Does the evolution of division of labour require accelerating returns from individual specialisation?

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 developed a conceptual framework and specific models to investigate the factors that can favour the initial evolution of reproductive division of labour. We found 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 is unlikely to evolve before division of labour, this limits the extent to which topological constraints could have 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 26
Division of labour, where cooperating individuals specialise to carry out distinct tasks, plays a 27 key role at all levels of biology (Bourke, 2011;Queller, 1997;Maynard Smith & Szathmáry, 28 1995;West et al., 2015). Cells are built by genes carrying out different functions (Bourke, 29 2011;Levin & West, 2017). In clonal groups of bacteria, cells specialise to produce and secrete 30 different factors that facilitate growth (Dragoš et al., 2018a;Veening et al., 2008;West & 31 Cooper, 2016). Pathogens rely on division of labour for protection from the host immune 32 response and competitors (Ackermann et al., 2008;Diard et al., 2013). Multicellular organisms 33 are composed of reproductive germ cells and sterile somatic cells that are not passed to the next 34 generation (Bourke, 2011;Maynard Smith & Szathmáry, 1995). The ecological dominance of 35 the social insects arises from division of labour between queens and the different types of 36 workers (castes) (Oster & Wilson, 1978;Wilson, 1978). 37 38 It has long been established that the evolution of division of labour requires an efficiency 39 benefit from individual specialisation ( Figure 1A & 1B for reproductive division of labour) 40 (Bourke, 2011;Cooper & West, 2018;Ispolatov et al., 2012;Michod, 2006;Oster & Wilson, 41 1978;Schiessl et al., 2019;Smith, 1776;Maynard Smith & Szathmáry, 1995;Solari et al., 42 2013). In particular, that there is an accelerating (convex) return when individuals commit more 43 effort to a particular task, such that twice the investment more than doubles the return (Bourke, 44 2011;Cooper & West, 2018;Ispolatov et al., 2012;Michod, 2006;Solari et al., 2013). An 45 accelerating return from individual investment can exist for several reasons. A task could 46 become more effective as more effort is put into it, or it could incur diminishing costs. This 47 could occur if there are larger upfront costs from performing a task. For instance, any 48 reproduction by a cell in Volvocine groups first requires individual growth to the size of a 49 daughter colony (Michod, 2006). Alternatively, there could be a disruptive cost to carrying out 50 multiple tasks at the same time, if the tasks don't mix well. For instance, in cyanobacteria the 51 enzymes that fix environmental nitrogen are degraded by oxygen, a bi-product of 52 photosynthesis (Flores & Herrero, 2010).  (Cooper 59 & West, 2018;Michod, 2006;Schiessl et al., 2019). (B) In contrast, an accelerating return from 60 more cooperation (or reproduction) favours reproductive division of labour, with some 61 individuals specialising in high levels of cooperation (helpers) and others in low levels of 62 cooperation (reproductives) (Cooper & West, 2018;Michod, 2006;Schiessl et al., 2019). We term this scenario 'between-individual differences' because it requires that there is pre-197 existing phenotypic or environmental variation between individuals in the group. For the 198 within-species case, ancestral groups are usually composed of clonal or highly related 199 individuals, who will be phenotypically similar or identical. Consequently, this mechanism 200 could be less important for the division of labour except when there are consistent differences 201 in the microenvironment experienced by different individuals (Tverskoi et al., 2018;Tverskoi 202 & Gavrilets, 2021). In contrast, this scenario is likely to be widespread in the evolution of non-203 reproductive division of labour between species, such as for mutualisms or symbioses (Kiers et 204 al., 2011;Rueffler et al., 2012;Wyatt et al., 2014). Individuals of different species often differ 205 in their abilities to perform certain tasks (Kiers et al., 2011). Rueffler et al. termed this scenario 206 'positional', but we avoid that term to prevent confusion with topological position (Rueffler et 207 al., 2012). 208 209 Between-individual differences provides a first-order fitness benefit to dividing labour, and so 210 it does not matter whether the subsequent benefits of increased cooperation or fecundity are 211 accelerating or diminishing (Figure 1), so long as these benefits are different for different 212 individuals ( Figure 2). When some individuals are predisposed to being either helpers or 213 reproductives, then individual specialisation provides an efficiency benefit to group fitness by 214 capitalising on these inherent differences. The final scenario that can favour division of labour is when reciprocal specialisation by both 218 helpers and reproductives provides a fitness benefit to the group (Figure 3). This scenario 219 requires two key conditions. First, simultaneous specialisation, where some individuals invest 220 more in cooperation (more viability benefits for the group), and others invest less in 221 cooperation (greater individual fecundity; but see below). Second, this reciprocal specialisation 222 must provide a group-level fitness benefit, because the increased benefits of cooperation are 223 preferentially directed towards reproductives. 224 225 226 Figure 3: Division of labour is favoured by reciprocal specialisation. We assume that there 227 are diminishing returns from specialisation in either viability or fecundity ( Figure 1A). (A) In 228 this case, a unilateral increase in cooperation by helpers or a unilateral decrease in cooperation 229 by reproductives leads to a diminishing fitness benefits to the group, which favours uniform 230 cooperation (no division of labour). (B) In contrast, a reciprocal increase in cooperation by 231 helpers (more viability benefits provided by helpers) and a decrease in cooperation by 232 reproductives (larger reproductive fecundity) can produce an accelerating return to the fitness 233 of the group if the benefits of increased cooperation are preferentially directed to reproductives. 234 Thus, reciprocal specialisation can still favour division of labour, even though the returns from 235 individual specialisation are diminishing. In the middle plots of (A) and (B), only the shape of 236 the benefits from increased specialisation are plotted.

