Information-Geometric Inequalities of Chemical Thermodynamics

Kohei Yoshimura1 and Sosuke Ito1, 2

1Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0031, Japan

2JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan(Dated: August 4, 2020)

We study a connection between chemical thermodynamics and information geometry. We clarifya relation between the Gibbs free energy of an ideal dilute solution and an information-geometricquantity called an f -divergence. From this relation, we derive information-geometric inequalitiesthat give a speed limit for a changing rate of the Gibbs free energy and a general bound of chemicalfluctuations. These information-geometric inequalities can be regarded as generalizations of theCramer–Rao inequality for chemical reaction networks described by rate equations, where unnor-malized concentration distributions are of importance rather than probability distributions. Theyhold true for damped oscillatory reaction networks and systems where the total concentration is notconserved so that the distribution cannot be normalized. We also formulate a trade-off relation be-tween speed and time on a manifold of concentration distribution by using the geometrical structureinduced by the f -divergence. Our results apply to both closed and open chemical reaction networks,thus they are widely useful for thermodynamic analysis of chemical systems from the viewpoint ofinformation geometry.

I. INTRODUCTION

The history of chemical thermodynamics originatesaround the middle of the 19th century [1, 2]. The chemi-cal reaction in an ideal dilute solution is one of the mainsubjects in chemical thermodynamics. For example, thestatic nature of an ideal dilute solution under near-equilibrium condition has been well studied since then.After the invention of mathematics called chemical reac-tion network theory (CRNT) around the 1970s [3, 4], itsdynamic properties have also been well investigated [5].One of the most important results of CRNT is that a classof chemical reaction networks called a complex balancednetwork has a Lyapunov function, which can be associ-ated with thermodynamic quantities such as the Gibbsfree energy [3, 6, 7]. This Lyapunov function is calledthe pseudo-Helmholtz function, and its time derivative isconnected to the entropy production rate [8, 9].

On the other hand, thermodynamics for stochastic pro-cesses have been well studied as stochastic thermody-namics [10–13]. In stochastic thermodynamics, phys-ical quantities are given by probabilities, and we candiscuss relations between thermodynamics and informa-tion theory [14–26] because probability plays a crucialrole in information theory [27]. For example, in re-cent years, stochastic thermodynamics met a branchof information theory called information geometry [28–34], and its importance has been verified in recentstudies of thermodynamic uncertainty relations [35–43].An information-geometric quantity called the Fisher in-formation gives several geometric bounds such as theCramer–Rao bound [27–29] and these bounds indicatethermodynamic uncertainty relations in stochastic ther-modynamics [33, 44–48].

Although analogy between stochastic thermodynam-ics and chemical thermodynamics has been studied [8,9, 11, 49–57], connections between chemical thermody-

namics and information theory are still vague becauserate equations which govern chemical reactions are basedon unnormalized concentration distributions rather thanprobability distributions. Nevertheless, a few researcheshave been conducted from the perspective of a connec-tion between chemical thermodynamics and informationtheory [9, 58], focusing on the fact that the pseudo-Helmholtz function has a similar form to the Kullback–Leibler divergence, which plays a fundamental role in in-formation theory.

In this paper, we clarify a connection between chemicalthermodynamics and information theory from the view-point of information geometry. In information geometry,f -divergence is well studied as a measure of the differ-ence between two positive measures. The two positivemeasures do not have to be necessarily normalized likeprobability distributions and the Kullback–Leibler diver-gence may not be well defined for them. We show that thepseudo-Helmholtz function is not given by the Kullback–Leibler divergence, but by an f -divergence. Because thepseudo-Helmholtz function is a representative quantity ofa chemical reaction system, this connection reveals howan information-geometric concept plays a fundamentalrole in chemical thermodynamics. Introducing a gener-alization of the Fisher information from an f -divergenceand an average-like quantity which is more suitable tochemical reaction networks than the ordinary average,we obtain a generalization of the Cramer–Rao inequalityto CRNT. This generalized Cramer–Rao inequality indi-cates the speed limit for the changing rate of the Gibbsfree energy in terms of the fluctuation of the chemicalpotential. We numerically confirm the inequalities fora damped oscillatory reaction network, specifically theBrusselator model [59], and a system where the sum ofconcentrations is not conserved so that the distributioncannot be normalized. We also examine the geometry ofconcentration distributions. We formulate trade-off re-

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lations between time and speed in terms of informationgeometry. The trade-off relations are numerically illus-trated by solving an association reaction.

This paper is organized as follows. In Sec. II, weformulate chemical reaction networks and introduce thepseudo-Helmholtz function. Sec. III extends the formu-lation to open CRNs. Sec. IV is an introduction to in-formation geometry of both probability spaces and pos-itive measure spaces. We discuss mathematical proper-ties of an f -divergence and the connection between theFisher information and an f -divergence. Sec. V givesinformation-geometric inequalities, which are the mainresults of this paper. We show that the speed limitsfor the changing rate of the Gibbs free energy and moregeneral observables are given by the intrinsic speed, theFisher information, for both closed and open cases. Also,we indicate the speed limit is regarded as a generalizationof the CramÃľr–Rao inequality. In addition, we examinethe geometrical structure of concentrations, and revealtrade-off relations between speed and time. In Sec. VI,we confirm our main results through three characteristicmodels of chemical reaction networks. A conclusion anda further perspective of researches in Sec. VII.

II. THERMODYNAMICS OF CLOSEDCHEMICAL REACTION NETWORKS

A. Kinetics of chemical reaction networks

In this paper, we consider the thermodynamics of adilute solution with the temperature and pressure keptconstant. Since the solvent is dominant, the volume isregarded as a constant. In this chapter, we focus onclosed systems.

We consider a chemical reaction network (CRN) con-sisting of N species of molecules {Xi}i=1,2,...,N in a closedvessel. A CRN is defined as a set of M reactions

N∑i=1

νiρXi

k+ρ

k−ρ

N∑i=1

κiρXi, (1)

where reactions are labelled with ρ = 1, 2, . . . ,M , stoi-chiometric coefficients νiρ, κiρ are nonnegative integers,and k±ρ are rate constants. The reactions are assumed tobe reversible. In the closed CRN, the time evolution ofXi’s concentration [Xi] is described by the following rateequation

d[Xi]dt

=M∑ρ=1

(κiρ − νiρ)Jρ, (2)

where Jρ is the reaction rate of the ρ-th reaction. Ac-cording to Waage–Guldberg’s law of mass-action [60], the

reaction rate Jρ is given by

Jρ = J+ρ − J−ρ , (3)

with J+ρ = k+

ρ

N∏i=1

[Xi]νiρ , J−ρ = k−ρ

N∏i=1

[Xi]κiρ , (4)

where J±ρ is the forward/reverse reaction rate. By thecoefficients in Eq. (2), we define the stoichiometric ma-trix of the CRN as a N×M matrix with its (i, ρ)-elementSiρ := κiρ − νiρ, which corresponds to the change of [Xi]when one unit of ρ-th reaction proceeds. The rate equa-tion can be written briefly in vector notation as

d[X]dt

= SJ . (5)

The rate equation Eq. (2) has a priori conserved quan-tities. If ` ∈ RN satisfies

`TS = 0T, (6)

i.e., ` ∈ ker ST = {v | STv = 0}, the time derivative of` · [X] is zero

d

dt(` · [X]) = `TSJ = 0, (7)

where the superscript T means transposition. Thus ` · [X]is conserved. We call a left null vector of a stoichiomet-ric matrix a conservation law. Note that a closed CRNhas at least one conservation law, corresponding to theconservation of the total mass.

