Abstract
Reproductive division of labor (e.g. germsoma specialization) is a hallmark of the evolution of multicellularity, signifying the emergence of a new type of individual and facilitating the evolution of increased organismal complexity. A large body of work from evolutionary biology, economics, and ecology has shown that specialization is beneficial when further division of labor produces an accelerating increase in absolute productivity (i.e. productivity is a convex function of specialization). Here we show that reproductive specialization is qualitatively different from classical models of resource sharing, and can evolve even when the benefits of specialization are saturating (i.e. productivity is a concave function of specialization). Through analytical theory and evolutionary individualbased simulations, we demonstrate that reproductive specialization is strongly favored in sparse networks of cellular interactions that reflect the morphology of early, simple multicellular organisms, highlighting the importance of restricted social interactions in the evolution of reproductive specialization.
Introduction
The evolution of multicellularity set the stage for unprecedented increases in organismal complexity (Szathmáry and Smith, 1995; Knoll, 2011). A key factor in the remarkable success of multicellular strategies is the ability to take advantage of withinorganism specialization through cellular differentiation (Queller and Strassmann, 2009; Brunet and King, 2017; CavalierSmith, 2017). Reproductive specialization, which includes both the creation of a specialized germ line during ontogeny (as in animals and volvocine green algae) and functional differentiation into reproductive and nonreproductive tissues (as in plants, green and red macroalgae, and fungi), may be especially important (Cooper and West, 2018; Michod et al., 2006; Ispolatov et al., 2012; Solari et al., 2013; Michod, 2007; West et al., 2015). Reproductive specialization is an unambiguous indication that biological individuality rests firmly at the level of the multicellular organism (Michod, 1999; Folse and Roughgarden, 2010), and is thought to play an important role in spurring the evolution of further complexity by inhibiting withinorganism (celllevel) evolution (Buss, 1988) and limiting reversion to unicellularity (Libby and Ratcliff, 2014). Despite the central importance of reproductive specialization, its origin and further evolution during the transition to multicellularity remain poorly understood (McShea, 2000).
The origin of specialization has long been of interest to evolutionary biologists, ecologists, and economists. A large body of theory from these fields shows that specialization pays off only when it increases total productivity, compared to the case where each individual simply produces what they need (Szathmáry and Smith, 1995; Smith and Szathmáry, 1997; Goldsby et al., 2012; Corning and Szathmáry, 2015; Hidalgo and Hausmann, 2009; Boza et al., 2014; Taborsky et al., 2016; Page et al., 2006; Rueffler et al., 2012; Szekely et al., 2013; Findlay, 2008; Amado et al., 2018). Certain types of trading arrangements maximize the benefits of specialization; highly reciprocal interactions, which facilitate exchange between complementary specialists, amplify cooperation (Allen et al., 2017; Pavlogiannis et al., 2018). Still, previous work finds that even when groups grow in an ideal spatial arrangement, increased specialization and trade is only favored by natural selection when productivity increases as an accelerating function of the degree of specialization (i.e., productivity is a convex, or superlinear, function of the degree of specialization). Conversely, saturating functional returns (i.e. productivity is a concave, or sublinear, function of the degree of specialization) should inhibit the evolution of specialization (Cooper and West, 2018; Michod et al., 2006; Ispolatov et al., 2012; Solari et al., 2013; Michod, 2007; West et al., 2015).
Reproductive specialization differs from classical models of trade in several key respects. Trade between germ (reproductive) and somatic (nonreproductive) cells is intrinsically asymmetric, because the cooperative action, multicellular replication, is not a product that is shared evenly. Selection acts primarily on the fitness of the multicellular group as a whole (Folse and Roughgarden, 2010). As a result, optimal specialization can result in behaviors that reduce the shortterm fitness of some cells within the multicellular group (Michod et al., 2006; Michod, 2007), often manifest as reproductive altruism.
Understanding the evolution of cellcell trade, a classic form of social evolution (Kirk, 2005), requires understanding the extent of betweencell interactions. Network theory has proven to be an exceptionally powerful and versatile technique for analyzing social dynamics (Wey et al., 2008; Lieberman et al., 2005), and indeed, is uniquely well suited to understanding the evolution of early multicellular organisms. When cells adhere through permanent bonds, sparse networklike bodies (i.e. filaments and trees) often result (Amado et al., 2018). This mode of group formation is not only common today among simple multicellular organisms (Umen, 2014; Claessen et al., 2014), but is the dominant mode of group formation in the lineages evolving complex multicellularity (i.e. plants, red algae, brown algae, and fungi, but not animals).
In this paper, we develop and investigate a model for how the network topology of early multicellular organisms affects the evolution of reproductive specialization. We find that under a broad class of sparse networks, complete functional specialization can be adaptive even when returns from dividing labor are saturating (i.e. concave/sub linear). Sparse networks impose constraints on who can share with whom, which counterintuitively increases the benefit of specialization (McShea, 2000). By dividing labor, multicellular groups can capitalize on high betweencell variance in behavior, ultimately increasing grouplevel reproduction. Further, we consider group morphologies that naturally arise from simple biophysical mechanisms and show that these morphologies strongly promote reproductive specialization. Our results show that reproductive specialization can evolve under a far broader set of conditions than previously thought, lowering a key barrier to major evolutionary transitions.
Model
Reproductive specialization can be modeled as the separation of two key fitness parameters, those related to either viability or fecundity, into separate cells within the multicellular organism (Michod, 2006; Folse and Roughgarden, 2010). The dichotomy of viability versus fecundity was originally used by Michod, 2006 to partition components of cellular fitness into actions that contribute to keeping a cell alive (viability), and actions that directly contribute to reproduction (fecundity). Multicellular organisms often have evolved to divide labor along these two lines (i.e. reproduction by germ cells and survival provided by somatic cells), while their unicellular ancestors had to do both. We define viability as activities keeping the cell alive (e.g. investing in cellular homeostasis or behaviors that improve survival), and fecundity as activities involved in cellular reproduction.
At the cellular level, there appears to be a fundamental asymmetry in how viability effort and fecundity effort can be shared among cells: while multicellular organisms readily evolve differentiated cells that are completely reliant on helper cells (i.e. glial cells that support neurons in animals or companion cells that support sieve tube cells in plants), no cell can directly share its ability to reproduce. To better understand the intuition behind this, consider a cell that elongates prior to fission. This cell must grow to approximately twice its original length. Two cells cannot elongate by 50% and then combine their efforts; elongation is an intrinsically single cell effort. We thus use a model in which viability can be shared across connected cells, but fecundity cannot be shared (note, in order to test the sensitivity of our predictions to this assumption, in a later section we will consider the more general case in which viability and fecundity can both be shared, but by different amounts).
We consider a model of multicellular groups composed of clonal cells that each invest resources into viability and fecundity. Because there is no withingroup genetic variation, withingroup evolution is not possible, though selection can act on grouplevel fitness differences. Specifically, we consider the pattern of cellular investment in fecundity and viability, and their sharing of these resources with neighboring cells within the group, to be the result of a heritable developmental program. Thus, selection is able to act on the multicellular fitness consequences of different patterns of cellular behavior within the group. We let v denote each cell’s investment into viability, and b denote each cell’s investment into fecundity. Each cell's total investment is constrained so that $v+b=1$. However, a cell's return on its investment is in general nonlinear. Here, we let α represent the ‘return on investment exponent': by tuning α above and below 1.0, we can simulate conditions with accelerating and saturating (i.e. convex and concave, or super and sublinear) returns on investment, respectively. We let $\stackrel{~}{v}$ and $\stackrel{~}{b}$ represent a cell’s return on viability and fecundity investments, respectively. Following Michod, 2006; Michod and Roze, 1997, we calculate a cell’s reproductive output as a multiplicative function of $\stackrel{~}{v}$ and $\stackrel{~}{b}$ (thus, both functions must be positive for a cell to grow). A single cell’s reproduction rate is $w=\stackrel{~}{v}\stackrel{~}{b}={v}^{\alpha}{b}^{\alpha}$. At the group level, fitness is the total contribution of all cells in the group toward the production of new groups (i.e. group level reproduction). The group level fitness is thus the sum of $\stackrel{~}{v}\stackrel{~}{b}$ over all cells.
Finally, cells may share the products of their investment in viability with other cells to whom they are connected.For a given group, the details about who may share with whom, and how much, is encoded in a weighted adjacency matrix $\mathbf{\mathbf{c}}$. The element ${c}_{ij}$ defines what proportion of viability returns cell i shares with cell j. Cells cannot give away all of their viability returns, as they would no longer be viable; mathematically, we count a cell among its neighbors and thus ensure that they always ‘share’ a positive portion of viability returns with themselves, so that ${c}_{ii}>0$. Furthermore, since a cell cannot share more viability returns than the total it possesses, we have $\sum _{i=1}^{N}{c}_{ji}=1$ for a group of N cells. For the networks we consider, each cell takes a fraction β of its viability returns and shares that fraction equally among all of its ${n}_{i}$ neighbors (including itself), and keeps the rest of its returns $1\beta $ for itself. Therefore cell i keeps a total fraction of $1\beta +\frac{\beta}{{n}_{i}}$ of its returns for itself and gives $\frac{\beta}{{n}_{i}}$ to each of its nonself neighbors. In other words, $c}_{ii}=1\beta +\frac{\beta}{{n}_{i}$, ${c}_{ij}=\frac{\beta}{{n}_{i}}$ if cells i and j are connected, and ${c}_{ij}=0$ if cells i and j are not connected. This means the total amount of returns kept by cell i depends on both the network topology and β. When $\beta =0$ there is no sharing, and when $\beta =1$ cells share everything equally among all connections and themselves. We refer to β as interaction strength. A given group topology (unweighted adjacency matrix) and β completely specify $\mathbf{\mathbf{c}}$.
