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Theoretical Biology and MedicalModelling

Open AccesResearch

Optimization of biotechnological systems through geometricprogramming

Alberto Marin-Sanguino*1

, Eberhard O Voit2

, Carlos Gonzalez-Alcon3

andNestor V Torres1

Address: 1Grupo de Tecnologia Bioqumica. Departamento de Bioquimica y Biologia Molecular, Facultad de Biologia, Universidad de La Laguna,38206 La Laguna, Tenerife, Islas Canarias, Spain, 2The Wallace H. Coulter Department of Biomedical Engineering at Georgia Institute ofTechnology and Emory University, 313 Ferst Drive, Atlanta, GA, 30332, USA and 3Grupo de Tecnologia Bioquimica. Departamento de EstadisticaInvestigacion Operativa y Computacion, Facultad de Fisica y Matematicas, Universidad de La Laguna, 38206 La Laguna, Tenerife, Islas Canarias,Spain

Email: Alberto Marin-Sanguino* - [email protected]; Eberhard O Voit - [email protected]; Carlos Gonzalez-Alcon - [email protected];Nestor V Torres - [email protected]

* Corresponding author

Abstract

Background: In the past, tasks of model based yield optimization in metabolic engineering were

either approached with stoichiometric models or with structured nonlinear models such as S-

systems or linear-logarithmic representations. These models stand out among most others,because they allow the optimization task to be converted into a linear program, for which efficient

solution methods are widely available. For pathway models not in one of these formats, an Indirect

Optimization Method (IOM) was developed where the original model is sequentially represented

as an S-system model, optimized in this format with linear programming methods, reinterpreted in

the initial model form, and further optimized as necessary.

Results: A new method is proposed for this task. We show here that the model format of a

Generalized Mass Action (GMA) system may be optimized very efficiently with techniques ofgeometric programming. We briefly review the basics of GMA systems and of geometric

programming, demonstrate how the latter may be applied to the former, and illustrate the

combined method with a didactic problem and two examples based on models of real systems. The

first is a relatively small yet representative model of the anaerobic fermentation pathway in S.cerevisiae, while the second describes the dynamics of the tryptophan operon in E. coli. Both models

have previously been used for benchmarking purposes, thus facilitating comparisons with the

proposed new method. In these comparisons, the geometric programming method was found to

be equal or better than the earlier methods in terms of successful identification of optima and

efficiency.

Conclusion: GMA systems are of importance, because they contain stoichiometric, mass action

and S-systems as special cases, along with many other models. Furthermore, it was previously

shown that algebraic equivalence transformations of variables are sufficient to convert virtually any

types of dynamical models into the GMA form. Thus, efficient methods for optimizing GMA

systems have multifold appeal.

Published: 26 September 2007

Theoretical Biology and Medical Modelling2007, 4:38 doi:10.1186/1742-4682-4-38

Received: 27 May 2007Accepted: 26 September 2007

This article is available from: http://www.tbiomed.com/content/4/1/38

2007 Marin-Sanguino et al; licensee BioMed Central Ltd.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Theoretical Biology and Medical Modelling2007, 4:38 http://www.tbiomed.com/content/4/1/38


BackgroundModel based optimization of biotechnological processesis a key step towards the establishment of rational strate-gies for yield improvement, be it through genetic engi-neering, refined setting of operating conditions or both.

As such, it is a key element in the rapidly emerging field ofmetabolic engineering [1,2]. Optimization tasks involv-ing living organisms are notoriously difficult, becausethey almost always involve large numbers of variables,representing biological components that dominate celloperation, and must account for multitudinous and com-plex nonlinear interactions among them [3]. The steadyincrease in the ready availability of computing power hassomewhat alleviated the challenge, but it has also,together with other technological breakthroughs, beenraising the level of expectation. Specifically, modelers aremore and more expected to account for complex biologi-cal details and to include variables of diverse types and

origins (metabolites, RNA, proteins...). This trend is to bewelcomed, because it promises improved model predic-tions, yet it easily compensates for the computer techno-logical advances and often overwhelms availablehardware and software methods. As a remedy, effort hasbeen expanded to develop computationally efficient algo-rithms that scale well with the growing number of varia-bles in typical optimization tasks.

The most straightforward attempts toward improved effi-ciency have been based, in one form or another, on thereduction of the originally nonlinear task to linearity,because linear optimization tasks are rather easily solved,

even if they involve thousands of variables. One variant ofthis approach is the optimization of stoichiometric fluxdistribution models [4]. The two great advantages of thismethod are that the models are linear and that minimalinformation is needed to implement them, namely fluxrates, and potentially numerical values characterizingmetabolic or physico-chemical constraints. The signifi-cant disadvantage is that no regulation can be consideredin these models.

An alternative is the use of S-system models within themodeling framework of Biochemical Systems Theory [5-7]. These models are highly nonlinear, thus allowing suit-

able representations of regulatory features, but have linearsteady-state equations, so that optimization under steady-state conditions again becomes a matter of linear pro-gramming [8]. The disadvantages here are that muchmore (kinetic) information is needed to set up numericalmodels and that S-systems are based on approximationsthat are not always accepted as valid. Linear-logarithmicmodels [9] similarly have the advantage of linearity atsteady state and the disadvantage of being a local approx-imation.

An extension of these linear approaches is the IndirectOptimization Method [10]. In this method, any type ofkinetic model is locally represented as an S-system. This S-system is optimized with linear methods, and the result-ing optimized parameter settings are translated back into

the original model. If necessary, this linearized optimiza-tion may be executed in sequential steps.

