Lecture 4

1. Lecture 4 Reaction system as ordinary differential equations Reaction system as stochastic process Metabolic network and stoichiometric matrix Graph spectral analysis/Graph spectral clustering and its application to metabolic networks

2. Introduction Metabolism is the process through which living cells acquire energy and building material for cell components and replenishing enzymes. Metabolism is the general term for two kinds of reactions: (1) catabolic reactions –break down of complex compounds to get energy and building blocks, (2) anabolic reactions—construction of complex compounds used in cellular functioning How can we model metabolic reactions?

3. What is a Model? Formal representation of a system using --Mathematics --Computer program Describes mechanisms underlying outputs Dynamical models show rate of changes with time or other variable Provides explanations and predictions

4. Typical network of metabolic pathways Reactions are catalyzed by enzymes. One enzyme molecule usually catalyzes thousands reactions per second (~102-107) The pathway map may be considered as a static model of metabolism

5. Dynamic modeling of metabolic reactions is the process of understanding the reaction rates i.e. how the concentrations of metabolites change with respect to time

6. An Anatomy of Dynamical Models Stochastic Deterministic -- Space -- -- Space -- --No Space -- --No Space -- Discrete Variables Stochastic Boolean Networks; Stochastic Cellular Automata Discrete Time Markov Chains Discrete Time Boolean Networks; Cellular Automata Finite State Machines Iterated Functions; Difference Equations Continuous Variables Iterated Functions; Difference Equations Discrete Time Markov Chains Coupled Discrete Time Markov Chains Coupled Continuous Time Markov Chains Continuous Time Discrete Variables Coupled Boolean Differential Equations Continuous Time Markov Chain Boolean Differential Equations Stochastic Ordinary Differential Equations Stochastic Partial Differential Equations Ordinary Differential Equations Partial Differential Equations Continuous Variables

7. Differential equations Differential equations are based on the rate of change of one or more variables with respect to one or more other variables

8. An example of a differential equation Source: Systems biology in practice by E. klipp et al

9. An example of a differential equation Source: Systems biology in practice by E. klipp et al

10. An example of a differential equation Source: Systems biology in practice by E. klipp et al

11. Schematic representation of the upper part of the Glycolysis Source: Systems biology in practice by E. klipp et al.

12. ODEs representing a reaction system The ODEs representing this reaction system Realize that the concentration of metabolites and reaction rates v1, v2, …… are functions of time

13. The rate equations can be solved as follows using a number of constant parameters

14. The temporal evaluation of the concentrations using the following parameter values and initial concentrations

15. Notice that because of bidirectional reactions Gluc-6-P and Fruc-6-P reaches peak earlier and then decrease slowly and because of unidirectional reaction Fruc1,6-P2 continues to grow for longer time.

16. The use of differential equations assumes that the concentration of metabolites can attain continuous value. But the underlying biological objects , the molecules are discrete in nature. When the number of molecules is too high the above assumption is valid. But if the number of molecules are of the order of a few dozens or hundreds then discreteness should be considered. Again random fluctuations are not part of differential equations but it may happen for a system of few molecules. The solution to both these limitations is to use a stochastic simulation approach.

17. Stochastic Simulation Stochastic modeling for systems biology Darren J. Wilkinson 2006

18. Molecular systems in cell

19. [m1(out)] Molecular systems in cell[ ]: concentration of ith object [m1(in)] [m2] [m3] [m5] [p1] [p4] [p2] [m4] [p3] [r1] [r４] [r2] [r3]

20. [m1(out)] Molecular systems in cellcj: cj’: efficiency of jth process c1 c3 c4 [m1(in)] [m2] [m3] [m5] c2 [p1] [p4] [p2] c5 [m4] c12 c6 c8 c10 [p3] [r1] [r４] [r2] [r3] c9 c13 c7 c11

21. [m1(out)] Molecular systems for small molecules in cell c1 h3=c3 [m2] h5=c4 [m3] c3  p2 ,r2 c4  p4 ,r4 c3 c4 [m1(in)] [m2] [m3] [m5] c2 c5 h1=c1 [m1(out)] h2=c2 [m1(in)] [m4] c2  p1 ,r1 h4=c5 [m2] c5  p3 ,r3 Stochastic selection of reaction based on(h1, h2, h3, h4, h5)

22. Molecular systems for small molecules in cell [m1(out)]=100 c1 h3=c3 [m2] h5=c4 [m3] c5  p2 ,r2 c4  p4 ,r4 c3 c4 [m1(in)] [m2] [m3] [m5] c2 c5 h1=c1 [m1(out)] = 100 c1 h2=c2 [m1(in)] [m4] c2  p1 ,r1 h4=c5 [m2] c5  p3 ,r3 Stochastic selection of reaction based on(100c1, h2, h3, h4, h5) Reaction 1