Within a group of N cells, the overall returns on viability that a given cell enjoys, then, comprises its own returns as well as whatever is shared with it by other members of the group. This can be written as $\stackrel{~}{v}}_{i}={v}_{i}^{\alpha}{c}_{ii}+\sum _{j\ne i}^{n}{v}_{j}^{\alpha}{c}_{ji$, or equivalently, $\stackrel{~}{v}}_{i}=\sum _{j}^{n}{v}_{j}^{\alpha}{c}_{ji$. Note that this is a column sum, since it describes the total incoming viability returns a cell receives as a result of its own effort and trade with neighboring cells. Therefore, we write the group level reproduction rate (i.e. the group fitness) for a group of N cells as
where all three of the above equations are equivalent. We investigate evolutionary outcomes under this definition of group level fitness for groups with different topologies (who shares with whom), and in scenarios with various return on investment exponents α.
Results
Fixed resource sharing
We first consider cases wherein cells within a group share across fixed intercellular interactions. In each case we vary the return on investment exponent, α, between 0.5 and 1.5, and the interaction strength, β, between 0.0 and 1.0, both in increments of 0.1. For each combination of topology, α, and β, the group investment strategy (${v}_{i}$ for all i) was allowed to evolve for 1000 generations.
We begin with simple topologies: groups with no connections and groups that are maximally connected. They represent, respectively, the case in which all cells within the group are autonomous and the case in which every cell interacts with all others (i.e. a ‘wellmixed’ group). In the absence of interactions, cells cannot benefit from functions performed by others and therefore must perform both functions v and b; hence specialization is not favored, and does not evolve. In the fully connected case, a high degree of specialization is observed for many values of α and β (Figure 1a). Consistent with classic results (Cooper and West, 2018; Michod et al., 2006; Ispolatov et al., 2012; Solari et al., 2013; Michod, 2007; West et al., 2015), specialization is only achieved in the fully connected case for $\alpha >1$.
Next, we consider a simple sparse network in which each cell within a group is connected to only two other cells, forming a complete ring (Figure 1b); we refer to this as the neighbor network. Surprisingly, preventing trade between most cells encourages division of labor. We find that specialization evolves even when $\alpha <1.0$, that is, when the returns on investment are saturating or concave. In our simulations, this topology leads to alternating specialists in viability and fecundity (Figure 1b). Analytically, we find that this topology always favors at least some degree of specialization whenever $\alpha >\frac{3}{4\beta}$.
We next study a network with cells that can be separated into two disjoint subgroups, where every edge of the network connects a cell in one subgroup to a cell in the other subgroup and no within subgroup connections exist, that is, a bipartite graph (Figure 1c). We refer to the specific network structure in Figure 1c as the ‘balanced bipartite’ network. We find that specialization evolves even when $\alpha <1.0$, similar to the neighbor network. However, we find that specialization evolves for a wider range of α and β values for the balanced bipartite network than for the neighbor network.
We can analytically determine under what conditions complete generalization is optimal. The complete generalist investment strategy is where every cell in the group invests equally into viability and fecundity, defined as: ${v}_{i}^{*}=\frac{1}{2}$ for all i. For these simple topologies, the complete generalist strategy is either a maximum or a saddle point, depending on the values of α and β. Complete generalization is only favored when the Hessian evaluated at the generalist investment strategy $\frac{{\mathrm{\partial}}^{2}W}{{\mathrm{\partial}}_{{v}_{k}}{\mathrm{\partial}}_{{v}_{\ell}}}{}_{{\overrightarrow{v}}^{\ast}}={\mathbf{H}}^{\ast}$ is negative definite, that is, all of its eigenvalues are negative. The largest eigenvalues of the Hessian for the complete, neighbor, and balanced bipartite networks are $\alpha {\left(\frac{1}{2}\right)}^{2\alpha 3}(1+\alpha \beta )$, $\alpha {\left(\frac{1}{2}\right)}^{2\alpha 3}(1+\frac{4}{3}\alpha \beta )$, and $\alpha {\left(\frac{1}{2}\right)}^{2\alpha 3}(1+\frac{2N}{N+2}\alpha \beta )$, respectively. When α and β are chosen so that the largest eigenvalue becomes nonnegative, complete generalization cannot maximize group fitness.
While we have not analytically shown where the fitness maximum occurs in cases where the generalist strategy becomes a saddle point, evolutionary simulations (Figure 1) suggest that when complete generalization is not a fitness maximum, a high degree of (or even complete) specialization typically does maximize fitness.
In all cases in which complete specialization is achieved in evolutionary simulations, $\stackrel{~}{v}\stackrel{~}{b}$ terms for viability specialists go to zero, as they cannot reproduce on their own. Furthermore, the fecundity specialists are entirely reliant on the viability specialists for their survival; if viability sharing were suddenly prevented, their $\stackrel{~}{v}\stackrel{~}{b}$ terms would also be zero. This amounts to complete reproductive specialization (Cooper and West, 2018; Kirk, 2005; Michod, 2006).
Evolving resource sharing
Until now, sharing has been included in every intercellular interaction within groups. Here, we consider the case in which there is initially no sharing, and sharing must evolve along with specialization. These simulations begin with no resource sharing (i.e. $\beta =0$); during every round, each group in the population has a 2% chance that a mutation will impact its developmental program, and the β value for one of its cells will change. The new β value is chosen from a truncated Gaussian with standard deviation of 10% of the mean, centered on the current value. Whatever is not retained is shared equally across all interactions, including the self term.
Evolutionary simulation results are similar to those from the fixedsharing model (Appendix 1—figure 1). Saturating specialization (i.e. specialization despite a concave return function) still occurs for the neighbor and balanced bipartite topologies. Thus, for both fixed and evolved resource sharing, we observe specialization for the largest range of parameters (including $\alpha <1$) not when the group is maximally connected, but rather when connections are fairly sparse. Therefore, a sparse group topology constitutes a cooperationprone physical substrate that can favor the evolution of cellular.
As an example of the benefit of evolving sharing, consider that the maximum fitness according to Equation 1 for a group of N disconnected cells scales as $N{\left(\frac{1}{2}\right)}^{2\alpha}$. On the other hand, for the balanced bipartite network with a complete specialization strategy (i.e. $\overrightarrow{v}=\u27e80,1,0,1,\mathrm{\dots}\u27e9$), the fitness scales as $\left(\frac{{N}^{2}\beta}{2N+2}\right)$. The ratio of these fitnesses is $\left(\frac{{N}^{2}\beta}{2{N}^{2}+2N}\right){2}^{2\alpha}\approx \beta {2}^{2\alpha 1}$, where the approximation is for large N. So for larger groups and when $\alpha >\frac{1}{2}\frac{\mathrm{log}\beta}{2\mathrm{log}2}$, if a group can evolve resource sharing (i.e. letting $\beta \to 1$ and adopting the specialist investment strategy) its maximum fitness will increase.
Benefit of specialization
We now consider a simple example to highlight why specialization can be adaptive despite saturating (i.e., concave) returns from trade. Consider groups of four cells, connected via the nearestneighbor topology (i.e. in a ring). We directly calculate the grouplevel fitness of generalists and specialists for two scenarios: $\alpha =0.9$ and $\alpha =1$ by summing the contributions of each cell within these groups (Figure 2). In this simple scenario, reproductive specialization strongly increases group fitness (33% for $\alpha =1$ and 16% for $\alpha =0.9$).
The benefit of specialization in neighbor networks increases with group size. For a ring of size N, fitness under the specialist strategy $\overrightarrow{v}=\u27e80,1,0,1\mathrm{\dots}\u27e9$ is $W=\frac{\beta}{3}N$. For a ring of generalists the fitness is $W=N{(\frac{1}{2})}^{2\alpha}$. Therefore, whenever $\alpha >\frac{\mathrm{log}3\mathrm{log}\beta}{2\mathrm{log}2}$, the ring of complete specialists enjoys a greater fitness than the ring of complete generalists. Again, note that complete generalization becomes disfavored when $\alpha >\frac{3}{4\beta}$, so there is a narrow regime where $\frac{3}{4\beta}<\alpha <\frac{\mathrm{log}3\mathrm{log}\beta}{2\mathrm{log}2}$ during which neither complete generalization nor complete specialization is optimal. Numerical optimization and evolutionary simulations suggest that even in this region, however, the specialization score of the optimal strategy is large (Figure 1).
Effect of sparsity
Surprisingly, saturating specialization appears to be the rule, rather than the exception, for sparsely connected graphs. We investigated ErdősRényi random graphs with varying degrees of connectivity to systematically examine the relationship between sparsity and the value of α at which specialization is favored. We find that many randomly assembled graphs obtain maximum fitness through complete reproductive specialization even when α is below 1 (Figure 3b,c). It is only at the extremes of sparsity and connectivity (near the fully connected or fully unconnected points) that generalists maintain superior fitness for all values of $\alpha <1$. We further show that this general trend is independent of the size of a group; saturating specialization is favorable for groups of size $N=10$, $N=100$, and $N=1000$. When network connectivity is at its minimum, the group consists solely of isolated cells that cannot interact. Under these conditions generalists are favored. Similarly, at maximum connectivity every cell interacts with every other cell. Under these conditions generalists are favored unless $\alpha \beta >1$. However, when connectivity is small but not zero, specialization arises most readily. We conjecture that the troughs in Figure 3b, where specialization occurs for the lowest values of α, occur when connectivity is just large enough so that the existence of a spanning tree is more likely than not.
