Optical tweezers

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(This page is the product of the literature project in TFY4335.)

An optical tweezer is a device using a focused laser beam to apply forces to microscopic objects, and this can be used for control and measurements in a large variety of applications. The manipulated objects can range from nanometer scale to micrometer scale, and the forces range from femtonewtons to nanonewtons.[1] Optical trapping of particles as small as 20-30nm has been demonstrated.[2] This article will both explain the physical principles behind behind the technique, as well as mention several of its applications. Optical tweezers are part of the TFY4335 - Bionanovitenskap curriculum.

Innhold

Physical principles

Ray optics explanation of the gradient force: a) The particle is displaced from the focal point, and will refract more light to one side than the other. It will then experience a net force along the intensity gradient. b) The particle is centered on the focal point, and will refract an equal amount of light to both sides. It is now in an equilibrium at the focal point, and experiences no net force. (Source: Wikimedia Commons)
Induced dipole explanation of the gradient force: At this moment in time, the gradient causes the force pulling the positive part to the left to be greater than the force pulling the negative part to the right, causing a net force to the left towards the focal point. At a later point in time, when the field and dipole has changed directions, the same consideration will still give a net force towards the focal point. (Source: Screenshot from Java applet. PhET Interactive Simulations, University of Colorado (2008)[Software]. Available from [1].)

An optical tweezer works by trapping a particle in the focal point of a focused laser beam. Although all the forces working to keep the particle trapped come from the same physical principles, the theory is usually explained by decomposing the forces into two separate effects. These are often called the scattering force and the gradient force. While these models give a qualitative understanding of the principles, the exact quantitative theoretical calculations can be so complex in practice that they cannot replace direct calibration of the equipment. Much effort has gone into the exact computation of optical forces, but quantitative agreements with experiments have generally been hard to obtain, with the exception of large particles.[1][2]

Scattering force

The component from the scattering force is the simplest one to explain and understand. This force pushes the particle away from the source and, assuming the gradient force is of similar magnitude, skews the equilibrium position of the trapped particle slightly down-stream from the exact focal point of the laser. The force originates in the transfer of momentum from absorbed and symmetrically back scattered photons. The exact explanations for this force are relatively similar for all particle sizes, and will not be shown in detail here. With unfocused lasers, where there is no significant intensity gradient, this component of the force dominates the entire overall force.[1]

Gradient force

When the beam at some point has a large intensity gradient, produced by a well focused laser through a lens with well corrected aberrations and a high numerical aperture, another component of the force becomes significant. This is the component which keeps the trapped particle laterally stable, as well as working against the scattering force, by always pulling it towards the center. To obtain a stable trap this gradient force must be large enough to prevent the scattering force from removing the particle from the focal point. In general the gradient force is within a small range proportional to the displacement from the center, and therefore acts similarly to an ideal spring obeying Hooke's Law. If one seeks a relatively simple explanation for the origins of this force, we can once again split the problem into two cases, now depending on the particle size. While this seems convenient, most object that are interesting for trapping are unfortunately in the intermediate size range between these two areas, where the physical explanations are much more complex.[1][3]

The ray optics model (d >> wavelength)

When the trapped particle is much larger than the wavelength of light, the gradient force is best explained by ray optics. The light gets refracted by the particle, which is acting like a lens, and continues on in a different direction. Since photons have a momentum, this change in direction corresponds to a change in the momentum of the light, which through conservation of momentum and Newton's laws give rise to a force on the particle. If the particle is in the exact center of the focal point, the intensities are symmetrical, the forces cancel each other out, and the gradient force will give no lateral motion.

If the particle is positioned slightly to one side of the focal point, it will experience an asymmetrical intensity gradient and the light refracted on one side will have a higher intensity than the light refracted on the other side. Since the rate of momentum change, and hence the force, is proportional to the light intensity, the particle will experience a net force in the direction of the intensity gradient. This direction of the force is assuming that the particle has a greater index of refraction than the surrounding medium. If the opposite is true, the force will be directed in the opposite direction of the gradient.

