Growth substrate availability and cell lineage segregation

We began with simulations in which the environment surrounding cell groups was altered by increasing or decreasing growth substrate concentration. These in silico experiments were initiated with equal numbers of randomly distributed red and blue cells, which did not differ in any way other than their color. The two neutral color markers were used to judge whether cell lineages remain randomly mixed or become spatially segregated as cell groups expand. Environmental substrate availability was decreased from saturating to sparse across multiple simulations, and we observed three different regimes in cell group structure:

1. Well mixed with smooth front.

When growth substrate was supplied to cell groups at saturating concentration, the red and blue cell lines appeared to remain well mixed relative to their random initial distributions. The advancing fronts of cell groups were smooth (Figure 1A,D).

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Figure 1. Dynamic simulations show that cell lineages segregate in a manner dependent on growth substrate availability.

Simulations began with a 1∶1 mixture of red and blue cells, where cell color served a neutral marker for lineage segregation. As bulk substrate concentration was decreased, we observed an increased propensity for cell lineages to segregate in space. This pattern held true under (A–C) surface growth and (D–F) radial growth conditions.

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3. Segregation with irregular front.

When substrate availability was sparse, we noted another qualitative shift in cell group structure. Red and blue cell lineages separated into adjacent sectors, just as described above. Additionally, the advancing fronts of cell groups became sensitive to small irregularities, which grew into tower clusters separated by open space (Figure 1C,F). Akin to the sector structures described above, each cell tower consisted of only one color, and thus appeared to contain the descendents of a single ancestral cell.

Further exploration with the simulation framework suggested that these three structure regimes represent qualitatively different regions within a continuum of possibilities. When we altered substrate availability by small increments over a sufficiently large range, we observed cell group structures that were intermediate between those shown in Figure 1. For simplicity and clarity in the remainder of the paper, we will focus only on the three distinct patterns of cell group spatial structure described above. Before proceeding, we also ruled out the possibility that our results were an artifact of simulating cell groups in two-dimensional space by repeating our simulations in three dimensions, which yielded qualitatively identical results (Supporting Information, Figure S1). All subsequent simulations were performed in two dimensions using the surface growth condition (as in Figure 1A–C).

We quantified lineage segregation in cell groups by performing replicate simulations under the three substrate availability conditions shown in Figure 1. At every time step of each simulation, we identified every actively growing cell and, within a 10 cell-length radius, measured the local frequency of other actively growing cells of the same color (Methods). The resulting segregation index directly measures the spatial assortment of cell lineages and ranges from 0 to 1, where 1 denotes complete lineage segregation on a spatial scale of 10 cell lengths. Fifty replicate simulations of each substrate availability condition were performed, and the average segregation index from each series was visualized as a function of cell group size (Figure 2). The results quantitatively confirm our observation that decreasing growth substrate availability leads to stronger lineage segregation in cell groups.

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Figure 2. Lineage segregation in growing cell groups, visualized as a function of increasing cell group size.

We ran 50 simulations under each of the three substrate availability conditions shown in Figure 1. Each simulation was initiated with 10% blue cells and 90% red cells, and we calculated the segregation index relative to the blue cell line (Methods). Dark lines are means; shaded regions are running 95% confidence intervals.

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A general model for lineage segregation

Our next goal was to describe why environmental substrate concentration affects lineage assortment in expanding cell groups. Under limited growth substrate availability, the majority of cell growth and division occurs along a group's advancing front in an active layer whose depth depends on substrate penetration (Figure 3). Previous work has hinted that active layer depth is a critical factor influencing cell group surface structure [39],[40], and we therefore hypothesized that it is not substrate concentration in particular, but more generally the depth of a cell group's active layer that controls cell lineage segregation. Because segregation increased as growth substrate supply decreased in our preliminary simulations, we predicted that thinner active layers would lead to stronger lineage segregation in expanding cell groups.

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Figure 3. The active layer of a cell group, illustrated for the surface growth condition.

Cells are colored according to their growth rate: green cells are growing and make up the cell group's active layer. Black cells have become inactive due to lack of available growth substrate. The left-hand panel illustrates the vertical profile of growth substrate concentration along the dashed blue line.

