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, the Newton’s equation of motion, is behind the molecular dynamics

That defines the biologically meaningful, discrete conformational state(s) of a protein unfoldedProtein: folded openChannel: closed enzyme: conformational change

Now onto cell biology …

We Know Many “Structures” (adrenergic regulation)

of Biochemical Reaction Systems (Cytokine Activation)

EGF Signal Transduction Pathway

What will be the “Equation” for the computational Cell Biology?

How to define a state or states of a cell?

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

levels of mRNA,

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

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

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.

The Stochastic Nature of Chemical Reactions

Single Channel Conductance

First Concentration Fluctuation Measurements (1972) (FCS)

Fast Forward to 1998

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

Michaelis-Menten Theory is in fact a Stochastic Theory in disguise…

Mean Product Waiting Time From S to P, it first form the complex ES with mean time 1/(k 1 [S]), then the dwell time in state ES, 1/(k -1 +k 2 ), after that the S either becomes P or goes back to free S, with corresponding probabilities k 2 /(k -1 +k 2 ) and k -1 /(k -1 +k 2 ). Hence, T kk k kk k kkS k T ][ 1          E ES k 1 [S] k -1 k2k2 E+E+

Mean Waiting Time is the Double Reciprocal Relation! max ][ 1 1 ][ vSv K kSkk kk T M     

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.

The Basic Markovian Assumption: X+YZ k1k1 The chemical reaction contain n X molecules of type X and n Y molecules of type Y. X and Y associate to form Z. In a small time interval of  t, any one particular unbonded X will react with any one particular unbonded Y with probability k 1  t + o(  t), where k 1 is the reaction rate.

A Markovian Chemical Birth- Death Process nZnZ k1nxnyk1nxny k 1 (n x +1)(n y +1) k -1 n Z k -1 (n Z +1) k1k1 X+YZ k -1 n x,n y

An Example: Simple Nonlinear Reaction System A+2X3X 11 22 X+B C 11 22

k number of X molecules 012 N k

Steady State Distribution for Number Fluctuations

Nonequilibrum Steady-state (NESS) and Bistability a=500, b=1, c=20 defining cellular states

The Steady State is not an Chemical Equilibrium! Quantifying the Driving Force: A+2X 3X, 11 22 X+B C 11 22 A+BC,

Without Chemical Potential Driving the System:

An Example: The Oscillatory Biochemical Reaction Systems (Stochastic Version) A X k1k1 k -1 BY k2k2 2X+Y 3X k3k3

The Law of Mass Action and Differential Equations dt d c x (t) =k 1 c A - k -1 c x +k 3 c x 2 c y k 2 c B - k 3 c x 2 c y = dt d c y (t)

uu a = 0.1, b = 0.1 a = 0.08, b = 0.1 The Phase Space

(0,0) (0,1) (0,2) (1,0) (1,1) (2,0) (1,2) (3,0) (2,1) k1nAk1nA k1nAk1nA k1nAk1nA k1nAk1nA k1nAk1nA k 2 n B 2k 3 k -1 2k -1 3k -1 4k -1 k -1 (n+1) (n,m)(n-1,m) (n+1,m) (n,m+1)(n+1,m+1) k1nAk1nA k1nAk1nA (n,m-1) k 2 n B (n-1,m+1) k 3 n (n-1)m k 3 (n-1)n(m+1) k 3 (n-2)(n-1)(m+1) k -1 m k -1 (m+1) k 2 n B (n+1,m-1) k1nAk1nA k 3 (n-2)(n-1)n

Stochastic Markovian Stepping Algorithm (Monte Carlo)  =q 1 +q 2 +q 3 +q 4 = k 1 n A + k -1 n+ k 2 n B + k 3 n(n-1)m Next time T and state j? (T > 0, 1< j < 4) q3q3 q1q1 q4q4 q2q2 (n,m) (n-1,m)(n+1,m) k1nAk1nA (n,m-1) k 2 n B k 3 n (n-1)m k -1 n (n+1,m-1)

Picking Two Random Variables T & n derived from uniform r 1 & r 2 : f T (t) = e - t, T = - (1/ ) ln (r 1 ) P n (m) = k m / , (m=1,2,…,4) r2r2 0 p1p1 p 1 +p 2 p 1 +p 2 +p 3 p 1 +p 2 +p 3 +p 4 =1

Concentration Fluctuations

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

Defining Biochemical Noise

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?

Continuous Diffusion Approximation of Discrete Random Walk Model

Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity  FPPD t tvuP    ),,(            vubvu vuvuua D             vub vuua F 2 2 StochasticDeterministic, Temporal Complexity

Time Number of molecules (A) (C) (D) (B)(E) (F) 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 X V (0), X V (t), approaches to x(t), the deterministic solution of the differential equations, based on the law of mass action, with initial condition x 0.

Therefore, the stochastic CME model has superseded the deterministic law of mass action model. It is not an alternative; It is a more general theory.

The Theoretical Foundations of Chemical Dynamics and Mechanical Motion Newton’s Law of MotionThe Schrödinger’s Eqn. ħ → 0 The Law of Mass ActionThe Chemical Master Eqn. V → x 1 (t), x 2 (t), …, x n (t) c 1 (t), c 2 (t), …, c n (t)  (x 1,x 2, …, x n,t) p(N 1,N 2, …, N n,t)

What we have and what we need? A theory for chemical reaction networks with small (and large) numbers of molecules in terms of the CME It requires all the rate constants under the appropriate conditions One should treat the rate constants as the “force field parameters” in the computational macromolecular structures.

Analogue with Computational Molecular Structures – 40 yr ago? While the equation is known in principle (Newton’s equation), the large amount of unknown parameters (force field) makes a realistic computation very challenging. It has taken 40 years of continuous development to gradually converge to an acceptable “set of parameters” The issues are remarkably similar: developing a set of rate constants for all the basic biochemical reactions inside a cell, and predict biological (conformational) states, extracting the kinetics between them, and ultimately, functions. (c.f. the rate of transformation into a cancerous state.)

Open-system nonequilibrium Thermodynamics

Recent Developments