Figure 5. Panel A shows a schematic of a... | Rockefeller University Press

Figure 5.

$Small-number effects cause stochasticity in sister chromatid separation time. (A) Schematic of the stochastic model of separase-mediated cohesin removal. Separase activation (through securin degradation) is represented by a constant increase in the cohesin cleavage rate, k(t). The rate increases for a period τ until it reaches its maximum value, kmax. Each chromosome initially has N cohesin complexes (N1, N2, and N3 for chromosomes I, II, and III, respectively), which are randomly cleaved by separase. Chromosomes separate once the number of remaining cohesins falls below a threshold n (n1, n2, and n3 for chromosome I, II, and III, respectively). See Materials and methods for model details. (B) Frequency distributions and Gaussian fit (continuous lines) of the time differences between the separation of centromeres 1 and 2 or centromeres 2 and 3, either from a simulation using optimal parameter values of the basic model (salmon) or determined experimentally in wild-type cells (gray/black, same data as in Fig. 1 F). Mean ± SD of the simulation results or the Gaussian fit (for experimental data); n = number of cells; P values from EMD. (C) Goodness of fit of the model variants measured by EMD (see Materials and methods) with and without constraining τ at low values (τ < 5 s). Points indicate the validation EMD values obtained in the fivefold cross-validation (see Materials and methods); boxplots show median, interquartile range, and range. Basic, basic model with time-dependent cleavage rate alone; processive, model with bursts of cohesin cleavage; steric hindrance, model in which only a surface fraction of cohesin is available for cleavage and inner layers of cohesin are progressively exposed (see Materials and methods). (D) Parameter values of the basic model fitted to the experimental data. Points indicate values obtained in each of the fivefolds (see Materials and methods); boxplots show median, interquartile range, and range; the y-axis spans the parameter range tested, except for the inset showing n1, n2, and n3. (E) One-at-a-time (OAT) sensitivity analysis of the basic model. Each parameter was varied individually across its allowed range while all other parameters were held fixed at their optimal values obtained by fitting. 10 batches of 10,000 simulations were performed for each parameter point. Each batch yielded a mean and SD of the separation time differences (Δt) between cen1 and centromere 2, and centromere 2 and 3. Plot shows mean of the Δt SD ± SEM across 10 batches. Dashed vertical lines indicate the optimal parameter values. (F) Stochastic cohesin loss trajectories near the time of separation in the basic model. Simulated cohesin counts for chromosomes 1, 2, and 3 are shown for three representative simulations, aligned to t = 0 at the final separation event. Lines indicate cohesin loss over time; points mark the moment of separation for each chromosome. (G) Schematic summarizing the results: separase becomes active upon securin degradation without positive feedback. If a small number of cohesin complexes suffice to maintain cohesion, stochasticity of the final cleavage events dominates the timing of sister chromatid separation, leading to asynchrony and no clear order between chromosomes. Refer to the image caption for details.$ Panel A shows a schematic of a stochastic model for cohesin cleavage. Each chromosome starts with an initial number of cohesin complexes, uppercase N. Chromosome separation occurs once a threshold number of cohesin complexes, lowercase n, is reached. Panel B shows histograms with Gaussian fits comparing simulated and experimental time differences between centromere separations, where the x-axis represents time difference in seconds and the y-axis represents percentage of cells. Panel C shows a box plot evaluating model goodness-of-fit using Earth Movers Distance, comparing different model variants and constraints. Panel D shows box plots of optimal parameter values from model fitting, where the x-axis represents parameter categories and the y-axis represents number or rate values within tested ranges. Panel E shows line graphs from sensitivity analysis of model parameters, where the x-axis represents parameter values and the y-axis represents standard deviation of time differences in seconds. Panel F shows line graphs of stochastic cohesin loss trajectories over time, where the x-axis represents time in seconds and the y-axis represents number of cohesin complexes remaining. Panel G shows a schematic summarizing that stochastic cohesin cleavage without feedback leads to variability and asynchrony in sister chromatid separation timing across chromosomes.

Small-number effects cause stochasticity in sister chromatid separation time. (A) Schematic of the stochastic model of separase-mediated cohesin removal. Separase activation (through securin degradation) is represented by a constant increase in the cohesin cleavage rate, k(t). The rate increases for a period τ until it reaches its maximum value, k_max. Each chromosome initially has N cohesin complexes (N₁, N₂, and N₃ for chromosomes I, II, and III, respectively), which are randomly cleaved by separase. Chromosomes separate once the number of remaining cohesins falls below a threshold n (n₁, n₂, and n₃ for chromosome I, II, and III, respectively). See Materials and methods for model details. (B) Frequency distributions and Gaussian fit (continuous lines) of the time differences between the separation of centromeres 1 and 2 or centromeres 2 and 3, either from a simulation using optimal parameter values of the basic model (salmon) or determined experimentally in wild-type cells (gray/black, same data as in Fig. 1 F). Mean ± SD of the simulation results or the Gaussian fit (for experimental data); n = number of cells; P values from EMD. (C) Goodness of fit of the model variants measured by EMD (see Materials and methods) with and without constraining τ at low values (τ < 5 s). Points indicate the validation EMD values obtained in the fivefold cross-validation (see Materials and methods); boxplots show median, interquartile range, and range. Basic, basic model with time-dependent cleavage rate alone; processive, model with bursts of cohesin cleavage; steric hindrance, model in which only a surface fraction of cohesin is available for cleavage and inner layers of cohesin are progressively exposed (see Materials and methods). (D) Parameter values of the basic model fitted to the experimental data. Points indicate values obtained in each of the fivefolds (see Materials and methods); boxplots show median, interquartile range, and range; the y-axis spans the parameter range tested, except for the inset showing n₁, n₂, and n₃. (E) One-at-a-time (OAT) sensitivity analysis of the basic model. Each parameter was varied individually across its allowed range while all other parameters were held fixed at their optimal values obtained by fitting. 10 batches of 10,000 simulations were performed for each parameter point. Each batch yielded a mean and SD of the separation time differences (Δt) between cen1 and centromere 2, and centromere 2 and 3. Plot shows mean of the Δt SD ± SEM across 10 batches. Dashed vertical lines indicate the optimal parameter values. (F) Stochastic cohesin loss trajectories near the time of separation in the basic model. Simulated cohesin counts for chromosomes 1, 2, and 3 are shown for three representative simulations, aligned to t = 0 at the final separation event. Lines indicate cohesin loss over time; points mark the moment of separation for each chromosome. (G) Schematic summarizing the results: separase becomes active upon securin degradation without positive feedback. If a small number of cohesin complexes suffice to maintain cohesion, stochasticity of the final cleavage events dominates the timing of sister chromatid separation, leading to asynchrony and no clear order between chromosomes.