Agent-based modeling of nuclear chromosome ensembles identifies determinants of homolog pairing during meiosis

Abstract

During meiosis, pairing of homologous chromosomes (homologs) ensures the formation of haploid gametes from diploid precursor cells, a prerequisite for sexual reproduction. Pairing during meiotic prophase I facilitates crossover recombination and homolog segregation during the ensuing reductional cell division. Mechanisms that ensure stable homolog alignment in the presence of an excess of non-homologous chromosomes have remained elusive, but rapid chromosome movements appear to play a role in the process. Apart from homolog attraction, provided by early intermediates of homologous recombination, dissociation of non-homologous associations also appears to contribute to homolog pairing, as suggested by the detection of stable non-homologous chromosome associations in pairing-defective mutants. Here, we have developed an agent-based model for homolog pairing derived from the dynamics of a naturally occurring chromosome ensemble. The model simulates unidirectional chromosome movements, as well as collision dynamics determined by attractive and repulsive forces arising from close-range physical interactions. Chromosome number and size as well as movement velocity and repulsive forces are identified as key factors in the kinetics and efficiency of homologous pairing in addition to homolog attraction. Dissociation of interactions between non-homologous chromosomes may contribute to pairing by crowding homologs into a limited nuclear area thus creating preconditions for close-range homolog attraction. Incorporating natural chromosome lengths, the model accurately recapitulates efficiency and kinetics of homolog pairing observed for wild-type and mutant meiosis in budding yeast, and can be adapted to nuclear dimensions and chromosome sets of other organisms.

Fig 1. Determinants of chromosome dynamics during the meiotic homology search.

While searching for their homologous partner, chromosomes move about the cell nucleus and while performing a continuous random walk (dashed arrows) within the confines of the nuclear envelope (bold circles), with velocity and changes in direction determined by interactions with other chromosomes as well as thermal noise. (A) When homologs enter each other’s attractive radii (i.e., centers are 400 nm apart), they exert attractive forces (Eq (3); j hom) on each other which move them closer until they reach their respective exclusive radii (purple; here 50 nm) keeping them at a constant distance of 100 nm. They subsequently continue their effective “random walk” in a paired status moving as a single entity non-homologous with respect to all other chromosomes. (B) When a chromosome enters the repulsive radius of a non-homolog (red; here 400 nm), the repulsive force in (Eq (3); j nhom) diverts their movement at the angle of their encounter with new velocity proportional to the minimum interaction distance. For illustration, ring colors match the terms in the Morse potential governing the interactions between chromosomes in Eqs (1)–(3); also see Fig 2.

Fig 2. Strength of exponentially decaying forces between non-homologous (short-range repulsive) and homologous (short-range attractive) chromosomes.

Vertical black lines represent the centers of mass, x_i, and the colored regions indicate each chromosome’s force modeled as a decaying exponential. The first column shows the strength of the forces. The second column shows a cartoon illustration of the corresponding chromosome dynamics when they meet, and the third column contains representative frames from Supplemental Movie 1 [45]. The solid circles on the righthand side indicate the effective interaction radius (see Table 1) beyond which the force is negligible. (A-C) Illustration of exponentially decaying forces when two non-homologous chromosomes meet. The repulsive force felt by a non-homologous chromosome corresponds to where its center of mass crosses into the other chromosome’s repulsive region (denoted by * in A). (D-F) Similar illustrations but for an attractive homologous interaction.

Fig 3. Effects of chromosome number on pairing efficiencies and kinetics.

Pairing frequencies for the indicated number of homolog pairs in 200 realizations are combined to compute the percent of homologs paired at the indicated time points. Average pairing levels of the indicated number of equally sized homolog pairs. Error bars are omitted for clarity.

Fig 4. The pairing model captures experimentally determined association kinetics between homologous and non-homologous chromosomes during wild-type meiosis.

