  For a biological system to grow, the biomass must be incorporated,
transferred, and accumulated into the underlying reaction network. There are
two perspectives for studying growth dynamics of reaction networks: one way is
to focus on each node in the networks and study its associated influxes and
effluxes. The other way is to focus on a fraction of biomass and study its
trajectory along the reaction pathways. The former perspective (analogous to
the "Eulerian representation" in fluid mechanics) has been studied extensively,
while the latter perspective (analogous to the "Lagrangian representation" in
fluid mechanics) has not been systematically explored. In this work, I
characterized the biomass transfer process for autocatalytic, growing systems
with scalable reaction fluxes. Under balanced growth, the long-term growth
dynamics of the systems are described by delay differential equations (DDEs).
The kernel function of the DDE serves as a unique pattern for the catalytic
delay for a reaction network, and in frequency domain the delay spectrum
provides a geometric interpretation for long-term growth rate. The DDE
formulation provides a clear intuition on how autocatalytic reaction pathways
lead to system growth, it also enables us to classify and compare reaction
networks with different network structures.
