Spatial segregation and cooperation in radially expanding microbial colonies under antibiotic stress

Antibiotic resistance in microbial communities reflects a combination of processes operating at different scales. In this work, we investigate the spatiotemporal dynamics of bacterial colonies comprised of drug-resistant and drug-sensitive cells undergoing range expansion under antibiotic stress. Using the opportunistic pathogen Enterococcus faecalis with plasmid-encoded β-lactamase, we track colony expansion dynamics and visualize spatial patterns in fluorescently labeled populations exposed to antibiotics. We find that the radial expansion rate of mixed communities is approximately constant over a wide range of drug concentrations and initial population compositions. Imaging of the final populations shows that resistance to ampicillin is cooperative, with sensitive cells surviving in the presence of resistant cells at otherwise lethal concentrations. The populations exhibit a diverse range of spatial segregation patterns that depend on drug concentration and initial conditions. Mathematical models indicate that the observed dynamics are consistent with global cooperation, despite the fact that β-lactamase remains cell-associated. Experiments confirm that resistant colonies provide a protective effect to sensitive cells on length scales multiple times the size of a single colony, and populations seeded with (on average) no more than a single resistant cell can produce mixed communities in the presence of the drug. While biophysical models of drug degradation suggest that individual resistant cells offer only short-range protection to neighboring cells, we show that long-range protection may arise from synergistic effects of multiple resistant cells, providing surprisingly large protection zones even at small population fractions.

dynamical regimes, though extrapolation of our data suggests a leader game resulting in stable coexistence between sensitive and resistant cells (though we cannot rule out marginal stability that would occur at the boundary). The analysis provides a convenient quantitative measure of cooperation (via entries of an e ective payo matrix) with a simple and intuitive interpretation. While a full elucidation of the game dynamics will require further study (most notably, time-dependent measures of composition across conditions), we anticipate this preliminary analysis will serve as a foundation for future investigations of cooperative resistance under di erent conditions. The idea, brie y, is to assume that populations of each cell type grow with a per capita rate (g i ) of the form, g R = Af r + B(1 f r ) where g R is the growth rate for resistant cells, g S the rate for sensitive cells, A, B, C , D are coe cients of an e ective payo matrix, and f r is the fraction of resistant cells.
Competition assays between sensitive and resistant cells are used to infer the parameters of the payo matrix, which fully determine the phase space of possible dynamical regimes: stability of purely resistant populations, stability of sensitive populations, bistability (where both types of homogeneous populations are possible, depending on initial conditions), and coexistence (71). More speci cally, the model reduces to a replicator-like equation for f r where G (f r ) is a gain function that depends on di erences in the payo matrix entries The steady state dynamics depend only on the sign of the di erences B D and C A, which correspond to the relative tness of a resistant invader and the relative tness of a sensitive invader, respectively.
Kaznatcheev et al show that the payo matrix can be estimated by quantifying the frequency dependence of sub-population growth rates at di erent (initial) population compositions (71), where the parameters correspond to the f r ! 0 and f r ! 1 intercepts of the single species data (as in Equation S1). While we do not have detailed time series data on growth rates for individual sensitive and resistant populations, we can apply a slightly coarser approach by estimating the gain function G (f r ) using initial and nal measurements of population composition (f r ) estimated from uorescence data. Speci cally, we calculate an e ective gain function s-similar to the selection coe cients often used to quantify pairwise competition experiments-as where T is the total length of the experiment (approximately 6 days). Equation S4 is just a two-point estimate of G (f r ) which assumes that the dynamics between time 0 and time T are determined by Equation S2 with G (f r ) = s a constant (which, however, will vary for each experiment that starts with a di erent initial value of f r ). By estimating s for each experiment (each starting from a di erent composition), we can estimate the e ective gain function, and the limits of that function (extrapolated to f R = 0 and f R = 1) provide estimates of C A and B D , respectively.