Or
Mathematically, this scenario involves the last term of the Taylor expansion 239 (2e; W ' ! ' " Δz " Δz # ). This term is generated by a between-individual, second-order fitness 240 effect, capturing how increased investment in viability by some individuals affects the returns 241 from increased investment in fecundity by others, and vice-versa. Rueffler at al. referred to this 242 as a 'synergistic benefit' to division of labour (Queller, 1985(Queller, , 2011Rueffler et al., 2012). 243 244 Critically, this scenario still involves an efficiency benefit to specialisation, but at the level of 245 group fitness rather than in each fitness component separately (appendix C.1). By this we mean 246 that there is an accelerating fitness benefit to the group when helpers and reproductives 247 reciprocally specialise, leading to a higher group fitness than in groups with uniform 248 cooperation (generalists). This occurs if the increased help given to reproductives is sufficiently 249 amplified by the increased fecundity of reproductives (Yanni et al., 2020). This synergistic 250 efficiency benefit can favour division of labour even if there are diminishing returns from 251 individual specialisation. 252 253 Division of labour by reciprocal specialisation can also evolve without a joint mutation in the 254 level of cooperation of both helpers and reproductives (no simultaneous specialisation). In this 255 case, the chance invasion (to fixation) of a slightly deleterious mutant that specialises in only 256 one phenotype ( Figure 5A) can destabilise uniform cooperation, creating a selection pressure 257 for the other phenotype to also specialise that is greater than the selection pressure to purge the 258 initial mutant. In this scenario, it is nevertheless the synergistic benefit from reciprocal 259 specialisation that makes division of labour more efficient. Our above analysis has shown that reproductive division of labour can be favoured for three 263 reasons: (1) accelerating returns make individual specialisation more efficient; (2) between-264 individual differences make individual specialisation more efficient; or (3) there is a synergistic 265 efficiency benefit from reciprocal specialisation. These results agree with previous analyses by 266 Rueffler et al. (Rueffler et al., 2012). We consider two spatial models, based on the group structures proposed by Yanni et al.,to 280 examine whether topologically constrained groups favour division of labour by: (a) between-281 individual differences; and / or (b) reciprocal specialisation (Yanni et al., 2020). 282 283 (a) Can topological constrains lead to division of labour by between-individual differences? 284 Consider a group in which cells alternately have either two or three neighbours, in a branching 285 structure ( Figure 4A). Such a group structure might have occurred for some early forms of 286 multicellular life (Yanni et al., 2020). We term cells with three neighbours "node" cells and 287 cells with two neighbours "edge" cells. We assume that cells investing an amount . into 288 cooperation produce an amount H(z) of a public good. We assume non-accelerating returns 289 from individual specialisation (i.e. H && (z) ≤ 0 or F′′ (z) ≤ 0). The cell keeps a fraction 1 − λ 290 of the public good that it produces, and the remaining fraction λ is shared equally between its 291 direct neighbours (the "shareability" of cooperation: 0 < @ ≤ 1 ). We assume that the viability 292 of a cell is equal to the sum of the public good that it absorbs. 293 294 For this model, we find that for all social traits (λ > 0), reproductive division of labour by 295 between-individual differences can evolve ( Figure 4D; appendix A.1). This occurs because 296 different cells have different viability-fecundity trade-offs depending on their position in the 297 group. Edge cells receive relatively less public good from their (fewer) neighbours, and so pay 298 a smaller opportunity cost from decreased fecundity (increased cooperation). In contrast, node 299 cells receive relatively more public good from their (more numerous) neighbours, and so pay 300 a larger opportunity cost from decreased fecundity (increased cooperation). Consequently, this 301 between-cell difference favours node cells to specialise in fecundity (reproductives) and edge 302 cells to specialise in increased cooperation (helpers). Importantly, because this pathway to 303 division of labour is driven entirely by a first-order effect (2a & 2b), it does not require a 304 second-order efficiency benefit from specialisation (2c, 2d or 2e). important at the onset of the evolution of multicellularity ( Figure 4B) (Yanni et al., 2020). We 314 assume arbitrarily that "odd" cells along the filament are putative helpers and "even" cells are 315 putative reproductives. We otherwise make the same assumptions as for the branching structure 316 model: there is a non-accelerating return from individual specialisation (i.e. H && (z) ≤ 0 or 317 F′′ (z) ≤ 0), and the cell keeps a fraction 1 − λ of the public good that it produces, with the 318 remaining fraction λ being shared equally by its direct neighbours. 319 320 If the amount of public good shared with neighbours is sufficiently large (high λ), then we find 321 that division of labour via reciprocal specialisation can evolve ( Figure 4E; appendix A. 3). For 322 instance, in the case of linear fecundity and public good returns (H && (z) = F && (z) = 0), division 323 of labour by reciprocal specialisation can evolve if helpers share more of the public good that 324 they produce with their neighbours than they keep for themselves (λ > * ( ). If there are 325 diminishing returns from specialisation (H && (z) < 0 or F && (z) < 0), then division of labour can 326 still be favoured but then the amount of the public good preferentially shared with neighbours 327 must be even greater still (higher λ; Figure 4E). 328 329 In appendix A.5, we show with a formal analysis of arbitrary group structures that division of 330 labour can be favoured due to reciprocal specialisation if: 331 332 λµ > d 333 where @ is the shareability of cooperation, B is a measure of how easily the group can be "bi-335 partitioned", and C is the average number of neighbours across all cells. Thus, reciprocal 336 specialisation can favour division of labour if: (1) groups are more sparse (low C); (2) groups 337 are structured such that helpers can be neighbours with reproductives more than with other 338 helpers, and vice-versa (high B); and/or (3) when the benefits of cooperation are preferentially 339 shared with neighbours (high λ). In combination, these three factors amplify the synergistic 340 benefits of reciprocal helper and reproductive specialisation, which can produce an accelerating 341 fitness return for the group, even when there are non-accelerating returns from individual 342 specialisation (appendix C. 1). 343