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Active layer depth is not solely a function of bulk growth substrate concentration. For example, higher substrate diffusivity increases active layer depth by allowing substrate to enter further into the cell group before being depleted. Faster cell growth rates, on the other hand, decrease active layer depth by raising the rate of substrate consumption at the cell group's outer surface. If we are correct that active layer depth is the underlying determinant of lineage segregation, all of the physical and biological factors that control active layer depth should also influence lineage segregation in cell groups.

Using an analytical technique from chemical engineering (Methods), we combined the factors that influence active layer depth into a dimensionless number, δ, which has the following form for our system:(1)

Here, G_bulk is the bulk liquid concentration of growth substrate, D_G is the growth substrate diffusion coefficient, Y is the yield with which cells convert substrate to biomass, μ_max is the maximum specific cell growth rate, ρ is the cell biomass density, and h is the height of the diffusion boundary layer (Figure 3). The smaller the value of δ, the thinner the cell group's active layer.

We performed three new sets of simulations to test the hypothesis that active layer depth controls cell lineage segregation. Within each set, we varied active layer depth (δ) by altering only one parameter from Equation 1: maximum cell growth rate (μ_max), bulk growth substrate concentration (G_bulk), or growth substrate diffusivity (D_G). At the end of each simulation, we calculated the segregation index. Our hypothesis makes two key predictions: 1) cell lineage segregation should be inversely related to δ, a proxy for active layer depth. 2) The relationship between cell lineage segregation and δ should be independent of which parameter from Equation 1 is altered.

The results are shown in Figure 4 and support both predictions. Lineage segregation within cell groups declines with increasing δ, regardless of how δ is altered. Using the dimensionless number δ renders our results independent of the exact values of G_bulk, D_G, Y, μ_max, ρ, and h used to run simulations. It is the relative magnitudes of these parameters in combination that ultimately matter.

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Figure 4. Lineage segregation within cell groups is inversely related to active layer depth.

The factors influencing cell group active layer depth were combined into a single dimensionless number, δ. This number was varied across 3 sets of simulations by independently altering μ_max (maximum cell growth rate, red), G_bulk (bulk substrate availability, black), or D_G (substrate diffusivity, blue). Cell groups were grown to 100 µm maximum height, and then the segregation index was calculated (filled circles are means, and bars denote 95% confidence intervals). The horizontal dotted line represents the final segregation index of simulations in which δ was infinitely large, allowing all cells to grow at the maximum rate at all times. The simulations show that cell lineage segregation is inversely related to active layer depth, independently of how active layer depth is altered.

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How does active layer depth influence cell lineage segregation? When growth substrate penetrates through most of a cell group before being depleted, all cells grow and divide, pushing each other into a homogeneous mixture. As active layer depth decreases below the total thickness of a cell group, however, cells that happen to fall below a critical distance from the group's front can no longer contribute to population expansion. Decreasing active layer depth thus reduces the cell group's effective population size, rendering it more susceptible to genetic drift along its advancing front. Because the cells are constrained in space, reductions in genetic diversity along the group's leading edge lead to localized clusters of individuals that all descend from a common progenitor [30]. This phenomenon – often referred to as sectoring or gene surfing [28]–[30] – has been observed in agar colonies of Paenibacillus dendritiformis [26], Escherichia coli and Saccharomyces cerevisiae [27].

Reducing active layer depth even further yields an additional qualitative shift in cell group structure: the expanding population becomes sensitive to small irregularities along its leading edge. Cells in the peaks of surface irregularities retain access to substrate and grow into tower projections, while cells in the troughs of surface irregularities lose access to substrate and cease growing. This process is related to viscous fingering at the interface of two fluids [39],[41], and it is known to generate rough surface structure along the leading edges of growing biofilms, bacterial colonies on agar [34],[35],[40], and moving fronts in general [36]. From a biological perspective, our analysis predicts that such surface roughness is accompanied by abrupt genetic lineage segregation along the front of an expanding population.

Bridging cell lineage segregation and social evolution

The spatial assortment of cell lineages is potentially critical for traits that affect the reproduction of other individuals in the population. It is increasingly recognized that cells express many such social phenotypes [4],[12], which are often involved in nutrient acquisition and pathogenesis [42]–[45]. A common example is the secretion of extracellular enzymes or nutrient-chelating molecules. Cells that synthesize these substances must forgo a fraction of their reproductive capacity [17]–[19], but if enough cells participate, all can gain a net benefit (to the detriment of their host, in the case of pathogens).

