure supplement 1 are characterized by J = 1.6 , near the expected threshold for biofilm formation; as a result, f > 0 in both cases. Furthermore, implementing the same simulation as Figure 1C (with the same D = 0.33 ) but for slower-moving cells, characterized by a larger J = 120 , yields a population that forms an even larger biofilm fraction f = 82% (Figure 3-figure supplement  1 and Video 5). Conversely, when cellular dispersal is faster as in Figure 3, characterized by a smaller J = 0.1 , the cells disperse away before sufficient autoinducer is produced to initiate biofilm formation, and f = 0% . Taken together, these results support our hypothesis that J ≳J * ∼ 1 is another necessary condition for biofilm formation.

A universal biophysical threshold for biofilm formation
Thus far, we have shown that the two conditions D ≳D * and J ≳J * are both necessary for biofilm formation. Is the combination of both sufficient to fully specify the conditions required for biofilm formation? To test this possibility, we implement Source data 1.

Figure supplement 1. Diminished motility enables autoinducer to accumulate, resulting in increased biofilm formation.
Figure supplement 1-source data 1.

Figure 3 continued
Video 4. Animated form of Figure 3: Results of the same simulation as in Video 1, but for faster-moving planktonic cells (larger D 1 and χ 1 ). Panels and colors show the same quantities as in Video 1. The inoculum initially centered about the origin consumes nutrient (purple), establishing a gradient that drives outward dispersal by chemotaxis (outward moving green curves); the cells also produce autoinducer (red) concomitantly. More rapid dispersal enables the planktonic cells to 'outrun' the growing autoinducer plume, as shown by the extended and magnified view in (B). As a result, the population continues to disperse in the planktonic state and the final biofilm fraction is 10,983 numerical simulations of Equations 1-4 exploring the full physiological ranges of the input parameters that describe cellular, nutrient, and autoinducer properties for diverse bacterial species/ strains and environmental conditions (Supplementary file 1). For each simulation, we compute D , J , and f . Remarkably, despite the extensive variability in the values of the underlying parameters, all the results cluster between two states parameterized by D and J , as shown in Figure 4A: motilitymediated dispersal without biofilm formation ( f = 0% , green points) when either D <D * or J <J * , and biofilm formation without dispersal ( f = 100% , blue points) when both D >D * and J >J * . Many different combinations of the input parameters yield the same (D , J ) ; yet, no matter the input values of these parameters, which vary over broad ranges for different cells and environmental conditions, (D , J ) uniquely specify the resulting biofilm fraction f for all points, as shown in Figure 4B-C-indicating that these two dimensionless parameters reasonably encompass all the factors determining biofilm formation within our model. We observe some exceptions at the boundary between these two states, likely because the simplifying assumptions underlying the derivation of the D and J parameters begin to break down. Nevertheless, the boundary between both states, summarized by the relation D * /D +J * /J ∼ 1 with D * and J * both ∼ 1 (black curve), thus specifies a universal biophysical threshold for biofilm formation.