B. Equilibrium conditions

In thermodynamics, it is postulated that a closed sys-tem relaxes to equilibrium, at which a certain free energyis minimized depending on the condition. In the presentcase, the function to be minimized is the Gibbs free en-ergy, thus the equilibrium distribution [X]eq is defined asa distribution that minimizes the Gibbs free energy.

Since the solution is supposed to be dilute, the Gibbsfree energy per unit volumeG and the chemical potentialsµi are expressed as

G =N∑i=1

µi[Xi]−RTN∑i=1

[Xi] +G0, (8)

µi = ∂G

∂[Xi]= µ◦i (T ) +RT ln[Xi], (9)

where G0 is a constant, R is the gas constant, and µ◦i arethe standard chemical potentials, which are independentof the concentration [2, 8, 9]. It is plausible to call G theGibbs free energy because the volume does not change.

Possible value of the concentration is restricted be-cause the concentration changes obeying the rate equa-tion. That can be seen by integrating the rate equation.

3

Let sρ be the ρ-th column vector of the stoichiometricmatrix, S = (s1, . . . , sM ). Then [X] at t is obtained as

[X] = [X]0 + S∫ t

0dt J (10)

= [X]0 +M∑ρ=1

(∫ t

0dt Jρ

)sρ, (11)

where [X]0 is the initial concentration. Thus a change ofconcentration [X]− [X]0 must be a linear combination of{sρ}ρ=1,2,...,M . So a set of concentrations that [X] mayreach is given by

S([X]0) := {[X]0 + Sξ | ξ ∈ RM} ∩ RN≥0, (12)

where RN≥0 is the set of N -dimensional vectors with non-negative elements. This set is called the stoichiometriccompatibility class [5], and ξ the extent of reaction.

A necessary condition for equilibrium can be obtainedas that the derivative of the Gibbs free energy with re-spect to ξρ vanishes for all ρ,

∂G

∂ξρ=

N∑i=1

∂[Xi]∂ξρ

∂G

∂[Xi]=

N∑i=1

Siρµeqi = ((µeq)TS)ρ = 0.

(13)

So the equilibrium distribution [X]eq satisfies

N∑i=1

(µ◦i +RT ln[Xi]eq)Siρ = 0. (14)

Note that it means that the equilibrium chemical poten-tial µeq is a conservation law.

On the other hand, an equilibrium state is often char-acterized by the detailed balance, which is based on ki-netics,

J+ρ = J−ρ . (15)

The consistency between the thermodynamic conditionof equilibrium (13) and this detailed balance conditionleads to a relation

k+ρ

k−ρ= exp

(− (µ◦TS)ρ

RT

), (16)

which is called the local detailed balance property. Itcan be regarded as a bridge between thermodynamicsand kinetics.

C. Gibbs free energy with divergence

From Eq. (13), µeq is a conservation law, thereforeµeq · [X] is time invariant

d

dt(µeq · [X]) = (µeq)TSJ = 0. (17)

Stoichiometric compatibility class

f-divergence

Rate equation

FIG. 1. Schematic of a stoichiometric compatibility class andthe f -divergence for N = 3. This f -divergence gives the dif-ference of the Gibbs free energy (G−Geq)/RT .

So the Gibbs free energy at equilibrium Geq can be ex-pressed as

Geq = µeq · [X]eq −RTN∑i=1

[Xi]eq +G0 (18)

= µeq · [X]−RTN∑i=1

[Xi]eq +G0 (19)

for a concentration [X](6= [X]eq) of an arbitrary time,namely, in the same stoichiometric compatibility class as[X]eq. As a result, we obtain the expression

G−Geq

= (µ− µeq) · [X]−RTN∑i=1

[Xi] +RT

N∑i=1

[Xi]eq (20)

= RT

N∑i=1

([Xi] ln [Xi]

[Xi]eq − [Xi] + [Xi]eq), (21)

by which a function is usually defined as

D([X]‖[X]eq) :=N∑i=1

([Xi] ln [Xi]

[Xi]eq − [Xi] + [Xi]eq).

(22)

This suggestive form has been known for more than a halfcentury, and D([X]‖[X]eq) called the pseudo-Helmholtz

4

function [3], Shear’s Lyapunov function [6, 7] or the rela-tive entropy [9] (see Fig. 1). In terms of information ge-ometry [28], it is regarded as an f -divergence of a positivemeasure space. As we see in Sec. IV, an f -divergence isnonnegative and equal to zero if and only if the two ar-guments coincide. Therefore G is greater than or equalto Geq, and G = Geq only in equilibrium, [X] = [X]eq.

It is also known that the Gibbs free energy of a closedsystem never increases under the mass-action kinetics,i.e., it is a Lyapunov function of a closed CRN. To showthis fact, we calculate the time derivative of the Gibbsfree energy

dG

dt= µ · d[X]

dt= µTSJ . (23)

From the local detailed balance property Eq. (16), µTScan be transformed as

(µTS)ρ = (µ◦TS)ρ +RT lnN∏i=1

[Xi]Siρ

= −RT lnk+ρ

∏Ni=1[Xi]νiρ

k−ρ∏Ni=1[Xi]κiρ

= −RT lnJ+ρ

J−ρ. (24)

By substituting it into Eq. (23), we have

dG

dt= −RT

M∑ρ=1

(J+ρ − J−ρ ) ln

J+ρ

J−ρ=≤ 0. (25)

The last inequality follows from the fact that the signsof J+

ρ − J−ρ and ln(J+ρ /J

−ρ

)are always the same for all

ρ. Since dG/dt is negative unless the detailed balanceis satisfied, and G is always not less than Geq, hence Gdecreases to Geq monotonically. Here, we point out twofacts. First, the left-hand side of the inequality Eq. (25)coincides with the opposite sign of the entropy produc-tion rate. Thus the inequality expresses the second lawof thermodynamics. Second, the left-hand side is alsorepresented by the f -divergence D between J+ and J−

dG

dt= −RT

(D(J+‖J−) +D(J−‖J+)

). (26)

D. Entropy production rate and affinity

We can formulate the second law of thermodynamicsin CRN [9]. Here we review a few important points. Theentropy production rate σ due to chemical reactions isgiven by

σ = R

M∑ρ=1

(J+ρ − J−ρ ) ln

J+ρ

J−ρ. (27)

So it coincides with −(1/T )dG/dt in closed CRNs. Aswe explained in the preceding section, the entropy pro-duction rate is always positive except for the equilibriumstate, and that indicates the second law of thermody-namics.

