# What reproduces asexually by fragmentation bullets

Reproduction is a defining feature of living systems. To reproduce, aggregates of biological units e. 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 *What reproduces asexually by fragmentation bullets* 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.

Mode of reproduction is a defining trait of all organisms, including colonial bacteria and multicellular organisms. To produce offspring, aggregates must fragment by splitting into two or more groups. The particular way that a given group fragments defines the life cycle of the organism.

For instance, insect colonies can reproduce by splitting or by producing individuals that found new colonies. Similarly, some colonial bacteria propagate by fission or by releasing single cells, while others split in highly sophisticated ways; in multicellular organisms reproduction typically proceeds via a single-cell bottleneck phase.

The space of possibilities for fragmentation is so vast that an exhaustive analysis seems daunting. Focusing on fragmentation modes What reproduces asexually by fragmentation bullets

a simple kind we parametrise all possible modes of group fragmentation and identify those modes leading to the fastest population growth rate.

Two kinds of What reproduces asexually by fragmentation bullets cycle dominate: The prevalence of these life cycles in nature is consistent with our null model and suggests that benefits accruing from population growth rate alone may have shaped the evolution of fragmentation mode.

A requirement for evolution—and a defining feature of life—is reproduction [ 1 — 4 ]. Perhaps the simplest mode of "What reproduces asexually by fragmentation bullets" is binary fission in unicellular bacteria, whereby a single cell divides and produces two offspring cells. In more complex organisms, such as colonial bacteria, reproduction involves fragmentation of a group of cells into smaller groups. Bacterial species demonstrate a wide range of fragmentation modes, differing both in the size at which the parental group fragments and the number and sizes of offspring groups [ 5 ].

For example, in the bacterium Neisseriaa diplococcus, two daughter cells remain attached forming a two-celled group that separates into two groups of two cells only after a further round of cell division [ 6 ]. Staphylococcus aureusanother coccoid bacterium, divides in three planes at right angles to one another to produce grape-like clusters of about 20 cells from which single cells separate to form new clusters [ 7 ]. Magnetotactic prokaryotes form spherical clusters of about 20 cells, which divide by splitting into two equally sized clusters [ 8 ].

These are just a few examples of a large number of diverse fragmentation modes, but why should there be such a wide range of life cycles? Do fragmentation modes have adaptive significance or are they simply the unintended consequences of particular cellular processes underpinning cell division?

If adaptive, what selective forces shape their evolution? Can different life cycles simply provide different opportunities to maximise population growth rate? A starting point to answer these questions is to consider benefits and costs of group living in cell collectives. Benefits may arise for various reasons. Cells within groups may be better able to withstand environmental stress [ 9 ], escape predation [ 1011 ], or occupy new niches [ 1213 ]. Also, via density-dependent gene regulation, cells within groups may gain more of a limiting resource than they would if alone [ 1415 ].

On the other hand, cells within groups experience increased competition and must also contend with the build up of potentially toxic waste metabolites [ 1617 ]. Thus, it is reasonable to expect *What reproduces asexually by fragmentation bullets* optimal relationship between group size and fragmentation mode that is environment and organism dependent [ 18 — 21 ]. Here we formulate and study a matrix population model [ 22 ] that considers all possible modes of group fragmentation.

By determining the relationship between life cycle and population growth rate, we show that there is, overall, a narrow class of optimal modes of fragmentation. When the process of fragmentation does not involve costs, optimal fragmentation modes are characterised by a deterministic schedule and binary splitting, whereby groups fragment into exactly two offspring groups.

Contrastingly, when a cost is associated with fragmentation, it can be optimal for a group to fragment into multiple propagules. Our results show that the range of life cycles observed in simple microbial populations are likely shaped by selection for intrinsic growth rate advantages inherent to different modes of group fragmentation.

While we do not consider complex life cycles, our results may contribute to understanding the emergence of life cycles underpinning the evolution of multicellular life. We consider a population in which a single type of cell or unit or individual can form groups or complexes or aggregates of increasing size by cells staying together after reproduction [ 18 ].

We assume that the size of any group is smaller than nand denote groups of size i by X i see the list of used variables in Table 1. Groups die at rate d i and cells within groups divide at rate b i ; hence groups grow at rate ib i. Groups produce new complexes by fragmenting or splittingi. We further assume that fragmentation is triggered by growth of individual cells within a given group.

Such a group can either stay together "What reproduces asexually by fragmentation bullets" fragment. If it fragments, it can do so in one of several ways.

Mathematically, such fragmentation patterns correspond to the five partitions of 4 a partition of a positive integer i is a way of writing i as a sum of positive integers without regard to order; the summands are called parts [ 23 ].

A mixed fragmentation mode is given by a probability distribution over the set of pure fragmentation modes. The relationship between What reproduces asexually by fragmentation bullets

and mixed fragmentation modes is hence similar to the one between pure strategies and mixed strategies in evolutionary game theory [ 24 ]. One of our main results is that mixed fragmentation modes are always dominated by pure fragmentation modes.

Hence, we focus our exposition on pure fragmentation modes, and leave the details of how to specify mixed fragmentation modes to the Supporting Information S1 TextAppendix A.

A Cells within groups What reproduces asexually by fragmentation bullets

size i divide at rate b ihence 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. Together with the fitness landscape given by the vectors of birth rates b and death rates deach fragmentation strategy specifies a set of biological *What reproduces asexually by fragmentation bullets.* A set of reactions. Finally, one reaction of the type.

The sets of reactions 12 and 3 give rise to the system of differential equations. This is a linear system that can be represented in matrix form as. In the long term, the solution of Eq 4 converges to that of an exponentially growing population with a stable distribution, i.

Indeed, under the assumption of no density limitation, the evolutionary dynamics are described by two uncoupled sets of differential equations of the form 4: Fitness landscapes capture the many advantages or disadvantages associated with group living. These advantages may come either in the form of additional resources available to groups depending on their size or as an improved protection from external hazards.

For our numerical examples, we consider two classes of fitness landscape, each representing only one of these factors. The three associated projection matrices are given by. We then say that ternary and binary fission are dominated by the unicellular propagule strategy. Although for simplicity we focus our exposition on pure fragmentation strategies, we also consider mixed fragmentation strategies, i.

A natural question to ask is whether a mixed fragmentation mode can achieve a faster growth rate than a pure mode. We find that the answer is no. For any fitness landscape and any maximum group size nmixed fragmentation modes are dominated by a pure fragmentation mode S1 TextAppendix B.