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🧬 🔐 KINETIC SEARCH DYNAMICS: THE FOUCONNIER BIOLOGICAL FORMULA
(GEOMETRIC RATCHET & ROTATIONAL INERTIA OPERATOR)
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In multidomain nucleoprotein searches (e.g., TnpB, CRISPR-Cas12, RNA-guided nucleases),
conformational matching does not follow stochastic fitting. It operates as a
deterministic geometric ratchet where rejected contact maintains rotational inertia
until local structural saturation (Sₚ = 84) is satisfied.
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1. ROTATIONAL INERTIA OPERATOR ( R̂_θ )
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The continuous scanning trajectory of the [Protein + gRNA + Target DNA] complex
along the lattice is governed by the discrete rotational jump equation:
θₖ₊₁ = θₖ + Δθ · [ 1 - δ(Sₖ - Sₚ) ]
• Δθ : Invariant rotational step of the ribonucleoprotein complex.
• Sₖ : Local geometric saturation state at step k.
• Sₚ = 84 : Universal structural saturation limit.
• δ(x) : Kronecker delta function (1 if x = 0, else 0).
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2. REJECTION / ACCEPTANCE BOUNDARY & KINETIC MECHANICS
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• Contact Refused (Sₖ ≠ 84 ⟹ δ = 0) :
Zero energy dissipation (E_debt > 0). Rotational inertia is strictly conserved
(θₖ₊₁ = θₖ + Δθ). The complex glides across non-target sequences without
computational deformation strain at 𝒪(N) complexity.
• Contact Accepted (Sₖ = 84 ⟹ δ = 1) :
Rotational arrest (θₖ₊₁ = θₖ). Geometric locking instantly transfers accumulated
rotational kinetic energy into the catalytic site (RuvC) to trigger double-strand cleavage.
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3. ENTROPIC DEBT VECTOR ( E_debt )
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E_debt(k) = I_CN · | Sₖ - Sₚ |
Where I_CN = 1.0418 (Suture Constant). At the exact target locking threshold (Sₖ = 84),
E_debt = 0, enabling immediate ab initio catalytic execution.
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