## Contents lists available at SciVerse ScienceDirect Journal of Theoretical Biology

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### Abstract

journal homepage: www.elsevier.com/locate/yjtbi A variational principle for computing nonequilibrium fluxes and potentials in genome-scale biochemical networks

### Citations

3706 | Convex Optimization - Boyd, Vandenberghe - 2004 |

3280 | Variational Analysis
- ROCKAFELLAR, WETS
- 1998
(Show Context)
Citation Context ...ponding set of thermodynamically feasible fluxes and potentials. More precisely, when S is row reduced (to have full row rank), problem (EP) represents a singlevalued surjective saddle point mapping (=-=Rockafellar and Wets, 1997-=-) between a parameter vector (c,b) and the corresponding primal-dual optimal vector ðv % f ,v% r ,y%Þ. That is, each parameter vector corresponds to a unique primal-dual optimal vector, and every opti... |

98 |
Network Flows and Monotropic Optimization
- Rockafellar
- 1984
(Show Context)
Citation Context ...r metabolic networks (Holzhütter, 2004). A comprehensive introduction to convex optimization is given in Boyd and Vandenberghe (2004). Problem (EP) is an example of a monotropic optimization problem (=-=Rockafellar, 1984-=-). Further discussion of such problems (and duality) can be found in Bertsekas (1999). 6.1. Strong duality and Tellegen’s theorem Since problem (EP) is convex with a finite attainable optimum objectiv... |

49 |
Systems Biology: Properties of Reconstructed Networks
- Palsson
- 2006
(Show Context)
Citation Context ...nding to the reverse reactions. The fluxes for these one-way reactions form the vectors vf and vr. The net flux is then v vf vr. Flux balance analysis has been implemented using linear programming (=-=Palsson, 2006-=-). We take the flux balance analysis problem to be maximize v f ,vr,ve d T ve subject to Svf Svr þSeve 0, vf ,vr Z0, ‘rve rh: ðFBAÞs74 R.M.T. Fleming et al. / Journal of Theoretical Biology 292 (201... |

22 |
The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks
- Holzhutter
- 2004
(Show Context)
Citation Context ...ium constants are used as weightingR.M.T. Fleming et al. / Journal of Theoretical Biology 292 (2012) 71–77 75 factors, has previously been explored as an optimality principle for metabolic networks (=-=Holzhütter, 2004-=-). A comprehensive introduction to convex optimization is given in Boyd and Vandenberghe (2004). Problem (EP) is an example of a monotropic optimization problem (Rockafellar, 1984). Further discussion... |

21 | Energy balance for analysis of complex metabolic networks
- Beard, Liang, et al.
- 2002
(Show Context)
Citation Context ...irchhoff’s current law in an electrical network. Recent work has sought to augment flux balance analysis with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (=-=Beard et al., 2002-=-; Price et al., 2006; Yang et al., 2005; Nagrath et al., 2007; Fleming et al., 2010; Schellenberger et al., 2011). The incorporation of thermodynamic constraints into genomescale models has produced m... |

19 |
M (2006) Putative regulatory sites unraveled by network-embedded thermodynamic analysis of metabolome data
- Kummel, Panke, et al.
(Show Context)
Citation Context ...o genomescale models has produced models that are biologically more realistic and reveal greater insight into the control mechanisms operating in these complex biological systems (Beard et al., 2002; =-=Kummel et al., 2006-=-; Xu et al., 2008; Garg et al., 2010; Liebermeister et al., 2010). See Soh and Hatzimanikatis (2010) for a recent broad review of the application of thermodynamic constraints to biochemical networks. ... |

16 | Treatise on thermodynamics - Planck - 1927 |

16 |
Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium, Biophys
- Qian, Beard
(Show Context)
Citation Context ...hemical potential is defined by u 2ry %. Up to scalar multiplication, b T y ð1=2rÞb T u corresponds to the rate of work being done by the environment to maintain the system away from equilibrium (=-=Qian and Beard, 2005-=-). Therefore, one may interpret the Lagrange dual problem as a minimization of this rate of work, balanced against minimization of the sum of thermodynamically feasible forward and reverse fluxes. We ... |

16 |
Network analysis of intermediary metabolism using linear optimization. II. Interpretation of hybridoma cell metabolism
- Savinell, Palsson
- 1992
(Show Context)
Citation Context ...ly by a network of chemicals (nodes) and reactions (edges). To analyze these networks at genome scale, systems biologists often use a linear optimization technique called flux balance analysis (FBA) (=-=Savinell and Palsson, 1992-=-). Flux balance requires that the sum of fluxes into and out of each node in the network be zero. This is equivalent to Kirchhoff’s current law in an electrical network. Recent work has sought to augm... |

