Optical Traps: An Introduction

Part 1: Optical Traps: An Introduction

00:00:11;09 Okay, so today we're going to have a lecture on the design,
00:00:15;12 principles, and applications of optical tweezers.
00:00:19;20 My name is Carlos Bustamante and I would like to tell you
00:00:24;06 about how it is that optical tweezers work, and what are the
00:00:29;17 principles on which they are based, and what kind of applications
00:00:33;13 you are able to use optical tweezers on. Now, the name optical
00:00:41;02 tweezer was given by Arthur Ashkin to a device that he invented
00:00:45;15 in the 1970s while he was working at Bell Labs. And the idea is
00:00:49;27 to use light in order to manipulate objects, physical objects.
00:00:54;09 It turns out that particles with high indices of refraction relative
00:00:58;28 to the medium, as you may imagine is the case with plastic
00:01:03;04 beads or glass beads, or oil droplets, are attracted to intense
00:01:08;17 regions of a laser beam, which is focused through a microscope
00:01:12;09 objective, and can be held permanently at that position.
00:01:17;19 At that focal point. And this is the discovery that Arthur
00:01:21;06 Ashkin made in the 1970s. In fact, here you see a paper that
00:01:27;16 was published by Arthur Ashkin in 1970, where he describes
00:01:32;06 the single-beam trap, or single-beam optical tweezers, instrument.
00:01:37;09 It's also called an optical trap. And the double-beam optical
00:01:41;09 trap, as indicated here. They found that if they collimated or
00:01:46;24 they focused a light beam through to a diffraction limited
00:01:51;29 spot in a cuvette that contained many, many small beads,
00:01:55;28 plastic beads, it was possible to actually trap momentarily those
00:02:00;17 beads. But for a single trap, they found that those beads tended to be
00:02:04;03 pushed along the direction of propagation of the light. As indicated here.
00:02:07;20 And so they developed the dual trap beam, in order to have
00:02:11;06 another force of light actually that was propagating in the opposite direction
00:02:16;10 in order to obtain a stable spot to trap the bead in the center
00:02:20;11 between the two beams. Later on in 1987, Ashkin continued
00:02:28;21 developing the idea of the optical trap. And in this case, with the idea
00:02:35;03 of using the optical trap, or the optical tweezers, to trap and manipulate
00:02:39;02 single cells. And so they were able to show that they were able to trap
00:02:43;23 a live bacteria. They were able to sort living cells from nonliving
00:02:49;26 cells. And even manipulate organelles inside the cells.
00:02:53;24 In this particular publication in '87, they also realized that instead
00:02:59;08 of using visible light, they were to use infrared light and it was possible
00:03:05;09 to in fact, decrease and diminish the damage that the radiation
00:03:10;05 could actually do on the biological systems. Now the principle
00:03:17;17 or the fundamental physical principle behind the idea of the
00:03:21;15 laser tweezer, or optical tweezers, is the fact that light exerts
00:03:27;04 force on matter. And in fact, in 1619 already, Johannes Kepler
00:03:34;11 suggested that the solar repulsion of the finely divided matter
00:03:39;26 that comes off the tails of a comet is due to the pressure of light
00:03:45;12 exerted by the light coming from the sun. And so, you see this
00:03:50;15 idea is very early. Newton considered this idea in his optics
00:03:55;17 book, and considered it possible. And later on, eventually,
00:04:01;06 James Clerk Maxwell, when he developed the four equations
00:04:04;05 that describe all the theories of electromagnetism, was able to
00:04:08;05 in fact predict theoretically that light should be able to exert
00:04:13;11 pressure and forces on matter. However, the first experimental
00:04:19;29 demonstration in the laboratory for the evidence of radiation
00:04:23;17 pressure, as it's also known, given the fact that this works
00:04:27;16 not only with light, namely with radiation that we can see with our
00:04:31;02 naked eye, but also with infrared light, with UV light. Radiation of any
00:04:36;23 wavelength is due to Levedeb who published a very influential
00:04:43;26 and important paper in the Annalen der Physik in the year 1901.
00:04:48;04 And you can see here that the title of his paper is in fact
00:04:53;12 First experimental evidence for pressure of light on solid bodies.
00:04:57;29 Now, in order to understand how these optical tweezers work,
00:05:11;01 we need to actually describe the trapping in two different regimes, the trapping
00:05:15;17 of objects by light in two different regimes. The first regime we're
00:05:19;14 going to consider is the regime in which the wavelength of light
00:05:22;03 is much larger than the size of the object you want to trap.
00:05:25;22 Imagine, for example, using wavelength of visible light and you are
00:05:29;19 trying to trap an atom or a molecule, a small molecule, so we
00:05:35;04 call "d" the size of the object, and I remind you that light
00:05:40;06 can be seen as nothing else but electromagnetic radiation that
00:05:45;23 travels in space and that electromagnetic radiation is
00:05:52;10 essentially, when it is polarized, means that you have an
00:05:55;14 electric field that is vibrating in a particular plane. In this case, the
00:05:59;18 vertical plane. And a magnetic field is vibrating in the horizontal
00:06:05;08 plane. Okay, so now what happens when you collimate or
00:06:13;10 you focus light, say through an objective? You take light from
00:06:18;27 a laser and you fill the back focal plane or your objective
00:06:22;01 and then you collimate it into a diffraction limited spot.
00:06:25;06 What's going to happen is you're going to have a situation
00:06:27;12 like this. Where you have the light, in fact, much more intense
00:06:30;18 at one particular region which happens to coincide with the
00:06:33;26 axis of this cylindric shape. And also, at the center of this cylindric
00:06:40;20 shape. So if you have an object, a small molecule for example,
00:06:44;13 or an atom, that is nearby as is indicated here. What is going to
00:06:49;00 happen is that the electric field of the electromagnetic radiation
00:06:51;19 is going to induce an electric dipole moment in the molecule.
00:06:55;13 And that's what we call here "mu." You see here, we are
00:06:58;28 calling mu as an electric dipole moment generated in the
00:07:03;08 molecule by the electric field of the light. And it turns out
00:07:07;02 that an induced dipole moment will be attracted toward the higher
00:07:12;25 intensities of the light, as indicated here. So a force, which I
00:07:17;21 indicate here with this arrow, will actually act on the molecule
00:07:23;05 and will bring the molecule, in fact, towards the center of
00:07:26;22 the beam. So, you see that in this small size limit, we can explain
00:07:33;14 the trapping of objects by light simply as the coupling between
00:07:38;16 an electric dipole moment, which is induced by the electric field of the
00:07:41;24 light, as indicated here, right? Here you can see the electric dipole
00:07:46;04 moment can be written as for an object, any object, an atom
00:07:50;10 or a molecule, any object that has polarizibility, or is polarizable.
