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Cell Cycle

Field: Cell Biology

The series of events that take place in a cell that cause it to divide into two daughter cells.

Sequence of Expressions

Define the state variable Mstate∈[0,1]M_{state} \in [0, 1] representing the commitment to mitosis. The transition from G2 to M is governed by the activation of the Cyclin B-CDK complex, CDKB\text{CDK}_{B}. Let CB(t)C_B(t) be the concentration of Cyclin B and DG2(t)D_{G2}(t) be the DNA damage status. The transition rate λG2→M\lambda_{G2 \to M} is modeled by a cooperative switch function: λG2→M=kmax⋅CB(t)nKMn+CB(t)n⋅exp⁡(−DG2(t)Dcrit)\lambda_{G2 \to M} = k_{max} \cdot \frac{C_B(t)^n}{K_M^n + C_B(t)^n} \cdot \exp\left(-\frac{D_{G2}(t)}{D_{crit}}\right) where nn is the Hill coefficient, KMK_M is the Michaelis constant, and DcritD_{crit} is the critical damage threshold. The state transition is initiated when λG2→M>λthreshold\lambda_{G2 \to M} > \lambda_{threshold}.
Let C(t)C(t) be the concentration of the Cyclin and D(t)D(t) be the concentration of the CDK. The instantaneous activity of the kinase, A(t)A(t), is defined by the mass-action kinetics: A(t)=kcat⋅C(t)⋅D(t)KM+C(t)⋅D(t)A(t) = k_{cat} \cdot \frac{C(t) \cdot D(t)}{K_M + C(t) \cdot D(t)} where kcatk_{cat} is the catalytic rate constant and KMK_M is the apparent Michaelis constant for the substrate. The phosphorylation rate of a target protein PP is then given by the differential equation: d[Pphos]dt=(A(t)Ksub+[Punphos])[Punphos]−kdephos[Pphos]\frac{d[P_{phos}]}{dt} = \left( \frac{A(t)}{K_{sub} + [P_{unphos}]} \right) [P_{unphos}] - k_{dephos} [P_{phos}] where [Punphos][P_{unphos}] and [Pphos][P_{phos}] are the concentrations of unphosphorylated and phosphorylated protein, respectively.
Consider the concentration of a specific cyclin, [C](t)[C](t). Its degradation is catalyzed by the Anaphase Promoting Complex/Cyclosome (APC/C), which acts as a protease. The rate of change of [C][C] is governed by the differential equation: d[C]dt=−kdeg⋅[APC/C]⋅[C]+Synthesis Rate\frac{d[C]}{dt} = -k_{deg} \cdot [APC/C] \cdot [C] + \text{Synthesis Rate} where kdegk_{deg} is the degradation rate constant, and [APC/C][APC/C] is the concentration of the active complex. The degradation rate is assumed to follow pseudo-first-order kinetics with respect to [C][C] when [APC/C][APC/C] is saturating, simplifying the rate law to: d[C]dt=−keff[C]\frac{d[C]}{dt} = -k_{eff} [C] where keff=kdeg[APC/C]satk_{eff} = k_{deg} [APC/C]_{sat}. This decay dictates the timing of cell cycle exit.
Let vv be the polymerization velocity of the replication fork. The synthesis of DNA is modeled by the rate of incorporation of nucleotides. For the leading strand, the rate of synthesis is constant: d[Dlead]dt=v\frac{d[D_{lead}]}{dt} = v For the lagging strand, synthesis occurs in discrete Okazaki fragments, requiring a processivity model. The rate of synthesis of a fragment ii is viv_i, and the gap filling rate is vgapv_{gap}. The overall fidelity and stability are maintained by the error rate ϵ\epsilon: ϵ=NmismatchNbases≤ϵcrit\epsilon = \frac{N_{mismatch}}{N_{bases}} \le \epsilon_{crit} where NmismatchN_{mismatch} is the number of incorrect base pairs and NbasesN_{bases} is the total number of bases synthesized. The fork movement is governed by the balance of polymerization and helicase unwinding rates: dPositiondt=vpoly−vunwind\frac{d\text{Position}}{dt} = v_{poly} - v_{unwind}
Let SS be the state of the cell cycle, where S→AnaphaseS \to \text{Anaphase} is the transition of interest. Define the attachment status function A:K→{0,1}\mathcal{A}: K \to \{0, 1\}, where KK is the set of kinetochores, and A(k)=1\mathcal{A}(k) = 1 if kinetochore kk is properly attached to a spindle microtubule, and A(k)=0\mathcal{A}(k) = 0 otherwise. The Spindle Assembly Checkpoint (SAC) enforces the condition that the transition S→AnaphaseS \to \text{Anaphase} is only permitted if the global attachment satisfaction function Φ\Phi is true:\nΦ=∏k∈KA(k)=1\Phi = \prod_{k \in K} \mathcal{A}(k) = 1
