From chemical reaction systems to cellular states: A

computational Approach

Hong Qian

Department of Applied MathematicsUniversity 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

unfolded Protein: folded

open Channel: 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, 737-741.

Protein Copy Numbers in Yeast

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

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, 1877-1882.

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/(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,

Tkk

kkk

kkkSkT

21

1

21

2

21101

][1

E ESk1[S]

k-1

k2

E+

Mean Waiting Time is the Double Reciprocal Relation!

maxmax

221

21

1][

1

1][

vSvK

kSkkkkT

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+Y Zk1

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 t, any one particular unbonded

X will react with any one particular unbonded Y with probability k1t + o(t),

where k1 is the reaction rate.

A Markovian Chemical Birth-Death Process

nZ

k1nxnyk1(nx+1)(ny+1)

k-1nZ k-1(nZ +1)

k1X+Y Zk-1

nx,ny

An Example: Simple Nonlinear Reaction System

A+2X 3X1

2

X+B C1

2

( )( )2k k 1 k 2

k

( )1ak k 1

2c1bk

number of X molecules

0 1 2 Nk

( )k 1 2v ak k 1 c

2c 2c

1b 12 b

( ) ( ) ( )k 2 1w k 1 k k 1 b k 1

kv

k 1w kw

k 1v

Steady State Distribution for Number Fluctuations

1

1k

1k

00

1k

00

1

2k

1k

1k

k

0

k

wv

1p

wv

pp

pp

pp

pp

,

Nonequilibrum Steady-state (NESS) and Bistability

.1 2 0 05

61 2 10

a=500, b=1, c=20

125c

ab

defining cellular states

The Steady State is not an Chemical Equilibrium! Quantifying

the Driving Force:

A+2X 3X,1

2X+B C

1

2

A+B C, /oB

eqG k T2 2

1 1

abe

c

ln lno 1 1B B

2 2

ab abG G k T k T

c c

Without Chemical Potential Driving the System:

: , ,2 2

1 1

abif G 0

c

1kθ

1kαaα

1kbβcβ

wv

then2

1

1

2

k

k

)()(:

θ0

k

0

k epkθ

pp

and ,!

:

An Example: The Oscillatory Biochemical Reaction Systems

(Stochastic Version)

A Xk1

k-1

B Yk2

2X+Y 3Xk3

The Law of Mass Action and Differential Equations

dtd cx(t) = k1cA - k-1 cx+k3cx

2cy

k2cB - k3cx2cy=dt

d cy(t)

u u

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)

k1nA k1nA

k1nA

k1nA

k1nA

k2 nB

k2 nB

k2 nB

k2 nB

k2 nB2k3

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

(n,m-1)

k2 nB

k2 nB

(n-1,m+1)

k3 n (n-1)m

k3 (n-1)n(m+1)k3 (n-2)(n-1)(m+1)

k-1m k-1(m+1)

k2 nB k2 nBk2 nB

(n+1,m-1)k1nA

k3 (n-2)(n-1)n

Stochastic Markovian Stepping Algorithm (Monte Carlo)

=q1+q2+q3+q4 = k1nA+ k-1n+ k2nB+ k3n(n-1)m

Next time T and state j? (T > 0, 1< j < 4)

q3q1

q4

q2

(n,m)(n-1,m) (n+1,m)k1nA

(n,m-1)

k2 nB

k3 n (n-1)mk-1n

(n+1,m-1)

Picking Two Random Variables T & n derived from uniform r1 & r2 :

fT(t) = e - t, T = - (1/) ln (r1)

Pn(m) = km/, (m=1,2,…,4)

r2

0 p1 p1+p2 p1+p2+p3 p1+p2+p3+p4=1

Concentration Fluctuations

Stochastic Oscillations: Rotational Random Walks

a = 0.1, b = 0.1 a = 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

)1,1()1)(2)(1(),1()1(

)1,(),1(

),()1(),,(

3

1

21

3211

YXYXX

YXX

YXBYXA

YXYXXBXAYX

nnPnnnknnPnk

nnPnknnPnknnPnnnknknknk

dttnndP

Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity

FPPDt

tvuP

),,(

vubvu

vuvuuaD 22

22

2

vubvuuaF

2

2Stochastic Deterministic, Temporal Complexity

Time

Num

ber o

f mol

ecul

es

(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 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.

.)(lim

;)()(supPrlim

0V1

V

V1

tsV

x0XV

0εsxsXV

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 Motion The Schrödinger’s Eqn.ħ → 0

The Law of Mass Action The Chemical Master Eqn.V →

x1(t), x2(t), …, xn(t)

c1(t), c2(t), …, cn(t)

(x1,x2, …, xn,t)

p(N1,N2, …, Nn,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