Background
Developing methodologies for quantitative analysis, both experimental and mathematical, of complex biochemical networks has become one of the central themes of post-genomic biochemistry and mathematical biology. Several disparate approaches, including metabolic control analysis (MCA) 12, flux balance analysis (FBA) 34, and correlation metric construction (CMC) 56, share many commonalities. The objective of this work is to provide a unifying mathematical framework and a thermodynamic basis for these approaches. The thermodynamic basis is the nonequilibrium steady-state (NESS) theory 78 originally developed to describe macromolecular, free-energy transduction 910111213, and the mathematical methods are based on linear analysis near a NESS.
The physical concept of a NESS (also known simply as a steady-state in the MCA community) applies to that of a driven system, such as a living organism, with a constant flow of matter (transforming nutrients to waste) through the system and a corresponding dissipation of free energy 9. Even though a NESS looks remarkably similar to a thermodynamic equilibrium, in that chemical concentrations reach stationary values, the concentrations are maintained constant in a NESS by balancing influxes and effluxes, rather than by balancing the forward and backward fluxes of each elementary reaction as in thermodynamic equilibrium. To an observer concerned only with the chemical concentrations, a NESS would seem to be a true equilibrium; but, in fact, it represents a pseudo-equilibrium, where the work done to drive the system appears as heat, and deserves further consideration 9.
We distinguish between two types of linear analysis: (i) linear stability analysis 14, and (ii) control analysis (or sensitivity analysis) that addresses how the steady-state shifts in response to certain perturbations to the system. These perturbations may be deterministic – leading to MCA – or stochastic – leading to CMC. As expected, the matrix A obtained in linear stability analysis, the flux control-coefficient matrix C in MCA, the stoichiometric matrix S of FBA, and the correlation matrix R from CMC are intimately related. We develop the relationships in the context of complex biochemical networks and their NESS thermodynamics. We show that all the key matrices in MCA and CMC can be computed if one knows the fluxes, chemical potential differences, and the stoichiometric matrix.
The theoretical concept of a NESS provides valuable mathematical and thermodynamic properties. Although many of the approaches mentioned above have provided a mathematical treatment of NESS, many have never explicitly taken the related thermodynamics into account. The key insight is to consider concentrations as measures of potentials, providing a relationship between potential differences (voltages), fluxes (currents), and resistances. It becomes immediately clear that the resistances are related to reaction effectors, such as the expression levels of enzymes, that affect both the forward and backward fluxes of a given reaction, but do not change the chemical potential difference. It is this thermodynamic perspective of a NESS that can be used to develop a comprehensive theory that unites these approaches.
In the context of the newly developed biochemical circuit theory (BCT) 1516, the flux J of each metabolic reaction in a NESS metabolic network is decomposed into J = φ+ - φ- and the chemical potential difference of the reaction in the NESS is Δμ = kBT ln(φ-/φ+), where kB is Boltzmann's constant and T is absolute temperature. Therefore, knowing the chemical potential difference and the flux of a reaction gives
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Furthermore, if the standard-state free energy of reaction Δμo is known from equilibrium thermodynamics, then e(Δμ−Δμo)/kBT
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyzau2aaWbaaSqabeaacqGGOaakcqqHuoarcqaH8oqBcqGHsislcqqHuoarcqaH8oqBdaahaaadbeqaaiabd+gaVbaaliabcMcaPiabc+caViabdUgaRnaaBaaameaacqWGcbGqaeqaaSGaemivaqfaaaaa@3C87@ determines the ratio of the metabolites' concentrations, products to reactants, in NESS.
In many theoretical approaches, metabolic networks and signaling pathways are often separated into disparate mechanisms with only peripheral interactions. This is far from reality, however, as these mechanisms are intimately intertwined and often indistinguishable. For example, in whole-body metabolism, glucose and fatty acids serve as the fuel for cellular metabolism, but also as signals for insulin secretion in pancreatic β-cells and ketogenesis in hepatocytes, where cellular organelles, such as mitochondria, play a key role in the handshake between metabolism and signaling. Ultimately, our goal is to understand how these cellular systems behave in response to stress and disease 17. It is in applying the unified mathematical framework that includes the thermodynamic perspective to these processes that we believe the most significant contributions will be made.
