3. lecture
Laser tweezers
Due to its high intensity and directionality, lasers can be preferentially used in opticaltweezers where cells and other small objects under microscopic observation can be caught and then be moved to desired position. The facility of micromanipulation by laser tweezer has fruitful presence and perspective future in biophysics.
Principles. The laser beam is focused onto the transparent object under manipulation. The transparency at the wavelength of the laser light is highly critical. If even a small fraction of the huge incident radiation was absorbed, the tiny object would evaporate immediately. Under illumination, the object will move to the center of the beam. When the beam is shifted away, the object moves together with it; the particle is trapped by the beam as if by tweezers.
What is the mechanism of this action? Consider a transparent bead of radius r illuminated with a laser beam having a Gaussian intensity profile (Fig. 3.1). The center of the sphere is located at distance xB from the position of the maximum light intensity of the laser beam. The bead has a refractive index nat the wavelength of the laser λ. For the sake of simplicity, consider only two symmetric rays of the laser field that lie in a plane with the center of the bead at positions x and
(2xB-x) for rays 1 and 2, respectively. The rays fall a=xB-x apart from the axis of the bead. The intensities of the rays passing the bead are I1 = ΔN1/Δx/Δt and I2 = ΔN2/Δx/Δt in ray 1 and in ray 2, respectively, whereΔN denote the number of photons in space (Δx) and time (Δt) elements.We will determine what is the integrated force F exerted on the bead along the x axis and how does this force depend on the position of the bead in the beamxB.
The photons of the two rays are deviating from their original directions due to light refraction while crossing the object, and thus their momentum changes. The conservation of momentum must hold, so the momentum of the bead changes too, i.e. a force originating from light gradient is exerted on the bead. It should be mentioned that a much smaller additional force is also acting on the bead which comes from the reflections of the photons at the boundaries (including the multiple reflections inside the bead). The force is attributed to the light pressure and shows primarily in the direction of the laser light (the laser pushes away the bead). As the reflected intensity on the transparent bead amounts a couple of percent of the incident intensity only, the force from light pressure will be neglected in our approximation.
Consider the paths of the rays through the bead. As the two rays are symmetric relative to each other, we shall discuss only the path of ray 1 (see Fig. 3.2). From the triangle OAB, we find
.(3.1)
Snell’s law of light refraction gives
,(3.2)
from which
.(3.3)
The deviation of ray 1 from its original direction is
.(3.4)
On leaving the bead, ray 1 is refracted again. Now β is the angle of incidence and α is the angle of refraction, and consequently the ray is deviated by the angle γ again. If we compare the directions of the ray before and after the bead, the deviation is altogether 2γ. The same holds for ray 2, but the deviation is in the opposite direction.
The momentum of a photon in the beam is
(3.5)
both before and after the bead, but the direction of the momentum vector changes. It is in the direction of the y axis before the bead, whereas afterwards it makes an angle of 2γ with the y axis, as shown in Fig. (3.3) for both rays. This means that the x component of the momentum vector of the photon is zero before the bead, but differs from zero after the bead. Thus, by applying the law of conservation of momentum for the interaction of one photon with the bead, for the x components we can write
for ray 1,
,(3.6)
and for ray 2,
,(3.7)
where p1x and p2x are the momenta that the bead “obtains” from the photons in the x direction.
The force exerted on the bead is
,(3.8)
where Δp is the change of the momentum of the bead in time Δt. If we take into account the number of photons that pass the bead in Δt in the two rays, and Eqs (3.5), (3.6) and (3.7), we obtain
,
,
.(3.9)
After introducing the intensity I = N/Δx/Δt instead of the photon number, we can get the elementary force of the light gradient for these two rays. The total force can be obtained by integration of the infinitesimal form of Eq. (3.9) for the entire laser radiation passing through the bead:
,(3.10)
where γ(x) can be expressed from Eqs. (3.1)-(3.4) as function of x:
.(3.11)
If the intensity profile of the laser field I(x) is known, then the force can be exactly calculated from the position of the bead by using Eqs. (3.10) and (3.11). Here, we will draw qualitative conclusions only. The force is directed towards the center of the beam, where the intensity is the highest (towards the „light gradient”). This is the reason why the bead moves to the center of the beam, and this explains why it shifts together with the beam. As our calculation shows, the more I1 and I2 differ from each other within a definite distance, i.e. the larger is the light gradient, the larger will be the force. If the bead lies in the center of the beam (xB = 0), no force appears in the x direction. If the bead is shifted away from the center, the force increases until the bead reaches the point where the Gaussian intensity profile has its inflection point. Then the force diminishes slowly to zero. Good trapping needs highly convergent rays going into a tight focus. Usually a microscope is used to get the laser beam which meets the requirement. After experimental determination of the relationship between force and deviation of the bead from the equilibrium position (xB = 0), one can use the tweezer for force measurement („single molecule force spectroscopy”).
