Machine Learning-Assisted Mathematical Modeling of Evolutionary Biochemical Processes

Abstract

Background: Evolutionary biochemical systems consist of multiscale interactions between biochemistry and reaction kinetics as they relate to mutation, selection, growth of populations, and feedback loops from metabolic products. Though the governing mechanisms of evolutionary biochemical systems are comprehensible, the kinetic parameters are often poorly identifiable. The objective of this study is to design a framework that combines mathematics and machine learning to estimate uncertain biochemical parameters and amend mechanistic dynamics that are incomplete, all while honoring constraints of biology. The methods of this study include a model composed of six coupled nonlinear ordinary differential equations which treat substrates, enzymes, products, population, the frequency of beneficial mutations, and mean fitness. The existence, uniqueness, positivity, boundedness, and local stability of this model were established, and synthetic trajectories were generated through adaptive Runge–Kutta integration. Sparse trajectories that were noisy were used to create a pure neural predictor that is complemented by a residual neural model in a neural framework of a misspecified ODE. The system converged to the positive equilibrium (0.2743,10.6019,7.8683,18.4012,0.8506,1.9291), and the eigenvalues of the six-dimensional Jacobian had all negative real components. The other system of residual learning in the misspecified ODE had an RMSE of 0.6790, and pure machine learning had an RMSE of 0.1262. The best hybrid model had an RMSE of 0.0513 and a relative L2 error of 0.0056. The residual neural model came from a mechanistic framework and helped the author to model a real evolutionary biochemical system when data were scarce. The biochemical interpretation was further developed by relating the catalytic flux to the elementary enzyme scheme E + S ⇌ ES → E + P and by identifying substrate, enzyme, product, cofactors, buffer, and biomass as the principal chemical or biochemical components. A physical-chemistry layer was also added to connect kinetic parameters with activation free energy, temperature, pH, ionic strength, reversibility, and transport assumptions.