SUPPLEMENTAL TEXT II: BIOPHYSICAL MODEL OF PROTECTIVE RESISTANCE
To estimate the length scale for protective resistance, we rst consider a single, xed, spherical, enzyme-producing (resistant) cell of radius a placed at the origin. Our goal is to nd the spatial pro le of antibiotic concentration A(r ) a distance r from the origin, assuming that A(1) = A ext is the external concentration far away from the resistant cell and the drug follows the di usion equation, where D is the di usion constant of drug. The surface of the cell contains N bound copies of the enzyme, each of which degrades nearby antibiotic according to Michaelis Menten kinetics at rate where k 0 ⇡ k 0 cat /K 0 m and the superscripts indicate that these are local rates valid when antibiotic is present at the cell surface r = a (i.e. these rates do not include the e ects of di usion of antibiotic to the enzyme). The last step in Equation S6 assumes A(a) ⌧ K 0 m , though it is straightforward to incorporate the full nonlinear rate equation as needed.
In nitely rapid degradation at the cell surface. When the number of enzymes (N ) is large-so that the entire surface of the cell is covered with enzyme-and the reaction rate (k 0 ) is in nitely fast, the concentration of drug is vanishingly small at the cell surface. This scenario corresponds to the classic Smoluchowski problem (75) from chemical kinetics. The steady state solution of Equation S5 with A(1) = A ext and A(a) = 0 is given by For a given pro le A(r ), the size R p of the "protection zone"-the region surrounding the resistant cell for which A is less than some critical value A c (i.e. the MIC)-is given by For very large drug concentrations A ext A c , the protection zone R p includes only the resistant cell (R p ! a). However, as A ext is reduced to A c , the protection zone grows rapidly, potentially encompassing many cell lengths. For example, if A ext = 1.1A c (i.e. the concentration exceeds the MIC by only 10 percent), the protection zone R p = 11a.
Aext a factor containing the dependence on A(a), which itself can be related to the local reaction rate constant k 0 by assuming a partially re ecting boundary condition at the cell surface (75) leading to and In the limit k 0 4⇡aD , we have ! 1 and the pro le reduces to the di usion-limited case (Equation S7). On the other hand, when the reaction rates are nite and of the same order as the di usion e ects, the protection zone R p will be reduced by a factor R p ! R p . Unfortunately, estimates of k 0 (the reaction rate given that enzyme and substrate are localized together) are not typically available. The fact that many -lactamases show near di usion-limited rates means that k 0 4⇡r E D , with r E the reaction radius of a single enzyme and D the di usion constant of drug in solution.
However, the radius of a single cell is at least 2-3 orders of magnitude larger (a r E ), so the di usion-limited regime for a single enzyme is not necessarily di usion-limited when the enzymes sit on the surface of a cell. More speci cally, covering a cell of radius a with a di usion-limited enzyme (i.e. su ciently large k 0 to yield A(r E ) = 0 at the reaction radius r E for a single enzyme) does not necessarily mean that the concentration A(a) at the surface of that cell will be zero. On the other hand, di usion of drug in a bacterial colony may be signi cantly slower than di usion in solution, which would favor reduced concentrations at the cell surface.