00:07:56;02 And its polarizability is described by this quantity alpha. Then
00:08:01;08 the electric field of the light will generate the dipole moment.
00:08:04;11 It turns out that the energy of this object in the field is given
00:08:08;06 by this quantity minus 1/2 the induced dipole moment, times
00:08:12;14 the electric field, the external electric field, and we can
00:08:16;03 of course write the dipole moment here as alpha times E.
00:08:19;11 So we'll obtain the square of this electric field, which is proportional to
00:08:23;17 the intensity of the electromagnetic radiation, as shown
00:08:27;13 here. In other words, the energy of the dipole in the field,
00:08:31;24 E, is proportional to the intensity of the light. And now,
00:08:39;18 because we have now this dipole moment, essentially
00:08:43;21 in a potential energy field due to the electric field, then it is possible
00:08:48;14 to calculate a force as minus the gradient of the intensity, as indicated
00:08:54;22 here. Okay, so essentially the force that the dipole experiences
00:09:01;24 is proportional to the gradient of the intensities. So, in fact,
00:09:06;09 if the beam is not homogeneous, but is being collimated by
00:09:10;18 an objective, right? Then what you will see is that the dipole moment
00:09:15;22 will be attracted to the most intense part of that beam.
00:09:19;09 And that's what we call the gradient force. Okay, so that's what we
00:09:23;11 call here, the gradient force. Okay, so to summarize. If you
00:09:29;08 have a dipole moment, a molecule, an atom, and you have
00:09:33;02 it close to one of these intense beams of light that have been collimated
00:09:39;04 and focused through an objective, what you will see is that the
00:09:42;26 force appears, that is essentially pulling the dipole moment
00:09:47;26 induced in the molecule, right, towards the center, the place
00:09:51;08 where the beam is most intense. But, besides this attracting
00:09:58;02 force, which we are observing here with the gradient force,
00:10:01;13 there are other kinds of forces that are also present when
00:10:04;17 the object is small compared to the wavelength of light.
00:10:07;05 For example, incident light, as shown here, will hit on the
00:10:10;21 molecule or on the atom, and it will be reflected by the molecule
00:10:14;11 or the atom. It will be essentially scattered away. And in turns out
00:10:19;22 that because light carries momentum, right? Remember light
00:10:25;14 carries momentum and each photon of the light carries
00:10:28;21 an amount of momentum that's equal to the Planck constant divided by
00:10:32;01 the wavelength of light. Then what is going to happen is that these
00:10:35;23 photons that were actually coming in this case from
00:10:40;05 left to right, right? And hit the atom, and will be bounced
00:10:44;06 back. So they have lost their right momentum, and the result
00:10:49;03 is that because momentum, linear momentum, is a quantity
00:10:52;16 that is conserved in nature, just like how energy is conserved
00:10:55;18 in nature, then somebody else has to gain momentum
00:10:58;22 in the right direction. And that somebody else turns out to be the
00:11:02;21 atom itself, okay? So imagine if you are actually playing
00:11:06;18 a game of pool, and then you come with some ball that hits
00:11:11;26 the white ball here, with one angle it gets reflected and of course,
00:11:19;03 the other ball, the white ball, will actually move in the opposite
00:11:22;15 direction, because it's trying to conserve momentum.
00:11:25;09 Okay, so this is therefore a reaction of recoiling force
00:11:30;13 that we must take into account, as well. So let's try to formalize a little
00:11:37;01 bit what we mean by this recoil force, that as I mention has to do with the fact
00:11:43;10 that light carries momentum. In fact, I want to remind you
00:11:46;29 first that according to Newton's Second Law, what we call force
00:11:50;09 is nothing but the rate of change of momentum of a system.
00:11:54;05 Right? Mathematically, it can be written in this way, where you can
00:11:58;18 see that the force is nothing by the derivative of the momentum
00:12:01;22 of the system, with respect to time. The rate of change
00:12:05;10 of momentum. Now let's assume that we have N photons
00:12:11;16 coming, and they are coming towards a surface per second.
00:12:16;17 And each one of them carries a momentum, h, I told you that
00:12:21;05 Planck constant divided by the wavelength of light. And
00:12:24;02 that these photons are going to be absorbed by this surface,
00:12:28;00 which they are actually hitting. Then what's going to happen
00:12:32;25 of course, is that each photon was bringing momentum p, equal to
00:12:37;01 h/lambda. And now, at the end of absorption, all the momentum of the
00:12:41;17 photon is absorbed into the object. So that the momentum of the
00:12:45;23 photon will go from p to 0. In other words, it has a change of
00:12:49;10 -p. Each photon has lost an amount of momentum that we call
00:12:54;24 -p, okay? So we can actually write that in this way. We can say that
00:13:02;01 the object, the surface in this case, is going to experience
00:13:06;29 a force, a pressure, due to the light that is equal to the
00:13:12;02 rate of change in here. Now this rate of change of the N
00:13:17;09 momenta, h/lambda, that was coming from each photon to
00:13:22;09 hit the surface. And of course, we can write the wavelength of light
00:13:26;14 as the frequency of light divided by the speed of light times
00:13:29;14 the index of refraction of the medium. So we have this
00:13:32;21 expression. And if you look carefully, here we have N
00:13:37;11 number of photons per unit times the h mu, which is the
00:13:42;25 energy of each photon. So that turns out to be actually the
00:13:46;01 power, W, of the beam. And so you see that the force
00:13:50;25 that is experienced by the surface due to the absorption
00:13:54;05 of the photons is actually equal to the power of the light
00:14:01;28 hitting the surface, which is energy per unit time, times the index
00:14:07;18 of refraction of the medium, divided by the speed of light.
00:14:09;13 Now, of course, this assumes that everything worked perfectly.
00:14:15;06 And no photon was scattered, for example. That all photons were
00:14:19;10 absorbed. So in practice, people often write this expression in
00:14:23;04 terms of some kind of empirical efficiency factor, epsilon,
00:14:26;08 which multiplies the quantity we have derived here. Because
00:14:31;05 we take into account the fact that some photons are not
00:14:33;18 absorbed, some photons were reflected or scattered or
00:14:36;07 so on, and so forth. But the important thing is the following.
