Introduction and applications Bending & twisting rigidity of DNA with Magnetic Traps.
What we will learn today Magnetic particle Magnetic Traps Tuesday: DNA: Want to twist it and compact it. --Essential for life. (1m à 5 µ m; supercoiled) Measure via Magnetic Tweezers: apply range of forces --Put a little magnetic particle on end of DNA: Pull & Twist it with magnets. Measure DNA twist and stretch via microscope. Tell how stiff it is. • DNA does NOT act like a Hookian Spring— (F ≠ -kx) acts like a Worm-Like Chain (WLC). • DNA twisted like Right-handed Helix—makes difference if you over- or under-wind it.
Magnetic Tweezers and DNA Can be conveniently used to stretch and twist DNA. What is force, energy? U = E = - µ . B : µ is proportional to B : U ~ - µ ο B 2 . F = - U (Force is always the slope of the energy) Δ 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. (either mechanically, or electrically move magnets) Forces ranging from a few fN to nearly 100 pN: Huge Range What is Torque? Dipole moment induced, and µ α B. τ = µ x B = 0 With Super-paramagnetic bead, no permanent dipole. • DNA also tends to twist if twist magnets (since µ follows B). (either mechanically, or electrically move magnets) More sophisticated experiments: Watch as a function of protein which interacts with DNA (polymerases, topoisomerases), as a function of chromatin: look for bending, twisting.
How stiff is DNA, longitudinally, laterally? Magnetic Trap movie (Web-browser, Safari: ADN.SWF) How to attach DNA: to glass; to paramagnetic bead Set-up of Experimental system Detect nanometer displacements with visible light Microscopy N S Video camera CCD Measureforce.3g2 Experimental Set-up
Magnetic Traps: Measuring DNA extension Can go to ~10,000 Hz! 1-5 µ m bead Find the (x, y) center to determine the Diffraction rings location within 1 nm! Move objectives up/down at (in 1 second) calibrated steps (~ 100 nm over several microns). Use as a reference for actual experiment. X,Y,Z are then Focal point used with Z objective fixed Z and something Z: Can get like 10 nm resolution is changing Z. (with visible or IR (500-1000 nm light!)
How to measure Force? Ideas? Δ F = - U (Force is always the slope of the energy) U = - µ . B : U ~ - µ ο B 2 To know F, you need to know size of magnetic field (B) and bead’s susceptibility µ ο F = kx Know F, measure x, get k. Difficult. Instead, use Brownian noise. DNA-bead system acts like a small pendulum pulled vertically (z) from it’s anchoring point, subject to Brownian fluctuations (x,y). [Mag Traps.x y motion.force.swf]
How to measure Force? Using Brownian noise of bead motion. DNA-bead system acts like a small pendulum pulled vertically (z) from it’s anchoring point, subject to Brownian fluctuations (x,y). [Mag Traps.x y motion.force.swf]
Force measurement- Magnetic Pendulum Looks like regular pendulum with force not equal to gravitational force but due to magnetic force. Do not need to characterize the magnetic field nor the bead susceptibility. Just use Brownian motion. Looks just like a pendulum acting under the force of gravity! T. Strick et al., J. Stat. Phys., 93 , 648-672, 1998
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 x 2 or v 2 has ½ k B T of energy. Derive the Force vs. side-ways motion. ½ k < δ x 2 > = ½ k B T F = k l Note: U vert. disp = ½ kl 2 U δ x displacement = ½ k(l 2 + δ x 2 ) Therefore, same k applies to δ x . ½ (F/ l) < δ x 2 > = ½ k B T F = k B T l < δ x 2 > T. Strick et al., J. Stat. Phys., 93 , 648-672, 1998
Force measurements- raw data Measure < δ x 2 >, L and get F F = k B TL < δ x 2 > Example: Take L = 7.8 µ m (4.04 pN-nm)(7800nm)/ 577 2 nm = 0.097 pN At higher F, smaller δ x; so does δ z. Lambda DNA = 48 kbp = 15 µ m At low extension, with length doubling, δ x ~ const., F doubles. At big extension (L: 12-14 µ m), Δ x decrease, F ↑ 10x. Spring constant gets bigger. Hard to stretch it when almost all stretched out! T. Strick et al., J. Stat. Phys., 93 , 648-672, 1998
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.
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