Discussion
The transition from the planktonic to biofilm state is known to depend on a large array of factors that describe cellular concentration, motility, and proliferation; nutrient availability and consumption; and autoinducer production, availability, and sensing-all of which can vary considerably for different strains/species of bacteria and environmental conditions. Therefore, quantitative prediction of the onset of biofilm formation is challenging. The biophysical model presented here provides a key step toward addressing this challenge. In particular, for the illustrative case we consider-in which cells can either disperse through active motility, retaining them in the planktonic state, or form an immobilized biofilm when exposed to sufficient autoinducer-we have shown that the onset of biofilm formation is uniquely specified by a biophysical threshold set by the two dimensionless parameters D (quantifying nutrient availability) and J (quantifying bacterial dispersal). Importantly, within the formulation of our model, this threshold is universal: many different combinations of cellular and environmental factors are described by the same (D,J ) , and thus, yield the same onset of biofilm formation. Therefore, given a bacterial strain and set of environmental conditions, extensions of our model could help provide a way to predict whether a biofilm will form a priori. Indeed, because the factors that define D and J can be directly measured, our work now provides quantitative principles and predictions (as summarized in Figure 4) to guide future experiments. For generality, our model also incorporates proliferation, nutrient consumption, and autoinducer production by cells after they have transitioned to the biofilm state. Hence, within our model, biofilm-produced autoinducer could also drive surrounding planktonic cells to transition to the biofilm state. In this case, we expect that the long-time fraction of the population in the biofilm state, f , will also depend on nutrient depletion and autoinducer production by the growing biofilm. Indeed, performing a similar calculation as that underlying the nutrient availability parameter, D , yields a third dimensionless parameter, S ≡ τ d,2 /τ a,2 ; here, τ d,2 and τ a,2 describe the times at which biofilm cells have depleted all the available nutrient and produced enough autoinducer to reach the threshold for biofilm formation, respectively (Appendix 2). Thus, we hypothesize that, while the onset of biofilm formation is specified by (D,J ) , the final extent of biofilm that has formed will also be described by S . The results shown in Figures 1-4C have a fixed S = 50 , which describes the case of a biofilm that produces autoinducer rapidly; repeating these simulations for the opposite case of slow autoinducer production by biofilm cells, with S = 1/50 , yields the state diagram shown in Figure 4D. In agreement with our hypothesis, while the transition to the biofilm state (black line) is not appreciably altered by the change in S , the transition to complete biofilm formation ( f = 1 ) is more gradual in this case (compare Figure 4A,B,D,E). Moreover, we note that our analysis thus far has focused on the case in which autoinducer production is nutrient-dependent; however, this process may sometimes be nutrient-independent (Narla et al., 2021). In this case, we expect that our overall analysis still applies, but with the onset of biofilm formation specified by only the dispersal parameter J -as confirmed in

Possible extensions of our work
The transition from the planktonic to biofilm state is highly complex and, in many cases, has features that are unique to different species of bacteria. Nevertheless, our model provides a minimal description that can capture many of the essential features of biofilm formation more generally-thereby providing a foundation for future extensions of our work, some of which are described below.
1. For simplicity, our model considers only one spatial dimension; however, fascinating new effects may arise in higher-dimensional implementations of our model. For example, in our prior work modeling the collective migration of planktonic bacteria in the absence of quorum sensingmediated biofilm formation, we found that variations in the shape of the cellular front orthogonal to the main propagation direction 'smooth out' over time Bhattacharjee et al., 2022). In particular, cells at outward-bulging parts of the front are exposed to more nutrient, which diminishes their ability to respond to the nutrient gradient via chemotaxis and thus slows them down. As a result, the migrating front eventually smooths to a flat shape whose subsequent dynamics can then be described using just one spatial dimension, just as in our treatment here. However, we expect that this behavior could be altered in interesting new ways when the cells can additionally produce and sense autoinducer and thereby transition to the biofilm state, as is the case here. In this case, we speculate that because cells at outward-bulging parts of the front are exposed to more nutrients and have a weaker chemotactic response, autoinducer production and accumulation will be more rapid relative to cellular dispersal. That is, at these parts of the front, τa and τc will be shorter and longer, respectively, causing the dispersal parameter J to be larger locally. Thus, our model would predict biofilm formation to occur first at these parts of the front, potentially also influencing subsequent dispersal and biofilm formation at other locations along the front. Therefore, while our conclusions here could be the same locally at different parts of the front, the global behavior of the population could be different-potentially giving rise to e.g., spatially-heterogeneous biofilm formation. 2. As an illustrative example, our model considers the case in which cells produce a single autoinducer; however, some quorum sensing systems utilize multiple autoinducers (Miller and Bassler, 2001;Miller et al., 2002;Pesci et al., 1997), which could be described using additional field variables and equations similar to Equation 4. Moreover, while we take the nutrient to be the sole chemoattractant, in some cases, autoinducers can also act as chemoattractants (Laganenka et al., 2016), which could also be described in our framework by e.g., introducing autoinducerdependent chemotaxis in the drift velocity in Equation 1. 3. Our model considers positive quorum sensing control in which planktonic cells transition to the biofilm state in a step-like fashion when the local autoinducer concentration exceeds a threshold be described by the relation D * /D +J * /J ∼ 1 , as shown by the black line; this relation combines the transition between the two states that occurs at both D * ∼ 1 and J * ∼ 1 . Away from this boundary, all simulations for the same D and J collapse to have the same biofilm fraction f , as shown by the points in (B) and examples (i)-(iii) and (v) in (C)-confirming the universality of our parameterization. Near the boundary, we observe some slight differences between simulations, as shown in (B) and examples (iv) and (vi) in (C). The values of the simulation parameters for the examples in (C) are given in Source Data file 1. The data in (A-C) correspond to a fixed value of the third dimensionless parameter S = 50 , which describes the case of biofilm cells that produce autoinducer rapidly; repeating these simulations for the opposite case of slow autoinducer production by biofilm cells ( S = 1/50 ) yields the state diagram shown in (D), but for 14,351 simulation runs; again, (E) shows the standard deviation of the corresponding values of f . As shown by (D-E), while the transition between the two states (black line) is unaffected by the change in S , the transition to complete biofilm formation is more gradual. Together, the three parameters D , J , and S provide a full description of the onset and extent of biofilm formation across vastly different conditions.
