The evolution of division of labour in structured and unstructured groups

Recent theory has overturned the assumption that accelerating returns from individual specialisation are required to favour the evolution of division of labour. Yanni et al., 2020, showed that topologically constrained groups, where cells cooperate with only direct neighbours such as for filaments or branching growths, can evolve a reproductive division of labour even with diminishing returns from individual specialisation. We develop a conceptual framework and specific models to investigate the factors that can favour the initial evolution of reproductive division of labour. We find that selection for division of labour in topologically constrained groups: (1) is not a single mechanism to favour division of labour—depending upon details of the group structure, division of labour can be favoured for different reasons; (2) always involves an efficiency benefit at the level of group fitness; and (3) requires a mechanism of coordination to determine which individuals perform which tasks. Given that such coordination must evolve prior to or concurrently with division of labour, this could limit the extent to which topological constraints favoured the initial evolution of division of labour. We conclude by suggesting experimental designs that could determine why division of labour is favoured in the natural world.


Introduction 25
Division of labour, where cooperating individuals specialise to carry out distinct tasks, plays 26 a key role at all levels of biology (Bourke, 2011;Queller, 1997 could become more effective as more effort is put into it, or it could be carried out with 46 diminishing costs. This could occur if there are large upfront costs from performing a task. 47 For instance, any reproduction by a cell in Volvocine groups first requires individual growth 48 to the size of a daughter colony (Michod, 2006). Alternatively, there could be a disruptive 49 cost to carrying out multiple tasks at the same time if the tasks don't mix well. For instance, 50 in cyanobacteria the enzymes that fix environmental nitrogen are degraded by oxygen, a bi-51 product of photosynthesis (Flores & Herrero, 2010). Without loss of generality, we assume that the mutant strain is composed of n h helpers that 118 invest z h ≥ 0 into cooperation and n r reproductives that invest z r ≥ 0 into cooperation 119 (where n h + n r = n > 2 and z h > z r ). We set individual fitness as the product of individual 120 fecundity, F > 0, and individual viability, where helpers and reproductives may in principle 121 have different viability functions, V h > 0 and V r > 0 (but see appendix C.2.) (Michod, 2006;122 Yanni et al., 2020). The fitness of the clonal group is given by the sum of individual fitness: 123 124 W(z h , z r )=n h F(z h )V h (z h ,z r )+n r F(z r )V r (z h ,z r ), ( 1 ) 125 where the first term on the right-hand side is the total fitness of the prospective helpers and 126 the second term on the right-hand side is the total fitness of the prospective reproductives. 127 128 Fecundity is determined by an individual's investment in cooperation (F = F(z), where is 129 the focal individual's level of cooperation) and viability is determined by the level of 130 cooperation at the level of the group (V h = V h (z h , z r ) and V r = V r (z h , z r )). We assume that 131 We determine the invadability conditions that favour reproductive division of labour by 140 applying the general approach of Rueffler et al. 2012. The key step is to approximate the 141 relative fitness of a reproductive division of labour mutant by taking a second-order Taylor 142 expansion of fitness, centred on the resident strategy of uniform cooperation, z * : 143 144 W(z * + Δz h , z * + Δz r ) − W(z * , z * ) ≈ W z h Δz h + W z r Δz r + 1 2 W z h z h Δz h 2 + 1 2 W z r z r Δz r 2 + W z h z r Δz h Δz h , where Δz h > Δz r captures the change in the level of cooperation for mutant helpers and 145 reproductives, respectively, which we assume are small in magnitude. The superscripts 146 represent first and second order partial derivatives, where all partial derivatives are evaluated 147 at the resident strategy of uniform cooperation (z h = z r = z * ). If a mutant strain exists such 148 that Equation 2 is positive (W(z * + Δz h , z * + Δz r ) > W(z * , z * )), then division of labour 149 between helpers and reproductives is favoured to evolve. Conversely, if for all possible 150 mutant strains, Equation 2 is negative (W(z * , z * ) > W(z * + Δz h , z * + Δz r )), then uniform 151 cooperation is evolutionarily stable. 152