The affinity Fρ of a reaction is defined as follows [61]

Fρ :=N∑i=1

νiρµi −N∑i=1

κiρµi = −(µTS)ρ. (28)

The affinity corresponds to the energy difference betweenthe reactant and product of the ρ-th reaction. From thelocal detailed balance condition Eq. (16), one can rewriteit as

Fρ = RT lnJ+ρ

J−ρ. (29)

Thus, the entropy production is expressed as the sum ofthe product between the reaction rate and the affinity

Tσ =M∑ρ=1

JρFρ. (30)

This expression shows there is an analogy with stochasticthermodynamics of master equations [62].

III. THEMODYNAMICS OF OPEN CHEMICALREACTION NETWORKS

We have already seen the connection between theGibbs free energy G and the f -divergence D in closedCRNs. On the other hand, in open CRNs, the Gibbsfree energy is not directly related to the f -divergence,even in a CRN called a complex balanced network wherethe f -divergence becomes a Lyapunov function. This isbecause the chemical potential at a steady state is nota conservation law in general. However, we can asso-ciate the Gibbs free energy with the f -divergence. Inthis chapter, we formulate open CRNs and associate theGibbs free energy with the f -divergence in two ways, themethod already known and the one we newly propose,respectively.

A. Setup for open CRNs

Let {Yj}j=1,2,...,N ′ be the chemical species that areexchanged with the environment, and {Xi}i=1,2,...,N bethe other internal species. The former is assumed to bechemostatted, that is, their concentrations are constant.

The CRN consisting of them is expressed asN∑i=1

νiρXi +N ′∑j=1

ν(N+j)ρYj

k+ρ

k−ρ

N∑i=1

κiρXi +N ′∑j=1

κ(N+j)ρYj . (31)

5

The stoichiometric matrix S = (κiρ − νiρ)ρ=1,...,Mi=1,...,N+N ′ can

be decomposed into an N rows of X part and an N ′ rowsof Y part

S =(

SX

SY

), (32)

where SXiρ = κiρ − νiρ (i = 1, 2, . . . , N) and SY

jρ =κ(j+N)ρ − ν(j+N)ρ (j = 1, 2, . . . , N ′). Hereafter, for anN + N ′ row quantity Q, we let QX be the first N rowsand QY the remainder as the above case. Since the con-centrations of the chemostatted species are assumed tobe constant, the dynamics are expressed by the followingrate equation

d[Z]dt

:= d

dt

([X][Y]

)=(

SXJ0

), (33)

where [Z]T =([X]T [Y]T

), Jρ = J+

ρ − J−ρ and

J+ρ = k+

ρ

N∏i=1

[Xi]νiρM∏j=1

[Yj ]ν(j+N)ρ (34)

J−ρ = k−ρ

N∏i=1

[Xi]κiρM∏j=1

[Yj ]κ(j+N)ρ . (35)

A steady state is defined as a state at which the concen-tration does not change in time, SXJ = 0.

In open CRNs, ` · [Z] is not necessarily conserved evenif ` is a conservation law, i.e., `TS = 0T holds. Since wehave

d

dt(` · [Z]) = (`X)TSXJ , (36)

` · [Z] is conserved if `X belongs to ker(SX)T. Thus aconservation law ` leads to a conserved quantity if ` is anelement of the linear space

(ker(SX)T × RN ′

)∩ker ST =:

LX, where×means the direct product between two linearspaces.

B. One way to associate G with D

One way to associate the Gibbs free energy with thef -divergence is to decompose a basis of the space of con-servation laws ker ST. This formalism is based on Raoand Esposito’s paper Ref. [9]. It can be used only whenthe steady state is detailed balanced.

To construct the desired basis, we exploit the lin-ear space LX. Let {`λx}λx=1,2,...,Λx be a basis of LX,then we obtain a basis of ker ST by adding some vectors{`λxy}λxy=1,2,...,Λxy . The former vectors lead to quanti-ties `λx · [Z] that are conserved in an open CRN. We callthem X-conservation laws and the latter XY-conservationlaws. XY-conservation laws `λxy are usually called bro-ken laws because the remainder, X-conservation laws, are

always true conservation laws. However, since an XY-conservation law might be a true conservation law, wedo not use the conventional terminology.

From the local detailed balanced condition Eq. (16),we have the following relation (see Eq. (24))

(µTS)ρ = −RT lnJ+ρ

J−ρ. (37)

So if we suppose the steady state to be detailed balancedJ+ρ = J−ρ , i.e., be an equilibrium, the chemical potential

at equilibrium µeq becomes a conservation law. Then wecan expand it with the prepared basis as

µeq =Λx∑λx=1

fλx`λx +

Λxy∑λxy=1

fλxy`λxy . (38)

A function G1 is defined by

G1 := G−Λxy∑λxy=1

fλxy`λxy · [Z], (39)

which is called the transformed Gibbs free energy inRef. [9]. From Eq. (8) and the expansion of µeq Eq. (38),we see

G1 =N+N ′∑i=1

([Zi]µi −RT [Zi]) +G0

−

(µeq · [Z]−

Λu∑λu=1

fλu`λu · [X]

).

(40)

Since the concentrations of the chemostatted species areconstant, [Yj ] coincide with [Yj ]eq. Therefore we have

G1 =N∑i=1

([Xi](µi − µeqi )−RT [Xi]) + const. (41)

= RT

N∑i=1

([Xi] ln [Xi]

[Xi]eq − [Xi] + [Xi]eq)

+ const.,

(42)

where we use the fact that RT [X]eq is constant. Sincethe last constant term is equal to G1 at the equilibrium,we write it Geq

1 . We finally obtain the following equationas in the closed CRN

G1 = Geq1 +RTD([X]‖[X]eq). (43)

The transformation from G to G1 Eq. (39) looks like theLegendre transformation, hence it might have a physi-cal meaning. However, we should note that G1 has anarbitrariness regarding the way to choose the basis andis only defined for open CRNs that relax to equilibrium,thus it has a limited scope of application.

6

C. Another way to associate G with D

We newly propose another way of association, which issimpler and more widely applicable than the precedingone. Letting [Z]ss be the concentration at a steady stateand µss = (µ◦i +RT ln[Zi]ss)i=1,...,N+N ′ the chemical po-tential, we define

G2 := G− µss · [Z]. (44)

Since [Yj ] coincide with [Yj ]ss as in the detailed balancedcase, we have

G2 =N∑i=1

([Xi](µi − µssi )−RT [Xi]) + const. (45)

=N∑i=1

([Xi](µi − µssi )−RT [Xi] +RT [Xi]ss) + Gss

2

(46)= Gss

2 +RTD([X]‖[X]ss), (47)

where Gss2 is G2 at the steady state.