Filaments and trees
Sparse topologies like the neighbor network configuration have significant biological relevance, and direct ties to early multicellularity. The first step in the evolution of multicellularity is the formation of groups of cells (Szathmáry and Smith, 1995; Kirk, 2005; Willensdorfer, 2008; Bonner, 1998; Fairclough et al., 2010). Simple groups readily arise through incomplete cell division, forming either simple filaments (Figure 4a) or treelike morphologies (Figure 4b; Bengtson et al., 2017b; Droser and Gehling, 2008; BermanFrank et al., 2007; Ratcliff et al., 2012). Filament topologies have been widely observed in independentlyevolved simple multicellular organisms, from ancient fossils of early red algae (Butterfield, 2000; Figure 4a) to extant multicellular bacteria (Claessen et al., 2014) and algae (Umen, 2014). Branching multicellular phenotypes have also been observed to readily evolve from baker’s yeast (Ratcliff et al., 2015; Figure 4b), and are reminiscent of ancient funguslike structures (Bengtson et al., 2017a) and early multicellular fossils of unknown phylogenetic position from the early Ediacaran (Droser and Gehling, 2008).
Simulations of populations of groups with filamentous and branched topologies reveal that specialization is indeed favored in the sublinear regime (Figure 4a and b) ; conversely, sublinear specialization is never observed for fully connected topologies (Figure 4c). While the generalist strategy is never a critical point for these networks (which have $\mathbf{\mathbf{c}}\ne {\mathbf{\mathbf{c}}}^{T}$, see Materials and methods), we conjecture that there is a nearby critical point which maximizes fitness at small values of α and becomes unstable at larger values of α. We introduce a new metric, ${\alpha}^{*}$, defined as the value of α such that the largest (least negative) eigenvalue of the Hessian evaluated at the complete generalist strategy is zero when $\beta =1$. For topologies in which each member has the same number of neighbors, ${\alpha}^{*}$ is a critical value at which generalization is no longer an optimal strategy. However, even for groups where the number of neighbors for each cell varies, we can still use ${\alpha}^{*}$ as a proxy for how amenable a topology is to saturating specialization. The smaller ${\alpha}^{*}$, the more specialization is likely to be favored. We plot vertical lines where $\alpha ={\alpha}^{*}$ (solid lines in Figure 4(a) Figure 4(b)), and dotted lines to indicate roughly where the simulation curves cross specialization of 0.5. These results show that, for these topologies, ${\alpha}^{*}$ acts as an effective metric for how amenable a network is to saturating specialization. This metric ${\alpha}^{*}$ only depends on topology and can in principle be calculated analytically given any network. We examined the value of ${\alpha}^{*}$ as filaments and a variety of treelike structures grow larger, and find that specialization becomes more strongly favored (Figure 4D ). While group size has no effect on specialization for some topologies, like the neighbor network, filaments and trees all see a decrease in ${\alpha}^{*}$ as group size increases; ${\alpha}^{*}$ eventually plateaus once groups are larger than a few tens of cells. Simple and easily accessible routes to multicellular group formation can readily evolve in response to selection for organismal size (Ratcliff et al., 2012), and this process may also strongly favor the evolution of cellular differentiation (McCarthy and Enquist, 2005; Heim et al., 2017; McClain and Boyer, 2009; Bonner, 1998).
Mean field model
Finally, to capture some general principles underlying this phenomenon, we consider a meanfield model with N cells (N >> 1), each of which is connected to z other cells. For simplicity we consider the case in which $\beta =1$ and $\alpha =1$. We pick $\alpha =1$ as at this point, if the fitness of specialists is greater than that of generalists, specialization will be favored for at least some values of $\alpha <1$. If the fitness of generalists is greater than or equal to that of specialists, specialization will only be favored if $\alpha >1$.
For generalists, the fitness is simply ${W}_{G}=N/4$, as each cell has $v=1/2$ and $b=1/2$ (before and after sharing). Viability specialists produce $v=1$ and $b=0$, while fecundity specialists produce $v=0$ and $b=1$. Viability specialists then share $v=1/(z+1)$ with each of their z neighbors. After sharing, fecundity specialists receive $v=1/(z+1)$ from each of their viability specialist neighbors. But how many of their neighbors are viability specialists? We label the fraction of cells connected to fecundity specialists that are viability specialists f, that is, f is the mean number of viability specialists connected to each fecundity specialist divided by z, averaged over all fecundity specialists. For a bipartite graph, $f=1$; for a randomly connected graph on which half of cells are viability specialists and half of cells are fecundity specialists, $f=1/2$. Group fitness is thus:
Here, $zf/(z+1)$ is the average viability returns each fecundity specialist has received after sharing, which is multiplied by the amount of fecundity each fecundity specialist has (1) and the number of fecundity specialists ($N/2$). Writing ${W}_{S}$ in terms of ${W}_{G}$:
Specialists will be favored if the ratio ${W}_{S}/{W}_{G}>1$. This will be true if:
which reduces to:
This inequality implies that specialization will only be favored if fecundity specialists are preferentially connected to viability specialists, that is, if $f>1/2$. Further, for a fully connected network $f=1/2$, so this inequality is never satisfied, that is, specialists cannot have larger fitness than generalists for $\alpha =1$ and fully connected topologies, as classically predicted.
Further, f cannot be more than 1, so if the threshold from the inequality in Equation 5 is greater than or equal to 1, specialization cannot be favored for $\alpha <1$. Thus, specialization for $\alpha <1$ is only possible if:
which reduces to: $z>1$. This again reproduces a classic result: specialization for $\alpha <1$ is not possible for disconnected cells.
This analysis allows us to interrogate specific cases. For example, if $z=3$, f must be greater than 2/3, while if $z=4$, f must only be greater than 5/8. Can such networks be constructed? The answer will depend on both the number of cells and how they are connected. Ultimately, the question of if a graph can be made with particular values of f and z is a graph coloring problem, and beyond the scope of this manuscript. However, this inequality presents a useful heuristic which can be used to determine if specialization is favored by measuring just a few properties of the graph.
Effect of varying ratios of specialists
We now allow the fraction of fecundity specialists to be X (rather than forcing $X=1/2$). For generalists, the group fitness is unchanged, ${W}_{G}=N/4$, while for specialists the group fitness is:
Writing ${W}_{S}$ in terms of ${W}_{G}$ gives:
Specialists will be favored if the ratio ${W}_{S}/{W}_{G}>1$. This will be true if:
Compared to the threshold value of f when $X=1/2$, if $X>1/2$, that is, more than half of cells are fecundity specialists, the value of f necessary for specialization to be favored is lower. If $X<1/2$, the threshold value of f is higher than if $X=1/2$. In other words, 1:2 is different from 2:1, and they both are different from 1:1. Once again, the question of if a particular configuration can be created–and how–is a graph coloring problem beyond the scope of this manuscript. However, this mean field heuristic gives us some information about how to expect graphs with different ratios of specialists to generalists to behave.
We again ask what must be true for f to be less than 1 (if $f>1$, specialization will not be favored). Thus, specialization is only possible if:
which reduces to:
For a mean field model, specialization with $\alpha <1$ is impossible if fewer than one fourth of cells are fecundity specialists. We stress here that this is a mean field model, and does not apply to scenarios in which cells have a wide range of values of z. If such networks do or do not favor specialization for $\alpha <1$ will again be a graph coloring problem.
Discussion
During the evolution of multicellularity, formerly autonomous unicellular organisms evolve into functionallyintegrated parts of a new higher level organism (West et al., 2015; Michod and Nedelcu, 2003). Evolutionary game theory (Corning and Szathmáry, 2015; Nash, 1950; Smith, 1988) argues that functional specialization should only evolve when increased investment in trade increases reproductive output. Conventionally, this requires returns from specialization to be accelerating, that is, convex or superlinear (Szathmáry and Smith, 1995; Smith and Szathmáry, 1997; Goldsby et al., 2012; Corning and Szathmáry, 2015; Boza et al., 2014; Taborsky et al., 2016; Page et al., 2006; Rueffler et al., 2012; Szekely et al., 2013). While this idea is intuitive, it is, in the case of fixed group topology, also overly restrictive. In this paper, we explore how social interactions within groups, measured by their network topology, affect the evolution of reproductive specialization. Indeed, when all cells within groups interact (with equal interaction strength), returns on investment must be an accelerating, that is, convex, function of investment for specialization to evolve (Figure 1a; Szathmáry and Smith, 1995; Smith and Szathmáry, 1997; Corning and Szathmáry, 2015; Cooper and West, 2018). Yet for a broad class of sparsely connected networks, complete specialization can evolve even when the viability and fecundity return on investment curves are saturating, that is, concave (Figure 3).
To understand how specialization can be favored despite concave return on investment (ROI) curves, consider Jensen's inequality. Jensen's inequality states that for a convex function $F(x)$, $\u27e8F(x)\u27e9>F(\u27e8x\u27e9)$, that is, the average value of $F(x)$, $\u27e8F(x)\u27e9$, is larger than $F(\u27e8x\u27e9)$, where $\u27e8x\u27e9$ is the average value of x. A corollary of Jensen's inequality is that the opposite is true for concave functions, that is, for a concave function $G(x)$, $\u27e8G(x)\u27e9<G(\u27e8x\u27e9)$. Jensen’s inequality guarantees that for concave ROI functions generalists produce more total viability and fecundity than specialists, and that for convex ROI functions specialists produce more total viability and fecundity than generalists.
Crucially, however, Jensen's inequality does not connect ROI convexity/concavity to group fitness. Jensen’s inequality relates the degree of specialization to the average viability and average fecundity produced, but does not itself say anything about group fitness, which is the product of viability and fecundity averaged across all cells. For fully connected topologies (i.e. Figure 4c), greater absolute productivity proportionally increases group fitness, and differentiation can only evolve with accelerating benefits of specialization. This is not the case for topologically structured organisms, where fitness also depends on how complementary specialist cells are connected. Natural selection acts on realized productivity, that is, average $vb$; mutations that increase average v or average b without increasing average $vb$ are not adaptive. The importance of connecting complementary specialists has long been appreciated in other contexts, such as metabolic crossfeeding, for which it has been shown that the spatial arrangement of unlike specialists plays a key role in determining their productivity (and thus fitness) (Co et al., 2020). Indeed, While Jensen's inequality ensures that generalists will produce more viability and fecundity than specialists given a concave ROI function, specialization can still increase the fitness of topologically structured groups by increasing realized productivity.
