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Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems

Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems. Jianghai Hu, Wei-Chung Wu Denise Wolf Shankar Sastry. Stanford University University of California at Berkeley Lawrence Berkeley National Laboratory

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Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems

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  1. Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems Jianghai Hu, Wei-Chung Wu Denise Wolf Shankar Sastry Stanford University University of California at Berkeley Lawrence Berkeley National Laboratory University of California at Berkeley

  2. Overview • Goal • To study quantitatively the subtilin production in B. subtilis • cells during different phases of population growth • Hierarchical Model • Microscopic: each cell is modeled as a stochastic hybrid system • Macroscopic: ODEs for population level variables • Simulations • SHS method vs. averaging method

  3. Bacillus subtilis • Bacillus subtilisATCC 6633 • Survival strategies under environmental stresses • Synthesis of antibiotics (e.g. subtilin) • Sporulation (forming spores) • Chemotaxis (moving towards chemical stimulus) courtesy of Adam Arkin

  4. Phosphate Metabolism Degradative Enzyme Synthesis Antibiotic Synthesis Competence Chemotaxis ENVIRONMENTAL SIGNALS Aerobic/anaerobic Respiration Starvation, high cell density, … Nitrogen Metabolism Sporulation Stress Response Network Courtesy of Denise Wolf

  5. Subtilin Synthesis Model Biosynthesis pathway of subtilin ([Banerjee & Hansen, 1988], [Entian & de Vos, 1996])

  6. Subtilin Synthesis Model spaS spaRK

  7. Subtilin Synthesis Model SpaRK spaS spaRK

  8. Subtilin Synthesis Model SpaS SpaRK spaS spaRK subtilin

  9. Subtilin Synthesis Model SigH SpaS SpaRK spaS spaRK S1

  10. Subtilin Synthesis Model SigH SpaS SpaRK spaS spaRK S2 S1

  11. Subtilin Synthesis Model SigH SpaS SpaRK spaS spaRK S2 S1

  12. absorption by medium at low cell density environment X: nutrient level subtilin Interaction with Environment SigH SpaS SpaRK spaS spaRK S2 S1

  13. Stochastic Hybrid System Model X SigH SpaS SpaRK spaS spaRK S2 S1 subtilin • Continuous States: [SigH], [SpaRK], [SpaS] • Discrete States: S1 , S2 • Input: X • Output: [SpaS] • Randomness in switching of S1 , S2

  14. Synthesis of SigH X SigH SpaS SpaRK spaS spaRK S2 S1 [SigH]: concentration level of SigH -1[SigH], if X  X0 d[SigH] dt = -1[SigH] + k3 ,if X < X0 natural decaying rate threshold

  15. SigH SpaRK SpaS spaS spaRK S2 S1 • Synthesis of SpaRK is controlled by the random switch S1 -2[SpaRK], ifS1is off(0) d[SpaRK] dt = -2[SpaRK] + k4 ,if S1is on(1) • S1switches every time according to a Markov chain with transition matrix • , where a0anda1depend on [SigH]. 1-a0a0 a1 1-a1 -Grk /RT -Grk /RT • Thermodynamics constraint [Shea & Ackers, 1985]: 1 + e e [SigH] [SigH] prk= Grk: Gibbs free energy when S1 is on R: Gas constant T: Temperature Synthesis of SpaRK

  16. SigH SpaRK SpaS spaS spaRK S2 S1 • Synthesis of SpaS is controlled by the random switch S2 -3[SpaS], ifS2is off(0) d[SpaS] dt = -3[SpaS] + k5 ,ifS2is on(1) • S2switches every time according to a Markov chain with transition matrix • , where b0andb1depend on [SpaRK]. 1-b0b0 b1 1-b1 -Gs /RT -Gs /RT • Thermodynamics constraint: 1 + e e [SpaRK] [SpaRK] ps= Gs: Gibbs free energy when S2 is on R: Gas constant T: Temperature Synthesis of SpaS

  17. (0,0) (0,1) (1,0) (1,1) Single B. subtilis Cell as a SHS • Continuous state ([SigH], [SpaRK], [SpaS])R3 • Discrete state (S1, S2)  {0,1}{0,1} • Input is nutrient level X • Output is [SpaS] X [SpaS]

  18. -2[SpaRK], ifS1is off(0) d[SpaRK] dt = -2[SpaRK] + k4 ,if S1is on(1) [SpaRK]  0, ifS1is off(0) [SpaRK]  k4/2,if S1is on(1) Assume S1 is in equilibrium distribution [SpaRK]n , n=0, 1, …, is a Markov chain [SpaRK]n  [SpaRK]n with probability 1-prk [SpaRK]n  [SpaRK]n + (1-) k4/2 with probability prk Analysis of [SpaRK] • Fix [SigH], analyze [SpaRK] Random switching between two stable equilibria

  19. Equilibrium Distribution of [SpaRK]n • Suppose  = 1/m, k4/2=1 • Distribution of [SpaRK]nas nconcentrates on [0,1] • Start rational, stay rational [SpaRK]n=0.d1…dl xxx… with probability (prk)d (1- prk)l-d, d=d1+…+dl (m-nary)

  20. consumption production Macroscopic Model • Population density D governed by the logistic equation • Maximal sustainable population size D= min (X, Dmax) • Dynamics of nutrient level X dD dt = rD(1 - D/D) dX dt = -k1D + k2 D [SpaS]avg

  21. b(X) …….. N 0 1 2  consumption production Birth-and-Death Chain for Population Dynamics • Population size, n, is governed by a birth-and-death chain • Birth rate, b(X), depends on the nutrient level X • Death rate, , is fixed • Dynamics of nutrient level Xbecomes dX dt = -k1n + k2 n [SpaS]

  22. Some Remarks • Hierarchical Model • Macroscopic model: ODEs • Microscopic model: SHS • A compromise between accuracy and complexity • More accurate model: • Population size as a birth-and-death Markov chain • Each cell has an exponentially distributed life span • Simpler model: averaging method

  23. Synthesis of SpaRK is controlled by the random switch S1 -2[SpaRK], ifS1is off(0) d[SpaRK] dt = -2[SpaRK] + k4 ,if S1is on(1) Averaging Dynamics of [SpaRK] SigH SpaRK SpaS spaS spaRK S2 S1 • In equilibrium distribution S1is on with probability prk • The averaged dynamics of [SpaRK] is d[SpaRK] dt = -2[SpaRK] +prkk4

  24. Simulations: A Typical System Trajectory

  25. Different Time Scales • Slower and more deterministic dynamics • D, X • [SigH] • Faster and more random dynamics • [SpaRK] • [SpaS] • Time scale analysis • Fixed slower variables, analyze the probabilistic property of the faster variables

  26. Observations • Slow variables follow a limit circle • Fast variables induce random fluctuations around it • Variance of fluctuations can be controlled by changing the transit probabilities of the switches • The less frequent the switches, the larger the variance

  27. t t t Simulations: Production vs. Max Density Production level of subtilin as a function of time under different maximal carrying capacity Dmax SHS Averaged

  28. Average Subtilin Production vs. Maximal Carrying Capacity There is a threshold of Dmax below which subtilin is not produced consistently, and above which it is produced at linearly increasing rate with Dmax [Kleerebezem & Quadri, 2001]

  29. Average Subtilin Production vs. Production Rate of SigH

  30. Conclusions and Future Directions • A stochastic hybrid system model of subtilin synthesis • Analysis of fast variable dynamics • Numerical simulations • A birth-and-death chain for population growth • More complete system dynamics • Model validation • Acknowledgement: • This work is in collaboration with Prof. Adam Arkin (LBNL)

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