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Cellular Dynamics From A Computational Chemistry Perspective

Cellular Dynamics From A Computational Chemistry Perspective. Hong Qian Department of Applied Mathematics University of Washington. The most important lesson learned from protein science is ….

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Cellular Dynamics From A Computational Chemistry Perspective

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  1. Cellular Dynamics From A Computational Chemistry Perspective Hong Qian Department of Applied Mathematics University of Washington

  2. The most important lesson learned from protein science is …

  3. The current state of affair of cell biology:(1) Genomics: A,T,G,C symbols(2) Biochemistry: molecules

  4. Experimental molecular genetics defines the state(s) of a cell by their “transcription pattern” via expression level (i.e., RNA microarray).

  5. Biochemistry defines the state(s) of a cell via concentrations of metabolites and copy numbers of proteins.

  6. Protein Copy Numbers in Yeast Ghaemmaghami, S. et. al. (2003) “Global analysis of protein expression in yeast”, Nature, 425, 737-741.

  7. Metabolites Levels in Tomato Roessner-Tunali et. al. (2003) “Metabolic profiling of transgenic tomato plants …”, Plant Physiology, 133, 84-99.

  8. But biologists define the state(s) of a cell by its phenotype(s)!

  9. How does computational biology define the biological phenotype(s) of a cell in terms of the biochemical copy numbers of proteins?

  10. Theoretical Basis: The Chemical Master Equations: A New Mathematical Foundation of Chemical and Biochemical Reaction Systems

  11. The Stochastic Nature of Chemical Reactions • Single-molecule measurements • Relevance to cellular biology: small copy # • Kramers’ theory for unimolecular reaction rate in terms of diffusion process (1940) • Delbrück’s theory of stochastic chemical reaction in terms of birth-death process (1940)

  12. Single Channel Conductance

  13. (FCS) First Concentration Fluctuation Measurements (1972)

  14. Fast Forward to 1998

  15. 0.2mM 2mM Lu, P.H., Xun, L.-Y. & Xie, X.S. (1998) Science, 282, 1877-1882. Stochastic Enzyme Kinetics

  16. Stochastic Chemistry (1940)

  17. The Kramers’ theory and the CME clearly marked the domains of two areas of chemical research: (1) The computation of the rate constant of a chemical reaction based on the molecular structures, energy landscapes, and the solvent environment; and (2) the prediction of the dynamic behavior of a chemical reaction network, assuming that the rate constants are known for each and every reaction in the system.

  18. A B 2 ¶ ¶ ¶ æ ö P P F ( x ) = - ç ÷ D P 2 ¶ ¶ h t x ¶ è ø x ¶ E ( x ) = - F ( x ) ¶ x dP ( A , t ) k1 = - + k P ( A , t ) k P ( B , t ) 1 2 A B dt k2 Kramers’ Theory, Markov Process & Chemical Reaction Rate

  19. But cellular biology has more to do with reaction systems and networks …

  20. Traditional theory for chemical reaction systems is based on the law of mass-action

  21. k1 A X k-1 k2 B Y k3 2X+Y 3X Nonlinear Biochemical Reaction Systems and Kinetic Models

  22. d cx(t) = k1cA - k-1 cx+k3cx2cy dt d cy(t) = k2cB - k3cx2cy dt The Law of Mass Action and Differential Equations

  23. a = 0.08, b = 0.1 a = 0.1, b = 0.1 u u Nonlinear Chemical Oscillations

  24. A New Mathematical Theory of Chemical and Biochemical Reaction Systems based on Birth-Death Processes that Include Concentration Fluctuations and Applicable to small systems.

  25. k1 X+Y Z The Basic Markovian Assumption: The chemical reaction contain nX molecules of type X and nY molecules of type Y. X and Y bond to form Z. In a small time interval of Dt, any one particular unbonded X will react with any one particular unbonded Y with probability k1Dt + o(Dt), where k1 is the reaction rate.

  26. k1(nx+1)(ny+1) k1nxny nZ k-1nZ k-1(nZ +1) k1 X+Y Z k-1 A Markovian Chemical Birth-Death Process

  27. Chemical Master Equation Formalism for Chemical Reaction Systems M. Delbrück (1940) J. Chem. Phys. 8, 120. D.A. McQuarrie (1963) J. Chem. Phys. 38, 433. D.A. McQuarrie, Jachimowski, C.J. & M.E. Russell (1964) Biochem. 3, 1732. I.G. Darvey & P.J. Staff (1966) J. Chem. Phys. 44, 990; 45, 2145; 46, 2209. D.A. McQuarrie (1967) J. Appl. Prob. 4, 413. R. Hawkins & S.A. Rice (1971) J. Theoret. Biol. 30, 579. D. Gillespie (1976) J. Comp. Phys. 22, 403; (1977) J. Phys. Chem. 81, 2340.