G2 is defined for general steady states and coincideswith G1 up to a constant if the steady state is detailedbalanced. In fact,

G2 = G− µeq · [Z] (48)

= G1 −Λx∑λx=1

fλx`λx · [Z] (49)

and `λx · [Z] are constants, thus G2 − G1 = const.. ThusG2 is a generalization of G1, then we denote G2 as G andcall it the transformed Gibbs free energy.

The transformed Gibbs free energy we introduce herecan be defined for open CRNs that do not satisfy detailedbalance. Not all CRNs have steady-state solutions, butour definition Eq. (44) enables us to study much broaderclass of open CRNs than the previous one Eq. (39) does.For open CRNs with multiple steady states, one can de-fine the transformed Gibbs free energy by choosing thesteady state in Eq. (44). Remarkably, the following re-sults hold regardless of the choice of a steady state. Wecomment that CRNs that do not have steady-state so-lutions are outside our framework. They include openCRNs that sustain oscillations. G2 also has the meritthat it removes the arbitrariness in the choice of the ba-sis as G1.

D. Entropy production rate and affinity

The entropy production rate σ and the affinity Fρof an open CRN have the same form as of closed one.The entropy production rate of an open CRN can be de-composed into the adiabatic and non-adiabatic parts σa,

σna [8, 9] which are defined as

σ = σa + σna (50)

σa := R

M∑ρ=1

Jρ lnJ+,ssρ

J−,ssρ

(51)

σna := R

M∑ρ=1

Jρ lnJ+ρ J−,ssρ

J−ρ J+,ssρ

, (52)

where J±,ssρ are the reaction rates at the steady state.The time derivative of the transformed Gibbs free en-ergy gives the minus sign of the non-addiabatic entropyproduction rate dG/dt = −Tσna. If the steady state isdetailed balanced, or equivalently, J+,ss

ρ = J−,ssρ holds,dG/dt can be written by the f -divergence between thereaction rates as in the closed case

−dGdt

= RT (D(J+‖J−) +D(J−‖J+)). (53)

IV. INFORMATION GEOMETRY

Information geometry deals with a manifold of proba-bility distributions p = (pi)i=1,2,...,N ∈ RN>0 that satisfythe normalization condition

∑Ni=1 pi = 1, or a manifold of

positive measures on a discrete set m = (mi)i=1,2,...,N ∈RN>0, which does not have to be normalized [28]. Theformer manifold is called a probability simplex, and thelatter a positive measure space. We use the term ”distri-bution” for either a probability distribution or a positivemeasure in this section.

A. f-divergence

A divergence D(·‖·) is a measure of the separation be-tween two distributions m and n that satisfies the fol-lowing conditions [28]:

1. D(m‖n) ≥ 0

2. D(m‖n) = 0 ⇐⇒ m = n

3. D(m‖m+ dm) = 12

N∑i,j=1

gijdmidmj + o(dm2),

and the matrix (gij) 1≤i,j≤N is positive definite.

Note that a divergence is similar to a distance functionbut it is not really because it is not symmetricD(m‖n) 6=D(n‖m).

One of well-known divergences is an f -divergence [63],which has the following form

D(m‖n) =N∑i=1

mif

(nimi

). (54)

7

The function f has to fulfill some conditions. For bothkinds of manifold, f should be a convex differentiablefunction which satisfies f(1) = 0. If one chooses f(x) =− ln x, D becomes the Kullback–Leibler divergence. Forpositive measure spaces, the condition f ′(1) = 0 is im-posed additionally. f is called a standard convex functionwhen f ′′(1) = 1 holds.

The nonnegativity of an f -divergence is easily proved.Let p and q be probability distributions in a probabilitysimplex. From Jensen’s inequality, we have

N∑i=1

pif

(qipi

)≥ f

(N∑i=1

piqipi

)= f(1) = 0. (55)

So an f -divergence on a probability simplex is nonnega-tive and equal to zero if and only if p = q. On the otherhand, for a positive measure manifold, since f is convexand f ′(1) = 0, f takes the minimum value 0 at x = 1.Thus f is nonnegative, and so is an f -divergence sincethe all coefficients of f are positive. An f -divergence iszero if and only if f(ni/mi) = 0 for all i. It is equivalentto that two distributions are the same.

B. Fisher information

Letting θ be the parameter of distributions and D adivergence, the Fisher information I(θ) [28] is defined as

I(θ) :=N∑

i,j=1gijdmi

dθ

dmj

dθ, (56)

gij = ∂

∂mi

∂

∂mjD(n‖m)

∣∣∣∣n=m

. (57)

In information geometry,

ds2 :=N∑

i,j=1gijdmidmj ' 2D(m‖m+ dm) (58)

is interpreted as the square of the line element be-tween two close distributions. If the distributions areparametrized by the time t, the Fisher information be-comes the square of the speed on the manifold ds/dt,

I(t) =N∑

i,j=1gijdmi

dt

dmj

dt= ds2

dt2. (59)

An f -divergence with a standard convex function leadsto gij = m−1

i δij , where δij is Kronecker’s delta, then theFisher information always has the unique form

I(θ) =N∑i=1

1mi

(dmi

dθ

)2. (60)

A significant fact related to the Fisher information isthe Cramer–Rao inequality for a probability distribution

p(θ) [29]

Var(θ) ≥ 1I(θ) , (61)

where θ is an unbiased estimator of θ, that is, 〈θ〉θ :=∑Ni=1 pi(θ)θi = θ holds, and Var(θ) is the variance of θ,

〈(θ − 〈θ〉θ)2〉θ.

V. INFORMATION GEOMETRY IN CHEMICALTHERMODYNAMICS

A. Geometrical structure of chemicalthermodynamics

A set of concentrations can be interpreted as a posi-tive measure space in both closed and open CRNs. Themeasures are concentrations [Z], and the measurable setis the index set of species. This space is thought to havethe a priori f -divergence, with its standard convex func-tion f(x) = − ln x+ x− 1,

D([X]‖[X′]) =N∑i=1

[Xi]f(

[X′i][Xi]

)

=N∑i=1

([Xi] ln [Xi]

[X′i]− [Xi] + [X′i]

), (62)

because G of a closed CRN and G of an open CRN aredescribed by the f -divergence as

G = Geq +RTD([X]‖[X]eq) (63)

and

G = Gss +RTD([X]‖[X]ss). (64)

Let us confirm that f is a standard convex function. It isobviously smooth, and convex because f ′′(x) = 1/x2 > 0.It is readily seen that the values at x = 1 are f(1) =f ′(1) = 0 and f ′′(1) = 1. Hence, f is a standard convexfunction.

As we pointed out in Sec. IV, a divergence gives ageometrical structure to a manifold, namely, a metricgij . An f -divergence always provides the metric of theform gij = δijm

−1i . Therefore, manifolds of concentra-

tion distributions are considered to be equipped with themetric gij = δij [Xi]−1. The metric is given naturallywith respect to only internal species. That is becausethe concentrations of the chemostatted species are keptconstant and do not appear in the divergence Eq. (64).It is possible to extend the metric as gij = δij [Zi]−1

for i, j ∈ {1, 2, . . . , N + N ′}, but we just neglect thechemostatted species in this paper when we consider thegeometrical structure of CRNs. Hence, we only say herethat the concentration distributions of closed and openCRNs have the completely same metric gij = δij [Xi]−1,where i, j ∈ {1, 2, . . . , N}.

