Biomass transfer on autocatalytic reaction networks a delay-differential equation formulation Wei-Hsiang Lin12

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Biomass transfer on autocatalytic reaction networks:
a delay-differential equation formulation
Wei-Hsiang Lin1,2
1Institute of Molecular Biology, Academia Sinica, Taiwan
2Genome and Systems Biology Degree Program, National Taiwan University, Taiwan
Mar 7, 2025
Abstract. For a biological system to grow and ex-
pand, mass must be transferred from the environment
to the system and be assimilated into its reaction net-
work. 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 dif-
ferential equation (DDE) for its long-term dynamics.
The kernel function of the DDE summarizes the over-
all 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 pro-
vides rigorous estimation scheme for growth rate upon
dimensional reduction of reaction networks.
0. Introduction. An essential feature of biologi-
cal systems is their ability to grow and expand, which
involves incorporating external resources into internal
biomass. This conversion process often involves a re-
action network that performs biosynthesis and energy
production. In many natural systems, reaction net-
works are autocatalytic in the sense that the system
generates its own (Blokhuis et al., 2020, Hordijk et al.,
2012, Letelier et al., 2006,Vassena and Stadler, 2024,
Despons, 2025, Despons et al., 2024, Marehalli Srini-
vas et al., 2024, Gagrani et al., 2024, M¨uller et al.,
2022). Understanding the general principles under-
lying metabolic conversion and biomass synthesis is
fundamentally important for all living systems.
To obtain useful insights, we consider the anal-
ogy between material transfer in physical systems
and biomass transfer in biochemical systems. In fluid
mechanics, Eulerian description inspects the assesses
influx and efflux in terms of absolute coordinates,
whereas Lagrangian description inspects movement of
a fluid parcel along a stream. In reaction networks, one
could focus on the influxes and effluxes of each nodes
and calculate the material change rate of each node,
which corresponds to the Eulerian view. Alternatively,
a small cohort of biomass may be tracked and transfer
along the reaction pathways, reflecting a Lagrangian
view.
For growing reaction networks, majority of theo-
retical studies adopt the Eulerian description (Kondo
and Kaneko, 2011,Maitra and Dill, 2015, Mb and S,
2000, Scott et al., 2010). The Lagrangian view has
been used to develop metabolic models of pulse-chase
experiments with isotope labeling (Antoniewicz et al.,
2007, Thommen et al., 2023), but it has not been for-
mulated for network growth and autocatalysis. In this
work, I demonstrate that analyzing biomass transfer
along reaction pathways is an intuitive and power-
ful way of studying autocatalytic growth dynamics.
The analysis focuses on nodes that regulate bound-
ary influxes of the entire system. These nodes, termed
gatekeepers, limit entry of the biomass into the system.
By adopting a Lagrangian view, biomass transfer from
boundary influxes to gatekeepers can be analyzed as an
autocatalytic process, enabling formulation of delay-
differential equations (DDEs) for gatekeeper biomass
(denoted as Z).
Specifically, the long-term growth dynamics of
a reaction network can be accessed via equation
dZ
dt =βZ +Rα(τ)Z(tτ), whereby the catalytic
kernel α(τ) describes the process by which gatekeeper
biomass autocatalyzes itself. Note that the formula-
tion here is distinct from the classical renewal equation
x(t) = Rα(τ)x(tτ), which is applied to age-
structured population growth (Britton, 2003). In this
work, α(τ) is calculated explicitly for scalable reaction
networks (SRNs), which is a general class of reaction
networks that includes linear and nonlinear flux func-
tions and allows long-term exponential growth (Lin
et al., 2020).
1
arXiv:2210.09470v3 [q-bio.MN] 7 Mar 2025
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, dja0 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 uj1and 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
Figure 1. Growing reaction networks and the modeling frameworks. (A) Schematic of a reaction network system (circular
nodes) and the surrounding environment (square nodes). The gatekeeper set G is a subset of the system, which controls the
boundary reaction influxes (red arrows). (B) Eulerian and Lagrangian views on reaction network and biomass. (C) Schematic
of reaction pathways in a network. For example, π1is a direct reaction pathway from x1to x4, and π2is another reaction
pathway from x1to x3, back to x2, and ultimately reaching x4. Blue paths represent the trajectory of biomass movement.
(D) Reaction networks with different flux function classes (defined in later sections). SAPs are a special cases of LRNs, which
are special cases of SRNs.
