biophysical chemistry applying polymer theory to...
TRANSCRIPT
Applying polymer theory to biomolecules
Biology is full of polymers: proteins, DNA, RNA, polysaccharides, lipids.
Q: Can they usefully be modelled using polymer theory?
Outline:
1. Review of freely-jointed chain
2. Applications to genomic DNA
3. Entropic elasticity and pulling DNA
4. Beyond the freely-jointed chain: the Worm-like chain model
Jonathan Doye, University of Oxford 2
Review of freely-jointed chains
Also called random-flight model, Gaussian or ideal chain.
N segments of length l. The direction of a segment is uncorrelated withadjacent segments, i.e. freely jointed. Each segment represented by a vectorli.
Contour length L = Nl.
We can characterise the polymer by its mean square end-to-end distance 〈R2〉:
〈R2〉 = 〈R ·R〉 =
⟨N∑i
li ·N∑j
lj
⟩=
N∑i
N∑j
〈li · lj〉 (1)
=
N∑i
〈li · li〉+N∑i
N∑j 6=i
〈li · lj〉 = Nl2 (2)
i.e. 〈R2〉1/2 = lN1/2
Jonathan Doye, University of Oxford 3
Can also characterize the polymer by its radius of gyration Rg, which is themean separation betwen monomers.
For a freely-jointed chain R2g = 〈R2〉/6 = Nl2/6
Although the "chemical" monomers of a real polymer chain will not be freely-jointed, on longer length scales of s monomers a polymer can still effectivelybehave like a freely-jointed chain.
i.e. 〈R2〉 = NKl2K, where lK = slc is the Kuhn length and NK = Nc/s. (c
stands for chemical unit).
Deviations in this scaling can occur because of
(i) self-avoidance ⇒ Rg ∼ N3/5
(ii) effective attractions between the polymer (i.e. poor solvent)⇒ Rg ∼ N1/3.
Jonathan Doye, University of Oxford 4
Genome sizegenome size contour Rg
organism type example (bases/base pairs) length (FJC)ssDNA virus STMV 1063 0.7µm 15nmdsDNA virus bacteriophage T2 150 000 50µm 0.9µmprokaryote E. coli 4.6× 106 15mm 5µmeukaryote human 2× 3.2× 109 2m 0.2mm
Rg was calculated for double-stranded DNA (dsDNA) using a Kuhn length of300 base pairs where the rise per base pair is 3.4 Å, and for single-strandedRNA (ssRNA) a Kuhn length of 20 Å and backbone separation between bases of6.4 Å.
However, these genome dimensions are significantly larger than their ‘‘con-tainers’’:
size of STMV capsid: 70 Åsize of T2 capsid: 100 nmsize of E. coli cell: 2µm long, 1µm widediameter of human nucleus: 6µm
Q. Are the Rg reasonable?
Jonathan Doye, University of Oxford 5
Freed DNA!
Jonathan Doye, University of Oxford 6
Packaging DNA: ssRNA virus
The active virus particle can sponta-neously self-assemble from a solutioncontaining the genome and the viruscapsid protein.
The compression of the RNA must becompensated by attractive interactionswith the inside of the capsid.
Jonathan Doye, University of Oxford 7
Packaging DNA: dsDNA virus
Double-stranded DNA is much stiffer than single-stranded RNA (or DNA) and soa different approach is needed.
Bacteriophages have a packaging motor located at one vertex of the capsidthat is able to push DNA into the empty capsid. It uses the cell’s ATP, as thesource of energy for the work that it does.
Jonathan Doye, University of Oxford 8
Single-molecule experiments have studied the forces that the motor is able toexert on DNA as the genome is packaged.
It has been estimated that the motor has to work against an internal force of∼ 50pN when packaging the final part of the genome, and that the internalpressure is 6MPa.
The pressure provides the driving force for the injection of the DNA into a cellduring infection.
