Mathematical Models of Bioevo-Cybernetics I: Entropy Dissipation and the Origins of Complexity

Life emerged from non-living matter through energy-driven, self-organizing processes constrained by thermodynamics, stabilized by dissipative structures, and coordinated by cybernetic feedback. We present a staged mathematical framework—Bioevo-Cybernetics I—that formalizes this progression from protocells to unicellular and multicellular systems. The model integrates entropy dissipation, free-energy throughput, and hierarchical feedback regulation, showing how organisms sustain order by channeling metabolic flows into dissipative pathways. Evolution is thus reframed as an active, self-regulating process in which variation, selection, and feedback interact under entropy–dissipative constraints, progressively transforming stochastic mutation into directed pathways of complexity. As a case study, we apply the framework to the evolutionary transition from heterotrophic flagellates to photosynthetic dinoflagellates. The model captures key drivers—environmental energy flow, cellular asymmetry, population processes, and internal integration—while explicitly incorporating thermodynamic gating functions that determine whether complexity is maintained or lost. Simulations reproduce major thresholds of evolutionary innovation, including plastid acquisition, nuclear dualism, and stable flagellar reconfiguration, with results aligning with fossil and geobiological records. The framework quantifies the trade-off between increasing complexity and decreasing adaptability, also predicts contingent outcomes such as bistability, hysteresis and environmental dependence, providing an explanation for both the early origin and later fossil appearance of dinoflagellates. By uniting entropy dissipation, cybernetic control, and evolutionary transitions in a single quantitative framework, Bioevo-Cybernetics I offers a predictive approach to reconstructing the origins and long-term dynamics of biological complexity. This work establishes a foundation for testing how thermodynamic and regulatory constraints interact to shape major evolutionary innovations, from the origin of life to the rise of eukaryotic complexity.