Linear Response Theory of Evolved Metabolic Systems Jumpei F. Yamagishi1and Tetsuhiro S. Hatakeyama1 1Department of Basic Science The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8902 Japan

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Linear Response Theory of Evolved Metabolic Systems

Jumpei F. Yamagishi1and Tetsuhiro S. Hatakeyama1

1Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan

Predicting cellular metabolic states is a central problem in biophysics. Conventional approaches, however,

sensitively depend on the microscopic details of individual metabolic systems. In this Letter, we derived a uni-

versal linear relationship between the metabolic responses against nutrient conditions and metabolic inhibition,

with the aid of a microeconomic theory. The relationship holds in arbitrary metabolic systems as long as the law

of mass conservation stands, as supported by extensive numerical calculations. It offers quantitative predictions

without prior knowledge of systems.

Metabolism is the physicochemical basis of life. Under-

standing its behavior has been a major goal of biophysics [1–

4]. At the same time, the prediction of cellular metabolic

states is a central problem in biology. In particular, predic-

tion of the responses of metabolic systems against environ-

mental variations or experimental operations is essential for

manipulating metabolic systems to the desired states in both

life sciences and in applications such as matter production in

metabolic engineering [5] and the development of drugs tar-

geting cellular metabolism [6–8].

Previous studies have mainly attempted to predict the

metabolic responses by predicting the metabolic states before

and after perturbations, and they require building an ad hoc

model for each speciﬁc metabolic system. In systems biol-

ogy, constraint-based modeling (CBM) has often been used to

predict the cellular metabolic states [9–11]. In this method,

the intracellular metabolic state is predicted by solving an op-

timization problem of models of metabolic systems, includ-

ing a detailed description of each metabolic reaction. To con-

struct the optimization problem, metabolic systems of cells

are assumed to be optimized through (sometimes artiﬁcial)

evolution for some objectives [9, 10, 12], e.g., maximiza-

tion of the growth rate in reproducing cells such as cancer

cells and microbes [13] and maximization of the production

of some molecules in metabolically engineered cells [14]. In-

deed, metabolic systems of reproducing cells exhibit certain

ubiquitous phenomena across various species, and those phe-

nomena can be explained as a result of optimization under

physicochemical constraints [13]. Although the assumption of

optimal metabolic regulation seems acceptable, knowing the

true objective function of cells, which is essential to making

a model for CBM, remains nearly impossible. Besides, even

with remarkable progress in omics research, fully reconstruct-

ing metabolic network models for each individual species or

cell of interest is still a challenge. Moreover, the numerical

predictions are sensitive to the details of the concerned con-

straints and the objective functions selected [15–18]. There-

fore, new methods independent of the details of metabolic sys-

tems are required.

Instead of metabolic states themselves, here, we focus on

the responses of metabolic systems to perturbations. At ﬁrst

glance, such prediction is seemingly more difﬁcult than pre-

dicting the cellular metabolic states because it seems to re-

quire information not only on the steady states but also on

their neighborhoods. However, from another perspective, to

predict only the metabolic responses, we may need to under-

stand the structure of only a limited part of the state space

of feasible metabolic states. In contrast, we must seek the

whole space to predict the metabolic states themselves. If op-

timization through evolution and some physicochemical fea-

tures unique to metabolic systems constrain the behavior in

the state space, there might be universal features in the re-

sponses of metabolic systems to perturbations, independent

of system details, as in the linear response theory in statistical

mechanics [19–21].

In this Letter, we demonstrate a universal property of intra-

cellular metabolic responses in the optimized metabolic reg-

ulation, using a microeconomic theory [22–24]. By intro-

ducing a microeconomics-inspired formulation of metabolic

systems, we can take advantage of tools and ideas from mi-

croeconomics such as the Slutsky equation that describes

how consumer demands change in response to income and

price. We thereby derive quantitative relations between

the metabolic responses against nutrient abundance and

those against metabolic inhibitions, such as the addition of

metabolic inhibitors and leakage of intermediate metabolites;

the former is easy to measure in experiments while the lat-

ter may not be. The relations universally hold independent of

the details of metabolic systems as long as the law of mass

conservation holds. Our theory is applicable to any metabolic

system and will provide quantitative predictions on the intra-

cellular metabolic responses without detailed prior knowledge

of microscopic molecular mechanisms and cellular objective

functions.