Rather than being unusual, networks favoring specialization readily arise as a consequence of physical processes structuring simple cellular groups (Allen et al., 2017). For example, septin defects during cell division create multicellular groups with simple graph structures (Figure 4a and b), where cells are connected only to parents and offspring (Bengtson et al., 2017b; Droser and Gehling, 2008; Ratcliff et al., 2012; Ratcliff et al., 2013). If cells share resources only with physicallyattached neighbors, then the physical topology of the group describes its interaction topology, and these sparse networks strongly favor reproductive specialization. Finally, we note that the primary benefit of sparsity is that sparse networks are likely to be at least somewhat bipartite. The more bipartitelike a network is, the less effort is wasted, and the easier it is for specialization to be favored.
Disentangling the evolutionary underpinnings of ancient events is notoriously difficult. Still, it is worth examining the independent origins of complex multicellularity, which are independent runs of parallel natural experiments in extreme sociality. Complex multicellularity (large multicellular organisms with considerable cellular differentiation) has evolved in at least five eukaryotic lineages, once each in the animals (King, 2004), land plants (Kenrick and Crane, 1997), and brown algae (Silberfeld et al., 2010), two or three times in the red algae (Cock and Collén, 2015; Yoon et al., 2006), and 8–11 times in fungi (Nagy et al., 2018). In all cases other than animals, these organisms form multicellular bodies via permanent cellcell bonds, creating longlasting highly structured cellular networks. Both fossil and phylogenetic evidence suggests that early multicellular organisms in these lineages were considerably less complex, growing as relatively simple graph structures. For example, 1.2 billion year old red algae formed linear filaments of cells (Butterfield, 2000), basal multicellular charophyte algae formed circular sheets of cells radiating from a common center (Kenrick and Crane, 1997), the ancestor of the brown algae likely formed a branched haplostichous thallus that was either filamentous or pseudoparenchymatous (Silberfeld et al., 2010), and hyphal fungi are primarily composed of linear chains of cells. Much less is known about the topology of animals prior to the evolution of cellular specialization. One hypothesis is that early metazoans resembled extant colonial choanoflagellates (Fairclough et al., 2013), the closestliving protistan relatives of the animals (Fairclough et al., 2010). Extant colonyforming choanoflagellates have evolved a variety of multicellular structures with sparse cellular topologies and permanent cellcell bonds. For example, many species form branched, treelike structures (Leadbeater, 2015), Choanoeca flexa grows as a sheet of cells (Brunet et al., 2019), and Salpingoceca rosetta can form either linear chains or rosettes in which the cells are connected via cytoplasmic bridges formed through incomplete cytokinesis (Dayel et al., 2011). While these growth forms are quite diverse, they all share characteristics (i.e. permanent cellular bonds and sparse topologies) that promote the evolution of cellular differentiation.
The main differences between our work and previous investigations of the effect of group topology on specialization is that we consider the productivity of groups as a whole, not the cells within them, and we consider situations of highly asymmetric sharing. Our approach is general, and can be applied to other systems of trade and specialization, so long as (1) only the aggregate productivity of the group (and not the particles within it) is maximized, (2) the productivity of each particle within the group is a multiplicative function of returns on investment into two (or more) tasks, and (3) there is an asymmetry in how products of those investments are shared. While in this work we have focused on reproductive division of labor, a process in which fecundity returns are not shared at all, we show in the supplement that as long as sharing of two goods is sufficiently asymmetric, specialization with saturating returns on investment can still be adaptive (Appendix 1—figure 2).
Finally, we note that alternative paths to specialization likely exist. For example, cells at different positions in a group may experience different local environments, which may produce cells with varied fecundityviability tradeoffs. A previous paper demonstrated that the evolution of specialization is favored if these ‘positional effects’ result in an initially heterogeneous population of cell types (Tverskoi et al., 2018). However, these positional effects were considered for the case of wellmixed groups (i.e. completely connected network topologies). We thus anticipate that future work examining the relationship between cellular interaction topology and cellular heterogeneity (as well as a wide range of complex and varied relationships between viability, fecundity, and multicellular fitness) will provide unique insight into the origin and diversity of multicellular forms.
Conclusion
We explored the evolution of reproductive specialization in multicellular groups with various cellular interaction topologies. Our results demonstrate that group topological structure can play a key role in the evolution of reproductive division of labor. Indeed, within a broad class of sparsely connected networks, specialization is favored even when the returns from cooperation are saturating (i.e. concave); this result is in direct contrast to the prevailing view that accelerating (i.e. convex), returns are required for natural selection to favor increased specialization (Cooper and West, 2018; Michod et al., 2006; Ispolatov et al., 2012; Solari et al., 2013; Michod, 2007; West et al., 2015). Our results underscore the central importance of life history tradeoffs in the origin of reproductive specialization (Michod et al., 2006; Michod, 2007; Hammerschmidt et al., 2014; van Gestel and Tarnita, 2017; Noh et al., 2018), and support the emerging consensus that evolutionary transitions in individuality are not necessarily highly constrained (Ratcliff et al., 2012; Ratcliff et al., 2017; Fairclough et al., 2010; Brunet and King, 2017; Pennisi, 2018; Black et al., 2019; Rose et al., 2020; van Gestel and Tarnita, 2017; Black et al., 2019; Staps et al., 2019; Grosberg and Strathmann, 2007).
Materials and methods
Analysis
The gradient of the fitness with respect to the group investment strategy $\overrightarrow{v}$, is
where ${\widehat{e}}_{k}$ is a unit vector in the $k}^{\mathrm{t}\mathrm{h}$ direction. First notice that if $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$, and $\overrightarrow{v}=\frac{1}{2}\overrightarrow{1}$ where $\overrightarrow{1}$ is a vector of ones, then the gradient is zero. This strategy, $\overrightarrow{v}=\frac{1}{2}\overrightarrow{1}$, corresponds to the ‘generalist’ strategy, where every cell invests equally into both tasks. Second, notice that if $\mathbf{\mathbf{c}}\ne {\mathbf{\mathbf{c}}}^{T}$ then the gradient is not zero under the generalist strategy, so at least some degree of specialization must be necessary to maximize fitness. To determine the stability of this solution we examine ${\mathbf{\mathbf{H}}}^{*}$, the Hessian (see SI Equation 3) evaluated at the generalist critical point. If ${\mathbf{\mathbf{H}}}^{*}$ is negative definite, then the generalist strategy is a fitness maximum and is therefore an optimal strategy. If, on the other hand, ${\mathbf{\mathbf{H}}}^{*}$ has both positive and negative eigenvalues then the generalist strategy lies at a saddle point within the fitness landscape, and therefore the optimal strategy must be somewhere else in (or on the boundary of) the domain (i.e. ${v}_{i}\in [0,1]$ for all $i\in 1,2,\mathrm{\dots}N$). Finally, note that ${\mathbf{\mathbf{H}}}^{*}$ is never positive definite since $\overrightarrow{1}$ is always an eigenvector with negative eigenvalue (when $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$).
We also use the zero crossing of the largest eigenvalue of ${\mathbf{\mathbf{H}}}^{*}$ evaluated at $\overrightarrow{v}=\frac{1}{2}\overrightarrow{1}$ and $\beta =1$ as an overall measure of how amenable a network is to specialization, even when $\mathbf{\mathbf{c}}\ne {\mathbf{\mathbf{c}}}^{T}$.
Evolutionary simulations
Request a detailed protocolOur evolutionary simulations maintain the same overall structure as the WrightFisher model: a discretetime Markov chain framework with fitnessweighted multinomial sampling between generations and constant population size. Therefore we refer to them as WrightFisher evolutionary simulations. We initialize a population of $\mathcal{N}=1000$ groups, each of group size $N=10$, with uniform random investment strategies. We then let them evolve for 1000 generations, selecting offspring according to the relative fitness of each group (see Equation 1). At each generation, there is a 2% chance for a mutation to a given group’s investment strategy $\overrightarrow{v}$. If a mutation occurs, a new investment strategy is selected from a truncated multivariate gaussian distribution centered at the current (premutation) investment strategy and with standard deviation equal to $\frac{1}{10}\overrightarrow{v}$. After mutations each group’s fitness is calculated according to Equation 1, and the population is ranked according to fitness. Finally, $\mathcal{N}$ groups are selected (with replacement) to populate the next generation, according to a multinomial distribution weighted by the groups’ fitness ranks.
Measuring specialization
Request a detailed protocolTo quantify the degree of specialization associated with a given group’s optimal investment strategy—the one which maximizes the fitness—we introduce the following metric, which we refer to simply as ‘Specialization’:
Specialization ranges from 0 (for groups consisting of cells investing equally in functions v and b) to 1 for groups consisting of cells investing exclusively in either function.
Code availability
Request a detailed protocolAll evolutionary simulations and other computations associated with this work are available at github.com/dyanni3/topologicalConstraintsSpecialization (Yanni, 2020; copy archived at https://github.com/elifesciencespublications/topologicalConstraintsSpecialization).
Appendix 1
Analysis
As described in the main text, the fitness for a group of $N$ individuals is defined as
and the gradient of the fitness with respect to the group investment strategy $\overrightarrow{v}$, is
where ${\widehat{e}}_{k}$ is a unit vector in the ${k}^{\text{th}}$ direction.
