Fragmentation modes and the evolution of life cycles

Abstract

Reproduction is a defining feature of living systems. To reproduce, aggregates of biological units (e.g., multicellular organisms or colonial bacteria) must fragment into smaller parts. Fragmentation modes in nature range from binary fission in bacteria to collective-level fragmentation and the production of unicellular propagules in multicellular organisms. Despite this apparent ubiquity, the adaptive significance of fragmentation modes has received little attention. Here, we develop a model in which groups arise from the division of single cells that do not separate but stay together until the moment of group fragmentation. We allow for all possible fragmentation patterns and calculate the population growth rate of each associated life cycle. Fragmentation modes that maximise growth rate comprise a restrictive set of patterns that include production of unicellular propagules and division into two similar size groups. Life cycles marked by single-cell bottlenecks maximise population growth rate under a wide range of conditions. This surprising result offers a new evolutionary explanation for the widespread occurrence of this mode of reproduction. All in all, our model provides a framework for exploring the adaptive significance of fragmentation modes and their associated life cycles.

Fig 1. Life cycles and fragmentation modes.

A Cells within groups of size i divide at rate b_i, hence groups grow at rate ib_i; groups die at rate d_i. The sequences b_i and d_i define the fitness landscape of the model. We consider an exhaustive set of possible fragmentation modes, comprising both pure and mixed life cycles. In general, when growing from size i to size i + 1, groups stay together with probability q_i+1, or fragment according to fragmentation pattern κ with probability q_κ. Each fragmentation pattern (determining the number and size of offspring groups) can be identified with a partition of i + 1, i.e., a way of writing i + 1 as a sum of positive integers, that we denote by κ ⊢ i + 1. B Pure fragmentation modes are strategies with degenerate probability distributions over the set of partitions (so that q_κ = 1 for exactly one fragmentation pattern, including staying together). Here we illustrate the pure fragmentation mode 2 + 1 + 1, for which q₂ = q₃ = q₂₊₁₊₁ = 1, and q_κ = 0 for all other κ.

Fig 2. The optimal fragmentation mode is pure and characterised by binary fragmentation.

A Mixed fragmentation strategies are dominated. Here we show the empirical probability distribution of the growth rate of mixed fragmentation modes for n = 4 (generated from a sample of 10⁷ randomly generated fragmentation modes) subject to the fitness landscape {b, d} = {(1, 2, 1.4), (0, 0, 0)}. The growth rates of all seven pure fragmentation modes for n = 4 are indicated by arrows. In this case, 2+2 achieves the maximal possible growth rate among all possible fragmentation modes. B Optimal fragmentation modes are characterised by binary splitting. Population growth rate (λ₁) for all seven pure fragmentation modes for n = 4 subject to the fitness landscape {b, d} = {(1, b₂, 1.4), (0, 0, 0)} as a function of the birth rate of groups of size 2, b₂. Each of the four fragmentation modes characterised by binary fragmentation (1+1, 2+1, 2+2, and 3+1) can be optimal depending on the value of b₂. Contrastingly, nonbinary fragmentation modes (1+1+1, 1+1+1+1, and 2+1+1) are never optimal.

Fig 3. Optimal fragmentation modes for fecundity and survival landscapes (costless fragmentation).

A Life cycles achieving the maximum population growth rate for n = 4 under fecundity landscapes (i.e., d₁ = d₂ = d₃ = 0). In this scenario, fragmentation mode 2+2 is optimal for most fitness landscapes. B Life cycles achieving the maximum population growth rate for n = 4 under survival landscapes (i.e., b₁ = b₂ = b₃ = 1). In this scenario, fragmentation modes emitting a unicellular propagule (1+1, 2+1, 3+1) are optimal for most parameter values. We use ratios of birth rates and differences between death rates as axes because one can consider b₁ = 1 and min(d₁, d₂, d₃) = 0 without loss of generality (S1 Text, Appendix D). Shaded areas are obtained from the direct comparison of numerical solutions, lines are found analytically (S1 Text, Appendix F).

Fig 4. Optimal fragmentation modes for fecundity and survival landscapes (costly fragmentation).

For proportional costs (panels A and B), splitting into π parts involves the loss of π − 1 cells. In this case, and for n = 4, only two pure modes are possible: 2+1 (whereby a 4-unit group splits into a pair of cells and a single cell and loses one cell) and 1+1 (whereby a group of three splits into two single cells and loses one cell). For fixed costs (panels C and D), splitting involves the loss of a single cell, no matter the kind of partition. In this case, and for n = 4, an additional mode is possible: 1+1+1 (whereby a 4-unit group splits into three single cells and loses one cell). This last nonbinary mode can be optimal under a wide range of parameters.

Fig 5. Advantages of group living.

Group size benefit g_i = [(i − 1)/(n − 2)]^α as a function of group size for different values of the degree of complementarity α. If α < 1, g_i is concave; if α = 1, g_i is linear; if α > 1, g_i is convex.

Fig 6. Optimal life cycles under monotonic fecundity landscapes.

Birth rates are given by b_i = 1 + Mg_i where g_i = [(i − 1)/(n − 2)]^α. A Costless fragmentation, n = 20. B Fragmentation with proportional costs, n = 21. C Fragmentation with fixed costs, n = 21. For costless fragmentation and fragmentation with proportional costs, only binary modes 19+1, 18+2, …, 10+10 are optimal. In these cases, diminishing returns (α < 1) make equal binary fragmentation (10+10) optimal. Also, optimality of the unicellular propagule strategy (19+1) requires increasing returns (α > 1). For fragmentation with fixed costs, nonbinary modes 7+7+6, …, 1+…+1 can also be optimal.

Fig 7. Optimal life cycles under monotonic survival landscapes.

Death rates are given by d_i = M(1 − g_i) where g_i = [(i − 1)/(n − 2)]^α. A Costless fragmentation, n = 20. B Fragmentation with proportional costs, n = 21. C Fragmentation with fixed costs, n = 21. For costless fragmentation and fragmentation with proportional costs, only binary modes 19+1, 18+2,…, 10+10 are optimal. In these cases, diminishing returns to scale (α < 1) make equal binary fragmentation (10+10) optimal. Also, optimality of the unicellular propagule strategy (19+1) requires increasing returns to scale (α > 1). For fragmentation with fixed costs, nonbinary modes 7+7+6,…,1+…+1 can also be optimal.