Microeconomic formulation of metabolic regulation.— We

ﬁrst provide a microeconomic formulation of optimized

metabolic regulation, which is equivalent to linear program-

ming problems in CBM (Fig. 1).

We denote the set of all chemical species (metabolites)

and that of all constraints by Mand C, respectively. C

can reﬂect every type of constraints such as the allocation

of proteins [25, 26], intracellular space [27], membrane sur-

faces [28], and Gibbs energy dissipation [18] as well as the

bounds of reaction ﬂuxes.

In the microeconomic formulation, variables to be opti-

mized are the ﬂuxes of metabolic pathways, whereas they are

ﬂuxes of reactions in usual CBM approaches; a metabolic

pathway is a linked series of reactions and thus comprises

multiple reactions. The sets of reactions and pathways are

denoted by Rand P, respectively. Let us then consider

two stoichiometry matrices for reactions and pathways, Sand

K, respectively (see also SM, Table I). For chemical species

arXiv:2210.14508v2 [q-bio.MN] 12 Jul 2023

α(∈ M),|Sαi|represents the number of units of species α

produced if Sαi >0and consumed if Sαi <0in reaction i;

whereas if αdenotes a constraint (α∈ C), Sαi is usually neg-

ative and |Sαi|represents the number of units of constraint

αrequired for reaction i. The stoichiometry matrix Kfor

metabolic pathways Pis also deﬁned similarly. Throughout

the Letter, we use indices with primes such as i′to denote

pathways and those without primes such as ito denote reac-

tions, and |Sαi|and |Kαi′|are called input (output) stoichio-

metric coefﬁcients of reaction iand pathway i′, respectively,

if Sαi and Kαi′are negative (positive).

Cells are assumed to maximize the ﬂux of some objec-

tive reaction o(∈ R)such as biomass synthesis in reproduc-

ing cells and ethanol or ATP synthesis in metabolically engi-

neered cells.

We deﬁne the set of the species consumed in and the

components required for reaction oas objective components

O(⊂ M ∪ C), and thus Sαo for each objective component

α(∈ O)is negative. Because the reactants of a reaction can-

not be compensated for each other due to the law of mass

conservation [24, 29, 30], the ﬂux of objective reaction o, i.e.,

the objective function, is limited by the minimum available

amount of objective components Oas follows:

Λ(f) := min

α∈O 

1

−Sαo 

X

j′∈P

Kαj′fj′+Iα



,(1)

where f={fi′}i′∈P represents the ﬂuxes of metabolic path-

ways. The arguments of the above min function represent bio-

logically different quantities: if αis a species (α∈ M), Iαis

its intake ﬂux and Pj′Kαj′fj′represents its total production

rate, while if αis a constraint (α∈ C), Iαis the total capacity

for constraint αand Pj′Kαj′fj′+Iαis the amount of αthat

can be allocated to the objective reaction.

The optimized solution ˆ

fis determined as a function of K

and Iwith the following constraints for the available pathway

ﬂuxes f:

−X

j′∈P

Kαj′fj′≤Iα.(α∈ E ∪ C)(2)

Here, E(⊂ M)denotes the set of exchangeable species that

are transported through the cellular membrane. That is, the

above constraints reﬂect that the total consumption of species

cannot exceed their intakes. If species αis produced by ob-

jective reaction o, the intake effectively increases and SαoΛis

added to the right-hand side of Eq. (2), although this is not the

case for most species.

This optimization problem (1-2) can be interpreted as a mi-

croeconomic problem in the theory of consumer choice [22–

24], considering Λ(f)as the utility function. By focusing on

an arbitrary component ν, one of inequalities (2) serves as the

budget constraint for νif Kνj′≤0for all pathways j′, while

the remaining inequalities in Eq. (2) then determine the solu-

tion space [Fig. 1(a)]: for example, if we choose glucose as

ν, the corresponding inequality in Eq. (2) represents carbon

allocation. Here, the maximal intake Iνof νcorresponds to

the income, and the input stoichiometric coefﬁcient for each

Equivalent

fi′

fj′

v̂

∂̂

fi′

∂pν

j′

=−̂

fj′

∂̂

fi′

∂Iν

Response to

Nutrient Condition

fi′

fj′

fi′

fj′

(a)