For the system, by assuming large number of ob-
jects and reaction events, Xand Jare approximated
as continuous variables (Figure 1D) and follow the sys-
tems of ordinary differential equation (ODE):
dXi
dt =
m
X
a=1
SiaJa,(3)
for k= 1,· · · , n. Coefficients for environmental nodes
(i.e., cia, dia) do not appear in the equation since it is
assumed that their amounts are unlimited.
In general, each flux function Ja(X) can be consid-
ered as a multivariate nonlinear function from Rnto
R. Without restricting flux function types, the system
can display nonphysical behaviors such as unbounded
growth unboundedly and blow up at a finite time, or
decay to negative values. Appropriate conditions will
be imposed such that the system grows exponentially
in the long term. Let N=X1+· · · +Xndenote the
total biomass, define the long-term growth rate λas
λ:= lim
t→∞
1
tlog N(t).(4)
Our focus is to analyze how λis determined by the
catalytic efficiency and biomass transfer time of the
system. This paper is structured in three parts. First,
I analyze biomass transfer in a simple autocatalytic
pathway (SAP) (Fiedler et al., 2013) to develop in-
tuition and concrete formulae. Then, I generalize the
analysis to linear reaction networks (LRNs) (Nandori
and Young, 2022, Unterberger and Nghe, 2022), before
extending to scalable reaction networks (SRNs), with
this latter enabling us to study many nonlinear sys-
tems that grow exponentially (Figure 1D).
3. Biomass transfer along simple autocat-
alytic pathways (SAPs). Let us consider a reac-
tion network with nsystem nodes and nreactions
(Figure 2A). The reactions are ϕk:xkxk+1 for
k= 1,· · · , n 1 and ϕn:E1x1. The internal fluxes
from xkto xk+1 follow linear functions Jk(X) = akXk,
while the boundary influx ϕnis catalyzed by the gate-
keeper node xn, with J0(X) = bXn. This type of net-
work is termed a simple autocatalytic pathway (SAP).
3
It grows exponentially and represents a primitive au-
tocatalytic structure. I calculate the long-term growth
rate λusing two different perspectives:
Eulerian view By inspecting the influx and efflux
of each node, we have a linear ODE: dX
dt =MX with
M=
a10· · · · · · b
a1a20· · · 0
0a2.......
.
.
.
.
.......an10
0· · · 0an10
.(5)
The solution is X(t) = eMtX(0). Let σmax represent
the principal eigenvalue of M(the eigenvalue with the
largest real part). In this case, it can be shown (by the
Perron-Frobenius Theorem) that σmax R. For large
t,N(t)eσmaxtand λ= limt→∞ 1
tlog N(t) = σmax
(see Supplementary Information (SI) for details).
Lagrangian view: In this case, I assume that the
biomass is discretized at microscopic level and can
be “tracked” in the system, enabling the underlying
stochastic process for biomass transfer to be evaluated.
The SAP behaves as an autocatalytic circuit, with the
last node xnpromoting entry of environmental mate-
rial E1into the system and this material later becoming
biomass on xn. This autocatalytic process involves two
important quantities: the amplification rate κ(that
measures the efficiency by which xnbrings in external
resources)
κ:= total mass flux on ϕn
total mass on xn
=bXn
Xn
=b. (6)
Figure 2. Simple autocatalytic pathways (SAPs). (A) Illustration of a SAP. The rate constants (akand b) for each reaction
flux are labeled under the arrows. The encircled xnfor boundary reactions represents the gatekeeper node of this influx.
(B) Left: SAPs with n=1, 2 or 3. Middle: the decay kernels and decay spectra of the three SAPs at left. Right: The decay
spectra An(s) of the three SAPs at far left. Note that the intersections between y=An(s) and y=scorrespond to the value
of λn. (C) An SAP of length ncan be bounded by two SAPs of length two; one as an upper bound (blue) and one as a lower
bound (green). The ordered relation is used to represent that one SAP has a higher growth rate than the other one, i.e.,
A1A2if λ2λ1.
4
In addition, we define the arrival function h(τ), which measures the waiting time for biomass to travel from
x1to xn:
h(tt) := mass arrived x1at time t,transferred along SAP, and arrived xnat time t
mass arrived x1at time t’ .(7)
Intuitively, the higher the κ, the more effective
node xnis at taking in biomass, and so the faster the
system grows. The shorter the waiting time (τ:= tt,
a random variable), the faster biomass can be incorpo-
rated into node xn, so the system grows more rapidly.
