A computational approach to developing mathematical models of polyploid meiosis

Abstract

Mathematical models of meiosis that relate offspring to parental genotypes through parameters such as meiotic recombination frequency have been difficult to develop for polyploids. Existing models have limitations with respect to their analytic potential, their compatibility with insights into mechanistic aspects of meiosis, and their treatment of model parameters in terms of parameter dependencies. In this article I put forward a computational approach to the probabilistic modeling of meiosis. A computer program enumerates all possible paths through the phases of replication, pairing, recombination, and segregation, while keeping track of the probabilities of the paths according to the various parameters involved. Probabilities for classes of genotypes or phenotypes are added, and the resulting formulas are simplified by the symbolic-computation system Mathematica. An example application to autotetraploids results in a model that remedies the limitations of previous models mentioned above. In addition to the immediate implications, the computational approach presented here can be expected to be useful through opening avenues for modeling a host of processes, including meiosis in higher-order ploidies.

Figure 1

(A–E) Models of meiotic recombination in diploids and autotetraploids. The left column shows diploids and the right column tetraploids. (A) Replication of chromosomes. (B) Pairing of chromosomes into bivalents (diploids, tetraploids) and quadrivalents (tetraploids). (C) Recombination of paired chromosomes. See also Figure 2. (D) Distribution of (possibly) recombined chromosomes into gametes. (E) Crossing of gametes into seeds. Shaded circles show overall numbers of possibilities in that step. 1, p, p₁, and p₂ on the left sides of steps denote probabilities from previous steps. Formulas on the right sides of steps denote probabilities up to and including the current step. Alternatives are separated by vertical bars in square brackets, r is the frequency of recombination between the two marker loci, q is the recombination frequency between the centromere and the centromere-proximal marker, τ is the quadrivalent-to-bivalent ratio, and p_DP and p_PC are the probabilities of having a partner switch between the markers and between the centromere-proximal marker and the centromere, respectively. f(p_DP, p_PC, r, q) denotes a function of the parameters indicated.

Figure 2

(A–E) Chromosome pairing and recombination events. (A) Quadrivalent with partner switch between centromere and centromere-proximal locus. (B) Quadrivalent with partner switch between loci. (C and D) Quadrivalents with partner switch outside the two regions in A and B, effectively equivalent to E. (E) Two bivalents (no partner switch). D, centromere-distal locus; P, centromere-proximal locus; open circles, centromeres. X’s denote possible recombination events (two events per X). Shaded text and arrows describe the order of recombination events taken account of in the computer program. Curly brackets indicate that chromatid partner choice is not independent. Annotation of recombination events is implicitly the same in C–E.

Figure 3

(A–D) Gamete mode 9 through a partner switch (but without double reduction). Partner switch is between the two loci, A and B. Circles denote centromeres. X denotes two recombinations. (A) A quadrivalent forms, and the two chromosomes (solid lines) recombine twice. (B) The recombined chromosomes. The distal markers, B, have been swapped. (C) Chromosomes move apart in anaphase I. (D) Chromatids move apart in anaphase II. Two gametes now have configuration A_iB_j/A_jB_i (gamete mode 9). A single recombination leads to one gamete of mode 9. A similar process is shown (with one or two recombinations) for the partner switch being between the centromere and the centromere-proximal locus (A′ and B′, continuing with C and D).

Figure 4

(A–D) Gamete mode 10 through a partner switch (but without double reduction). Partner switch is between the two loci, A and B. Circles denote centromeres. X’s denote two recombinations each. (A) A quadrivalent forms, and the chromosomes (solid lines) recombine twice in each of the positions indicated by X. (B) The recombined chromosomes. The distal markers, B, have been swapped. (C) Chromosomes move apart in anaphase I. (D) Chromatids move apart in anaphase II. Two gametes have now configuration A_iB_j/A_jB_k (gamete mode 10). A single recombination leads to one gamete of mode 10.