In many cases the evolution of simple cooperative phenotypes depends on three factors: 1) c, the cost incurred by cooperative individuals 2) b, the benefit gained by the receivers of cooperative behavior, and 3) r, the correlation between genotypes of givers and receivers of cooperation. Cooperation is predicted to evolve when rb>c, a condition known as Hamilton's Rule [9]. The cost and benefit factors are measured in terms of reproductive fitness. When cooperation is genetically determined, relatedness may be thought of as the degree to which the benefits of cooperation are preferentially distributed to other cooperative individuals.

The segregation index depicted in Figures 2 and 4 is equivalent to a form of the relatedness coefficient in Hamilton's Rule: both measure the degree of biased interaction among relatives (here, physical proximity amounts to biased interaction). As such, our segregation index forms a bridge between social evolution theory and the emergence of lineage segregation in cell groups, allowing us to extend our prediction from the previous section. Because thin active layer conditions generate lineage segregation, we predict that decreasing active layer depth will promote interaction among clonemates (increasing r in Hamilton's Rule) and favor the evolution of cooperation [9],[12],[23]. Positive spatial assortment of related cells does not guarantee that cooperation will be favored, however, as the same segregation that allows cooperators to preferentially interact also increases the strength of competition between them [24].

We tested our prediction by implementing a cooperative phenotype in our model framework and competing cooperative cells against exploitative cells that devote all resources to growth. Cooperative individuals secrete a diffusible compound that benefits all other cells in the local area (we will refer to the compound as an extracellular enzyme). Local availability of the secreted enzyme increases cell growth rate by a fold factor B, but only after the enzyme's concentration passes a threshold value, τ. Cooperative cells constitutively secrete the enzyme and incur a fold decrease in growth rate of C x R_E, where C is a cost scaling factor and R_E is the enzyme production rate. In our main analysis, B = 3, C = 0.3, and R_E ranges from 0 to 2. We derived these values from experimental data on elastase, a secreted enzyme and virulence factor of the bacterial pathogen Pseudomonas aeruginosa [19],[46].

Lineage segregation favors cooperation in cell groups

We asked whether a cooperative cell line, which pays a cost to produce a diffusible, publicly beneficial compound, could outcompete an exploitative cell line that invests all of its resources into growth. Each competition simulation began with a randomly distributed 1∶1 mixed monolayer of the two cell types, and cell groups were grown to a maximum height of 100 µm. We then calculated the evolutionary fitness of the cooperative cell line, relative to that of the exploitative cell line (Methods). This competition pairing was repeated over a range of extracellular enzyme production rates on the part of cooperative cells. The higher the enzyme production rate, the more rapidly cells accrue its benefit, but the larger the cost suffered by cooperative cells. Finally, all competition pairings were repeated across three active layer depth conditions (δ = 10, 2, 1), representing the three cell group structure regimes described in Figure 1.

Figure 5 summarizes the results of our competition simulations. When active layers are thick (δ = 10), leading to well mixed cell lineages, the extracellular enzyme is homogenously distributed through cell groups. The non-cooperative cell line is therefore able to consistently exploit and outcompete the cooperative cell line (Figure 5A). This result is consistent with numerous observations that exploitative mutants outcompete enzyme-secreting bacteria when they are inoculated together in liquid culture, in which cell lineages largely remain mixed [17]–[20].

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Figure 5. Cooperation is favored as cell group active layer depth decreases and lineage segregation increases.

We examined competition between enzyme-secreting cells (cooperative, labeled blue) and non-secreting cells (exploitative, labeled red) under three different active layer conditions: δ = 10 (A, well mixed lineages), δ = 2 (B, lineage sectoring), and δ = 1 (C, lineage tower formation). Each empty black circle denotes the relative fitness of the cooperative cell type at the end of a single simulation (40 replicates per column). Sample images (drawn from simulations indicated by black arrows) are shown in the corner of each plot, along with concentration [g/L] profiles of the extracellular enzyme. (A) When cell lineages remain mixed, cooperative cells are always outgrown by exploitative cells. (B) When cell lineages segregate into sectors, there is a narrow range of enzyme production rates at which cooperative cells outcompete exploitative cells. (C) When lineages are strongly segregated into cell tower projections, there is a large range of enzyme production rates at which cooperative cells outcompete exploitative cells.

