Constraint satisfaction problems for metabolic networks

Transcription

1 Constraint satisfaction problems for metabolic networks Alessandro Seganti Università La Sapienza October 14, 213 Supervisors: F. Ricci-Tersenghi, A. De Martino A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

2 1 Introduction Reaction network Static Picture Dynamic Picture 2 Boolean activity Coarse Grained dynamics Message passing Results on Random Graphs The Mean Field Approximation 3 Real network The hypothesis Mean Field Solutions Correlation Matrix Role of the modules Preliminary results 4 Conclusions A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

3 Reaction network Introduction Reaction network What is a reaction network? A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

4 Static Picture Introduction Static Picture The most connected metabolites are removed and many topological groups are found. Ravasz et al, Science (22),297:1551. Reaction network divided in pathways that are functionally related. A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

5 Dynamic Picture Introduction Dynamic Picture In principle we should study the kinetics: ẋ µ = i ξ µ ν i i (x, k, t) u µ + noise Non equilibrium steady states of the fluxes of the reactions. Flux Balance Analysis FBA, Von Neumann growth model VN) B.O. Palsson, Systems biology (book) (26), A. De Martino et al, PNAS (29), A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

6 FBA Introduction Dynamic Picture Non equilibrium steady states of the fluxes of the reactions. Flux Balance Analysis FBA, Von Neumann growth model VN) v = v 1... = v = Solution space optimum A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

7 Boolean activity Coarse Grained dynamics Coarse Grained Dynamics Dynamical states on longer times scales than chemical processes Interested in states of the system satisfying minimal constraints inspired by the study of the fluxes. Looking at the fluxes, many reactions are never functioning. Is it possible to describe the system with boolean variables? A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

8 Boolean activity Coarse Grained dynamics Coarse Grained Dynamics Dynamical states on longer times scales than chemical processes Interested in states of the system satisfying minimal constraints inspired by the study of the fluxes. Looking at the fluxes, many reactions are never functioning. Is it possible to describe the system with boolean variables? FBA A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

9 Boolean activity Coarse Grained dynamics Coarse Grained Dynamics Dynamical states on longer times scales than chemical processes Interested in states of the system satisfying minimal constraints inspired by the study of the fluxes. Looking at the fluxes, many reactions are never functioning. Is it possible to describe the system with boolean variables? VN A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

10 Boolean activity Message passing Message Passing Constraints are written in a boolean form. We derived the equations and solved it by a message passing algorithm Solved on random reaction networks (topology similar to real network) A. Seganti, A. De Martino and F. Ricci-Tersenghi - Boolean constraint satisfaction problems for reaction networks JSTAT (213), P99 A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

11 Boolean activity Message passing Message Passing Constraints are written in a boolean form. We derived the equations and solved it by a message passing algorithm Solved on random reaction networks (topology similar to real network) A. Seganti, A. De Martino and F. Ricci-Tersenghi - Boolean constraint satisfaction problems for reaction networks JSTAT (213), P99 A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

12 Boolean activity Message passing Message Passing Constraints are written in a boolean form. We derived the equations and solved it by a message passing algorithm Solved on random reaction networks (topology similar to real network) A. Seganti, A. De Martino and F. Ricci-Tersenghi - Boolean constraint satisfaction problems for reaction networks JSTAT (213), P99 A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

13 Boolean activity Message passing Message Passing Constraints are written in a boolean form. We derived the equations and solved it by a message passing algorithm Solved on random reaction networks (topology similar to real network) A. Seganti, A. De Martino and F. Ricci-Tersenghi - Boolean constraint satisfaction problems for reaction networks JSTAT (213), P99 A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

14 Boolean activity Results on Random Graphs Results on Random Graphs θ is used to vary the fraction of reactions functioning. A typical phase transition is observed. It is possible to understand the type of the phase transition by looking at θ + and θ <µ >.6.4 λ=1, q =.8 S-MB + S-MB + H-MB + H-MB + <ν >.6.4 λ=1, q =.8 S-MB + S-MB + H-MB + H-MB θ θ A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

15 Boolean activity Results on Random Graphs Results on Random Graphs θ is used to vary the fraction of reactions functioning. A typical phase transition is observed. It is possible to understand the type of the phase transition by looking at θ + and θ. Hard-MB, θ + Soft-MB, θ + Soft-MB, θ - Hard-MB, θ - Hard-MB, θ + Soft-MB, θ + Soft-MB, θ - Hard-MB, θ q λ q λ A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

