Reproduction costs can drive the evolution of groups

A fascinating wealth of life cycles is observed in biology, from unicellularity to the concerted fragmentation of multi-cellular units. However, the understanding of factors driving the evolution of life cycles is still limited. We investigate how reproduction costs influence this process. We consider a basic model of a group structured population of undifferentiated cells, where groups reproduce by fragmentation. Fragmentation events are associated with a cost expressed by either a fragmentation delay, a fragmentation risk, or a fragmentation loss. The introduction of such fragmentation costs vastly increases the set of potentially optimal life cycles. Based on these findings, we suggest that the evolution of life cycles and the splitting into multiple offspring can be directly associated with the fragmentation cost. Moreover, the impact of this cost alone is strong enough to drive the emergence of multicellular groups, even under scenarios that strongly disfavour groups compared to solitary individuals.

In the case of the fragmentation delay, the process of fragmentation is not immediate and where T it the fragmentation delay. Consequently, this scenario can be captured by changing 106 the fitness landscape in terms of the birth rate at the size prior to fragmentation.
Again, this scenario corresponds to a change of the fitness landscape.
Here, Eqs. (3a) and (3b) describe the dynamics of the abundances of groups x i that grow 141 without fragmentation, because they do not reach the maturity size m. The first two terms in integer π i (κ) is the number of groups of size i that emerge in a single act of fragmentation 146 according to the partition κ, and mb m is the growth rate prior to fragmentation (see Eq. (1)).

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Eq. (3c) describes the dynamics of groups of maturity size m, which will inevitably frag-148 ment according to the partition κ upon the next cell division. For fragmentation with de-149 lay, the rate of transition to the next state (fragmentation) is smaller than the cell birth rate (2)).

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The equation system (3) is linear with respect to x i . Thus, it can be written in a form of 154 matrix differential equation In the long run, the solution of Eq. (4) converges to that of an exponentially growing popula-157 tion with a stable distribution, i.e., The leading eigenvalue λ gives the total population growth rate, and its associated right eigen- For a given deterministic life cycle associated to fragmentation at size m according to the 169 partition κ, the characteristic equation (7) reduces to (see Appendix A.2 for a derivation) where The parameter In the set of random fitness landscapes, each element of the birth and death rates vector (b and d) was sampled independently from the uniform distribution U (0, 1).

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In the set of random detrimental fitness landscapes, for each landscape, we initially sam-186 pled two sequences of n = 19 random numbers, each using the uniform distribution U (0, 1).

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Then, the first sequence has been sorted in descending order to form the vector of the birth 188 rates b and the second sequence has been sorted in ascending order to form the vector of death  τ 1 j, τ 2 j, τ 1 = τ 2 and τ 1 + τ 2 ⊂ κ, For any fitness landscape and any fragmentation cost scenario, the life cycle employing such

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The total amount of allowed life cycles is still too large to track each of them individually.

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Therefore, a classification is necessary. We focus on three significant subsets: binary frag-

Evolutionary optimal life cycles under random fitness landscapes 244
The previous section introduced the range of potentially optimal life cycles, but it did not 245 give any insight about interconnection between life cycles and fitness landscapes. Some life 246 cycles may be evolutionary optimal under a larger set of fitness landscapes than others. To 247 study the distribution of optimal life cycles for costly fragmentation, we generated a large      new colonies [Stein, 1958]. Since the maturity size for G. pectorale is 16 cells, but the frag- provide some height to the fruiting body, so spores can be distributed across larger territory 405 [Bonner, 1959]. Cells in the stalk die without contributing to the spores, thus the stalk is 406 the cost of the fragmentation. Both organisms, support the hypothesis as well, but only on a 407 conceptual level.

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Another aspect of our study is that all three considered scenarios of the fragmentation  The evolution of groups from unicellular ancestors is often considered to be driven by 431 some ongoing benefits provided by the group membership such as better protection [Stanley,432 1973], access to novel resources [Rainey and Travisano, 1998] and the opportunity to coop-433 erate (reviewed in [Kaiser, 2001] and in [Grosberg and Strassmann, 2007]). In our work we x -birth, death and fragmentation -occur with a constant rate. Thus, the dynamics of the 458 population state can be described by a set of linear differential equations or, equivalently, by a matrix differential equation where A is a projection matrix defined by demographics of the population [Caswell, 2001].

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An element a i,j of the projection matrix describes the rate of change of the number of groups 462 of size i caused by processes occurring with groups of size j.