  28. k1 A X k-1 k2 B Y k3 2X+Y 3X Nonlinear Biochemical Reaction Systems: Stochastic Version

  29. k2 nB k2 nB k2 nB (n-1,m+1) (n,m+1) (n+1,m+1) k2 nB k3 (n-1)n(m+1) k3 (n-2)(n-1)(m+1) k1nA k1nA (n+1,m) (n-1,m) (n,m) k-1(m+1) k-1m k2 nB k3 n(n-1)m k3 (n-2)(n-1)n k1nA (n,m-1) (n+1,m-1) k2 nB k-1(n+1) k1nA (0,2) (1,2) k2 nB k1nA (0,1) (1,1) (2,1) 2k3 k2 nB k2 nB k2 nB k1nA k1nA k1nA (3,0) (0,0) (1,0) (2,0) k-1 2k-1 3k-1 4k-1

  30. k2 nB k1nA (n,m) (n-1,m) (n+1,m) q3 k-1n q1 k3 n(n-1)m q2 (n,m-1) (n+1,m-1) q4 Next time T and state j? (T > 0, 1< j < 4) Stochastic Markovian Stepping Algorithm (Monte Carlo) l =q1+q2+q3+q4 = k1nA+ k-1n+ k2nB+ k3n(n-1)m

  31. p1 p1+p2 p1+p2+p3+p4=1 p1+p2+p3 0 Picking Two Random Variables T & n derived from uniform r1& r2: fT(t) = l e -l t, T = - (1/l)ln (r1) Pn(m) = km/l , (m=1,2,…,4) r2

  32. Concentration Fluctuations

  33. a = 0.1, b = 0.1 a = 0.08, b = 0.1 Stochastic Oscillations: Rotational Random Walks

  34. Defining Biochemical Noise

  35. An analogy to an electronic circuit in a radio If one uses a voltage meter to measure a node in the circuit, one would obtain a time varying voltage. Should this time-varying behavior be considered noise, or signal? If one is lucky and finds the signal being correlated with the audio broadcasting, one would conclude that the time varying voltage is in fact the signal, not noise. But what if there is no apparent correlation with the audio sound?

  36. dP n n t ( , , ) [ ] X Y = - k n k n k n k n n n + + + - P n n ( 1 ) ( , ) A X B X X Y - 1 1 2 3 X Y dt k n P n n k n P n n + - + - ( 1 , ) ( , 1 ) A X Y B X Y 1 2 k n P n n + + + ( 1 ) ( 1 , ) X X Y - 1 k n n n P n n + - - + - + ( 1 )( 2 )( 1 ) ( 1 , 1 ) X X Y X Y 3 Continuous Diffusion Approximation of Discrete Random Walk Model

  37. P u v t ¶ ( , , ) ( ) D P FP = Ñ × Ñ - t ¶ s æ ö a u u v u v + + - 2 2 ç ÷ D = ç ÷ u v b u v - + 2 2 2 è ø Stochastic Deterministic, Temporal Complexity æ ö a u u v - + 2 ç ÷ F = ç ÷ b u v - 2 è ø Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity

  38. (D) (A) (B) (E) Number of molecules (C) (F) Time Temporal dynamics should not be treated as noise!

  39. A Second Example: Simple Nonlinear Biochemical Reaction System From Cell Signaling

  40. We consider a simple phosphorylation-dephosphorylation cycle, or a GTPase cycle:

  41. ATP ADP k1 S k-1 A A* k2 I k-2 Pi with a positive feedback Ferrell & Xiong, Chaos, 11, pp. 227-236 (2001)

  42. Two Examples From Cooper and Qian (2008) Biochem., 47, 5681. From Zhu, Qian and Li (2009) PLoS ONE. Submitted

  43. NTP NDP E R R* P Pi Simple Kinetic Model based on the Law of Mass Action

  44. hyperbolic delayed onset 1 activation level: f bistability 1 4 activating signal: q Bifurcations in PdPC with Linear and Nonlinear Feedback c = 0 c = 1 c = 2

  45. K R* R P NR* … (N-1)R* 0R* 1R* 2R* 3R* Markov Chain Representation v0 v1 v2 w0 w1 w2

  46. Steady State Distribution for Number Fluctuations

  47. Large V Asymptotics

  48. Beautiful, or Ugly Formulae

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