16 Boolean activity The Mean Field Approximation The Mean Field Approximation The Mean Field Approximation (MF) of VN model is another known biological approach: Network Expansion In the random case it is possible to see that MF solutions are a subset of the complete problem step 1 step 2 step 3 step 4 A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

17 Boolean activity The Mean Field Approximation The Mean Field Approximation The Mean Field Approximation (MF) of VN model is another known biological approach: Network Expansion In the random case it is possible to see that MF solutions are a subset of the complete problem <ν> + <ν> MAX + <ν> + <ν> MAX MF <ν> <ν> + <ν> MAX + <ν> + <ν> MAX MF <ν>.6.6 <ν> <ν> <µ> <µ> A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

18 Real network Hypothesis on the real network The hypothesis In real networks some hypothesis have to be made: All reactions can take place all metabolites are accessible (well mixed reactors) infinite enzymes no external regulation The direction of all the reactions is given 242 reversible reactions 565 irreversible reactions 7 uptakes A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

19 Real network Mean Field Solutions The solutions for the metabolites 6 5 #metabolites on at convergence (scope) #metabolites on at t= (seed) A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

20 Real network Mean Field Solutions The solutions for the reactions.1.8 θ=.5 θ=1 θ=1.5 θ=2 θ=2.5 θ.1.8 ALL θ.6.6 freq freq REA ON REA ON A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

21 Real network Correlation Matrix Correlation Matrix We focused on the reactions (without reversed). We defined a correlation matrix : C ij = ν i ν j + (1 ν i )(1 ν j ). ν i = 1() if reaction i is (not) operating A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

22 Real network Correlation Matrix Correlation Matrix We focused on the reactions (without reversed). We defined a correlation matrix : C ij = ν i ν j + (1 ν i )(1 ν j ). ν i = 1() if reaction i is (not) operating module sizes module sizes MLM NSP OXP PPB PPP PTR PUT PYR TCA TLM TRA TTP UNA VLI MGO AAM ACM ANA APM CEB CPG CYS FOL GLU GLY GOX GSM HISMET MLM NSP OXP PPB PPP PTR PUT PYR TCA TLM TRA TTP UNA VLI MGO AAM ACM ANA APM CEB CPG CYS FOL GLU GLY GOX GSM HISMET A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

23 Real network Role of the modules Are the modules really functional? A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

24 Real network Preliminary results Solution on E.Coli metabolic network (complete case) 1 VN <µ> + + BP <ν> + + BP θ θ 1 FBA <µ> BP + BP <ν> BP + BP θ θ A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

25 Conclusions Conclusions Modelization of metabolism in term of constraint satisfaction problem. Solved the problem on random system and understood the phase space organization. Applied the model to the real system. Modularization of the MF solutions seems correlated with the function. Preliminary solutions using the complete algorithm are promising. Work in progress: Introduction of reversibility. Connection with flux analysis/experiments. Modularization of complete problem solutions. A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

26 Conclusions Conclusions Modelization of metabolism in term of constraint satisfaction problem. Solved the problem on random system and understood the phase space organization. Applied the model to the real system. Modularization of the MF solutions seems correlated with the function. Preliminary solutions using the complete algorithm are promising. Work in progress: Introduction of reversibility. Connection with flux analysis/experiments. Modularization of complete problem solutions. A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

27 Last Conclusions Thank you for your attention. A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

28 Conclusions The Network of Ecoli # metabolites 1 1 in Degreee out Degree # reactions in Degree out Degree degree degree A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

29 The Network of Ecoli II Conclusions The stoichiometric matrix A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

30 Conclusions Finding the modules Take the correlation matrix Choose a cutoff ( C) Make an adiacency matrix: A ij = 1 if C ij > C, otherwise Find the connected components in this correlation network A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

31 Conclusions Finding the modules Take the correlation matrix Choose a cutoff ( C) Make an adiacency matrix: A ij = 1 if C ij > C, otherwise Find the connected components in this correlation network A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

32 Conclusions Finding the modules Take the correlation matrix Choose a cutoff ( C) Make an adiacency matrix: A ij = 1 if C ij > C, otherwise Find the connected components in this correlation network A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

33 Conclusions Finding the modules Take the correlation matrix Choose a cutoff ( C) Make an adiacency matrix: A ij = 1 if C ij > C, otherwise Find the connected components in this correlation network A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22

34 Conclusions Choosing the cutoff Which is the right cutoff to search for the modules? It is possible to find it using an entropy measure M. Marsili, I. Mastromatteo, Y. Roudi, Arxiv: (213). 4 y=x 4 y=x A. Seganti (Università La Sapienza) CSPs for metabolic networks October 14, / 22