153
The three pathways to division of labour 154 We found that reproductive division of labour could be favoured for three distinct reasons, 155 corresponding to different subsets of terms on the right-hand side of Equation 2. Our results 156 for reproductive division of labour, where fitness is partitioned as the product of fecundity 157 and viability, align with those found by Rueffler et al. 2012 for division of labour more 158 generally. We now go through these three distinct scenarios. 159 160 Scenario 1: Accelerating returns from individual specialisation 161 The first and most studied scenario that can favour division of labour is when there are 162 accelerating returns from individual specialisation. This occurs if there is an accelerating 163 fitness return from either helper specialisation in cooperation and / or reproductive 164 specialisation in fecundity (Figure 1) (Cooper & West, 2018;Michod, 2006;Oster & Wilson, 165 1978). 166 167 Mathematically, this scenario is a consequence of the third and fourth terms of the Taylor 168 expansion (2c & 2d), which capture the second-order fitness effect of a small, unilateral 169 change in cooperation by either prospective helpers or reproductives, respectively. Division 170 of labour is favoured to evolve whenever at least one of 2c and 2d is greater than zero 171 (W z h z h > 0 or W z r > 0), and where we have assumed that the first two terms are both 172 zero (W z h = 0 and W z r = 0; see between-individual differences below; Figure 2A). In either 173 scenario, an efficiency benefit to group fitness arises from individual specialisation because 174 the more effort that each individual puts into a task, the better they can perform that task. The second scenario that can favour reproductive division of labour is when there are pre-180 existing differences between individuals in the group, such that some individuals are 181 predisposed to one task or the other. For example, if some individuals can secure larger 182 viability benefits for the group at the same fecundity cost as others ( Figure 2B). 183 We term this scenario 'between-individual differences' because it requires that there is pre-199 existing phenotypic or environmental variation between individuals in the group. For the 200 within-species case, ancestral groups are usually composed of clonal or highly related 201 individuals, who will be phenotypically similar or identical. Consequently, this mechanism 202 could be less important for the division of labour except when there are consistent differences 203 The final scenario that can favour division of labour is when reciprocal specialisation by both 220 helpers and reproductives provides a fitness benefit to the group (Figure 3). This scenario 221 requires two key conditions. First, simultaneous specialisation, where some individuals invest 222 more in cooperation (more viability benefits for the group), and others invest less in 223 cooperation (greater individual fecundity; but see below). Second, this reciprocal 224 specialisation must provide a group-level fitness benefit, because the increased benefits of 225 cooperation are preferentially directed towards reproductives.

Or
Mathematically, this scenario involves the last term of the Taylor expansion 241 (2e; W z h z r Δz h Δz r ). This term is generated by a between-individual, second-order fitness 242 effect, capturing how increased investment in viability by some individuals affects the returns 243 from increased investment in fecundity by others, and vice-versa. Rueffler at al. referred to 244 this as a 'synergistic benefit' to division of labour (Queller, 1985(Queller, , 2011Rueffler et al., 2012). 245 246 Critically, this scenario still involves an efficiency benefit to specialisation, but at the level of 247 group fitness rather than in each fitness component separately (appendix C.1). By this we 248 mean that there is an accelerating fitness benefit to the group when helpers and reproductives 249 reciprocally specialise, leading to a higher group fitness than in groups with uniform 250 cooperation (generalists). This occurs if the increased help given to reproductives is Division of labour by reciprocal specialisation can also evolve without a joint mutation in the 256 level of cooperation of both helpers and reproductives (no simultaneous specialisation). In 257 this case, the chance invasion (to fixation) of a slightly deleterious mutant that specialises in 258 only one phenotype ( Figure 5A) can destabilise uniform cooperation, creating a selection 259 pressure for the other phenotype to also specialise that is greater than the selection pressure to 260 purge the initial mutant. In this two-step scenario, it is nevertheless the synergistic benefit 261 from reciprocal specialisation that makes division of labour more efficient. 262 263 Group structure in the general framework 264 Our above analysis has shown that reproductive division of labour can be favoured for three 265 reasons: (1) accelerating returns make individual specialisation more efficient; (2) between-266 individual differences make individual specialisation more efficient; or (3) there is a 267 synergistic efficiency benefit from reciprocal specialisation. These results agree with 268 previous analyses by Rueffler et al. 2012. . We term cells with three neighbours "node" 289 cells and cells with two neighbours "edge" cells. We assume that cells investing an amount 290 ≥ 0 into cooperation produce an amount H(z) of a public good. We assume non-291 accelerating returns from individual specialisation (i.e. H ′′ (z) ≤ 0 or F′′ (z) ≤ 0). The cell 292 keeps a fraction 1 − λ of the public good that it produces, and the remaining fraction λ is 293 shared equally between its direct neighbours (the "shareability" of cooperation: 0 < ≤ 1 ). 294 We assume that the viability of a cell is equal to the sum of the public good that it absorbs. group. Edge cells receive relatively less public good from their (fewer) neighbours, and so 300 pay a smaller opportunity cost from decreased fecundity (increased cooperation). In contrast, 301 node cells receive relatively more public good from their (more numerous) neighbours, and 302 so pay a larger opportunity cost from decreased fecundity (increased cooperation). 303 Consequently, this between-cell difference favours node cells to specialise in fecundity 304 (reproductives) and edge cells to specialise in increased cooperation (helpers). Importantly, 305 because this pathway to division of labour is driven entirely by a first-order effect (2a & 2b), 306 it does not require a second-order efficiency benefit from specialisation (2c, 2d or 2e). We 316 assume arbitrarily that "odd" cells along the filament are putative helpers and "even" cells are 317 putative reproductives. We otherwise make the same assumptions as for the branching 318 structure model: there is a non-accelerating return from individual specialisation (i.e. 319 H ′′ (z) ≤ 0 or F′′ (z) ≤ 0), and the cell keeps a fraction 1 − λ of the public good that it 320 produces, with the remaining fraction λ being shared equally by its direct neighbours. 321

