Optical Tweezers F scatt F grad 1

Velocity autocorrelation function from the Langevin model kinetic property property of equilibrium fluctuations For 3-dimensional model Green-Kubo relationship 2 Fluctuation – Dissipation Theorem

x-axis F(x) Periodic asymmetric potential (Randomly fluctuating) A simple Brownian ratchet Random diffusion of protein (Gaussian probability distribution) Probability for protein to move across the potential barrier to the right (+x) is higher than to move to the left 0 a -b 3

Manipulating single molecules Attach molecule to magnetic particle and use magnetic field Magnetic tweezers Attach molecule to dielectric particle and use laser light Optical tweezers mirror AFM tip photodiode position detector cantilever laser imaging surface sample laser beam focus of optical trap trap F external F optical trap force balances the external force magnetic bead external magnets DNA surface Atomic Force Microscope Optical Tweezers Magnetic Tweezers objective F 1

The physics behind optical tweezers The change in momentum can be calculated by the difference of momentum flux between entering and leaving a dielectric object is the Poynting vector for an electromagnetic wave Momentum flux of photons is given by 5

Optical Trapping - a>> Conditions for Mie scattering when the particle radius a is larger than the wavelength of the light. We can use a ray optics argument and look at the transfer of momentum a 6

Optical Tweezers Lateral gradient force in non-uniform light High intensitylow intensity 7

Axial gradient forces towards focus of laser light Force due to reflection 8

Scattering force and gradient force are separable n m = refractive index trapping medium n p = refractive index particle m = n p /n m (in the F scatt, F grad equation) Optical Trapping - a<< Condition for Rayleigh scattering when the particle radius a is smaller than the wavelength of the light. F grad > F scatt requires tight focusing a 9

The scales Can trap 0.1 to 10’s  m 1  m is….. …the same as 1/100th diameter of a hair. In water, you can move a particle at about  m per sec. Sensitivity ~ 1 – 100pN Require 10mW per trap. Can rotate at 100’s of Hz. 10

Optical Trap Dynamics Equation of motion of particle in a potential well restoring force Brownian motion Newtonian force drag force 11

Particle in fluid Solution is of exponential decay Damping provided by water  12

Particle in ideal trap Spring constant  or trap stiffness) 13 Solution is a simple harmonic motion t

Trapped particle in fluid 14 Solution is of damped simple harmonic motion t

The whole picture Time averaged effect is 0 Stochastic events introduce fluctuations in the particle’s position Add in the effect of Brownian motion 15

Trap dynamics Look at the movement of the particle in x and y 16

Collecting data How can we collect this data? Moving 100s nm at a few kHz!!! 17

Quadrant Photodiode

Intensity distribution signals D x and D y

Quadrant Photodiode linear response for small displacements 

Quadrant Photodiode Quadrant photodiode collects the laser light transmitted through the condenser lens. Small changes in the transmitted and scattered light are measured. Advantages Large bandwidth 100s kHz Very fast compared to f 0 High light level as collecting laser light Disadvantages Complex arrangement Single particle 21

Trap strength or stiffness Fourier transform to get the power spectrum Lorenzian Calibration using the Power spectrum 22

Real data 23

Calibration (viscous drag force calibration) Vibrate container with liquid with known amplitude x o and frequency  AA 24

Calibration (viscous drag force calibration) AA AA Double frequency Signal is Get A from fitting A  against  for different  ’s. calibration constant 25

Bio-applications The size of particles that can be trapped is ~0.1  m to 10’s  m Approximately the same size as many biological specimen. e.g. Blood cells, stem cells, DNA molecules Either trapped directly, or beads used as handles to reduce optical damage. Ashkin et al. Nature. 330, 768 (1987) Block et al. Nature. 338, 514 (1989) 26

Measuring force/motion Molloy et al. Biophys J. 68, S298 (1995) biological object trapped bead quadrant detector imaging lens Image trapped bead (handle) onto quadrant detector Measure movement of shadow –nm accuracy! –kHz response Adjust trap to maintain position gives measurement of force –pN accuracy! 27

RNA Polymerase 28

RNA-Polymerase 29

e.g. Stretching/twisting of DNA Perkins et al. Science. 264, 822 (1994) Wang et al. Science. 282, 902 (1998) Attach handles to ends of DNA molecule Pull, let go and observe what happens! –understanding of protein folding 30

DNA mechanics Unzipping a DNA double strand 31