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How to use
Choose a preset or build your own cell. The picture shows a cross-section: the nucleus with its dashed envelope, the two centrosomes, spindle microtubules, chromosomes with coloured kinetochores (red free, green correct, purple wrong) and the contractile ring. The phase bar at the bottom shows progress. The graphs track cyclin B / securin / Cdk1, free and wrong kinetochores, the checkpoint signal and APC/C activity, and the geometry of anaphase and cytokinesis. Fast-forward completes the division, Space pauses, R restarts. Things to try: (1) normal division and read the time from NEBD to anaphase; (2) switch to 46 chromosomes and compare the capture time with H(2n)/ka in the readout; (3) nocodazole arrest: the cell sits in prometaphase with the checkpoint on, then slide the poison back to 0 (washout) and watch division resume; (4) turn the checkpoint off and see anaphase start with free kinetochores; (5) raise the error rate and weaken Aurora B to create lagging chromosomes and micronuclei; (6) cool the cell to 25 C; (7) slow the furrow.
The science
Mitosis copies one cell's chromosomes into two daughter cells. Before it starts the DNA has been replicated, so every chromosome is a pair of identical sister chromatids held together by cohesin. Mitosis has to pull exactly one chromatid of each pair into each daughter, and the cell uses a small molecular timer plus a quality-control check to do it.
Phases. In prophase the chromatin condenses and the two centrosomes move apart. At nuclear-envelope breakdown (NEBD) microtubules from the poles can reach the chromosomes (prometaphase). Chromosomes congress to the spindle equator (metaphase), the sisters split (anaphase A moves the chromatids toward the poles, anaphase B pushes the poles apart) and an actomyosin ring pinches the cell in two (cytokinesis).
Kinetochore capture. Each chromosome carries two kinetochores. In the model every free kinetochore is captured at a rate ka (about 0.25 per minute at 37 C, scaled by the microtubule supply). With N free kinetochores the time until all are captured is the maximum of N exponential waiting times, whose mean is the harmonic sum
⟨tcapture⟩ = (1/ka) (1 + 1/2 + … + 1/N) ≈ (ln N + 0.577)/ka
so doubling the chromosome number adds only a fixed delay (about 0.69/ka). A captured kinetochore is wrong (merotelic, attached to both poles) with probability perr and is repaired at rate kc by Aurora B kinase, which destabilises attachments that are not under tension.
Spindle assembly checkpoint. Unattached kinetochores make the mitotic checkpoint complex (MCC), which binds and blocks Cdc20, the activator of the ubiquitin ligase APC/C. With u the fraction of free kinetochores and S the checkpoint strength,
dM/dt = kon S u (1 − M) − koff M, M* = konS u/(konS u + koff), a = 1/(1 + (M/Mc)²)
One free kinetochore is enough to keep a low (switch-like inhibition). When the last one is captured M decays with a half-life of ln 2/koff (under a minute here; real cells need a few minutes), APC/C switches on, and it tags securin and cyclin B for destruction: d(securin)/dt = −kA a · securin and d(cyclin B)/dt = −kB a · cyclin B. When securin falls below 10 % separase is free to cut cohesin and anaphase begins after ln 10/kA ≈ 5.8 minutes of full APC/C activity. Cdk1 activity (a steep function of cyclin B) must fall before the furrow can start, which is why cytokinesis follows anaphase by roughly ten minutes.
Movements. Bioriented chromosomes are pulled by both kinetochore fibres and relax to the equator like a damped spring, x(t) = x0 e−κt; polar ejection forces set the width of the plate. In anaphase A kinetochore fibres shorten at vA (about 1 µm/min), in anaphase B the spindle lengthens at vB, and the furrow ingresses at vf; the dividing cell stays at constant volume, a b² = R³. All rates scale with temperature with Q10 = 2. Lagging merotelic chromosomes that were never corrected travel slowly and end up as micronuclei, a route to aneuploidy.
Numerical checks run in the page. (1) Monte-Carlo mean capture time of 40 kinetochores against H(40)/ka; (2) cyclin B decay with APC/C fully on against the exponential; (3) the securin timer ln 10/kA when the checkpoint is off; (4) anaphase A kinematics; (5) the MCC steady state; (6) volume conservation of the elongating cell; (7) congression relaxation against the exponential.
Simplifications. Rates are round, plausible values for a cultured mammalian cell, not fitted data; the spindle is a bipolar fan, not a self-organising microtubule network; the model treats attachment as a Poisson process and ignores the search-and-capture geometry, spindle forces on single chromosomes, the Mad1/Mad2 template, cohesin cleavage kinetics and cell-size-dependent scaling of the spindle; chromosome positions are a 3D cartoon constrained to the correct kinematics; the 2D drawing is a section through the cell.