00:14:38;26 You see how the fact that light carries momentum means that if
00:14:42;28 these photons either are absorbed, and therefore they lose
00:14:46;28 completely their momentum on a surface or on an object,
00:14:51;19 the object or that surface are going to gain that momentum that
00:14:55;09 the light had. And therefore is actually going to experience a force,
00:14:58;08 even if that force is nothing but the rate of change of momentum.
00:15:02;03 In other words, how the momentum changes in time.
00:15:06;27 Okay, in the low small size regime that we're discussing here,
00:15:13;19 when the object is much smaller than the wavelength of light.
00:15:15;28 It's still true that you have two types of forces, the gradient force
00:15:19;27 I talked about before, which is proportional pretty much to the
00:15:23;02 radius cubed of the object. And you have the scatter force,
00:15:27;14 or the pressure force, or the momentum force, as you wish.
00:15:31;13 That is proportional to the radius of the sixth power of the object.
00:15:35;11 And really if you want to trap stably any object that is very small compared to
00:15:41;00 a wavelength of light, let's say an atom or molecule, you have
00:15:44;12 to essentially balance these two forces to make sure that
00:15:48;22 you obtain stable trapping. One solution, for example, is
00:15:53;15 to increase the numerical aperture of the lens. You will see
00:15:56;17 in the other regime, and this will tend to reduce the focal spot
00:16:01;14 making it smaller and increase the gradient force to the point in which
00:16:05;12 you can overcome the scattering force. Anyway, so let me
00:16:12;04 now go to the second case. The second case is where the object
00:16:17;10 is much larger than the wavelength of light, as indicated
00:16:20;10 here. So d is much larger than lambda. And in this case,
00:16:24;01 it is not possible to describe interactions in terms of gradient
00:16:29;04 force or scattering force, but it is possible instead to describe
00:16:32;11 the interaction using ray optics. And so what happens when
00:16:36;19 light hits, say, a plastic bead? A polystyrene bead or a glass bead.
00:16:42;25 What you know is that the object in this case, the bead,
00:16:47;24 is going to then reflect the light, refract the light, or scatter the light.
00:16:53;15 Okay? So we must consider all these 3 effects. As I said
00:16:59;28 before in this regime, it is no longer possible to separate
00:17:03;18 the scattering force and the gradient force, but we can still
00:17:07;20 understand how the trapping takes place by using conservation
00:17:14;24 of linear momentum, which we have already been talking about.
00:17:17;12 So ultimately, of course, stable trapping requires that all these
00:17:22;10 forces that arise from all these phenomena, namely the
00:17:28;01 scattering force, the refraction, and the reflection, would be
00:17:31;13 balanced so that you obtain an object physically and mechanically trapped.
00:17:36;00 But, let's remember some of the ideas that we've been talking about.
00:17:40;13 So, what happens when a photon meets a refractive object?
00:17:44;18 Imagine in this case, imagine that we have a bead and this bead
00:17:49;16 is placed in the beam, in the path of a beam. And so there is a number of
00:17:54;16 photons that are coming towards the bead, as is indicated here,
00:17:57;28 and each photon brings a momentum of h/lambda. So that's the
00:18:02;27 incident momentum of the light. It hits the bead at this position. It
00:18:08;06 gets refracted, as indicated here, and again gets refracted
00:18:12;28 as it comes out of the bead in a different direction for what we
00:18:16;13 call now "momentum out" of the beam. So notice that
00:18:21;20 the incident momentum of the light has now changed into
00:18:25;10 the outgoing momentum of the light, and there's a change in
00:18:29;03 momentum of the light corresponding to deltaP.
00:18:32;24 So, the light has essentially lost an amount of momentum
00:18:38;10 that is proportional to this arrow here, deltaP. So because
00:18:45;20 momentum is conserved, the object, which is the bead, has to gain
00:18:49;18 momentum now in the same amount in the direction opposite
00:18:54;29 to the original way that the light had lost. And so, you see that
00:18:59;25 essentially, because momentum is a conserved quantity in nature,
00:19:04;12 then the rate of change of momentum of the light actually
00:19:11;11 is equal to the rate of change of momentum of the object, with
00:19:14;22 an opposite sign. And the rate of change of momentum, of course
00:19:17;10 is a force that acts on the object. In this case, this arrow
00:19:23;15 that actually tends to kick the bead upwards. In this case,
00:19:27;29 it's forward. Okay, so now that we can see that we can use ray optics to
00:19:37;24 understand how light acts on objects, it is possible then to see
00:19:42;21 two different regimes. Imagine that you have a beam, this beam
00:19:47;04 actually has a Gaussian profile. So I want you to imagine
00:19:50;27 that this beam is more intense in the center, just like any
00:19:54;09 laser pointer. The center of the beam is much more intense
00:19:58;15 than the outer layers of the beam, the periphery of the beam.
00:20:03;11 Okay, so. Here, imagine now that the center of the beam is
00:20:08;02 up here. So this is the profile of intensity of the beam. Note there is
00:20:12;04 not much like coming from the periphery, but then as you get
00:20:15;24 to the center of the beam, the beam becomes more and more intense.
00:20:19;21 And then presumably, you can imagine that this continues
00:20:21;26 symmetric toward the other side, right? So it will eventually
00:20:25;08 decay. So there is a Gaussian profile of intensity in the beam.
00:20:29;19 Okay, so now let's assume that a peripheral ray actually first
00:20:33;28 hits the object, in this case, this plastic bead. So what is going to
00:20:38;28 happen is that this bead is going to refract the light along this line,
00:20:42;20 and then refract it again as you see here coming out.
00:20:46;19 The light used to bring momentum downwards. Only had
00:20:50;17 momentum downwards, right? But now, notice that the
00:20:53;20 light ends up with a momentum that is less downwards
00:20:57;17 and partly towards the right, okay? So the light has lost
00:21:02;10 forward momentum and has actually gained right momentum.
00:21:10;19 So the result of that is that the bead, because of that beam,
00:21:14;26 is going to gain momentum forward to compensate for the momentum
00:21:20;01 lost by the light. And towards the left, in order to compensate
00:21:25;03 for the momentum gained by the light in the rightward direction.