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When active layer depth is decreased (δ = 2), there is a narrow range of extracellular enzyme production rates at which cooperative cells outcompete exploitative cells (Figure 5B). The critical difference is that cooperative cells and exploitative cells no longer remain well mixed; rather, they segregate into clonal regions. As a result, the benefit of extracellular enzyme released by cooperative cells accrues asymmetrically to other cooperative cells. The range of enzyme production rates at which cooperative cells prevail is narrow, however, because the benefits of lineage segregation (increasing r in Hamilton's Rule) can be outweighed by the cost of higher extracellular enzyme production (increasing c in Hamilton's Rule).

Further decreasing active layer depth (δ = 1) leads to the growth of spatially isolated, clonal cell towers. Under these conditions, the benefits of a cooperative secreted enzyme are distributed even more asymmetrically to other cooperative cells. Consistent with our predictions, this allows cooperative cells to outcompete exploitative cells over a larger range of enzyme production rates (Figure 5C). We also noted the sizable variation between simulation runs when δ = 1, particularly if extracellular enzyme production rates were low (Figure 5C, enzyme production rate  = 0, 0.25, 0.5). This variation reflects a founder effect; it manifests most strongly when there is no or little difference between the competitive abilities of cooperative and exploitative cell lines, rendering the outcome of each simulation subject to chance events that determine which cells seed the few tower structures that emerge from an expanding cell group.

Our results show that thin active layer conditions allow cells expressing cooperative phenotypes to outcompete exploitative cells within a single cell group. To better account for the long-term evolution of a metapopulation comprising many cell groups, we performed an invasion analysis to determine whether a novel cooperative mutant can spread through a metapopulation otherwise containing only exploitative cells (Supporting Information, Text S1). We also examined the reciprocal case to determine if a rare exploitative mutant can invade a metapopulation otherwise containing only cooperative cells [32],[33]. We found that cooperation can invade under a large swath of parameter space, but only under thin active layer conditions that promote lineage segregation can cooperative cells eliminate exploitative cell types on a metapopulation scale (Supporting Information, Figure S2).

The results of both our local competition and invasion analyses are robust to the cost/benefit ratio of cooperation, with one partial exception when cells invest very heavily into an expensive cooperative phenotype (Supporting Information, Figure S3).

Conclusion

Our study indicates that an order of magnitude change in nutrient availability, nutrient diffusivity, cell metabolic efficiency, cell growth rate, or biomass density can shift cell groups from a regime of lineage mixing to a regime of pronounced lineage segregation. The number δ defined in Equation 1 relates these parameters to the depth of a cell group's active layer, which governs how cell lineages become spatially assorted over time. Thick active layers promote lineage mixing, while decreasing active layer depth generates increasingly strong lineage segregation. Cell lineage segregation, in turn, favors the evolution of cooperative phenotypes.

Previous work performed with bacteria in liquid planktonic culture has concluded that cooperative cell phenotypes cannot be selectively favored within a single population also containing exploitative cells [17],[19],[20]. Our study shows that this conclusion will not always hold because cooperative cells can spontaneously segregate from exploitative cells when they are constrained in space. Our results also imply that, given realistic parameters for a cooperative cell phenotype, the benefits of preferential interaction between cooperators can outweigh the costs of increased competition between related cells that are clustered together in space [24].

Like all models, ours uses simplifying assumptions. We deliberately omit some physical processes, such as shear stress, that may be applied to cell groups in the real world [47]. Our simulations also do not consider active cell motility, which in reality could influence cell group structure and evolution. We have additionally assumed that cell phenotypes of interest, such as extracellular enzyme secretion, are expressed constitutively or not at all. In nature, the expression of many social phenotypes is adjusted in response to environmental cues [48]–[50]. Though these simplifications should be assessed theoretically and empirically, they were critical in allowing us to identify basic physical and biological parameters that control cell group structure and evolution.

In summary, our model suggests that clusters of genetically related cells can emerge quite easily in spatially constrained cell groups, even when cells possess no mechanism for actively gathering with clonemates. Lineage segregation allows cooperative cells to outcompete exploitative cells, and accordingly we predict that localized cooperation will evolve more readily in cell groups than suggested by models and experiments that only consider liquid environments.