The online version of this article includes the following source data and figure supplement(s) for figure 4: Source data 1.
Figure supplement 1. In the case of 'protected' nutrient-independent autoinducer production, the transition from planktonic to biofilm states occurs at, J ∼ 1 independent of D .    value. That is, when planktonic cells encounter sufficiently concentrated autoinducer, the diffusivity and chemotactic coefficient transition in a step-like fashion from the constant values D 1 and χ 1 , respectively, to zero after the time duration τ , for simplicity. In real systems, the change in cellular motility may not be as temporally abrupt. Future work could address a more gradual loss of motility in our theoretical framework by, for example, considering a cellular diffusivity and chemotactic coefficient that gradually transition from their planktonic values to zero over a non-zero time scale. Given that the same cells would be transitioning from the motile planktonic to immotile biofilm state-but in this case with the introduction of a time-varying diffusivity and chemotactic coefficient-we expect that the long-time biofilm fraction f will be similar, and only the spatial profile of the biofilm population may be altered. Hence, we expect that our main findings summarized in Figure 4 will be unaffected by such a change. Indeed, performing the same representative simulation shown in Figure 1C, but with both motility parameters D 1 and χ 1 smoothly transitioning to zero in time, shows nearly identical results (Figure 1-figure  supplement 1)-confirming our expectation that the temporal nature of the arrest in motility does not appreciably influence our model results and conclusions. 4. While we take the transition to the biofilm state as being irreversible, this is often not the case (Barraud et al., 2006;Kaplan, 2010;Abdel-Aziz, 2014). Longer-time transitions back to the planktonic state could be described using additional terms similar to the last terms of Equations 1; 2, but with the opposite sign. Similar modifications could be made to describe other species of bacteria (e.g., Vibrio cholerae) that utilize the opposite case of negative quorum sensing control, in which biofilm cells instead transition to the planktonic state when the autoinducer accumulates above a threshold value (Hammer and Bassler, 2003;Bridges and Bassler, 2019). 5. Biofilms are often formed by multiple different microbial species, whereas our model describes biofilm formation by a single species, for simplicity. Nevertheless, we expect that our theoretical framework can be extended by following reasoning similar to that described in this paper, but with the introduction of additional equations and variables in the governing Equations 1-4 to describe the distinct cell and chemical types, as appropriate. For example, if the different species i consume and respond to distinct nutrients c i , and secrete and respond to distinct autoinducers a i , each species could be described in isolation using our same governing Equations 1-4, but now extended to incorporate the distinct variables c i , a i , b 1,i , and b 2,i . Then, directly following our approach, each species would be described by its own dimensionless parameters D i and J i , with D * /D i +J * /J i ∼ 1 again specifying the threshold for biofilm formation for each. We hypothesize that the composition of the final two-species biofilm community would then be given by the combination of each single-species biofilm. Alternatively, in the case that the different species consume and respond to the same nutrient c , and secrete and respond to the same autoinducer a , our Equations 1-4 could again be extended to consider the cellular parameters specific to each species i. In this approach, however, biofilm formation by each of the species cannot be described in isolation, because they are coupled through the nutrient and autoinducer dynamics. Instead, the calculations of the characteristic time scales τ d , τa , and τc would need to be extended, following our approach, to now reflect contributions from all the different species. We hypothesize that the overall multi-species community would then be described by one set of governing dimensionless parameters (D,J ) , and D * /D +J * /J ∼ 1 would again specify a universal biophysical threshold for the onset of biofilm formation for the overall community-but the composition of the final multi-species biofilm that results above this threshold may not be uniquely specified by (D,J ) . 