Jonathan Doye, University of Oxford 9
Packaging DNA: eukaryotic cells
Free energy cost for bending DNAaround histones is compensated bybinding free energy.
Jonathan Doye, University of Oxford 10
Elasticity in the FJC model: Hooke
A freely-jointed chain can show elastic behaviour that is purely entropic.
When subject to a force there is a free energy cost to pay associated withforcing the chain to adopt an entropically less likely conformation.
It can be shown that probability distribution for the end-to-end vector R fora freely-jointed chain is a Gaussian:
p (R) ∝ exp
(−3R2
2Nl2
)(3)
Hence, the entropy is S = k log(p(R)) + c.
As all conformations of a freely-jointed chain have the same energy
F = −dAdz
= kTd
dzlog (p(R)) = kT
d
dz
(− 3z2
2Nl2
)= −3kTz
Nl2(4)
A freely-jointed chain obeys Hooke’s law with a force constant that increaseswith temperature.
Jonathan Doye, University of Oxford 11
Elasticity in the FJC model: Beyond Hooke
The previous derivation assumed that the chain continues to have a Gaussianprobability distribution even when subject to a force.
Here we calculate the partition function of a freely-jointed chain subject to aforce:
Z =
∫dl1
∫dl2 . . .
∫dlN exp
(−E
({li}
)/kT
)(5)
We can make progress by noting that the energy is separable
E({li}
)= −Fz = −FR · z = −F
N∑i
li · z = −FlN∑i
cos θi. (6)
As each link is independent Z = ZN1 where
Z1 =
∫ 2π
0
dφ1
∫ π
0
sin θ1eFl cos θ1/kTdθ1 = 2π
[−kTF leFl cos θ1/kT
]π0
(7)
= 4πkT
F lsinh
(Fl
kT
)(8)
Jonathan Doye, University of Oxford 12
This can then be used to obtain a expression for the force-extension curve:
〈z〉 = −∂A∂F
= kT∂ logZ
∂F= NkT
∂
∂F
(log
(sinh
(Fl
kT
))− logF
)(9)
= Nl
(coth
(Fl
kT
)− kTF l
)(10)
At low force (F � kT/l) this simplifies to Hooke’s Law:
F =3kT
Nl2z (11)
and at high force (F � kT/l) to
F =kT
l
1
1− z/L(12)
The force diverges as the polymer approaches its contour length.
Jonathan Doye, University of Oxford 13
Stretching DNA
The force-extension curve can beobtained for a single moleculeof DNA can be obtained using anoptical trap set-up.
The freely-jointed chain providesa good description of the stretch-ing, but there are deviations athigher force.
The worm-like chain model pro-vides an excellent description.
Jonathan Doye, University of Oxford 14
Worm-like chain model
The worm-like chain model usually provides a better representation of rel-atively stiff polymers, such as double-stranded DNA, actin fibres and micro-tubules.
t(s)
t(0)
s
The polymer is represented by a continuous flexible rod that can be describedin terms of a bending modulus or a persistence length lp.
The persistence length describes the decay in correlations of the tangentvectors:
〈t(0) · t(s)〉 = e−s/lp (13)
Jonathan Doye, University of Oxford 15
This allows the mean square end-to-end distance to be calculated:
〈R2〉 = 〈R ·R〉 =
⟨∫ L
0
t(s)ds ·∫ L
0
t(s′)ds′
⟩(14)
=
∫ L
0
ds
∫ L
0
〈t(s) · t(s′)〉ds′ =∫ L
0
ds
∫ L
0
e−|s−s′|/lpds′ (15)
= 2lpL
(1 +
lpL
(e−L/lp − 1
))(16)
For L� lp this simplifies to the freely-jointed chain result where lK = 2lp.
The force-extension relation can also be derived:
F =kT
lp
[1
4 (1− z/L)2− 1
4+z
L
](17)
Jonathan Doye, University of Oxford 16