For a greater qualitative understanding of the ray optics model, readers should experiment with the Java applet given here or in the links section of this article.[1][3]

Dipole model (d << wavelength)

When the trapped particle is much smaller than the wavelength of light, the gradient force is best explained by seeing the particle as a small induced dipole in an inhomogeneous electrical field. While the dielectric particle alone is not a dipole, the time-varying electrical field from the laser will induce a fluctuating dipole. Since there is a strong gradient present in the electrical field, the dipole will be pulled along this gradient towards the focal point. Through this model the gradient force can be expressed quantitatively as[1]

F_{grad} = \frac{2\pi \alpha}{cn_m^2}\nabla I_0

where c is the speed of light in vacuum, nm is the index of refraction of the medium, I0 is the intensity of the incident light, and α is the polarizability of the sphere expressed by[1]

\alpha=n_m^2a^3\frac{m^2-1}{m^2+2}

where a is the radius of the sphere and m is the ratio of the index of refraction of the particle to the index of refraction of the medium.

For a greater qualitative understanding of the dipole model, readers should experiment with the Java applet given here or in the links section of this article.[1][3]

Applications

Optical tweezers can provide a powerful macroscale interface with the world on a nanoscale. This section presents some of it's important and less important applications.

Manipulation

Dynamic position control and optical trap arrays

Tetris: A scanning beam optical tweezer used to control the position of 42 glass microspheres of 1μm diameter in a 25μm x 20μm area under a microscope in a game of tetris.[4]

A single beam tweezer can be used to trap a single particle near it's focal point. In this subsection, methods by which it is possible to control the relative positions, in three dimensions, of a number of particles through a dynamic array of optical traps are described. The array can be set up in several ways.

Both techniques can be used for high precision, independent control of microparticle positions. For the scanning laser tweezer these favourable characteristics were demonstrated by Theodoor Pielage, Bram van den Broek and Joost van Mameren when they used 42 glass microspheres to create a tetris game.[4] Follow link for video demonstration [2]. The scanning technique can also be used for dynamic assembly and actuation of microscopic mechanical devices, as demonstrated by Terray et al.[5]

Structure fabrication and assembly

The previous subsection described how an array of optical traps can be produced and that these traps can be individually moved about in three dimensions. Here it is outlined how such an arrangement can be used to arrange and fix particles in an arbitrary structure and how structures can be made through optically induced photopolymerization. The assembly process will be discussed in terms of holographic optical tweezers as used by Sinclair et al. and by Chapin et al.[8][6] In principle, however, other techniques, like time shared tweezers, may be used.

3D-assembly: Nine 2μm particles assembled using HOTs from a single line into three axially aligned triangles. The time taken for each stage of this process to be invoked is indicated in the top left corners of the images.[8].

For relatively simple and planar arrangements of particles, a static hologram can be used. The hologram defines trap positions in the focal plane as described in the previous subsection and the whole arrangement of traps can be moved such as to trap particles in all foci.

For more complex and three dimensional structures, such a procedure is impossible or impractical. Especially in arrangements where traps are closely surrounded by other traps in many directions. Instead, one can use a fully automated procedure that only requires the input of final particle destinations. First, the process detects the arbitrary initial positions of the required particles. Then it calculates and applies holograms to trap the particles at these positions. Hologram calculation algorithms were mentioned in the previous subsection. Next, a path planning module derives the trajectories each particle must take in order to arrive at their prespecified positions. Finally, a sequence of holograms is obtained and applied.[6]

Particles that are arranged in three dimensions using HOTs, can be permanently fixed. For example, this can be done by having the particles immersed in a gel solution which sets after a certain time.[8]

Photopolymerization: A sculpture made by photopolymerization with the help of optical tweezers. The finest features are in the order of 100nm across[3].

Optical tweezers can also be used to assemble permanent microstructures through photopolymerization. Photopolymerization refers to processes where a substance in solution polymerize into a solid structure when exposed to light. The intense illumination from the focal point in an optical tweezer is ideal for driving photochemical reactions, and the fact that the focal point can be moved in three dimensions with great precision makes it possible to create microscopic objects of arbitrary shape. Under the right conditions the technique can yield features smaller than the wavelength of light. The ability to fabricate and stitch together any kind of small structures can have great applications for microelectromechanical systems such as sensors and lab-on-a-chip technology.[3]

Photopolymerization by optical tweezers has been demonstrated on many occasions. This includes the creation of microscopic rotors[9] and detailed three dimensional plastic sculptures.[3]

Microparticle and cell sorting

Microparticle and cell sorting: Single cells or particles are aligned to flow along the vertical axis of the setup. Optical reconition in the analysis region determines which particles that are to be directed right or left by the optical swicth.[10]

