Introduction & applications Part II

1. No HW assigned (HW assigned next Monday).2. Quiz today3. Bending & twisting rigidity of DNA with

Magnetic Traps.

“MT”

Quiz #4 on (page 119-143) Chpt. 4 of ECB1. _______________ bonds between neighboring regions of the polypeptide backbone often give rise to regular folding patterns, known as _______________ and _________________.

2. Molecules that assist in the folding of proteins in vivo are called __________________________. 3. Proteins that couple the chemical energy from ATP hydrolysis into mechanical work are called ________________________.

4. A protein generally folds into the shape in which (what function) _________________________ is minimized.5. At present, the only way to discover the precise folding pattern of any protein is by experiment, using either ______________________ or __________________________________________.

Hydrogen

α helices β sheets

molecular chaperones

molecular motors

the free energy (G)

X-ray crystallographyNuclear magnetic resonance (NMR) spectroscopy

Magnetic Tweezers and DNA

Watch as a function of protein which interacts with DNA (polymerases, topoisomerases), as a function of

chromatin: look for bending, twisting.

Can be conveniently used to stretch and twist DNA.

• DNA tends to be stretched out if move magnet up.• DNA also tends to twist if twist magnets (since follows B).(either mechanically, or electrically move magnets)

Forces ranging from a few fN to nearly 100 pN: Huge Range

Dipole moment induced, and B. = x B = 0

U = - . B

F= ( . B) : U ~ -B2.

Δ

It is the gradient of the force, which determines the direction, the force is up. (i.e. where B is highest)

With Super-paramagnetic bead, no permanent dipole.

Force measurement- Magnetic Pendulum

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

The DNA-bead system behaves like a small pendulum pulled to the vertical of its anchoring point & subjected to Brownian fluctuations

Each degree of freedom goes as x2 or v2 has ½kBT of energy.

Do not need to characterize the magnetic field nor the bead susceptibility, just use Brownian motion

Equipartition theorem: what is it?

Derive the Force vs. side-ways motion.

F = kB T l

< x2 >

½ k < x2 > = ½ kBT

F = k l

½ (F/ l) < x2 > = ½ kBT

Note: Uvert. disp = ½ kl2

Ux displacement = ½ k(l2+x2)Therefore, same k applies to x .

Force measurements- raw data

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

F = kB Tl< x2 >

(4.04 pN-nm)(7800nm)/ 5772 nm = 0.097 pN

Measure < x2 >, l and have F!

At higher F, smaller x; so does z.

Example: Take l = 7.8 m

Lambda DNA = 48 kbp = 15 m

At low extension, with length doubling, x ~ const., F doubles.

At big extension (l: 12-14 m),x decrease, F ↑10x.

Spring constant gets bigger. Hard to stretch it when almost all stretched out!

Z = l

X

Measure z, measure x

Find F by formula.

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.

FJCWLC

FJC vs. WLCBottom line

At very low (< 100 fN) and at high forces (> 5 pN), the FJC does a good job.

In between it has a problem.

There you have to use WJC.

The Freely Jointed Chain Model

bF

bF bcos()2b2 sin()d e

Fbcos( )

kBT

0

2b2 sin()d eFbcos( )

kBT

0

b

Where -F x b cos() is the potential energy acquired by a segment aligned along the direction with an external force F. Integration leads to:

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)

F

bF b coth(Fb

kBT) kBT

Fb

b(

Fb

kBT)

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.