(b) Response to

Metabolic Inhibition

FIG. 1. Schematic illustration. (a) (left) Metabolic CBM formu-

lation with reaction ﬂuxes vas variables. The solution subspace

(convex set of possible allocations), called the ﬂux cone, is shown

in pink. (right) Microeconomic formulation with pathway ﬂuxes f

as variables and an objective ﬂux Λ. The pink area in f-plane (bot-

tom surface) represents the solution subspace, whereas the blue plane

vertical to f-plane is the budget constraint for a component ν. The

blue points ˆ

vand ˆ

frepresent the optimized ﬂuxes of reactions and

pathways, respectively. Given v=Pfwith pathway matrix P, both

formulations are equivalent optimization problems (see Supplemen-

tal Material (SM), Sec. S1 for details and Sec. S2 and Fig. S1 for

a simple example). (b) Liner relation between the metabolic re-

sponses against changes in nutrient conditions (yellow) and those

against metabolic inhibitions (green) [Eq. (3)).

pathway, pν

j′: = −Kνj′, serves as the price of pathway j′in

terms of ν.

Relation between responses of pathway ﬂuxes to nutrient

abundance and metabolic inhibition.— Because Eqs. (1-2)

can be interpreted as a microeconomic optimization problem,

we can apply and generalize the Slutsky equation in the theory

of consumer choice [22]. The equation shows the relationship

between changes in the optimized demands for goods in re-

sponse to income and price. In metabolism, it corresponds

to the relationship between the responses of optimal pathway

ﬂuxes ˆ

f(see SM, Sec. S4 for derivation):

∂ˆ

fi′(K, I)

∂pν

j′

=−ˆ

fj′(K, I)∂ˆ

fi′(K, I)

∂Iν

.(3)

The right-hand side represents the responses of pathway i′

against increases in Iν, whereas the left-hand side represents

those against metabolic inhibitions in pathway j′because the

metabolic price pν

j′=−Kνj′quantiﬁes the inefﬁciency of

conversion from substrate νto endproducts in pathway j′[24].

The derivation of Eq. (3) relies solely on the law of mass

conservation, i.e., the reactants of a reaction cannot be com-

pensated for each other. Because the law of mass conservation

stands in every chemical reaction, the relation (3) of the two

Pathway response to inhibition

Pathway response to

IGlc

(a) (b) Glc

−5

FIG. 2. Responses of the optimized pathway ﬂuxes ˆ

f. (a) Re-

sponses to metabolic inhibitions, ∆ˆ

fi′(∆KGlc,j′)/∆pGlc

j′, are plot-

ted against the nutrient responses, ˆ

fj′∆ˆ

fi′(∆IGlc)/∆IGlc. All dif-

ferent shapes and colors of markers represent different i′and j′, re-

spectively. IGlc = 5 [mmol/gDW/h]. (b) 13 active extreme path-

ways, computed using efmtool [31], are shown. Colors correspond

to those of the markers for manipulated pathways j′in panel (a). The

whole metabolic network of the E. coli core model is shown in gray.

measurable quantities must hold in arbitrary metabolic sys-

tems as long as their metabolic regulation is optimized for a

certain objective. In particular, the case i′=j′will be useful:

it indicates that measuring the responses of a pathway ﬂux to

changes in the nutrient environment provides quantitative pre-

dictions of the pathway’s responses to metabolic inhibition or

activation, and vice versa.

To conﬁrm the validity of Eq. (3), we numerically solved

the optimization problems (1-2) with pathway ﬂuxes fas vari-

ables using the E.coli core model [9, 32] and randomly chosen

stoichiometric coefﬁcients for the single constraint (Fig. 2).

In this numerical calculation, metabolic pathways from ex-

changeable species to objective components are chosen as

linear combinations of extreme pathways or elementary ﬂux

modes [33] for stoichiometry without objective reaction o

[Fig. 2(b)], although the above arguments do not depend on

the speciﬁc choices of metabolic pathways (see SM, Sec. S3

for details). As shown in Fig. 2(a), the linear relation (3) be-

tween metabolic responses is indeed satisﬁed. Notably, it is

satisﬁed regardless of the number and type of constraint(s) C,

whereas the metabolic states themselves can sensitively de-

pend on the concerned constraints and environmental condi-

tions.