6. Biofilm formation may be regulated by other, non-quorum sensing-based, processes not considered in our model. For example, the intracellular accumulation of secondary signaling molecules such as cyclic di-GMP can also regulate biofilm formation (Valentini and Filloux, 2016;Simm et al., 2004;Jenal et al., 2017;Römling et al., 2013;Hengge, 2009;Krasteva et al., 2010;Baraquet and Harwood, 2013;Trampari et al., 2015;Davis et al., 2013;Boehm et al., 2010;Russell et al., 2013). In some cases, this process may be controlled by quorum sensing (Waters et al., 2008) and thus could be described by our model, while in others, it is controlled by other cues such as e.g., contact with surfaces, which would need to additionally be incorporated into our theoretical framework. 7. Finally, we note that our model is deterministic-describing the cellular processes of motility, nutrient consumption, proliferation, and autoinducer production, availability, and sensing using the parameters {D 1 , χ 1 , c − , c+, κ 1 , κ 2 , c char , γ 1 , γ 2 , k 1 , k 2 , α, τ , a * } , respectively. Each of these is taken to be single-valued in each of our simulations. However, these parameters can have a distribution of values arising from e.g., inherent cell-to-cell variability. Because these values define the governing D and J that specify the threshold for biofilm formation in our model, we expect that variability in the parameter values would broaden the planktonic-to-biofilm transition predicted by our model. That is, we expect the transition specified by the black curve in Figure 4A to be smeared out, similar to what is seen in Figure 4D, though due to a fundamentally different reason-with biofilm formation arising in some cases at lower (D,J ) than predicted by the black curve. Indeed, similar behavior was recently observed in a distinct model of biofilm formation on flat surfaces (Sinclair et al., 2022). Exploring the influence of such variations by using a more probabilistic approach in our theoretical framework will thus be a useful direction for future research.

Materials and methods
To numerically solve the continuum model described by Equations 1-4, we follow the experimentally validated approach used in our previous work (Bhattacharjee et al., 2021;Amchin et al., 2022). Specifically, we use an Adams-Bashforth-Moulton predictor-corrector method in which the order of the predictor and corrector are 3 and 2, respectively. Because the predictor-corrector method requires past time points to inform future steps, the starting time points must be found with another method; we choose the Shanks starter of order 6 as described previously (Rodabaugh and Wesson, 1965;Shanks, 1966). For the first and second derivatives in space, we use finite difference equations with central difference forms in rectilinear coordinates. The temporal and spatial resolution of the simulations are δt = 0.1 s and δx = 20 µm , respectively; furthermore, we constrain our analysis to simulations for which the peak of the overall bacteria population moves slower than δx/δt . Repeating representative simulations with different spatial and temporal resolution indicates that even finer discretization does not appreciably alter the results (Figure 4-figure supplement 2). Thus, our choice of discretization is sufficiently finely-resolved such that the results in the numerical simulations are not appreciably influenced by discretization. Furthermore, performing the same representative simulation shown in Figure 1C, but with the shape of the initial inoculum changed from a Gaussian profile to a step function with the same maximum cellular concentration and width, shows nearly identical results (Figure 1-figure supplement 2)-suggesting that our results are robust to variations in this initial condition chosen. Further probing the mathematical structure of our biophysical model to examine additional influences of initial conditions and explore the possibility of oscillatory solutions, closed orbits, or singularities would be a fascinating direction for future work.