00:21:30;05 Okay, so you might tell me at this point, so now you have shown to me
00:21:35;13 that you have a bead that is not centered in a beam of light
00:21:40;19 the bead will be kicked out of the beam, right? Okay, but that's not
00:21:46;05 the complete story yet. You see, now we have to do the analysis on what happens
00:21:51;14 with photons that come very close to the center of the beam.
00:21:55;24 There are many more photons here because it's much more
00:21:59;10 intense, right? The intensity of the beam is much higher at the center
00:22:02;28 of the beam. And these photons will be refracted and we will have
00:22:07;23 more of an asymmetric situation here, in which the light, these
00:22:11;20 photons also have lost momentum downward. And have gained
00:22:16;09 momentum rightward, right? So now, what is going to happen is
00:22:21;07 that the bead is going to gain, in order to compensate for that lost
00:22:24;17 momentum, you're going to have momentum downward
00:22:28;23 and also towards the left, in order to compensate for the
00:22:35;01 momentum gained by the photons toward the right.
00:22:37;27 So, notice you have this force that appears due to the peripheral
00:22:43;13 beam, and then you have this force that appears due to the
00:22:46;21 central beam rays. And you can see that the net force is going
00:22:52;22 to be essentially in this direction, because there are many more
00:22:56;04 photons actually coming this way than coming that way.
00:22:59;07 So, the net force is going to grab this bead and center it
00:23:04;07 like this toward the center of the beam. And this is why
00:23:09;25 plastic beads or glass beads that are actually in the periphery of a laser beam,
00:23:15;29 are immediately centered into the center of the laser beam.
00:23:21;02 This is the trapping that we are talking about. But notice one thing.
00:23:24;27 That the bead is not only centered at the center of the beam, but
00:23:29;08 for example, you also push a little bit downward. Because
00:23:32;22 the net force, remember, is going to be also downward, partly.
00:23:37;04 So that means that the bead is going to be trapped in the center,
00:23:40;12 in this case, but it's going to be pushed along the direction of
00:23:44;17 the propagation of the light. And that's not good because
00:23:47;24 we want to trap these beads stably. So, it turns out that
00:23:51;22 this case is a case when you use light that comes parallel
00:23:56;09 from the beam that is not focused. So that's what happens when you have
00:24:03;03 very poorly focused or not focused light. The case we want
00:24:07;27 to consider then is the other case, which is shown here on the right.
00:24:12;10 Where you see that now we have the beam, you have again
00:24:16;28 a Gaussian profile of the beam, right? It's more intense in the center than in
00:24:21;28 the periphery, right? And now what we have done is
00:24:24;26 we have centered the bead and we have put it exactly
00:24:27;14 at the center of the beam. And so you can see a peripheral ray
00:24:32;15 now is going to be refracted in this fashion. And on the other side,
00:24:38;04 also the peripheral ray is going to be refracted in this fashion.
00:24:42;01 The light was bringing this beam, for example, was bringing
00:24:49;19 momentum from left to right. And now ends up with a momentum
00:24:55;22 from left to right, but lifts momentum from left to right, and instead
00:25:00;12 it has gained momentum downward. Compared to what it had
00:25:03;28 originally. So the result of that is that the bead is going to experience
00:25:13;09 a momentum in this direction, towards the focus of the beam.
00:25:18;16 So the bead is going to be, as you wish, sucked up towards the
00:25:23;28 focus of the beam, because having opened the numerical aperture of the
00:25:28;04 lens the way that we have opened it in this case, we essentially
00:25:31;13 have inverted the direction of the force acting on the bead.
00:25:34;24 So you see that now if you're using a very high numerical
00:25:37;29 aperture lens, and you put the bead in the center, that bead
00:25:41;10 is going to experience forces from both sides that actually
00:25:45;15 are trying to levitate the bead towards the focus of the beam.
00:25:48;27 Of course you will also have some light that will be reflected
00:25:52;20 like shown here in the dashed lines for both beams.
00:25:56;05 And that light is now gaining momentum backward
00:26:00;23 and is going to produce a force that is pushing a little bit
00:26:03;11 the bead along the original propagation of the light.
00:26:06;29 So in the end, all these forces have to actually balance each
00:26:12;17 other in order to produce a stable trapping of the object in the beam.
00:26:18;04 So, to summarize. What I want to say, therefore, is that
00:26:22;08 if you want to trap an object that is larger than the wavelength of
00:26:27;13 the light and you want to trap it in the beam in a stable fashion,
00:26:32;08 near the focus, then what you want to do is to use a high
00:26:36;25 numerical aperture lens, as shown here, which will produce forces
00:26:40;27 that both tend to pull the bead toward the focus of the
00:26:44;19 light, and will also push the bead a little bit due to scattering,
00:26:49;13 due to reflection. And that hopefully, you can actually balance
00:26:53;05 those two forces and of course, in that way you can obtain
00:26:57;15 stable trapping. Notice that the stable trapping position that I
00:27:04;21 showed you in that diagram is located just below the focus, because
00:27:08;08 of horizontal scattering force, which is actually pushing the
00:27:11;07 bead a little bit downward, right? Now, if you're still having
00:27:15;16 a plastic bead or a glass bead, you use a bubble. Now it turns
00:27:21;17 out that the index of refraction of the particle is smaller than the index of
00:27:25;08 refraction of the medium, because air has an index of refraction
00:27:27;29 smaller than water. And then what happens is that everything I told you
00:27:32;18 is still true, except that all the directions of the forces
00:27:36;03 invert. And so you have what is called a metastable system
00:27:39;27 and it's not very good for actually trapping objects. Experimentally,
00:27:46;14 we find that if you place a 1um bead in an infrared laser,
00:27:51;13 essentially, about 3% of the power of that laser actually
00:27:57;09 can be considered converted into a trapped force, as indicated
00:28:02;24 here by this expression. Okay, so in fact, this demonstration
00:28:11;17 that having a high numerical aperture lens allows you to stably trap
00:28:16;22 an object without having to use two counter populating beams
00:28:20;17 of light, as originally proposed by Ashkin, was also published
00:28:25;04 in the 1980's by Arthur Ashkin in collaboration with Steve Chu,
00:28:31;14 as you see here. In this important paper, they show that it is
00:28:36;13 possible to correct for the scattering force, or the reflection force,
00:28:40;10 by actually using high numerical aperture that will essentially
00:28:46;10 produce a force, as indicated here, that will actually pull the bead
00:28:52;04 towards the focus of the beam. And here is a movie that actually
00:29:01;03 shows that, where we have a bead that is trapped. You can see the bead
00:29:06;08 in the center is trapped in an optical trap, and on the side
00:29:10;21 there is a pipet. The pipet carries another bead or you can
00:29:16;13 actually use that pipet to actually manipulate that bead in the optical
00:29:20;13 trap. And what I want you to see is the relationship, the
00:29:26;23 inverted relationship there is between the position of the bead
00:29:29;08 and the position of the beam, due to this conservation of momentum.