Relation between responses of reaction ﬂuxes.— Although

Eq. (3) generally holds for arbitrary metabolic pathways, it

may be experimentally easier to manipulate a single metabolic

reaction. Manipulation of a single reaction can affect multi-

ple pathways because they are often tangled via a common

reaction in the metabolic network. Thus, we should consider

the contributions of multiple pathways. The simplest way for

this is to sum up Eq. (3) for all the pathways that include the

perturbed reaction i. However, to precisely conduct this sum-

mation, we need to know the whole stoichiometry matrix or

metabolic network. Hence, another relation closed only for

the reaction ﬂuxes vis required for application without the

need to know the details of the metabolic systems.

To derive such a relation, we consider effective changes in

the stoichiometric coefﬁcients Sαi for reaction ias metabolic

inhibitions: e.g., inhibition of enzymes, administration of

metabolite analogs, leakage of metabolites, and inefﬁciency

in the allocation of some resource. We then obtain an equality

on the optimized reaction ﬂuxes ˆ

v, formally similar to Eq. (3)

(see SM, Sec. S4 for derivation):

∂ˆvi(S, I)

∂qν

=−ˆvi(S, I)∂ˆvi(S, I)

∂Iν

(4)

by deﬁning the metabolic price qν

iof reaction iin terms of νas

a function of S, instead of the metabolic price pν

i′of pathway

i′as a function of K,

qν

i:= X

α∈M∪C

−Sαi

∂ˆvi

∂Iα∂ˆvi

∂Iν

.(5)

The coefﬁcient ∂ˆvi

∂Iα.∂ˆvi

∂Iν=: cν

α(i)quantiﬁes the number of

units of component νthat can compensate for one unit of αin

reaction iand is experimentally measurable. For example, if ν

is glucose and αis another metabolite such as an amino acid,

cν

αindicates how many units of glucose are required to com-

pensate for one unit of the amino acid, similar to the “glucose

cost” in previous studies [34].

For the linear response relation (4), it is sufﬁcient to cal-

culate only the change in metabolic price (not the metabolic

price itself), which depends on the type of manipulations in

concern: (I) manipulations leading to the loss of a single com-

ponent and (II) those leading to the loss of multiple compo-

nents.

If experimental manipulation causes the loss of a single

component α(∈ M∪C)in reaction i,Sαi effectively changes

only for that α[Fig. 3(a)]. In such a case, the metabolic price

change is just given by ∆qα

i= ∆Sαi. An example of such

experimental manipulations is the administration of an analog

to a reactant of a multibody reaction: if αand βreact [see

Fig. 3(a)], the metabolic analog of βcan produce incorrect

metabolite(s) with α, leading to the loss of α, and thus, reac-

tion irequires more αto produce the same number of prod-

ucts, causing effective increases in the input stoichiometric

coefﬁcient |Sαi|. Another example is the changes in the total

capacity and effective stoichiometry for a constraint: for ex-

ample, the mitochondrial volume capacity will work as such a

constraint and can be genetically manipulated [35–37]. Equa-

tion (4) for case (I) is numerically conﬁrmed in Fig. 3(a).

If metabolic inhibition of multiple reactant species of a re-

action iis simultaneously caused, the stoichiometric coefﬁ-

cients Sµi for multiple reactants µof reaction iwill effectively

change [Fig. 3(b)]. Accordingly, the reaction price changes

by ∆qν

i=Pµ∆Sµicν

µ(i). In experiments, such cases would

correspond to the inhibition of enzymes, leakage of the inter-

mediate complex of reaction i, and so forth. Even in this case

(II), the linear relation (4) is veriﬁed by numerically calculat-

ing the price changes of reaction jdeﬁned in Eq. (5) with the

E.coli core model including 77 reactions [Fig. 3(b)] as well as

a larger-scale metabolic model including 931 reactions [38]

(SM, Fig. S2). Here, although the precise calculation of the

coefﬁcients cν

µ(i)requires information regarding not only the

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LinearResponseTheoryofEvolvedMetabolicSystemsJumpeiF.Yamagishi1andTetsuhiroS.Hatakeyama11DepartmentofBasicScience,TheUniversityofTokyo,3-8-1Komaba,Meguro-ku,Tokyo153-8902,JapanPredictingcellularmetabolicstatesisacentralprobleminbiophysics.Conventionalapproaches,however,sensitivelydependonthemicrosco...

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Linear Response Theory of Evolved Metabolic Systems Jumpei F. Yamagishi1and Tetsuhiro S. Hatakeyama1 1Department of Basic Science The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8902 Japan

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