15 |
Formulating genome-scale kinetic models in the post-genome era. Molecular Systems Biology, 4:171, 2008. SH Lam and DA Goussis. The CSP method for simplifying kinetics
- Jamshidi, Palsson
- 1994
(Show Context)
Citation Context ...left in the model. We see this property of the model as an advantage rather than a disadvantage, as thermodynamic feasibility is necessary but not sufficient for satisfaction of mass action kinetics (=-=Jamshidi and Palsson, 2008-=-). In the objective of problem (EP), one could replace cTðvf þvrÞ by cT f vf þcT r vr. Assuming the existence of a mass action kinetic non-equilibrium steady state, Section 7 demonstrates that this st... |

15 |
What Is Flux Balance Analysis
- Orth, Thiele, et al.
- 2010
(Show Context)
Citation Context ...thousand. Flux balance analysis (FBA) computes a set of fluxes that satisfy steady state mass-conservation constraints and are optimal for a biological objective function (Savinell and Palsson, 1992; =-=Orth et al., 2010-=-). A flux is a reaction rate; it represents flow through the network and is analogous to current in an electrical circuit. We denote the net flux of the jth reaction by the variable vj AR. The concent... |

14 | A computational study of the homogeneous algorithm for large-scale convex optimization
- Andersen, Ye
- 1998
(Show Context)
Citation Context ...ticular: Problem (EP) is convex and includes only linear constraints. It is feasible whenever problem (FBA) is feasible, and its optimal solution is then unique. Efficient polynomial-time algorithms (=-=Andersen and Ye, 1998-=-) exist for solving convex optimization problems of this form, based on interior methods (Ye, 1997). For both (FBA) and (EP), these algorithms are guaranteed to return an optimal solution, or a certif... |

12 |
Probability Theory—The Logic of Science
- Jaynes
- 2003
(Show Context)
Citation Context ... between flux and change in potential, for each reaction. From an information theoretic perspective, by maximizing the entropy of the internal fluxes, problem (EP) gives the most unbiased prediction (=-=Jaynes, 2003-=-) of internal elementary fluxes, subject to mass-conservation constraints and boundary conditions imposed by exchange fluxes. The next theorem proves that if there is a set of thermodynamically feasib... |

12 | A Treatise on Electricity and Magnetism Vol - Maxwell |

10 | ED: Thermodynamically feasible kinetic models of reaction networks
- Ederer, Gilles
(Show Context)
Citation Context ... potentials. For an elementary reaction, one would expect that forward flux should depend only on substrate chemical potential(s) and reverse flux should depend only on product chemical potential(s) (=-=Ederer and Gilles, 2007-=-). According to mass action kinetics, the rate of an elementary forward reaction only depends on a forward kinetic parameter and substrate concentration(s) (Cornish-Bowden, 2004). In the objective of ... |

10 |
Candidate states of Helicobacter pylori’s genome-scale metabolic network upon application of ‘‘loop law’’ thermodynamic constraints
- Price, Thiele, et al.
- 2006
(Show Context)
Citation Context ...aw in an electrical network. Recent work has sought to augment flux balance analysis with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; =-=Price et al., 2006-=-; Yang et al., 2005; Nagrath et al., 2007; Fleming et al., 2010; Schellenberger et al., 2011). The incorporation of thermodynamic constraints into genomescale models has produced models that are biolo... |

10 |
A protocol for generating a high-quality genome-scale metabolic reconstruction
- Thiele, Palsson
- 2010
(Show Context)
Citation Context ...hat there exists a feasible solution ðv,veÞ for (1). Ensuring the existence of a steady state flux is part of a quality control process during reconstruction of S and Se from experimental literature (=-=Thiele and Palsson, 2010-=-). Any reversible reaction Aþ2B"C may be split into two oneway reactions: forward (Aþ2B-C) and reverse (Aþ2B’C). To distinguish between forward, reverse, and exchange reactions we split the augmented ... |

9 |
Thermodynamicsbased metabolic flux analysis
- Henry, Broadbelt, et al.
- 2007
(Show Context)
Citation Context ... nonconvex continuous optimization problem (Beard et al., 2002; Nagrath et al., 2007; Fleming et al., 2010), solving an NP-hard problem (Yang et al., 2005), or solving a mixed integer linear program (=-=Henry et al., 2007-=-; Schellenberger et al., 2011). Mixed integer programs have unpredictable computational complexity. The purpose of this work is to show that Kirchhoff’s loop law and the second law of thermodynamics a... |

