Biophysical Techniques

Optical Trap

The optical tweezers truly became a revolutionary biological tool when it was enhanced with several biological and optical innovations. Techniques were developed which allowed the force exerted by the trap on a bead as well as the displacement of the bead from the center of the trap to be measured with high accuracy. At the same time, biologists had developed methods of coating latex spheres with antibodies that bind to a specific epitope. Such activated beads could be used as a purification tool to pull a specific protein out of solution. It was soon realized that if such a bead were captured in an optical trap, force could be applied to an individual protein molecule. In one of the pivotal experiments such a bead was attached to a kinesin molecule that was 'walking' on a microtubule. This allowed researchers to determine that kinesin moves in discrete 8 nm steps on a microtubule and to and measure its force-velocity curve. Subsequent developments have dramatically increased the accuracy and flexibility of optical tweezers.

Principles of operation

If a particle is much smaller than the wavelength of light the electric field experienced by the particle is approximately uniform in space but oscillating in time. The finite dielectric constant of the particle causes it to develop a uniform volume polarization in response to the external electric field which partially cancels the electric field in its interior (see J.D. Jackson, Classical Electrodynamics, Second edition, p. 149). This reduction of the electric field lowers the total electromatnetic energy of the system, which is given by the integral of the electric and magnetic fields over all space. It is therefore energetically favorable for the particle to go the the position of maximum field where the energy reduction is greatest.

The center of the trapping beam therefore becomes a potential well into which the particle is drawn. If the particle is at the peripheral part of the trap beam, as illustrated in the figure above at left, it will be drawn towards the center, where it causes a greater depletion of the electric field. This is closely related to the elementary problem is electro-statics, where a dielectric slab is drawn into the gap of a parallel plate capacitor held at constant charge. As in the case of the optical trap, a uniform volume polarization of the dielectric generates a surface charge that partially cancels the external field.

The work required to pull the slab out of the capacitor corresponds to the increase in increase in energy stored in the capacitor, whose capacitance is increased by the removal of the dielectric.

Another way of understanding the optical trap is to treat the trapped particle as a lens which deflects the trapping beam as it passes through. In the limit that the bead is large compared with the wavelength (the opposite of the limit considered above) we can estimate the effect of the particle on the trapping beam by determining how the bundle of light rays representing the beam is affected by the particle. If the particle system is totally symmetric, with the particle at the center of the trap each ray will pass through the particle in a symmetrical manner and there will be no pertubation to the beam. This results in an output beam which is identical to the input beam, as illustrated below.

However, if the particle is pulled sideways the trap beam will pass through the edge of the particle. Just as rays that pass through the edge of a lens, the rays will be refracted towards the optical axis, as shown below.

Since light itself carries momentum, this deflection of the laser beam requires the particle to exert a force on the trap beam. The conservation of momentum dictates that an equal and opposite force is exerted on the particle by the trapping beam. In the figure above the red arrows indicate the force exerted on the beam of light and the blue arrows represent the equal and oppposite force that is exerted on the particle. It is seen that the blue arrows point towards the center of the trap field, i.e., the force tends to restore the particle to its position at the center of the trap.

The same is true if the particle moves up or down from the center of the trap, as illustrated below.

If the particle moves up, it will tend to focus the outgoing beam into a narrower bundle, increasing their net upward momentum. This net upward force on the laser beam results in a downward force on the particle. Conversely, if the particle is moved upward it tends to cause the output beam to splay out, decreasing the upward momentum of the outgoing beam. This causes an equal and opposite upward force on the particle. Again, the force tends to restore the particle to the center of the trap.

This second way of describing the action of an optical trap is particularly useful in understanding how force and position detection are done in an optical trap. If we can detect this change in momentum in the trap beam we could measure the instantaneous force on the particle, as well as its displacement from the center of the trap. This is done as illustrated below.

After passing through the condenser, the output beam passes through an aperture and is then detected by a position sensitive detector (PSD) positioned at the rear focal plane of the condensor. Any transverse motion of the particle will cause the center of the output beam to shift sideways on the PSD as illustrated above, giving rise to a measurable signal. Similarly, any motion of the particle upwards or downwards with make the output beam larger or smaller, and will result in a greater of smaller fraction of the output beam reaching the detector. In this way, the three dimensional vector force on the particle may be determined, as well as the three dimensional displacement of the particle from the trap center.

Remarkably, this technique allows forces to be measured with picoNewton accuraccy and displacements to be measured with Angstrom accuracy. The classic paper describing Optical Tweezers technology is Svoboda and Block [4].

Optical Torque Wrench

The basic physical idea behind the OTW is the same as for the optical trap, a particle will go to the position where the energy is lowest. To trap the particle in a specific orientation as well as in a specific position we make use of a particle which has an anisotropic polarizability. For instance, we can use quartz, whose z axis is easier to polarize than its x or y axes. The polarizability as a function of direction takes the form of an elipsoid, as illustrated below.
This means that if quartz aligns its z axis with the polarization vector of the trapping field, as shown above, its dielectric constant effective becomes larger and the potential well created by the trap field becomes deeper. It is therefore energetically favorable for the particle to go to the center of the trap, and to align its z axis with the trap polarization vector. By rotating the polarization of the laser beam, the particle may be rotated in the trap. You can follow this link to see a short video of a particle being rotated.

However, for the optical torque wrench be truely analogous to the optical tweezers we need a way of measuring the torque being applied to the particle by the optical trap. This can be done in a manner entirely analogous force detection. The conservation of angular momentum requires that the torque exerted on the particle by the trapping beam is equal and opposite to the torque exerted on the trapping beam by the particle. If we can measure the change in angular momentum of the trapping beam as it leaves the sample chamber we will know the instantaneous torque exerted on the particle, as illustrated below.

Since in this case the torque is generated by an interaction between the polarization of the laser and the anisotropic polarizability of the particle, the change in angular momentum manifests itself as a change in the polarization state of the trapping beam. The linear polarization of the input trapping beam consists of equal parts right circular and left circular polarized light, and the polarization of the output beam is eliptical, for which the left and right circular components are out of balance. A polarization analyzer on the output trap beam detects this imbalance of right and left circular polarization components, which is a direct measure of the torque.

Since we can measure the instantaneous torque, it is a relatively simple matter to use this information to actively stabilize the torque exerted on the particle. The measured value of the torque is used to correct to polarization angle to clamp the torque at a constant value. The plot below shows the torque on a trapped particle, initially fluctuating due to Brownian rotation motion.

At about 0.4 seconds the torque clamp is activated, which stabilizes the applied torque at 100 pN-nm. The thermally driven torque fluctuations are seen to be dramatically reduced by the feedback loop. At 0.8 seconds the set point of the torque clamp is set to 0 pN-nm. This work is described in reference [5] below. A typical experimental geometry for the OTW is shown below. Following this link will display a brief movie of a DNA tether being wound up 15 turns, then spontaneously unwinding after the trap is turned off.

This general configuration will make it possible to determine how the activity of nucleic acid enzymes is affected by super-helical torsional stress on the DNA polymer.

References