23. [m1(out)]=99 Molecular systems for small molecules in cell c1 h3=c3 [m2] =0 h5=c4 [m3] =0 c3 c4 [m1(in)]=1 [m3]=0 [m5]=0 c2 [m2]=0 c5 h1=c1 [m1(out)] = 99 c1 h2=c2 [m1(in)] = 1 c2 [m4]=0 h4=c5 [m2] =0 Stochastic selection of Reaction based on (99 c1, 1 c2, 0, 0, 0)  Reaction 1

24. [m1(out)]=98 Molecular systems for small molecules in cell c1 h3=c3 [m2] =0 h5=c4 [m3] =0 c3 c4 [m1(in)]=2 [m3]=0 [m5]=0 c2 [m2]=0 c5 h1=c1 [m1(out)] = 98c1 h2=c2 [m1(in)] = 2 c2 [m4]=0 h4=c5 [m2] =0 Stochastic selection of Reaction based on (98 c1, 2 c2, 0, 0, 0)  Reaction 1

25. [m1(out)]=97 Molecular systems for small molecules in cell c1 h3=c3 [m2] =0 h5=c4 [m3] =0 c3 c4 [m1(in)]=3 [m3]=0 [m5]=0 c2 [m2]=0 c5 h1=c1 [m1(out)] = 97c1 h2=c2 [m1(in)] = 3 c2 [m4]=0 h4=c5 [m2] =0 Stochastic selection of Reaction based on (97 c1, 3 c2, 0, 0, 0)  Reaction 2

26. h1=c1 [m1(out)] = 97c1 [m1(out)]=97 Molecular systems for small molecules in cell c1 h3=c3 [m2] =1 c3 h5=c4 [m3] =0 c3 c4 [m1(in)]=2 [m3]=0 [m5]=0 c2 [m2]=1 c5 h2=c2 [m1(in)] = 2 c2 [m4]=0 h4=c5 [m2] =1 c5 Stochastic selection of Reaction based on (97 c1, c2, 1 c3, 0, 1 c5)  Reaction 1

27. [m1(out)]=96 Molecular systems for small molecules in cell c1 h3=c3 [m2] =1 c3 h5=c4 [m3] =0 c3 c4 [m3]=0 [m5]=0 c2 [m1(in)]=3 [m2]=1 c5 h1=c1 [m1(out)] = 97c1 h2=c2 [m1(in)] = 3 c2 [m4]=0 h4=c5 [m2] =1 c5 Stochastic selection of Reaction(96 c1, 3 c2, 1 c3, 0, 1 c5) Reaction 3

28. [m1(out)]=96 Molecular systems for small molecules in cell c1 h3=c3 [m2] =0 h5=c4 [m3] =1 c4 c3 c4 [m3]=1 [m5]=0 c2 [m1(in)]=3 [m2]=0 c5 h1=c1 [m1(out)] = 97c1 h2=c2 [m1(in)] = 3 c2 [m4]=0 h4=c5 [m2] =0 Stochastic selection of Reaction based on (96 c1, 3 c2, 0, 1 c4 , 0) …

29. c2 c3 c1 c4 m1(out) m1(in) m1(in) m2 m2 m3 m3 m5 c5 m2 m5 [m1(out)] Input data c1 c3 c4 [m1(in)] [m2] [m3] [m5] c2 c5 [m4] Reaction parameters and Reactions [m1(out)] Initial concentrations [m1(in)] [m2] [m3] [m4] [m5]

30. Gillespie Algorithm Step 0: System Definition objects (i = 1, 2,…, n) and their initial quantities: Xi(init) reaction equations (j=1,2,…,m) Rj: m(Pre)j1 X1 + ...+ m(Pre)jn Xn =m (Post) j1 X1 +...+ m (Post) jnXn reaction intensities: ci for Rj Step 1: [Xi]Xi(init) Step 2:hj: :probability of occurrence of reactions based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n) Step 3: Random selection of reaction Here a selected reaction is represented by index s. Step 4: Quantities for individual objects are revised base on selected reaction equation [Xi] ← [Xi] – m(Pre)s + m(Post)s

31. Gillespie Algorithm (minor revision) Step 0: System Definition objects (i = 1, 2,…, n) and their initial quantities Xi(init) reaction equations (j=1,2,…,m) Rj: m(Pre)j1 X1 + ...+ m(Pre)jn Xn =m (Post) j1 X1 +...+ m (Post) jnXn reaction intensities: ci for Rj Step 1: [Xi]Xi(init) Step 2:hj: :probability of occurrence of reactions based on cj (j=1,2,..,m) and [Xi] (i=1,2,..,n) Step 3: Random selection of reaction Here a selected reaction is represented by index s. Step 4: Quantities for individual objects are revised base on selected reaction equation X’i = [Xi] – m(Pre)s + m(Post)s X’iXimax X’i 0 No No Yes Yes Step 5: [Xj] X’i

