1. Introduction and applications
magnetic tweezers II
Bending & twisting rigidity of DNA with Magnetic Traps.

2. Magnetic Tweezers and DNA
Can be conveniently used to stretch and twist DNA.
With Super-paramagnetic bead, no permanent dipole.
Dipole moment induced, and m a B. t = m x B = 0
U = - m . B : U ~ -moB2. Δ
F = - U (Force is always the slope of the energy)
It is the gradient of the force, which determines the direction. The force is up, i.e., where B is highest.
DNA tends to be stretched out if move magnet up.
DNA also tends to twist if twist magnets
(since m follows B).
(either mechanically, or electrically move magnets)
Forces ranging from a few fN to nearly 100 pN: Huge Range
More sophisticated experiments: Watch as a function of protein which interacts with DNA (polymerases, topoisomerases), as a function of chromatin: look for bending, twisting.

3. Force measurement- Magnetic Pendulum
The DNA-bead system behaves like a small pendulum pulled to the vertical of its anchoring point & subjected to Brownian fluctuations
Do not need to characterize the magnetic field nor the bead susceptibility.
Just use Brownian motion.
Equipartition theorem:
Each degree of freedom goes as x2 or v2 has ½kBT of energy.
Derive the Force vs. side-ways motion.
½ k < x2 > = ½ kBT
F = k l
Note: Uvert. disp = ½ kl2
Udx displacement = ½ k(l2+dx2)
Therefore, same k applies to dx .
½ (F/ l) < x2 > = ½ kBT
F =
kB T l
< x2 >
T. Strick et al., J. Stat. Phys., 93, , 1998

4. Force measurements F = kbT L/ <dx>2 Fx F F L L
F cos(da) da
F cos(da) Fx L
Fx = ?
= [F cos(da)] sin(da)
F cos(da)
Fx = F sin(da) cos(da) da L
sin(da) = ?
= dx / L
½ kb T = ½ kx <dx>2
Equipartition theorem in x direction:
Hooke’s Law: Fx = kx dx : kx =?
kx = Fx / dx = F sin(da) cos(da)/ dx
For small angle ?
cos(da) = 1
kx = Fx / dx = F sin(da)/dx = F [dx / L]/dx = F/L
Thus: kb T = kx <dx>2 = F/L <dx>2
F = kbT L/ <dx>2
Strick et al Science

5. Force measurements- raw data
Measure < x2 >, L
and get F
Z = l X
Measure z, measure dx
Find F by formula.
F =
kB TL
< x2 >
Example: Take L = 7.8 mm
(4.04 pN-nm)(7800nm)/ 5772 nm = pN
At higher F, smaller dx; so does dz.
Lambda DNA = 48 kbp = 15 mm
At low extension, with length doubling, dx ~ const., F doubles.
At big extension (L: mm),
Dx decrease, F ↑10x.
Spring constant gets bigger.
Hard to stretch it when almost
all stretched out!
T. Strick et al., J. Stat. Phys., 93, , 1998

6. Analysis: The Worm-Like Chain for elasticity
F= - kx
Hooke’s Law: You apply a force on something and it increases in length linearly. Proportional constant = k. Minus sign because it’s a restoring force.
Force related to fractional increase (x/L). What if force isn’t proportional to distance?
DNA is much longer than it is wide – λ DNA (virus DNA) is about 16.5 µm upon full extension, but the molecule’s diameter is only about 2 nanometers, or some four orders of magnitude smaller. Can expect some simplification where don’t need to get into details of molecular bonding.

7. Length vs. Superhelicity of DNA
For forces below 0.4 pN, the molecule displays a symmetric response to positive and negative supercoiling: plectonemes are formed, reducing the extension of the tether.
For forces between 0.4 pN and 3 pN, no shortening in the tether is observed for negative supercoiling. In this case, the increased torsional stress on the molecule
leads to local denaturation to relieve this stress.
For positive supercoiling, plectonemens are still formed.
For forces higher than 3 pN, no shortening is observed for both positive and negative supercoiling. This can be explained because the molecule undergoes a transition to P-DNA, which is characterized by winding of the phosphate-sugar backbone inside the structure, exposing the bases to the solution.

8. Torsionally stressed single DNA molecule
Playing with phone cord: can you explain graphs?
When the force is
increased above 0.5 pN, the curve becomes asymmetric: supercoils still form for positive coiling while local denaturation adsorbs the torsional stress for negative s.
Low F: symmetric under s  - s
The shortening
corresponds to the formation of plectonemes upon writhing.
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.
Extension vs. supercoiling at constant force
Three regimes
T. Strick et al., J. Stat. Phys., 93, , 1998

9. Over- and under-stretching
Upon twisting a DNA molecule it takes a number of turns, before the DNA length reduces significantly and plectonemes are formed. The point (Nbuckling) where DNA starts to form
plectonemes with a constant length reduction per turn is called buckling instability
Rotation extension curves
for different forces. At higher forces one cannot induce supercoils but denature the DNA molecule.