An example of a setup for microparticle and cell sorting, as used in an experiment by Wang et al, is shown in Figure "Microparticle and cell sorting".[10] This particular setup applies optical forces along with the laminar flow characteristics of a microfluidic system. A low Reynolds number environment makes fluid flow analogous to a conveyor belt. The role of the optical switch is then simply to direct particles left or right, corresponding to waste and collection respectively, based on the judgement of the optical recognition stage. The switch may be implemented as an optically actuated mechanical one, but it can also be a right increasing light intensity gradient stretching the full width of the channel that is turned on for target cells or particles that are to be part of the collection, as shown in the figure to the right and below. Also, to allow for left or right direction of several particles at once, it is possible to utilise HOTs. It is possible for the optical recognition stage to identify a variety of particle or cell properties. For example, Wang et al. used such a setup to sort cells based on the fluorescence of a protein[10] and Chapin et. al. demonstrated how microparticles can be sorted by size.[6]

Optical actuation of micromachines

Microscopic rotor which was both created and drived by optical tweezers. a) Illustration. b) Image of the rotor tumbling freely in solution. c) Illustration. d) Image of a trapped rotor, prevented from movement and rotation. e) Image of a rotor trapped in focus while being rotated by optical forces[9].

Optical tweezers can not only be used for the assembly and fabrication of tiny mechanical devices, but have also shown great promise as actuators for such systems. Because of the high surface area to volume ratio, friction has been a large problem for micromechanical devices, and the need for precise and relatively strong forces are apparent. The optical tweezers can solve the problem of driving the devices by applying precise forces exactly where they are needed, in a very customizable and controllable manner. [3]

An interesting application is the assembly and actuation of tiny pumps and valves for use in microfluidics. Optical tweezers can both assemble, position and actuate multiple microscopic devices and particles inside microfluidic channels, while at the same time eliminating physical contact with the outside world. More details about these applications in microfluidics will follow in the next section.[11][5]

Optical tweezer setups can even be used to apply torques around the beam axis instead of linear forces, creating a so called optical vortex. This is achieved by sending a laser beam with simple parallel wavefronts through a certain phase profile, transforming it into a helical phase profile where the photons carry an orbital angular momentum. A torque can also be achieved from the helical shape of the trapped object itself in a way analogous to windmills, meaning that the light will deflect in certain directions causing momentum to be transferred to the object in a way that produces a torque around a rotational axis.[3][9]

Both the above methods can be used to drive the rotational motion of different microdevices. As an example, a helical beam can be aimed and scaled to drive the teeth of a microscopic gear whose torque can then be transferred through a micromechanical system for other uses. These uses can be many things, from measuring the properties of structures like biopolymers to helping with the precise fabrication of nanotechnological tools through microelectromechanical systems. The principle of optically exerted torque has been demonstrated in practice by creating microscopic rotors, which were then rotated by use of an optical tweezers setup. The principle behind this torque was not the optical vortex, but simply shaping the rotor in a way that made the light deflect in certain directions. It was also demonstrated that this rotor could be set up to transfer its torque to microscopic gear wheels. This, combined with photopolymerization technique able to make microscopic objects of arbitrary shape, demonstrates the big possibilities in micromechanical devices brought by optical tweezers.[9]

Microfluidics and Lab-on-a-chip

Many of the techniques and applications discussed above can provide a huge asset when developing microfluidics and lab-on-a-chip systems. Microfluidics are systems for precise control of manipulation of fluid at a micrometer scale, and lab-on-a-chip specifically refers to the scaling of the functions of a chemical laboratory down to this size range. This can be very useful in chemistry and biology for many reasons, including increased analysis speed and the extremely reduced need of sample quantities.

Still there are many challenges in constructing a small and versatile lab-on-a-chip system. While traditional lithography is very effective for making the simple channels, further miniaturization has been halted by lack of techniques for fabrication of smaller valves, pumps and mixers. Actuation of these devices has also been a challenge, with the need of a quick, flexible and noninvasive solution. Earlier attempts at pumps and valves based on intricate systems of gears and cantilevers have so far failed to be implemented into microfluidics in a practical way. Optical tweezers can potentially solve many of these problems, with its ability to precisely and dynamically control large amounts of microparticles at the same time, and assemble them into needed devices like valves and pumps. Also as discussed earlier, particles or cells running through such a system can be sorted automatically, and the problem of mixing in laminar flow can be solved through microparticles spinning in an optical vortex or by using a microscopic rotor.