Model Framework

We simulate cell groups using an individual-based model described in detail previously [31]. Simulation parameters are listed in Table S1 (Supporting Information). Cell growth is a function of the local microenvironment, namely the concentrations of solutes such as growth substrate (G) and extracellular enzyme (E) (Supporting Information, Table S2). The uptake of growth substrate by each cell is considered when calculating the spatial gradients of substrate concentration. We achieve this by solving a reaction-diffusion equation, where r is a growth rate expression:(2)

Following the common assumption that reaction-diffusion is much faster than cell growth and division [31], our simulations proceed according to the following steps:

• Cell growth and division
  1. 1) Every cellular agent grows according to local substrate concentration and (for competition simulations) extracellular enzyme availability. Agents that exceed a critical radius are divided into two new agents.
  2. 2) Agents that now overlap due to their growth and/or division in the previous step are moved so as to eliminate overlap throughout the cell group. This process causes the cell group's front to advance in space.
• Update solute concentration fields
  1. 3) Bulk concentrations of all solutes (growth substrate or extracellular enzyme) are held constant throughout the simulation. Thus, the bulk liquid (the region outside the boundary layer) acts as an infinite source, in the case of substrate, or a perfect sink, in the case of extracellular enzyme.
  2. 4) The new spatial concentration fields of all solutes are determined by solving Equation 2 (and an analogous equation for extracellular enzyme concentration) to steady state at each iteration.

Computation

The individual-based simulation framework was written in the Java programming language, and its related numerical methods are detailed elsewhere [31]. Briefly, they include the Euler method to grow cells at each iteration, a hard-sphere collision detection method to identify pushing events between neighboring agents, and the FAS multigrid to solve reaction-diffusion equations to steady state [51]. The 3D images in Figure S1 where rendered using POV-Ray. All other figures were prepared using Matlab (the Mathworks, Inc.). The computations in this paper were run on the Odyssey cluster supported by the Harvard University FAS Research Computing Group.

Calculation of the segregation index

To obtain the segregation index for a cell group at a single point in time, we first identify every actively growing cell. These M cells are indexed by A_i: A₁, A₂, …, A_M. To measure segregation with respect to a single focal cell A_i, we identify all other individuals within a distance of 10 cell lengths. The N cells in this neighborhood are indexed by a_j: a₁, a₂, …, a_N.

We define a genetic identity function, g(a_j):(3)and a metabolic activity function, m(a_j):(4)where [G] is the local concentration of growth substrate, and K_G is the half-saturation constant for cell growth rate.

Segregation with respect to a focal cell, s(A_i), is calculated as the mean product of the g and m functions for every cell in its neighborhood:(5)

Finally, we define the segregation index for the entire cell group as the mean value of s(A_i) across all metabolically active cells:(6)

Our segregation index measures the degree to which co-localized, metabolically active cells are clonally related to each other. The index is equal to a form of the relatedness coefficient from social evolution theory under the following assumptions: 1) A cell expressing the cooperative phenotype equally benefits all other individuals within a 10 cell-length radius; 2) Each cell within range of receiving cooperative benefits makes a contribution to mean relatedness proportional to its growth rate; 3) Cell groups are seeded randomly from a large population pool.

Derivation of the number δ

The dimensionless number, δ, is a proxy for the depth to which growth substrate penetrates into a cell group before being depleted by cell metabolic activity. δ is derived by non-dimensionalizing Equation 2. We normalize growth substrate concentration by its bulk liquid concentration, , and local biomass by cell biomass density, x = X/ρ. We then normalize the space coordinates by the height of the boundary layer, h. The steady state, dimensionless version of Equation 2 becomes:(7)

Note that the factor multiplying the Laplacian of , , is the square of δ as defined in the main text. δ is also the inverse of the Thiele modulus [52], a number commonly used in chemical engineering to quantify the activity of solid catalysts.

Calculation of evolutionary fitness

We calculate the competitive fitness of each cell line as the mean number of rounds of cell division per unit time that each achieves over the course of a simulation:(8)where N_S,t is the number of cells of strain S present within the cell group at time t. The relative fitness of a strain S₁ in local competition with another strain S₂ is defined as: .