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 …

The current state of affair of cell biology:

(1) Genomics: A,T,G,C symbols

(2) Biochemistry: molecules

Experimental molecular genetics defines the state(s) of a cell by their “transcription pattern” via

expression level (i.e., RNA microarray).

Biochemistry defines the state(s) of a cell via

concentrations of metabolites and copy numbers of proteins.

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

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

How does computational biology define the biological phenotype(s)

of a cell in terms of the biochemical copy numbers of proteins?

Theoretical Basis:

The Chemical Master Equations: A New Mathematical

Foundation of Chemical and Biochemical Reaction Systems

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)

Single Channel Conductance

First Concentration Fluctuation Measurements (1972)

(FCS)

Fast Forward to 1998

Stochastic Enzyme Kinetics

0.2mM

2mM

Lu, P.H., Xun, L.-Y. & Xie, X.S. (1998) Science, 282, 1877-1882.

Stochastic Chemistry (1940)

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.

Kramers’ Theory, Markov Process & Chemical Reaction Rate

PxF

xx

PD

tP

)(

2

2

xxE

xF )(

)(

A Bk2

k1

),(),(),(

21 tBPktAPkdt

tAdP

A B

But cellular biology has more to do with reaction systems

and networks …

Traditional theory for chemical reaction systems is based on

the law of mass-action

Nonlinear Biochemical Reaction Systems and Kinetic Models

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

Nonlinear Chemical Oscillations

A New Mathematical Theory of Chemical and Biochemical

Reaction Systems based on Birth-Death Processes that Include

Concentration Fluctuations and Applicable to small systems.

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 bond 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)

k1

X+Y Zk-1

Chemical Master Equation Formalism for Chemical

Reaction SystemsM. 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.

Nonlinear Biochemical Reaction Systems: Stochastic Version

A Xk1

k-1

B Yk2

2X+Y 3Xk3

(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 nB

2k3

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-1mk-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+p3p1+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

nnPnnnk

nnPnk

nnPnknnPnk

nnPnnnknknknkdt

tnndP

Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity

FPPDt

tvuP

),,(

vubvuvuvuuaD 22

22

2

vubvuua

F2

2

Stochastic Deterministic, Temporal Complexity

Time

Num

ber

of m

olec

ules

(A)

(C)

(D)

(B) (E)

(F)

Temporal dynamics should not be treated as noise!

A Second Example: Simple Nonlinear Biochemical Reaction

System From Cell Signaling

We consider a simple phosphorylation-dephosphorylation

cycle, or a GTPase cycle:

A A*

S

ATP ADP

I

Pi

k1

k-1

k2

k-2

Ferrell &

Xiong, C

haos, 11, pp. 227-236 (2001)with a positive feedback

Two ExamplesF

rom

Coo

per

and

Qia

n (2

008)

Bio

chem

., 4

7, 5

681.

From

Zhu, Q

ian and Li (2009) PLoS

ON

E. S

ubmitted

Simple Kinetic Model based on the Law of Mass Action

NTP NDP

Pi

E

P

R R*

].][[

],)[]][[(

,][

*

*

*

RPβJ

RREαJ

JJdt

Rd

2

χ1

21

activating signal:

acti

vati

on

leve

l: f

1 4

1

Bifurcations in PdPC with Linear and Nonlinear Feedback

= 0

= 1

= 2

hyperbolic delayed onset

bistability

R R*

K

P

2R*0R* 1R* 3R* … (N-1)R* NR*

Markov Chain Representation

v1

w1

v2

w2

v0

w0

Steady State Distribution for Number Fluctuations

1

1k

1k

00

1k

00

1

2k

1k

1k

k

0

k

w

v1p

w

v

p

p

p

p

p

p

p

p

,

Large V Asymptotics

)(exp

)(

)(logexp

logexp

xφV

xw

xvdxV

w

v

w

v

11

Beautiful, or Ugly Formulae

Bistability and Emergent Sates

Pk

number of R* molecules: k

defining cellular attra

ctors

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

We Prove a Theorem on the CME for Closed Chemical Reaction Systems• We define closed chemical reaction systems

via the “chemical detailed balance”. In its steady state, all fluxes are zero.

• For ODE with the law of mass action, it has a unique, globally attractive steady-state; the equilibrium state.

• For the CME, it has a multi-Poisson distribution subject to all the conservation relations.

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

The Semiclassical Theory.

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)

Chemical basis of epi-genetics:

Exactly same environment setting and gene, different internal

biochemical states (i.e., concentrations and fluxes). Could

this be a chemical definition for epi-genetics inheritance?

Chemistry is inheritable!

Emergent Mesoscopic Complexity• It is generally believed that when systems become

large, stochasticity disappears and a deterministic dynamics rules.

• However, this simple example clearly shows that beyond the “infinite-time” in the deterministic dynamics, there is another, emerging stochastic, multi-state dynamics!

• This stochastic dynamics is completely non-obvious from the level of pair-wise, static, molecule interactions. It can only be understood from a mesoscopic, open driven chemical dynamic system perspective.

A B

discrete stochastic model among attractors

ny

nx

chemical master equation cy

cx

A

B

fast nonlinear differential equations

appropriate reaction coordinate

ABpr

obab

ility

emergent slow stochastic dynamics and landscape

(a) (b)

(c)

(d)

In a cartoon fashion

The mathematical analysis suggests three distinct time scales,

and related mathematical descriptions, of (i) molecular

signaling, (ii) biochemical network dynamics, and (iii) cellular

evolution. The (i) and (iii) are stochastic while (ii) is deterministic.

The emergent cellular, stochastic “evolutionary” dynamics follows not

gradual changes, but rather punctuated transitions between

cellular attractors.

If one perturbs such a multi-attractor stochastic system:

• Rapid relaxation back to local minimum following deterministic dynamics (level ii);

• Stays at the “equilibrium” for a quite long tme;

• With sufficiently long waiting, exit to a next cellular state.

alternative attractor

localattractor

Relaxation process

abrupt transition

Relaxation, Wating, Barrier Crossing: R-W-BC of Stochastic Dynamics

• Elimination

• Equilibrium

• Escape

In Summary

• There are two purposes of this talk:

• On the technical side, a suggestion on computational cell biology, and proposing the idea of three time scales

• On the philosophical side, some implications to epi-genetics, cancer biology and evolutionary biology.

Into the Future:Toward a Computational

Elucidation of Cellular attractor(s) and inheritable epi-

genetic phenotype(s)

What do We Need?

• It requires a theory for chemical reaction networks with small numbers of molecules

• The CME theory is an appropriate starting point

• 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 Protein 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 essentially impossible.

• It has taken 40 years of continuous development to gradually converge to an acceptable “set of parameters”

• But the issues are remarkably similar: defining biological (conformational) states, extracting the kinetics between them, and ultimately, functions.

Thank You!