Robust Geometric Confinement of Apoptotic Signaling Pathways: Bridging Stochastic Differential Equations and the Chrysene Formalism via H-infinity Inspired Molecular Splinting

Abstract

Classical pharmacology often represents cellular apoptosis as a stochastic dynamical system governed by noise-induced transitions within biochemical cascades. These systems can be modeled using stochastic differential equations (SDEs) to capture thermodynamic fluctuations, conformational drift, and solvent-mediated perturbations in the intracellular environment. Classical computer-aided drug design (CADD) typically optimizes ligand-target interactions through thermodynamic binding models, empirical force fields, molecular docking, molecular dynamics, and statistical sampling of conformational ensembles. These methods are powerful, but they may not directly address the problem of robustly confining a target protein to a prescribed conformational neighborhood under persistent stochastic forcing.

This paper develops a mathematical framework for robust geometric confinement under the Chrysene Formalism. Instead of treating a molecular therapeutic as a time-dependent active controller, we model a rigid chrysene-based molecular brace as a passive static potential field or geometric constraint. The stochastic conformation of an apoptotic target protein is represented by a multidimensional state vector, and its structural deviation from a desired target conformation is quantified by an Entropic Strain Tensor, which measures the overall quadratic spatial deviation. An H-infinity-inspired minimax formulation is used as a design principle for selecting static splinting potentials that reduce worst-case conformational strain under bounded perturbations, while residual Brownian noise is retained as an irreducible stochastic input. In the reflected stochastic differential equation idealization, a rigid geometric brace confines the trajectory to a compact tether domain, ensuring the conformational deviation remains strictly bounded by a specified finite radius. Thus, the principal mathematical result is finite-radius stochastic confinement. Exact zero strain arises only in the singular mathematical limit where this tether radius shrinks to zero, not as a literal physical claim for thermally fluctuating molecular systems.