MPA 2013 Thermodynamic in metabolism workshop

Physical constraints and counting problems

in metabolic networks:exploiting duality

Daniele De Martino

daniele.demartino@iit.it, CLNS-IIT,Chimera group, Dipartimento di Fisica, La Sapienza

Synopsis

1) Introduction to the second law of thermodynamics:

from heat engines to metabolic networks

2) Exploiting duality:

3) Methods: Relaxation and Montecarlo

4) A different instance of the same (computational) issue:

5) Some results in E.Coli and human metabolic networks

6) Conclusions

Existence (calculation) of free energies

Infeasibility (elimination)of closed reaction loops

Identification ofconserved metabolic pools

Assessment of production profiles <=>

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An example

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A “waste” of ATP(futile cycle)Hexokinase

Glucose-6-phosphatase

PDB 3O08 PDB 1IDQ

They don't act together in vivo (allosteric regulation)

= Work!

ATP pump, PDB 2zxe

What if they work backward? Can they produce ATP?What if they work backward? Can they produce ATP?

What if they work backward? Can they produce ATP?

Infeasible cycle

“Such a creation is absolutely contrary to all accepted ideas, to the lawsof mechanics and physics: it is unjustifiable” (Carnot, 1824)

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The second law of thermodynamics:heat engines and entropy

A cyclic engine thatonly converts heat in work is infeasible (Kelvin)

reversible engineMax efficiency (Carnot)

S is a state function!(Clausius)

Isolated system

E.Fermi “Thermodynamics” (1937)D.Kondepudi & I.Prigogine “Modern Thermodynamics” (1999)

S defineddefined from the infeasibility of “perpetual motion of second kind”

Metabolic networks

Core E coli iAF1260 network, JD Orth, RMT Fleming, BØ Palsson - 2010 - EcoSal

Not isolated

A complex web of interacting

processes - O(103) for

genome-scale reconstructions of

simple bacteria

The second law of thermodynamics:Gibbs free energy and chemical reactions

Conservation of energy for reversible transformation

Isolated Constant T,P (Gibbs)

P,T constants: typical conditions of chemical systems

E.Fermi “Thermodynamics” (1937)D.Kondepudi & I.Prigogine “Modern Thermodynamics” (1999)

The second law of thermodynamics:metabolic networks and duality

Given a flux model, are the Gibbs inequalities consistent?

Gordan (1873) theorem of alternative: Gordan (1873) theorem of alternative:

1) 2)

D.Beard et al, Biophys.J. (2002)N.Price et al, Biophys. J. (2002)D.Beard & H.Qian “Chemical biophysics” (2008)Soh & HatzimanikatisCurr.op.microbio. (2010) Either 1) or 2) have real (non trivial) solutionsEither 1) or 2) have real (non trivial) solutions

Statement of the problem

Given a metabolic network stoichiometric matrixS and a flux model v (e.g. from FBA)

1) 2)

Either there is 1) a chemical potential vector or there are 2) infeasible loops Implementation of thermodynamics: given a flux model get rid of 2) and/or calculate 1) with experimental prior Computationally dual problems!Computationally dual problems!

Reaction i, metabolite μ

D.De Martino et al Plos cbio (2012))D.De Martino phys. rev. E (2013, arxiv 2012)See also E.Noor et al BMC bioinfo. (2012)

Apart from uptakes, biomass reaction...

Many works exploit the connection between flux feasibility and calculation of free energies:A.muller & A.Bockmayr BMC Bioinformatics (2013), J.Schellenberger et al, biophys.J. (2011), A.kummel et al, mol. sys. bio. (2006), C.Henry et al Biophys.J. (2007), A.Hoppe et al BMC sys. Bio. (2007)

With this theorem the connection is explicit (free energies are defineddefined if no loops)

Solve sys 1: RelaxationStarting from prior g(0), correct step by step the least unsat constraint (luc)

α const. (minOver) slower, nearer to prior

α prop. to luc(Motzkin relaxation) faster

Even minOver convergesin times sublinear with respect to the network size

Relaxation converges iffsys 1) is feasibleotherwise...sys 2) should have a solution

Key point: if relaxation doesn't convergesKey point: if relaxation doesn't convergesIt “cycles” among a subset of constraintsIt “cycles” among a subset of constraintsA.schrijver “theory of linear and integer programming” (1986)

E.Amaldi & M.Mattavelli discr.appl.math(2002)

W.Krauth & M.Mezard J.Phys. A (1987)

Solve sys 2: Monte Carlo One route: solve for the unconstrained kernel and then solve for

non-negativity constraints Alternatively: solve directly for positive integer variables

