METHODOLOGY ARTICLE Open Access

Thermodynamically consistent model calibrationin chemical kineticsGarrett Jenkinson and John Goutsias*

Abstract

Background: The dynamics of biochemical reaction systems are constrained by the fundamental laws ofthermodynamics, which impose well-defined relationships among the reaction rate constants characterizing thesesystems. Constructing biochemical reaction systems from experimental observations often leads to parametervalues that do not satisfy the necessary thermodynamic constraints. This can result in models that are notphysically realizable and may lead to inaccurate, or even erroneous, descriptions of cellular function.

Results: We introduce a thermodynamically consistent model calibration (TCMC) method that can be effectivelyused to provide thermodynamically feasible values for the parameters of an open biochemical reaction system. Theproposed method formulates the model calibration problem as a constrained optimization problem that takesthermodynamic constraints (and, if desired, additional non-thermodynamic constraints) into account. By calculatingthermodynamically feasible values for the kinetic parameters of a well-known model of the EGF/ERK signalingcascade, we demonstrate the qualitative and quantitative significance of imposing thermodynamic constraints onthese parameters and the effectiveness of our method for accomplishing this important task. MATLAB software,using the Systems Biology Toolbox 2.1, can be accessed from http://www.cis.jhu.edu/~goutsias/CSS lab/software.html. An SBML file containing the thermodynamically feasible EGF/ERK signaling cascade model can be found inthe BioModels database.

Conclusions: TCMC is a simple and flexible method for obtaining physically plausible values for the kineticparameters of open biochemical reaction systems. It can be effectively used to recalculate a thermodynamicallyconsistent set of parameter values for existing thermodynamically infeasible biochemical reaction models of cellularfunction as well as to estimate thermodynamically feasible values for the parameters of new models. Furthermore,TCMC can provide dimensionality reduction, better estimation performance, and lower computational complexity,and can help to alleviate the problem of data overfitting.

BackgroundPhysical systems are constrained to operate according tothe fundamental laws of thermodynamics. The conserva-tion of mass and energy and the production of entropy(or heat dissipation) dictate that certain events are phy-sically impossible. A broken glass, for example, will notspontaneously reassemble, and a bar of gold will not for-tuitously appear from thin air. Not all physical con-straints imposed by thermodynamics are intuitivelyobvious. As a matter of fact, thermodynamic constraintsimposed on biochemical reaction systems are routinelyoverlooked in the literature, either due to ignorance of

their existence or difficulties in understanding the impli-cations of modern non-equilibrium thermodynamics.There is an increasing consensus, however, that caremust be taken to ensure that the kinetic parameters of abiochemical reaction system meet these thermodynamicconstraints [1-6].There are many publications discussing the problem

of estimating the kinetic parameters of a biochemicalreaction system from experimental data of molecularconcentrations, when the underlying stoichiometry isknown [7-9]. Essentially, all approaches to this problem,which is often referred to as model calibration, arebased on deriving a cost function and choosing an opti-mization algorithm to minimize that function [10]. Thecost function provides a measure of ‘goodness of fit’ of

* Correspondence: goutsias@jhu.eduWhitaker Biomedical Engineering Institute, The Johns Hopkins University,Baltimore, MD 21218, USA

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

© 2011 Jenkinson and Goutsias; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

the estimated biochemical reaction system dynamics toavailable observations, and may be designed by a varietyof statistical inference techniques, such as maximum like-lihood, Bayesian inference, etc., or by simply employingan appropriate distance metric, such as least-squares. Theoptimization procedure used must be characterized bysuperior performance for finding global minima, due tothe highly non-convex and multi-modal nature of thecost function [10]. The vast majority of published modelcalibration methods, however, produce biochemical reac-tion systems that may not be physically realizable, sincethey do not take into account the fact that the underlyingkinetic parameters may be constrained by the fundamen-tal laws of thermodynamics.Recently, there have been several attempts to address

the issue of thermodynamic constraints in chemicalkinetics [2-5]. Among proposed methods, the thermody-namic-kinetic modeling (TKM) approach [5] enjoyssome benefits over other techniques. However, we havepreviously noted in [11] that this approach is unnecessa-rily complicated and can be cumbersome, especiallywhen dealing with molecular perturbations (commonlyused in systems biology) or when merging estimatedTKM models [5].To address these problems, we have recently proposed

two techniques for estimating the kinetic parameters ofclosed biochemical reaction systems from availableobservations of molecular concentrations in a thermody-namically consistent fashion [11,12]. In [12], we modelbiochemical reaction systems by mass-action kinetics,use maximum-likelihood estimation, and employ a pro-jection step that allows us to appropriately choosekinetic parameter values so that the final system is ther-modynamically feasible. In [11], we employ a Bayesianinference approach, eliminate the projection step,and derive a biophysically based cost function overparameters that can be chosen independently withoutviolating the underlying thermodynamic constraints.Unfortunately, both methods are limited, being appliedto closed biochemical reaction systems using standardmass action kinetics.In this paper, we propose a method for calibrating the

kinetic parameters of biochemical reaction models ofcellular function so that the resulting systems are ther-modynamically feasible. The method, which we refer toas Thermodynamically Consistent Model Calibration(TCMC), works with any iterative parameter estimationalgorithm of choice and can be applied to open bio-chemical reaction systems, which, in problems of sys-tems biology, are more realistic than the closed systemswe considered previously. Furthermore, TCMC is cap-able of handling any kinetic rate laws in ideal mixtures,such as non-mass action rate laws commonly used todescribe complex enzymatic reaction schemes.

We exemplify practical aspects of the proposed tech-nique by recalculating the kinetic parameters of a well-known model of the EGF/ERK signaling cascade [13],which is thermodynamically infeasible. This allows us topropose a thermodynamically feasible model for thisimportant signaling pathway that is physically realizableand better matches available densitometric data. Com-puter simulations reveal a number of qualitative andquantitative differences of possible biological significancebetween the thermodynamically feasible and the ther-modynamically infeasible published model, which needto be validated experimentally. Moreover, we discuss anumber of important advantages gained by TCMC overestimating the kinetic parameters using a collective fit-ting approach that does not consider the underlyingthermodynamic constraints. Besides producing physi-cally realizable and thermodynamically consistent mod-els, TCMC may result in dimensionality reduction,better estimation performance, and lower computationalcomplexity. MATLAB software, using the Systems Biol-ogy Toolbox 2.1 http://www.sbtoolbox2.org, can beaccessed from http://www.cis.jhu.edu/~goutsias/CSSlab/software.html. An SBML file containing the thermody-namically feasible EGF/ERK signaling cascade model canbe found in the BioModels database http://www.ebi.ac.uk/biomodels-main.We believe that TCMC can be effectively used to

recalculate the parameter values of any existing thermo-dynamically infeasible biochemical reaction model ofcellular function, as well as to estimate the parametersof new biochemical reaction models from availableexperimental data, thus producing physically plausibleversions of these models compatible with the fundamen-tal laws of thermodynamics. Finally, as more chemicalspecies or reactions are discovered, TCMC can be usedto easily extend existing models of cellular activity in athermodynamically consistent and computationally effi-cient fashion.