344
If one or two of these factors are particularly favourable for reciprocal specialisation, then the 345 condition(s) on the remaining factor(s) can be relaxed. For instance, cells in a filament have 346 only 2 neighbours (C = 2), and the potential alternation of helpers and reproductives in the 347 filament means that helpers can share their cooperative public goods with reproductives 348 exclusively (maximal B). Consequently, reciprocal specialisation is possible even when the 349 shareability of cooperation is reasonably low (e.g. λ > * ( for linear benefits). We considered a well-mixed social group of n cells, where all cells share the benefits of 376 cooperation with one another, and so there are no topological constraints ( Figure 4C). We then 377 examined whether division of labour could be favoured by: (a) between individual differences; 378 and / or (b) reciprocal specialisation. In both cases, we assume that when a cell invests z into 379 cooperation, it produces an amount H(z) of a public good. A cell keeps a fraction 1 − λ of the 380 public good that it produces and the remaining fraction λ is shared by the rest of the social 381 group members equally. We again consider the case where there is a non-accelerating return 382 In the well-mixed group of identical cells, we find that division of labour cannot arise by 387 between individual differences (appendix A. 4). This is because all cells have the same number 388 of neighbours, which we have shown more generally can never produce between-individual 389 differences. This prediction could be violated if one of our assumptions do not hold: for 390 instance, if there are consistent differences in the microenvironment that predispose some cells 391 to one task or the other (Tverskoi et al., 2018;Tverskoi & Gavrilets, 2021;Yanni et al., 2020). 392 393 (b) Can reciprocal specialisation favour division of labour without a topological constraint? 394 In the well-mixed group of identical cells, if the amount of public good shared with neighbours 395 is sufficiently large (high λ), then we find that division of labour via reciprocal specialisation 396 can evolve ( Figure 4F; appendix A. 4.). If there are linear returns from increased specialisation 397 (H && (z) = F && (z) = 0), then division of labour can evolve when the public good produced by 398 an individual benefits an average group-member more than the producer (λ > .,* . ; Figure 5F). 399 These results are like those found for a filament of cells ( Figure 4E). In both cases, more 400 generous sharing (higher λ) means that the synergistic benefits of reciprocal specialisation can 401 be great enough to compensate for the non-accelerating returns from individual specialisation. 402 In well-mixed groups, very generous sharing (λ ≈ 1) also compensates for the fact that helpers 403 are neighbours with all other helpers (no sparsity and minimally "bi-partionable"). 404