00:29:35;01 Of light momentum. So please, let's see that bead now.
00:29:39;25 You can see that the bead is here in the center of the trap,
00:29:43;21 and now here is a pipet that we can use to suck the bead out
00:29:46;28 of this optical trap. So there, you see. And you can see how we apply
00:29:52;02 force on the bead, the light moves up and we actually
00:29:55;04 apply force up on the bead and the light moves out.
00:29:58;12 That's precisely the conservation of momentum
00:30:01;00 between the beads and the light that is the principle
00:30:04;29 of optical trapping.

Part 2: Optical Traps: Building and Calibrating

00:00:11;15 The design of an instrument that can actually carry out optical trapping, therefore,
00:00:15;11 is quite simple.
00:00:16;11 All you need is a source of light -- in this case, a laser.
00:00:19;12 You can actually, then, expand that laser through a number of optical... optical pieces,
00:00:25;20 and eventually you will actually fill the back focal plane of an objective lens,
00:00:30;14 as shown here.
00:00:31;14 Then, that objective lens will collimate the light and form a diffraction-limited spot
00:00:36;17 at that position, which is what is your trap, properly, right?
00:00:40;24 And of course, in addition to those optics, which are the basic optics for trapping,
00:00:45;20 what you want is to have an eyepiece so that you have, essentially, another source of light
00:00:50;07 that allows you to maybe see the bead that you are trapping through... through... through
00:00:57;10 an ocular -- through some kind of lens -- in your microscope.
00:01:00;05 Or you can... you may want to actually project it into a video camera in order to be able
00:01:06;04 to record the trapping.
00:01:09;15 Okay.
00:01:11;27 What wavelengths of light can be used for laser trapping?
00:01:15;20 Okay.
00:01:16;09 What you see here is a spectrum that is the spectrum of absorption of light by different...
00:01:24;24 different materials.
00:01:28;02 Most importantly, water, which is all-present in biological applications, of course.
00:01:32;19 And so you see that water actually has an increasing spectrum absorption towards
00:01:41;12 the higher wavelengths, right?
00:01:43;00 So, if you go from the visible to the infrared, essentially, you have an increase in the absorption
00:01:49;00 of water across that section, right?
00:01:50;22 On the other hand, of course we want to take into account the fact that when you are studying
00:01:54;23 biological samples we also have proteins and... and molecules that will absorb light.
00:02:00;06 And notice that something like hemoglobin, for example, will have instead an
00:02:04;16 increasing spectrum towards a lower wavelength, towards the blues, for example... the wavelengths
00:02:09;09 in the blue.
00:02:10;09 And so basically, what I want to point out here is that if you look at this spectrum
00:02:15;11 you see that, actually, there is a point where we have a minimum of absorption for both water,
00:02:22;02 which is our background, of course, in biology, and all kinds of biological molecules
00:02:27;23 and samples.
00:02:28;23 And so that minimum of light absorption actually occurs at 1060 nanometers, approximately.
00:02:34;25 So, this is what we call the absorption window of biological materials, assuming that biological
00:02:42;13 materials of course contain water, right?
00:02:45;16 And so this is important why?
00:02:48;06 Because when we are trying to manipulate molecules or when we manipulate biological molecules,
00:02:53;09 in fact... or cells, or organelles inside cells, we want to minimize the amount of light
00:02:59;02 absorbed by those organelles and by those molecules in order to minimize the damage
00:03:05;11 done by the radiation, right?
00:03:07;14 And so, now, these days, actually most laser optical tweezers applications actually
00:03:12;26 use diode lasers.
00:03:13;26 These are solid-state lasers.
00:03:15;22 And typically these actually function in the wavelength regime between 835 and 840 nanometers.
00:03:23;03 And they are pretty good.
00:03:24;20 Actually, it's a good compromise to trap with minimal damage to biological samples.
00:03:31;08 Now, what kind of forces can we then generate with optical tweezers?
00:03:36;10 Okay.
00:03:37;10 It turns out that with optical tweezers, typically you can obtain forces that are of the order
00:03:42;08 of a tenth of picoNewton, or 10^-13 Newtons, to about a hundred picoNewtons,
00:03:48;00 or about 10^-10 Newtons.
00:03:50;13 Okay?
00:03:51;13 So, now we have, therefore, three orders of magnitude.
00:03:54;16 Essentially, we have three orders of magnitude to play, between 0.1 picoNewtons all the way
00:04:00;07 to a 100 picoNewtons.
00:04:03;09 Now, we are not accustomed in biology to actually talk about Newtons, so maybe it's a good idea
00:04:09;14 to remember what a Newton is.
00:04:11;23 A Newton is about... approximately about the weight of an apple on the surface of the Earth,
00:04:18;07 right?
00:04:19;07 Because an apple is just about 100 grams, and 100 grams, or 0.1 kilograms,
00:04:24;26 multiplied by the acceleration of gravity, which instead of 9.8 we are going to take it by just...
00:04:31;13 as a hand-waving argument, as 10 meters per second squared, that gives you exactly 1 Newton.
00:04:37;13 In other words, 100 grams accelerated by an acceleration of 10 meters per second squared
00:04:42;11 gives you 1 Newton.
00:04:43;18 So, that's a weight of an apple on the surface of the Earth.
00:04:46;28 Of course, the unit that I have been mentioning here is picoNewtons.
00:04:50;28 A picoNewtons is 10^-12 Newtons.
00:04:54;07 I believe I once calculated that that corresponds approximately to the weight of a red blood cell
00:05:00;22 on the surface of the Earth.
00:05:03;00 So you, of course, cannot sense or experience the weight of a red blood cell in your hand,
00:05:09;28 but at least you can have a sense of the scale we are talking about, right?