ResultsBiochemical Reaction SystemsMost biological processes of interest to systems biologyare modeled by means of open biochemical reaction sys-tems; i.e., systems that exchange mass with their sur-roundings [14]. Living cells, for example, are opensystems maintaining themselves by exchanging materialswith their environment. Mass exchange is modeledeither explicitly or implicitly. Examples of explicit mod-eling include: clamped species, reactions with null spe-cies as reactants or products, and irreversible reactions[15]. Clamped species are chemicals whose concentra-tions are held fixed. They are often used to model mole-cular species whose concentrations are affected byunknown reactions. It is apparent that these chemicals

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 2 of 13

must be supplied or removed from the system at appro-priate rates to ensure that their concentrations do notdeviate from their fixed values. On the other hand, reac-tions with null reactants or products model mass trans-fer in and out of the system, respectively. Finally, theuse of an irreversible reaction is based on the assump-tion that the concentration of at least one of its pro-ducts is clamped to zero (otherwise, the reaction can bereversed at sufficiently high product concentrations),which implies mass transfer as well. An example ofimplicitly modeling mass exchange is a reaction withreactants or products not modeled by the system. Acommon case would be the phosphorylation of a proteinwithout explicitly modeling conversion of ATP to ADP[5].An open biochemical reaction system is comprised of

N molecular species X1, X2,..., XN that interact throughM coupled reactions, given by∑

n∈NvnmXn �

∑n∈N

ν ′nmXn, m ∈ M,

where N := {1, 2, . . . , N}, ℳ: = {1, 2,..., M}, and νnm,ν ′nm ≥ 0 are the stoichiometries of the reactants and pro-ducts. We can characterize an open biochemical reac-tion system at time t ≥ 0 by the concentrations{xn(t), n ∈ N } of all molecular species at t. Clampedmolecular species Nc ⊂ N have concentrations that donot vary with time, whereas, the concentrations of theremaining ‘dynamic’ species Nd = N \Nc evolve as afunction of time. We will assume that the system char-acterizes reactions in an ideal and well-stirred (homoge-neous) mixture at constant temperature and volume andthat the concentrations {xn(t), n ∈ Nd} vary continu-ously in time. In this case, we can describe the dynamicevolution of the molecular concentrations in the systemby the following chemical kinetic equations:

dxn(t)dt

=

{ ∑m∈M

snmφm(t, k), t ∈ T , n ∈ Nd

0, t ∈ T , n ∈ Nc,(1)

initialized by setting xn(0) = qn, n ∈ N , for some initialconcentrations qn, n ∈ N , where jm(t, k) is the net fluxof the mth reaction, snm := ν ′

nm − vnm is the net stoichio-metry coefficient of the nth molecular species associatedwith the mth reaction, T := [0, tmax] is an observationtime window of interest, and k is a vector of kineticparameters {kj, j ∈ J }, where J = {1, 2, . . . , J}. Theparameters k characterize the biochemical reaction sys-tem at hand and are independent of the molecular con-centrations {xn(t), n ∈ N }. Moreover, we assume thatthese parameters do not vary with time.By appropriately pruning and modifying an open bio-

chemical reaction system, we can derive a closed

subsystem (i.e., a system that does not exchange masswith its surroundings) that lends itself to thermody-namic analysis since external or unknown thermody-namic forces no longer exist. To do so, we first removeall null and irreversible reactions, as well as partiallymodeled reactions (i.e., reactions with incomplete stoi-chiometries). Next, we remove clamping of molecularspecies involved in reversible reactions. Subsequently,we keep only reactions that are thermodynamically inde-pendent. Thermodynamic independence among reac-tions means that a reaction is only driven by its ownthermodynamic force, which implies that the affinity ofthe reaction will be zero if and only if there is nochange in its degree of advancement. This condition isusually fulfilled if we keep only elementary reactions (i.e., reactions that take place in one single step). As aconsequence, we obtain a closed reaction set ℳ0 ⊆ ℳ.Finally, we remove all species that are no longerinvolved with any of the reactions in ℳ0, leaving onlythe molecular species N0 ⊆ N associated with theclosed subsystem.The main rationale behind the second step is that the

kinetic parameters k considered in this paper areassumed to be independent of the molecular concentra-tions. As a consequence, the values of these parameterswill not change if the concentrations of the clampedspecies are allowed to vary. Therefore, we can constructa (possibly artificial) situation in which the concentra-tions of the clamped species vary as if they weredynamic species. Because our goal in creating the closedsubsystem is to discover and enforce thermodynamicconstraints on the kinetic parameters, we must includethe clamped species in our model. This is contrary tosimply removing all clamped species and the associatedreactions, since this approach will not allow us to deter-mine thermodynamically feasible values for the kineticparameters of the removed reactions.The third step is due to a simplification imposed on

us by the current state of non-equilibrium thermody-namics. Thermodynamically dependent reactions influ-ence each other, since the thermodynamic force of onereaction may drive the other reaction and vice versa.Unfortunately, it is not clear at this point how to dealwith thermodynamically coupled reactions. Futureresearch may be necessary to address this issue.The resulting closed biochemical reaction subsystem is

comprised of N0 molecular species {Xn, n ∈ N0} thatinteract through M0 coupled reversible reactions. Thedynamic evolution of the molecular concentrations inthis system is governed by:

dxn(t)dt

=∑

m∈M0

snmφm(t,k), t ∈ T , n ∈ N0, (2)

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 3 of 13

initialized by xn(0) = qn, for n ∈ N0. We will character-ize all reactions in ℳ0 by the generalized mass-actionrate law [16]. In this case, the net fluxes are given by

φm(t, k) = fm[x(t), π]

×⎛⎝r2m−1

∏i∈N0

[xi(t)]νim − r2m

∏i∈N0

[xi(t)]ν′im

⎞⎠ ,

(3)

for m Î ℳ0, where r2m-1, r2m are the (generalized)rate constants of the mth forward and reverse reactions,respectively. The quantity fm[x(t), π] can be any positiveand finite function of the concentrations x(t) and maydepend on a set of kinetic parameters π. For usual massaction kinetics, fm[x(t), π] = 1. However, for more com-plex schemes, this function usually takes a rational orpolynomial form. It is known that all reversible reactionrate laws in ideal mixtures (including reversible Michae-lis-Menten kinetics) can be described by (3) [5]. Notethat k includes the kinetic parameters π as well as therate constants {r2m-1, r2m, m Î ℳ0}. It also includes thekinetic parameters of all reactions in ℳÎℳ0.It is a direct consequence of thermodynamic analysis

that a closed biochemical reaction system will asympto-tically reach a unique non-zero state {xn > 0, n ∈ N0}of chemical equilibrium at which all concentrationsbecome stationary (assuming, of course, we have non-zero initial conditions), satisfying the following detailedbalance equations:

r2m−1

∏n∈N0

xνnmn = r2m

∏n∈N0

xν′nmn , for all m ∈ M0. (4)

As a consequence,

zm := lnr2m−1

r2m=

∑n∈N0

snm ln xn, for all m ∈ M0. (5)