At small displacements, the restoring force is proportional to the distance between the center of thesphere and the focus of the lasers (harmonic approximation) and it can be described as
.(3.12)
If we can determine the constant k (referred to as the trap stiffness) and the position of the bead in the trap xB, then the force on the bead in the trap can be derived. Modern image analysis techniques are able to measure the position of a micron-sized sphere with an accuracy of 10 nm. The trap stiffness varies considerably depending on the design of the optical tweezers and the size of the sphere, but a value of 50 pN/μm is reasonable. This gives a resolution of 0.5 pN in force measurement.
The simplest way to carry out mechanical micromanipulation with molecules is depicted in Fig. 3.4. Each end of the molecule is attached to different dielectric microscopic beads, the surfaces of which are chemically active and provides reactive groups, for example NH2, COOH or SH. The activated microscopic beads serve as “handles” for molecular manipulation. While one of the beads is trapped by the laser tweezer (it has a fixed position), the other bead is grabbed by the tip of a glass micropipette and can be moved freely (the extension of the molecule can be adjusted). As the small displacement of the fixed bead from the focus of the laser measures the force (see above), the two quantities are available to construct the force vs. extension characteristics of the molecule.
Applications in biophysics. The ability to manipulate single molecules with nanometer precision and to measure forces on these molecules with piconewton accuracy using optical tweezers has opened up several important new areas in biophysics. By stretching the DNA or other biomolecules (e.g. proteins), physical models describing the mechanical properties of the molecule can be tested. The activity of a single enzyme acting on a DNA molecule can be determined and the effects of DNA binding proteins on the properties of DNA can be directly observed. We can measure the forces generated by molecular motors as they move cargos along microtubules or actin filaments (simply grab the cargo, and measure the force required to stop the motion).
Stretching DNA. The individual double stranded dsDNA molecule is characterized by at least four different force/extension regimes (Fig. 3.5).
1. Entropic elasticity regime of the B-form. A dsDNA molecule in solution bends and curves locally as a result of thermal fluctuations. Such fluctuations shorten the end-to-end distance of the molecule, even against an applied force. This elastic behavior is thus purely entropic in origin. The inextensible worm-like chain (WLC) model is often used to describe the entropic elasticity of DNA. In WLC model, the moleculeis treated as a flexible rod of length L that curves smoothlyas a result of thermal fluctuations. The rod’s localdirection decorrelates at distance s along the curve accordingto e–s/P, where the decay length, P, is the persistencelength of the chain. The stiffer is the chain, the longer is thepersistence length. For dsDNA in physiological salt, thepersistence length is approximately P ≈ 50 nm. The exact force(F) required to induce an end-to-end distance extension of x in a chain of contour length L must be obtained numerically, but a useful approximation is
.(3.13)
At the lowest forces (F < 10 pN), the molecule behaves as a Hookian spring and its extension is
proportional to the force applied at its end: F = kDNA·x. Thus, dsDNA behaves as a linear spring with a Hooke’s constant kDNA = 3·kBT/2·P·L, that is, inversely proportional to the length of the molecule and its persistence length. A 10 μm dsDNA molecule, for example, has a spring constant of approximately 10–5 pN·nm–1.
2. Enthalpic elasticity regime of the B-form of dsDNA. Above 10 pN, the end-to-end distance becomes longer than its theoretical B-form contour length L, indicating the existence of a finite stretch modulus. Thus, at these high forces, the chemical structure of DNA is being altered and the elastic response is not merely entropic but becomes enthalpic. The molecule behaves as a stretchable solid. Assuming that the contour length of the molecule increases linearly with the applied force, the following formula can be used between 5 and 50 pN:
.(3.14)
where S is the stretch modulus of the molecule. The stretch modulus of a simple elastic rod is related to its intrinsic persistence length, Pi, as:
.(3.15)
where r is the rod’s radius. An intrinsic persistence length of 60 nm is thus obtained for dsDNA, assuming its radius is 1 nm, in fair agreement with the value obtained from the entropic elasticity measurements.
3. The overstretching transition. When the dsDNA is subjected to forces of 65 pN or more, it suddenly changes form, stretching up to 70% beyond its canonical B-form contour length. The structure of this so-called S-form DNA is not known. The B-to-S overstretching transition occurs within a narrow range of forces (see the flat plateau inFig. 3.5), suggesting a cooperative process.
