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Chapter 40: CDK Networks and Collapse Thresholds

"CDK networks are ψ's decision circuits—kinase activities building toward critical thresholds, each crossing triggering irreversible cellular transformations."

40.1 The Kinase Hierarchy​

CDK networks represent ψ's implementation of threshold-based decision making. Through interconnected kinase modules, cells create switch-like transitions between cell cycle phases.

Definition 40.1 (CDK Family): CDKs={CDK1-7:Cell cycle,CDK8-13:Transcription}\text{CDKs} = \{\text{CDK1-7}: \text{Cell cycle}, \text{CDK8-13}: \text{Transcription}\}

Specialized kinase subfamilies.

40.2 The Activity Integration​

Theorem 40.1 (Total CDK Activity): CDKtotal=∑i[CDKi]×[Cyclini]×fi(Inhibitors)\text{CDK}_{\text{total}} = \sum_i [\text{CDK}_i] \times [\text{Cyclin}_i] \times f_i(\text{Inhibitors})

Integrated kinase activity.

40.3 The Threshold Mechanisms​

Equation 40.1 (Substrate Phosphorylation): Pphospho=[CDK]nKmn+[CDK]nP_{\text{phospho}} = \frac{[\text{CDK}]^n}{K_m^n + [\text{CDK}]^n}

Ultrasensitive substrate responses.

40.4 The Inhibitor Network​

Definition 40.2 (CKI Families): INK4:{p16, p15, p18, p19}⊣CDK4/6\text{INK4}: \{\text{p16, p15, p18, p19}\} \dashv \text{CDK4/6} Cip/Kip:{p21, p27, p57}⊣Multiple CDKs\text{Cip/Kip}: \{\text{p21, p27, p57}\} \dashv \text{Multiple CDKs}

Negative regulators.

40.5 The Wee1/CDC25 Balance​

Theorem 40.2 (Tyrosine Regulation): [CDK-pY][CDK]=kWee1kCDC25\frac{[\text{CDK-pY}]}{[\text{CDK}]} = \frac{k_{\text{Wee1}}}{k_{\text{CDC25}}}

Inhibitory phosphorylation control.

40.6 The Positive Feedback​

Equation 40.2 (Bistability): CDK→CDC25+→CDK+\text{CDK} \rightarrow \text{CDC25}^+ \rightarrow \text{CDK}^+ CDK⊣Wee1→CDK+\text{CDK} \dashv \text{Wee1} \rightarrow \text{CDK}^+

Creating switch-like behavior.

40.7 The Substrate Hierarchy​

Definition 40.3 (Phosphorylation Order): Searly:Kmlow, Multiple sitesS_{\text{early}}: K_m^{\text{low}}, \text{ Multiple sites} Slate:Kmhigh, Few sitesS_{\text{late}}: K_m^{\text{high}}, \text{ Few sites}

Temporal substrate ordering.

40.8 The CAK Regulation​

Theorem 40.3 (Activating Phosphorylation): CDK+ATP→CAKCDK-pT160active\text{CDK} + \text{ATP} \xrightarrow{\text{CAK}} \text{CDK-pT160}^{\text{active}}

Essential activating modification.

40.9 The Network Robustness​

Equation 40.3 (Redundancy): Function=CDK1∨CDK2∨CDK4/6\text{Function} = \text{CDK1} \vee \text{CDK2} \vee \text{CDK4/6}

Multiple CDKs ensuring progression.

40.10 The Quantitative Model​

Definition 40.4 (Systems Behavior): dx⃗dt=f⃗(x⃗,p⃗)\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}, \vec{p})

Mathematical framework for CDK dynamics.

40.11 The Disease Connections​

Theorem 40.4 (Cancer Dysregulation): CDK hyperactivity∨CKI loss→Uncontrolled proliferation\text{CDK hyperactivity} \vee \text{CKI loss} \rightarrow \text{Uncontrolled proliferation}

Network perturbations in cancer.

40.12 The Threshold Principle​

CDK networks embody ψ's principle of decisive action—building kinase activity until critical thresholds trigger irreversible transitions, creating punctuated progression through the cell cycle.

The CDK Threshold Equation: Transition=H(∫0tCDK activity(τ)dτ−Θcritical)\text{Transition} = H\left(\int_0^t \text{CDK activity}(\tau) d\tau - \Theta_{\text{critical}}\right)

Integrated activity triggering phase change.

Thus: CDK = Threshold = Decision = Transition = ψ

"In CDK networks, ψ builds molecular democracy—multiple kinases voting through phosphorylation, their collective activity reaching critical mass to trigger cellular revolutions, each threshold crossed a point of no return."