The catalytic kernel α(τ) is not merely a mathe-
matical tool but also represents a biophysical quantity
connecting microscopic kinetic constants to macro-
scopic growth dynamics. It illustrates the timescale
for biomass transfer and the efficiency of autocatalysis
and can be experimentally measured in the future. By
adopting the DDE formulation, one can compare re-
action networks of differing dimensions and topologies
and perform coarse-graining on complex reaction net-
works. Overall, I envision using DDEs to analyze the
dynamics of biomass transfer provides a deeper and
systems-level understanding of autocatalysis.
1. Reaction networks, reaction pathways,
and gatekeepers. Let us consider an open sys-
tem in which nutrients and waste can be exchanged
with the environment (Sughiyama et al., 2022). The
reaction network structure (Figure 1a) includes sys-
tem nodes {x1,· · · , xn}:= X, environmental nodes
{E1,· · · , E(n′)}:= E, and reactions between nodes
{ϕ1,· · · , ϕm}. Each node represents one chemical
species. A reaction ϕais represented by
ϕa:
n
X
i=1
ciaxi+
n′
X
j=1
c′
jaEj→
n
X
i=1
diaxi+
n′
X
j=1
d′
jaEj,(1)
where cia, cia′, dja, dja′≥0 are stoichiometric coeffi-
cients. We say xiis an upstream (or downstream)
node of ϕaif cia −dia is less (or greater) than 0, and ϕa
is the influx (or efflux) of node xiif xiis a downstream
(or upstream) node of ϕa. A node can have multiple
influxes and effluxes, and each reaction has at least one
upstream node and one downstream node.
To describe biomass transfer, a nonnegative num-
ber m(xk) is associated to each node xk. In prac-
tice, this number could be a molecular weight, or a
carbon atom count of biochemical molecules. The
biomass-weighted stoichiometry matrix is defined as
Sia := (dia −cia)m(xk). In this way, stoichiometry can
be interpreted by the unit of biomass. Mass conserva-
tion is required to avoid anomalous behaviors of system
growth (i.e., autocatalysis from void). In the frame-
work of this study, mass conservation can always be
achieved by introducing fictitious environmental nodes.
Finally, we denote in(xk) and out(xk) as the collections
of reactions ϕawith Ska >0 and Ska <0, respectively,
i.e., reactions that provide influx and efflux for node xk.
To adopt the Lagrangian view, we assume biomass
can be discretized in microscopic level and the dis-
cretized biomass units are tractable. In this way, a
Markov process which is consistent with ODE flux
model can be established, and the DDE formulation
can be derived (Figure 1B). We describe the rigorous
formulation in Supplementary Information (SI).
Reaction pathways are natural structures to model
biochemical reactions. To achieve sustainable growth,
an open system must convert environmental materials
into internal biomass, with reaction pathways being in-
dispensable for these interconversion processes. I define
a reaction pathway by
π(u, ω) = u0ω0· · · ωLuL+1,(2)
where nodes {u0,· · · , uL+1}=: {u}and reaction
{ω0,· · · , ωL}:= {ω}are ordered sets. For each re-
action ωj, the nodes uj−1and ujare its upstream
and downstream nodes, respectively. All nodes in {u}
are system nodes, except that the first or last nodes
can be environmental nodes. I allow {u}and {ω}
to have repeated members, and the length Lcan be
infinite. Where there is no confusion, {ω}can be
omitted and the reaction pathway can be written as
π:u0u1· · · uL+1.
Boundary influx reactions are reactions in which
the upstream comprises environmental nodes and the
downstream has system nodes. Here, each boundary
influx reaction ϕa(red arrows in Figure 1A) is assumed
to be catalyzed by one system node (labeled by the en-
circled nodes next to the red arrow). This node is
called the gatekeeper of ϕa. The collection of all gate-
keepers is denoted G. Intuitively, since gatekeepers
catalyze all of the boundary influxes collectively, their
rates of increase control the rate of system expansion.
Gatekeeper growth dynamics will be characterized in
the following sections.
2. Network dynamics and flux functions. Our
discussion so far focused on reaction network structure.
To construct a dynamical system, we need to define
variables and flux functions quantitatively. We de-
fine X:= (X1,· · · , Xn)Tas the biomass vector, where
Xk:= xk×(number of xk-type objects in the system).
Moreover, J:= (J[ϕ1],· · · , J[ϕm])Tdenotes the flux
function vector, equal to the number of reaction events
that happen per unit time. Note that the units for
Xand Jare [biomass] and [1/time], respectively, and
the unit for the biomass-weighted stoichiometry ma-
trix (described in the previous section) is [biomass].
Together, the biomass flux of reaction ϕaassociated
with node xkis SkaJa, with unit [biomass/time].
We assume that substances in the environment are
unlimited and are maintained at constant densities sur-
rounding the system (i.e., similar to a particle reservoir
of a grand canonical ensemble in statistical mechanics).
2