Fig 1.
Examples of multiple fragmentation in nature and their interpretation in terms of our model.
(A) M. polyspora grows multiple endopsora, released after the maternal cell lysis (picture adopted from [27], Copyright (1998) National Academy of Sciences). (B) segmented filamentous bacteria grows two holdfast-bearing cells inside a maternal cell. These cells are released in the result of the maternal cell lysis (picture adopted from [29]). The death of maternal cells is the fragmentation cost in these two examples. (C) genus Stanieria grows multiple cells within a single extracellular matrix. These cells are released simultaneously upon the break of the matrix (Picture adopted from [30]). The loss of the extracellular matrix corresponds to a fragmentation cost here.
Fig 2.
(A) Multicellular units increase in size and fragment to produce new units. Cell division (bi) and unit death (di) rates depend on the size of the unit and are determined by the environment. If the fragmentation process is costly, the division rate at the maturity size may be smaller than prescribed by the environment alone (see Eq (1)), the death rate at the maturity size may be larger than prescribed by the environment alone
(see Eq (2)), and some cells may be lost upon the fragmentation (highlighted in blue). In this example, m = 4, so the unit fragments when it reaches five cells, one cell is lost in fragmentation and the remaining four are split by the fragmentation mode 2+2. (B) The fragmentation mode of cell clusters can be described by a sum of integers. All possible fragmentations of units of size 2, 3, and 4 are presented here. Different life cycles have different growth rates and we are looking for the fastest growth in this context.
Fig 3.
The majority of life cycles cannot win growth competition and are thus called forbidden.
(A) All possible fragmentation modes with combined fragments size equal to 7. Allowed modes are further broken into binary, (nearly) equal split, seeding, and other classes, according to the definitions in the main text. For each of the forbidden modes, a couple of different subsets of parts with the same sum are underlined. (B) Proportions of different classes of fragmentation modes for different combined size at fragmentation. For sizes 2 and 3, all partitions are allowed. Starting from size 4, some are forbidden (for 4, it is 2+1+1). The proportion of forbidden modes grows rapidly with the size. For even sizes, more forbidden modes exist. Among the allowed modes, the proportions of binary, (nearly) equal split and seeding classes rapidly declines.
Fig 4.
Fragmentations by binary and (nearly) equal split partitions are likely to evolve in random environments.
The top panels present the ten most frequent fragmentation modes for (A) fragmentation with delay, (B) fragmentation with increased death risk and (C) fragmentation with cell loss, respectively. Each bar shows the frequency of the corresponding life cycle to be evolutionarily optimal. The colour of the bar represents the class of the life cycle, see Fig 3. (Nearly) equal split life cycles are represented in the form of 1+ … + 1 and the total number of cells Σ. The bottom panels (D—F) present the fractions of each of binary fragmentation, (nearly) equal split, seeding, and other allowed fragmentation modes as functions of fragmentation cost for the same scenarios. The majority of random environments promote the evolution of (nearly) equal split and binary fragmentation modes.
Fig 5.
Multicellular units can evolve in detrimental environments.
Each panel presents the fraction of the minimal, binary life cycle 1+1 (green) and equal split life cycles with a larger maturity size (orange) for random detrimental environments, where division rates decrease and death rates increase with cluster size. Panel (A) shows fragmentation with delay, (B)—fragmentation with risk, and (C)—fragmentation with cell loss. The increase in fragmentation costs, drives the evolution of life cycles involving formation of multicellular units. All evolutionarily optimal life cycles found have the form 1+ … + 1.
Table 1.
List of species considered in this work and their life cycles.