These constraints must be satisfied by the rate con-stants in order for the closed biochemical reaction sys-tem to be thermodynamically feasible.The constraints implied by (5) correspond to the reac-

tion ‘cycles’ in the system. A reaction cycle is comprisedof those reactions corresponding to the nonzero elementsof a vector in the null space null (S0) of the stoichiometrymatrix S0 = [snm, n ∈ N0, m ∈ M0] of the closed sys-tem. Clearly, (2) will be at a fixed point whenever the netfluxes of the underlying reactions are set equal to thecorresponding elements of a vector in null (S0). If wedenote by s the N0 × 1 vector whose nth element is thelog steady-state concentration ln x̄n and by z the M0 × 1vector whose mth element is the log equilibrium constantzm = ln(r2m-1/r2m), then we can write (5) in a matrix-vector form as ST

0s = z. Now, if b is a vector in the null

space of S0, then S0b = 0 and bTz = bTST0s = (S0b)Ts = 0,

or

∏m∈M0

(r2m−1

r2m

)bm

= 1, for all b ∈ null(S0), (6)

which are the well-known Wegscheider conditions [6].These conditions express necessary and sufficient con-straints on the reaction rate constants of a closed bio-chemical reaction system to be thermodynamicallyfeasible. Thus, if we denote the set of all thermodynami-cally feasible parameters k by W, then any k ∈ W satis-fies (6) and, likewise, any k that satisfies (6) is a memberof W.We want to emphasize that, in open biochemical reac-

tion systems, the rate constants of the reversible reac-tions must also be constrained by the Wegscheiderconditions, even if the system is far from equilibrium.To identify these constraints, we need to prune an openbiochemical reaction system into a closed subsystem, byemploying the technique discussed previously, and usethe resulting stoichiometry matrix S0 to calculate theconstraints given by (6).An equally important observation is that the rate con-

stants of the reactions pruned from an open system arenot constrained by the Wegscheider conditions, since(4) must only be satisfied by the reactions in the closedsubsystem. Furthermore, if a reaction m in a closed sys-tem is not part of a cycle, then bm = 0, for every b Înull (S0), and its forward and reverse rate constants willnot be thermodynamically constrained, since these con-stants trivially satisfy (6).It can be shown (see Additional file 1) that the entropy

production rate of an open biochemical reaction systemat chemical equilibrium in which the net fluxes of thereactions in ℳ0 equal to the elements of a vector in thenull space of the stoichiometry matrix S0, is given by

σ (b) = AVkB ln∏

m∈M0

(r2m−1

r2m

)bm

, for all b ∈ null(S0), (7)

where A = 6.02214084(18) × 1023mol-1 is the Avoga-dro number, V is the system volume, and kB =1.3806504 × 10-23JK-1 is the Boltzmann constant.According to the second law of thermodynamics, theentropy production rate must always be greater than orequal to zero, with equality if and only if the system isat thermodynamic equilibrium. It is therefore clear from(7) that the Wegscheider conditions imply that theentropy production rate must be zero in this case (i.e.,the system must be at thermodynamic equilibrium). Asa consequence, the chemical motive force (which is theamount of energy added to the system per unit time

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 4 of 13

due to mass exchange through its boundary) and the heatdissipation rate must also be zero. This makes intuitivesense, since a reaction cycle leaves all molecular concen-trations unchanged and, therefore, there is no change ininternal energy or mass flow through the system bound-ary. Clearly, we can think of the Wegscheider conditionsas being a direct consequence of the thermodynamicrequirement that s (b) = 0, for every b Î null (S0).

Linear constraintsUnfortunately, (6) imposes a possibly infinite numberof non-linear constraints on the rate constants of aclosed biochemical reaction system. However, it is suf-ficient to satisfy (6) for M2 = M0 - M1 basis vectors{b(i), i = 1, 2,..., M2} of the null space of S0, where M0

is the number of reactions and M1 = rank (S0) (seeAdditional file 1). By using this observation and bytaking logarithms on both sides of (6), we obtainthe following linear constraints on the log-rate con-stants {�2m−1 := ln r2m−1,�2m := ln r2m,m ∈ M0}:∑

m∈M0

b(i)m (�2m − �2m−1) = 0, for i = 1, 2, . . . , M2, (8)

where b(i)m is the mth component of the ith basis vectorb(i). In Additional file 1 we derive an analytical formulafor the basis vectors of the null space of S0 [see Equa-tion (S.1)]. As a consequence, the possibly infinite non-linear Wegscheider conditions given by (6) are equiva-lent to much more manageable finite linear constraintson the log-values of the parameters of the closed subsys-tem, given by

Wκ = 0, (9)

where � := ln(k) and W is an M2 × J matrix that canbe easily constructed from knowledge of S0 using (8).Hence, k ∈ W if and only if W ln(k) = 0.We should note that there might be additional linear

constraints that we may wish to impose on the loga-rithms of the kinetic parameters of a biochemical reac-tion system. Here are some examples:

• By using an appropriate experimental procedure,we may be able to accurately measure the equili-brium constant Rm of the mth reversible reaction.For a (generalized) mass-action reaction, we haveRm := r2m-1/r2m and thus we obtain a linear constraint

�2m−1 − �2m = z(meas)m on the log rate constants of the

mth reaction, where z(meas)m is the log value of the

measured equilibrium constant.• For a reversible Michaelis-Menten reaction, theHaldane relationship implies a linear constraintbetween the logarithms of the kinetic parameters of

a reversible Michaelis-Menten reaction and

z(meas)m [17].• By using experimental techniques, such as plasmonresonance or atomic force microscopy, it may be pos-

sible to obtain a highly accurate measurement k(meas)j

of an individual kinetic parameter kj. In this case, we

must impose the (trivial) constraint eTj κ = ln [k(meas)j ],

where ej is the jth column of the J × J identity matrix.• To reduce the dimensionality of parameter estima-tion, we may employ a sensitivity analysis approach,such as the one proposed in [18,19], to identify para-meters that do not appreciably influence the cost ofestimation. Determining accurate values for theseparameters is inconsequential to the behavior of thebiochemical reaction system at hand. Therefore, we

can fix these parameters to some nominal values k(0)j

throughout model calibration, resulting again in lin-

ear constraints of the form eTj κ = ln [k(0)j ].

• We may want to expand an existing (validated)thermodynamically feasible model to include addi-tional reactions and molecular species. We can dothis by fixing the parameters of the existing model

using linear constraints eTj κ = ln [k(exist)j ], where k(exist)j

is the jth parameter value of the existing model. Thenestimation takes place only on the parameters asso-ciated with the new reactions.

For a given biochemical reaction system, we can com-bine all possible linear constraints on the logarithms �of the kinetic parameters k into a single matrix equationof the form:

Aκ = c, (10)

where A is an appropriately constructed L × J matrixand c is an L × 1 vector of known values determined bythe constraints, with L being the number of constraints.Note that if there are no constraints other than theWegscheider conditions, we would simply have A = Wand c = 0.