405
To conclude, the well-mixed model shows that a topological constraint is not required for the 406 evolution of division of labour with non-accelerating returns from individual specialisation. 407 This is in direct contradiction to the results of Yanni et. al., in which a helper was never allowed 408 to benefit less than any of its neighbours from its own public good production (@ ≤ +,* + ) (Yanni 409 et al., 2020). In our model with a group of well-mixed cells, division of labour is favoured 410 because it provides an efficiency benefit at the group level, via reciprocal specialisation 411 We hypothesised that the benefits of between-individual differences (scenario 2) or reciprocal 417 specialisation (scenario 3) rely on the implicit assumption that cells are coordinating which 418 individuals specialise to become reproductive and helpers. This matters because mechanisms 419 for coordinating division of labour, such as between cell signalling, might not be expected to 420 exist before division of labour has evolved. Consequently, if coordination was required, then 421 this could limit the extent to which topological constrains favour the initial evolution of 422 division of labour. 423 424 20 We investigated this hypothesis by repeating our above analyses, while assuming that cells do 425 not have access to information that allows them to coordinate their phenotypes. Specifically, 426 cells do not know if they are "odd" or "even", or if they are "edge" or "node". We assumed 427 instead that a reproductive division of labour mutant induces each cell in the group to adopt the 428 role of a helper or reproductive with a uniform probability (random specialisation). Random 429 specialisation has been observed in a number of microbes (Ackermann et al., 2008;Diard et 430 al., 2013;Veening et al., 2008). For filaments, branching group structures, and well-mixed 431 groups, we found that division of labour can no longer evolve with non-accelerating returns 432 from individual specialisation (appendices B. 1. and B 2.). In appendix B. 3, we have shown 433 that this result holds for any group structure. 434

