1. Lecture 4: Metabolism Reaction system as ordinary differential equations Reaction system as stochastic process

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 (~10 2 - 10 7 ) 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 Discrete Time Discrete Variables Continuous Variables Deterministic --No Space -- -- Space -- Stochastic --No Space -- -- Space -- Finite State Machines Boolean Networks; Cellular Automata Discrete Time Markov Chains Stochastic Boolean Networks; Stochastic Cellular Automata Iterated Functions; Difference Equations Iterated Functions; Difference Equations Discrete Time Markov Chains Coupled Discrete Time Markov Chains Continuous Time Discrete Variables Continuous Variables Boolean Differential Equations Ordinary Differential Equations Coupled Boolean Differential Equations Partial Differential Equations Continuous Time Markov Chain Stochastic Ordinary Differential Equations Coupled Continuous Time Markov Chains Stochastic Partial Differential Equations

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

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

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

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. Molecular systems in cell [ ]: concentration of ith object [m 1(in) ] [m 2 ][m 3 ] [m 4 ] [m 5 ] [m 1(out) ] [r 1 ] [r 2 ][r 3 ] [r ４ ] [p 1 ] [p 2 ] [p 3 ] [p 4 ]

20. Molecular systems in cell c j : c j’ : efficiency of jth process [m 1(in) ] [m 2 ][m 3 ] [m 4 ] [m 5 ] [m 1(out) ] [r 1 ] [r 2 ][r 3 ] [r ４ ] [p 1 ] [p 2 ] [p 3 ] [p 4 ] c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 c8c8 c9c9 c 10 c 11 c 12 c 13

21. Molecular systems for small molecules in cell [m 1(in) ] [m 2 ][m 3 ] [m 4 ] [m 5 ] [m 1(out) ] c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ]h 2 =c 2 [m 1(in) ] h 4 =c 5 [m 2 ] h 3 =c 3 [m 2 ]h 5 =c 4 [m 3 ] c 2  p 1,r 1 c 5  p 3,r 3 c 3  p 2,r 2 c 4  p 4,r 4 Stochastic selection of reaction based on(h 1, h 2, h 3, h 4, h 5 )

22. Molecular systems for small molecules in cell [m 1(in) ] [m 2 ][m 3 ] [m 4 ] [m 5 ] [m 1(out) ]=100 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 100 c 1 h 2 =c 2 [m 1(in) ] h 4 =c 5 [m 2 ] h 3 =c 3 [m 2 ]h 5 =c 4 [m 3 ] c 2  p 1,r 1 c 5  p 3,r 3 c 5  p 2,r 2 c 4  p 4,r 4 Stochastic selection of reaction based on(100 c 1, h 2, h 3, h 4, h 5 )  Reaction 1

23. Molecular systems for small molecules in cell [m 1(in) ]=1 [m 2 ]=0 [m 3 ]=0 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=99 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 99 c 1 h 2 =c 2 [m 1(in) ] = 1 c 2 h 4 =c 5 [m 2 ] =0 h 3 =c 3 [m 2 ] =0 h 5 =c 4 [m 3 ] =0 Stochastic selection of Reaction based on (99 c 1, 1 c 2, 0, 0, 0)  Reaction 1

24. Molecular systems for small molecules in cell [m 1(in) ]=2 [m 2 ]=0 [m 3 ]=0 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=98 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 98 c 1 h 2 =c 2 [m 1(in) ] = 2 c 2 h 4 =c 5 [m 2 ] =0 h 3 =c 3 [m 2 ] =0 h 5 =c 4 [m 3 ] =0 Stochastic selection of Reaction based on (98 c 1, 2 c 2, 0, 0, 0)  Reaction 1

25. Molecular systems for small molecules in cell [m 1(in) ]=3 [m 2 ]=0 [m 3 ]=0 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=97 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 97 c 1 h 2 =c 2 [m 1(in) ] = 3 c 2 h 4 =c 5 [m 2 ] =0 h 3 =c 3 [m 2 ] =0 h 5 =c 4 [m 3 ] =0 Stochastic selection of Reaction based on (97 c 1, 3 c 2, 0, 0, 0)  Reaction 2

26. Molecular systems for small molecules in cell [m 1(in) ]=2 [m 2 ]=1 [m 3 ]=0 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=97 c1c1 c2c2 c3c3 c4c4 c5c5 h 2 =c 2 [m 1(in) ] = 2 c 2 h 4 =c 5 [m 2 ] =1 c 5 h 3 =c 3 [m 2 ] =1 c 3 h 5 =c 4 [m 3 ] =0 h 1 =c 1 [m 1(out) ] = 97 c 1 Stochastic selection of Reaction based on (97 c 1, c 2, 1 c 3, 0, 1 c 5 )  Reaction 1