Model calibrationWe will now assume that we have obtained noisy mea-

surements y = {y(p)n (tq), n ∈ Ns, p ∈ P , q ∈ Q} of the

concentration dynamics of selected molecular speciesn ∈ Ns ⊆ N in an open biochemical reaction system ofknown stoichiometry, obtained by a set of distinctexperiments p ∈ P = {1, 2, . . . , P} at discrete timepoints{tq, q ∈ Q}, Q = {1, 2, . . . , Q}, within the obser-vation time window T . The problem of model calibra-tion we consider in this paper is to determinethermodynamically consistent values for the kinetic

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 5 of 13

parameters {kj, j ∈ J }, so that the concentration

dynamics x = {x(p)n (tq), n ∈ Ns, p ∈ P , q ∈ Q} producedby the estimated system match y in some well-definedsense. Note that, for a given p ∈ P, the dynamics

{x(p)n (t), n ∈ N } are computed via (1) using initial con-

ditions x(p)n (0) = q(p)n that correspond to the pth experi-mental condition. Data y may be obtained byappropriately designed in vivo or in vitro experiments,or by simulating an established and experimentallyvalidated biochemical reaction model whose kineticparameters are thermodynamically infeasible. Instead offocusing on the quality of the estimated values of thekinetic parameters, it has been argued in [20] thatmatching predicted and experimental observations ofmolecular concentrations is the right thing to do dueto the ‘sloppiness’ of biochemical reaction systems (dif-ferent combinations of parameter values may essen-tially lead to the same concentration dynamics).There are many estimation techniques we can use to

address the previous problem, such as maximum likeli-hood or Bayesian inference. The final product of thesetechniques is a cost function C(� |y) used to quantifythe overall error between the predicted concentrationmeasurements x, obtained by simulating the biochemicalreaction system with kinetic parameter values k = exp{�}, and the noisy observations y. In an effort to reducethe typically large dynamic range of kinetic parametervalues, it is customary to estimate the log values �instead of k. As a consequence, the problem of interesthere is to compute an estimate κ̂ of the log kinetic para-meters �, so that

κ̂ = argminκ∈A

C(κ |y), (11)

where A is the set of all �’s satisfying the linear con-straints given by (10); i.e.,A := {κ : Aκ = c}. For simpli-city, we consider in this paper the least-squares errorcost criterion, given by

C(κ |y) =∑n∈Ns

∑p∈P

∑q∈Q

[y(p)n (tq) − x(p)n (tq)]2. (12)

This error criterion is a consequence of a maximumlikelihood approach to parameter estimation under theassumption of normally distributed observation errors.Note that the cost C depends on the log kinetic para-

meters � through the molecular concentrations x(p)n (tq).In this paper, we refer to the constrained optimization

problem given by (11) as Thermodynamically ConsistentModel Calibration (TCMC). The importance of TCMClies on the formulation of the model calibration problemas one of constrained optimization via (11), with con-straints that ensure at least the thermodynamic feasibility

of the resulting model. A useful observation is that TCMCis agnostic to the choice of the cost function used and thealgorithm employed for optimization. Moreover, we caneasily transform the constrained optimization problemgiven by (11) to a standard unconstrained problem.Indeed, a well known result of linear algebra implies thatA := {κ : Aκ = c} = {κ : κ = κ0 + Bv, v ∈ �d}, whered = J − rank (A), �0 is a J × 1 vector that satisfiesAκ0 = c, B is a J × d matrix whose columns form a basisfor the null space of matrix A, and ℜd is the space of all d× 1 real valued vectors. Thus, if we can find a particularsolution �0 to the constraints Aκ = c, we can reformulatethe constrained optimization problem given by (11) as thefollowing lower dimensional unconstrained problem:

κ̂ = κ0 + Bv̂

v̂ = argminv

C0 (v |y), (13)

where

C0(v |y) := C(κ0 + Bv |y). (14)

Note that we assume here that there is more than onesolution to Aκ = c. If only one solution exists, optimiza-tion is not necessary, since this solution will be ourparameter estimate. On the other hand, if Aκ = c has nosolution, then we cannot find a � that will simulta-neously satisfy all necessary constraints, indicating thatwe must reformulate the problem.The objective function C0 is non-convex with possibly

many local minima. As a consequence, a gradient-basedoptimization algorithm for solving (13) may prematurelyterminate at a local minimum with much larger costthan the globally minimum cost. To ameliorate this pro-blem, we have decided in this paper to use a stochasticoptimization algorithm, namely simulated annealing(SA) [21]. Stochastic optimization algorithms can moveaway from premature local minima, thus resulting inbetter solutions to optimization problems than whenusing deterministic techniques. Although many choicesexist for optimization, such as the simultaneous pertur-bation stochastic approximation (SPSA) methodemployed in our previous work [11] and genetic algo-rithms, SA is a stochastic search optimization algorithmthat enjoys several advantages over other algorithms. Inparticular, the most important features of SA are ease ofimplementation and the ability to avoid premature con-vergence by jumping away from local minima en routeto finding a global minimum. In SA, a new value of v isproposed nearby the current value. The proposed valuebecomes the new value with a certain probability basedon cost improvement. If the proposed value is notaccepted, then the current value is used. The proposedvalue is usually drawn from an appropriately chosen

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 6 of 13

probability distribution around the current value (e.g., aGaussian distribution centered at the current value). SeeAdditional file 1 for a detailed description of the SAalgorithm used.A natural question that arises here is whether different

choices for �0 and B affect the final result of optimiza-tion. If we had an algorithm that could always find theglobal solution to a non-convex optimization problem,then the choice of �0 and B would have no effect on thesolution. Since however global minima are difficult tofind, we expect that different choices for �0 and B willhave some impact on the final solution. Note that itwould be advantageous to choose �0 as close as possibleto the globally optimal solution. We attempt to do so inour subsequent example by taking �0 to be a solution toAκ0 = c that is closest (in the least-squares sense) topublished values. On the other hand, we expect that thechoice of B will have only a minor effect on optimiza-tion, since different matrices B amount to scaling orrotating the axes of the parameter space being searched.Good optimization algorithms, such as SA, are expectedto be robust to such alterations.

ExampleWe now demonstrate the proposed TCMC method byre-estimating the kinetic parameters of a classical modelof the EGF/ERK signaling cascade [13]. This model con-sists of three compartments (extracellular space, cyto-plasm, and endosomal volume), N = 100 biochemicalspecies, and M = 125 reactions. Moreover, it is charac-terized by 90 different kinetic parameter values, a num-ber that is smaller than the total number of individualreactions, due to the fact that some reactions share thesame kinetic parameters, whereas, other reactions arenot associated with any parameters.Although the Schoeberl model has provided valuable

insights into the biological mechanisms underlying EGF/ERK signaling, the values of the kinetic parameters pub-lished in the literature are thermodynamically infeasible.As a consequence, the concentration dynamics producedby the published model are physically impossible andcould not occur in nature. By using TCMC to recom-pute thermodynamically feasible values for the kineticparameters, we can construct a physically realizablemodel whose dynamics are expected to reflect the truebehavior of EGF/ERK signaling more accurately thanthe dynamics produced by the published model.We use the version of the Schoeberl model published

in the BioModels database [22]. Moreover, we employthe same experimental time series data of ERK-PP activ-ity used for creating the original model [13]. To simplifyimplementation of TCMC, we assume here that theSchoeberl model is characterized by J = 2M = 250kinetic parameters (two parameters per reaction). For an

irreversible reaction, we constrain the rate constant ofthe reverse reaction to be equal to zero. For those reac-tions not associated with any kinetic parameters, weassign two artificial parameters (one for the forward andone for the reverse reaction) and constrain both theirvalues to be zero.We implement the following TCMC procedure to re-

estimate the values of the kinetic parameters in a ther-modynamically consistent manner. First, we find theclosed subsystem of the Schoeberl model (see Additionalfile 1). This subsystem consists of N0 = 93 molecularspecies and M0 = 83 reversible elementary (monomole-cular and bimolecular) reactions. Then, we determinethe thermodynamic constraints by using the 93 × 83stoichiometry matrix S0 of the closed subsystem. Itturns out that the rank of the stoichiometry matrix S0 is65. As a consequence, the closed subsystem contains M2