8

Before stating our main results, we denote a few di-rect consequences of the geometrical structure. First,the square of the line element between the equilibriumconcentration and a concentration close to it is relatedto the Gibbs free energy. If a concentration is close toequilibrium [X] = [X]eq + δ[X], the Gibbs free energy isexpressed as G = Geq + δG with a small deviation δG.From Eq. (63), we obtain the expression of the square ofthe line element ds2 = 2D([X]‖[X]eq) by the Gibbs freeenergy

ds2 = 2RT

δG. (65)

So the Gibbs free energy difference corresponds to thesquare of the distance from the equilibrium concentra-tion under near-equilibrium condition. It is true for openCRNs, that is, the fluctuation of the transformed Gibbsfree energy from the steady-state value δG corresponds tods2 between the steady state and a concentration nearbyds2 = (2/RT )δG. Note that although the relations holdonly under near-equilibrium or near-steady-state condi-tion, our results in the following subsections apply tofar-from-equilibrium systems except for Eqs. (73)–(75).

Second, we can consider the Fisher information ofchemical reaction networks. Since the metric is δij/[Xi]for closed and open CRNs and d[Yj ]/dt = 0, the Fisherinformation I(t) is defined for both types of CRN as

I(t) =N∑i=1

1[Xi]

(d[Xi]dt

)2. (66)

It is also represented as

I(t) = − 1RT

M∑ρ=1

JρdFρdt

. (67)

We prove this formula that is true in both cases. Because(d/dt) ln[Xi] = (1/RT )(dµi/dt), I(t) is transformed asfollows

I(t) = 1RT

N∑i=1

d[Xi]dt

dµidt

(68)

= 1RT

N∑i=1

M∑ρ=1

dµidt

SiρJρ. (69)

If the CRN is closed, that ends the proof since Fρ =−(µTS)ρ. On the other hand, if open

I(t) = 1RT

N+N ′∑i=1

M∑ρ=1

dµidt

SiρJρ (70)

= − 1RT

M∑ρ=1

dFρdt

Jρ, (71)

where we used the fact that the chemical potentials ofthe chemostatted species do not change in time.

From the expression of the entropy production rateEq. (27), which is valid for closed and open CRNs,Eq. (67) is also written as

I(t) = − 1R

dσ

dt+ 1RT

M∑ρ=1

dJρdt

Fρ. (72)

Therefore the Fisher information is associated with theentropy production rate with the additional term. Fur-thermore, under near-equilibrium conditions, I(t) is di-rectly given by dσ/dt with an error of higher order in δJ .That is,

I(t) = − 12R

dσ

dt+ o(δJ2) (73)

is obtained. This is because the affinity is proportionalto the reaction rate under near-equilibrium condition. Itcan be proved as follows. J±ρ can be decomposed intothe equilibrium values J+,eq

ρ = J−,eqρ =: Jeq

ρ and thefluctuations δJ±ρ as J±ρ = Jeq

ρ + δJ±ρ . Then the affinitiesare obtained as Fρ = RTJρ/J

eqρ + o(Jρ), where Jρ =

δJ+ρ − δJ−ρ . So, we have

M∑ρ=1

dJρdt

Fρ 'M∑ρ=1

RTJρJeqρ

dJρdt

(74)

' 12d

dt

(M∑ρ=1

JρFρ

)= T

2dσ

dt. (75)

By substituting it into Eq. (72), we obtain the equalityEq. (73).

B. Speed limit on the Gibbs free energy

We describe the main result in this section as follows.We state, discuss, and prove our assertion for closedCRNs. Then we obtain the same result in subsystemsand open CRNs. That is because the discussion becomesconcise without loss of generality in closed CRNs. Thisway of description is followed in the next section.

To state our main results, we define the concentrationintegral 〈〈·〉〉 of a quantity q = (q1, q2, . . . , qN )T as

〈〈q〉〉 :=N∑i=1

qi[Xi]. (76)

A concentration integral coincides with an average ifthe weight is normalized. It might be possible to usenot the concentration but the normalized concentration[Xi]/

∑Ni=1[Xi] or the mole fraction as the weight. How-

ever, usually both of them do not obey any tractabledifferential equations like the rate equation. Since wewould like to consider time evolution, we use a concen-tration integral.

9

We further define the chemical variance of a chemicalpotential

⟨⟨∆µ2⟩⟩ as

⟨⟨∆µ2⟩⟩ :=

⟨⟨(µ− µeq)2⟩⟩ =

N∑i=1

(µi − µeqi )2[Xi], (77)

where (µ−µeq)2 means the vector((µi − µeq

i )2)i=1,2,...,N .

It is a variance-like quantity, but differs from a variancein two aspects. One is that the weight is not normalized,namely, it is defined by the concentration integral. Theother is that what is subtracted from the chemical po-tential µ is not an average but the equilibrium chemicalpotential µeq.

One of our main results is the fact that the time deriva-tive of the Gibbs free energy is bounded above by theproduct between the Fisher information and the chem-ical variance of a chemical potential. First, for closedCRNs, it is the inequality∣∣∣∣dGdt

∣∣∣∣ ≤√I(t)√〈〈∆µ2〉〉, (78)

where |·| means the absolute value. This inequality givesan upper bound of the speed at which the Gibbs free en-ergy decreases with the information geometric quantity,the Fisher information I(t). The inequality is rewrittenequivalently as

1I(t)

(dG

dt

)2≤⟨⟨

∆µ2⟩⟩ , (79)

which is similar to the Cramer–Rao inequality Eq. (61).We define a function of a quantity q, a reference value q,and the time t as

vq(t, q) :=√I(t)

√〈〈(q − q)2〉〉, (80)

where (q− q)2 =((qi − qi)2)

i=1,2,...,N , then Eq. (78) canbe rewritten as ∣∣∣∣dGdt

∣∣∣∣ ≤ vµ(t,µeq). (81)

The proof of Eq. (78) is straightforward. FromEq. (63), we have

dG

dt= RT

N∑i=1

d[Xi]dt

ln [Xi][Xi]eq (82)

=N∑i=1

d[Xi]dt

(µi − µeqi ), (83)

then, using the Cauchy–Schwarz inequality, the inequal-ity is obtained as follows∣∣∣∣dGdt

∣∣∣∣ =

∣∣∣∣∣N∑i=1

1√[Xi]

d[Xi]dt

√[Xi](µi − µeq

i )

∣∣∣∣∣ (84)

≤

√√√√ N∑i=1

1[Xi]

(d[Xi]dt

)2√√√√ N∑

i=1[Xi](µi − µeq

i )2. (85)

Here we do not use any approximation or assumptionof near-equilibrium, thus the speed limit holds far fromequilibrium.