An alternative to using S-system models is the GeneralMass Action (GMA) representation within BST. GMA sys-tems are very interesting for several reasons. First, theycontain both stoichiometric and S-system models asdirect special cases, which would allow the optimizationof combinations of the two. Second, mass action systemsare special cases of GMA models, so that, in some sense,Michaelis-Menten functions and other kinetic rate lawsare special cases, if they are expressed in their elemental,non-approximated form. Third, it was shown that virtu-

ally any system of differential equations may be repre-sented exactly as a GMA system, upon equivalencetransformations of some of the functions in the originalsystem. Thus, GMA systems, as a mathematical represen-tation, are capable of capturing any differentiable nonlin-earity that one might encounter in biological systems. Weshow here that GMA systems, while highly nonlinear, arestructured enough to permit the application of efficientoptimization methods based on geometric programming.

Formulation of the optimization task

Pertinent optimization problems in metabolic engineer-ing can be stated as the targeted manipulation of a system

in the following way:

maxormin f0(X) (1)

subject to:

opearation in steady state (2)

metabolic and physico-chemical constraints (3)

cell viability (4)

In this generic representation, (1) usually targets a flux or

a yield. The optimization must occur under several con-straints. The first set (2) ensures that the system will oper-ate under steady-state conditions. Other constraints (3)are imposed to retain the system within a physically andchemically feasible state and so that the total protein ormetabolite levels do not impede cell growth. Yet otherconstraints (4) guarantee that no metabolites are depletedbelow minimal required levels or accumulate to toxic con-centrations. These sets of constraints are designed to allowsustained operation of the system.
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Biochemical Systems Theory (BST)

Biological processes are usually modeled as systems of dif-ferential equations in which the variation in metabolitesXis represented as:

The elements ni,j of the stoichiometric matrixNare con-stant. The vectorvcontains reaction rates, which are ingeneral functions of the variables and parameters of thesystem. This structure is usually associated with metabolicsystems, but it is similarly valid for models describinggene expression, bioreactors, and a wide variety of otherprocesses in biotechnology. In typical stoichiometric anal-

yses, the reaction rates are considered constant. Further-more, the analysis is restricted to steady-state operation,

with the consequence that (5) is set equal to 0 and thereby

becomes a set of linear algebraic equations, which areamenable to a huge repertoire of analyses.

In analyses accounting for regulation, the reaction ratesbecome functions that depend on system variables andoutside influences. Even at steady state, these may be verycomplex, thereby rendering direct analysis of the system aformidable task [11]. As a remedy, BST suggests to repre-sent these rate functions with power laws:

In analogy with chemical kinetics, i is called the rate con-stant and fi,j are kinetic orders, which may be any realnumbers. Positive kinetic orders indicate augmentation,

whereas negative values are indicative of inhibition.Kinetic orders of 0 result in automatic removal of the cor-responding variable from the term. In the notation of BST,the firstnvariables are often considered the dependent var-iables, which change dynamically under the action of thesystem, while the remaining variablesXi fori = n + 1 ... m+ n are considered independent variables and typicallyremain constant throughout any given simulation study.

Thus, metabolites, enzymes, membrane potentials orother system components can easily be made dependent

or independent by the modeler without requiring altera-tions in the structure of the equations. BST is very compactand explicitly distinguishes variables from parameters.

Because we will later introduce concepts of geometric pro-gramming, it is noted that the power-law term in Eq. 6 isalso called a monomial. If this monomial is an approxima-tion of reaction rate V, its parameters can be directlyrelated to V, by virtue of the fact that the monomial is infact a Taylor linearization in logarithmic space [12]. Thus,choosing an operating point with index 0, one obtains:

Thus, it follows directly from 7 that the parameters of a

power-law (monomial) term can be computed as

System equations in BST may be designed in slightly dif-ferent ways. For the GMA form, each reaction is repre-

sented by its own monomial, and the result is therefore

Note that this is actually a spelled-out version of Eq. 5,

where the reaction rates are monomials as in Eq. 6. As an

alternative to the GMA format, one may, for each depend-

ent variable, collect all incoming reactions in one term

and do the same with all outgoing fluxes, which are

collectively called . These aggregated terms are nowrepresented as monomials, and the result is

Thus, there are at most one positive and one negative termin each S-system equation.

The conversion of a GMA into an S-system will becomeimportant later. It is achieved by collecting the aggregatedfluxes into vectors

whereN+ andN- are matrices containing respectively thepositive and negative coefficients ofNsuch thatN=N+ -N-. With these definitions, we can derive the matrices ofkinetic orders of S-systems from those of the correspond-ing GMA representation. Namely,

ddt

NX v= (5)

v Xi i jf

j

n mi j=

=

+

,1

(6)

ln lnln

lnln ln

ln

lnln lnv V

V

XX X

V

XX Xi

m n

m n m= +

( ) + +

++0

1 0

1 1 00

++( )n 0

(7)

ii

j

f

j

n m

v

X i j=

=+

0

10

,(8)

fv

X

v

X

X

vi j

i

j

i

j

j

i,

ln

ln=

=

0 0

(9)

dX

dtn X i ni i j j

j

p

k

f

k

n mj k= =

= =

+

, , ...1 1

1 (10)

Vi+

Vi

dX

dtV V X X i i i i j

g

j

n m

i j

h

j

n mi j i j= = +

=

+

=

+

, ,1 1

(11)

V v

V v

+ +

=

=

N

N(12)
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whereV,V+ andV- are square matrices of zeros having the

corresponding vectors as their main diagonals. G and Hcontain the kinetic orders of the S-system while F containsthose of the GMA [13]. GMA systems may be constructedin three manners [11]. First, given a pathway diagram,each reaction rate is represented by a monomial, andequations are assembled from all reaction rates involved.Second, it is possible (though not often actually done) todissect enzyme catalyzed reactions into their underlyingmass action kinetics, without evoking the typical quasi-steady-state assumption. The result is directly the specialcase of a GMA system where most kinetic orders are zero,one, or in some cases 2. Third, it has been shown that vir-tually any nonlinearity can be represented equivalently as

a GMA system [14]. As an example for this recastingtech-nique, consider a simple equation where production anddegradation are formulated as traditional Michaelis-Menten rate laws:

whereX0 is a dependent or independent variable describ-ing the substrate for the generation ofX1. To effect thetransformation into a GMA equation, define auxiliary var-iables asX2 = KM,2 +X1 andX3 = KM,1 +X0. The equationthen becomes

For simplicity of discussion, suppose thatX0 is a constant,independent variable. Thus,X3 is also constant and doesnot need its own equation. By contrast, X2 is a newdependent variable and from its definition we can calcu-late its initial value and see that its derivative must beequal to that ofX1.Therefore the equations:

form a system that is an exact equivalent of the originalsystem but in GMA format.