322
If the amount of public good shared with neighbours is sufficiently large (high λ), then we 323 find that division of labour via reciprocal specialisation can evolve ( Figure 4B; appendix A. 324 3). For instance, in the case of linear fecundity and public good returns (H ′′ (z) = F ′′ (z) = 0), 325 division of labour by reciprocal specialisation can evolve if helpers share more of the public 326 good that they produce with their neighbours than they keep for themselves (λ > 1 2 ). If there 327 are diminishing returns from specialisation (H ′′ (z) < 0 or F ′′ (z) < 0), then division of 328 labour can still be favoured but then the amount of the public good preferentially shared with 329 neighbours must be even greater still (higher λ; Figure 4E). 330

331
For an arbitrary group structure, our analysis in the previous section implies that division of 332 labour can evolve by between-individual differences, unless every cell in the group has the 333 same number of neighbours. Consider a group in which every cell has exactly neighbours. 334 In this case, we show (appendix A.5) that division of labour can still be favoured due to 335 reciprocal specialisation if: 336 is the shareability of cooperation as defined previously, and captures how easily the 339 group can be "bi-partitioned". That is, is a measure of the extent that the group can be  Figure 4C). These results are like those found for a filament of cells 414 ( Figure 4B). In both cases, more generous sharing (higher λ) means that the synergistic 415 benefits of reciprocal specialisation can be great enough to compensate for the non-416 accelerating returns from individual specialisation. In well-mixed groups, very generous 417 sharing (λ ≈ 1) also compensates for the fact that helpers are neighbours with all other 418 helpers (no sparsity and minimally "bi-partionable"). Consequently, if coordination was required, then this could limit the extent to which 447 topological constrains favour the initial evolution of division of labour. 448

449
We investigated this hypothesis by repeating our above analyses, while assuming that cells do 450 not have access to information that allows them to coordinate their phenotypes. Specifically, 451 cells do not know if they are "odd" or "even", or if they are "edge" or "node". We assumed 452 instead that a reproductive division of labour mutant induces each cell in the group to adopt 453 the role of a helper or reproductive with a uniform probability (random specialisation).   test whether division of labour is favoured by between-individual differences we must 544 determine whether an increase in cooperation by helpers produces a different group fitness 545 benefit than an increase in cooperation by reproductives. (C) To test whether division of 546 labour is favoured by reciprocal specialisation, we must determine whether there exists at 547 least one relative degree of helper-to-reproductive specialisation for which group fitness is 548 greater than the fitness of uniform cooperation. 549 550 551