10. DNA Packaging: Nucleosomes In Eukaryotic DNA, packaged into nucleosomes: about 2 loops around 4 histone proteins, making about 10 nm “disks”. Disks must be removed when DNA is active.
Histones can be acetylated to make nucleosomes less stable
(We can study what forces hold them together via Magnetic Traps
Keq = 0.9; H3K56Ac increases open fraction 4x
John Van Noort

11. Probing conformational changes
Recall:
Apply some Force, F: React diff.
DG = function (F) dz
Two states, differ by DG
Free energy
Extension dz ΔG
Fill in diagram
(w L.H.S. more stable)
-F dz
Free energy
Extension ΔG
Fraction open:
Force spectroscopy can measure condensation directly and interaction energies indirectly

12. Unwinding a Nucleosome Seems to do it in two steps, one 2-3 pN, unwinds first wrap, then at 6pN, unwraps the second wrap.
Van Noort, Biophysical J, 2009
Van Noort, Biophys. J.

13. Nucleosome unwinds in 2 steps: 1st ½ at 3 pN, then other ½ at ~ 14 pN
P(E)…1/(1 + exp[(FDx – DG)/kBT] :
DGunwrap = kJ/mole
Mihardja, Bustamante, PNAS, 2006

14. Two Models of DNA (simple) Freely Jointed Chain (FJC)
& (more complicated) Worm-like Chain (WLC)
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]
Idealized FJC:
Realistic Chain:
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.
WLC: Have a correlation length

15. The Freely jointed Chain (FJC)
The molecule as a chain of perfectly rigid subunits of length b joined
by perfectly flexible hinges. Segment-to-segment angle = q
You can think of b as the length of the repeating subunits, and is called the Kuhn length (= 2 x Persistence length).
Persistence length: you start out going in some direction: how long will you tend to keep going: for DNA—about 50 nm, or 150bp.
If the molecule is under an applied force f as above, the effective energy for the chain is given by
If there were no applied force, all configurations have equal energy (and therefore the system has large configurational entropy), and the chain orients itself in any which way—analogous to a random walk.

16. Force vs. Extension for DNA
F=-kx works well at very low force;
at higher force, DNA is extended (> 50%), need FJC or better is WLC
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.
You measure the Persistence length
Strick, J. Stat. Physics,1998.

17. WLC: A slight extension of FJC
A new phenomenological parameter has entered the energy term, A, which is a measure of the persistence length of the chain, or how long a segment of the chain will have tangent vectors all pointing in nearly the same direction. Indeed, the tangent-tangent correlation function for the wormlike chain at zero stretching force is given by
or, the similarity in directionality for the chain decays as an exponential in the persistence length.
An analytic solution to WLC is not currently known, but the above equation has been solved numerically.
At low force it again displays a Hookean linear relation, but as the extension nears the contour length of the molecule, it scales not as 1/f as predicted by freely jointed chains, but as 1/f1/2, in significantly better agreement with the data.

18. The Worm-Like Chain (WLC) for elasticity
Increase in end-to-end length (x) of polymer:
Worm Like Chain (WLC) of entropic elasticity.
Force related to fractional increase (x/L)
where A = Lp: persistence length, a measure of the chains bending rigidity
= 2x Kuhn Length
L = contour length
x = extension
Each unfolding event increases the contour length of the homopolymer by a constant value, ∆L.

19. WLC Fits very well at all stretches

20. Answer, and turn in at the end of class.
Class evaluation
1. 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.

21. Dekker, PLOS One, 2012

23. Undoing a loop
Manosas, 2010

24. WLC: A slight extension of FJC
A more realistic model for dsDNA is actually quite similar to the freely jointed chain, but changes from discrete segments to a continuous elastic medium. This can be done because DNA is actually a rather stiff molecule, with successive segments displaying a sort of cooperativity—all pointing in roughly the same direction. The figure below is a cartoon of the wormlike chain (WLC) model, where now we define r(s) as the position as a fuction of the relaxed-state contour length, s. Also shown is the tangent vector t(s), which is the first derivative of r(s) with respect to a line segment ds.
There is now another added term to the effective energy of the chain which is related to the curvature (itself proportional to the square of the tangent vector), and the summation is now replaced by integration along the entire contour length:

25. Answer, and turn in at the end of class.
Class evaluation
1. 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.

26. Force Calibration
Manosas, 2010

27. Torsionally stressed single DNA molecule
Playing with phone cord: can you explain graphs?
When the force is
increased above 0.5 pN, the curve becomes asymmetric: supercoils still form for positive coiling while local denaturation adsorbs the torsional stress for negative s.
Low F: symmetric under s  - s
The shortening
corresponds to the formation of plectonemes upon writhing.
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.
Extension vs. supercoiling at constant force
Three regimes
T. Strick et al., J. Stat. Phys., 93, , 1998

28. DNA Packaging In Eukaryotic DNA, packaged into nucleosomes: about 2 loops around 4 histone proteins, making about 10 nm “disks”. Disks must be removed when DNA is active.
Histones can be acetylated to make nucleosomes less stable
(We can study what forces hold them together via Magnetic Traps
Keq = 0.9
H3K56Ac increases open fraction 4x
John Van Noort