8 |
DA: Ab initio prediction of thermodynamically feasible reaction directions from biochemical network stoichiometry. Metab Eng 2005
- Yang, Qian, et al.
(Show Context)
Citation Context ...network. Recent work has sought to augment flux balance analysis with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; Price et al., 2006; =-=Yang et al., 2005-=-; Nagrath et al., 2007; Fleming et al., 2010; Schellenberger et al., 2011). The incorporation of thermodynamic constraints into genomescale models has produced models that are biologically more realis... |

7 |
Thermodynamics of biochemical reactions
- Alberty
- 2003
(Show Context)
Citation Context ...antify transformed reaction Gibbs energy (in vivo change in chemical potential) using quantitative measurement of absolute concentrations (Bennett et al., 2009), combined with experimentally derived (=-=Alberty, 2003-=-, 2006) or group contribution estimates (Henry et al., 2006, 2007; Jankowski et al., 2008; Finley et al., 2009) of standard transformed reaction Gibbs energy. However, obtaining in vivo chemical poten... |

7 |
Phosphorylation energy hypothesis: Open chemical systems and their biological functions
- Qian
(Show Context)
Citation Context ...ze or degrade that chemical. So far we have considered net flux for a stoichiometric matrix of reactions, each of which conserves mass. A living biochemical system operates in a nonequilibrium state (=-=Qian, 2007-=-) and exchanges mass with its surroundings. This can be modeled by including exchange reactions that do not conserve mass. The model is augmented with a matrix Se AR m k and a corresponding set of exc... |

6 |
Integrated energy and flux balance based multiobjective framework for large-scale metabolic networks
- Nagrath, Avila-Elchiver, et al.
(Show Context)
Citation Context ...k has sought to augment flux balance analysis with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; Price et al., 2006; Yang et al., 2005; =-=Nagrath et al., 2007-=-; Fleming et al., 2010; Schellenberger et al., 2011). The incorporation of thermodynamic constraints into genomescale models has produced models that are biologically more realistic and reveal greater... |

6 | Thermodynamics and fluctuations far from equilibrium. Springer-Verlag Berlin Heidelberg 2008 - Ross |

6 |
Quantitative prediction of cellular metabolism with constraint-based models
- Schellenberger, Que, et al.
- 2011
(Show Context)
Citation Context ...s with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; Price et al., 2006; Yang et al., 2005; Nagrath et al., 2007; Fleming et al., 2010; =-=Schellenberger et al., 2011-=-). The incorporation of thermodynamic constraints into genomescale models has produced models that are biologically more realistic and reveal greater insight into the control mechanisms operating in t... |

5 | Absolute metabolite concentrations and implied enzyme active site occupancy in Escherichia coli - Bennett - 2009 |

5 |
Nonlinear Networks IIa
- Duffin
- 1947
(Show Context)
Citation Context ...ictly) monotonically increasing function of change in chemical potential, then one obtains flux and potential vectors satisfying Kirchhoff’s laws with a single (strictly) convex optimization problem (=-=Duffin, 1947-=-). The challenge is to pose an optimization problem with optimality conditions that enforce a constitutive relationship between flux and potential matching established theory in chemical kinetics. 7. ... |

5 |
Some general Theorems for Nonlinear Systems Possesing Resistance
- Millar
- 1951
(Show Context)
Citation Context ...lem (QP) and a variational principle in operation. We believe there must also be a variational principle in operation in biochemical networks. In an abstract analysis of nonlinear resistive networks (=-=Millar, 1951-=-), one would refer to the objective in (EP) as the resistive content, as it is the sum of the areas under the constitutive relationships between flux and change in potential, for each reaction. From a... |

4 | Enzymes in context: kinetic characterization of enzymes for systems biology. The Biochemist 27:11 - Cornish-Bowden, Hofmeyr - 2005 |

4 | Primal-dual solution perturbations in convex optimization
- Dontchev, Rockafellar
- 2001
(Show Context)
Citation Context ...e primal-dual optimal vector, and every optimal vector is associated with at least one parameter (c,b). Moreover, under the same conditions, this saddle point mapping is locally Lipschitz continuous (=-=Dontchev and Rockafellar, 2001-=-). That is, the optimal vector of fluxes and potentials is smoothly perturbed by a smooth perturbation of the parameter vector. The properties discussed in the previous paragraph motivate future effor... |