In application, the concepts presented here allows one who has the stoichiometric matrix, S, and measures the correlation matrix, R, to obtain the control coefficients, and vice versa (Table 1). Furthermore, by incorporating the thermodynamic aspect, one is able to take full advantage of the thermodynamic properties, such as standard free energies of formation, which are typically more complete, more consistent, and more well analyzed than the kinetic information 17. This is advantageous in a field where the availability of kinetic parameters is a limiting factor. However, available data remains incomplete, and missing thermodynamic and kinetic data must continually be obtained 17.
<p>Table 1</p>
Summary and comparison table listing the relationship between the elasticity and control coefficient matrices and the stoichiometric (S) and correlation (R) matrices, where A = SRT and B = A-1
Symbol
Coefficient
Relationship
ϵ
local elasticity
(dg J)-1 RT
ε
steady-state elasticity
(dg J)-1 RT B (dg B)-1
C
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbeGaf83qamKbaKaaaaa@2CFA@
concentration control
-BS (dg J)
C
flux control
I - (dg J)-1 RT BS (dg J)
ϵ
Δμ
biochemical potential local elasticity
kBT (dg Δμ)-1 ST
ε
Δμ
biochemical potential steady-state elasticity
kBT (dg Δμ)-1 ST B (dg B)-1
C
Δμ
biochemical potential control
-kBT (dg Δμ)-1 ST BS (dg J)
Basic kinetic equations for biochemical networks
Let us consider a network of M biochemical reactions involving N biochemical species Xi (i = 1, 2, ..., N). The jth biochemical reaction (j = 1, 2, ..., M) is characterized by a set of stoichiometric coefficients ν={νij}
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGae8xVd4Maeyypa0Jaei4EaSNaeqyVd42aa0baaSqaaiabdMgaPbqaaiabdQgaQbaakiabc2ha9baa@3642@ and κ={κij}
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaaccmGae8NUdSMaeyypa0Jaei4EaSNaeqOUdS2aa0baaSqaaiabdMgaPbqaaiabdQgaQbaakiabc2ha9baa@3636@ such that
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Some of the integer ν's and κ's can be zero. The N × M matrix S = κ - ν is known as the stoichiometric matrix. Eq. (2) assumes that the forward and backward reaction kinetics are characterized by the constants k+j
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabgkHiTaqaaiabdQgaQbaaaaa@2FAB@. In cases where the kinetic scheme involves intermediate steps (e.g., enzyme binding), each of the intermediate steps can be incorporated as reactions in the form of Eq. (2), requiring additional kinetic parameters, and the stoichiometric matrix can be constructed to include elementary reactions representing actual interaction events. Segel 18, for example, details how the Michaelis-Menten mechanism is expressed in terms of Eq. (2).
For the reaction system (2), kinetic equations of the time-dependent concentration changes for each biochemical species can be written according to the law of mass action as 1920
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where we use xℓ to denote the concentration of respective Xℓ, and Jie
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOsaO0aaWbaaSqabeaacqWGQbGAaaGccqGH9aqpcqaHvpGzdaqhaaWcbaGaey4kaScabaGaemOAaOgaaOGaeyOeI0Iaeqy1dy2aa0baaSqaaiabgkHiTaqaaiabdQgaQbaaaaa@38F6@20:
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Then Eq. (3) becomes dx/dt = SJ + Je, in which x and Je are N-dimensional column vectors, and J is an M-dimensional column vector.
In a closed reaction system, Je = 0. In this case, it can be shown that thermodynamic equilibrium is the only positive stationary solution to Eq. (3): the internal fluxes J = φ+ - φ- = 0 for each and every reaction. For a closed biochemical reaction system, since all the fluxes are necessarily zero in its unique equilibrium (for each given set of parameters; unique in the dynamic sense), all the control coefficients are necessarily zero. Therefore, in a NESS, at least one injection flux and one efflux are nonzero (Jie
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOsaO0aa0baaSqaaiabdMgaPbqaaiabdwgaLbaaaaa@2FCD@ ≠ 0), or certain concentrations xi are held at constant levels. The former case is referred to as "external flux injection," while the latter is referred to as "external concentration clamping." A distinction between these two "forcing" mechanisms greatly clarifies many controversial issues concerning NESS.