= 83 - 65 = 18 independent reaction cycles, determinedby the columns of matrix B0, given by Equation (S.1) ofAdditional file 1 (see the file for details on the reactioncycles). Therefore, the rate constants of the closed sub-system must satisfy 18 independent Wegscheider condi-tions. It turns out that the published Schoeberl modelsatisfies only 10 of these conditions. As a matter of fact,the entropy reaction rates, given by (7), associated with5 independent reaction cycles are negative, in direct vio-lation of the second law of thermodynamics, whereas,the entropy reaction rates of 3 reaction cycles are posi-tive, with the remaining being equal to zero (see TableS.2 in Additional file 1).Next, we construct matrix A and vector c by combin-

ing the 18 thermodynamic constraints with 167 linearequality constraints originally built into the model thatrelate various parameters across reactions (see Addi-tional file 1 for details on these constraints), thus produ-cing the linear constraints Aκ = c. In this case, A is a 185× 250 matrix, whereas, c is a 250 × 1 vector with ele-ments 0 or -∞ (this corresponds to kinetic parameterswhose values are fixed to zero). It turns out that rank(A) = 171, which implies that the dimension of the nullspace of A is d = 79. Subsequently, we find a basis forthe null space of A and form the 250 × 79 matrix B.Moreover, we find a particular solution �0 of Aκ = cthat is closest, in the least-squares sense, among allother solutions to the published thermodynamicallyinfeasible parameter values. We accomplish this byusing a well-known approach for solving constrainedleast-squares problems [23]. Finally, by using (12) on theaforementioned experimental time series data and SA,we calculate the 79 × 1 vector v̂ by minimizing the costfunction C0(v |y), given by (14), and set κ̂ = κ0 + Bv̂.We take the thermodynamically feasible log kinetic

parameter values �0 to be close to the published kineticparameter values, since the published values already

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 7 of 13

produce a good match between the experimentally avail-able and predicted molecular dynamics. In this case,TCMC provides a thermodynamically consistent correc-tion of �0, by means of Bv̂, that reduces the cost of esti-mation, thus further improving the match between theexperimentally obtained and predicted dynamics.In Figure 1, we depict the concentration dynamics of

ERK-PP obtained by the published model (dotted curves),for two different input EGF concentrations, namely 50ng/mL and 5 ng/mL, whereas, in Figure S1 of Additionalfile 1 we depict the concentration dynamics of ERK-PPobtained by the published model for three additional con-centrations of input EGF, namely 0.5 ng/mL, 0.125 ng/mL, and 0.0625 ng/mL. Clearly, the resulting dynamicsmatch the available densitometric data (indicated by thecircles) rather poorly.We should note here that the ERK-PP dynamics ori-

ginally published in [13] seem to match the availabledata pretty well - see Figure 2F in [13]. However, themodel detailed in the original publication cannot beused to reproduce the results. On the other hand, themodel available in the BioModels database does notreproduce the results published in the Schoeberl paperbut produces dynamics that are very similar to the onesreported in that paper.The solid curves in Figure 1 and Figure S1 of Addi-

tional file 1 depict the ERK-PP concentration dynamicsestimated by TCMC. TCMC results in a thermodynami-cally consistent model of the EGF/ERK signalling cas-cade that produces ERK-PP concentration dynamicswhich match the available experimental data noticeablybetter than the dynamics produced by the published

model. As a matter of fact, model fit between predictedand experimentally measured ERK-PP concentrationdynamics, quantified by the least-squares error given by(12), is reduced by 69%. TCMC simultaneously adjuststhe values of the kinetic parameters in order to mini-mize the cost of fitting the ERK-PP response to theavailable densitometric data. This ‘collective fitting’strategy has been recognized in the literature [20] asbeing more desirable than constructing a biochemicalreaction system model from individual parameter esti-mates in a piecewise fashion, which is the case with thepublished Schoeberl model.To separate the effect of collective fitting versus impos-

ing the underlying thermodynamic constraints on thekinetic parameters, we use the same simulated annealingalgorithm employed by TCMC to estimate the kineticparameters without including the thermodynamic con-straints. The resulting (thermodynamically infeasible) esti-mated dynamics are depicted by the dashed curves inFigure 1 and Figure S1 of Additional file 1. As expected,these dynamics fit the densitometric data better than thedynamics obtained by the published model. As a matterof fact, model fit between predicted and experimentallymeasured ERK-PP concentration dynamics is reduced by70% in this case, which is slightly better than the oneobtained by TCMC. It will shortly become clear howeverthat the solution obtained without imposing the thermo-dynamic constraints leads to unrealistic system behavior.In the following, we discuss a number of advantagesgained by TCMC over estimating the kinetic parametersusing a collective fitting approach that does not considerthe underlying thermodynamic constraints.

time (min)

50 ng/mL EGF

published

estimated

TCMC

time (min)

5 ng/mL EGF

0 10 20 30 40 50 600

0.004

0.008

0.012

0.016

0.02

0 10 20 30 40 50 60

published

estimated

TCMC

Figure 1 ERK-PP concentration dynamics, measured in mol/m3, under two different input EGF concentrations. The Additional file 1reports additional dynamics.

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 8 of 13

DiscussionQualitative/quantitative value of thermodynamicconsistencyDue to lack of thermodynamic consistency in the para-meter values of the published Schoeberl model, the mole-cular dynamics produced by this model cannot possiblyoccur in nature. Because the values estimated by the pro-posed TCMC method satisfy all necessary thermody-namic constraints, it is expected that the resulting TCMCmodel will provide a more accurate representation ofEGF/ERK signaling than the published Schoeberl model.To provide an example of a potentially important dif-

ference between the published model and the TCMCmodel calculated in this paper, we consider the inte-grated response of ERK-PP activity (i.e., the cumulativeERK-PP concentration). It has been argued in the litera-ture that the integrated response provides an appropri-ate metric for quantifying dependence of DNA synthesison ERK activation in certain cells [24]. As a conse-quence, this feature of the ERK-PP concentrationdynamics can influence a number of diverse biologicallyoutcomes, such as cell cycle progression, cell prolifera-tion, and cell differentiation.In view of the fact that differences in the integrated