Integer minima of a quadratic function

MonteCarlo simulated annealing:1) Define a stochastic process whose equilibrium distribution is the Boltzmann2) Perform an annealing schedule with increasing β,β --> ∞ solution

a) extract i b) propose a change c) calculate energy variation d) accept it with probability that respect detailed balance

Detailed balance

Key hint: we don't have to look to all Key hint: we don't have to look to all variables: coupling with relaxationvariables: coupling with relaxationW.Krauth “Introduction to Montecarlo algorithms” (1996)

A.Sokal “Montecarlo methods: foundations and new algorithms” (1996)

Exploiting duality by coupling the methods

Legenda: μ ≡ g Ω ≡ ξ

D.Demartino, F.Capuani, M.Mori, A.De Martino & E.Marinari, submitted

The loop removal should preserve other constraints

See also Avis & Kaluzny, Am.Math.montly (2004)

Results (I): Refinement of experimental reactions free energy

1) Refine experimental data on reactions' free energy from feasible flux configurations (human red cell)

Human red cell metabolic network: Wiback & Palsson Biophys.J (2002)IAF1260: A.Feist et al, mol.sys.bio (2007)Also BiGG Database

2) Identify the loops of a given network(E coli) ----> consistency requiresthat some reversible reactions areirreversible

D.De Martino. M.Figliuzzi, A.De Martino & E.Marinari Plos cbio (2012)

3 loops in E coli metabolic network iAF1260

D.Demartino, F.Capuani, M.Mori, A.De Martino & E.Marinari, submitted

Results for iAf1260

Results (I): Identification of the loops

Problem: thermodynamically feasible solutions but infeasible conditions in vivo e.g. positive free energy of hydrolysis of ATP (detected by computing free energies) ---> human metabolic networks from Recon2 lack ATP maintenance

D.Demartino, F.Capuani, M.Mori, A.De Martino & E.Marinari, submitted

Cell specific subnetworks of Recon2: ATP hydrolysis free energy consistent with the FBA solution (biomass)

An infeasible loop that annoys FBA solutions of some Recon2 subnetworks(the ATP maintenance is added to the models)

To remove loops is not necessary to count them! Fast strategy: minimize Lp norm of fluxes compatibly with constraints (e.g. L1)(pFBA) N.Lewis et al mol.sys.bio. (2010), Holzutter EJB (2004)

Recon2: Thiele et al nat.biotech, (2013)www.humanmetabolism.org

Results (II)

Same math, different problem:Finding conserved metabolic pools

The left kernel of the stoichiometric matrix defines conservation laws

Positive integer solutions: conserved metabolic pools

S.Schuster & T Hofer J.chem.soc Far.trans. (1991), RR Vallabhajosyula et al, Bioinformatics (2006), Sauro & Ingalls biophys.chem. (2004), Famili & Palsson Biophys.J. (2003), E.Nikolaev et al, Biophys.J (2005), A.Corsnish-Bowden & J.Hofmeyr, J.theor.bio. (2002)

A.De Martino, D.De Martino, R.Mulet & A.Pagnani arxiv (2013), submitted

An exampleCompartmentalized glycolysis inTrypanosoma brucei (glycosomes)

ρ=V(cytosol)/V(glycosomes)

Useful information in order to understand: -Metabolic productive capabilities-Response to perturbations(environmental and/or genetical)-Correlations among concentration levels

Exploiting duality on genome-scale models

Bakker et al, J.bio.chem. (1999), PNAS (2000)

1

2

Duality conservation/production:The Motzkin theorem

1) 2)

Production profiles problem:Characterize the space

Is it possible to reconstruct objective functions from stoichiometry?Von Neumann constraints (problem originally formulated for economic networks)

It is possible to apply the same algorithm used for the thermodynamics

Von Neumann (1954) “A model of general economic equilibrium” (1937)B.Palsson “System biology” (2005),Imielinsky et al, Biophys J (2006),C.Martelli et al PNAS (2009)A.De Martino et al, JSTAT (2004), JSTAT (2007), EPL (2009)J.phys.conf (2010)

A.De Martino, D.De Martino, R.Mulet & A.Pagnani arxiv (2013), submitted

Results (I)

The irreducible conserved pools of the complete iAF1260 E.Coli metabolic network

They generate all the conservation lawsMainly conservation of proteins and/or t-RNAProbably some small errors: e.g. selenium compounds (no uptakes of selenium compounds)

It is possible to perform the same analysis in specific media (not shown)

A.De Martino, D.De Martino, R.Mulet & A.Pagnani arxiv (2013), submitted

The irreducible pools of iJR904 (Reed et al gen.bio (2003))

One conservation law is not a pool!5prdmbz[c] + dmbzid[c] + rdmbzi[c] − adocbi[c] − adocbip[c] − agdpcbi[c] − cbi[c].

