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.
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.
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
2
P
  F ( x) 
P
=D
- 

P
t
x 2 x  h

A
E ( x)
F ( x) = x
dP( A, t ) = k1P ( A, t ) + k2 P ( B, t )
dt
B
k1
A
k2
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
B
2X+Y
k1
k-1
k2
k3
X
Y
3X
The Law of Mass Action and
Differential Equations
d c (t)
2c
=
k
c
k
c
+k
c
1 A
-1 x
3 x y
dt x
d c (t)
2c
=
k
c
k
c
y
2 B
3 x y
dt
Nonlinear Chemical Oscillations
a = 0.1, b = 0.1
u
a = 0.08, b = 0.1
u
A New Mathematical Theory of
Chemical and Biochemical
Reaction Systems based on BirthDeath Processes that Include
Concentration Fluctuations and
Applicable to small systems.
The Basic Markovian Assumption:
X+Y
k1
Z
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.
A Markovian Chemical BirthDeath Process
k1(nx+1)(ny+1)
k1nxny
nZ
k-1nZ
X+Y
k-1(nZ +1)
k1
k-1
Z
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.
Nonlinear Biochemical Reaction
Systems: Stochastic Version
A
B
2X+Y
k1
k-1
k2
k3
X
Y
3X
k 2 nB
k 2 nB
(n-1,m+1)
(n,m+1)
(n-1,m)
(n,m)
k-1m
k 2 nB
k 1 nA
(0,2)
k 2 nB
k 1 nA
(n+1,m+1)
k 2 nB
k3 (n-1)n(m+1)
k 1 nA
k3 (n-2)(n-1)(m+1)
k 1 nA
k3 (n-2)(n-1)n
k 2 nB
(n+1,m)
k-1(m+1)
k 2 nB
k3 n (n-1)m
k 1 nA
(n,m-1)
(n+1,m-1)
k-1(n+1)
(1,2)
(0,1)
(1,1)
(2,1)
k 2 nB
k 1 nA
(0,0) k
-1
2k
k 2 nB
k 2 nB 3
k 1 nA
k 1 nA
(1,0) 2k (2,0) 3k (3,0)
-1
-1
4k-1
Stochastic Markovian Stepping
Algorithm (Monte Carlo)
(n-1,m)
k 2 nB
k 1 nA
(n,m)
k-1n
(n+1,m)
k3 n (n-1)m
(n,m-1) (n+1,m-1)
Next time T
and state j?
(T > 0, 1< j < 4)
l =q1+q2+q3+q4 = k1nA+ k-1n+ k2nB+ k3n(n-1)m
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)
0
p1 p1+p2
p1+p2+p3
r2
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
dP ( n X , nY , t )
dt
= - [ k1 n A + k -1 n X + k 2 nB + k 3 n X ( n X - 1) nY ] P ( n , n )
X
Y
+ k1 n A P ( n X - 1, nY ) + k 2 nB P ( n X , nY - 1)
+ k -1 ( n X + 1)P ( n X + 1, nY )
+ k 3 ( n X - 1)( n X - 2)( nY + 1)P ( n X - 1, nY + 1)
Stochastic Dynamics: Thermal
Fluctuations vs. Temporal Complexity
P ( u , v , t )
=   (DP - FP )
t
s  a + u + u2v - u2v 

D = 
2
2 
Stochastic
b + u v
2 Deterministic,
 -u v
Temporal Complexity
 a - u + u 2v 

F = 

2
b
u
v


Number of molecules
Temporal dynamics should not
be treated as noise!
(A)
(D)
(B)
(E)
(C)
(F)
Time
A Second Example: Simple
Nonlinear Biochemical Reaction
System From Cell Signaling
We consider a simple
phosphorylation-dephosphorylation
cycle, or a GTPase cycle:
ATP
ADP
k1
S
k-1
A
A*
k2
I
k-2
Pi
Ferrell & Xiong, Chaos, 11, pp. 227-236 (2001)
with a positive feedback
From Zhu, Qian and Li (2009) PLoS ONE. Submitted
From Cooper and Qian (2008) Biochem., 47, 5681.
Two Examples
Simple Kinetic Model based on the
Law of Mass Action
NTP
NDP
*
E
R
R*
d[ R ]
= J1 - J2 ,
dt
* χ
J 1 = (α[ E ][ R ] )[ R],
J 2 = β[ P ][ R ].
*
P
Pi
activation level: f
Bifurcations in PdPC with Linear
and Nonlinear Feedback
hyperbolic
delayed onset
1
c=0
bistability
c=1
c=2
1
4
activating signal: q
Markov Chain Representation
K
R*
R
P
v0
0R*
v1
1R*
w0
v2
2R*
w1
3R*
w2
…
(N-1)R*
NR*
Steady State Distribution for
Number Fluctuations
k -1
pk
pk pk - 1
v
p1
=

=
,
p0
pk - 1 pk - 2
p0  = 0 w
 k -1

v 
p0 = 1 +  
 k = 1  =0 w 
-1
Large V Asymptotics
v
v
= exp  log

w
 =1
 =1 w
v( x)
 exp V  dx log
w( x)
= exp (- Vφ( x) )
Beautiful, or Ugly Formulae
Bistability and Emergent Sates
Pk
number of R* molecules: k
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.


-1
lim Pr sup V X V ( s) - x( s)  ε  = 0;
V 
 s t

-1
lim V X V (0) = x0 .
V 
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
Newton’s Law of Motion
ħ→0
The Schrödinger’s Eqn.
y (x1,x2, …, xn,t)
x1(t), x2(t), …, xn(t)
V→
The Law of Mass Action
c1(t), c2(t), …, cn(t)
The Chemical Master Eqn.
p(N1,N2, …, Nn,t)
The Semiclassical Theory.
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 nonobvious from the level of pair-wise, static, molecule
interactions. It can only be understood from a
mesoscopic, open driven chemical dynamic
system perspective.
In a cartoon fashion
ny
cy
chemical master equation
B
A
(a)
(b)
fast nonlinear differential equations
nx
A
(d)
B
probability
discrete stochastic model
among attractors
cx
emergent slow stochastic dynamics
and landscape
A
B
(c)
appropriate reaction coordinate
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 multiattractor 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.
Relaxation, Wating, Barrier Crossing:
R-W-BC of Stochastic Dynamics
alternative
attractor
abrupt
transition
Relaxation
process
local
attractor
• 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 epigenetic 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!