Download Lecture 7. Two-State Systems

Lecture 7. Two-State Systems
Zhanchun Tu ( 涂展春 )
Department of Physics, BNU
Email: [email protected]
Homepage: www.tuzc.org
Main contents
●
Macromolecules with 2 states
●
State variable description of binding
●
Cooperative binding of Hemoglobin
●
RNA folding and unfolding
§7.1 Macromolecules with 2 states
Internal state variable idea
●
Examples of the internal state variable
description of macromolecules
●
Current trajectories and open probability for
Na+ channel subjected to different voltages
Ion channel
●
Energy landscape
incompressible
 Rout =
R
R
Rout
F =  s
Increasing driving
force
=−  A
0, closed
σ=
1, open
●
Open probability
[Nat. Struc. Biol. 9: 696 (2002)]
Δ ε=εclosed −εopen =−5 k B T
Problem: prove that
〈  〉= p open
Phosphorylation ( 磷酸化 )
●
Phosphorylation can alter the relative energies
of the active and inactive states of enzymes
The addition of a phosphate group
introduces a favorable electrostatic
interaction which lowers the active
state free energy with respect to the
inactive state free energy
σS = 0 inactive state
σS = 1 active state
σP = 0 unphosphorylated state
σP = 1 phosphorylated state
●
Probability in active states
Probability of the enzyme in active state, but
not phosphorylated.
Probability of the enzyme in active state
when phosphorylated.
when
§7.2 State variable description of
simple binding
Gibbs distribution
●
Open system with particle and
energy exchanges
N sN r =N u =const.
E sE r =E u
=const.
When the system stays a given state (Es(i), Ns(i)), the
number of states that the universe (=system+reservoir)
W u  E si , N is =
× W r  E u −E is  , N u −N si 
1
states of system
states of reservoir
Probability of finding a given state of the system
i 
i 
s
i 
s
p  E , N =
i 
W u E s , N s 
∑i W u  E si , N is 
Given Es(i) and Ns(i), we have
∝W r  E u− Esi , N u− N si 
S r  E u −E is  , N u −N is =k B lnW r  E u −E is  , N u −N is 
●
Gibbs distribution & grand partition function
i 
s
i
s
p  E , N ∝e
 i
 i
S r  Eu− E s , N u− N s / k B
∂ S r  i ∂ S r  i
S r  E u −E , N u −N =S r  E u , N u −
E −
N
∂Er s ∂N r s
i 
s
i 
s
i 
s
i 
s
i
i
−  E s − N s 
p  E , N ∝e
i
i
−  E s − N s 
=
e
Z
Simple ligand-receptor binding
revisited with Gibbs distribution
●
two states
–
Empty state σ = 0
–
Occupied state σ = 1
−  b− 
〈 N 〉=0× p 01× p1 = p1 =
e
−  b−
1e
§7.3 Cooperative binding of
Hemoglobin
Toy Model of a Dimeric Hemoglobin
●
Cooperativity parameter
Ising like model
p 0=
1
Z
−  −
p 1=
2e
Z
e−  2  J −2 
p 2=
Z
〈 N 〉=0× p 01× p1 2× p 2
J
J=0, non-cooperativity
Homework
Use the canonical distribution to redo the problem of dimoglobin binding. For
simplicity, imagine a box with N oxygen molecules which can be distributed
amongst Ω sites.
Calculate the probabilities p0, p1, and p2 corresponding to occupancy 0, 1, and 2,
Respectively. Draw the binding curves (i.e., the relations between p0, p1, p2 and
concentration of oxygen).
●
Monod–Wyman–Changeux (MWC) model
(1) Protein can exist in two distinct conformational states labeled T and R,
the energy of R state is higher than T state in amount of ε
(2) Ligand binding reaction has a higher affinity for the R state
 = R−T 0
0, site1 is empty
 1=
1, site1 is occupied
0, site2 is empty
2 =
1, site2 is occupied
0, protein in T state
m=
1, protein in R state
Please confirm:
Note: binding energy<0, higher affinity <=> lower binding energy
parameter
x
Remark: The first model contains more mechanistic details. However, in
many practical cases the coupling parameter (J) cannot be easily measured.
In such cases the MWC approximation allows quantitative treatments of
cooperative protein behavior using only two states and a few parameters.
Hierarchical models of 4 binding sites
●
Non-cooperative Model
states
●
Pauling model
states
exclude terms in the sum when α = γ
●
Adair model
states
fit
p 0=
1
2
3
4
14 K 1 x6 K 1 K 2 x 4 K 1 K 2 K 3 x K 1 K 2 K 3 K 4 x
p 1=
p 2=
p 3=
p 4=
4 K1 x
14 K 1 x6 K 1 K 2 x 24 K 1 K 2 K 3 x 3 K 1 K 2 K 3 K 4 x 4
6 K1 K 2 x
2
14 K 1 x6 K 1 K 2 x 24 K 1 K 2 K 3 x 3K 1 K 2 K 3 K 4 x 4
4 K1 K 2 K 3 x
3
2
3
14 K 1 x6 K 1 K 2 x 4 K 1 K 2 K 3 x K 1 K 2 K 3 K 4 x
4
K 1 K 2 K 3 K 4 x4
14 K 1 x6 K 1 K 2 x 24 K 1 K 2 K 3 x 3K 1 K 2 K 3 K 4 x 4
1
p 0=
4
1K d x
p 3=
4 K 3d x 3
1K d x
4
p 1=
p 4=
4 Kd x
1K d x 
4
K 4d x 4
1K d x4
p 2=
6 K 2d x 2
1K d x4
§7.4 RNA folding and unfolding
RNA folding as a two-state system
Probability of folding and unfolding state
Without force
 F0
Free energy
●
With force
 F 0− f  z
folded
unfolded
z
Free energy
states
weights
1
With force
 F 0− f  z
folded
unfolded
−   F 0− f  z 
e
z
p fold =
1
−  F 0 − f  z
1e
Fit: ΔF0=79kBT, Δz=22nm
Observed: Δz≈22nm
RNA folding and unfolding
can be described indeed by
the two-state model!
[Science 292 (2001) 733]
§ Summary & further reading
Summary
●
Gibbs distribution & grand partition function
i 
s
i 
s
p  E , N =
●
i 
i
−  E s − N s 
e
Z
Simple ligand-receptor binding
●
Toy Model of a Dimeric Hemoglobin
MWC model
Ising-like model
●
Hierarchical models of 4 binding sites of Hb
●
RNA folding and unfolding
p fold =
1
−  F 0 − f  z
1e
Further reading
●
●
●
Phillips et al., Physical Biology of the Cell, ch7
Graham, & Duke (2005) The logical repertoire
of ligand-binding proteins, Phys. Biol. 2, 159
Imai (1990) Precision determination and Adair
scheme analysis of oxygen equilibrium curves
of concentrated hemoglobin solution, Biophys.
Chem. 37, 1.