The number of irreducible pools scales linearly with the network size(a null model of Poissonian random networks reproduces this feature)

A.De Martino, D.De Martino, R.Mulet & A.Pagnani, arxiv (2013), submitted

Results (II)

Example: a photochemical modelof the Martian atmosphere

Nair et al “A photochemical model of the martian atmosphere” Icarus (1994)

Only one intake(a photon)4 irreducible poolscorresponding to1)Oxygen, 2)Nitrogen3)Hydrogen and4)Carbon monoxide

There is a conservation lawthat is not reducible to a pool!#O(+) + #O2(+) + #CO2(+) + + #CO2 - #electrons = const.

Charge conservation!

www.science.unsw.edu.auwww.science.unsw.edu.au

Conclusions Theorems of the alternative encode the dual nature of two

computational problems for the physical modeling of metabolic networks

Implementation of thermodynamic constraints (Gordan)

Analysis of production profiles/conservation laws (Motzkin)

Some alternative methods to linear (integer-linear, quadratic) programming: relaxational algorithms and MonteCarlo methods

Existence (calculation) of free energies

Infeasibility (elimination)of closed reaction loops

Identification ofconserved metabolic pools

Assessment of production profiles <=>

ReferencesThermodynamics:Thermodynamics:

E.Fermi “Thermodynamics” (1937), D.Kondepudi & I.Prigogine “Modern Thermodynamics” (1999).

Thermodynamic constraints in metabolic networks:Thermodynamic constraints in metabolic networks:D.Beard et al, Biophys.J. (2002), N.Price et al, Biophys. J. (2002), D.Beard & H.Qian “Chemical biophysics” (2008), Soh & Hatzimanikatis Curr.op.microbio. (2010),A.muller & A.Bockmayr BMC Bioinformatics (2013), J.Schellenberger et al, biophys.J. (2011), A.kummel et al, mol. sys. bio. (2006), C.Henry et al Biophys.J. (2007), A.Hoppe et al BMC sys. Bio. (2007), E.Noor et al BMC sys.bio. (2012), Jankowsky et al, Biophys J. (2008), D.De Martino et al, Plos cbio (2012), D.De Martino PRE (2013).

Theorems & Algorithms:Theorems & Algorithms:A.Schrijver “Theory of linear and integer programming” (1986), A.Solodovnikov “Systems of linear inequalities” (1980), Avis & Kaluzny, Am.Math.montly (2004),W.Krauth & M.Mezard J.Phys. A (1987), W.Krauth “Introduction to Montecarlo algorithms” (1996)A.Sokal “Montecarlo methods: foundations and new algorithms” (1996).

Conserved pools and production in metabolic networks:Conserved pools and production in metabolic networks:S.Schuster & T Hofer, J.chem.soc Far.trans. (1991), RR Vallabhajosyula et al, Bioinformatics (2006), Sauro & Ingalls biophys.chem. (2004), Famili & Palsson Biophys.J. (2003), E.Nikolaev et al, Biophys.J (2005), A.Corsnish-Bowden & J.Hofmeyr, J.theor.bio. (2002), Bakker et al, J.bio.chem. (1999), PNAS (2000),Von Neumann (1954) “A model of general economic equilibrium” (1337) Imielinsky et al, Biophys J (2006) C.Martelli et al PNAS (2009), B.Palsson “System biology” (2005),A.De Martino et al, JSTAT (2004), JSTAT (2007), EPL (2009), J.phys.conf (2010) A.De Martino et al. Arxiv (2013).

AcknowledgementsAndrea De MartinoEnzo MarinariMatteo FigliuzziMatteo MoriFabrizio CapuaniSimona Colabrese

Andrea Pagnani

Roberto Mulet

Mayya Sundukova

Chimera groupDipartimento di FisicaLa Sapienza Rome

Politecnico di TorinoHuman Genetics Foundation, Torino

Universida de la Habana, Institut Henri Poincaré, Paris

European Molecular Biology Laboratory

Appendix ADemonstration of the Gordan theorem

(5)

(6)

D.Demartino PRE (2013), see also Solodovnikov(1980)

Appendix A (II)Demonstration of the Gordan theorem

(5)

(6)

D.Demartino PRE (2013), see also Solodovnikov(1980)

Appendix B:“Phase transitions” in computational problems

A random network of N reactions among M metabolites

Each reaction has one substrate and one product randomly chosen

Average number of loops of length L

Average totalnumber of loops

Exponential number of loopsfor α>1The size of the basis is linear in N

See also M.Weigt, A.Hartmann “Phase transition in combinatorial optimization problems” (2006)