Pump in microfluidics: Image of microspheres being moved like a 2D analogue of a screw pump by a scanning optical tweezer. The series of images show the tracer particle(1.5µm) being pumped progressively to the left. The spheres making up the pump have a size of 3µm, and the channel is 6µm wide. [5]
Pump in microfluidics: Image of the "dumbbell" pump in different stages of rotation. The four 3µm microspheres making up the pump are moved independently by a scanning optical tweezer, one pair clockwise and the other pair counter-clockwise. The tracer particle is 1.5µm and the channel is 6µm wide.[5]
Valves in microfluidics: Image of the valves made from microspheres attached in a linear shape through photopolymerization. The size of the large sphere is 3µm, and the rest of the valve is around 1.5µm thick. a) The large sphere is held in place while the passive valve automatically blocks the flow of particles to the right. b) When the flow goes to the left the passive valve opens up automatically. c/d) The large sphere is held in place by the wall with a tweezer while the active valve is actuated by the same tweezer to direct the flow of particles up or down.[11]

Terray et al. demonstrated various applications in microfluidics by showing that one can assemble, position and actuate microscopic pumps and valves made from silica microspheres all with one single optical tweezer. The valve was made of microspheres connected to each other in rigid linear structures by use of photopolymerization. The structure was assembled inside the microfluidic system, where the solution needed for the laser to induce polymerization was present. The chain of particles was then transported to where it was needed, held in place, and sometimes actuated, all with the same single optical tweezer. Both passive and active valves were demonstrated, and it was shown that they could restrict and direct the flow of large particles while smaller particles and the fluid could pass.

Two kinds of active pumps were also demonstrated, which showed great ability to quickly control flow in both directions. These were also made from silica microspheres, but in this case they were all moved independently by a single beam in scanning mode instead of physically attached to each other by photopolymerization. One pump was made from four microspheres in a widened cavity of a channel, which were divided in pairs into two "dumbbells" rotating in opposite directions. The other was made by six microspheres arranged in a row, moved by a scanning tweezer like a two-dimensional analogue of the screw pump. All these valves and pumps can be seen in action in several videos available here.

It is apparent that optical tweezers used in combination with colloidal spheres have a great potential in microfluidics. It eliminates physical contact with the outside world, while being able to dynamically assemble, position and actuate many needed mechanical devices all inside the system itself, and will allow for an integration density far beyond what was previously available. The use of computer controlled holographic or scanning methods together with a large amount of microspheres has a huge potential for making up highly flexible, complex and integrated systems for chemical analysis and medical diagnostics in the future.[11][5][3]

Cell surgery

Cell surgery: Microscope image and illustration of organelle extraction surgery on a yeast cell using a single optical tweezer. Femtosecond-pulses (mode-lock) is used for the first four seconds only, to dissect the cell wall. After 17 seconds continuous-wave mode was used to trap and extract a single organelle (black arrow) from within the cell.[12]

By inducing photo-oxidation of biological materials with an optical tweezer setup, one can make a precise optical scalpel useful for surgery in single living cells.[3] This has already been used in the process of human assisted reproduction, by using a laser to cut through glycoprotein membranes. This has an advantage over the conventional method, which uses an acidic medium and can potentially harm neighbouring cells.[13]

Optical tweezers can also be used for intracellular organelle extraction. The process of trapping and controlling the cell, and the process of dissecting cell walls and cell membranes can both be achieved by use of a single laser. This is done by switching between two different operation modes of the laser. For the noninvasive trapping and manipulation, a continuous-wave mode is used. When the light is in the near-infrared part of the spectrum, cell damage is avoided while still maintaining a high enough gradient force to trap the cell. In this mode a cell can be trapped for hours with no effect on the cell viability. However, when switching to a mode-lock femtosecond-pulsed mode, the cell wall or membrane can be dissected in just a few seconds. The damage is not a product of thermal accumulation, because the deposited energy is still low. Instead the ultrashort laser pulses introduce nonlinear absorption and photochemical effects causing a build-up of structural changes. The fact that a cell can be dissected in such a short amount of time just by changing the operation mode implies a great potential for controlled and precise cell surgery using only a single optical tweezer.[12]

In 2008, Ando et al.[12] demonstrated the optical trapping and surgery of a living yeast cell using a single laser. They not only dissected the outer cell wall, but by alternating between the two laser modes they trapped and extracted a single organelle from inside the cell through the opening. Yeast cells are often treated as model organisms of eukaryotic cells, so the ability to do organelle analysis like this can be very useful for the investigation of organelle malfunctions causing human disease.