Recasting can be useful with equations that are difficult tohandle otherwise or for purposes of streamlining a model

structure and its analysis. One must note though thatoften the number of variables increases significantly. Inthe case shown, the number of equations rises from oneto two ifX0 is independent or to three if it is a dependent

variable.

Current optimization methods based on BST

The overall task is to reset some of the independent varia-bles so that some objective is optimized. The independent

variables in question are typically enzyme activities, which are experimentally manipulated through geneticmeans, such as the application of customized promotersor plasmids. The objective is usually the maximization ofa metabolite concentration or a flux. Three approacheshave been proposed in the literature.

Pure S-systems

Among a number of convenient properties, the steady

states of an S-system can be computed analytically bysolving a system of algebraic linear equation [6]. EquatingEq. 11 to zero and rearranging one obtains:

which is a monomial of the form

Monomial equations become linear by taking logarithmson both sides thus reducing the steady-state computationto a linear task:

Ay= b (19)

where

Ai,j =gi,j - hi,j

yi = InXi

Monomial objective functions become linear by takinglogarithms and so holds for many constraints on metabo-lites or fluxes. Therefore, constrained optimization ofpathways modeled as S-systems becomes a straightfor-

ward linear program [8].

Any other relevant constraint or objective function that isnot a power law can also be approximated using the

G V N F

H V N F

=

=

+ +

( )

( )

1

1

V

V(13)

dX

dt

V X

K X

V X

K X

max

M

max

M

1 1 0

1 0

2 1

2 1

=+

+

,

,

,

,

(14)

dX

dtV X X V X X max max

11 0 3

12 1 2

1= , , (15)

dX

dt

V X X V X X

dX

dtV X X V X

max max

max max

11 0 3

12 1 2

1

21 0 3

12 1

=

=

, ,

, , XX

X t X

X t K X M

21

1 0 10

2 0 2 10

=

= +

( )

( ) ,

(16)

i j

g

j

n

i j

h

j

n

X

X

i j

i j

,

,

=

=

=1

1

1 (17)

i

i

j

g h

j

n

X i j i j, , .

=

=1

1 (18)

bii

i

= ln
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abovementioned methods. Then logarithms can be takenand Eqns 14 can be rewritten as:

maxormin F(y)

Subject to:

Ay= b (20)

By= d (21)

Cye (22)

yLyyU (23)

Where Fis the logarithm of the flux or variable to be opti-mized, and superscripts L and Urefer to lower and upperbounds. Eq. 20 assures operation at steady state. MatrixB

and vectord account for additional equality constraintsand C and e are analogous constraints for additional ine-qualities, which could, for instance, limit the magnitudeof a metabolite concentration or flux, and improve thechances of viability. Optimization problems of this typeare called linear programs (LPs) and can be solved very effi-ciently for large numbers of variables and constraints [15].

The advantage of the pure S-system approach is its greatspeed combined with the fact that S-system models haveproven to be excellent representations of many pathways.

The disadvantage is that the optimization process, bydesign, moves the system away from the chosen operating

point, so that questions arise as to how accurate the S-sys-tem representation is at the steady state suggested by theoptimization.

Indirect Optimization Method

If the pathway is not modeled as an S-system, the reduc-tion of the optimization task to linearity is jeopardized. Acompromise solution that has turned out to be quite effec-tive is the Indirect Optimization Method (IOM) [10]. Thefirst step of IOM is approximation of the alleged model

with an S-system. This S-system is optimized as shownabove. The solution is then translated back into the origi-nal system in order to confirm that it constitutes a stable

steady state and is really an improvement from the basalstate of the original model. The S-system solution typi-cally differs somewhat from a direct optimization result

with the original model, but since it is obtained so fast, itis possible to execute IOM in several steps with relativelytight bounds, every time choosing a new operating pointand not deviating too much from this point in the nextiteration [16]. The speed of the process is slower than inthe pure S-system case, but still reasonable. Variations onIOM are to search for subsets of independent variables to

be manipulated for optimal yield at lower cost and formulti-objective optimization tasks [17,18].

Global GMA optimization

A global optimization method for GMA systems [19] has

been recently proposed based on branch-and-reducemethods combined with convexification. These methodsare interesting because of the variety of roles that GMAmodels can play (see above). The disadvantage of the glo-bal method is that it quickly leads to very large systemsthat are non-convex, even though they allow relativelyefficient solutions.

Geometric programming

Geometric programming (GP) [20] addresses a class ofproblems that include linear programming (LP) and othertasks within the broader category of convex optimizationproblems. Convex problems are among the few nonlinear

tasks where, thanks to powerful interior point methods,the efficient determination of global optima is feasibleeven for large scale systems. For example, a geometric pro-gram of 1,000 variables and 10,000 constraints can besolved in less than a minute on a desktop computer [21];the solution is even faster for sparse problems as they arefound in metabolic engineering. Furthermore, easy to usesolvers are starting to become available [22,23].

GP addresses optimization programs where the objectivefunction and the constraints are sums of monomials, i.e.,power-law terms as shown in Eq. 6. Because of theirimportance in GP, sums of monomials, all with positive

sign, are called posynomials. If some of the monomialsenter the sum with negative signs, the collection is calledasignomial. The peculiarities of convexity and GP methodsrender the difference between posynomials and signomi-als crucial.