response of ERK-PP activity may cause distinct biologi-cal outcomes, it is reasonable to believe that a keyobjective of EGF/ERK signaling is to maintain robustintegrated response to changes in input EGF concentra-tion while producing a quick and sharp ‘switch-like’transition between states of differing biological out-comes. In Figure 2, we provide a log-log plot of the inte-grated response of ERK-PP activity predicted by thepublished (dotted curve) and TCMC (solid curve) mod-els, for a wide range of input EGF concentrations.Although the integrated response predicted by bothmodels is indeed robust for input EGF concentrationslarger than 10-2 ng/mL, when the EGF concentrationdecreases below 10-2 ng/mL, the integrated response ofERK-PP activity predicted by the TCMC model exhibitsa sharper transition from large to small values than theone predicted by the published model. As a matter offact, the TCMC model predicts seven orders of magni-tude decrease in the integrated response, when theinput EGF concentration decreases from 10-2 ng/mL to10-3 ng/mL, whereas, the published model predicts onlyfour orders of magnitude decrease. By considering thediscussion in [24], we believe that this behavior by theTCMC model may turn out to be a biologically impor-tant property of the EGF/ERK signaling pathway thatcannot be effectively captured by the published model.We will now show that the TCMC model may result

in a biologically plausible prediction of ERK-PP activitythat can also be different than the one produced by the

estimated thermodynamically infeasible model using col-lective fitting. In Figure 3, we show the long-term beha-vior of the estimated thermodynamically infeasiblemodel. Under certain normal biological conditions (e.g.,typical EGF receptor concentration, such as the oneconsidered in our paper), ERK-PP activity is expected todecay to zero at steady-state [25]. However, one can seefrom Figure 3 that, even after 4 hours, the concentrationof ERK-PP predicted by the estimated thermodynami-cally infeasible model is steadily increasing - possibly

EGF concentration (ng/mL)

published

TCMC

10-3

10-2

10-1

100

101

102

10-10

10-8

10-6

10-4

10-2

100

Figure 2 Integrated response of ERK-PP activity, measured inmol · min/m3, as a function of input EGF concentrations.

0 50 100 150 2000

0.002

0.004

0.006

0.008

0.01

time (min)

0 10 20 30 40 50 600

0.002

0.004

0.006

0.008

0.01

0.012

250

0.012

Figure 3 Long-term ERK-PP concentration dynamics, measuredin mol/m3, predicted by the estimated thermodynamicallyinfeasible model (dashed curves) and the TCMC model (solidcurves) under 0.125 ng/mL (blue curves) and 0.0625 ng/mL(red curves) input EGF concentrations. The inset shows the short-term behavior predicted by the two models as well as thecorresponding densitometric data (blue and red circles).

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 9 of 13

being driven by the chemical perpetuum mobiles thatoccur when the Wegscheider conditions are violated.This unrealistic behavior appears despite the fact thatthe transient dynamics, depicted in the inset of Figure 3by the dashed curves, fit the data better than the TCMCdynamics, denoted by the solid curves. Such a sustainedand long lasting response may lead to different biologi-cal outcomes than the ones resulting from ERK-PPactivity that decays to zero at steady-state [25]. Thus,the estimated thermodynamically infeasible model canlead to erroneous biological predictions, despite its rea-sonable fit to the available densitometric data.Our previous examples show that thermodynamic

consistency may result in model behavior that is differ-ent than the one predicted by thermodynamically infea-sible models of cellular function. However, moreresearch is needed to experimentally validate observeddifferences and demonstrate that lack of thermodynamicconsistency may indeed result in inaccurate (or evenfalse) biological predictions.

Flux analysisFlux-based analysis of biochemical reaction systems is awidely used method for understanding the principlesunderlying the production and regulation of mass flowin cellular systems, such as signaling or metabolic path-ways. It turns-out that the Wegscheider conditions,given by (6), constrain the reaction fluxes. If flux analy-sis does not take into account these constraints, then itmay lead to inaccurate or misleading conclusions aboutthe behavior and properties of mass flow in biochemicalreaction systems.If {b(i), i = 1, 2,..., M2} is a set of basis vectors of the

null space of the stoichiometry matrix S0 of the closedsubsystem, and φ+

m(t, k), φ−m(t, k) are respectively the

forward and reverse fluxes of the mth reaction at time t,then (see Additional file 1 and [6])

∑m∈M0

b(i)m lnφ+m(t,k)

φ−m(t,k)

= 0,

for i = 1, 2, . . . , M2, t ∈ T ,

(15)

where b(i)m is the mth element of the basis vector b(i).Note that this equation must be satisfied at each timepoint t ∈ T , even far away from equilibrium. Moreover,it is satisfied for any vector b in the null space of S0.Since TCMC always leads to a thermodynamically fea-

sible biochemical reaction system with parameters satis-fying the Wegscheider conditions, the flux constraintsimposed by (15) are satisfied as well. Thus, thermodyna-mically consistent flux analysis can be performed on theresulting system without any additional considerations,and the behavior of the system is always physicallyrealizable.

Bias-variance tradeoff and overfittingIn addition to the previous advantages, there is animportant statistical benefit for thermodynamically con-straining the parameters of a biochemical reaction sys-tem. By searching for kinetic parameter values within athermodynamically consistent subset of the parameterspace, we may reduce the variance of estimation andthus lower the estimation error through the well-knownbias-variance tradeoff.The mean-square error (MSE) E[[C(κ̂ |y) − C(κ true|y)]2]

in cost, where κ̂ is an estimator of the ‘true’ parameters�true, satisfies:

E[[C(κ̂ |y) − C(κ true|y)]2]︸ ︷︷ ︸MSE

= [C(κ true|y) − E[C(κ̂ |y)]︸ ︷︷ ︸Bias

]2

+ E[[C(κ̂ |y)n − E[C(κ̂ |y)]]2]︸ ︷︷ ︸Variance

.

Generally speaking, imposing constraints on the para-meters may increase the bias term but decrease the var-iance. However, since the true parameter values mustsatisfy the thermodynamic constraints, we expect adecrease in variance without an increase in bias. As aconsequence, searching for parameter values within thethermodynamically consistent subspace of the parameterspace may lead to a lower mean square error in costdue to smaller variance. Since the volume of a searchspace grows exponentially in the dimension of thespace, gains in variance (and hence improvements in themean square error) are expected to be large.A related statistical notion in estimation problems is

data overfitting. Overfitting refers to situations wheremodel complexity (i.e., number of parameters) is highand the amount and quality of available data is com-paratively low. In this case, it is often possible that wematch the data so well that, in addition to matching theunderlying physical phenomena of interest, the modelfits the measurement noise as well. This situationreduces the predictive power of the estimated model.The relationship of overfitting to the bias-variance tra-deoff is clear: complex models are characterized by lowbias and are able to describe a wide range of phenom-ena but suffer from high variance in parameter estima-tion (i.e., different data sets may lead to wildly differentparameter estimates via overfitting).Most often, the behavior of biochemical reaction

systems is only influenced by a small number of para-meters (due to robustness of such systems to the under-lying kinetic parameters). This reduces the actualnumber of parameters that must be estimated with pre-cision. Moreover, the thermodynamic constraints further

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 10 of 13

reduce the number of parameters to be estimated, alle-viating some overfitting concerns. Imposing additionalparameter constraints, such as the ones employed bythe Schoeberl model, may further be used to combatthis issue. Unfortunately, model complexity is muchhigher than the amount and quality of available data inmost problems of systems biology and overfittingremains a major concern even when using TCMC. Inthe example considered in this paper, time series data isonly available for one crucial chemical species. As aconsequence, it is natural to expect that the dynamicsproduced by the TCMC model overfit the available datato a certain extent. Thus, when more experimental databecome available, TCMC must be rerun in order to pro-duce a better calibration of the model, with a new costfunction that includes the additional data.In light of these concerns, some may argue that col-

lective fitting of model parameters is not the correctapproach, and that a reductionist approach is moreappropriate (i.e., attempting to measure parameters indi-vidually and then combine the results to determine anappropriate model calibration). Unfortunately, thereductionist approach is time consuming, extremelyexpensive, and, in most cases, impossible with currentexperimental techniques. Moreover, incorrect imple-mentation of a reductionist approach may lead to athermodynamically infeasible model calibration. This isclearly the case with the Schoeberl model (and mostprobably with other models published in the literature).TCMC is a collective fitting procedure, but offers a prag-

matic compromise to the reductionist approach. In light ofthe fact that some parameters may be measured individu-ally with extreme precision, TCMC allows for these para-meters to be fixed to their measured values using matrixA. In addition, a more advanced Bayesian cost functioncan be used (e.g., see [11]) to factor in prior experimentalknowledge when parameters have been previously esti-mated with less precise experimental techniques.

