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

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

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)

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, 170601 (2004) Barzel and Biham, ApJ 658, L37 (2007)

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

Macroscopic surface Test tube Sub-micron grain Cell

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,…

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)

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

The probability distribution

n P(n) n

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

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

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

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

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

Monte Carlo methods no

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

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

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

The Multiplane Method Lipshtat and Biham, PRL 93, 170601 (2004) Barzel, Biham and Kupferman, PRE 76, 026703 (2007)

The multiplane method

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

The Reduced Multiplane Method

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) 026703 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

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

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

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

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., 115 20941 (2007) B. Barzel and O. Biham, J. Chem. Phys. 127, 144703 (2007)

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…

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

Results: the methanol network

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

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

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

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

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

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

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)

The Exclusive Switch

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

The Repressilator Monte Carlo Simulations Rate equations

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?

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

The Switch General Switch Exclusive Switch A A A A A A A A b a b a A A

The Switch This does not hold for a high plasmid copy number 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)

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

The Switch General Switch Exclusive Switch Weakens A Does not affect B Weakens A and Strengthens B

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)

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

Toxin-Antitoxin Module

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.

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

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

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.