Reverse-time analysis and boundary classification of directional biological dynamics with multiplicative noise

The dynamics of living systems often serves the purpose of reaching functionally important target states. We previously proposed a theory to analyze stochastic biological dynamics evolving towards target states in reverse time. However, a large class of systems in biology can only be adequately described using state-dependent noise, which had not been discussed. For example, in gene regulatory networks, biochemical signaling networks or neuronal circuits, count fluctuations are the dominant noise component. We characterize such dynamics as an ensemble of target state aligned (TSA) trajectories and characterize its temporal evolution in reverse-time by generalized Fokker-Planck and stochastic differential equations with multiplicative noise. We establish the classification of boundary conditions for target state modeling for a wide range of power law dynamics, and derive a universal low-noise approximation of the final phase of target state convergence. Our work expands the range of theoretically tractable systems in biology and enables novel experimental design strategies for systems that involve target states.