A GP problem has the generic form:

min P0(x) (24)

Subject to:

Pi(x) 1 i = 1...n (25)

Mi(x) = 1 i = 1...p (26)

where Pi(x) andMi(x) must fulfill strict conditions. EveryfunctionMi(x) must be a monomial, while the objectivefunction P0(x) and the functions Pi(x) involved in ine-qualities must be posynomials. Signomials are not per-mitted, and optimization problems involving themrequire additional effort.
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The equivalence between monomials and power lawsimmediately suggests the potential use of GP for optimi-zation problems formulated within BST. In the next sec-tions, several methods will be proposed to develop suchpotential.

Results and discussionIt is easy to see that steady-state equations of S-systems arereadily arranged as monomials as shown in Eq 18 andthat optimization tasks for S-systems directly adhere tothe format of a GP, except that GP mandates minimiza-tion. However, this is easily remedied for maximizationtasks by minimizing the inverse of the objective, whichagain is a monomial. By contrast, steady-state GMA equa-tions as shown in Eq. 10 do not automatically fall withinthe GP structure, because GMA systems usually includenegative terms, thus making them signomials. Further-more, inversion of an objective that contains more than

one monomial is not equivalent to a monomial.

When the objective or some restriction falls outside theGMA formalism, it can be recast into proper form as hasbeen discussed above and will be shown in one of the casestudies.

Two strategies

The proposed solutions for adapting GP solvers to treat

GMA systems rely on condensation [24], but they do it in

different ways. Condensation is a standard procedure in

GP which is exactly equivalent to aggregation in BST.

Namely, the sum of monomials is approximated by a sin-

gle monomial. In the terminology of GP, the condensa-

tion is generically denoted as

and, in the terminology of Eqs. 10 and 11, defined as:

where i

andgi,j

are chosen such that equality holds at achosen operating point; thus, the result is equivalent tothe Taylor linearization that is fundamental in BST as wasshown in eqn. 7 [5,7,12]. As in the Taylor series, the con-densed form is equal to the original equation at the oper-ating point. For any other point, as it can be shown thatthe left and right hand side of eqn. 29 are equivalent tothose of the Arithmetic-Geometric inequality:

and therefore, the condensed form is an understimationof the original.

Objective functions can only be minimized in GP, this isseldom a problem given that the functions to maximize

are often monomials that can be inverted: a variable, areaction rate or a flux ratio. Posynomial objectives are usu-ally entitled for minimization, like the sum of certain var-iables. Nonetheless, it is also relevant in metabolicengineering to consider the maximization of posynomi-als, such as the sum of variables or fluxes. In such cases,condensation or recasting can be used. For en extensiveintroduction on GP modelling see [25].

A local approach: Controlled Error Method

The steady-state equation of a GMA system may be writtenas the single difference of two posynomials:

P(x) - Q(x) = 0 (31)

If both posynomials are condensed, every equation willbe reduced to the standard form for monomial equations:

Because the division of a monomial by another is itself amonomial.

Since the steady state equations of the GMA have beencondensed to those of an s-system, this method could be

regarded as a direct generalization of classical IOM meth-ods. One of the advantages of this approach is the possi-bility of keeping posynomial inequalities and objectivesas they are and therefore reduce the amount of condensa-tion (approximation) needed, but there is another inter-esting possibility. When a posynomial is approximated bycondensation, the A-G inequality, Eq. 30, guarantees thatthe monomial is an underestimation of the constraint.Furthermore, the posynomial structure is not altered

when divided by a monomial so the quotient between aposynomial and its condensed form is always greater thanor equal to 1 and provides the exact error as a posynomialfunction. Therefore the problem can be constrained to

allow a maximum error per condensed constraint:

So the original problem is solved as a series of GPs in which the GMA equations are successively condensedusing the previous solution as the reference point. Toassure validity an extra set of constraints is added to

()C

( ( ) ( ( ) ( )) ( )C P C M M Mnx x x x = + + =1 0 (28)

,

, ,C n X X i j jj

k

k

f

k

n m

i j

g

j

n j k i j

= =

+

=

=1 1 1

(29)

aa

wi

i

i

w

i

n

i

n i

==

11

(30)

( ( ))( ( ))

C P

C Q

x

x= 1 (32)

j k

b

kj

j k

b

kj

X

C X

j k

j k

,

,

( ) +1 (33)
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ensure that every iteration will only explore the neighbor-hood of the feasible region in which error due to conden-sation remains below an arbitrary tolerance set by theuser.

A global approach: Penalty TreatmentA similar yet distinct strategy that minimizes the use ofcondensation is an extension of the penalty treatmentmethod [26], a classic algorithm for signomial program-ming. In this method, a signomial constraint such as

P(x) - Q(x) = 0 (34)

where P and Q are posynomials, is replaced by two posyn-omial equalities through the creation of an ancilliary var-iable t:

These are not valid GP constraints, so the followingrelaxed version is used:

Upon dividing byt, the feasible area of the original prob-lem is contained in the feasible area of the new relaxed

version and aproximation by condensation is not needed.In order to force these inequalities to be tight in the finalsolution, the objective function is augmented with pen-

alty terms that grow with the slackness of the constraints,namely the inverses of the condensation of the relaxedconstraints. The result of this procedure is a legal GP:

Where the condensed terms are calculated at the basalsteady state. If the obtained solution falls within the feasi-ble area of the original problem, it is taken as a solution,if it does not (any of the relaxed inequalities is below 1,the solution is used as the next reference point: condensa-tions are calculated again, the weights of the violated con-straints are increased and the new problem is solved. Thisprocedure is repeated until a satisfactory solution isobtained. The original method used 1 as the initial valueof the weights and increased them all in every iteration,some modifications are useful for our purposes:

The initial weights are selected such that the overall pen-alty terms are just a fraction of the total objective in theinitial point. In the case studies explored in this paper,such fraction was 10%.