435
Consequently, for division of labour to evolve with non-accelerating returns from individual 436 specialisation, there must exist some mechanism to coordinate which cells specialise to perform 437 which tasks. A clear example of coordinated division of labour in topologically constrained 438 groups is the use of between-cell signalling in some cyanobacteria filaments to determine 439 which cells become nitrogen fixing heterocysts (Flores & Herrero, 2010;Meeks & Elhai, 440 2002). However, a signal to coordinate distinct phenotypes is unlikely to exist prior to division 441 of labour, and so a topological constraint is less likely to have favoured the initial evolution of 442 division of labour in cyanobacteria. Instead, division of labour could have been favoured by an 443 accelerating return from individual specialisation (scenario 1), with coordination only evolving 444 subsequently. Empirically, an accelerating return seems likely, as the key tasks performed by 445 reproductives and helpers do not mix well (photosynthesis and nitrogen fixation) (Flores & 446 Herrero, 2010;Meeks & Elhai, 2002). 447 448 These analyses do not suggest that topological constraints could never favour the initial 449 evolution of division of labour. As an alternative to signalling, a pre-existing cue could allow 450 division of labour to initially evolve with a metabolically cheaper form of coordination. For 451 example, phenotype could be determined in response to the number of neighbours or the local 452 concentration of some resource. However, the biological plausibility of any mechanism based 453 on pre-existing cues would need to be explicitly justified and modelled on a case-by-case basis 454 (Duarte et al., 2011). This would include modelling the metabolic cost and effectiveness of any 455 coordination mechanism (Cooper et al., 2020;Duarte et al., 2012;Liu et al., n.d.). Empirically, reproduction of each phenotype (Ackermann et al., 2008;Diard et al., 2013;Dragoš et al., 466 2018a;Dragoš et al., 2018b;Mavridou et al., 2018;Mridha & Kummerli, 2021;van Gestel et 467 al., 2015). These are rough suggestions for the kind of experiments required, as details and 468 possibilities will vary system from system, depending upon factors such as the degree of 469 specialisation, the mechanism by which labour is divided, and what manipulations are possible. 470 In addition, these experiments would need to follow from key first steps, such as demonstrating 471 division of labour and a trade-off between reproduction and cooperation (Diard et al., 2013;472 Dragoš et al., 2018a;Dragoš et al., 2018b;Veening et al., 2008;Zhang et al., 2020). 473 Testing for accelerating returns from individual specialisation (scenario 1): In at least three 475 treatments, vary the level of cooperation performed by the helpers (Figure 5A top-left), to test 476 whether the benefits of increased cooperation are accelerating ( Figure 5A right). Across at least 477 three other treatments, vary the level of reproduction by the reproductives (Figure 5A bottom-478 left), to test whether the benefits of increased fecundity are accelerating ( Figure 5A right). At 479 least three treatments are required to be able to test for non-linear (accelerating) benefits.  23 test whether division of labour is favoured by between-individual differences we must 486 determine whether an increase in cooperation by helpers produces a different group fitness 487 benefit than an increase in cooperation by reproductives. (C) To test whether division of 488 labour is favoured by reciprocal specialisation, we must determine whether there exists at 489 least one relative degree of helper-to-reproductive specialisation for which group fitness is 490 greater than the fitness of uniform cooperation. Division of labour can be favoured to evolve without accelerating returns from individual 507 specialisation. Nevertheless, for this to occur requires: (a) between-individual differences in 508 task-efficiency or synergistic benefits from reciprocal specialisation; and (b) a mechanism to 509 coordinate which individuals perform which tasks. In contrast, accelerating returns can favour 510 division of labour without a mechanism to coordinate task allocation, possibly making it more 511 likely to favour the initial evolution of division of labour. Ultimately, determining the relative 512 importance of these different pathways to division of labour is an empirical question, requiring 513 experimental studies of the type we have outlined above. 514

516
Resident strategy of uniform cooperation 517 We start by solving for the ESS strategy where both types of individuals invest the same amount 518 in cooperation (z h = z r = z; uniform cooperation). This is the level of uniform investment in 519 cooperation, z ⇤ , for which there is no selection for a uniform change in the amount of cooper-520 ation by all individuals in the group: More explicitly, we can write this as: where we have suppressed the functional dependencies for ease of presentation. The first term 523 gives the group fitness change due to a marginal increase in "helper" cooperation, and the 524 second term gives the group fitness change due to a marginal increase in "reproductive" coop-525 eration. So, at the uniform strategy, z ⇤ , any increase in the fitness caused by the increased co-526 operation of one subgroup of the population is balanced by a commensurate decrease in fitness 527 caused by the same increase in cooperation for the other subgroup (@W (z h , z r )/@z h | z h =zr=z ⇤ = 528 @W (z h , z r )/@z r | z h =zr=z ⇤ ).