Computational advantagesAccording to the ‘curse of dimensionality,’ which refersto an exponential increase in the volume of the para-meter space as its dimension grows, estimation becomessubstantially harder in high dimensional spaces. A‘naïve’ search of that space, in an effort to find the ‘true’parameter values of a biochemical reaction system[assuming that these values minimize the cost functiongiven by (12)], is hopeless. As a matter of fact, the prob-ability of obtaining parameter values that satisfy theWegscheider conditions and other underlying log-linearconstraints by uniformly sampling the entire parameterspace (which is an ‘easier’ problem than finding the‘true’ parameter values) is zero. As a consequence, theconstraints must be explicitly considered by the

optimization problem at hand to have any hope of suc-cessfully solving the problem of model calibration.As a matter of fact, since the feasible manifold is of

lower dimension than the entire parameter space, meth-ods that do not consider the underlying thermodynamicand non-thermodynamic constraints will spend mosttime searching the immense infeasible portions of theparameter space. The imposition of constraints amongthe kinetic parameters of a biochemical reaction systemreduces the dimensionality of the parameter space to asmaller feasible region and make parameter estimationcomputationally easier. TCMC makes this explicit, byperforming optimization over a lower dimensionalspace, spanned by the lower dimensional vectors v,instead of the entire parameter space, spanned by thehigher dimensional vectors �.

ConclusionsFor a biochemical reaction system to be physically realiz-able, it is required that the underlying kinetic parameterssatisfy certain thermodynamic constraints, known asWegscheider conditions. This issue has been largelyignored in the literature, as evidenced by the fact thatmany published models violate these constraints. Themodel calibration method we have proposed in thispaper can be effectively used to determine thermodyna-mically consistent values for the kinetic parameters ofany set of reactions in an ideal homogeneous mixture atconstant temperature and volume. Our method is simpleto understand and implement. Moreover, it can be easilyincorporated into any existing or newly proposed calibra-tion technique in order to make sure that the resultingmodel satisfies the fundamental laws of thermodynamicsas well as other desirable conditions and constraints.There are two major issues associated with calibrating

biochemical reaction systems:

1. The quality and quantity of available data areinadequate to allow sufficient estimation of allunderlying parameter values.2. Biochemical reaction models contain many para-meters whose numbers dramatically increase withmodel size and detail. As a consequence, the curseof dimensionality seriously hampers estimationalgorithms.

The first issue is primarily associated with current lim-itations of experimental methods and approaches. Toaddress this issue, we need substantial improvements inexperimental equipment and methodologies. However,TCMC scales well with future improvements in dataquality and quantity. Matrix A can handle arbitrary para-meter measurements (equilibrium constants, Haldanerelations, direct kinetic parameter measurements, etc.).

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 11 of 13

Moreover, TCMC can employ any cost function ofchoice, so additional concentration data can be incorpo-rated seamlessly therein.The second issue is the largest obstacle facing model

calibration techniques. To reduce dimensionality, wemust attempt to exploit mathematical structure particu-lar to the biological problem at hand. TCMC attemptsto address this problem in two ways:

• First, TCMC uses the fact that there are funda-mental physical principles underlying biochemicalreaction systems that may constrain the set of possi-ble kinetic parameter values. As a consequence ofthe fundamental laws of thermodynamics, mostcomplex biochemical networks contain reactioncycles that constrain the kinetic parameters accord-ing to (9). These constraints allow TCMC to reducedimensionality by restricting the estimation problemon a smaller thermodynamically feasible subset ofthe parameter space.• Second, experimental data and mathematical analy-sis can often provide other forms of constraints onthe underlying parameters (e.g., through directlymeasuring rate or equilibrium constants, by deter-mining Haldane relationships between enzymaticparameters, etc.). In particular, sensitivity analysismay reveal non-influential kinetic parameters thatcan be set to some nominal values without appreci-ably affecting system behavior. All these additionalconstraints can be accounted for by the A matrix in(10), further reducing the dimensionality of theensuing parameter estimation problem.

As we mentioned before, dimensionality reduction ismade explicit by TCMC, since optimization takes placeover a smaller dimensional vector v, instead of thehigher dimensional vector � specified by the model. Asa consequence, TCMC does not entirely rely on findingthe globally optimal parameter values that best fit avail-able dynamic data of molecular concentration. Imposingthermodynamic (and other log-linear) constraints allowsTCMC to restrict its search for appropriate parametervalues over a smaller subspace of the entire parameterspace in order to reach a compromise between optimaldata fit and biophysical feasibility. It has been recentlypointed out in the literature that this approach to para-meter estimation should be considered as an importantpart of determining the parameter values of complexbiological models [26].Recently, a method has been proposed in the literature

for inferring a complete and consistent set of kineticparameter values from incomplete and inconsistent data

[27]. This method, known as ‘parameter balancing,’employs a Bayesian estimation approach based entirelyon published data pertaining the values of the underly-ing kinetic parameters. Although parameter balancingcan be used to provide thermodynamically consistentvalues for the kinetic parameters of a biochemical reac-tion system, the method does not include quantitativedynamic measurements of molecular concentrations. Asa consequence, parameter balancing may result in athermodynamically feasible biochemical reaction modelthat does not adequately predict experimental observa-tions of dynamic system behavior. Future research mayfocus on combining TCMC with parameter balancing toutilize published parameter sets as well as dynamicexperimental data.A problem that we have not addressed in this paper is

the influence of ions, such as K+, and Ca2+, and certainenvironmental factors, such as the temperature and pH,on the thermodynamic behavior of a biochemical reac-tion system [28]. Our objective in this paper is not toaddress biochemical reaction systems with this level ofcomplexity, but to focus on the widely reported simplermodels of cellular function that consider only interac-tions among biochemical reactants in a fixed environ-ment. Note, however, that temperature and pHdependence of parameters has been accounted for in[27] via parameter balancing using log-linear equationsbetween biochemical parameters. Since arbitrary log-lin-ear constraints between parameters can be enforced byTCMC via (10), we suspect that TCMC can be useddirectly or appropriately modified to handle additionalbiochemical complexities that have not been addressedin this paper.Another problem that we have not addressed here is

constructing new biochemical reaction models of cellu-lar function. Since, in this paper, we only address themodel calibration problem, we take (1) as given andproceed to determine the parameter vector k from data.In general, determining the structure (i.e., the stoichio-metry) of a biochemical reaction network is an extre-mely laborious task. Preliminary work indicates thatthermodynamics can also play a key role in estimatingthe structural complexity of biochemical reaction sys-tems [29]. Future scientific investigations are necessaryto further examine this open problem.