32. Software: Simple Stochastic Simulator1.Create stoichiometric data file and initial condition file R1 [X1] = [X2] R2 [X2] = [X1] Reaction Definition: REQ**.txt Objects used are assigned by [ ] . Stoichiometetric data and ci: REACTION**.txt Reaction Parameter ci [X1] [X1] [X2] [X2] R1 1 1 0 0 1 R2 1 0 1 1 0 ci is set by user Initial condition: INIT**.txt [X1] 1000 [X2] 1000 max number (for ith object, max number is set by 0 for ith , [Xi]0 Initial quantitiy http://kanaya.naist.jp/Lecture/systemsbiology_2010

33. Software: Simple Stochastic Simulator2. Stochastic simulation Stoichiometetric data and ci: REACTION**.txt Initial condition: INIT**.txt Simulation results: SIM**.txt Reaction Parameter c: 1.0 1.0 // time [X1] [X2] 0.00 100.0 100.0 0.0015706073545097992 101.0 99.0 0.015704610011372147 100.0 100.0 0.01670413203960951 101.0 99.0 …. ….

34. Example of simulation results# of type of chemicals =2

35. [X1][X2] c=1, [X1]=1000, [X2]=0

36. [X1][X2] [X2][X1]c1=c2=1[X1]=1000

37. # of type of chemicals =3

38. [X1][X2][X3], [X1]=1000, c=1

39. [X1] [X2][X3], [X1]=1000, c=1

40. [X1][X2][X3], [X1]=1000, c=1

41. [X1][X2][X3],[X1]=1000, c=1

42. loop reaction [X1][X2][X3][X1], [X1]=1000, c=1

43. Representation of Reaction Data Set Reaction Data Initial Condition c1 [X1]= X1(init) [X1] 2[X1] [X2]= X2(init) c2 [X1] + [X2] 2[X2] c3 [X2] Φ

44. Example 2 EMP glcK ATP + [D-glucose] -> ADP + [D-glucose-6-phosphate] glcK ATP + [alpha-D-glucose] -> ADP + [D-glucose-6-phosphate] pgi [D-glucose-6-phosphate] <-> [D-fructose-6-phosphate] pgi [D-fructose-6-phosphate] <-> [D-glucose-6-phosphate] pgi [alpha-D-glucose-6-phosphate] <-> [D-fructose-6-phosphate] pgi [D-fructose-6-phosphate] <-> [alpha-D-glucose-6-phosphate] pfk ATP + [D-fructose-6-phosphate] -> ADP + [D-fructose-1,6-bisphosphate] fbp [D-fructose-1,6-bisphosphate] + H(2)O -> [D-fructose-6-phosphate] + phosphate fbaA [D-fructose-1,6-bisphosphate] <-> [glycerone-phosphate] + [D-glyceraldehyde-3-phosphate] fbaA [glycerone-phosphate] + [D-glyceraldehyde-3-phosphate] <-> [D-fructose-1,6-bisphosphate] tpiA [glycerone-phosphate] <-> [D-glyceraldehyde-3-phosphate] tpiA [D-glyceraldehyde-3-phosphate] <-> [glycerone-phosphate] gapA [D-glyceraldehyde-3-phosphate] + phosphate + NAD(+) -> [1,3-biphosphoglycerate] + NADH + H(+) gapB [1,3-biphosphoglycerate] + NADPH + H(+) -> [D-glyceraldehyde-3-phosphate] + NADP(+) + phosphate pgk ADP + [1,3-biphosphoglycerate] <-> ATP + [3-phospho-D-glycerate] pgk ATP + [3-phospho-D-glycerate] <-> ADP + [1,3-biphosphoglycerate] pgm [3-phospho-D-glycerate] <-> [2-phospho-D-glycerate] pgm [2-phospho-D-glycerate] <-> [3-phospho-D-glycerate] eno [2-phospho-D-glycerate] <-> [phosphoenolpyruvate] + H(2)O eno [phosphoenolpyruvate] + H(2)O <-> [2-phospho-D-glycerate]

45. Example 2 EMP alpha-D-glucose D-glucose D-glucose-6-phosphate alpha-D-glucose-6-phosphate D-fructose-6-phosphate [D-fructose-1,6-bisphosphate] [glycerone-phosphate] [D-glyceraldehyde-3-phosphate] [1,3-biphosphoglycerate] [3-phospho-D-glycerate] [2-phospho-D-glycerate] [phosphoenolpyruvate]

47. Metabolic network and stoichiometric matrix

48. Typical network of metabolic pathways Reactions are catalyzed by enzymes. One enzyme molecule usually catalyzes thousands reactions per second (~102-107) The pathway map may be considered as a static model of metabolism

49. What is a stoichiometric matrix? For a metabolic network consisting of m substances and r reactions the system dynamics is described by systems equations. The stoichiometric coefficients nij assigned to the substance Si and the reaction vj can be combined into the so called stoichiometric matrix.

50. Example reaction system and corresponding stoichiometric matrix There are 6 metabolites and 8 reactions in this example system stoichiometric matrix