Topic 3: Transition theory and the single molecule

1
Topic 3 Transition theory and the single molecule
transition theory
Single-molecule technique Optical Tweezers
Chemical reactions
Thermal fluctuations revisited
Example RNA folding
Bi-molecular reactions
Acid-base equilibrium
catalysis
2
Reading for Topic 3

• Howard, Chapters 4,5
• Nelson, Chapters 8,10.3,10.4
• (Research papers)

3
Reaction co-ordinates
Hydrogen atoms swapping places in a molecule, in
a straight line
4
Free Energy (G)
G2
G1
Reaction Co-ordinate
Principle of DETAILED BALANCE
5
Chemical Potential
In general
Using detailed balance for the rate constants
Take concentration terms into the exponent
6
(Equal rates from the top)
Free Energy (G)
G
G1
G2
7
Arrhenius plot
ln(k12) ln k0 (DG/k) (1/T)

8

• What is the frequency factor ?

Appropriate for single-bond transitions,
electronic transitions, light absorption
Appropriate for protein shape changes except
that potentials are not really quadratic.
9
(No Transcript)
10
Live demo optical tweezers
11
x
Equipartition of energy ½ k ltx2gt ½
kT Therefore spring constant K kT /
ltx2gt
12
The thermal force
Basically its F ma, but including the
thermal force F(t)
The Langevin Equation m (d2x/dt2) g (dx/dt)
K x F(t) 0
13
Magnitude of the thermal force
Displacement at frequency w due to thermal force
Average force and displacement are zero.
Consider mean-squares
Definition of power spectral density G(w) of
the thermal force
This is the mean-square displacement due to
components of the thermal force in the frequency
range w to w dw
14
The thermal displacement at ALL frequencies is
By equipartition of energy
For white noise force, G(w) G0
( An alternative derivation uses G(f) df instead
of G(w) dw, where w 2pf. This gives G(f)
4kTg )
15
Using Optical tweezers to measure transitions in
Single Molecules
Reference Liphardt J et al, 2001, Reversible
unfolding of single RNA molecules by mechanical
force. Science 292733-737
16
(No Transcript)
17
(No Transcript)
18
Transient kinetics
Probability of particle still being in state
A at time t
( b.c. P(0) , so A1 )
( mean lifetime ? t.Pt dt 1/k )
19
Bi-molecular Reactions Simple Binding
20
Water acid-base equilibrium
Conjugate acid/base pair
base proton acceptor
acid proton donor
d
21
general acid-base equilibrium
acid
base
d
22
Catalysed Exothermic (downhill) reaction
23
(No Transcript)