The weights are only increased if their correspondingconstraint was violated in the last iteration. In such cases,the weight would be multiplied times a fixed value. Forthe case studies considered here, the choice in the value ofsuch multiplier didn't have a significant impact in the per-formance of the method.

These variations on the original method serve to preventthe penalty terms from dominating the objective functionand pushing the relaxed problem towards the boundariesof the feasible region from the very beginning.

Case studies

In order to illustrate the combination of GP with BST,some optimization tasks were explored. The first exampledemonstrates the procedure with a very simple two varia-ble GMA system. The second example is a model of theanaerobic fermentation pathway in Saccharomyces cerevi-siae. The third example revisits an earlier case study con-cerned with the tryptophan operon in E. coli. Thesesystems were optimized using the Matlab based solverggplab [23] running on an ordinary laptop (1.6 GHz Pen-tium centrino, 512 Mb RAM). Matlab scripts were writtenin order to perform all the transformations required bythe two methods described. For comparison, the models

were also optimized using IOM [10] as well as Matlab's

optimization toolbox. The function used in this toolbox,fmincon(), is based on an iterative algorithm calledSequential Quadratic Programming, which uses the BGFSformula to update the estimated Hessian matrix duringevery iteration [27,28].

A seemingly simple problem

A very distinctive difference between the alternative meth-odsfor GMA optimization can be ilustrated by a problemmodified from [24], which presents the simplest possiblefragmented feasible region (see Fig. 1).

P t

Q t

( )

( )

x

x

=

= (35)

P t

Q t

( )

( )

x

x

(36)

min ( )[ ( )] [ ( )]

:

P wt

C Pw

t

C Q

P

i

i

i

i

i

0 xx x

+ +

+ subject to

(( )

( )

x

x

t

Q

ti n

i

=

1

1 0

(37) min

:

X

X X X X

X X

1

1 2 12

22

12

22

1

4

1

2

1

16

1

161 0

1

14

1

141

subject to

+ =

+ + =

3

7

3

70

1 5 5

1 5 5

1 2

1

2

X X

X

X

.

.

(38)
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The feasible region of this problem consists of two points(1.178,2.178) and (3.823,4.823), of which clearly the firstsolution is superior, because X1 is to be minimized. Asthese points are not connected, local methods are not ableto find one solution using the other as a starting point.

The problem was solved using IOM, controlled error andpenalty treatment methods. The initial point was set to be

(3.823,4.823), which is disconnected from the true opti-mal solution. While both IOM and the Controlled-Errormethod reported the initial point as the solution, the pen-alty treatment algorithm found the global optimum at(1.178,2.178).

In this case, most methods failed to find the optimal solu-tion because the approximated s-system had the operatingpoint as the only feasible solution while the relaxed prob-lem for the penalty treatment algorithm had a feasiblearea (shadowed in Fig. 1) that included and connectedboth feasible solutions.

Anaerobic fermentation in S. cerevisiaeThis GMA model [29] (see also appendix) is derived froma previous version [30] formulated with traditionalMichaelis Mentem kinetics to explain experimental data,and has been used to illustrate other optimization meth-ods [10,17,19]. It has the following structure (see Fig. 2):

The model was already formulated [29] as a GMA system,so that all its fluxes are monomials:

X v v

X v v v

X v v v

X v

in HK

HK PFK POL

PFK GAPD GOL

GA

1

2

3

4

1

2

2

= = =

=

= PPD PK GAPD PK HK PFK POL ATP

v

X v v v v v v

= + 5 2

(39)

Anaerobic fermentation in S. cerevisiaeFigure 2Anaerobic fermentation in S. cerevisiae.

Feasible area of the first exampleFigure 1Feasible area of the first example. The lines show the

nullclines of each of the two equations of the system. Theyintersect at two (unconnected) points, which constitute theonly feasible solutions. The feasible area of the relaxed prob-lem in the penalty treatment is marked in grey.
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The objective is (constrained) maximization of the etha-nol production rate, vPK. Together with the upper andlower bounds of the variables, two extra constraints will

be studied. The first is an upper limit to the total amountof protein. This is especially important for pathways of thecentral carbon metabolism as they represent a significantfraction of the total amount of cell protein and increasingthe expression of its enzymes by large amounts mightcompromise cell viability. As a first example, we assumethat the activity to protein ratio is the same for everyenzyme and set an arbitrary limit of four times theamount of enzymes in the basal state. As an alternative,

we explore the effect of limiting the total substrate pool.This constraint will later be subject to tradeoff analysis inorder to see its influence in the optimum steady state (seeFig3). Being posynomial functions, the constraints will be

supported by GP without any transformation. The Appen-dix contains a complete formulation of the optimizationproblem.

The results are sumarized in Table 1. Both GP methodsand the SQP found the same solution, although GP fin-ished in 0.5 s while SQP was significantly slower, taking1.5 s for the calculation. The IOM method was as fast asGP but it's solution violated one constraint.

Tryptophan operon

The third example addresses the tryptophan operon in E.coli, as illustrated in Fig. 4. This is an appealing benchmark

system, because it has already been optimized with othermethods [16,31].

A model of the system was recently presented by [32] andincludes transcription, translation, chemical reactions andtryptophan consumption for growth. It is thus more thana simple pathway model and demonstrates that GP andBST are applicable in more complex contexts. Finally, thismodel doesn't follow the structure of any standard for-malism so it will be a good example on how recasting wid-

ens the applicability of the method to a higher degree ofgenerality. The model takes the form

HereX1,X2 andX3 are dimensionless quantities represent-ing mRNA, enzyme levels and the tryptophan concentra-

tion, respectively. The rate equations are:

v X X

v X X X

v

in

H K

PFK

=

=

=

0 8122

2 8632

0 52

20 2344

6

10 7464

50 0243

7

.