30. Flory et al. 1989; Bustamante et al. 1994
force measurements Fx F F
F cos(da)
Worm-like chain:
Flory et al. 1989; Bustamante et al. 1994 da
F cos(da) Fx L
Fx = [F cos(da)] sin(da)
F cos(da)
Fx = F sin(da) cos(da) da L
sin(da) = dx / L
Equipartition theorem in x direction:
½ kb T = ½ kx <dx>2
Hooke’s Law: Fx = kx dx
kx = Fx / dx = F sin(da) cos(da)/ dx
For small angle, cos(da) = 1
kx = Fx / dx = F sin(da)/dx = F [dx / L]/dx = F/L
Thus: kb T = kx <dx>2 = F/L <dx>2
F = kbT L/ <dx>2
Strick et al Science

31. Nucleosome unwinds in 2 steps: 1st ½ at 3 pN, then other ½ at ~ 14 pN
P(E)…1/(1 + exp[(FDx – DG)/kBT] :
DGunwrap = kJ/mole
Mihardja, Bustamante, PNAS, 2006

32. N S S N
NAP1-Assisted Nucleosome Assembly on DNA Measured in Real Time by Single-Molecule Magnetic Tweezers
Slide 1 illustrates DNA in the magnetic tweezers, and when a nucleosome is formed, this reduces the DNA end-to-end length, as well as change the linking number of the free DNA.
Vlijm et al. (2012) PLoS ONE, 7(9)

33. Nucleosome H2A H2B H4 dimer H3 nucleosome dimer tetramer octamer
Binding in specific order: first tetramer (H3-H4)2 then two (H2A-H2B) dimers
DNA (-) and histones (+) are oppositely charged
Assembly changes linking number
dimer H3
nucleosome
dimer
Slide 2 illustrates that nucleosomes are only formed when the histones assemble in the right order. Therefore, flushing in just he histones will never result in proper nucleosomes. Assembly proteins or salt dialysis are needed to maintain the binding order of first two copies of H3-H4 as a tetramer, followed by two H2A-H2B dimers.
tetramer
octamer
720°
Twist=-2
Writhe=-2
L=Twist+Writhe
Crystal structure Luger et all. Nature 1997

34. Experiment coilable DNA nicked DNA force from bead position N S
Magnets
2.8 µm Magnetic bead
8kb DNA
Glass surface
Slide 3 is a picture of my setup, an illustrating that a nick in the DNA results in uncoilable DNA, and thus unsuitable for measuring linking number changes by the nucleosome (dis) assembly
coilable DNA
nicked DNA
force from bead position

35. Rotation Curve ∆Lk = ∆Lk,nuc+ ∆Lk,DNA End-to-end length (µm)
Flush in NAP1
and core
histones
Rotate
magnets
Rotation Curve
End-to-end length (µm)
Slide 4 shows the assay of nucleosome assembly in the flow cell, illustrating that nucleosome (dis)assembly also induces linking number changes. When the DNA molecule is tethered such that the DNA is rotationally constrained, the magnetic tweezers is an excellent tool to measure this linking number change by making rotation curves.
∆Lk = ∆Lk,nuc+ ∆Lk,DNA
Magnet turns

36. Disassembly of nucleosomes
Move magnet 5 mm in 1 sec, going from 0.3 pn to 14 pN applied force. (Bare DNA goes from 1.8um to 2.4um.
With nucleosomes present, see stepwise length increase for every nucleosomes disrupted
24 nm F
56nm~2*24nm
■ before
● after assembly
▲ after disassembly
Nucleosome wraps 147 bp, 50nm. Disruption happens in two steps, the first ~3pN. This is a reversible step and unwraps the first loop. The second step occurs at ~16pN and is irreversible, this removes the complete nucleosome. Shown above is at constant force the disruption steps. Most steps are ~25nm, thus half the lengh of the complete length of DNA in a nucleosome.
I moved my magnet 5mm in ~1 second to go from 0.3pN to 14pN applied force,
My bare DNA then normally goes from ~1.8micrometer to ~2.4 micrometer, instantly.
Since the nucleosomes are present, I see a slow, stepwise length decrease for every disrupted nucleosome.
Slide 5 shows the length increase due to disruption of nucleosomes. At a constant force of 14pN steps of ~24 nm appear. This because a nucleosome disrupts in two steps:
- The first step is the disruption of the outer histones H2A and H2B. When this occurs at ~3pN, this step is reversible when the force is lowered again.
- The second step is irreversible and occurs from 14pN (most of the times even higher forces are needed for the last nucleosomes),now all histones are falling of the DNA.
Both steps have a length increase around 24nm.
Vlijm et al. (2012) PLoS ONE, 7(9)

37. Assembly steps 27nm steps Vlijm et al. (2012) PLoS ONE, 7(9) Nap1+
core histones
1pN
For comparison: the assembly steps are similar to he disassembly steps.
Slide 6 shows the opposite: I assembled nucleosomes within my flow cell using an (ATP independent) assembly protein NAP1. Here one can see that the assembly steps are comparable to the disassembly steps.
Vlijm et al. (2012) PLoS ONE, 7(9)