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From chemical reaction systems to cellular states: A computational Approach. Hong Qian Department of Applied Mathematics University of Washington. The Computational Macromolecular Structure Paradigm:. From Protein Structures. To Protein Dynamics. In between,.
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From chemical reaction systems to cellular states: A computational Approach Hong Qian Department of Applied Mathematics University of Washington
In between, the Newton’s equation of motion, is behind the molecular dynamics
Protein: folded unfolded Channel: closed open That defines the biologically meaningful, discrete conformational state(s) of a protein enzyme: conformational change
We Know Many “Structures” (adrenergic regulation)
of Biochemical Reaction Systems (Cytokine Activation)
What will be the “Equation” for the computational Cell Biology?
Biochemistry defines the state(s) of a cell via concentrations of metabolites and copy numbers of proteins.
Protein Copy Numbers in Yeast Ghaemmaghami, S. et. al. (2003) “Global analysis of protein expression in yeast”, Nature, 425, 737-741.
Metabolites Levels in Tomato Roessner-Tunali et. al. (2003) “Metabolic profiling of transgenic tomato plants …”, Plant Physiology, 133, 84-99.
We outline a mathematical theory to define cellular state(s), in terms of its metabolites concentrations and protein copy numbers, based on biochemical reaction networks structures.
(FCS) First Concentration Fluctuation Measurements (1972)
0.2mM 2mM Lu, P.H., Xun, L.-Y. & Xie, X.S. (1998) Science, 282, 1877-1882. Stochastic Biochemical Kinetics
Michaelis-Menten Theory is in fact a Stochastic Theory in disguise…
k1[S] k2 ES E+ E k-1 1 1 k k = + + + - 2 1 T 0 T k + + + [ S ] k k k k k k 1 - - - 1 2 1 2 1 2 Mean Product Waiting Time From S to P, it first form the complex ES with mean time 1/(k1[S]), then the dwell time in state ES, 1/(k-1+k2), after that the S either becomes P or goes back to free S, with corresponding probabilities k2 /(k-1+k2) and k-1 /(k-1+k2). Hence,
+ k k 1 = + - 1 2 T k k [ S ] k 1 2 2 K 1 1 = + M v [ S ] v max max Mean Waiting Time is the Double Reciprocal Relation!
Traditional theory for chemical reaction systems is based on the law of mass-action and expressed in terms of ordinary differential equations (ODEs)
The New Stochastic Theory of Chemical and Biochemical Reaction Systems based on Birth-Death Processes that Include Concentration Fluctuations and Applicable to small chemical systems such as a cell.
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 associate 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.
k1(nx+1)(ny+1) k1nxny k-1nZ k-1(nZ +1) k1 X+Y Z k-1 A Markovian Chemical Birth-Death Process nx,ny nZ
a1 A+2X 3X a2 b1 C X+B b2 An Example: Simple Nonlinear Reaction System
number of X molecules k N 0 1 2 k
defining cellular states a=500, b=1, c=20 Nonequilibrum Steady-state (NESS) and Bistability
a1 b1 3X, C A+2X X+B a2 b2 A+B C, The Steady State is not an Chemical Equilibrium! Quantifying the Driving Force:
k1 A X k-1 k2 B Y k3 2X+Y 3X An Example: The Oscillatory Biochemical Reaction Systems (Stochastic Version)
d cx(t) = k1cA - k-1 cx+k3cx2cy dt d cy(t) = k2cB - k3cx2cy dt The Law of Mass Action and Differential Equations
a = 0.08, b = 0.1 a = 0.1, b = 0.1 u u The Phase Space
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
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
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
a = 0.1, b = 0.1 a = 0.08, b = 0.1 Stochastic Oscillations: Rotational Random Walks
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?
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
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
(D) (A) (B) (E) Number of molecules (C) (F) Time Temporal dynamics should not be treated as noise!
A Theorem of T. Kurtz (1971) In the limit of V →∞, the stochastic solution to CME in volume V with initial condition XV(0), XV(t), approaches to x(t), the deterministic solution of the differential equations, based on the law of mass action, with initial condition x0.