.

.

.

. .

332

0 011

20 7318

50 3941

8

30 6159

40 1308

9 140 6

X X X

v X X X XGAPD

. .

. . ..

= 0088

30 05

40 533

50 0822

10

28 6107

0 0945

0 0009

v X X X X

v X

PK

PO L

=

=

.

.

. . .

.XX

v X X X X

v X X

G O L

ATP

11

30 05

40 533

50 0822

12

5 13

0 0945==

. . . .

(40)

X v v

X v v

X v v v v

1 1 2

2 3 4

3 5 6 7 8

= = =

(41)

Tradeoff curve for the anaerobic fermentation pathway if thetotal substrate pools are kept fixedFigure 3Tradeoff curve for the anaerobic fermentation pathway if thetotal substrate pools are kept fixed. No upper limit for totalenzyme was used in this case.

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Flux

Substrates Pool (times basal)

Table 1: Optimization results for the GMA glycolitic model in S.

cerevisiae. Constraint violations are shown in boldface. GP

column stands for both methods

variable basal IOM GP & SQP

(times basal)

X1 0.03456 2.1946 2.0000

X2 1.0110 1.5801 2.0000

X3 9.1876 1.5294 2.0000

X4 0.009532 1.1936 2.0000

X5 1.1278 0.2803 0.5000

X6 19.7 7.4873 7.3343

X7 68.5 3.8583 3.7794

X8 31.7 2.9176 2.8577

X9 49.9 6.4799 4.7179

X10 3440 5.7195 4.1642

X11 14.31 0.0100 0.0100

X12 203 0.0100 0.0100

X13 25.1 27.0452 14.0396

X14 0.042 1.0000 1.0000

Flux 30.2231 214.6250 198.8542
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The GMA format is obtained by defining the followingancillary variables:

which turns the rates into power laws:

The objective function consists simply ofv8, which may beregarded as an aggregate term for growth and tryptophanexcretion.

A recurrent feature of previously found IOM solutions was

the noticeable violation of a constraint retaining a mini-mum tryptophan concentration. This discrepancy is a fea-ture for comparisons between methods beyondcomputational efficiency. The Appendix contains a com-plete formulation of the optimization problem.

In order to test the effectiveness of the controlled errorapproach, two variants were used in this model:

Fixed tolerance. The standard method in which everyiteration is limited to a maximum condensation error of10% by constraints described in Eq. 33.

Fixed step. No limit on the condensation error. The var-iation of the variables in every iteration is limited to 10%distance from the reference state.

When the constraints were absent (fixed step), the varia-tion of the variables was restricted to a fraction of the totalrange in every iteration, in order to prevent them frommoving too far from the operating point. Fig. 5 shows theevolution of the objective function and condensationerrors through iterations, both for fixed step and fixed tol-erance. Though both methods find the same solution, thefixed tolerance method is much faster and keeps the error

within a limit specified a priori. The fixed step method

remains within a lower margin of error in this case due tothe good quality of the condensed approximation but thismargin is not under direct control and will depend on thesize of the subintervals and on the model in an unforesee-able way. When the error tolerance was lowered to matchthe values observed for the fixed step method, both per-formed very similarly with a slight advantage of the fixedtolerance.

Both the controlled error and penalty treatment methods yielded the same results while SQP returned a solution

v XX X

v X X

v X

v X X

vX X

X

1 35 3

2 4 1

3 1

4 4 2

52 6

2

11 1

0 9

0 02

= ++ += +== +

=

( )

( . )

( . )

662

32

6 3 4

73 5

3

8 4 4 3 7

3

0 0022

1

1 7 50 005

+=

=+

= +

X

v X X

vX X

X

v X X X X X

.

( . ).

(42)

X X

X X

X X X

X X

X X

X X X

X

8 5

9 3

10 8 3

11 4

12 4

13 62 32

1

1

1

1

0 9

0 02

= += += += += +

= +

.

.

44 3

15 4

0 005

1 7 5

= +=

X

X X

.

.

(43)

v X X

v X X

v X

v X X

v X X X

v X X

v

1 9 101

2 11 1

3 1

4 12 2

5 2 62

131

6 3 4

7 0 00

===

===

=

. 222 3 5 91

8 15 3 7 141

X X X

v X X X X

=

(44)

A model of the tryptophan operonFigure 4A model of the tryptophan operon. Adapted from [32].
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that was feasible but yielded a lower flux. As can be seenin Table 2 no constraint violations occurred with GP.

When the lower bound was extended to include the levelsreached by other methods, all previous results were repro-duced. The tradeoff curve resulting from solving the prob-lem for different tryptophan lower bounds is depicted asFig6. SQP and error controlled method took about 1 s tofind the solution while the penalty tratment took 0.3 s.

ConclusionThe main challenge of non-linear optimization is dealingwith non-convexities. In some cases, like GP, there is anelegant transformation that convexifies the problem with-out adding undue complexity. But this is seldom the caseand dealing with non-convexities usually implies devel-opingad hoc tricks such as subdividng the system in manysubsystems, finding convex relaxations of the constraints,

adding extra variables or a combination of several of thesestrategies.

Geometric programming provides a simple and efficienttool for the optimization of biotechnological systems thattakes advantage of the structural regularity and flexibilityof GMA systems. In this work we have presented two dif-ferent strategies to do so, of which the penalty treatmentseems to be the most promising. The methods are quite

general, as this treatment of GP and recasting can beapplied to any rational function, which in fact includealmost all rate functions used in representations of meta-bolic processes.

The use of geometric programming also provides a solu-tion for the problem of constraint violations in the twostrategies considered. The possibility of keeping an arbi-

Tradeoff analysis for tryptophan model showing flux againstlower bound for tryptophanFigure 6

Tradeoff analysis for tryptophan model showing flux againstlower bound for tryptophan.