4. Entropic elasticity of the S-form. Above these forces, the S-DNA melts into single strands that exhibit the characteristicforce/extension behavior of ssDNA.
Elasticity of single protein molecules.Some examples will be listed where laser tweezer-related techniques monitored the folding/unfolding of single protein molecules. 1) The helices of the bacteriorhodopsin molecule are anchored to the bacterial membrane and a force of 100-200 pN is required to remove a helix from the membrane. 2) The spider dragline silk protein molecule unfolds through a number of rupture events, indicating a modular structure within single silk protein molecules. The minimum unfolding module size of 14 nm indicates that the modules are composed of 38 amino acids residues. 3) Some skeletal muscle proteins can withstand drags of 600 at (= 6·107 Pa). Among the muscle proteins, the titin is the most frequent player in the force measurements.
The titin is a ~3.5 MDa filamentous protein that constitutes about 10% of the total mass of vertebrate muscle proteins. This giant molecule spans the half sarcomere (unit of the muscular structure) from the Z-line to the M-line (Fig. 3.6). Titin is anchored to the Z-line and to the myosin-containing thick filaments of the A-band via its strong myosin-binding property. The I-band segment of the molecule consists of serially-linked tandem immunoglobulin (Ig) domains interrupted with a proline (P)-, glutamate (E)-, valine (V)- and lysine (K)-rich segment (named PEVK segment) and other unique sequences. Upon stretch of the sarcomere, passive force is generated by the extension of the I-band segment of the titin. On the other hand, the A-band segment of the titin is composed of super-repeats of Ig and fibronectin (FN) domains. The A-band segment of the titin does not participate in the generation of passive force under physiological conditions, but remains attached to the stiff thick filaments. This portion of the molecule is thought to provide a structural scaffold for the thick filament.
In the raw force vs. extension curve of the titin (Fig. 3.7), the initial rise shows nonlinear but elastic behavior during stretch (between points A and B) before a force transition begins. This stretch transition takes place until the maximum experimental force is reached (in point C). Upon release, the force drops rapidly, and the initially observed characteristic non-linear force response is established (at point D). The non-linear elastic behavior of the titin can be well described by the wormlike chain (WLC) model (see Eq. 3.13) both during stretch and during release indicating the entropic polymer nature of the titin. Indeed, the titin can be extended to lengths exceeding the ~1 µm length of the native, folded (but straightened) molecule, showing that part of the extension most likely occurs at the expense of unfolding in the titin (mainly in the PEVK region).
Force due to DNA replication in a single polymerase-DNA complex. The protein DNA polymerase catalyzes the replication of the DNA which reaction requires a single-stranded DNA (ssDNA) as a template. A complementary strand of ssDNA is synthesized to the original ssDNA. During the polymerization, both strands coil around each other, resulting in shortening of the end-to-end distance of the DNA. Therefore, if strain (force) is exerted on the DNA strand during polymerization, the reaction can be stopped or even reverted (Fig. 3.8). At a force of less than 34 pN between the beads, the ssDNA has low tension (is loosely attached to the bead) and the polymerase can fulfill the polymerization against a low resistance. By extension of the polymerase, the tension increases due to the shortening of the ssDNA by coiling of the dsDNA. At a force of 34 pN, the polymerase stops working. In other words, the polymerase catalyzing DNA replication can work against a maximum force of about 34 pN. If the force is further increased by increase of the distance between the beads, shortening of the dsDNA and therefore a reversion of the extension can be observed due to exonuclease activity of the polymerase. The exonuclease activity can increase the template tension about 100-fold above 34 pN.
Take-home messages. Optical tweezers use a highly focused laser beam to provide an attractive or repulsive force depending on the refractive index mismatch of the microscopic dielectric objects. The force shows to the light gradient and is typically on the order of piconewtons. Laser tweezers are used to micromanipulation by physically grabbing and moving small objects. They have been particularly successful in studies of single biomolecules and a variety of biophysical (mechanical) and biochemical reactions of biological significance.
Home works
1. By describing the thermal equilibrium of a dielectric bead in water, estimate the maximum allowed absorption of the bead at the wavelength of the laser.
2. Make reasonable estimation of the force on the bead originating from the pressure of the high intensity laser light.
3. How the relative refractive index of the bead can influence the direction of the forces in the optical tweezer?
References
Bustamante C, Smith S.B, Liphardt J, Smith D (2000) Current Opinion in Structural Biology, 10:279–285.
Kellermayer M. (2007) DSc. Thesis, University of Pécs, Hungary.
Maróti P, Berkes L, Tölgyesi F (1998) Problems in Biophysics, Akadémiai Kiadó, Budapest.
Nölting B (2004) Methods in Modern Biophysics, Springer
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