529
The constrained optimum, (z h , z r ) = (z ⇤ , z ⇤ ), computed using Equation 4 then the fitness function satisfies W (z h , z r ) = W (z r , z h ), and it can be seen that this implies 533 25 that the point (z h , z r ) = (z ⇤ , z ⇤ ) is actually a critical point of W (z h , z r ). give here the associated partial differentials of fitness (Equation 1): These expressions capture the fitness consequences of a marginal increase in cooperation by 540 helpers and reproductives, respectively. The first term of each captures the fecundity cost to 541 own type of producing more public good, whereas the second term and third term are the via-542 bility benefits that accrue to both types from this increased cooperation. If directional selection 543 in both traits is zero ( @W/@z h | z ⇤ = 0 and @W/@z r | z ⇤ = 0), then z h = z r = z ⇤ is a critical 544 point, and Equations 6 and 7 imply that These equations mean that, if z h = z r = z ⇤ is a critical point, then any marginal viability 546 benefit to the group of increased cooperation by one subgroup is cancelled by the fecundity 547 cost to that same subgroup. Moreover, Equations 8 and 9 together imply that 548 If this equation does not hold, then z h = z r = z ⇤ is not a critical point: i.e. there is a difference 549 in the viability-fecundity tradeoffs between subgroups such that some individuals (without loss 550 of generality, helpers) can secure larger benefits for the group at the same fecundity cost as 551 others (reproductives). This gives our first condition for division of labour being able to evolve:

552
The between-individual differences condition for division of labour If individuals are indistinguishable when both types invest equally in cooperation (z h = z r = 554 z), then the viability functions satisfy V h (z, z) = V r (z, z). In this case, Condition 11 can be 555 restated as This says that the contribution to total viability from the increased specialisation of helper indi-557 viduals is strictly larger than the contribution to total viability from the increased specialisation 558 of reproductives. As a result, helpers are pre-disposed to become more helper-like as they can 559 gain larger viability gains for the group than the other type of individual. 27 The terms of Equations 13 and 14 capture the second order effects of increased investment 567 in cooperation. The first term of each captures the decline in the fitness benefit of increased 568 cooperation due to the cross-interaction between fecundity and viability. For instance, as a 569 helper invests more in cooperation (higher x), it increases its own viability (higher V h ), but its 570 fecundity declines as well (lower F ) and so the relative benefit of this increased viability is 571 lessened (the cross term F 0 V z h h is negative). This represents a kind of decelerating return from 572 cooperation. The second term of each captures the second-order effect of decreased investment 573 in fecundity. If this term is positive, then this means that there is a diminishing fecundity cost 574 to increased investment in cooperation, which can favour division of labour. The third and 575 fourth terms capture the second order effect of increased investment in viability, that is: does 576 each successive investment in the public good lead to a larger or smaller increase in viability 577 than the previous investment of the same size? The return on investment (ROI) in viability is Thus if either Equations 13 or 14 are positive, than division of labour is favoured to evolve.