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To construct the projection matrix elements, consider groups of a certain size j. We 464 denote by q j,κ the probability that upon the growth from size j to j + 1, the group will 465 fragment by a partition κ j ≤ j + 1 (where the "≤" indicates that cells can be lost The main diagonal a i,i describes the changes in groups numbers due to growth and frag-493 mentations as well as the death of groups. The first component of a i,i is given by the fact that 494 once a group of size j grows or fragments, the number of groups of that size decreases. The 495 rates of decrease are equal to jb j q j,(j+1) due to the growth and jb j κ q j,κ due to the fragmen- ber of groups of size j at rate jb j q j,j+1 π j (j + 1), where π 1 (1 + 1) = 2 and π j (j + 1) = 1 if 499 j > 1. The last component of a i,i comes from the death of groups, which leads to a decrease 500 in their number at rate d j q j,(j+1) + d j κ q j,κ , where the first term describes the death rate in 501 the absence of the fragmentation and the second term describes the death rate of fragmenting 502 groups. Combined, the diagonal elements of projection matrix are 503 a j,j = −jb j q j,(j+1) − jb j κ q j,κ + jb j q j,j+1 π j (j + 1) − d j q j,(j+1) − d j κ q j,κ .
All elements of the projection matrix given by Eqs.
The population growth rate is given by the leading eigenvalue λ 1 of A, i.e., the largest Dividing both sides by To move the first multiplier with λ into the product, we rewrite it as Thus, Simplifying this, we finally obtain that the characteristic equation (17) can be written as and and These transformations allow us to set b 1 = 1 and min(d) = 0 without loss of generality. partitions τ 1 j and τ 2 j such that τ 1 = τ 2 , and an arbitrary partition φ k, and the 537 following three deterministic fragmentation modes: 538 1. κ 1 = τ 1 + τ 2 + φ 2j + k ≤ m + 1, whereby a complex fragments upon growth from 539 size m into a number of offspring given by partitions τ 1 , τ 2 , and φ. 540 2. κ 2 = τ 1 + τ 1 + φ 2j + k ≤ m + 1, whereby a complex fragments upon growth from 541 size m into a number of offspring given by two partitions τ 1 and one partition φ. 542 3. κ 3 = τ 2 + τ 2 + φ 2j + k ≤ m + 1, whereby a complex fragments upon growth from 543 size m into a number of offspring given by two partitions τ 2 and one partition φ.

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Denoting by λ(κ i ) the leading eigenvalue of the projection matrix induced by fragmenta-545 tion mode κ i , we can show that, for any fitness landscape, either λ(κ 1 ) ≤ λ(κ 2 ) or λ(κ 1 ) ≤ 546 λ(κ 3 ) holds. This means that a fragmentation mode with two different subsets of offspring 547 with the same combined size is dominated by a mode where one of these subsets repeats 548 twice, while another one is not present.

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As a consequence, for high fragmentation delay, under the optimal life cycle, group fragments 580 after reaching the most protected state with the minimal d i .

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To find which of many fragmentation modes available to the group reproducing at the 582 most protected state is evolutionary optimal, we consider the first order approximation of the 583 growth rate given by Then we use expressions of F i (λ) from Eq. (21) and discard all terms smaller than 1 Thus, In the optimal life cycle under high delay of fragmentation, groups fragment according to the 588 partition that provides the highest value of λ 1 .

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For the special case of the constant death rate, the optimal life cycle can be found explic-590 itly. In this case, the death rate can be set to d = 0 (see Eq. (23)), so At d = 0, F i (0) = 1, so: the right hand side of this expression is the number of produced offspring groups minus one.

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For the set of fitness landscapes detrimental to larger groups used in section 3.3, the 602 minimum of d i is achieved always at d 1 . Therefore, the maturity size for large delay is 1, 603 which corresponds to the unique fragmentation pattern 1 + 1. In our simulations the initial 604 increase in T resulted in the gradual decrease of the fraction of fitness landscapes promoting 605 1 + 1 to zero. However, further increase of T make some fitness landscapes promote 1 + 1 606 again, and above some intermediary value of T , the fraction of fitness landscapes promoting 607 1 + 1 begin to increase, see Fig. 4a.
Therefore, Eq. (8) becomes Or, after dividing by R, To analyse the solutions of obtained equation, we first discard all terms containing 1 R and get For m = 1, this equation has no solution, instead the proliferation rate of the population 620 undergoing 1+1 life cycles (the only life cycle with m = 1) is given by Thus, for κ = 1 + 1, the proliferation rate is given by

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To distinguish between such life cycles, we consider the first order approximation of the 628 growth rate given by We substitute λ in the form of Eq. (38) into Eq. (36) and discard all terms smaller than 1 Note, that offspring groups of size larger than i * do not contribute to sum at the end of the 631 expression at the left hand side, because F i>i * (λ 0 ) = 0. The term linear with respect to 1 R is 632 equal to i∈(1,··· ,m−1)\i * 1 + λ 0 +d i ib i The optimal life cycle maximizes this expression.

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For any given maturity size m, the life cycle producing more offspring groups with size 635 not exceeding than i * has higher λ 1 . Thus, under the optimal life cycle, the size of offspring 636 cannot be larger than i * .

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ib i + d i monotonically increases. Hence, i * = 1, so the optimal life cycle is the fragmentation 639 into unicellular propagules.

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For the set of random fitness landscapes used in section 3.2, the expression ib i + d i tend 641 to grow with i, so its minimum i * is more likely to be achieved at small values of i. Since, i * 642 establishes an upper limit on the size of offspring groups, our analysis suggests that this size 643 should decrease with R.