Hessian
The Hessian $\frac{{\partial}^{2}W}{\partial {v}_{k}\partial {v}_{\mathrm{\ell}}}$ is
Of particular interest for us is the value of the Hessian at the generalist strategy when $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$. In that case
where $\mathbf{\mathbf{a}}$ is the rownormalized adjacency matrix of the network. If $\mathbf{\mathbf{A}}$ is the network’s adjacency matrix then
The case when $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$
As noted above, when $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$, the generalist strategy is always a critical point where $\frac{\partial W}{\partial \overrightarrow{v}}=0$. To determine the stability of this solution we examine ${\mathbf{\mathbf{H}}}^{*}$ (Equation 4). If ${\mathbf{\mathbf{H}}}^{*}$ is negative definite, then the generalist strategy is a fitness maximum and is therefore an optimal strategy. If, on the other hand, ${\mathbf{\mathbf{H}}}^{*}$ has both positive and negative eigenvalues then the generalist strategy lies at a saddle point within the fitness landscape, and therefore the optimal strategy must be somewhere else in (or on the boundary of) the domain (i.e. ${v}_{i}\in [0,1]$ for all $i\in 1,2,\mathrm{\dots}N$). Finally, note that ${\mathbf{\mathbf{H}}}^{*}$ is never positive definite (when $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$). Consider $\mathbf{H}}^{\ast}\overrightarrow{1$:
Note $\mathbf{a}\overrightarrow{1}=\overrightarrow{1}$ since $\mathbf{\mathbf{a}}$ is rownormalized. Furthermore, $\alpha >0$, so $\overrightarrow{1}$ is always an eigenvector of ${\mathbf{\mathbf{H}}}^{*}$ with a negative eigenvalue.
We can next ask, under what conditions is ${\mathbf{\mathbf{H}}}^{*}$ negative definite? This will depend on the group topology, the nonlinear returns on investment $\alpha $, and the interaction strength $\beta $. We examine three cases: the neighbor graph, the balanced bipartite graph, and the complete graph.
When $\mathbf{\mathbf{c}}={\mathbf{\mathbf{c}}}^{T}$, the matrix ${\mathbf{\mathbf{H}}}^{*}$ is a special type of matrix called a circulant matrix, with well known properties. Its eigenvalues are given by the discrete Fourier transform of its first row. The ${k}^{\mathrm{th}}$ eigenvalue is
For the ring topology with $N=10$, for example
which has its maximum when $k=5$,
The maximum eigenvalue for the balanced bipartite and complete graphs can be computed similarly.
Evolution of resource sharing
Here we model the coevolution of sharing and specialization. We start with generalists that do not share at all. We then allow the amount of sharing and the degree of specialization to evolve. As described in the main text, during every round, each group in the population has a 2% chance that one if its cells will mutate and change how much ‘viability’ it shares. When this occurs, the fraction of its output to retain is chosen from a Gaussian with standard deviation of 10% centered on the current value. Whatever is not retained is shared equally across its interactions. The degree of specialization evolves as in simulations described in the main text.
Results are shown in Appendix 1—figure 1, for neighbor topologies, balanced bipartite topologies, and for a complete network.
General case of sharing two resources
We have so far focused on reproductive specialization, wherein the returns from one type of task (reproduction) are completely unshared while returns from another task (viability) are shared according to some functional interaction strength $\beta $. Here, we generalize somewhat to consider the returns from two arbitrary tasks which may each be shared to some extent, given by functional interaction strengths $({\beta}_{1},{\beta}_{2})$. For notational continuity we will continue to refer to the investment in those tasks as $b$ and $v$, and for tractability we will continue to assume that ${\alpha}_{1}={\alpha}_{2}$ and that there is a single topology governing who can trade with whom within the group. Of course, further generalizations could be made — e.g. each task could experience different returns on investment, there could be an arbitrary number of tasks, the availability of trading partners could differ between tasks, etc. However, we hope to show by this relatively modest generalization that there is nothing unique to reproductive tasks whose fruits are totally unshared that leads to specialization under regimes of sublinear return on investment.
The fitness function is modified so that
which yields the following Hessian at the generalist critical point (for the neighbor, balanced bipartite, and complete networks)
where
and,
where, as above, $\mathbf{\mathbf{A}}$ is the graph’s adjacency matrix (including self loops).
We see that for a given topology the adjacency matrix is fixed, so that ${\mathbf{\mathbf{c}}}_{\mathrm{\U0001d7cf}}$ and ${\mathbf{\mathbf{c}}}_{\mathrm{\U0001d7d0}}$ differ only in their functional interaction strengths ${\beta}_{1}$ and ${\beta}_{2}$. Therefore the maximum fitness strategy, specified by the vector ${\overrightarrow{v}}^{*}$, for a given group will depend under our model on the following parameters:
We demonstrate the effect of these parameters on the optimal strategy by finding the minimum value of $\alpha $ for which specialization becomes favored, which we denote ${\alpha}^{*}$, for a given pair $({\beta}_{1},{\beta}_{2})$ and given topology. The results are shown in Appendix 1—figure 2.
Jensen’s inequality and sparse topologies and asymmetric sharing
To understand how average fitness decouples from average $v$ and average $b$ for sparse topologies and asymmetric sharing, consider a ring of four cells in three different configurations: one that alternates between viability and fecundity specialists, one in which likespecialists are connected to each other, and one in which all cells are generalists (pictured below). For simplicity, we will set $\beta =1$, and we will initially consider the case when $\alpha =1$. When $\alpha =1$, Jensen's inequality tells us that generalists and specialists will be equally productive. Classically, this would suggest that specialists and generalists should have the same fitness.
And, indeed, all three cases have the same average v and the same average b (Â½ for each). However, the average fitnesses are all different.
Next, we consider the same three configurations, but with $\alpha =0.9$. Jensen's inequality tells us that for this value of $\alpha $, generalists should have a higher average v and average b. Indeed, the average v and b is higher for generalists than for specialists: 0.536 versus 0.5. However, the average fitness of generalists, 0.287, is still lower than the average fitness of alternating specialists, 0.333.
These examples show that Jensen's inequality still holds, and still correctly tells us which configuration has the highest average v and average b. However, average v and average b are no longer directly proportional to average fitness. Therefore Jensen's inequality does not directly inform average fitness, and we should not expect convex ROI functions to be required for specialists to be favored.
Star graphs
Let ${W}_{g}$ be the fitness for the star shaped network group of generalists ($v=0.5$, $b=0.5$) and ${W}_{s}$ be the fitness of specialists (all of the points of the star get $v=1$, $b=0$ and the central point gets $v=0$, $b=1$).
Next assume there are $N$ cells on the points of the star and 1 cell in the center of the star. We then have:
as the only individual with nonzero fitness is the central individual (all others have $b=0$). The central individual’s fecundity returns are ${1}^{\alpha}=1$, and it’s own viability returns are 0. However, the central individual gets shared $\frac{\beta}{2}$ of each of the $N$ other individuals’ viability returns (which are ${1}^{\alpha}=1$ each).
Next, for generalists, we have
The term on the left of Equation 9 comes from the fact that there are $N$ individuals each sharing $\frac{\beta}{2}$ of their viability returns (which is ${(\frac{1}{2})}^{\alpha}$ each) with themselves, and are getting $\frac{\beta}{N}$ of the central individual’s ${(\frac{1}{2})}^{\alpha}$ viability returns shared with them. Additionally, they are getting $1\beta $ of their own viability returns (withheld from sharing). Finally, each of their fecundity returns is ${(\frac{1}{2})}^{\alpha}$.
The term on the right of Equation 9 represents the contribution to the group fitness of the single central individual. That individual gets $\frac{\beta}{2}*{(\frac{1}{2})}^{\alpha}$ of viability returns shared to it $N$ times, and it also shares with itself and keeps a portion of its returns for itself. And it has a fecundity return of ${(\frac{1}{2})}^{\alpha}$.
Star topologies in the limit of large N
We first examine Equation 9 in the limit where $N\gg 1$:
which reduces to
and finally
To understand if generalists or specialists are favored we examine the ratio of generalist to specialist fitness $\frac{{W}_{g}}{{W}_{s}}$.
This means ${W}_{g}>{W}_{s}$ if $\beta >{2}^{\alpha +1}$. Since $\beta $ and $\alpha $ are both bounded between 0 and 1, this is never achievable. Therefore, at large $N$, specialists are always favored.
Star topologies in general
Alternately, if do not assume $N$ is large, but set $\beta =1$ for simplicity, we can solve for the largest $\alpha $ which would yield equal fitness to generalists and specialists:
Rings with even and odd numbers of cells
In the main text we plot ${\alpha}^{*}$ for ring graphs with even numbers of cells. We made this choice as specialization is slightly frustrated when there are odd numbers of cells. Rings with odd numbers of cells must have at least one location at which like specialists are connected, thus slightly increasing the value of ${\alpha}^{*}$ compared to a ring with an even number of cells (Appendix 1—figure 3). However, as the size of the graph increases, a single frustrated pairing matters less and less, and the value of ${\alpha}^{*}$ for rings with odd numbers and large N appears to approach the value of ${\alpha}^{*}$ for rings with even numbers of cells.
Data availability
All evolutionary simulations and other computations associated with this work are available at https://github.com/dyanni3/topologicalConstraintsSpecialization (copy archived at https://github.com/elifesciencespublications/topologicalConstraintsSpecialization); all parameters used in the current study are specified so all simulations can be repeated exactly.
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Decision letter

Raymond E GoldsteinReviewing Editor; University of Cambridge, United Kingdom

Diethard TautzSenior Editor; MaxPlanck Institute for Evolutionary Biology, Germany

Pierrick BourratReviewer; MacQuarie University, Australia
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
The evolution of germsoma differentiation is one of the most fundamental questions in evolutionary biology, and the present paper investigates the consequences of altering one of the most basic assumptions: the traditional (symmetric) division of labor that has been studied from biology to economics. The authors consider a diversity of network structures and fitness functions and they find that sparser networks lead to higher levels of specialization.
