Lecture 7. Two-State Systems

Zhanchun Tu (涂展春 )

Department of Physics, BNU

Email: tuzc@bnu.edu.cn

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

● Energy landscape incompressible

F= s

Rout=R

Rout

R

=− A

σ=

0, closed

1, open

Ion channel

Increasing driving force

● Open probability

[Nat. Struc. Biol. 9: 696 (2002)]

Δε=εclosed −εopen=−5 k BT

Problem: prove that ⟨⟩=popen

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 exchangesN sN r=N u

E sE r=E u

=const.

=const.

When the system stays a given state (Es(i), N

s(i)), the

number of states that the universe (=system+reservoir)

W uE s i , N s

i = 1 × W r Eu−E si , N u−N s

i

states of system states of reservoir

Probability of finding a given state of the system

p E si , N s

i =W uE s

i , N si

∑iW uE s

i , N si ∝W r Eu−E s

i , N u−N s i

Given Es(i) and N

s(i), we have Sr Eu−E s

i , N u−N si =k B lnW r Eu−E s

i , N u−N si

● Gibbs distribution & grand partition function

p E si , N s

i∝eS r Eu−E s

i , N u−N s

i/ kB

Sr Eu−E si , N u−N s

i =S rEu , N u−

∂ S r

∂E r

E s i−

∂ S r

∂N r

N s i

p E si , N s

i ∝e−E s

i− N s

i=

e−E s

i− N s

i

Z

Simple ligand-receptor binding revisited with Gibbs distribution

● two states– Empty state σ = 0

– Occupied state σ = 1

⟨ N ⟩=0×p01× p1=p1=e−b−

1e− b−

§7.3 Cooperative binding of Hemoglobin

Toy Model of a Dimeric Hemoglobin

● Ising like modelCooperativity parameter

p0=1Z

p1=2 e−−

Z

p2=e−2 J −2

Z

⟨ N ⟩=0× p01× p12×p2

JJ=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, p

1, and p

2 corresponding to occupancy 0, 1, and 2,

Respectively. Draw the binding curves (i.e., the relations between p0, p

1, p

2 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−T0 Note: binding energy<0, higher affinity <=> lower binding energy

1=0, site1 is empty

1, site1 is occupied

2=0, site2 is empty

1, site2 is occupied

m=0, protein in T state

1, protein in R state

Please confirm:

x

parameter

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

p0=1

14 K 1 x6 K 1 K 2 x24 K1 K 2 K 3 x3

K1 K 2 K 3 K4 x4

p1=4 K1 x

14 K1 x6 K 1 K 2 x24 K1 K2 K 3 x3

K1 K 2 K 3 K 4 x4

p2=6 K1 K 2 x2

14 K 1 x6 K1 K2 x24 K1 K 2 K 3 x3

K1 K 2 K 3 K4 x4

p3=4 K1 K 2 K 3 x3

14 K1 x6 K 1 K 2 x24 K 1 K 2 K 3 x3

K1 K 2 K 3 K4 x4

p4=K1 K2 K 3 K 4 x4

14 K 1 x6 K1 K2 x24 K1 K 2 K 3 x3

K1 K 2 K 3 K4 x4

p1=4 Kd x

1Kd x 4p0=

1

1K d x4

p3=4 K d

3 x3

1K d x4

p2=6 K d

2 x2

1K d x4

p4=K d

4 x4

1K d x4

§7.4 RNA folding and unfolding

RNA folding as a two-state system● Probability of folding and unfolding state

F 0

F 0− f zFre

e en

ergy

z

Without force

With force

folded unfolded

Fit: ΔF0=79k

BT, Δz=22nm

Observed: Δz≈22nm

p fold =1

1e− F 0− f z

weights

1

e− F 0− f z

F 0− f z

Fre

e en

erg y

z

With force

folded unfolded

states

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

p E si , N s

i =

e−E s

i − N s

i

Z

● Simple ligand-receptor binding

● Toy Model of a Dimeric Hemoglobin

Ising-like model

MWC model

p fold =1

1e− F 0− f z

● Hierarchical models of 4 binding sites of Hb

● RNA folding and unfolding

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.