DNA as a Polymer Flashcards
Degree of Polymerisation
Definition
-number of monomers, N
Total Mass of Population
Definition
M = N*m_mon
-where m_mon is the mass of a monomer unit
Homopolymer/Heteropolymer
Definition
- a homopolymer is made up of the same monomer
- a heteropolymer is in a random or block structure
Polymer Flexibility
- coil or rod like
- this is length scale dependent, characterised by the persistance length
Polymer Isomerisation
- isotactic, chemical groups on the same side
- syndiotactic, chemical groups on opposite sides
- atactic, chemical groups randomly distributed
Polymer Topology
- linear
- ring/circular
- branched
Polymer - Molecular Conformations
- atoms and molecules in a thermal bath follow random walks
- for a distribution of ensemble isotropic chains, every angle between steps is equally probable
- because each angle is equally probable, over a large number of steps, the ensemble average is zero so take the means square instead
Polymers - Ideal Chain Model
-assume fixed bond length: l = |ri| = |rj| ri . rj = l²cos(θij) -mean square distance : = l²ΣΣ -summed over I and j from 1 to n
Freely Jointed Chain Model
-no correlations between any θij’ and θij are allowed, so:
= 0 for I not equal j
-only N non-zero terms in the double sum since cos(θij)=1 only for I=j
=> = Nl²
-individual chain distance away from start point scales with sqrt(N) steps of step length l
Freely Jointed Chain Model
Real Chains
-as there are no angular restrictions from one bond to the next, the model allows the chain to cross or go through itself which obviously isn’t possible for a real chain
-this leads to the concept of excluded volume, the chain can’t move into the space already occupied by the chain
-for real chains:
not equal 0
Flory-Characteristic
-even after removing crossing over, after many many bonds, the pointing vector will be completely decorrelated from the final vector in the chain:
Ci = Σ
-summed over j from 1 to n
= CnNl²
-where N is the number of bonds and l is the bond length
-Cn is the Flory characteristic ratio, the average of all Ci
Contour Length
-the total length of the polymer
L = Nl
Freely Rotating Chain Model
Description
- bond lengths l and bond angles θ are both fixed
- but chain segment can take any torsional angle with no correlations:
- π < φi ≤ π
- the only correlation between bond vectors is that transmitted down the chain by the previous bond vectors component lcosθ
Persistence Length
-the range over which correlations between bond vectors has decayed
Number of Bonds in a Persistence Length
sp = 1 / -ln(cosθ)
Freely Rotating Chain Model
Mean Square Distance
-mean square distance is now also modulated by the fixed angle between bond vectors:
= nl² [1+cosθ]/[1-cosθ]
-where
C∞ = [1+cosθ]/[1-cosθ]
Worm Like Chain Model
Description
-used for very stiff polymers, such as double stranded DNA
-as the polymer is stiff, θ is very small, so we can expand:
cosθ ≈ 1 - θ²/2
-since θ is small, the correlation length is longer so sp is larger
Worm Like Chain Model
sp
ln(cosθ) ≈ - θ²/2
=>
sp = - 1/ln(cosθ) ≈ 2/θ²
Worm Like Chain Model
Persistane Length
lp = sp*l = 2l/θ²
Worm Like Chain Model
Flory Ratio
C∞ ≈ 4/θ²
Worm Like Chain Model
Limits
-two limits can be taken from the WLC model:
1) ideal chain limit for very long chains:
≈ 2lpl, L»lp, perfect coil
2) rod like limit for very short chains:
≈ L², L<
What is a typical persistence length for double stranded DNA?
~50nm
-this is quite long as double stranded DNA is a double helix making it a relatively stiff polymer
Suitable Models for Nucleic Acid Polymers
RNA and ssDNA
- RNA and ssDNA may be best modelled by FJC or FRC models but we can also use a WLC model with bond length l of nucleotide repeat along the chain
- each nucleotide acts independently ad can rotate around the sugar phosphate backbone single bond
Suitable Models for Nucleic Acid Polymers
dsDNA
- dsDNA is well modelled by the WLC model with lp~50nm
- the double helix makes dsDNA more rigid, the backbone is able to curve but is more or less fixed in place
Bending a Rod Like Section of DNA
Continuum Model
-as the rod is rigid we have small bend angles: r = l/θ -bending rigidity: K = YI -moment of inertia for a rod of circular cross section, diameter d: I = [πd^4]/64 -bending energy per unit length: ε = YI/2r² = YIθ²/2l² -for length l of DNA polymer
Surface Binding Mechanisms
Mica and Metal Cations
- M2+ ions in solution bind to negatively charged DNA and to the negatively charged mica surface
- as the M2+ ions are in solution and not fixed to any surface the DNA is able to slide over the surface once attached
- as they are mobile, the molecules tend to stretch out further
Surface Binding Mechanisms
AP-Mica
- positive charge is immobilised on the mica surface
- negatively charged DNA is attached to this electrostatically
- chains are less spread out and may overlap with each other
2D Equilibration
-molecules diffuse to minimum energy
Kinetic Trapping
-2D projection of 3D conformation
Stretching Individual Biomolecules - AFM
- ssDNA behaves as an entropic spring
- dsDNA is more complicated
- force vs extension graph
- ssDNA follows an exponentially increasing curve
- dsDNA follows this line at small and large extensions but there is a plateau in the middle
Optical Tweezers
-use laser to generate force on a dielectric head through momentum transfer of photons
-usually only exert forces up to ~100pN
-then try to pull the molecule out of the trap and observe the results
F = k_trap * x
Mechanical Stretching of Single DNA Molecules
ssDNA
- obeys WLC model
- force stretches the molecule out against entropy until it is straight
- force rises rapidly upon bond extension until breakage
Mechanical Stretching of Single DNA Molecules
dsDNA
- obeys WLC model
- force plateau reached just after full extension of entropic coil configuration
- overstretch transition occurs ~~65pN
- after overstretch transition behaves like ssDNA
Overstretching of dsDNA
- overstretching DNA melts the double helix
- if only one of the DNA strands is attached to each surface, the double helix is not torsionally constrained so the local topology can change
- one strand is only held to the other by intermolecular forces, stretching the molecule introduces mechanical work which melts off one strand relative to the other
- this phase transition occurs at constant force just as thermally induced transition occur at constant temperature