Conclusion 552
Division of labour can be favoured to evolve without accelerating returns from individual 553 specialisation. Nevertheless, for this to occur requires: (a) between-individual differences in 554 task-efficiency or synergistic benefits from reciprocal specialisation; and (b) a mechanism to 555 coordinate which individuals perform which tasks. In contrast, accelerating returns can 556 favour division of labour without a mechanism to coordinate task allocation, possibly making 557 it more likely to favour the initial evolution of division of labour. Ultimately, determining the 558 relative importance of these different pathways to division of labour is an empirical question, 559 requiring experimental studies of the type we have outlined above. More explicitly, we can write this as: where we have suppressed the functional dependencies for ease of presentation. 575 The first term gives the group fitness change due to a marginal increase in "helper" 576 cooperation, and the second term gives the group fitness change due to a marginal 577 increase in "reproductive" cooperation. So, at the uniform strategy, * , any increase 578  This says that the contribution to total viability from the increased specialisation of 637 helper individuals is strictly larger than the contribution to total viability from the 638 increased specialisation of reproductives. As a result, helpers are pre-disposed to 639 become more helper-like as they can gain larger viability gains for the group than the 640 other type of individual.
(13) 650 651 2 2 | ℎ = = * = 2n r F ′ V r z r + n r F ′′ V r + n h FV h z r z r + n r FV r z r z r (14) 652 653 The terms of Equations 13 and 14 capture the second order effects of increased 654 investment in cooperation. The first term of each captures the decline in the fitness 655 benefit of increased cooperation due to the cross-interaction between fecundity and 656 viability. For instance, as a helper invests more in cooperation (higher ), it increases 657 its own viability (higher ℎ ), but its fecundity declines as well (lower ) and so the 658 relative benefit of this increased viability is lessened (the cross term ′ ℎ ℎ is 659 negative). This represents a kind of decelerating return from cooperation. The 660 second term of each captures the second-order effect of decreased investment in 661 fecundity. If this term is positive, then this means that there is a diminishing fecundity 662 cost to increased investment in cooperation, which can favour division of labour. The 663 third and fourth terms capture the second order effect of increased investment in 664 viability, that is: does each successive investment in the public good lead to a larger 665 or smaller increase in viability than the previous investment of the same size? The 666 return on investment (ROI) in viability is accelerating if ℎ ℎ ℎ > 0, ℎ ℎ > 0, ℎ > 667 0, and > 0. The ROI is diminishing if these second derivates are negative: 668 ℎ ℎ ℎ < 0, ℎ ℎ < 0, ℎ < 0, and < 0.

Division of labour by reciprocal specialisation 690
The last remaining scenario for division of labour is that the resident strategy of 691 uniform cooperation is unstable to mutations in both traits, which can occur 692 depending on the value of the last term of the Taylor expansion (Equation 2; 693 ℎ Δ ℎ Δ ). This kind of instability can arise if there is a joint mutation that affects 694 the level of cooperation of both helpers and reproductives at the same time (Δ ℎ ≠ 0 695 and Δ ≠ 0). However, it could also occur if a slightly deleterious mutation in one 696 trait invades by drift and destabilises the other trait so much that the population 697 evolves away from the critical point. In either case, it will be found that same 698 condition must be satisfied in order for division of labour to evolve. In the rest of this 699 section, we give the general analysis of whether ( ℎ , ) = ( * , * ) is unstable to 700 two-trait mutations, and then consider a simplifying special case to clarify the 701 biological interpretation of this analysis. Here, we have used that ℎ = ℎ . Substituting this, and Equations 13 and 14, 718 into the Hessian condition gives 719 720 The reciprocal specialisation condition for division of labour 721 722 Assume that neither Condition 15 nor 16 is satisfied: i.e. the individual ROI is non-725 accelerating. Then the left-hand side of the inequality is strictly positive, which 726 means that Condition 19 is nontrivial. We will see in examples that Condition 19 can 727 be satisfied, which means that division of labour can evolve by reciprocal 728 specialisation even when the individual ROI is diminishing.
which holds for every positive λ > 0.
where n h is the number of edge cells and n r is the number of node cells. Note that for any 909 graph in which n h cells have degree a, and n r cells have degree b, we have that n h /n r = b/a. condition for division of labour by between-individual differences: So, since b > a, we find that division of labour by between-individual differences occurs for all 914 values of λ > 0 (any social trait). 915 In order to calculate the benefit of division of labour in Figure 4 of the main text, we evaluate Moreover, A ii = 1, for each i. Notice that the column sums of the matrix A are zero: The fitness function of the group associated with this graph is which is maximised for some z = z * between 0 and 1, by our assumption that F is decreasing 926 in z and H is increasing in z. 927 We now restrict to the 'marginal' case that F (z) and H(z) are both linear functions. In 928 this case we can, without loss of generality, take F (z) = 1 − z, and H(z) = z. The uniform 929 cooperation strategy is given by z * = 1/2. The first derivatives of W at the uniform strategy 930 are given by: Notice that, for fixed i, this tree is: Using Equation 29, it follows that +2/3 i = 4, 5, 6, 7 So vertices 2 and 3 are 'predisposed' towards helping less, whereas the more peripheral ver-940 tices (i = 1, 4, 5, 6, 7) are predisposed towards helping more. The vertices with the fewest 941 neighbours (i = 4, 5, 6, 7) are also the most predisposed towards helping more. 'even' cells are reproductives. This gives the following fitness for the group: The viability of helpers is V h (z h , z r ) = (1−λ)H(z h )+λH(z r ) and the viability of reproductives In the case that H = 0, division of labour is favoured only if λ > 1/2. If H is less than 958 zero, then the returns on investment are decelerating, and λ must be larger than 1/2 in order 959 for division of labour to be favoured. 960 In order to calculate the fitness benefit of division of labour in Figure 4 of the main text, we 961 evaluate max −π≤θ≤π W (z * + ∆z cos(θ), z * + ∆z sin(θ)) − W (z * , z * ), where we approximate 962 fitness using a second order Taylor expansion and setting ∆z = 0.01. In this case, we can 963 show analytically that the direction of mutation giving the maximal increase in fitness satisfies 964 θ = {−π/4, 3π/4}.