Decision letter after peer review:
Thank you for submitting your article "Topological constraints in early multicellularity favor reproductive division of labor" for consideration by eLife. Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Diethard Tautz as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Pierrick Bourrat (Reviewer #2).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
This is a very interesting paper on the evolution of germsoma differentiation in which the authors consider the topology of interactions between the cells that make up the whole. They find that classical considerations about the convexity or concavity of certain functions characterizing the advantages of specialization no longer hold when the network topology is nontrivial. And such topologies are indeed found in nature. The reviewers were generally very supportive of the work but raised a number of points that need to be addressed in a revised manuscript.
Essential revisions:
1) The authors use a notion of fitness in which clonal cells can have different fitnesses, or more accurately, clonal groups can have different fitness. We know that there are some precedents in the literature, but this notion of fitness does not correspond to the notion of fitness one can associate with natural selection. To illustrate why, consider the following plant example. Take a single genet with two ramets in two environmental patches, one rich and one poor. Each ramet might adopt a very different developmental strategy from the other, considering the ecological constraints it is subjected to. These two strategies would nevertheless not be heritable in the sense that two offspring ramets put in the same environmental patch would develop the same developmental strategies (excluding noise). Thus, the differential success of each ramet is not an evolutionary success that can be associated with natural selection. This is a case analogous to the one presented by the authors. The notion of fitness they refer to seems to be rather the notion of realized fitness. This has no implications for the author's results per se but instead leads to an interpretation in which natural selection is not at work for explaining the division of labor in situations of concavity.
2) Related to the previous point, there seems to be a tension between, on the one hand, the claim that a concave function can lead to an increase in reproductive specialization, and on the other hand, claims that it has something to do with fitness. Fitness is about expected values, and in a situation of concave function, two or more cells specializing would yield a lower collective fitness than when not specializing. From a purely analytical point of view (i.e., Fisher's fundamental theorem), this seems impossible. So my question to the authors is whether there is not hidden somewhere a convex function, which is the relevant one for the evolutionary dynamics observed. Otherwise, what is the ecological explanation of such a result? There must be some ecological constraints that give rise to this phenomenon, and it would be good to know what the authors think they are.
3) There are wellknown cases presented in the population genetics literature in which Fisher's fundamental theorem seems violated, but this is because of the environment (including the social environment) changes over time, such as frequencydependent effects on an individual's success. We wonder if the results of the authors could not related to this literature in some way.
4) The model description is a bit abstract and occasionally hard to follow. It would be great to have fecundity and viability defined, and even better to have some real biological example of what returns on viability might mean and how they might be shared (I don't find the filamentous fungi example informative, at least not in the way it is written). That would also help the reader understand why there are returns on viability but not on fecundity. That the v_{i} vector is the "group investment strategy" also comes as a surprise and takes a bit for the reader to put it all together. Similarly, the existence of both a general adjancecy matrix and of a special case one that uses the β, is somewhat confusing the way it's described. If the authors anyway only work with the special case of equal sharing with the nonself neighbors then why not define the 1β+β/ni quantities as c_{ij} when they appear in the text, and then write a fourth eqn for W in [1] that explicitly uses the β. That would certainly help the reader a lot.
5) Results subsection “Fixed resource sharing” first paragraph, we may be getting confused, but how can you vary β in the case when, as is now written, individual i "shares equally among interaction and self terms"? Doesn't this mean that β = 1?
6) “We conjecture that the troughs in Figure 3C, where specialization occurs for the lowest values of, occur when connectivity is just large enough so that a spanning tree is more likely to connect all individuals in the group than not”, we don't fully understand that conjecture: do the authors simply mean that the troughs occur when the random graph becomes connected with probability > 50%? (A spanning tree connects all individuals by definition.)
7) The authors suggest sparsity is the main determinant of whether a network supports reproductive specialization. But, their examples in 1B and 1C (where a ring is sparser than the bipartite network) to us suggest that it is not so much about "sparsity" as it is about "bipartiteness" – or how easy it is to subdivide the nodes into two classes such that most edges go between these two classes (that's what you'd want for specialization anyway, we guess), and that sparse graphs simply have a tendency to be close to bipartite. We suspect that a ring graph with an odd number of vertices will be less conducive to specialization (although you could still alternate germ/soma cells except at one point), and that a star graph where there is one node of degree n1 and all the others have degree 1 may be an example of a sparse graph where evolving specialization is not so easy (because for this graph it's not clear how to divide the vertices into germ and soma).
8) Related to the previous point: we would be interested if the authors have considered what happens when the optimal strategy is not 1:1 but, say, 1:2. Does that make specialization more difficult? Here we think that, with a few additional simulations, the authors could add a lot to the paper in terms of the ability to connect properties of the graph (beyond comparing some explicit topologies and random graphs of varying sparsity) to its ability to support the evolution of reproductive specialization.
9) Finally, it would be nice to see how the different specialists are distributed on these networks (at least when the specialization is equal to 1). One can infer it, but we think it would visually help the reader to get the gist of how the model works very quickly.
https://doi.org/10.7554/eLife.54348.sa1Author response
Essential revisions:
1) The authors use a notion of fitness in which clonal cells can have different fitnesses, or more accurately, clonal groups can have different fitness. We know that there are some precedents in the literature, but this notion of fitness does not correspond to the notion of fitness one can associate with natural selection. To illustrate why, consider the following plant example. Take a single genet with two ramets in two environmental patches, one rich and one poor. Each ramet might adopt a very different developmental strategy from the other, considering the ecological constraints it is subjected to. These two strategies would nevertheless not be heritable in the sense that two offspring ramets put in the same environmental patch would develop the same developmental strategies (excluding noise). Thus, the differential success of each ramet is not an evolutionary success that can be associated with natural selection. This is a case analogous to the one presented by the authors. The notion of fitness they refer to seems to be rather the notion of realized fitness. This has no implications for the author's results per se but instead leads to an interpretation in which natural selection is not at work for explaining the division of labor in situations of concavity.
Thank you for the thoughtprovoking comment. We are not completely sure if we understand your point correctly (and if not, we are happy to continue the discussion), but we are using the concept of fitness in the standard sense. While cells within groups can have different numbers of surviving offspring, in our simplified modeling world, selection does not act on these differences. Instead, we assume that the cells within each group are clonal, and that there can be genetic differences between groups. These genetic differences are responsible for cellular behavior, namely the extent to which they specialize in viability or fecundity tasks. Selection acts at the group level, in a way that is simply proportional to cellular productivity within these groups. Thus, selection acts between groups on differences in group fitness, which is caused by heritable variation in cellular behaviors underlying differentiation that vary between the groups.
However, we appreciate the reviewer’s point. We previously described mutations as affecting individual cells, which would of course mean that groups are no longer clonal. It is more accurate to say that in our simulation model, mutations change the pattern of specialization and sharing at different positions in the group, which we think of as being driven by a heritable developmental program. As with our analytical results, selection only acts on grouplevel fitness. When model parameters favor specialization, we see initially uniform groups in which every cell is a generalist gradually evolve developmental programs featuring complete reproductive specialization, providing a simulation test of our analytical results. We have thus revised and clarified the description of these simulations. We also stopped referring to the “fitness” of individual cells in the paper, since it is potentially confusing, instead describing the direct consequences of cellular specialization on their productivity (each cell’s 𝑣 ∗ 𝑏).
In addition to the changes noted in the paragraph above, we modified the model description to read:
“We consider a model of multicellular groups composed of clonal cells that each invest resources into viability and fecundity. Because there is no withingroup genetic variation, withingroup evolution is not possible, though selection can act on grouplevel fitness differences. Specifically, we consider the pattern of cellular investment in fecundity and viability, and their sharing of these resources with neighboring cells within the group, to be the result of a heritable developmental program. Thus, selection is able to act on the multicellular fitness consequences of different patterns of cellular behavior within the group.”
We modified the simulation description to read:
“These simulations begin with no resource sharing (i.e., 𝛽 = 0); during every round, each group in the population has a 2% chance that a mutation will impact its developmental program, and the 𝛽 value for one of its cells will change.”
2) Related to the previous point, there seems to be a tension between, on the one hand, the claim that a concave function can lead to an increase in reproductive specialization, and on the other hand, claims that it has something to do with fitness. Fitness is about expected values, and in a situation of concave function, two or more cells specializing would yield a lower collective fitness than when not specializing. From a purely analytical point of view (i.e., Fisher's fundamental theorem), this seems impossible. So my question to the authors is whether there is not hidden somewhere a convex function, which is the relevant one for the evolutionary dynamics observed. Otherwise, what is the ecological explanation of such a result? There must be some ecological constraints that give rise to this phenomenon, and it would be good to know what the authors think they are.
Thank you again for this question. We believe the reviewers have found a very important point that lacked clarity in the previous version of our manuscript. We appreciate the deep question regarding the presence of a hidden convexity. To be frank, we were surprised by this result at first as well. However, this surprise stems from the fact that the mathematical rule connecting specialization and the second derivative of the return on investment (ROI) function does not apply to asymmetric trade on sparse networks.
The following response explains the logic underlying this result and shows why there are no hidden convexities in our model. We apologize for its length, but since it is central to the paper we wanted to make the math absolutely clear.
In our model we make the simple assumption that group fitness is directly proportional to the sum of cellular productivity (each cell’s 𝑣 ∗ 𝑏) within the group. We see this as having the simplest biological interpretation, groups that generate a larger number of cells can make more progeny (i.e., more groups), all else equal. As a result, while holding the number of cells per group constant, the group with the highest fitness will be composed of cells with the highest average fitness.