Additional material

Additional file 1: In this document, we provide supplementarymathematical and computational details required to fullyunderstand the material presented in the Main Text.

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 12 of 13

AcknowledgementsThis research was supported in part by DoD, High Performance ComputingModernization Program, National Defense Science and Engineering Graduate(NDSEG) Fellowship, 32 CFR 168a, and in part by the National ScienceFoundation (NSF), GRANT CCF-0849907.

Authors’ contributionsGJ developed the general methodology and coded a substantial portion ofthe software. JG derived many theoretical results and ideas and wrotesubstantial portions of the final document. GJ and JG both interpreted theobtained computational results and approved the final version of the paper.

Received: 7 February 2011 Accepted: 6 May 2011 Published: 6 May 2011

References1. Alberty RA: Principles of detailed balance in kinetics. J Chem Educ 2004,

81(8):1206-1209.2. Colquhoun D, Dowsland KA, Beato M, Plested AJR: How to impose

microscopic reversibility in complex reaction mechanisms. Biophys J2004, 86(6):3510-3518.

3. Liebermeister W, Klipp E: Bringing metabolic networks to life:convenience rate law and thermodynamic constraints. Theor Biol MedModel 2006, 3:41.

4. Yang J, Bruno WJ, Hlavacek WS, Pearson JE: On imposing detailed balancein complex reaction mechanisms. Biophys J 2006, 91(3):1136-1141.

5. Ederer M, Gilles ED: Thermodynamically feasible kinetic models ofreaction networks. Biophys J 2007, 92(6):1846-1857.

6. Vlad MO, Ross J: Thermodynamically based constraints for ratecoefficients of large biochemical networks. WIREs Syst Biol Med 2009,1:348-358.

7. Crampin EJ, Schnell S, McSharry PE: Mathematical and computationaltechniques to deduce complex biochemical reaction mechanisms. ProgBiophys Mol Bio 2004, 86:77-112.

8. Maria G: A review of algorithms and trends in kinetic modelidentification for chemical and biochemical systems. Chem Biochem EngQ 2004, 18(3):195-222.

9. Chou IC, Voit EO: Recent developments in parameter estimation andstructure identification of biochemical and genomic systems. Math Biosci2009, 219(2):57-83.

10. Mendes P, Kell DB: Non-linear optimization of biochemical pathways:applications to metabolic engineering and parameter estimation.Bioinfomatics 1998, 14(10):869-883.

11. Jenkinson G, Zhong X, Goutsias J: Thermodynamically consistent Bayesiananalysis of closed biochemical reaction systems. BMC Bioinformatics 2010,11:547.

12. Jenkinson WG, Goutsias J: On constructing thermodynamically consistentparametrizations of kinetic models. Proc 10th IEEE Int Conf Bioinf Bioeng,IEEE Computer Society 2010, 214-219.

13. Schoeberl B, Eichler-Jonsson C, Gilles ED, Müller G: Computationalmodeling of the dynamics of the MAP kinase cascade activated bysurface and internalized EGF receptors. Nat Biotechnol 2002, 20:370-375.

14. Bertalanffy L: The theorey of open systems in physics and biology.Science 1950, 111:23-29.

15. Qian H, Beard DA: Thermodynamics of stoichiometric biochemicalnetworks in living systems far from equilibrium. Biophys Chem 2005,114(2-3):213-220.

16. Heinrich R, Schuster S: The Regulation of Cellular Systems New York:Chapman & Hall; 1996.

17. Beard DA, Qian H: Chemical Biophysics. Qnatitative Analysis of CellularSystems New York: Cambridge University Press; 2008.

18. Zhang HX, Dempsey WP, Goutsias J: Probabilistic sensitivity analysis ofbiochemical reaction systems. J Chem Phys 2009, 131:094101.

19. Jenkinson WG, Goutsias J: A screening method for dimensionalityreduction in biochemical reaction system calibration. Proc IEEE IntWorkshop on Genomic Signal Processing and Statistics Cold Spring Harbor,New York; 2010.

20. Gutenkunst RN, Waterfall JJ, Casey FP, Brown KS, Myers CR, Sethna JP:Universally sloppy parameter sensitivities in systems biology models.PLoS Comput Biol 2007, 3(10):1871-1878.

21. Kirkpatrick S, Gelatt CD, Vecchi MP: Optimization by simulated annealing.Science 1983, 220(4598):671-680.

22. Li C, Donizelli M, Rodriguez N, Dharuri H, Endler L, Chelliah V, Li L, He E,Henry A, Stefan MI, Snoep JL, Hucka M, Le Novère N, Laibe C: BioModelsDatabase: An enhanced, curated and annotated resource for publishedquantitative kinetic models. BMC Syst Biol 2010, 4:92.

23. Coleman TF, Li Y: A reflective Newton method for minimizing a quadraticfunction subject to bounds on some of the variables. SIAM J Optim 1996,6(4):1040-1058.

24. Asthagiri AR, Reinhart CA, Horwitz AF, Lauffenburger DA: The role oftransient ERK2 signals in fibronectin- and insulin-mediated DNAsynthesis. J Cell Sci 2000, 113(4):4499-4510.

25. Traverse S, Seedorf K, Paterson H, Marshall CJ, Cohen P, Ullrich A: EGFtriggers neuronal differentiation of PC12 cells that overexpress the EGFreceptor. Curr Biol 1994, 4(8):694-701.

26. Slezak DF, Suárez C, Cecchi GA, Marshall G, Stolovitzky G: When theoptimal is not the best: Parameter estimation in complex biologicalmodels. PLoS One 2010, 5(10):e13283..

27. Lubitz T, Schulz M, Klipp E, Liebermeister W: Parameter balancing inkinetic models of cell metabolism. J Phys Chem B 2010, 114:16298-16303.

28. Vinnakota KC, Wu F, Kushmerick MJ, Beard DA: In Methods in Enzymology.Volume 454. San Diego: Elsevier; 2009:(chap 2):29-68.

29. Kümmel A, Panke S, Heinemann M: Systematic assignment ofthermodynamic constraints in metabolic network models. BMCBioinformatics 2006, 7:512.

doi:10.1186/1752-0509-5-64Cite this article as: Jenkinson and Goutsias: Thermodynamicallyconsistent model calibration in chemical kinetics. BMC Systems Biology2011 5:64.

Submit your next manuscript to BioMed Centraland take full advantage of:

• Convenient online submission

• Thorough peer review

• No space constraints or color figure charges

• Immediate publication on acceptance

• Inclusion in PubMed, CAS, Scopus and Google Scholar

• Research which is freely available for redistribution

Submit your manuscript at www.biomedcentral.com/submit

Jenkinson and Goutsias BMC Systems Biology 2011, 5:64http://www.biomedcentral.com/1752-0509/5/64

Page 13 of 13