1. From Inter-stellar Chemistry to Intra-cellular Biology
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
רשתות ריאקציה הן דרך נפוצה לתאר מערכות שבהן מתרחשות תגובות בין הרכיבים השונים.
דוגמאות – רשת כימית / מטבולית, גנטית, אקולוגית, מודלים בסוציולוגיה (SIR)
נניח לרגע שאנו יודעים את קצבי התהליכים השונים במערכת, ורוצים לקבל פרדיקציות לגבי גדלים שונים – מספר המופעים של רכיב מסוים, קצב היצור של רכיב אחר, גודל ממוצע של הפלוקטואציות.
לא לדבר על המודלים הספציפיים
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 2

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
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, (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 Flux Desorption Recombination H+HH2
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. 12

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

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

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) 15

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

17. Monte Carlo methods no

18. The Gillespie Algorithm
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.
D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977) no

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

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

21. The Multiplane Method Lipshtat and Biham, PRL 93, 170601 (2004)
Barzel, Biham and Kupferman, PRE 76, (2007)

22. The multiplane method

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

24. The Reduced Multiplane Method

25. B. Barzel, O. Biham, and R. Kupferman Phys. Rev. E 76 (2007) 026703
The Multiplane Method
The original method:
B. Barzel, O. Biham, and R. Kupferman Phys. Rev. E 76 (2007)
We now extended it to include:
Dissociations:
Self Interactions:
Multiple products:
Branching ratios
Reaction products that are reactive
Feedback
להגיד שבמקור הזנחנו רק את הקורלציות של מינים שלא קשורים לתגובה הספציפית
כאן, לדוגמה אם מין מתפרק לשני מינים מיד נוצרת קורלציה ביניהם, שאם אנחנו רוצים להזניח... 25

26. System Size [S] System Size [S] Population Size (#/s) Population Size
לעשות X במקום קווים
Relative Errors
Relative Errors
System Size [S]
System Size [S] 26

27. And a more Complex one…
על רשתות אמיתיות יש הרבה מחקר אמפירי כיום בקונטקסט של אקולוגיה ותאים למשל
להוציא קצבי ריאקציה וכו'
ואנחנו יכולים לתת מענה לכל השיטות ביחד 27

28. Population Size Production Rate (#\s) System Size [S]
בקצה כבר קשה להריץ משוואת מאסטר בגלל זמן חישוב
לתת סקאלות זמן
Production Rate (#\s)
System Size [S] 28

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

30. But the equations are valid far beyond that…
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. Protein A acts as a repressor to its own gene
Biological networks: gene regulation networks
The Auto-repressor
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. The Auto-repressor Rate equations – Michaelis-Menten form
Rate equations – Extended Set
n = Hill Coefficient
= Repression strength

36. The Auto-repressor The Master Equation P(NA,Nr) :
Probability for the cell to contain NA free 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, (2006)
Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, (2007)

41. The Exclusive Switch

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

43. The Repressilator
Monte Carlo Simulations
Rate equations

44. How does the number of plasmids affect the dynamical behavior?
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
Loinger and Biham, Phys. Rev. E, 76, (2007)

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

47. The Switch This does not hold for a high plasmid copy number
Warren and ten Wolde
[PRL 92, (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, (2009)

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

49. The Switch General Switch Exclusive Switch 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)

51. Toxin-Antitoxin Module
phip
hipB
hipA A B
HipA – Stable toxin
HipB – Unstable Antitoxin, Neutralizes HipA
Mutation in HipA increases the persistence level

52. Toxin-Antitoxin Module

53. Toxin-Antitoxin Module
Stationary
phase
Time
Fresh
medium A
If A is present in large number in the cell, it will enter to a dormant state in which it is immune to antibiotic
This state is characterized by a long lag time.

54. Toxin-Antitoxin Module
HipA expression level
(mCherry fluorescence) [a.u.]
Lag time [min]
Lag time [a.u.]
HipA expression level (Atot) [a.u.]
Threshold behavior
Dormant/persister state if free A is large enough

55. Toxin-Antitoxin Module
Fraction of persisters = Prob(A>A0) [Threshold]
Dependence on A and B production
Fraction of persisters
Theory
Experiment
HipA expression level
(mCherry fluorescence) [a.u.]
HipA expression level (Atot) [a.u.]
P(Free A > A0)
Rotem, Loinger, Ronin, Levin-Reisman, Gabay, Biham and Balaban, preprint

56. Summary
Direct integration of the master equation and Monte Carlo simulations are not feasible for the simulation of complex reaction networks.
The moment equations provide an efficient method for the evaluation of reaction rates. The multiplane method also provides a good approximation of the distribution.
The number of equations is dramatically reduced compared to the master equation. The moment equations include one equation for each species and one equation for each reaction.
The moment equations are easy to construct using a diagrammatic method.