27. Molecular systems for small molecules in cell [m 1(in) ]=3 [m 2 ]=1 [m 3 ]=0 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=96 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 97 c 1 h 2 =c 2 [m 1(in) ] = 3 c 2 h 4 =c 5 [m 2 ] =1 c 5 h 3 =c 3 [m 2 ] =1 c 3 h 5 =c 4 [m 3 ] =0 Stochastic selection of Reaction(96 c 1, 3 c 2, 1 c 3, 0, 1 c 5 )  Reaction 3

28. Molecular systems for small molecules in cell [m 1(in) ]=3 [m 2 ]=0 [m 3 ]=1 [m 4 ]=0 [m 5 ]=0 [m 1(out) ]=96 c1c1 c2c2 c3c3 c4c4 c5c5 h 1 =c 1 [m 1(out) ] = 97 c 1 h 2 =c 2 [m 1(in) ] = 3 c 2 h 4 =c 5 [m 2 ] =0 h 3 =c 3 [m 2 ] =0 h 5 =c 4 [m 3 ] =1 c 4 Stochastic selection of Reaction based on (96 c 1, 3 c 2, 0, 1 c 4, 0)  …

29. Input data [m 1(in) ] [m 2 ][m 3 ] [m 4 ] [m 5 ] [m 1(out) ] c1c1 c2c2 c3c3 c4c4 c5c5 c1c1 m 1(out) m 1(in) c2c2 m2m2 c3c3 m2m2 m3m3 m3m3 m5m5 c4c4 m2m2 m5m5 c5c5 [m 1(out) ] [m 1(in) ][m 2 ][m 3 ][m 4 ][m 5 ] Initial concentrations Reaction parameters and Reactions

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

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

32. Software: Simple Stochastic Simulator 1.Create stoichiometric data file and initial condition file Reaction Definition: REQ**.txt R1[X1] = [X2] R2[X2] = [X1] Reaction Parameterci[X1][X1][X2][X2] R111001 R210110 Stoichiometetric data and c i : REACTION**.txt c i is set by user [X1]1000 [X2]1000 Initial condition: INIT**.txt max number (for ith object, max number is set by 0 for ith, [Xi]  0 Initial quantitiy Objects used are assigned by [ ].

33. Software: Simple Stochastic Simulator 2. Stochastic simulation Stoichiometetric data and c i : REACTION**.txt Initial condition: INIT**.txt Reaction Parameter c:1.01.0 // time[X1][X2] 0.00100.0100.0 0.0015706073545097992101.099.0 0.015704610011372147100.0100.0 0.01670413203960951101.099.0 …. Simulation results: SIM**.txt

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 3. Gene Expression and Regulation Transcription (prokaryotes) promotergene RNAP mRNA promoter + RNAP promoter ・ RNAP promoter + RNAP + gene promoter ・ RNAP # of free promoter is generally 0 (promoter ・ RNAP) or 1 !

44. Stochastic simulation 3. Gene Expression and Regulation Transcription (prokaryotes)

45. Representation of Reaction 3. Gene Expression and Regulation Transcription (prokaryotes) promoter1gene RNAP mRNA1 promoter1 + RNAP promoter1 ・ RNAP promoter1 + RNAP + mRNA1 promoter1 ・ RNAP # of free promoter is 0 (promoter ・ RNAP) or 1 ! promoter2gene RNAP mRNA2 promoter2 + RNAP promoter2 ・ RNAP promoter2 + RNAP + mRNA2 promoter2 ・ RNAP

46. Stochastic Simulation 1. Stoichiometric chemical reaction Reaction Data [X 1 ]2[X 1 ] c1c1 [X 1 ] + [X 2 ] 2[X 2 ] c2c2 [X 2 ] c3c3 Stochastic modeling for systems biology Darren J. Wilkinson 2006

47. Representation of Reaction Data Set [X 1 ]2[X 1 ] c1c1 [X 1 ] + [X 2 ]2[X 2 ] c2c2 [X 2 ]Φ c3c3 Reaction DataInitial Condition [X 1 ]= X 1 (init) [X 2 ]= X 2 (init)

48. Example 2 EMP glcKATP + [D-glucose] -> ADP + [D-glucose-6-phosphate] glcKATP + [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] pfkATP + [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 pgkADP + [1,3-biphosphoglycerate] ATP + [3-phospho-D-glycerate] pgkATP + [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]

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