00:05:14;17 So, it turns out that one of the things that is wonderful about optical tweezers as a technique
00:05:20;19 to manipulate and study biological processes is that they not only allow you to exert forces
00:05:28;01 on molecules, as we have seen, using the light exerted by light... the force exerted by light
00:05:34;00 on... on matter, but they are also able to tell you how much force you are exerting
00:05:40;11 on those objects, okay?
00:05:41;21 So, in other words, you can not only apply force, but you know how much force you are applying.
00:05:47;19 And that's one of the wonderful things about analytical optical tweezers.
00:05:51;24 And so, for that, of course, you need to calibrate your optical tweezers, and we are going
00:05:55;27 to now talk a little bit about the calibration of optical tweezers.
00:05:59;18 Okay.
00:06:01;02 The most popular method to calibrate an optical tweezer is to do the so-called virtual spring method,
00:06:07;01 which assumes that the light that is trapping your bead, okay, is a harmonic
00:06:13;20 potential, alright?
00:06:14;21 That is, a potential that is essentially... goes... has the shape of a harmonic potential
00:06:21;15 -- in other words, quadratic -- and that as you displace an object -- say, a bead --
00:06:27;04 that is in the middle of the trap, of that potential, and you displace it laterally by an amount
00:06:31;25 delta x, then that object -- the bead, in this case, that had been displaced by that amount,
00:06:37;16 delta x, will experience a restoring force that is trying to bring it back to the center,
00:06:42;22 which is exactly proportional to the displacement that you have generated.
00:06:49;06 In other words, the force -- the restoring force -- is nothing but a constant, kappa,
00:06:54;17 which is essentially the stiffness of the trap, times the... the displacement from the
00:07:01;10 center of the trap.
00:07:02;24 You may recall... those of you who remember your freshman physics may recall that
00:07:08;02 this is nothing but Hooke's law.
00:07:09;02 Hooke's law that says that the restoring force is proportional to the displacement that we create,
00:07:14;21 in this case, in the... in the bead away from the... from the optical trap.
00:07:19;27 It turns out that the position of the bead inside the optical trap can be measured
00:07:25;06 these days with nanometer resolution, and even with sub-nanometer resolution.
00:07:31;03 So it's possible to measure this delta x very precisely, and so we can measure this force
00:07:37;13 quite precisely if we happen to know kappa, which is the stiffness of the trap.
00:07:42;13 Now, how do we determine kappa, or the stiffness of the trap?
00:07:47;20 In order to do that, one very popular method is to fix the bead to a coverslip and
00:07:54;14 then translate this bead across the beam of light, and then see, what is the response of a photodetector?
00:08:03;15 Namely, the beam of light eventually will end up hitting a photodetector, right?
00:08:08;14 And you can imagine... you can see that if you translate the bead across the beam, right,
00:08:15;18 then the beam is going to be deflected.
00:08:17;24 And you want to measure that deflection in a photodetector, okay?
00:08:22;02 So now, in order to determine kappa, or the stiffness of the trap, then what we want to
00:08:29;12 do is we want to calibrate the trap stiffness using some known external force acting on
00:08:35;05 the bead, okay?
00:08:36;26 That can be done by different... in different ways.
00:08:40;21 One of them is the Stokes' law approach, in which you essentially use beads that are trapped
00:08:49;24 in the beam and then are displaced from the center of the beam by actually passing
00:08:53;23 some kind of flow, of just the liquid around the... the bead.
00:08:58;13 This flow will exert a drag force on the bead, the bead will be displaced from the center
00:09:03;28 of the trap by a given amount, and you can calculate how much is that force that
00:09:08;10 you are applying on the bead if you know the flow -- the velocity of the... of the water passing
00:09:13;00 around the bead -- and if you know the dimensions of the bead.
00:09:16;08 Why?
00:09:17;08 Because that's what is called Stokes' law, in fact.
00:09:19;04 Stokes' law says that the force experienced by the bead due to the flow of liquid
00:09:24;23 around the bead is equal to the frictional coefficient of the bead times the velocity of the water
00:09:30;01 passing around the bead.
00:09:31;12 If you know the velocity passing around the bead and you know that you have a sphere,
00:09:35;23 then this frictional coefficient is going to be nothing but six times pi times the viscosity
00:09:41;01 of light... of water, which you know, times the radius of the bead, which presumably
00:09:45;24 you also know.
00:09:46;24 So, as you can see here, everything that you need in order to calculate the force is...
00:09:52;00 is known by you, namely the viscosity, the radius of the bead, and the velocity of the...
00:09:57;24 of the flow, right?
00:10:00;02 So, that is a way to actually apply a known force and see how much displacement that produces
00:10:06;11 in the bead in the trap.
00:10:08;02 And remember, you already know that that displacement, in nanometers, of the bead in the trap produces
00:10:14;25 also a given displacement in the photodetector.
00:10:17;18 So eventually, you can relate the forces that you are applying and the displacement
00:10:22;21 that you are applying on the bead to a displacement in the photodetector.
00:10:25;25 So, you essentially have a way of calibrating your optical trap.
00:10:31;08 The other method is the so-called equipartition theorem method, which we will talk about
00:10:35;14 in more detail in a minute.
00:10:36;26 But let me then summarize, how do you calibrate the trap?
00:10:41;13 Here is your trap; there is your bead.
00:10:43;18 You displace the bead from the center of the trap by a given amount, delta x.
00:10:47;15 And we assume that the system is a linear spring and that there is a linear force restoring,
00:10:53;23 or trying to restore, the position of the bead towards the center of the trap.
00:10:56;19 In other words, we assume that the optical trap is a harmonic potential trapping
00:11:02;17 the bead, okay?
00:11:04;03 Next, what we are going to do is we are going to grab the bead and move it relative to the
00:11:09;08 trap center by different amounts.
00:11:11;22 And we know exactly by how much we actually move the bead relative to the trap center.
00:11:16;23 And at the same time, we are going to determine what is the response in the photodetector.
00:11:22;27 In other words, the light that you see is going to be moving in response to you applying
00:11:27;15 force on the bead.
00:11:29;12 The light is going to experience a force, if you wish, that will actually deflect it
00:11:34;02 on the photodetector.
00:11:35;19 And you record that response in the photodetector.
00:11:39;06 And then you determine the trap stiffness simply by using some known force, right?
00:11:44;14 And so, you see how much you displace the bead with that force that you apply.
00:11:50;19 You know the force that you are applying because, say, it's Stokes' law, so you know the force.
