Introduction & applications Part III

1. HW assigned later today (Due next Monday, 3/8/10).

2. March 15th, 17th, night of 18th: PresentationsReports due night of 18th.

“MT”

Today’s take-home lessons(i.e. what you should be able to answer at end of lecture)

1. FJC vs. WLC:

What’s the difference/ why WLC is better (but more complicated)

2. Twisting vs. Writhe

3. Twisting– anharmonicity of DNA

The Elasticity of a Polymer (DNA) Chain

In the presence of a force, F, the segments tend to align in the direction of the force.

Opposing the stretching is the tendency of the chain to maximize its entropy. Extension corresponds to the equilibrium. Point between the external force and the entropic elastic force of the chain.

Do for naked DNA; then add proteins and figure out how much forces put on DNA.

Two Models of DNA (simple) Freely Jointed Chain (FJC)

& (more complicated) Worm-like Chain (WLC)

Idealized FJC:

Realistic Chain:

FJC: Head in one direction for length b, then turn in any direction for length b.

[b= Kuhn length = ½ P, where P= Persistence

Length]

WLC: Have a correlation length

FJC: Completely straight, unstretchable. No thermal fluctuations away from straight line are allowed

The polymer can only disorder at the joints between segments

FJC: Can think of DNA as a random walk in 3-D.

The Freely Jointed Chain Model

bF

bF bcos()2b2 sin()d e

Fbcos( )

kBT

0

2b2 sin()d eFbcos( )

kBT

0

Where -Fb cos() is the potential energy acquired by a segment aligned along the direction with an external force F. Other terms are geometric terms. The integral is over all space, where it is much more convenient to do it in spherical coordinates, rather than in Cartesian coordinates.

Langevin Function

And for a polymer made up of N statistical segments its average end-to-end distance is:

R(F) Nb(Fb

kBT) L(

Fb

kBT)

b

F

bF b coth(Fb

kBT)

kBT

Fb

b(

Fb

kBT)

),,(),,(0

2

0

2

0

rfdrrdddxdydzzyxf

Integration leads to:

The Freely Jointed Chain ModelAt low forces:

x /L coth(Fb

kBT) kBT

Fb, for Fb << kBT :

coth(a) = 1

a

a

3 ..., therefore :

x/L =1

a a

3 1

a .... Fb

3kBT

F 3kBT

b

x

L

Thus, at low forces, the chain behaves as a Hookian spring with a spring constant, 3kBT/b = 3kBT/2P, where P is the persistence length of the chain. Lets see its physical meaning…..

F /x

The spring constant is inversely proportional to the size of the molecule, and to the persistence length. This dependence makes sense. Its easier to align a stiff polymer than a polymer that is very flexible. This is a manifestation of the nature of polymer elasticity: entropic!

Fitting the FJC Model to the Data

FJC

Small (< 100 fN) and large (>5 pN) forces FJC is good

In between have to use WJC.

The Worm-like Chain ModelUses a continuum description of the chain to address these limitations:

- The entropic elasticity of the DNA chain involves small deviations of the molecular axis due to thermal fluctuations.

- The direction of the chain is correlated over a distance, called the persistence length (= 2x Kuhn length) of the chain. For DNA, at 10 mM NaCl, PDNA = 150 bp or 550 nm.

- Bottom line: forces of the order of kBT/P are needed to align and straighten elastic units of these dimensions.

Worm-Like Chain

Fixed contour length, L. It has a tangent (of fixed length) which varies as you proceed along a contour (s)

U = C ds |dt/ds|2 |t(s)=1|s=0

s=L

∫

Lp=persistence length

If dt/ds large, curves a lot.

If C big, persistence length is big.

C = Lp·T·kB. 

(Large C means large penalty for bending –

and this means large Lp.) 

If used with Boltzmann weight, gives these decaying correlations—non-trivial

The Worm-like Chain Model

For forces > kBT/P, the force-extension behavior of the chain can be obtained from the effective energy of a stretched WLC:

r(s)z

ˆ z

E elastic

kBT

1

2EI

2r(s)

s2

2

ds Fz0

L

F

At high forces, the extension of the WLC approaches its contour length, L, and the tangent vectors fluctuate only slightly around the pulling axis, :

ˆ z

ˆ t

FP

kBT

1

4 1z

L

2 z

L

1

4

“This interpolation formula is an excellent approximation tothe exact solution throughout most of the range of forces investigated by the magnetic bead experiment.”

Bustamante et al. Science, 265, 1599-1600 (1994)

The Worm-like Chain Model

Fitting the WLC Model to the Data

FJC

WLC

DNA may be Supercoiled:Has both Twist and Writhe

Twist

Writhe

Relaxed DNA: Twist =1 turn/10.4 bp. Writhe = 0. ∆Tw=0

Can’t just twist up DNA and have it all go into twist.

Example: Phone cord.

Pull on DNA and writhe comes out.

Measure relaxed (non super-coiled) DNA and figure out length vs. force.

Linking Number = Twist + WritheLk = Tw + Wr

Twist (Tw), Writhe (Wr), Linking Number (Lk)

Tw: # of times the two strands wrap around each other

Wr: # of times C crosses itself.

Charvin, Contemporary Physics,2004

Tw=2 Tw=0

Wr=0 Wr=2

Linking Number = Twist + WritheLk = Tw + Wr

Ex: Hold DNA out straight so that it has no Writhe, add of take out twist, then let fold up (Twist goes into Writhe).

= (Lk – Lko)/Lko =∆Lk/ Lko

Supercoiled DNA (): is the deviation from relaxed linking number.

Normal DNA is negatively supercoiled, -0.06 = 6 turns for every 100 taken out. Why?

Positively supercoiled

Helps unwind DNA– makes it easier to uncoil, separate strands. Enzymes which do this called Topoisomerases.

Makes DNA more stable

Why? What about archebacteria that lives in hot springs?

Movie measureforce.

Some Examples of Tw and WrLinear DNA with Constrained Ends

Application to Eukaryotic Cells

In response to supercoiling, they will assume an amount of writhe, just as if their ends were joined.

In Eukaryotic cells: When a chromosome wants to express a particular gene, it becomes single stranded. Requires a tremendous amount of energy (remember partition function and genes, typically 10-100k a.a., require many H-bonds to be broken: DNA is a very stable structure). Requires enzymes– topoisomerases.

More Examples of Writhe and Twist

http://commons.wikimedia.org/wiki/Image:Circular_DNA_Supercoiling.png

DNA’s helical nature (natural twist) not shown.

Plectonemes more common: shape

bacterial plasmids take.

Circular DNA

Torsionally stressed single DNA molecule

T. Strick et al., J. Stat. Phys., 93, 648-672, 1998

Extension vs. supercoiling at constant forceThree regimes

Low F: symmetric under - The shorteningcorresponds to the formation of plectonemes upon writhing.

When the force isincreased above 0.5 pN, the curve becomes asymmetric: supercoils still form for positive coiling while local denaturation adsorbs the torsional stress for negative .

At forces larger than 3 pN no plectonemes are observed: the torsional stress is adsorbed not by writhe but in local structural changes of the molecule.

Playing with phone cord: can you explain graphs?

Class evaluation1. What was the most interesting thing you learned in class today?

2. What are you confused about?

3. Related to today’s subject, what would you like to know more about?

4. Any helpful comments.

Answer, and turn in at the end of class.