Reaction Networks with Fluctuations: From Inter-stellar Chemistry to Intra-cellular Biology

1. Reaction Networks with Fluctuations: From Inter-stellar Chemistry to Intra-cellular Biology Ofer Biham The Hebrew University Azi Lipshtat Gian Vidali Baruch Barzel Eric Herbst Adina Lederhendler Raz Kupferman Adiel Loinger Nathalie Balaban Hagai Perets Joachim Krug Hilik Frank Evelyne Roueff Amir Zait Jacques Le Bourlot Nadav Katz Franck Le Petit Israel Science Foundation The Adler Foundation for Space Research The Center for Complexity Science US-Israel Binational Science Foundation

2. Complex Reaction Networks Methanol Production on Interstellar Dust Grains Genetic Regulatory Network of A. thaliana flower morphogenesis (Gambin et al., In Silico Biol. 6, 0010 (2006)) Freshwater marsh food web University of Maine Tanglewood 4-H Camp and Learning Center

3. Laboratory Experiments Requirements: samples (silicates,carbon,ice) low temperatures vacuum low flux – long times… efficient detection of molecules Sample temperature Time detector Pirronello et al., ApJ 475, L69 (1997) Katz et al., ApJ 522, 305 (1999) Perets et al., ApJ 627, 850 (2005) Perets et al. ApJ 661, L163 (2007)

4. Reaction Networks The methanol network H+HH2 CO+HHCO HCO+HH2CO H2CO+HH3CO H3CO+HCH3OH CO+OCO2 HCO+OCO2+H O+OO2 OH+HH2O H H H H CO  HCO  H2CO  H3CO  CH3O Stantcheva, Shematovich and Herbst, A&A 391, 1069 (2002) Lipshtat and Biham, PRL 93, 170601 (2004) Barzel and Biham, ApJ 658, L37 (2007)

5. Rate Equations: The Production Rate: i,j : H,O,OH,CO,HCO,… Rate equations are useful for macroscopic systems The Problem: they do not account for fluctuations

6. Macroscopic surface Test tube Sub-micron grain Cell

7. Rate Equation  Master Equation For small grains under low flux the typical population size of H atoms on a grain may go down to n ~ 1 Thus: the rate equation (mean field approximation) fails One needs to take into account: The discreteness of the H atoms The fluctuations in the populations of H atoms on grains → Master Equation for the probability distribution P(n), n=0,1,2,3,…

8. Master Equation H+HH2 Flux Desorption Recombination Direct Integration: Biham, Furman, Pirronello and Vidali, ApJ 553, 595 (2001) Green, Toniazzo, Pilling, Ruffle, Bell and Hartquist, A&A 375, 1111 (2001) Monte Carlo: Charnley, ApJ 562, L99 (2001)

9. H2 Production Rate vs. Grain Size Rate equation Master equation Grain Size

10. The probability distribution

11. n P(n) n

12. Single-Species Reaction Network Competition between two processes: reaction and desorption. Two characteristic length scales, independent of grain size: X 2 X X+X X 2 H + H H 2 Number of sites visited before desorption. Number of empty sites around each atom.

13. Rate equations Master equation (integration) MC simulation Single-Species Reaction NetworkReaction-Dominated System Reaction Rate in Steady State vs. Grain Size.

14. Rate equations Master equation (integration) MC simulation Single-Species Reaction NetworkDesorption-Dominated System Reaction Rate in Steady State vs. Grain Size.

15. Two-Species Reaction Network Condition for accuracy of rate equations: For each species : Grain size must be larger than both. Reaction rate in steady state vs. grain size. Lederhendler and Biham, Phys. Rev. E (2008)

16. Master Equation for two Reactive Species }flux }desorption H+H→H2 O+O→O2 H+O→OH

17. no Monte Carlo methods

18. 1. Calculate the rates of all the possible processes, wi 2. Pick one process randomly, with probability proportional to its rate: 3. The expectation value of the elapsed time is: 4. Draw the elapsed time from the distribution: 5. Advance the time by Dt. no D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977) The Gillespie Algorithm

19. Complex Reaction Networks Number of equations increases Exponentially with the number of reactive species…

20. The number of variables grows exponentially with the number of reactive species Does not account for stochasticity Infeasible for complex networks Highly efficient Always valid Simulation Methodologies The Rate Equations: The Master Equation:

21. The Multiplane Method Lipshtat and Biham, PRL93, 170601 (2004) Barzel, Biham and Kupferman, PRE 76, 026703 (2007)

22. The multiplane method

23. Approximation: neglect correlations between X1 and X3 After tracing out:

24. The Reduced Multiplane Method

25. The Multiplane Method • The original method: • B. Barzel, O. Biham, and R. Kupferman Phys. Rev. E 76 (2007) 026703 • We now extended it to include: • Dissociations: • Self Interactions: • Multiple products: • Branching ratios • Reaction products that are reactive • Feedback

26. Population Size (#/s) Population Size Relative Errors Relative Errors System Size [S] System Size [S]

27. And a more Complex one…

28. Population Size Production Rate (#\s) System Size [S]

29. For example: Node: Edge: Loop: The Moment Equations The population size and the production rate are given by the moments: B. Barzel and O. Biham, Astrophys. J. Lett., 115 20941 (2007) B. Barzel and O. Biham, J. Chem. Phys. 127, 144703 (2007)

30. = + + + The Moment Equations The truncation scheme: Automation of the equations: The “miracle”: The truncation is based on a low population assumption. But the equations are valid far beyond that…

31. Moment Equations The number of equations is MINIMAL: For example: in the methanol network: Master equation: about a million equations Multiplane: few hundred equations Moments: 17 equations

32. Results: the methanol network

34. The Auto-repressor Biological networks: gene regulation networks • Protein A acts as a repressor to its own gene • It can bind to the promoter of its own gene and suppress the transcription

35. = Repression strength The Auto-repressor • Rate equations – Michaelis-Menten form • Rate equations – Extended Set n = Hill Coefficient

36. The Auto-repressor P(NA,Nr) : Probability for the cell to contain NAfree proteins and Nr bound proteins The Master Equation

37. The Genetic Switch • A mutual repression circuit. • Two proteins A and B negatively regulate each other’s synthesis

38. The Switch • Stochastic analysis using master equation and Monte Carlo simulations reveals the reason: • For weak repression we get coexistence of A and B proteins • For strong repression we get three possible states: • A domination • B domination • Simultaneous repression (dead-lock) • None of these state is really stable

39. The Exclusive Switch • An overlap exists between the promoters of A and B and they cannot be occupied simultaneously • The rate equations still have a single steady state solution

40. The Exclusive Switch • But stochastic analysis reveals that the system is truly a switch • The probability distribution is composed of two peaks • The separation between these peaks determines the quality of the switch k=1 k=50 Lipshtat, Loinger, Balaban and Biham, Phys. Rev. Lett. 96, 188101 (2006) Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, 021904 (2007)

41. The Exclusive Switch

42. A genetic oscillator synthetically built by Elowitz and Leibler Nature 403 (2000) It consist of three proteins repressing each other in a cyclic way The Repressilator

43. The Repressilator • Monte Carlo Simulations • Rate equations

44. Plasmids • The repressilator and synthetic toggle switch were encoded on plasmids in E. coli. • Plasmids are circular self replicating DNA molecules which include only few genes. • The number of plasmids in a cell can be controlled. How does the number of plasmids affect the dynamical behavior?

45. The Repressilator • Monte Carlo Simulations (50 plasmids) • Rate equations (50 plasmids) Loinger and Biham, Phys. Rev. E, 76, 051917 (2007)

46. A A A A A A A A A A B B The Switch General Switch Exclusive Switch a a b b

47. The Switch Warren and ten Wolde [PRL 92, 128101 (2004)] showed that for a single plasmid with h = 2, the exclusive switch is more stable than the general switch. • This does not hold for a high plasmid copy number Loinger and Biham, Phys. Rev. Lett., 103, 068104 (2009)

48. A A A A A A A A A A B B The Switch General Switch Exclusive Switch a a b b

49. A A A A A A A A A A B B A A A A The Switch General Switch Exclusive Switch a b a b Weakens A Does not affect B Weakens A and Strengthens B

50. Toxin-Antitoxin Module Bacterial persistence is a phenomenon in which a small fraction of genetically identical bacteria cells survives after an exposure to antibiotics What is the mechanism? Figure taken from: N.Q Balaban et al, Science 305, 1951 (2004)