00:11:56;15 You know the displacement.
00:11:57;15 Then, you can calculate kappa, or the trap stiffness.
00:12:04;01 So, if you use the Stokes' law, which I just told you before, this is a typical example
00:12:08;24 of how do you actually carry out this experiment.
00:12:12;26 You have your optical trap.
00:12:15;05 You can see that we are using two objectives, okay, to essentially collimate the light
00:12:21;22 onto a glass chamber, which is our optical trap light chamber.
00:12:25;19 And then we have a bead that is trapped in the center of the beam, here.
00:12:32;04 We have a bead.
00:12:33;04 The bead is normally moving in solution in the glass chamber, but this bead is trapped
00:12:38;06 by the beam.
00:12:39;06 And now, what we are going to do is we are going to use a motorized stage to actually
00:12:44;17 move at different speeds to create, essentially, a flow of liquid around the bead.
00:12:52;01 And we can actually do that in a computer controlled manner.
00:12:56;09 And so, explicitly, what is going to happen with the bead, now, is this, right?
00:13:01;13 The bead was originally in the center of the trap, but you are moving the whole... the...
00:13:07;22 your whole glass chamber is being moved in this fashion, back and forth at different
00:13:14;06 speeds.
00:13:15;06 And that means that, actually, the bead is going to experience a drag force that is
00:13:19;04 going to displace it from the center of the beam.
00:13:21;13 And so that's how we actually determine the displacement.
00:13:25;02 And because we know the velocity at which we actually are moving the motorized stage,
00:13:29;05 then we have velocity, we have the force, and therefore we can calculate the trap stiffness
00:13:35;25 of the trap.
00:13:37;17 So, here is the typical data that you obtain that way.
00:13:42;10 You have drag force here in the x-axis, right, which essentially is your... results from
00:13:50;17 the fact that you are moving the stage.
00:13:53;06 And you detect, at the level of your... of your photodetector, what is the volts,
00:13:59;09 or the change in volts, in the photodetector.
00:14:01;20 And you obtain a curve such as this one, here, that as you can see is very linear.
00:14:06;20 Right?
00:14:07;20 Now remember, you had previously done, also, another curve where you have what happens
00:14:15;00 when you vary the position of the fixed bead and you pass it through the beam,
00:14:19;18 and you look at what happens to the photodetector.
00:14:21;06 So, you have another plot of detector output versus position.
00:14:26;00 Since you just obtained, now, a new plot, which is shown here, of detector output versus
00:14:31;13 force, it is possible to combine those two to obtain a force versus displacement graph,
00:14:40;24 which is essentially your calibration curve.
00:14:42;15 And notice that it is very linear indeed.
00:14:44;22 So, the assumption that the optical trap is a harmonic potential is a very good assumption.
00:14:54;06 The other method of doing calibration is the so-called equipartition method.
00:15:00;01 And that uses a very important result in statistical mechanics that says that if you have a linear
00:15:06;25 system and that that linear system is in a harmonic potential -- just like we are assuming
00:15:12;02 is the case of a bead inside a laser... an optical trap -- then the mean quadratic fluctuation,
00:15:19;20 the mean quadratic movement, displacement, just by fluctuation, just by thermal fluctuation,
00:15:26;18 of this object along one direction in space -- in this case, x, okay -- is given by
00:15:35;23 this expression, namely that with each degree of freedom -- x -- associated with the system,
00:15:43;04 with this linear system, okay, the potential energy of the system due to these fluctuations
00:15:49;22 has associated a value of 1/2 kT, where k is the Boltzmann constant and T is the temperature
00:15:57;01 in degrees Kelvin, okay, absolute temperature.
00:16:00;14 So, notice that I know the temperature of my experiment, of course, so I know that.
00:16:05;08 I know the Boltzmann constant, right?
00:16:07;04 So, if I were able to have a bead, say, in an optical trap, and look at the fluctuations
00:16:13;18 of the bead in the optical trap, okay... in other words, that I could determine the
00:16:18;22 mean quadratic displacement of the bead in the optical trap, just due to thermal energy,
00:16:25;03 it's possible for me to obtain, therefore, this quantity.
00:16:27;27 And since I know this quantity, that constant, and the temperature of the laboratory,
00:16:32;27 then I can solve for the stiffness of the trap.
00:16:36;17 Because they larger the stiffness of the trap, the smaller would be the fluctuations that
00:16:41;27 your bead has, right?
00:16:43;23 So in fact, by looking at the fluctuations, you are able to calculate the stiffness of
00:16:48;20 the trap if you know the temperature and, of course, the Boltzmann constant,
00:16:52;17 which is a well-known quantity.
00:16:56;00 There are other ways of doing this.
00:16:58;21 It is not a trivial calibration.
00:17:05;05 Oftentimes, people use what is called the spectral density of fluctuations of the bead
00:17:12;18 in the trap.
00:17:13;18 And that's because of the following.
00:17:15;12 If you have a bead in a... in an optical trap or in optical tweezers, what's going to happen
00:17:20;27 is that the bead is going to fluctuate and vibrate due to thermal energy.
00:17:24;13 But it's going to fluctuate by a given amplitude for every different... at every different
00:17:32;15 frequency.
00:17:33;15 In other words, the bead will have what is called a spectrum of fluctuations, okay?
00:17:38;07 And so, what people often do is they actually obtain what is called the fluctuation power spectrum,
00:17:44;09 or the spectral density of fluctuation, of these beads in the trap in order to calculate
00:17:51;26 the stiffness of the trap.
00:17:53;27 And for that they use a quantity called the corner frequency, which we are going to
00:17:57;16 see just in a minute.
00:17:58;16 The corner frequency which is given by the stiffness of the trap divided by the friction coefficient
00:18:03;13 experienced by the bead in the liquid, which again is from Stokes' law,
00:18:07;18 and we know.
00:18:08;18 So, in principle, knowing this quantity called the corner frequency, we can in principle
00:18:12;25 calculate the stiffness of the trap again.
00:18:16;11 Okay.
00:18:17;12 So, to go back to the case of a bead in the optical trap.
00:18:22;06 Remember, we have a bead in the optical trap; this is our experiment.
00:18:25;10 This bead is fluctuating due to the fact that, actually, it's being hit... hit by many, many
00:18:31;25 forces due to the molecules of water that are hitting it.
00:18:34;13 This is the Brownian motion, if you wish, okay?
