Transcript From chemical reaction systems to cellular states: A computational Approach Hong Qian Department of Applied Mathematics University of Washington.

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
Protein: folded
Channel: closed
unfolded
open
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,
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.
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
k1[S]
E
k-1
ES
k2
E+
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,
T =
1
1
k2
k -1
+
+
+
0
T
k 1 [S ] k -1 + k 2 k -1 + k 2
k -1 + k 2
Mean Waiting Time is the
Double Reciprocal Relation!
k -1 + k 2
1
+
T =
k 1k 2 [ S ] k 2
KM 1
1
=
+
v max [ S ] v max
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 BirthDeath Processes that Include
Concentration Fluctuations and
Applicable to small chemical
systems such as a cell.
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 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.
A Markovian Chemical BirthDeath Process
k1(nx+1)(ny+1)
k1nxny
nx,ny
nZ
k-1nZ
X+Y
k-1(nZ +1)
k1
k-1
Z
An Example: Simple Nonlinear
Reaction System
A+2X
a1
a2
b1
X+B
b2
3X
C
a1ak (k - 1)
b 1bk
k
b 2c
vk = a1ak (k - 1) + b 2 c
wk = a2 (k + 1)k(k - 1) + b1b(k + 1)
a 2 k (k - 1)(k - 2)
b 2c
0
b 2c
1
b 1b
vk - 1
2
2 b 1b
vk
N
k
wk -1
wk
number of X molecules
Steady State Distribution for
Number Fluctuations
k -1
pk
pk pk - 1
v
p1
=
 =
,
p0 pk - 1 pk - 2
p0  = 0 w

v 
p0 = 1 +  
 k = 1  = 0 w 

k -1
-1
Nonequilibrum Steady-state
(NESS) and Bistability
a1 = a2 = 10 -6
b1 = b 2 = 0.05
a=500, b=1, c=20
ab
= 25   1
c
The Steady State is not an
Chemical Equilibrium! Quantifying
the Driving Force:
a1
b1
C
A+2X a 3X, X+B
b2
2
a2b2
 ab 
- DG o / kBT
 c  =a b =e
 
1 1
eq
A+B
C,
a1 b 1 ab
ab
DG = DG + kBT ln = kBT ln
c
a2 b 2 c
o
Without Chemical Potential
Driving the System:
ab a 2 b 2
if :
=
, DG = 0,
c a1 b 1
vk
β2 c
α1a
θ
then :
=
=
=
wk β1b( k + 1) α2 ( k + 1) k + 1
pk θ
-θ
and :
=
, p0 = e
p0 k!
k
An Example: The Oscillatory
Biochemical Reaction Systems
(Stochastic Version)
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
The Phase Space
a = 0.1, b = 0.1
u
a = 0.08, b = 0.1
u
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 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 P r sup V X V ( s ) - x( s )  ε  = 0;
V 
 s t

-1
lim V X V (0 ) = x0 .
V 
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
y (x1,x2, …, xn,t)
x1(t), x2(t), …, xn(t)
V→
The Law of Mass Action
c1(t), c2(t), …, cn(t)
The Schrödinger’s Eqn.

The Chemical Master Eqn.
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