To understand why a hidden convexity need not be present here, we will first discuss the mathematical rule that typically connects specialization and the second derivative of the ROI function: Jensen’s inequality. Jensen’s inequality states that for a convex function F(x), the average value of 𝐹(𝑥), ⟨𝐹(𝑥)⟩, is larger than 𝐹(⟨𝑥⟩), where ⟨𝑥⟩ is the average value of 𝑥. In other words, ⟨𝐹(𝑥)⟩ > 𝐹(⟨𝑥⟩). A corollary of Jensen’s inequality is that the opposite is true for concave functions, i.e., for a concave function 𝐺(𝑥), ⟨𝐺(𝑥)⟩ < 𝐺(⟨𝑥⟩). We assume this is the rule the reviewers refer to when they say “in a situation of concave function, two or more cells specializing would yield a lower collective fitness than when not specializing.”
For the traditional case of fully connected topologies and symmetric sharing, Jensen’s inequality provides a mathematical connection between a group’s fitness and its degree of specialization. With fully connected topologies, each cell shares its returns on 𝑣 and 𝑏 equally with all cells. As a result, all cells end up with 𝑣 equal to the average 𝑣 and 𝑏 equal to the average 𝑏. As a result, each cell’s 𝑣 ∗ 𝑏 is thus the average of 𝑣 multiplied by the average of 𝑏. Thus, group fitness is directly proportional to average 𝑣 and average 𝑏. Jensen’s inequality guarantees that for convex ROI functions, the average v produced by specialists (⟨𝐹(𝑥)⟩) is higher than the average 𝑣 produced by generalists (𝐹(⟨𝑥⟩)). The same is true for 𝑏. Since specialists must have larger average 𝑣 and 𝑏 than generalists, group fitness must be higher as well. The same argument holds in reverse for concave functions, i.e., if the ROI function is concave, the average 𝑣 and average 𝑏 are lower for specialists than for generalists so group fitness is lower, too. Thus, for fully connected topologies and symmetric sharing, Jensen’s inequality allows one to connect group fitness to average 𝑣 and average 𝑏.
Crucially, however, the connection between ROI convexity/concavity and fitness is indirect. Jensen’s inequality directly relates the degree of specialization to the average 𝑣 and average 𝑏; Jensen’s inequality does not itself say anything about group fitness. Group fitness is proportional to the average of the product 𝑣 ∗ 𝑏, which does not have to be directly proportional to average 𝑣 or average 𝑏 (even though for fully connected topologies and symmetric sharing ⟨𝑣𝑏⟩ is directly proportional to ⟨𝑣⟩ and ⟨𝑏⟩).
That Jensen’s inequality connects the degree of specialization to average 𝑣 and average 𝑏, but not group fitness is a crucial distinction. Jensen’s inequality is a mathematical truism, and cannot be violated. Indeed, for any concave ROI function, generalists will produce more 𝑣 and 𝑏 than specialists. This fact is just as true for sparse topologies and asymmetric sharing as it is for fully connected topologies and symmetric sharing. However, for sparse topologies and asymmetric sharing, group fitness is not directly proportional to average 𝑣 and average 𝑏; instead, group fitness strongly depends on network structure as well.
To understand how group fitness decouples from average 𝑣 and average 𝑏 for sparse topologies and asymmetric sharing, consider a ring of four cells in three different configurations: one that alternates between viability and fecundity specialists, one in which likespecialists are connected to each other, and one in which all cells are generalists (pictured in Author response image 1; red cells are viability specialists, blue cells are fecundity specialists, and purple cells are generalists). For simplicity, we will set 𝛽 = 1, and we will initially consider the case when 𝛼 = 1. When 𝛼 = 1, Jensen’s inequality tells us that generalists and specialists will be equally productive. Classically, this would suggest that specialists and generalists should have the same fitness.
And, indeed, all three cases have the same average 𝑣 and the same average 𝑏 (½ for each). However, the group fitnesses are all different. Note, depending on the configuration, ⟨𝑣⟩⟨𝑏⟩ can be greater than, less than, or equal to ⟨𝑣𝑏⟩.Next, we consider the same three configurations, but with 𝛼 = 0.9. Jensen’s inequality tells us that for this value of 𝛼, generalists should have a higher average 𝑣 and average 𝑏. Indeed, the average 𝑣 and 𝑏 is higher for generalists than for specialists: 0.536 versus 0.5. However, the group fitness of generalists, 1.15, is still lower than the group fitness of alternating specialists (i.e., leftmost configuration in the schematic in Author response image 1), 1.333.
These examples show that Jensen’s inequality still holds, and still correctly tells us which configuration has the highest average 𝑣 and average 𝑏. However, average 𝑣 and average 𝑏 are no longer directly proportional to group fitness. Therefore, Jensen’s inequality does not directly inform group fitness, and we should not expect convex ROI functions to be required for specialists to be favored.
Ultimately, the difference between the classic fully connected topologies and the asymmetric sharing / sparse topologies we consider is the ability of viability specialists to preferentially share viability to fecundity specialists. In fully connected topologies, some of the potential benefit from specialization is wasted. Consider the case where viability specialists share viability with other viability specialists: this is entirely unhelpful, two viability specialists (with 0 fecundity) that help each other survive still both have zero fitness. When the return on investment function is linear (and the graph is fully connected), the effect of sharing with the same cell type (as opposed to a complementary cell type) exactly cancels out the benefits of specializing; trade is only beneficial once the return on investment function becomes convex. In this case, specialists make enough extra v and b that groups of specialists do better than groups of generalists.
Again, these results were surprising. To help establish intuition for the role of preferentially connecting unlike specialists, we developed a mean field model. Based on the average number of connections per cell, this model determines the fraction of connections that must be between unlike specialists for the network to support specialization with concave ROI. We find that if unlike specialists are preferentially connected, specialization despite concave ROI should be expected for a wide range of networks. These results are similar to our simulations of randomly generated graphs (see Figure 3 in the main text), in which we also observed specialization for a wide range of parameters. Combined, these observations suggest that specialization, despite concave ROI, does not require precisely designed topologies, but is a general principle applicable to many different network structures. We agree that this was not clear in the original manuscript, and have added this model to the main text as described below.
We added Discussion paragraphs on Jensen’s inequality:
“To understand how specialization can be favored despite concave return on investment (ROI) curves, consider Jensen’s inequality. […] With asymmetric sharing and sparse topologies, Jensen’s inequality still informs the average viability and fecundity produced, but does not directly inform the group fitness.”
We also added a mean field model to help the reader develop intuition for these results:
“Mean field model
Finally, to capture some general principles underlying this phenomenon, we consider a meanfield model with cells, each of which is connected to other cells. […]However, this inequality presents a useful heuristic which can be used to determine if specialization is favored by measuring just a few properties of the graph.”
We also added a Discussion paragraph on directed/wasted effort:
“Finally, we note that the primary benefit of sparsity is that sparse networks are likely to be at least somewhat bipartite. The more bipartitelike a network is, the less effort is wasted, and the easier it is for specialization to be favored.”
Finally we added a section on the four cell network:
“Jensen's inequality and sparse topologies and asymmetric sharing
To understand how average fitness decouples from average 𝑣 and average 𝑏 for sparse topologies and asymmetric sharing, consider a ring of four cells in three different configurations: one that alternates between viability and fecundity specialists, one in which likespecialists are connected to each other, and one in which all cells are generalists. […] Therefore, Jensen’s inequality does not directly inform average fitness, and we should not expect convex ROI functions to be required for specialists to be favored.”
3) There are wellknown cases presented in the population genetics literature in which Fisher's fundamental theorem seems violated, but this is because of the environment (including the social environment) changes over time, such as frequencydependent effects on an individual's success. We wonder if the results of the authors could not related to this literature in some way.
This question seems to be related to the central issue raised above – is there a hidden feature in the model that makes specialization beneficial despite a concave investment function (i.e., by adding convexity somewhere or introducing some frequencydependent or environmental effect)? Above, we explained the logic of why specialization can be favored despite concave investment functions.
Also, to be clear, we don’t see our work as violating Fisher’s fundamental theorem, which states that selection will increase the mean fitness of a population at a rate proportional to the genetic variation in fitness in the population. Because our groups are clonal, all the genetic variation in our population occurs between groups, not within groups, and selection is acting only on grouplevel fitness. Thus, Fisher’s fundamental theorem is at a limit within groups (as there is no withingroup genetic variation, there is no withingroup evolution).
4) The model description is a bit abstract and occasionally hard to follow. It would be great to have fecundity and viability defined, and even better to have some real biological example of what returns on viability might mean and how they might be shared (I don't find the filamentous fungi example informative, at least not in the way it is written). That would also help the reader understand why there are returns on viability but not on fecundity. That the v_{i} vector is the "group investment strategy" also comes as a surprise and takes a bit for the reader to put it all together. Similarly, the existence of both a general adjancecy matrix and of a special case one that uses the β, is somewhat confusing the way it's described. If the authors anyway only work with the special case of equal sharing with the nonself neighbors then why not define the 1β+β/ni quantities as c_{ij} when they appear in the text, and then write a fourth eqn for W in [1] that explicitly uses the β. That would certainly help the reader a lot.
These are great points, and we agree that it is crucial that the model and its biological meaning are clear to the reader.
The dichotomy of viability vs. fecundity was originally used by Michod to partition components of cellular fitness into actions that contribute to keeping a cell alive (viability), and actions that directly contribute to reproduction (fecundity). The intuition underlying this is that multicellular organisms often have evolved to divide labor along these two lines (i.e., reproduction of the organism by germ cells and survival provided by somatic cells), while their unicellular ancestors had to do both. We define viability as an activity (Michod uses the term “effort”) that keeps the cell alive (e.g., investing in cellular homeostasis or behaviors that improve survival), and fecundity more narrowly as effort involved in cellular reproduction itself. At the cellular level, there appears to be a fundamental asymmetry in how viability effort and fecundity effort can be shared among cells: while multicellular organisms readily evolve differentiated cells that are completely reliant on helper cells for survival (i.e., glial cells that support neurons in animals or companion cells that support sieve tube cells in plants), no cell can directly share its ability to reproduce. To better understand the intuition behind this, consider a cell that elongates prior to fission. This cell must grow to approximately twice its original length. Two cells cannot elongate by 50% and then combine their efforts; elongation is an intrinsically single cell effort.