965
A.4 A well-mixed group of cells 966 We now consider a group that is "well-mixed" such that cells share the benefits of cooperation 967 with all other group members. That is, we let 1 − λ be the amount of benefits produced by an 968 individual that it keeps for itself and λ is the amount of benefits that is shared equally by all 969 other cells in the group. This produces the following expected fitness function: where n = n h + n r and we have that the viability of a helper is V h (z h , z r ) = (1 − λ)H(z h ) + Otherwise, if the returns from cooperation are diminishing (say, H < 0, with F linear), then 979 we can use the general condition 17 to find, Notice that, for diminishing returns (H < 0), we have that x > 0, and in this case it follows 981 that:

982
(1 + n h x)(1 + n r x) 1 + In other words, if H < 0, then λ must be even higher for division of labour to be favoured, 983 than when H = 0.

984
In order to calculate the fitness benefit of division of labour in Figure 4 of the main text, we 985 evaluate max −π≤θ≤π W (z * + ∆z cos(θ), z * + ∆z sin(θ)) − W (z * , z * ), where we approximate 986 fitness using a second order Taylor expansion and setting ∆z = 0.01.

987
A.5 General graph analysis: reciprocal specialisation 988 We now return to the case of a general graph model, and consider when division of labour is 989 possible via reciprocal specialisation. Once again, fix some graph G. Assume that the first 990 derivatives, ∂ i W , vanish at the strategy of uniform cooperation (z i = z * ). Recall (Equation 29) 991 that this is equivalent to assuming that all the vertices of the graph G have the same degree, d.

992
The matrix of second derivatives of W is 993 Hess = ∂ 2 W ∂z i ∂z j q * = 2F (z * )H (z * ) (δ ij − λM ij ) , where M is a symmetrized version of the A matrix: All vertices have the same degree, say d, so that the matrix M is given by where L is the Laplacian matrix: The uniform strategy (z = z * ) is unstable if and only if the matrix Hess has one or more where the characteristic polynomial P(x) is a degree n polynomial defined by: 1004 P(x) = det (L − xI) .
The roots of P(x) are all non-negative (this is because L is a symmetric matrix). Moreover, Recall that d is the number of neighbours that an individual cell has in the graph. The eigen-1009 value µ can be thought of (see below) as a measure of how 'bipartitionable' the graph is. The 1010 inequality can then be interpreted loosely as follows: if the graph is made "more bipartite", λ 1011 can decrease. If the number of neighbours, d, is increased, λ must increase. 1012 To further understand µ, a useful property of L is that, for any vector of numbers x = 1013 (x 1 , ..., x N ): The largest eigenvalue of L is On the other hand, It follows that µ ≤ 2d.
The total expected fitness is then If λ ≤ 1/2, the function has a critical point at z h = z r = z = 1/(2(1 − λ)), where its first 1113 derivatives vanish. We compute that the determinant of the Hessian of W at the critical point 1114 is 1115 which is positive. So