To study how the collective structure of cell groups arises from the activity of many individual cells, we used a computational model that employs mechanistic descriptions of solute diffusion and cell growth 
. Our framework is derived from the latest generation of agent-based models that have been developed over the last decade using biochemical engineering principles (Methods
, Supporting Information, Table S1
). The model's underlying assumptions are described and justified in detail elsewhere 
, and empirical tests have demonstrated the framework's ability to make accurate predictions for real biological systems 
Briefly, each cell is implemented as a circular agent in explicit two-dimensional space, and each simulation is set on one of two possible conditions. The first consists of cells growing on a flat surface with growth substrate (nutrients) diffusing from above. The second condition represents a cell cluster immersed in a resource pool, such that substrate diffuses into the cluster from all directions. The transport of all solutes occurs exclusively through diffusion. Each cell grows according to a Michaelis-Menten function of substrate concentration in its local environment and divides once it reaches a maximum radius (Methods
, Supporting Information, Table S2
). Cells move passively due to the forces exerted between neighboring individuals as they grow and divide.
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 ().
Dynamic simulations show that cell lineages segregate in a manner dependent on growth substrate availability.
2. Segregation with smooth front
When substrate availability was decreased to a moderate concentration, the surfaces of cell groups remained smooth, but their internal structures were substantially altered. Cell lineages segregated as group fronts advanced, creating adjacent red and blue cell sectors (). This segregation occurred because many cell lineages were cut off from advancing fronts and ceased growing, while the few remaining lineages proliferated into adjoining zones containing only one cell type.
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 (). 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 . 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 ).
We quantified lineage segregation in cell groups by performing replicate simulations under the three substrate availability conditions shown in . 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 (). The results quantitatively confirm our observation that decreasing growth substrate availability leads to stronger lineage segregation in cell groups.
Lineage segregation in growing cell groups, visualized as a function of increasing cell group size.
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 (). Previous work has hinted that active layer depth is a critical factor influencing cell group surface structure 
, 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.
The active layer of a cell group, illustrated for the surface growth condition.
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:
Here, Gbulk is the bulk liquid concentration of growth substrate, DG is the growth substrate diffusion coefficient, Y is the yield with which cells convert substrate to biomass, μmax is the maximum specific cell growth rate, ρ is the cell biomass density, and h is the height of the diffusion boundary layer (). 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 (μmax), bulk growth substrate concentration (Gbulk), or growth substrate diffusivity (DG). 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 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 Gbulk, DG, Y, μmax, ρ, and h used to run simulations. It is the relative magnitudes of these parameters in combination that ultimately matter.
Lineage segregation within cell groups is inversely related to active layer depth.
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 
. This phenomenon – often referred to as sectoring or gene surfing 
– has been observed in agar colonies of Paenibacillus dendritiformis 
, Escherichia coli
and Saccharomyces cerevisiae 
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 
, and it is known to generate rough surface structure along the leading edges of growing biofilms, bacterial colonies on agar 
, and moving fronts in general 
. 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 
, which are often involved in nutrient acquisition and pathogenesis 
. 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 
, 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
, a condition known as Hamilton's Rule 
. 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 and 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 
. 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 
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
, where C
is a cost scaling factor and RE
is the enzyme production rate. In our main analysis, B
0.3, and RE
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 
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 .
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 (). 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 
Cooperation is favored as cell group active layer depth decreases and lineage segregation increases.
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 (). 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 (). We also noted the sizable variation between simulation runs when δ
1, particularly if extracellular enzyme production rates were low (, 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 
. 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
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 
. 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 
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 
. 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 
. 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.