Measurement

Use of handles and calibration of optical tweezers

In order to conduct measurements on for example long strands of biopolymers such as DNA and RNA, so-called handles are being used. Biological macromolecules may have insufficient refractive abilities to be stably trapped by an optical force. Handles, however, are microspheres of polystyrene or silica that due to large refractiveness are subject to a strong force in an optical trap. If one end of, for example a DNA molecule, is chemically attached to a handle and the other is fixed or attached to another handle, measurements on the properties of the DNA molecules can be made.[14]

Usually, an optical tweezer is modelled as a spring in which particle displacement from the focus is proportional to the restoring force exerted. In such a model, the particle is subject to a harmonic potential. Calibration usually involves determining the equivalent spring constant k of the trap. This can be done, for example, by the equipartition theorem as follows.[1]

\frac{1}{2}k_BT = \frac{1}{2}k\langle x^2 \rangle

By measuring the variance in position of a trapped particle \langle x^2 \rangle and temperature T, the above equation can be solved for k. Adjusting this parameter can be done by varying the overall intensity or intensity gradient of the trapping laser.[1] Another calibration methods involves measuring the mean displacement \langle x \rangle from the focus point caused by a viscous drag force. The drag force can be created by moving the sample stage at a known and constant velocity vdrift while holding a trapped particle stationary. Since the trapped particle is stationary, the viscous drag force and the optical trapping force must cancel and for a spherical particle we have the following equation.

k \langle x \rangle = v_{drift}6 \pi \eta R

If the particle radius R and solution viscosity is η also is known, k can be calculated.

Measuring transcription by RNA polymerase and behaviour of biological motors

Measurement on DNA transcription: An RNA polymerase enzyme is fixed to a bead that is optically trapped. As the DNA strand is transcribed, the stage moves in order to hold the bead in a fixed position relative to the trap. The movement of the enzyme along the strand can then be found from the movement of the stage.[1]

A single RNA polymerase enzyme can be chemically fixed to an optical handle and make a transcript of a DNA molecule attached to a moveable stage, as shown in Figure "Measurement on DNA transcription". The bead is held at a constant distance from the focus of the optical trap and is thus able to exert a constant force suspending the DNA strand. As the enzyme transcribes the DNA, it moves along the strand. The stage to which the DNA molecule is fixed is then moved about by piezoelectrics such as to maintain the position of the particle within the trap. From recordings of this movement, the enzyme's movement along the strand can be deduced. Such an experiment was performed by Neuman et al. in 2003. They sought to explain why periods of constant RNA polymerase motion are interrupted by frequent pauses.[15]

Using a similar setup, it is achievable to measure the motion of biological motors moving along a strand in general. Information that can be obtained from these kinds of experiments are for example, if the motor moves constantly or in steps, what the size of the steps might be and how much force the motor is able to exert. Also, it is possible to measure if there is any dependence of ATP concentration in solution on this motion and and what this relationship might be.

Measuring kinetics of folding

Strand length vs. time recordings for different applied forces.[16]
Fraction of time spent in the unfolded state vs. applied force with probability density function. F1 / 2 refers to the force at which the RNA strand is in the folded and unfolded states for an equal amount of time.[16]

By attaching a folded molecule of DNA or RNA molecule to a handle, the biopolymer can be unfolded through application of a mechanical force in an optical tweeer setup. Liphardt, Onoa, Smith, Tinoco and Bustamante, unfolded RNA molecules in this way 2001. They applied a gradually increasing force to the strand and found that at a certain force f = 14.5pN, there was a clearly defined discontinuity in the force vs. extension curve. They concluded that this was due to the sudden unfolding of the strand. Also, holding the force constant and not too far from f = 14.5pN, the strand would make sudden jumps back and forth between the folded and the unfolded state and spend a certain total and force dependent time in each state. Subsequently, they made recordings of strand length z vs. time for different applied forces. A force vs. fraction of time spent in the folded state plot was also made. See Figures "Strand length vs. time" and "Fraction of time spent in the unfolded state vs. applied force".[16] To this, a force dependent probability density function for the fraction of time spent in the unfolded state could be fitted. The form of the density function is given below.[17]

P(f) = \frac{1}{1 + e^{- (\Delta F_0 - f \Delta z)/k_B T}}

Knowing f and Δz in the above equation, it is possible to calculate the free energy difference associated with the folding ΔF0. By considering the number of times the strand makes the jump from folded to unfolded state and the other way around, rate constants for given applied forces can be obtained.