To connect the simulations to our previous experiments (Bhattacharjee et al., 2021), we choose a total extent of 1.75 × 10 4 µm for the size of the entire simulated system, with no-flux conditions for the field variables b 1 , b 2 , c , and a applied to both boundaries at x = 0 and 1.75 × 10 4 µm . As in the experiments, we initialize each simulation with a starting inoculum of planktonic cells with a Gaussian profile defined by the maximum concentration b 1,0 at x = 0 and a full width at half maximum of 100 µm . Nutrient is initially uniform at a fixed concentration c 0 , and the autoinducer and biofilm concentrations are initially zero, throughout. Furthermore, following previous work Dell'Arciprete et al., 2018;Volfson et al., 2008;Farrell et al., 2013;Klapper and Dockery, 2002;Head, 2013), we also incorporate jammed growth expansion of the population in which growing cells push outward on their neighbors when the total concentration of bacteria is large enough. In particular, whenever the total concentration of bacteria (planktonic and biofilm) exceeds the jamming limit of 0.95 cells µm −3 at a location x i , the excess cell concentration is removed from x i and added to the neighboring location, x i + δx , where δx represents the spatial resolution of the simulation, retaining the same ratio of planktonic to biofilm cells in the new location. We repeat this process for every location in the simulated space for each time step.
We run each simulation for a total simulated duration of t sim = 20 h . At this final time, we use the simulation data to directly compute f ≡´b 2 dx b2dx+´b1dx , the total fraction of the population in the biofilm state. We also compute the values of the dimensionless parameters D , J , and S using the equations presented in the main text. We note that the autoinducer production time τa (Equation 6) is only finite for η ≡ αa * /k 1 < 1 ; when η ≥ 1 , the rate of autoinducer loss exceeds that of autoinducer production, and thus the time required to reach the threshold for biofilm formation diverges. Because both D and J are defined as τ d /τa and τc/τa , respectively, for simulations with η ≥ 1 , we represent them on the state diagrams in Figure 4 and Figure 4-figure supplement 1 at (D, J ) = (10 −2 , 10 −3 ) , the smallest values shown on the diagrams. All of these simulations have f = 0 , as expected. Furthermore, to ensure that t sim is sufficiently long, we (i) only perform simulations with τa and τ a,2 smaller than t sim , and (ii) do not include simulations with f = 0 but τc = τ sim , for which sufficient time has not elapsed for planktonic cells to chemotactically disperse.

Additional files
Supplementary files • Transparent reporting form • Source data 1. Data for Figure 4C: Excel book summarizing the inputs used for each of the ten simulations shown in Figure 4C (i-vi). Different sheets within the book correspond to the individual bar graphs-Sheet "Bar Graph (i)" corresponds to all ten runs shown in (i), Sheet "Bar Graph (ii)" corresponds to all ten runs shown in (ii), and this follows for the remaining sheets. The first column identifies the input, its variable, and its units. The following columns correspond to each of the ten runs shown in the bar graph.

Data availability
All data generated or analyzed during this study are included in the manuscript and supporting file; source data files have been provided for all figures.
Planktonic : X 2 /D 1 , X 2 /chi 1 , X 2 /Dc, X 2 /Daγ −1 1 , γ −1 2 , τ , co b1, 0 k1 , co b1, 0 k2 , a * b1, 0 k1 , a * b1, 0 k2 , ⋋ −1 , (αβ 1,0 ) −1 (S4) where g(c) ≡c c+c char . Given that the characteristic autoinducer concentration a * arises in the argument of the Heaviside step function in Equations S1 and S2, we choose A = a * . Moreover, given that the planktonic cells have a characteristic concentration b 1,0 defined by the initial inoculum, we choose B 1 = b 1,0 . The fraction of the population in the biofilm state is defined as ; thus, to ensure that f = f for simplicity, we also choose B 2 = B 1 = b 1,0 . Finally, given that the nutrient has a characteristic concentration c 0 defined by the initial saturation, we choose C = c 0 . With these choices of characteristic quantities, multiple length and time scales emerge as possible choices for X and T , respectively: Time scale: X 2 /D 1 , X 2 /chi 1 , X 2 /Dc, X 2 /Da, γ −1 1 , γ −1 2 , τ , co b1, 0 k1 , co b1, 0 k2 , a * b1, 0 k1 , a * b1, 0 k2 , ⋋ −1 , (αβ 1,0 ) −1 Each such choice will lead to the emergence of many different dimensionless groups characterizing this problem. Nevertheless, all these different groupings are accounted for in the dimensionless parameters D , J , and S described in the main text, with the exception of quantities involving the nutrient diffusivity Dc , planktonic-to-biofilm transition rate τ −1 , and the natural autoinducer degradation rate λ , which have corresponding time scales that are much smaller than the other time scales of the systems considered here and are neglected from our analysis for simplicity.