To calculate h(τ), each step of the internal reac-
tions ϕk, k 1 is considered. During a small time
interval ∆t, the total biomass transferred across phik
is Jkt=akXkt, which corresponds to a Poisson
process with a “reaction event” of rate ak. Given
an xk-type molecule, the waiting time τfor the “re-
action event” to happen follows an exponential dis-
tribution Tkakeakτ. Therefore, the total wait-
ing time for biomass to arrive x1and xnis equal to
T=T1+· · · +Tn1. Since h(τ) denotes the probabil-
ity density function of T, it also acts as a delay kernel
between the first and last flux functions:
Jn1(t) = Zt
τ=0
h(τ)J0(tτ). (8)
Here, Jk(X(t)) is simplifed as Jk(t). Note that
Jn1(t) = dXn
dt and and J0(t) = bXn(t) = κXn(t), so
Eq (8) can be written as a delayed differential equation
(DDE) of Xn:
dXn
dt =Zt
0
κh(τ)Xn(tτ). (9)
This DDE is uniquely determined by κand h(τ), with
the error term is neglictable for large Xn. Importantly,
by considering the Laplace transform L[.] and defining
A(s) := L[κh(τ)], the long-term growth rate satisfies
the algebraic relation (see SI)
λ=A(λ).(10)
We called A(s)catalytic spectrum of the network. For
SAPs, the explicit formula is
A(s) = ba1
s+a1
× · · · × an1
s+an1
.(11)
Note that A(s) on a positive real axis decreases mono-
tonically. Hence, Eq. (10) can be visualized geomet-
rically, with λrepresenting the intersection between
y=sand y=A(s). Qualitatively, this facilitates
analysis of how λvaries according to system parame-
ters (see below for examples).
Examples 3.1. Consider SAPs of n= 1,2,3 and
use both Eulerian and Lagrangian perspectives to an-
alyze λ(Figure 2B).
For n= 1, the ODE is dX1
dt =bX1and λ=b.
Using the DDE approach, since biomass is transferred
from E1to x1directly, there is no time delay and the
arrival function is a delta function, i.e., h(τ) = δ(τ).
The autocatalytic spectrum A(s) = L[δ(τ)] = bso, by
Eq.(10), λ=b, i.e., the same result as for the ODE
method.
For n= 2, the ODE is dX
dt =MX with M=
a b
c d,and λis the largest zero of the characteristic
polynomial pM(t) = t(t+a1)a1b. Using the DDE
approach, the arrival function is h(τ) = a1ea1τ, hence
A(s) = L[bh(τ)] = ba1
s+a1. By Eq.(10), we have the al-
gebraic relation λ=ba1
λ+a1, generating the same result
as for the ODE method.
For n= 3, the ODE is dX
dt =MX with M=
a10b
a1a20
0a20
,and λis the largest zero of the charac-
teristic polynomial pM(t) = (t)(t+a1)(t+a2)+a1a2b.
Using the DDE approach, h(τ) = a1e(a1τ))a2ea2τ)
and A(s) = L[bh(τ)] = ba1
s+a1
a2
s+a2. Eq.(10) produces
the algebraic relation λ=ba1a2
(s+a1)(s+a2), resulting in the
same outcome as for the ODE method.
From these examples, it is apparent that increasing
values of n(i.e., the number of nodes) causes A(s)
to have more factors in the form ak
s+akin Eq.(11),
and hence it decays faster. This scenario implies that
y=A(s) and y=sintersect at smaller s(Figure 2B),
which conforms to geometric intuition.
Example 3.2. The exact formula for a SAP with
n= 2 is λ=a1
2q1 + 4b
a11, whereas for SAPs with
n > 2 the formulae become cumbersome. An interest-
ing question to consider is whether multiple reactions
can be “coarse-grained” into one reaction to obtain an
“equivalent reaction pathway”, in a fashion similar to
the law of resistors in an electric series circuit (Figure
2C). This possibility can be assessed by comparing the
catalytic spectra between multiple-reaction SAPs and a
single-reaction SAP. Mathematically, no exact equiva-
5
摘要:

Biomasstransferonautocatalyticreactionnetworks:adelay-differentialequationformulationWei-HsiangLin1,21InstituteofMolecularBiology,AcademiaSinica,Taiwan2GenomeandSystemsBiologyDegreeProgram,NationalTaiwanUniversity,TaiwanMar7,2025Abstract.Forabiologicalsystemtogrowandex-pand,massmustbetransferredfrom...

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