Synergistic protection in cell ensembles with multiple resistant cells. The up-
per bound for the length scale of the protection zone surrounding a single resistant cell is set by a, the radius of the cell. For drug concentrations on the order of 2x the MIC-similar to those used in our experiments-one would only expect protection to extend for a few cell lengths, but experiments suggest protection extends considerably farther in cellular communities. One explanation could be that the collective e ects of multiple resistant cells are synergistic, producing a protection zone that re ects more than a simple accumulation of single cell e ects.
To investigate this possibility, we consider a large spherical ensemble of radius R a that contains many resistant cells. For simplicity, we assume that each resistant cell expresses a su cient quantity of enzyme to completely degrade any drug that reaches the cell surface. In this case, the concentration of drug approaches zero at the surface of the ensemble, and the problem is again equivalent to the Smoluchowski problem, but the length scale is now set by R (S13) A large ensemble of resistant cells is therefore expected to exhibit protection zones with length scales set by the size of the ensemble. As before, this serves as an upper bound, and nite reaction rates / rapid di usion may decrease this range.
Surprisingly, however, these large-scale protection zones can be achieved even when the ensemble contains only a small fraction of resistant cells. Speci cally, let us assume the ensemble is made up of both sensitive and resistant cells. Furthermore, assume a total of N r resistant cells, each represented by a circle of radius a, are uniformly distributed over the surface of the sphere. The total fraction of the ensemble surface covered by resistant cells is given by f ⇡ Nr a 2 4R 2 . How does the size of the protection zone depend on this fraction (or equivalently, on N r )?
To solve the problem, we exploit the analogy with electrodynamics used in (76) in the context of di usion-limited cellular signaling. Speci cally, the steady state di usion equation D + 2 A = 0 has the same form as the equation for the electrostatic potential in a charge-free space, + 2 = 0. As a result, the di usive current density is analogous to the electric eld vector, and the total di usive current J entering a closed surface is analogous to the total charge Q on a surface. The condition A = 0 at the spherical surface is equivalent, in the electrodynamics analog, to the surface being isopotential.
The analogy allows us to equate the total di usive ux J through any closed surface to the electrical capacitance C (in cgs units) of an isolated conductor of the same shape (76) according to This analogy is particularly convenient since the capacitance, C , has been calculated for conductors of many di erent sizes and shapes. In the case of our spherical ensemble of cells, the capacitance of interest is that of an insulating sphere of radius R covered uniformly with N r conducting disks of radius a, all connected with in nitesimal wires to form a single conductor. The capacitance C of this object is derived in (76) as an expression that is valid when the distance between neighboring disks (in this case, resistant cells) is large compared to the radius of a single disk (cell). The total di usive ux for the cell ensemble is therefore given by where J max = 4⇡R D A ext is the ux through a spherical ensemble containing only resistant cells. Assuming the ux is uniform over the sphere (no angular dependence), the drug pro le outside of the ensemble (r R ) is Similarly, the size of the protection zone becomes When the number of resistant cells is small (N r ⌧ R /a), the length scale of the protection zone scales as ↵ ⇠ N r a; that is, the net e ect of N r resistant cells is approximately equal to N r times the e ect of each cell. In the other extreme, as the number of resistant cells is large, R p approaches that for a purely resistant ensemble (↵ ! R ).
But interestingly, the e ects of multiple resistant cells saturate raplidly (as in the case for cell receptors on the surface of a single cell (76)). As an example, for a spherical ensemble of R = 1000 microns, the protection zone radius R p reaches 90 percent of its maximum value with only N r ⇡ 30000, corresponding to f ⇡ Nr a 2 4R 2 ⇡ 0.007-that is, when less than 1 percent of the surface is covered with resistant cells.

SUPPLEMENTAL FIGURES
FIG S1 Expansion velocity (left) and appearance time (right) of mixed colonies at di erent concentrations of ampicillin (AMP). Each row corresponds to a particular inoculum density. We note that particularly at low inoculum density, it becomes di cult to track colony expansion because the colonies do not have well de ned boundaries on the scanner images. At these densities, we cannot rule out weak concentration dependence of expansion velocity.

FIG S2
Spatial patterns for example colonies starting from di erent initial densities (columns) and di erent initial compositions (rows). For each condition, colonies were grown in the presence or absence of super-MIC AMP (1 mug/mL).

FIG S3
The estimated fraction is insensitive to the thresholding process a. Raw images of green and red channel acquired by imaging the mixed colonies (left) and binary image for di erent threshold (right). Middle images show the binarization using Otsu's method and other images show ± 25% ,± 12.5% change in the threshold. b. The fraction of sensitive cells estimated from images with di erent threshold. c and d. Similar to panel a and b, but for colonies with narrow monoallelic regions, which makes the binarization di cult and thereby prone to over/underestimation of cells.