4 |
Genome-scale thermodynamic analysis of Escherichia coli metabolism
- Henry, Jankowski, et al.
- 2006
(Show Context)
Citation Context ...ge in chemical potential) using quantitative measurement of absolute concentrations (Bennett et al., 2009), combined with experimentally derived (Alberty, 2003, 2006) or group contribution estimates (=-=Henry et al., 2006-=-, 2007; Jankowski et al., 2008; Finley et al., 2009) of standard transformed reaction Gibbs energy. However, obtaining in vivo chemical potentials is not a limitation to validation of the practical ut... |

4 |
Modular rate laws for enzymatic reactions: thermodynamics, elasticities and implementation
- Liebermeister, Uhlendorf, et al.
(Show Context)
Citation Context ...cally more realistic and reveal greater insight into the control mechanisms operating in these complex biological systems (Beard et al., 2002; Kummel et al., 2006; Xu et al., 2008; Garg et al., 2010; =-=Liebermeister et al., 2010-=-). See Soh and Hatzimanikatis (2010) for a recent broad review of the application of thermodynamic constraints to biochemical networks. The second law of thermodynamics may be applied to n Correspondi... |

4 |
Optimization of productivity and thermodynamic performance of metabolic pathways
- Xu, Smith, et al.
- 2008
(Show Context)
Citation Context ...has produced models that are biologically more realistic and reveal greater insight into the control mechanisms operating in these complex biological systems (Beard et al., 2002; Kummel et al., 2006; =-=Xu et al., 2008-=-; Garg et al., 2010; Liebermeister et al., 2010). See Soh and Hatzimanikatis (2010) for a recent broad review of the application of thermodynamic constraints to biochemical networks. The second law of... |

3 | Biochemical thermodynamics - Alberty - 2006 |

3 |
Group contribution method for thermodynamic analysis of complex metabolic networks
- Jankowski, Broadbelt, et al.
- 2008
(Show Context)
Citation Context ...using quantitative measurement of absolute concentrations (Bennett et al., 2009), combined with experimentally derived (Alberty, 2003, 2006) or group contribution estimates (Henry et al., 2006, 2007; =-=Jankowski et al., 2008-=-; Finley et al., 2009) of standard transformed reaction Gibbs energy. However, obtaining in vivo chemical potentials is not a limitation to validation of the practical utility of problem (EP). Even if... |

3 |
Network thermodynamics
- Oster, Perelson, et al.
- 1971
(Show Context)
Citation Context ...onsistent potentials and currents for the circuit. Biochemical networks are significantly more complicated than linear resistive networks. However, some of the same underlying network concepts apply (=-=Oster et al., 1971-=-). In this work we construct an optimization problem in a form similar to problem (QP), where the potentials are Lagrange multipliers for an equality constraint on the flow variables, and the optimali... |

3 | Network thermodynamics in the postgenomic Era - Soh, Hatzimanikatis - 2010 |

2 |
Existence of positive equilibria for mass conserving and mass-action chemical reaction networks with a singleterminal-linkage class. , (in preparation
- Akle, Dalal, et al.
- 2011
(Show Context)
Citation Context ...ing steady-state mass action kinetics, namely S expðlogðk f ÞþF T logðxÞÞ S expðlogðkrÞþR T logðxÞÞ b: ð16Þ Ongoing work aims to establish the necessary and sufficient conditions for this to be so (=-=Akle et al., 2011-=-). Nonetheless, assuming (16) is satisfiable, there exist cf and cr such that cf logðkf ÞþR T y % , cr logðkrÞþF T y % and optimality conditions (10)–(11) become logðv % f Þlogðkf Þ F T y % , log... |

2 |
Thermodynamic analysis of biodegradation pathways
- Finley, Broadbelt, et al.
- 2009
(Show Context)
Citation Context ...rement of absolute concentrations (Bennett et al., 2009), combined with experimentally derived (Alberty, 2003, 2006) or group contribution estimates (Henry et al., 2006, 2007; Jankowski et al., 2008; =-=Finley et al., 2009-=-) of standard transformed reaction Gibbs energy. However, obtaining in vivo chemical potentials is not a limitation to validation of the practical utility of problem (EP). Even if one could find free ... |

2 |
von Bertalanffy 1.0: a COBRA toolbox extension to thermodynamically constrain metabolic models
- Fleming, Thiele
- 2011
(Show Context)
Citation Context ...ce analysis as additional linear inequality constraints (Fleming et al., 2009). Software packages to quantitatively assign reaction directionality for genome-scale models of metabolism are available (=-=Fleming and Thiele, 2011-=-; Schellenberger et al., 2011). In contrast to the addition of constraints arising from the second law of thermodynamics, the addition of energy conservation constraints has been problematic because t... |

