Biomass transfer on autocatalytic reaction network: a delay differential equation formulation

TL;DR

This work formulates biomass transfer in autocatalytic reaction networks as a delay differential equation by tracking biomass along reaction pathways through gatekeepers. Central to the method are the catalytic kernel $\alpha(\tau)$ and its spectrum $A(s)$, which connect microscopic reaction kinetics to macroscopic growth rates, enabling cross-comparisons of networks with different topology and dimensionality. The approach extends from simple SAPs to linear LRNs and further to scalable SRNs, providing both exact relations $\lambda=A(\lambda)$ (LRNs) and effective kernels $\alpha_{eff}$, $A_{eff}$ for nonlinear cases, together with ergodic averaging results (Theorems C and D) that justify coarse-graining and phase/time averaging. The framework yields practical bounds and coarse-graining strategies for growth-rate estimation across complex networks and suggests experimental avenues to measure the catalytic kernel via isotope labeling. Collectively, the paper offers a unified, Lagrangian-based toolkit to quantify and compare autocatalytic growth across network topologies and nonlinearities in biological systems.

Abstract

For a biological system to grow and expand, mass must be transferred from the environment to the system and be assimilated into its reaction network. Here, I characterize the biomass transfer process for growing autocatalytic systems. By track biomass along reaction pathways, an n-dimensional ordinary differential equation (ODE) of the reaction network can be reformulated into a one-dimensional delay differential equation (DDE) for its long-term dynamics. The kernel function of the DDE summarizes the overall amplification and transfer delay of the system and serves as a signature for autocatalysis dynamics. The DDE formulation allows reaction networks of various topologies and complexities to be compared and provides rigorous estimation scheme for growth rate upon dimensional reduction of reaction networks.