200 400 600 800 1000 1200 14003.5

4

4.5

5

5.5

6

6.5

Flux

Min Trp

Table 2: Comparison of results obtained for the tryptophan model with different methods. All the results that violate the lower bound

forX3were reproduced with GP by relaxing such bound. Constraint violations are shown in boldface.

iterative Modified

basal IOM IOM IOM GP SQP

X1 0.18465 1.198 |X1|0 1.198 |X1|0 1.198 |X1|0 1.199 |X1|0 1.2 |X1|0X2 7.9868 1.071 |X2|0 1.095 |X2|0 1.055 |X2|0 1.148 |X2|0 1.180 |X2|0X3 1418 0.347 |X3|0 0.465 |X3|0 0.273 |X3|0 0.8 |X3|0 0.825 |X3|0X4 0.00312 0.0058 0.0053 0.062 0.00414 0.0035

X5 5 4 4 4 4 4

X6 2283 5000 5000 5000 5000 2384

X7 430 1000 1000 1000 1000 1000

Vtrp 1.310 4.26 |Vtrp|0 3.884 |Vtrp|0 4.54 |Vtrp|0 3.062 |Vtrp|0 2.61 |Vtrp|0

Effect of the error constraints in the optimization algorithmFigure 5Effect of the error constraints in the optimization algorithm.

Results of optimizing the model of the tryptophan operonusing fixed step and fixed tolerance.

0 5 10 15 201

2

3

4

5Fixed step

Flux

0 5 10 15 201

2

3

4

5Controlled Error

Flux

5 10 150

0.002

0.004

0.006

0.008

0.01

%E

rror

Iteration5 10 15

0

0.02

0.04

0.06

0.08

%E

rror

Iteration
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trarily small approximation error in every iteration pre-vents the buildup of discrepancies in the Controlled ErrorMethod which results in a "safer" condensation while thePenalty treatment doesn't rely on condensation to definethe feasible area. It has been shown elsewhere [21] that

GP can deal with big systems, and the sparse nature of theproblems in metabolic engineering improves the capabil-ities of the approach. It is therefore reasonable to expectboth strategies considered here to scale well for big prob-lems but it is yet to be seen which one of the two behavesbetter in such cases.

Geometric programming is a relatively recent and activearea in operations research, which implies that furtherimprovements and refinements for the optimization ofGMA systems are to be expected. But even with existingmethods, the optimization of this large class of systems,

which is further expanded by the technique of recasting,has become feasible for execution of moderately sizedtasks even on simple desktop computers.

A Optimization problems

Table 3: A.1 Anaerobic fermentation by error controlled method

min

Subjectto:

Steadystate

Errortolerances

( . ). . .0 0945 30 05

40 533

50 0822

101 X X X X

0 8122

2 86321

2 8632

20 2344

6

10 7464

50 0243

7

10 7464

.

.

.

.

. .

.

X X

X X X

X X

=

550 0243

7

20 7318

50 3941

8 28 6107

110 5232 0 0009

.

. . .( . . )

X

C X X X X X +

= 11

0 5232

0 011

20 7318

50 3941

8

3

0 6159

5

0 1308

9 14

0

.

( .

. .

. .

X X X

C X X X X

.. . . .

.

. )

.

6088

3

0 05

4

0 533

5

0 0822

12

30 6159

1

20 0945

1

2 0 011+

=

X X X X

X XX X

X X X X

C X

50 1308

9

30 05

40 533

50 0822

10

30

0 09451

2 0 011

.

. . .

.

.

( .

=

66159 50 1308

9 30 05

40 533

50 0822

100 0945

2 8632

X X X X X X

C X

. . . .. )

( .

+

110 7464

50 0243

7 20 7318

50 3941

8 28 6100 5232 0 0009. . . . .. .X X X X X X+ + 77 11 5 13

1X X X+

=)

0 5232 0 0009

0 5232

20 7318

50 3941

8 28 6107

11

20 73

. .

( .

. . .

.

X X X X X

C X

+118

50 3941

8 28 6107

11

30 6159 50 1308

0 00091

0 011

X X X X

X X

+ +

. .

. .

. )

.

XX X X X X X

C X

9 140 6088 30 05 40 533 50 0822 12

30

12

0 0945

0 011

+. . . ..

( . .. . . . . ..6159 50 1308

9 140 6088

30 05

40 533

50 08221

20 0945X X X X X X X + 112

30 6159

50 1308

9 30 05

40 533

50

1

2 0 011 0 0945

)

. .. . . . .

+

+

X X X X X X00822

10

30 6159

50 1308

9 30 05

40 533

52 0 011 0 0945

X

C X X X X X X( . .. . . . +

+

+

0 082210

10 7464

50 0243

7 20 7318

5

1

2 8632 0 5232

.

. . .

)

. .

X

X X X X X

00 39418 2

8 610711 5 13

10 7464

50 0243

0 0009

2 8632

. .

. .

.

( .

X X X X X

C X X

+ +

XX X X X X X X X7 20 7318

50 3941

8 28 6107

11 5 130 5232 0 00091

+ + + +. . ). . .
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13/16



Table 4: A.2 Anaerobic fermentation by penalty treatment

min

Subject to:

Steady state

( . )

.

. . .

.

0 0945

2 8632

30 05

40 533

50 0822

101

11

10 7464

50

X X X X

wt

X X

++..

. . .( . .

02437

11

20 7318

50 3941

8 28 61070 5232 0 0009

X

wt

C X X X X X

++

111

22

20 7318

50 3941

8

22

30 6159

0 5232

0 011

)

.

( .

. .

.

