Cell division dynamics

Two sets of equations are used subsequently to represent (i) the cell division dynamics and (ii) the methylation dynamics during cell divisions. We assume that transgenerational effects, when present, occur during the fertilization step at the end of each generation as they reflect environmental effects on rates of methylation coming from one or both parents and are transferred into the zygote through the gametes. During the maternal-to-zygotic transition, the initiation time of methylation dynamics during development is assumed to be delayed as a consequence of the dependence on the nucleocytoplasmic ratio ν [35, 36] and the parental repressor ρ, which is assumed to speed up cell divisions during the early stages of development and to delay the achievement of full methylation rates after the initiation of methylation changes as a consequence of the conflict between maternal and zygotic genomes. Although there is no empirical evidence for the cause of this delay, it is plausible to speculate that it might have evolved as a way to solve the male-female epigenomic conflict before the zygotic epigenome takes full control over development. However, this question needs to be properly addressed in future empirical studies.

Cell division dynamics follow a logistic growth process (Eq 1), as an approximation of division rates based on empirical data from the Northern leopard frog Rana pipiens and the loach Misgurnus fossilis [37, 38], with a variable growth rate (Eq 2) that depends on the amount of parental repressor in individual cells:

with

where N_g,d is the number of cells in generation g and cell division d, K is the equilibrium developmental state in terms of number of cells, μ is the realized cell division rate, ρ is the amount of parental repressor and r is the intrinsic cell division rate.

The parental repressor ρ changes across cell divisions at a constant degradation rate ρ_deg (Eq 3), and its repressing effect ρ_c on methylation dynamics is a function of its current value ρ_g,d and initial value ρ₀ (Eq 4), when ρ₀ > 0. ρ is a dimensionless variable that represents the effect of the repressor on methylation rates and cell division rate, and decreases across cell divisions due to the degradation rate ρ_deg. The effect of the repressor on cell division rate is one of the hypotheses that can explain the fast cell divisions during the early stages of development (as reviewed by [39]) and it has support from experimental studies (e.g. [40]). By assigning ρ₀ = 1.0, we assume that the parental repressor causes an initial twofold increase in the speed of cell division, as indicated in Eq (2), and its effect decreases throughout development.

Given that cell divisions during the early stages of development happen without cell growth and hence the initial total cytoplasmic volume is conserved, the nucleocytoplasmic ratio ν can be calculated as a function of the number of cells (Eq 5).

According to experimental evidence [15], the maternal-to-zygotic transition (here assumed to be represented by methylation levels) is initiated when the nucleocytoplasmic ratio reaches a specific threshold (ν_t). The process of ZGA assumes that the initial methylation level is conserved across cell divisions [28, 41, 42] and only changes when ν ≤ ν_t. Once that is achieved, methylation dynamics towards lineage-specific methylation levels (e.g. gametic cell lineage) are initiated.