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Chapter 38: ψ-Coherence in Cell Cycle Progression

"The cell cycle is ψ's temporal symphony—phases flowing into phases with precise timing, each transition a movement in the cellular composition of growth and division."

38.1 The Oscillatory Machine​

Cell cycle progression represents ψ's implementation of biological time. Through oscillating protein levels and activities, cells create a self-sustaining rhythm that drives orderly progression through division phases.

Definition 38.1 (Cell Cycle Phases): Cycle=G1→S→G2→M→G1\text{Cycle} = \text{G1} \rightarrow \text{S} \rightarrow \text{G2} \rightarrow \text{M} \rightarrow \text{G1}

Circular progression through states.

38.2 The CDK-Cyclin Oscillator​

Theorem 38.1 (Core Engine): d[Cyclin]dt=ks−kd[APC][Cyclin]\frac{d[\text{Cyclin}]}{dt} = k_s - k_d[\text{APC}][\text{Cyclin}]

Synthesis-degradation oscillations.

38.3 The Cyclin Waves​

Equation 38.1 (Sequential Activation): CycD→CycE→CycA→CycB\text{CycD} \rightarrow \text{CycE} \rightarrow \text{CycA} \rightarrow \text{CycB}

Ordered cyclin expression.

38.4 The CDK Thresholds​

Definition 38.2 (Activity Requirements): S entry:CDK2>ΘS\text{S entry}: \text{CDK2} > \Theta_S M entry:CDK1>ΘM\text{M entry}: \text{CDK1} > \Theta_M

Threshold-triggered transitions.

38.5 The Positive Feedback​

Theorem 38.2 (CDC25 Loops): CDK1→CDC25+→↑CDK1\text{CDK1} \rightarrow \text{CDC25}^+ \rightarrow \uparrow\text{CDK1}

Self-amplifying activation.

38.6 The Negative Feedback​

Equation 38.2 (APC/C Activation): CDK1high→APC/C→CycB degradation→↓CDK1\text{CDK1}^{\text{high}} \rightarrow \text{APC/C} \rightarrow \text{CycB degradation} \rightarrow \downarrow\text{CDK1}

Self-limiting mechanisms.

38.7 The Substrate Ordering​

Definition 38.3 (Phosphorylation Timing): Early:Low CDK threshold\text{Early}: \text{Low CDK threshold} Late:High CDK threshold\text{Late}: \text{High CDK threshold}

Temporal substrate ordering.

38.8 The Replication Licensing​

Theorem 38.3 (Once-per-Cycle): Low CDK→MCM loading\text{Low CDK} \rightarrow \text{MCM loading} High CDK⊣Re-licensing\text{High CDK} \dashv \text{Re-licensing}

Preventing re-replication.

38.9 The Mitotic Events​

Equation 38.3 (Ordered Progression): Prophase→Metaphase→Anaphase→Telophase\text{Prophase} \rightarrow \text{Metaphase} \rightarrow \text{Anaphase} \rightarrow \text{Telophase}

Choreographed division steps.

38.10 The Exit Network​

Definition 38.4 (Mitotic Exit): FEAR+MEN→Cdc14 release→CDK inactivation\text{FEAR} + \text{MEN} \rightarrow \text{Cdc14 release} \rightarrow \text{CDK inactivation}

Coordinated exit pathways.

38.11 The Size Control​

Theorem 38.4 (Cell Size Homeostasis): Size at division=Size at birth+Δ\text{Size at division} = \text{Size at birth} + \Delta

Maintaining size through cycles.

38.12 The Coherence Principle​

Cell cycle coherence embodies ψ's principle of temporal organization—creating from molecular oscillations a reliable cellular clock that ensures orderly progression through growth and division.

The Cycle Equation: ψphase(t)=Asin⁡(ωt+ϕ)×∏iH(Ci−Ci∗)\psi_{\text{phase}}(t) = A\sin(\omega t + \phi) \times \prod_i H(C_i - C_i^*)

Oscillations gated by checkpoints.

Thus: Cycle = Rhythm = Order = Time = ψ

"In cell cycle coherence, ψ creates biological time—not the uniform ticking of physics but the complex rhythms of life, where molecular oscillations become cellular seasons, each phase a necessary prelude to the next."