581
This gives the second condition for division of labour.

582
The accelerating returns from indiviudal specialisation condition for division of labour or This condition is satisfied if the strength of directional selection pushing the population back to 607 the critical point along either of the trait-value directions is less than the strength of directional 608 selection on a trait when moved off of the critical point along the other trait direction. To 609 evaluate the Hessian condition, we first compute the second order cross derivatives: The reciprocal specialisation condition for division of labour In the case that the viability functions are symmetric (Equation 5), this condition further sim-623 plifies to This inequality is satisfied if the viability of reproductives increases faster with increased coop-625 eration from helpers, than it does from increased cooperation from reproductives. This makes 626 clear that reciprocal specialisation can evolve if reproductives stand to gain more from help 627 from helpers than they would gain by helping themselves. 628 30 Figure A1: A star.
either a, or b neighbours, with a < b. We term the cells with a neighbours as edge cells and 648 cells with b neighbours as node cells. 1 This gives the following expected fitness: where n h is the number of edge cells and n r is the number of node cells. Note that for any 650 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. 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 664 in z and H is increasing in z. 665 We now restrict to the 'marginal' case that F (z) and H(z) are both linear functions. In 666 this case we can, without loss of generality, take F (z) = 1 z, and H(z) = z. The uniform 667 cooperation strategy is given by z ⇤ = 1/2. The first derivatives of W at the uniform strategy 668 are given by: Notice that, for fixed i, So vertices 2 and 3 are 'predisposed' towards helping less, whereas the more peripheral ver-678 tices (i = 1, 4, 5, 6, 7) are predisposed towards helping more. The vertices with the fewest 679 neighbours (i = 4, 5, 6, 7) are also the most predisposed towards helping more. Let us assume that the group is composed of n cells that form a one dimensional filament 682 wrapped into a ring, where n is an even number ( Figure A3). Label the cells 1 through n, and 683 assume that 1 is neighbours with n. We assume that each cell in the filament shares social 684 benefits with only its direct neighbours. Suppose that 'odd' cells are putative helper cells, and 685 Figure A3: A ring of cells.
'even' cells are reproductives. This gives the following fitness for the group: The viability of helpers is V h (z h , z r ) = (1 ) In the case that H 00 = 0, division of labour is favoured only if > 1/2. If H 00 is less than 696 zero, then the returns on investment are decelerating, and must be larger than 1/2 in order 697 for division of labour to be favoured.

698
A.4 A well-mixed group of cells 699 We now consider a group that is "well-mixed" such that cells share the benefits of cooperation 700 with all other group members. That is, we let 1 be the amount of benefits produced by an 701 individual that it keeps for itself and is the amount of benefits that is shared equally by all 702 40 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 )H(z h ) + Otherwise, if the returns from cooperation are diminishing (say, H 00 < 0, with F linear), then 712 we can use the general condition 17 to find, 713 > n 1 n Notice that, for diminishing returns (H 00 < 0), we have that x > 0, and in this case it follows 714 that:

715
(1 + n h x)(1 + n r x) In other words, if H 00 < 0, then must be even higher for division of labour to be favoured, 716 than when H 00 = 0. A.5 General graph analysis: reciprocal specialisation 718 We now return to the case of a general graph model, and consider when division of labour is 719 possible via reciprocal specialisation. Once again, fix some graph G. Assume that the first 720 derivatives, @ i W , vanish at the strategy of uniform cooperation (z i = z ⇤ ). Recall (Equation 29) 721 that this is equivalent to assuming that all the vertices of the graph G have the same degree, d.

722
The matrix of second derivatives of W is 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 the characteristic polynomial P(x) is a degree n polynomial defined by: 734 P(x) = det (L xI) .
The roots of P(x) are all non-negative (this is because L is a symmetric matrix). Moreover, 735 x = 0 is a root of P(x). Let µ be the largest root of P(x); i.e. µ is the largest eigenvalue of Recall that d is the number of neighbours that an individual cell has in the graph. The eigen-739 value µ can be thought of (see below) as a measure of how 'bipartitionable' the graph is. The 740 inequality can then be interpreted loosely as follows: if the graph is made "more bipartite", 741 can decrease. If the number of neighbours, d, is increased, must increase.

43
To further understand µ, a useful property of L is that, for any vector of numbers x = 743 (x 1 , ..., x N ): The largest eigenvalue of L is On the other hand, It follows that µ  2d.
Moreover, if the graph is bipartite, choose any bipartite colouring of its vertices. Then, assign-747 ing x = +1 to vertices of one colour, and x = 1 to vertices of the other colour, we find 748 that 749 X edges i j (x i x j ) 2 = X edges 4 = 2dn.
This means that µ achieves its maximum value, µ = 2d, for bipartite graphs. It can be shown 750 that µ = 2d if and only if the graph is bipartite. µ can be regarded, therefore, as a meaure For n even, division of labour is possible if > 1/2. For large odd n, µ is approximately 4, 757 and so division of labour is again possible for > 1/2. However, for small odd n, µ is less 758 than 4. This means that, for small n, division of labour is "easier" for even numbers of cells 759 than for odd numbers of cells.
For some examples, see Figure A5.

46
We repeat the above analyses where we now assume that individuals adopt their phenotypes