+

+

+

wt

X X X

wt

C X X550 1308

9 140 6088

1 30 05

40 533

50 0822

121

20 0945. . . . .. )X X w X X X X + +

++ +

+w

t

C X X X X X X3

3

30 6159

50 1308

9 30 05

40 533

52 0 011 0 0945( . .. . . . 00 0822 10

3 310 7464

50 0243

7 20 7312 8632 0 5232

.

. . .

)

( . .

X

w

t

C X X X X+ + 88 50 3941

8 28 6107

11 5 130 0009X X X X X X + +. .. )

0 8122

2 86321

2 8632

20 2344

6

10 7464

50 0243

7

10 7464

.

.

.

.

. .

.

X X

X X X

X X

=

550 0243

7

1

20 7318

50 3941

8 28 6107

11

1

0 5232 0 0009

.

. . .. .

X

t

X X X X X

t

+

11

20 7318

50 3941

8

2

30 6159

50 1308

9 1

1

0 52321

0 011

.

.

. .

. .

X X X

t

X X X X 440 6088

30 05

40 533

50 0822

12

2

30

1

20 0945

1

2 0 011

+

. . . .

.

.

.

X X X X

t

X66159

50 1308

9

30 05

40 533

50 0822

10

30

0 09451

2 0 011

X X

X X X X

X

.

. . ..

.

=

.. . . . ..

.

615950 1308

9 30 05

40 533

50 0822

10

3

0 09451

2 8632

X X X X X X

t

+

XX X X X X X X

1

0 7464

5

0 0243

7 2

0 7318

5

0 3941

8 2

8 610 5232 0 0009. . . . .. .+ +

007

11 5 133

1X X X

t +


14/16



Table 5: A.3 Tryptophan by error controlled method

min

Subject to:

Steady state

Ancilliary variables

Error tolerances

( )X X X X 15 3 7 141 1

X X

X X

X

X X

X X X

C x x X X X X

9 101

11 1

12 2

2 62

131

3 5 91

15

1

1

3 4 0 0022

=

=

+ +( . X X X X3 7 141

1

=)

(( ) )

(( ) )

(( ) )

(( .

C x X

C X X

C X x X

C

1 5 1

1 1

1 3 1

0 9

81

3 91

8 101

+ =

+ =+ =

++ =

+ =

+ =

X X

C X X

C X X X

C X

4 111

4 121

62

32

131

1

0 02 1

1

) )

(( . ) )

(( ) )

(( 33 141

4 151

0 005 1

1 7 5 1

+ =

=

. ) )

(( . ) )

X

C X X

x x X X X X X X X

C x x X X X X

3 4 0 00223 4 0 0022

3 5 91

15 3 7 141

3 5 91

15

+ ++ +

.

( . XX X X

x

C x

X

C X

X x

C

3 7 141

3

3

8

1

1 5

1 51

1

11

1 3

1

+

+

+ +

+

+ +

+

+

)

( )

( )

(

XX x

X

C X

X

C X

X X

8

4

4

4

4

62

3

31

0 9

0 91

0 020 02

1

)

.

( . )

.( . )

+

+

+ +

++

+

+

22

62

32

3

3

4

4

1

0 005

0 0051

1 7 5

1 7 5

( )

.

( . )

.

( .

C X X

X

C X

X

C X

+ +

+

+ +

)) +1


15/16



Competing interests

The author(s) declare that they have no competing inter-ests.

AcknowledgementsThis work was supported by a research grant from the Spanish Ministry of

Science and Education ref. BIO2005-08898-C02-02.

References1. Stephanopoulos G, Aristidou A, Nielsen J:Metabolic Engineering: Prin-

ciples and Methodologies Academic Press; 1998.2. Torres N, Voit E: Pathway Analysis and Optimization in Metabolic Engi-

neeringCambridge University Press; 2002.

3. Mendes P, Kell D: Non-linear optimization of biochemical

pathways: applications to metabolic engineering and param-eter estimation. Bioinformatics 1998, 14(10):869-83.

4. Varma A, Boesch BW, Palsson BO: Metabolic flux balancing:Basic concepts, scientific and practical use. Bio-Technology1994:994-998.

5. Savageau M: Biochemical systems analysis. I. Some mathemat-ical properties of the rate law for the component enzymaticreactions.J Theor Biol1969, 25(3):365-9.

6. Savageau M: Biochemical systems analysis. II. The steady-statesolutions for an n-pool system using a power-law approxima-tion.J Theor Biol1969, 25(3):370-9.

7. Voit E: Computational Analysis of Biochemical Systems. A Practical Guidefor Biochemists and Molecular Biologists Cambridge University Press;2000.

Table 6: A.4 Tryptophan penalty approach

min

Subject to:

Steady state

Ancilliary variables

( )

( .

X X X X

wt

X X X

wt

C x x X X X

15 3 7 1 41 1

11

2 6

2

13

1 11

3 5 93 4 0 0022

+

+ +

+

++

++

++

+

1

15 3 7 1 4

1

28

39

34

10

81 5 1 1 3

X X X X

wX

C xw

X

C Xw

X

C X x

)

( ) ( ) ( ))

( . ) ( . ) ( )+

++

++

+

+

wX

C Xw

X

C Xw

X

C X X

wX

511

46

12

47

13

62

32

84

0 9 0 02

CC Xw

X

C X( . ) ( . )39

15

40 005 1 7 5++

X X

X X

XX X

X X X

t

x x X X X X

9 101

11 1

1

12 2

2 62

131

1

3 5 91

1

1

1

3 4 0 0022

=

=

+ +. 115 3 7 141

1

1X X X

t

( )

( )

( )

( . )

( .

1 5 1

1 1

1 3 1

0 9 1

0 0

81

3 91

8 101

4 111

+

+

+ +

x X

X X

X x X

X X

22 1

1

0 005 1

1 7 5

4 121

62

32

131

3 141

4 15

+

+

+

X X

X X X

X X

X X

)

( )

( . )

( . ) 1 1
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