While we believe this was also Michod’s interpretation of “reproductive effort”, this concept, like many in biology, is subject to interpretation. While it is clear the cellular behaviors underlying replication cannot be shared, resources required for reproduction could have been provided by another cell. Fortunately, our main conclusions are general and can accommodate various definitions of reproductive effort, which include sharing. While we present the simplest case (no sharing of reproductive effort) to help explain our paper’s central idea as clearly as possible, we do not need to forbid sharing of reproductive effort in order for specialization to evolve with concave ROI functions. As long as there is a significant asymmetry in how much 𝑣 and 𝑏 are shared, specialization with concave ROI functions will evolve. We explore this generalization in Figure 2. Because much of the effort involved in reproduction is, by its nature, unshareable, we believe that asymmetry in sharing between 𝑣 and 𝑏 is a general feature of multicellular systems. Of course, our results are far more general than the evolution of multicellularity, and should apply to many systems in which entities trade along networks (biological, economic, etc.). We have updated the description in the text of how cells can share viability, illustrating the general point with additional examples.
Finally, we have updated our description of the model to clarify the roles of 𝛽 and 𝑐_{𝑖𝑗}.
We modified the “Model” section to read:
“Reproductive specialization can be modeled as the separation of two key fitness parameters, those related to either viability or fecundity, into separate cells within the multicellular organism (13,35). […]Thus, selection is able to act on the multicellular fitness consequences of different patterns of cellular behavior within the group.”
And added clarifying statements like:
“In other words, 𝑐_{𝑖𝑖} = 1 − 𝛽, 𝑐_{𝑖𝑗} = 𝛽/𝑛_{𝑖} if cells 𝑖 and 𝑗 are connected, and 𝑐_{𝑖𝑗} = 0 if cells 𝑖 and 𝑗 are not connected.”
5) Results subsection “Fixed resource sharing” first paragraph, we may be getting confused, but how can you vary β in the case when, as is now written, individual i "shares equally among interaction and self terms"? Doesn't this mean that β = 1?
We do not allow cells to give all of their viability returns away, as this would result in cells that are not viable. Cells keep (1 − 𝛽) of their viability returns, and designate β of their viability returns for sharing. They then split their viability returns designated for sharing into 𝑁 + 1 equally sized portions, give one portion to each of their 𝑁 neighbors, and keep one portion for themselves. In other words, the β portion that is designated “for sharing” is not all given away, but is split equally among the connected cells and the cell itself. We have updated the explanation accordingly.
We removed the sentence “Individual 𝑖 shares 𝑣_{𝑖}^{𝛼} equally among interaction and self terms.” that was on line 146. The model was explained in the previous section, so this statement was ultimately redundant and unclear. Instead, we modified the model explanation to read:
“Cells cannot give away all of their viability returns, as they would no longer be viable; mathematically, we count a cell among its neighbors and thus ensure that they always “share” a positive portion of viability returns with themselves, so that 𝑐_{𝑖𝑖} > 0.”
And
“…and when 𝛽 = 1 cells share everything equally among all connections and themselves.”
6) “We conjecture that the troughs in Figure 3C, where specialization occurs for the lowest values of, occur when connectivity is just large enough so that a spanning tree is more likely to connect all individuals in the group than not”, we don't fully understand that conjecture: do the authors simply mean that the troughs occur when the random graph becomes connected with probability > 50%? (A spanning tree connects all individuals by definition.)
The reviewers are correct that a spanning tree connects all individuals. We were attempting to state this fact using technical language that would be clear to experts (50% probability that a spanning tree was present) as well as with more general language accessible to nonexperts (50% probability that all individuals connected). We have updated this statement to be clear to all.
We modified this statement to read:
“We conjecture that the troughs in Figure 3B, where specialization occurs for the lowest values of 𝛼, occur when connectivity is just large enough so that the existence of a spanning tree is more likely than not.”
7) The authors suggest sparsity is the main determinant of whether a network supports reproductive specialization. But, their examples in 1B and 1C (where a ring is sparser than the bipartite network) to us suggest that it is not so much about "sparsity" as it is about "bipartiteness" – or how easy it is to subdivide the nodes into two classes such that most edges go between these two classes (that's what you'd want for specialization anyway, we guess), and that sparse graphs simply have a tendency to be close to bipartite. We suspect that a ring graph with an odd number of vertices will be less conducive to specialization (although you could still alternate germ/soma cells except at one point), and that a star graph where there is one node of degree n1 and all the others have degree 1 may be an example of a sparse graph where evolving specialization is not so easy (because for this graph it's not clear how to divide the vertices into germ and soma).
Thank you for this point, with which we largely agree; “bipartiteness” is more important than sparsity. Sparsity is important for two reasons. First, as the reviewers suggest, sparse graphs tend to be close to bipartite. Second, sparse topologies are common in nature. Thus, sparsity may be a common natural route to being somewhat close to bipartite. We have modified the text to make this point clear. However, as will be highlighted by our discussion below, additional features of the graph, such as connectivity, can play a role in determining if specialization is favored as well.
We also appreciate the star graph suggestion, which is a very interesting topology. This graph is also relevant to understanding the evolution of asymmetric investment in germ and somatic cells, as is common in among independentlyevolved multicellular organisms, like animals, plants, and volvocine green algae. Hopefully after clarifying our model in response to previous questions it is now clear that the star graph can strongly favor specialization. We now work this example in detail in the supplement. Briefly, we will discuss a five cell star here.
If the central cell were a viability specialist, specialists would have a lower fitness than generalists. However, if the central cell is a fecundity specialist, then the four surrounding viability specialists provide the central cell with enough viability to make this configuration favored. In fact, for large groups (𝑁 > > 1), specialists are always favored – for any 𝛽 and any 𝛼. For the five cell examples shown above, specialists are favored for α values as low as 0.222; a heat map relating degree of specialization to 𝛼 and 𝛽 is shown in Author response image 3.
Finally, the reviewers are correct about rings with even and odd numbers of cells. The effect is small, but it is true that specialization is frustrated in rings with odd numbers of cells. In fact, we cut a discussion of this phenomenon from our manuscript as we worried that it would be a distraction. However, it is now clear that its absence in fact raises questions, so we have restored it.
Here, 𝛼^{∗}, the lowest value of 𝛼 for which specialization is favored, is plotted versus the number of cells in the ring, for rings with even and odd numbers of cells.As discussed above, we added a Discussion paragraph that reads:
“Finally, we note that the primary benefit of sparsity is that sparse networks are likely to be at least somewhat bipartite. The more bipartitelike a network is, the less effort is wasted, and the easier it is for specialization to be favored.”
We added a section analytically working out when specialists and generalists are favored for star topologies with various N, 𝛼, and 𝛽.
We added a section discussing even and odd numbered rings.
8) Related to the previous point: we would be interested if the authors have considered what happens when the optimal strategy is not 1:1 but, say, 1:2. Does that make specialization more difficult? Here we think that, with a few additional simulations, the authors could add a lot to the paper in terms of the ability to connect properties of the graph (beyond comparing some explicit topologies and random graphs of varying sparsity) to its ability to support the evolution of reproductive specialization.
Thank you for this interesting suggestion. First, we return to the mean field model introduced in comment 2, but now allow the fraction of fecundity specialists to be 𝑋 (rather than forcing 𝑋 = ½). The model itself is presented in detail in the text changes below; here, we summarize the results.
Our mean field model suggests that a larger proportion of fecundity specialists makes concave specialization easier to achieve. Further, we find that concave specialization is only possible if more than one fourth of cells are fecundity specialists.
We stress here that this is a mean field model, and does not apply to scenarios like the star network, in which cells have very different individual values of 𝑧. If such networks do or do not favor specialization for 𝛼 < 1 will again be a graph coloring problem.
We added a section on varying ratios of specialists:
“Effect of varying ratios of specialists
We now allow the fraction of fecundity specialists to be 𝑋 (rather than forcing 𝑋 = 1/2). […] If such networks do or do not favor specialization for 𝛼 < 1 will again be a graph coloring problem.”
9) Finally, it would be nice to see how the different specialists are distributed on these networks (at least when the specialization is equal to 1). One can infer it, but we think it would visually help the reader to get the gist of how the model works very quickly.
We have visualized investment in different tasks on fully connected, ring, and bipartite graphs. We agree that these images are instructive for the reader.
In Author response image 4 is an image of specialists in a nearest neighbor topology:
In Author response image 5 for the bipartite network, complete specialization happens readily:
Author response image 6 for the complete network, generalists dominate with some fluctuations:
We modified Figure 1 to incorporate these images:
Article and author information
Author details
Funding
National Science Foundation (IOS1656549)
 William C Ratcliff
 Peter J Yunker
National Institutes of Health (GM138030)
 William C Ratcliff
National Science Foundation (BMAT2003721)
 Peter J Yunker
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Senior Editor
 Diethard Tautz, MaxPlanck Institute for Evolutionary Biology, Germany
Reviewing Editor
 Raymond E Goldstein, University of Cambridge, United Kingdom
Reviewer
 Pierrick Bourrat, MacQuarie University, Australia
Publication history
 Received: December 11, 2019
 Accepted: September 17, 2020
 Accepted Manuscript published: September 17, 2020 (version 1)
 Version of Record published: November 3, 2020 (version 2)
Copyright
© 2020, Yanni et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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