(A) Modeling of pairing between 16 uniformly-sized partner chromosomes. For each modeling run, chromosome distances at t = 3h are used to uniquely index homologous chromosome pairs, which are arranged along the x-axis based on ascending initial distances. The y-axis indicates average distances from 200 realizations of the agent-based model Eqs (1)–(3) at the time points indicated by the color code. Error bars indicate standard deviations. The inset shows experimentally determined distances between a single pair of GFP-labeled yeast chromosome III measured in a synchronized meiotic culture in fixated nuclei (n > 100) (data from [13]). Nuclei are arranged based on distances between homologous GFP signals at a given time point. Note that x-axes are different in the inset due to the fact that in the experiment cells were observed at a given time point and then discarded, whereas in the simulation the same cell was tracked over time. (B) Results from modeling of distances between non-homologous chromosome pairs in the same nuclei analyzed in (A). For non-homologous pairs, each homolog partner is matched with a non-homologous chromosome that at t = 3h exhibits a distance optimally matched to that with its cognate homolog partner. Inset (B) shows experimentally determined distances between non-homologous GFP-tagged budding yeast chromosomes II and III (error bars indicate SD). For details on experimental conditions see [13]. The pairing distance is highlighted in gray at 400 nm. *Note the experimental data do not include information for t = 9h.

Fig 5. Effects of chromosome size on pairing kinetics and efficiency.

(A) Average distances over time between homolog pairs where repulsive and attractive radii are adjusted proportionally to the sizes of actual yeast chromosomes. Note that the three smallest homolog pairs are markedly slower in achieving pairing than all other chromosomes. (B) Non-homologous chromosomes equidistant to each of the two homologs were identified at t = 3h and their distances were monitored throughout the simulation in the same set of model nuclei monitored in (a). (200 realizations, error bars indicate SD). The gray rectangle highlights pairing distances at or below 400 nm.

Fig 6. Modeling identifies a critical threshold of chromosome movement velocity for efficient homolog pairing.

(A) Dots indicate the average pairing levels over time of all 32 true-sized chromosomes. Black indicates the velocity of chromosome movements in the wild-type model in Figs 4 and 5 (300 nm/s). Chromosomes fail to pair at velocities around 150 nm/s and only increases above 240 nm/s have a substantial impact. For a more detailed analysis of pairing kinetics of (a) see S1 Fig. (B) Effects of movement velocity for actual chromosome sizes. The graph shows pairing levels of chromosomes of increasing sizes at t = 9h, indicating that pairing efficiencies of larger chromosomes are more dramatically affected by changes in velocity than those of smaller chromosomes.

Fig 7. Contributions of attractive and repulsive forces on pairing efficiencies and kinetics.

(A) Pairing wild-type model using true chromosome lengths and a standard translational movement velocity of 300 nm/s as primarily studied herein. In (B) the attractive strength C_a = 0 in the WT model is used to highlight the effect of repulsive interactions alone. In (C) the reverse is true, C_r = 0 in the WT model is used to highlight the effect of attraction alone. (200 realizations, error bars indicate SD). The pairing distance is highlighted by a gray rectangle.

Fig 8. Polymer chain model for chromosomes incorporates flexibility and orientation.

(A) Polymer chain model as an active dumbbell where each end is represented by the single bead model (1) with an additional term to ensure they stay together (4). (B) Representative still from a simulation of the active dumbbell movie, see Supplemental Movie 3 [45].

Fig 9. Polymer chain model as an active dumbbell.

(A) Average distances over time between homolog pairs where repulsive and attractive radii are adjusted proportionally to the sizes of actual yeast chromosomes. Note that the three smallest homolog pairs are markedly slower in achieving pairing than all other chromosomes. (B) Non-homologous chromosomes equidistant to each of the two homologs were identified at t = 3h and their distances were monitored throughout the simulation in the same set of model nuclei monitored in (A). (200 realizations, error bars indicate SD). The gray rectangle highlights pairing distances at or below 400 nm.

Fig 10. Modeling of pairing kinetics at reduced DSB abundance.

(a) Wild-type model with true-sized chromosomes. Results differ from Fig 4A) due to use of true sized rather than uniformly sized chromosomes. The inset shows experimental data for yeast chromosome III in hypomorphic spo11-HA/yf. Note the x-axis in the inset shows different nuclei derived from aliquots at the indicated time points, whereas the same cells were tracked through time during simulations. (b) Scenario I for hypomorphic spo11 with 3-fold reductions of the wild-type levels of both attractive and repulsive forces. For non-homologous chromosome distances see S3 Fig. (c) Scenario II for spo11 hypomorphic mutant with reduced movement speed and attractive force strength, with experimental observations for spo11-HA/yf shown in the inset. Translational movement velocity is reduced to 77% of wild type levels (230 nm/s), consistent with slower chromosome movements observed in spo11 [15], and attractive force is reduced 3-fold representing decreased attractive forces exerted by fewer DSBs (see Table 1). The pairing distance is highlighted in gray at 400 nm. (200 realizations, error bars indicate SD).