2 |
Quantitative assignment of reaction directionality in constraint-based models of metabolism: application to Escherichia coli
- Fleming, Thiele, et al.
- 2009
(Show Context)
Citation Context ...ection of a negative change in chemical potential (Burton et al., 1957). Constraints on net reaction flux can be incorporated within flux balance analysis as additional linear inequality constraints (=-=Fleming et al., 2009-=-). Software packages to quantitatively assign reaction directionality for genome-scale models of metabolism are available (Fleming and Thiele, 2011; Schellenberger et al., 2011). In contrast to the ad... |

2 |
Integrated stoichiometric, thermodynamic and kinetic modeling of steady state metabolism
- Fleming, Thiele, et al.
- 2010
(Show Context)
Citation Context ...t flux balance analysis with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; Price et al., 2006; Yang et al., 2005; Nagrath et al., 2007; =-=Fleming et al., 2010-=-; Schellenberger et al., 2011). The incorporation of thermodynamic constraints into genomescale models has produced models that are biologically more realistic and reveal greater insight into the cont... |

2 |
Thermodynamic analysis of regulation in metabolic networks using constraint-based modeling
- Garg, Yang, et al.
- 2010
(Show Context)
Citation Context ...ls that are biologically more realistic and reveal greater insight into the control mechanisms operating in these complex biological systems (Beard et al., 2002; Kummel et al., 2006; Xu et al., 2008; =-=Garg et al., 2010-=-; Liebermeister et al., 2010). See Soh and Hatzimanikatis (2010) for a recent broad review of the application of thermodynamic constraints to biochemical networks. The second law of thermodynamics may... |

2 | Tellegen's Theorem and Thermodynamic Inequalities - Oster, Desoer - 1971 |

2 |
2011a. Elimination of thermodynamically infeasible loops in steady-state metabolic models
- Schellenberger, Lewis, et al.
(Show Context)
Citation Context ...s with Kirchhoff’s loop law for energy conservation as well as the second law of thermodynamics (Beard et al., 2002; Price et al., 2006; Yang et al., 2005; Nagrath et al., 2007; Fleming et al., 2010; =-=Schellenberger et al., 2011-=-). The incorporation of thermodynamic constraints into genomescale models has produced models that are biologically more realistic and reveal greater insight into the control mechanisms operating in t... |

2 |
Duality, thermodynamics, and the linear programming problem in constraint-based models of metabolism
- Warren, Jones
- 2007
(Show Context)
Citation Context ...eld equations that enforce Kirchhoff’s loop law and the second law of thermodynamics. Previous work noted an ‘‘analogy’’ between Lagrange multipliers and chemical potentials in flux balance analysis (=-=Warren and Jones, 2007-=-). The key limitation of that work, concerned with duality in linear optimization, is that the optimality conditions of a linear optimization problem cannot enforce any relationship between net flux a... |

1 |
Fundamentals of Enzyme Kinetics, third ed
- Cornish-Bowden
- 2004
(Show Context)
Citation Context ...mical potential(s) (Ederer and Gilles, 2007). According to mass action kinetics, the rate of an elementary forward reaction only depends on a forward kinetic parameter and substrate concentration(s) (=-=Cornish-Bowden, 2004-=-). In the objective of problem (EP), for simplicity of exposition, consider replacing cTðvf þvrÞ with cT f vf þcT r vr. When cf cr is constrained to lie in the range of S T , this provides another mec... |

1 |
Introduction to
- Strang
- 1986
(Show Context)
Citation Context ...Maxwell, 1873) is a variational principle that underlies this circuit. It seeks a set of currents that minimize the heat (or power) dissipated subject to KCL. This is the convex optimization problem (=-=Strang, 1986-=-) minimize FðxÞ 1 2 xT Rx b T x subject to Ax f : y, ðQPÞ where the node variables y are Lagrange multipliers for the equality constraints. The optimality conditions rFðxÞA T y yield equations that... |

1 |
Genome-scale reconstruction of E. coli’s transcriptional and translational machinery: a knowledgebase, its mathematical formulation, and its functional characterization
- Thiele, Jamshidi, et al.
- 2009
(Show Context)
Citation Context ...enome-scale modeling tool. The theorems of this paper provide the theoretical basis for efficiently computing thermodynamically feasible fluxes, even for the largest genome-scale biological networks (=-=Thiele et al., 2009-=-). In particular: Problem (EP) is convex and includes only linear constraints. It is feasible whenever problem (FBA) is feasible, and its optimal solution is then unique. Efficient polynomial-time alg... |