Phyfold: environment aware mathematical modeling for protein folding dynamics integrated with physics informed neural network

Protein folding remains a fundamental problem in computational biology, where understanding folding trajectories and dynamic behavior under physiological conditions is critical for linking structure to function. Recent advances such as AlphaFold have revolutionized sequence to structure prediction, establishing a powerful and robust foundation for computational protein modeling. Building upon this success, we introduce a novel physics informed computational framework that expands static structure prediction toward dynamic and physically consistent modeling of protein folding processes. The proposed framework unifies a mathematically grounded reaction diffusion model of folding kinetics with deep sequence embeddings through a physics informed neural network (PINN), ensuring consistency with governing physical laws while remaining biologically grounded. We refer to this unified framework as PhyFold. Our results illustrate that the proposed approach effectively captures environment dependent folding dynamics and shows strong agreement with both numerical reference solutions and experimental fluorescence measurements under physiologically relevant conditions. By explicitly integrating environmental factors such as temperature, pressure, and pH, PhyFold enables realistic, physics aware simulation of protein folding dynamics. This approach offers a natural extension of modern structure prediction methods and opens new directions for protein design, structure guided drug discovery, and personalized medicine. The code to reproduce the work is available at https://github.com/jamshaidwarraich/PhyFold.git .Scientific contributionPhyFold introduces a hybrid Bio-Physics Informed Neural Network framework that models protein folding kinetics as a reaction diffusion partial differential equation with environmentally modulated kinetic coefficients, integrated with ProtT5 sequence embeddings for sequence specific spatiotemporal prediction an approach not previously demonstrated in the literature. Unlike purely data driven or static structure prediction methods, PhyFold explicitly encodes temperature, pH, and pressure dependence into governing kinetic coefficients, enabling physically interpretable simulation of cooperative folding transitions across diverse physiological conditions. Dual validation against finite difference reference solutions and experimental GFP fluorescence measurements confirms mathematical consistency and biological relevance, establishing PhyFold as a computationally efficient surrogate for environment aware protein folding dynamics.