The above discussion can be extended to subsystems.If we are concerned with some specific species S ={Xi}i∈AS ⊂ {X1,X2, . . . ,XN}, where AS is the indexset of S, we can define the partial Gibbs free energy GSas

GS := RT∑i∈AS

([Xi] ln [Xi]

[Xi]eq − [Xi] + [Xi]eq). (86)

The inequality Eq. (78) also holds for this partial Gibbsfree energy∣∣∣∣dGSdt

∣∣∣∣ ≤√IS(t)√〈〈∆µ2〉〉S =: vµ,S(t,µeq), (87)

where

IS(t) :=∑i∈AS

1[Xi]

(d[Xi]dt

)2(88)

⟨⟨∆µ2⟩⟩

S:=∑i∈AS

(µi − µeqi )2[Xi]. (89)

The partial Gibbs free energy can show nontrivial be-havior, e.g., oscillation, even in a closed CRN, while theGibbs free energy of a total system decreases monoton-ically. Hence, this bound Eq. (87) also becomes a non-trivial one. The proof is almost the same as of Eq. (78).

Furthermore, the speed limit obtains in even openCRNs for the transformed Gibbs free energy G. The timederivative of G is bounded above as the Gibbs free energyof closed systems was by the Fisher information and thedeviation of chemical potential∣∣∣∣dGdt

∣∣∣∣ ≤√ISX(t)√〈〈(µ− µss)2〉〉SX

, (90)

where SX = {X1, . . . ,XN} and (µ − µss)2 is the vector((µi − µss

i )2)i=1,2,...,N+N ′ . Since the transformed Gibbs

free energy is given by the f -divergence, one can provethe inequality Eq. (90) easily. This inequality obtains inany open CRN which has a steady-state solution.

We comment an experimental importance of the speedlimit. The inequality Eq. (78) can be written as follows

1√〈〈∆µ2〉〉

∣∣∣∣dGdt∣∣∣∣ ≤ ds

dt. (91)

We define the fluctuation-response ratio by the left-handside. Even far from equilibrium, the fluctuation-responseratio can be interpreted as a measure of the speed atwhich the deviation from the equilibrium or steady statevanishes. The inequality Eq. (91) shows that it is sup-pressed by the intrinsic speed on the stoichiometric com-patibility class ds/dt. Since the intrinsic speed ds/dtneeds only the current concentration and the concentra-tion’s time derivative, it is experimentally obtained more

10

Stoichiometric compatibility class

FIG. 2. Geometric picture of q and other quantities appear-ing in Sec.V C for N = 3. qmin, which makes the right handsides of the inequalities Eq. (94) and Eq. (95) smallest, isobtained as a projection of q onto ker ST, which is perpendic-ular to the stoichiometric compatibility class. It is not theorthogonal projection in the Euclidean space. Instead, X1/2qis orthogonally projected to X1/2qmin.

easily than the fluctuation-response ratio, which requiresinformation of equilibrium concentration. Our result as-sures the validity of the intrinsic speed as a measure ofconvergence speed. ds/dt is also defined in CRNs withmultistability and sustained oscillation. Although theyare out of the range of our study, the application to thosesystems is implied.

C. Generalized Cramer–Rao inequality forchemical reaction networks

We consider the result in the previous section further.If we use the original definition of the Gibbs free energyEq. (8), the time derivative of G is given by

dG

dt=

N∑i=1

d[Xi]dt

µi, (92)

while in Eq. (83) there is the additional term−∑Ni=1(d[Xi]/dt)µeq

i . This is because µeq satisfiesEq. (13), thus is orthogonal to d[X]/dt = SJ . Hence,it is crucial for the speed limit Eq. (78) that µeq is aconservation law. This suggests that we can make useof conservation laws to evaluate the time derivative of aconcentration integral.

For example, letting q be a conservation law, we canadd 0 = −q · (d[X]/dt) to the time derivative of 〈〈q〉〉

d

dt〈〈q〉〉 = q · d[X]

dt= (q − q) · d[X]

dt. (93)

Then, we obtain the following inequality in the same wayas the proof of Eq. (78)∣∣∣∣ ddt〈〈q〉〉

∣∣∣∣ ≤√I(t)√〈〈(q − q)2〉〉 = vq(t, q), (94)

or equivalently

1I(t)

(d

dt〈〈q〉〉

)2≤⟨⟨

(q − q)2⟩⟩ . (95)

We may call this inequality the generalized Cramer–Raoinequality for CRNs. Eq. (94) reveals the fact that theFisher information does not only acts as a speed limit onthe Gibbs free energy, but also on general quantities.

We can construct the q(=: qmin) that minimizesvq(t, ·). We introduce a diagonal matrix X

Xij := δij [Xi], (96)

then we can rewrite⟨⟨

(q − q)2⟩⟩ as

N∑i=1

(qi − qi)2[Xi] =∥∥∥X1/2(q − q)

∥∥∥2(97)

=∥∥∥X1/2q − X1/2q

∥∥∥2, (98)

where ‖·‖ is the Euclidean norm. Since the linear spacethat X1/2q belongs to is

X1/2 ker ST := {X1/2` | ` ∈ ker ST}, (99)

we see that Eq. (98) is smallest when X1/2q is the or-thogonal projection of X1/2q onto X1/2 ker ST (see Fig. 2).Therefore, letting {X1/2`λ}λ=1,2,...,Λ be an orthonormalbasis of X1/2 ker ST, qmin is given by

qmin = X−1/2Λ∑λ=1

(X1/2`λ · X1/2q)X1/2`λ (100)

=Λ∑λ=1

((`λ)TXq)`λ. (101)

These results can be restricted to subsystems asEq. (87). For a subset of species S, we have∣∣∣∣ ddt 〈〈q〉〉S

∣∣∣∣ ≤ vq,S(t, q) (102)

:=√IS(t)

√∑i∈AS

(qi − qi)2[Xi]. (103)

The right-hand side is minimized when qi = qmini for all

i ∈ AS .

11

Under near-equilibrium condition, it can be provedthat µeq coincides with µmin with an error of order ofthe deviation ∆[X] := [X]− [X]eq. For any ` ∈ ker ST,(

X1/2(µ− µeq))· X1/2` (104)

= RT

N∑i=1

[Xi] ln [Xi][Xi]eq `i (105)

= RT

N∑i=1

`i([Xi]− [Xi]eq) +O(∆[X]2) (106)

= RT∆(` · [X]) +O(∆[X]2), (107)

and `·[X] does not change in time, therefore X1/2(µ−µeq)is orthogonal to X1/2 ker ST. This means that X1/2µ isorthogonally projected to X1/2µeq ∈ X1/2 ker ST, there-fore we see µeq ' µmin. Then the speed limit Eq. (78) isthe tightest Cramer–Rao bound under near-equilibriumconditions with an error of second order of the concen-tration deviation.