Define the DNA unwinding process by the helicase enzyme H\mathcal{H} acting on a double helix segment of length LL. Let N(t)N(t) be the number of base pairs unwound at time tt. The rate of unwinding is governed by the kinetic equation:\ndNdt=kunwind⋅[MCM]⋅(11+KM/[ATP])−kreanneal⋅(N−Neq)\frac{d N}{d t} = k_{unwind} \cdot [MCM] \cdot \left( \frac{1}{1 + K_M / [ATP]} \right) - k_{reanneal} \cdot (N - N_{eq})\nwhere kunwindk_{unwind} is the maximum unwinding rate, [MCM][MCM] is the concentration of the MCM complex, and kreannealk_{reanneal} models the rate of spontaneous reannealing, ensuring the process is thermodynamically favorable (ΔG<0\Delta G < 0 for unwinding).
Consider the concentrations of key regulatory proteins: Cyclin B (CBC_B), CDK1 (DD), and the inhibitory phosphatase PP\text{PP}. The G2/M transition is modeled by the activation of CDK1, which requires the formation of the active complex D∙CBD \bullet C_B. The rate of complex formation and subsequent phosphorylation (P\text{P}) is described by the following system of coupled differential equations:\nd[D∙CB]dt=kform[D][CB]−kinact[D∙CB]⋅(1−[P][P]max)\frac{d [D \bullet C_B]}{d t} = k_{form} [D][C_B] - k_{inact} [D \bullet C_B] \cdot \left( 1 - \frac{[P]}{[P]_{max}} \right) \nwhere kformk_{form} is the association rate, kinactk_{inact} is the inactivation rate, and the term (1−[P][P]max)\left( 1 - \frac{[P]}{[P]_{max}} \right) represents the inhibitory effect of phosphatases, driving the system towards the mitotic state when [P][P] drops below a critical threshold.
Define the DNA damage state D\mathcal{D} as a binary variable, D=1\mathcal{D}=1 if damage is present, and D=0\mathcal{D}=0 otherwise. The DDR pathway activates a signaling cascade involving kinases (e.g., ATM, ATR). Let PactiveP_{active} be the concentration of the activated checkpoint protein. The activation rate is modeled by a Michaelis-Menten type function dependent on damage: \nd[Pactive]dt=kon⋅D⋅[Pinactive]−koff[Pactive]⋅(1+[Pactive]KD)\frac{d [P_{active}]}{d t} = k_{on} \cdot \mathcal{D} \cdot [P_{inactive}] - k_{off} [P_{active}] \cdot \left( 1 + \frac{[P_{active}]}{K_D} \right) \nIf D=1\mathcal{D}=1, the resulting high [Pactive][P_{active}] concentration triggers the activation of CDK\text{CDK} inhibitors (e.g., p21\text{p21}), mathematically enforcing a cell cycle arrest by setting the effective transition rate λG1→S=0\lambda_{G1 \to S} = 0 until D\mathcal{D} returns to 00.
Let S∈Rk\mathbf{S} \in \mathbb{R}^k be the vector of environmental and internal signaling metrics (e.g., nutrient levels, growth factor concentrations, DNA damage markers). Define the progression probability PG1→SP_{G1 \to S} as a function of these metrics: PG1→S=H(S)=11+e−(wTS−θ)P_{G1 \to S} = \mathcal{H}(\mathbf{S}) = \frac{1}{1 + e^{-(\mathbf{w}^T \mathbf{S} - \theta)}} where w∈Rk\mathbf{w} \in \mathbb{R}^k are the weight vectors representing the sensitivity to each signal, and θ\theta is the critical threshold for progression. The cell state X(t)∈{G1,S,G2,M,G0}\mathbf{X}(t) \in \{G1, S, G2, M, G0\} transitions according to the condition: X(t+Δt)=S if PG1→S>τG1\mathbf{X}(t+\Delta t) = S \text{ if } P_{G1 \to S} > \tau_{G1} and X(t)=G1 otherwise\mathbf{X}(t) = G1 \text{ otherwise}, where τG1\tau_{G1} is the minimum required progression probability.
Let X(t)=(XG1,XS,XG2,XM)T\mathbf{X}(t) = (X_{G1}, X_{S}, X_{G2}, X_{M})^T be the population density vector across the four major cell cycle phases at time tt. Synchronization is achieved by introducing a chemical inhibitor I\mathcal{I} that arrests the transition rate λi→j\lambda_{i \to j} between phases ii and jj. The modified flow dynamics are given by:\ndXdt=X⋅T⋅(1−I(X))\frac{d \mathbf{X}}{d t} = \mathbf{X} \cdot \mathbf{T} \cdot (1 - \mathcal{I}(\mathbf{X})) \nwhere T\mathbf{T} is the transition matrix, and I(X)\mathcal{I}(\mathbf{X}) is the inhibition function. For G1 arrest, I(X)→1\mathcal{I}(\mathbf{X}) \to 1 when X\mathbf{X} reaches a critical density, forcing the transition rate λG1→S→0\lambda_{G1 \to S} \to 0, resulting in a stable equilibrium state X∗\mathbf{X}^* where dXdt=0\frac{d \mathbf{X}}{d t} = \mathbf{0}.