Steady-state concentrations
In the equilibrium state of a closed reaction system, φ+ = φ- and the ratio of chemical concentrations is independent of the amount of material and the initial condition:
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giving rise to the concept of chemical equilibrium constants. Such a system of biochemical reactions is non-dissipative and is associated with a number of conserved quantities. An essential mathematical characteristic of the closed system is that its equilibrium point is degenerate (neutrally stable on a certain manifold), i.e., there is no unique stationary solution for different initial conditions.
For an open system that approaches a NESS, the stationary solution is not usually degenerate. If we assume {xi∗
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aa0baaSqaaiabdMgaPbqaaiabgEHiQaaaaaa@2FC5@} is an asymptotically stable NESS, small differences in the initial condition should lead to the same NESS. We use the notations S
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbiGae83uamfaaa@2D0B@ to denote the set of concentrations of internal species and I
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbiGae8xsaKeaaa@2CF7@ to denote the concentrations of external species which are clamped or independently varied.
In the linear regime near the NESS 17,
d
d
t
(
δ
x
i
)
=
∑
j
=
1
N
A
i
j
(
δ
x
j
)
x
j
∗
,
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGKbazaeaacqWGKbazcqWG0baDaaGccqGGOaakcqaH0oazcqWG4baEdaWgaaWcbaGaemyAaKgabeaakiabcMcaPiabg2da9maaqahabaGaemyqae0aaSbaaSqaaiabdMgaPjabdQgaQbqabaqcfa4aaSaaaeaacqGGOaakcqaH0oazcqWG4baEdaWgaaqaaiabdQgaQbqabaGaeiykaKcabaGaemiEaG3aa0baaeaacqWGQbGAaeaacqGHxiIkaaaaaaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdGccqGGSaalaaa@4EAB@
in which δxj = xj - xj∗
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aa0baaSqaaiabdQgaQbqaaiabgEHiQaaaaaa@2FC7@ and
A
i
j
=
∑
ℓ
=
1
M
(
κ
i
ℓ
−
ν
i
ℓ
)
(
ν
j
ℓ
ϕ
+
ℓ
−
κ
j
ℓ
ϕ
−
ℓ
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqae0aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpdaaeWbqaaiabcIcaOiabeQ7aRnaaDaaaleaacqWGPbqAaeaacqWItecBaaGccqGHsislcqaH9oGBdaqhaaWcbaGaemyAaKgabaGaeS4eHWgaaOGaeiykaKIaeiikaGIaeqyVd42aa0baaSqaaiabdQgaQbqaaiabloriSbaakiabew9aMnaaDaaaleaacqGHRaWkaeaacqWItecBaaGccqGHsislcqaH6oWAdaqhaaWcbaGaemOAaOgabaGaeS4eHWgaaOGaeqy1dy2aa0baaSqaaiabgkHiTaqaaiabloriSbaakiabcMcaPaWcbaGaeS4eHWMaeyypa0JaeGymaedabaGaemyta0eaniabggHiLdGccqGGUaGlaaa@57ED@
A = {Aij} is called the linear stability matrix. We introduce the N × M matrix R = {Rij=νijϕ+j−κijϕ−j}
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaei4EaSNaemOuai1aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqaH9oGBdaqhaaWcbaGaemyAaKgabaGaemOAaOgaaOGaeqy1dy2aa0baaSqaaiabgUcaRaqaaiabdQgaQbaakiabgkHiTiabeQ7aRnaaDaaaleaacqWGPbqAaeaacqWGQbGAaaGccqaHvpGzdaqhaaWcbaGaeyOeI0cabaGaemOAaOgaaOGaeiyFa0haaa@46B2@, such that A = SRT. We also denote A-1 = B.
If the nonlinear dynamics scheme of Eq. (3) conserves certain quantities, such as total element, motif, or enzyme concentrations, S does not have full rank and A is singular 171920. In such cases, it is possible to transform A into a nonsingular matrix by replacing its linearly dependent rows with vectors in its left null space (see Ref. 18 of 20). By doing so, one removes the redundancies from the original dynamics scheme. A more detailed mathematical analysis will be published elsewhere. Both here and below, A is understood to be the transformed nonsingular matrix.