Finally, let us extend the results to open CRNs. For aquantity q ∈ RN+N ′ , a speed limit on the change of 〈〈q〉〉is given by

d

dt〈〈q〉〉 =

N∑i=1

qi

M∑ρ=1

SiρJρ (108)

because d[Yj ]/dt = 0. Thus it is bounded by quantitiesrelated to the internal species∣∣∣∣ ddt〈〈q〉〉

∣∣∣∣ ≤√ISX(t)√〈〈(q − q)2〉〉SX

, (109)

where q is an element of ker(SX)T×RN ′ . The generalizedCramer–Rao bound for an open CRN is also obtained as

1ISX(t)

(d

dt〈〈q〉〉

)2≤⟨⟨

(q − q)2⟩⟩SX. (110)

D. Geometry of stoichiometric compatibility class

In addition to the above results, we indicate an infor-mation geometric aspect of stoichiometric compatibilityclasses. Both closed and open CRNs are considered in thesame notation because the chemostatted species do notaffect the geometrical structure. A concentration distri-bution is confined to a stoichiometric compatibility class.We can consider this confinement is a consequence of con-straints `λ · [X] = Lλ(const.), besides the discussion inSec. II. This is reminiscent of the situation of probabilitydistributions where they are restricted by the normal-ization condition

∑i pi = 1. In fact, this constraint is

a key to deduce basic results in information geometry,such as Cencov’s theorem [28, 64]. On the other hand,the more generic restrictions of chemical reactions needa nontrivial extension of information geometry.

We indicate an information-geometric characterizationof stoichiometric compatibility classes. Let ri :=

√[Xi]

and aλi :=√Lλ/|`λi |. Note that Lλ can be always non-

negative by choosing `λ properly. Then the metric withrespect to ri is the Euclidean because

ds2 =N∑i=1

1[Xi]

d[Xi]2 =N∑i=1

(2dri)2. (111)

If Lλ 6= 0, the constraint on concentration distributionsbecomes

N∑i=1

sign(`λi )(riaλi

)2= 1, (112)

which is an equation of a quadric surface, like an ellipsoidor a hyperboloid. If Lλ = 0, the constraint represents acone that contains the origin [X] = 0 as a specific caseof quadric surfaces. This consideration shows that thestoichiometric compatibility class can be considered asthe intersection between such quadric surfaces and thepositive orthant.

We also consider the geometry of reaction dynamics ona stoichiometric compatibility class. The length L of thepath between the concentration at t = 0, [X]t=0, and att = τ , [X]t=τ is defined as

L :=∫ τ

0

ds

dtdt =

∫ τ

0

√1

[Xi]

(d[Xi]dt

)2dt. (113)

From the speed limit on the Gibbs free energy Eq. (78),the fluctuation-rate ratio gives a lower bound to L

L ≥∫ τ

0

1√〈〈∆µ2〉〉

∣∣∣∣dGdt∣∣∣∣dt. (114)

Also from the Cauchy–Schwarz inequality, we have

L2 ≤ 2τC, (115)

where C is called the thermodynamic cost and defined as

C := 12

∫ τ

0dtI(t). (116)

As shown in Sec. V A, under near-equilibrium conditions,the Fisher information of a closed CRN is expressed as

I(t) = − 12R

dσ

dt+ o(∆[X]2), (117)

thus its integral is∫ τ

0dtI(t) ' σ(0)− σ(τ)

2R . (118)

Since this quantity has the same dimension as reactionrates Jρ and d[Xi]/dt, it may represent how fast the sys-tem relaxes in the time interval τ even far from equilib-rium. Hence, we can interpret C as the mean relaxation

12

0 10 20 30 40 50 60Time

0

2

4

6

8

10

12

14C

once

ntra

tion

FIG. 3. The time evolutions of X and Y’s concentrationsobtained by integrating Eq. (122) with the parameters k+

1 =1× 10−3, k−

1 = k+2 = k−

2 = 1, k+3 = 1× 10−2, k−

3 = 1× 10−4,[X]0 = 1, [Y]0 = 6, [A]0 = [B]0 = 1× 103. They oscillate, thenrelax to the equilibrium.

rate. Then we have more informative expression of theinequality Eq. (115)

τ ≥ L2

2C . (119)

It shows a trade-off relation between time and the meanrelaxation rate in CRNs.

Apart from the dynamics, one can define the short-est distance D = D([X], [X′]) between two distributions[X], [X′] under the geometry of stoichiometric compati-bility classes. Since the shortest distance always serves asa lower bound of the length L, another trade-off relationobtains

τ ≥ D2

2C . (120)

Although this inequality is a weaker bound thanEq. (119), it gives a lower bound to the mean relaxationrate C ≥ D2/2τ which needs only information of the timeinterval and the initial and final distributions.

VI. EXAMPLES

Through two characteristic examples of CRN, we checkthe inequalities Eq. (78), (87), and (94).

A. Speed limit in damped Brusselator

The first example is the Brusselator [59, 65], whichis a notable model of oscillating reactions such as theBelousov–Zhabotinsky reaction. We consider the follow-

0 10 20 30 40 50 60Time

100

101

102

103

104

Ener

gy /

Tim

e

FIG. 4. The speed limit Eq. (78) with respect to the reactionsystem (121). The speed of the Gibbs free energy changecannot exceed vµ(t,µeq).

ing CRN

A X2X + Y 3X

X + B Y + A.(121)

Then the concentrations obey the rate equation below

d[X]dt

= J1 − J3

d[Y]dt

= −J2 + J3

d[A]dt

= −J1 + J3

d[B]dt

= −J3,

(122)

where

J1 = k+1 [A]− k−1 [X] (123)

J2 = k+2 [X]2[Y]− k−2 [X]3 (124)

J3 = k+3 [X][B]− k−3 [Y][A]. (125)

In a usual Brusselator model, the change of two species,X and Y, are of interest, so the others are assumed to beconstant because of the abundance. On the other hand,we consider the system as closed, hence we do not set[A] and [B] constant, but sufficiently large to observe thedamped oscillation of [X] and [Y].

While the Brusselator exhibits a damped oscillationof concentration as in Fig. 3, |dG/dt| is suppressed byvµ(t,µeq) as shown in Fig. 4. The speed limit Eq. (78)is verified for the damped oscillatory CRN. Also the in-equality for the partial Gibbs free energy Eq. (87) can beconfirmed. Focusing on S = {X,Y}, we show its appear-ance in Fig. 5.

There are situations where two curves are close, thatis the inequality is tight. That occurs when the chemical

13

0 10 20 30 40 50 60Time

10−3

10−2

10−1

100

101

102

103

Ener

gy /

Tim

e

FIG. 5. The speed limit Eq. (78) for a subset S = {X,Y}.The speed of the partial Gibbs free energy change also doesnot exceed the corresponding function vµ,S(t,µeq).

potential changes exponentially. Because we used theCauchy–Schwarz inequality to prove the speed limit inEq. (85), equality holds when there exists a constant αsuch that for all i,

1√[Xi]

d[Xi]dt

= α√

[Xi](µi − µeqi ). (126)

It is equaivalent to

d

dt(µi − µeq

i ) = α′(µi − µeqi ), (127)

where α′ = RTα. Thus the equality holds when thedeviation of the chemical potential µi − µeq

i is propor-tional to the exponential function eα

′t. If the constantα′ is negative, this condition means an exponential de-cay to the equilibrium. Under near-equilibrium condi-tions, the system is assumed to relax to the equilibriumexponentially. Hence, µeq should be the minimizer of thespeed limit. We asserted that µeq ' µmin under near-equilibrium conditions in Sec. V C. Hence, the equalitycondition is consistent with the previous discussion.