Links

Java applet demonstrating the ray optics model of the gradient force

Java applet demonstrating the dipole model of the gradient force

Video demonstration of Real-life μ-Tetris

Videos of microfluidic valves and pumps assembled and actuated by optical tweezers

References

  1. ↑ 1,00 1,01 1,02 1,03 1,04 1,05 1,06 1,07 1,08 1,09 1,10 Neuman KC, Block SM, "Optical trapping", Review of Scientific Instruments (2004); 75(9): 2787-2809.
  2. ↑ 2,0 2,1 Svoboda, K. and Blocks S.M. Optical trapping of metallic Rayleigh particles. Opt. Lett 19, 930-932 (1994).
  3. ↑ 3,00 3,01 3,02 3,03 3,04 3,05 3,06 3,07 3,08 3,09 3,10 Grier, D. G. (2003). A Revolution in Optical Manipulation. Nature, 424, 810-816.
  4. ↑ 4,0 4,1 Pielage, T., van den Broek, B. and van Mameren, J. Real-life μ-Tetris. 31.01.2009, from http://www.nat.vu.nl/~joost/tetris/
  5. ↑ 5,0 5,1 5,2 5,3 5,4 5,5 Terray, A., Oakey, J. and Marr, D.W.M. Microfluidic control using colloidal devices. Science 296, 1841-1844 (2002)
  6. ↑ 6,0 6,1 6,2 6,3 Chapin, Stephen C., Germain, Vincent and Dufresne, Eric R. (2006) Automated trapping, assembly and sorting with holographic optical tweezers. Optics express, 14, 26, 13095-13100.
  7. ↑ Sinclair, G. et. al. (2004). Interactive Application in holographic optical tweezers og multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping. Optics Express, 12, 8, 1665-1670.
  8. ↑ 8,0 8,1 8,2 Sinclair, G., Jordan, P., Courtail, M. P., Cooper, J. and Laczik, Z. J. (2004) Assemply of 3-dimensional structures using programmable holographic optical tweezers. Optics Express, 12, 22, 5475-5480.
  9. ↑ 9,0 9,1 9,2 9,3 Galajda, P. and Ormos, P. Complex micromachines produced and driven by light. Appl.Phys.Lett. 78, 249-251 (2001)
  10. ↑ 10,0 10,1 10,2 Wang, M. M. et al. (2004). Microfluidic sorting of mammalian cells by optical force switching. Nature Biotechnology, 23, 83-87.
  11. ↑ 11,0 11,1 11,2 Terray, A., Oakey, J. and Marr, D.W.M. Fabrication of linear colloidal structures for microfluidic applications. Appl. Phys. Lett. 81, 1555-1557 (2002)
  12. ↑ 12,0 12,1 12,2 Ando, J., Bautista, G., Smith, N., Fujita, K., Daria, VR., Optical trapping and surgery of living yeast cells using a single laser. Review of scientific instruments (2008) 79, 103705
  13. ↑ Wright, G., Tucker, M.J., Morton, P.C., Sweitzer-Yoder, C.L and Smith, S.E., Micromanipulation in assisted reproduction: A review of current technology. Curr.Opin.Obstet.Gyn. 10, 221-226 (1998)
  14. ↑ Sovaboda, K. and Block, S. M. (1994). Biological Applications of Optical Forces. Annual Review of Biophysics and Biomolecular Structures, 23, 247-285.
  15. ↑ Neuman. K. C. et al. (2003). Ubiquitous Transcriptional Pausing Is Independent of RNA Polymerase Backtracking. Cell, 115(4), 437-447.
  16. ↑ 16,0 16,1 16,2 Liphardt, J. et al. (2001). Reversible Unfolding of Single RNA Molecules by Mechanical Force. Science, 292(5517), 733-737.
  17. ↑ Nelson, P. (2008). Biological Physics. New York: W. H. Freeman and Comany.
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