B. Generalized CramÃľr–Rao inequality in a modelwhere the total concentration does not conserve

We confirm the validity of Eq. (94) furthermore byobserving the following CRN.

2A BA + B B + C

(128)

This CRN is simple but sufficient to break the conserva-tion of total concentration as shown in Fig. 6. Note thatthe preceding Brusselator model looks complicated butpreserves the total concentration, [X] + [Y] + [A] + [B],

10−2 10−1 100 101 102

Time

0.0

0.2

0.4

0.6

0.8

1.0

Con

cent

ratio

n

FIG. 6. The time evolution of the concentrations in the CRNEq. (128) calculated with the parameters k+

1 = k−1 = k+

2 =1, k−

2 = 1× 10−5, [A]0 = 1, [B]0 = [C]0 = 1× 10−5. The totalconcentration [A] + [B] + [C] (dashed line) is not constant.

thus the concentration can be normalized by dividing thisconstant.

The stoichiometric matrix is

S =

−2 −11 00 1

, (129)

so that the conservation law is only

` =

121

(130)

up to a scale factor. Therefore [A] + 2[B] + [C] =: Lbecomes a constant instead of the total concentration.

If we set q = (1, 1, 1)T, 〈〈q〉〉 is the total concentration

〈〈q〉〉 = [A] + [B] + [C]. (131)

The projection qmin is given by

qmin = `TXq`TX` ` = L

[A] + 4[B] + [C]`, (132)

where a denominator `TX` appears unlike Eq. (101) be-cause X1/2` is not a unit vector here. Then the changeof the total concentration is bounded as in Fig. 7. Tocompare with the tightest bound vq(t, qmin), the boundgiven by the trivial conservation law q = 0, vq(t, 0), ispresented together.

C. Trade-off relations on stoichiometriccompatibility class

We illustrate our discussion in Sec. V D by consideringthe following association reaction

2A B. (133)

14

10−2 10−1 100 101 102

Time

10−6

10−4

10−2

100

102

104

FIG. 7. Bounds on the changing rate of the total concentra-tion 〈〈q〉〉. The bound given by qmin is much tighter than thatgiven by q = 0.

0.2 0.4 0.6 0.80.0

0.5

1.0

1.5

2.0

2.5

3.0

Con

cent

ratio

n

Time

FIG. 8. The time evolution of the concentrations in the as-sociation reaction. The parameters are set to k+ = k− = 1,[A]0 = 2.9 and [B]0 = 0.05.

Here we can obtain the distance D analytically. For theconservation law ` = (1, 2)T, let [A] + 2[B] = L. For newvariables r1 = 2

√[A], r2 = 2

√[B], we have the metric

and the constraint as

ds2 = (dr1)2 + (dr2)2 (134)(r1

2√L

)2+(

r1√2L

)2= 1. (135)

Because the stoichiometric compatibility class is an ellip-tic, the coordinate can be parametrized by a parameterθ as (r1, r2)t=0 = (2

√L cos θ1,

√2L sin θ1), (r1, r2)t=τ =

(2√L cos θ2,

√2L sin θ2). Then the distance is obtained

0.0 0.2 0.4 0.6 0.8

0.080

0.085

0.090

0.095

0.100

FIG. 9. The trade-off relations Eq. (119) and Eq. (120) areverified for the CRN Eq. (133). For the fixed τ = 0.1, L2/2Cand D2/2C give a lower bound. Because the concentrationchange is monotonic, L and D give the same bound.

as

D =

∣∣∣∣∣∣∫ θ2

θ1

dθ

√(dr1

dθ

)2+(dr2

dθ

)2∣∣∣∣∣∣ (136)

= 2√L

∣∣∣∣∣∫ θ2

θ1

dθ

√1− 1

2 cos2 θ

∣∣∣∣∣ (137)

= 2√L

∣∣∣∣E(θ2; 1√2

)− E

(θ1; 1√

2

)∣∣∣∣, (138)

where E(x; k) :=∫ x

0 dθ√

1− k2 cos2 θ is the incompleteelliptic integral of the second kind.

We introduce the following notation to check the trade-off relation for time t and time interval τ

Lτ (t) :=∫ t+τ

t

dt′ds

dt′(139)

Cτ (t) := 12

∫ t+τ

t

dt′I(t′) (140)

Dτ (t) := D([X]t, [X]t+τ ). (141)

For the fixed time interval τ = 0.1, the trade-off relationsare shown in Fig. 9. Since the relaxation is monotonic,the length L and the distance D should coincide witheach other. In Fig. 9, Lτ (t)2/2Cτ (t) and Dτ (t)2/2Cτ (t)are actually the same.

VII. CONCLUSION

We have studied thermodynamics of chemical reactionnetworks in terms of information geometry. We have re-vealed that geometrical structure and the Fisher infor-mation can be obtained in CRNs by using informationgeometry. Then we have derived speed limits in CRNs,

15

e.g., Eq. (78), (90), and (94). Our results are not re-stricted to near-equilibrium conditions and hold even ifthe CRN is open or far from equilibrium. This broadrange of application shows the universality of the speedlimit. It has been shown that the speed limit can be inter-preted as a generalization of the Cramer–Rao inequalityEq. (95) outside probability spaces. These results aremainly based on the form of the Gibbs free energy thatincludes the f -divergence D([X]‖[X]eq) and the conser-vation quantities dwelling in a CRN. We have furtherobtained a trade-off relation between time and speed inCRNs examining the geometry of stoichiometric compat-ibility classes.

Our study provides a new perspective on chemicalthermodynamics in terms of information geometry. Thephysical quantities obtained from information geometrylike the intrinsic speed ds/dt or the length of reactionsL would be helpful to investigate biological systems as

CRNs. Though the information geometry of chemicalthermodynamics is brought by the f -divergence, one canask whether there are more fundamental reasons why in-formation geometry is applicable to CRNs. CRNs arejust a kind of dynamical systems and have nothing to dowith probability in its formulation. The fact that how-ever they can be studied by using information geometry,which is usually useful in probability theory, would bea clue to investigate the link between various dynamicalsystems in nature and information.

ACKNOWLEDGMENTS

We thank Keita Ashida, Kiyoshi Kanazawa, AndreasDechant, Takahiro Sagawa and Shin-ichi Sasa for fruit-ful discussions. S. I. is supported by JSPS KAKENHIGrant No. 19H05796 and JST Presto Grant No. JP-MJPR18M2.

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