00:18:37;15 So, the bead is experiencing a Brownian noise behavior trapped in the middle of the optical trap.
00:18:45;00 Well, the equation that describes the movement of the bead that is surrounded by a viscous medium
00:18:53;26 -- water in this case -- subjected to a fluctuating force due to the hitting
00:19:01;22 of the bead by the molecules of water -- the Brownian force -- and that is also placed
00:19:07;19 in the center of a beam of light that is trapping the object in a harmonic potential
00:19:14;12 is the so-called Langevin equation.
00:19:17;14 And in this equation you have three terms.
00:19:19;16 one is the friction experienced by the bead in the viscous medium, the velocity of the bead,
00:19:24;23 the instantaneous velocity of the bead.
00:19:27;10 Then you have the fluctuating force due to the Brownian hitting of the molecules
00:19:32;24 against the surface of the bead, whose average is zero, but whose quadratic value is not zero,
00:19:38;06 and depends on the temperature of the... of the solution and the friction coefficient
00:19:44;02 of the bead.
00:19:45;03 And finally you have the force that is trapping the bead in the center of the trap, that is
00:19:51;13 trying to hold it in the center of the trap, which is the restoring force that is trying
00:19:55;20 every time you move the bead away from the trap tries to restore the bead back to its position
00:20:01;24 in an amount proportional to the displacement that you produced times the stiffness
00:20:07;24 of the trap.
00:20:08;24 Well, you can solve this equation for delta x, and in fact you can solve this equation
00:20:14;06 for delta x squared, which is important.
00:20:16;25 The mean quadratic fluctuation as a function of frequency.
00:20:20;06 And the solution is given here, in this expression.
00:20:23;04 And you see that, actually, it's a very simple solution.
00:20:25;24 It says that if you want to know by how much amplitude is the bead displaced from the center
00:20:32;26 of the trap at a particular frequency -- omega, in this case, right? -- then you just have
00:20:40;02 to use this expression: four times the Boltzmann constant times the temperature divided by
00:20:44;26 the friction coefficient of the bead, parentheses that frequency squared plus a constant called
00:20:51;09 omega-c, which is what we call the corner frequency.
00:20:54;13 And I'll remind you that I said that the corner frequency is nothing but the stiffness of the trap
00:20:59;12 divided by gamma.
00:21:00;25 Okay.
00:21:01;25 So, what this equation predicts is that if you plot the mean quadratic fluctuation of
00:21:08;08 the bead as a function of frequency, what you're going to have is what is called a
00:21:13;00 Lorentzian power spectrum, because this is an equation of a Lorentzian, and in fact here it is.
00:21:18;14 Okay.
00:21:19;14 Here is your red... take a look at the red curve, here.
00:21:21;11 It is flat for very low frequencies.
00:21:24;19 Here is power in nanometers squared, because it's displacement squared.
00:21:29;05 Remember, delta x squared, that is the displacement squared, so nanometers squared per...
00:21:34;19 per Hertz.
00:21:35;19 And then this is a power spectrum or is a power density spectrum.
00:21:39;16 And it's constant.
00:21:40;18 Notice... notice that the system is constant for all these frequencies, for all...
00:21:44;23 for low frequencies all the way to the point where now, all of a sudden, you see a corner, here,
00:21:52;11 where the spectrum has started decaying.
00:21:54;28 This point, where the actual turning occurs, is called the corner frequency.
00:22:00;24 And I told you that that's equal to the... the stiffness of the trap divided by the
00:22:05;17 friction coefficient of the... of the bead... experienced by the bead.
00:22:08;26 We know the friction coefficient of the bead because we know it's a sphere, and we know
00:22:12;13 the... the radius of the bead, and we know the viscosity of the medium.
00:22:17;00 So, if we can measure the power spectrum, as shown here, for a bead in an optical trap,
00:22:24;07 in principle we can calculate the corner frequency.
00:22:26;13 And from the corner frequency we can obtain the stiffness of the trap, which is what we
00:22:30;28 need in order to measure forces with the optical trap.

Part 3: Optical Traps: Measuring Steps by Molecular Motors

00:00:11;07 Okay.
00:00:12;07 Next, we are going to talk about the resolution, the spatial resolution, of optical traps.
00:00:17;15 And this is an important consideration because, of course, ultimately what we are interested in
00:00:23;08 is in being able to actually follow processes in biology, molecular processes that we
00:00:29;17 can follow in real time.
00:00:31;27 And if we want to see those molecular processes, typically those molecular processes involve
00:00:37;02 a conformational change, or the movement of a motor on a track, or some kind of displacement
00:00:44;00 of... of a given magnitude that we want to be able to detect with the optical tweezers.
00:00:50;05 And so, how well can we resolve a given displacement due to a conformational change, as I said,
00:00:58;00 or a movement of a motor in a... on a track is an important consideration.
00:01:04;16 So, imagine that we actually are studying a motor.
00:01:09;03 For example, it could be kinesin, and it's moving on a track.
00:01:13;07 In this case, it’s... you can imagine this is a microtubule.
00:01:18;04 And what we've done is we have attached our motor, which is shown here, like this
00:01:22;20 little red guy, here... we have attached it by some kind of tether, shown here in black,
00:01:28;24 to a bead that is in an optical trap.
00:01:31;02 So essentially, this tether becomes the transducer between the movement of the motor and the
00:01:38;00 bead in the trap.
00:01:39;00 And remember, since we have a good optical tweezer machine, we know exactly what is happening
00:01:43;26 to the bead in the trap: how much it's moving, what force is being exerted on the bead
00:01:49;07 and therefore on the guy, here, trying to do the moving.
00:01:52;13 And so... but we have this tether in between that we need to actually take into account.
00:01:59;05 So, the question -- the rhetorical question -- we are posing here is, how small of a step
00:02:04;12 can we detect?
00:02:05;14 First of all, let's see what happens when we actually have a geometry such as this.
00:02:10;26 And here you see an experiment, a typical experiment where actually we are studying
00:02:15;27 one such motor -- in this case, it's RNA polymerase -- which is attached to a surface.
00:02:20;10 This surface can be a glass surface or it can be another bead that is held on a pipette,
00:02:26;17 for example.
00:02:27;17 And then what you have is the piece of DNA that the RNA polymerase is going to transcribe,
00:02:32;13 right?
00:02:32;27 And the end of the DNA, now, is attached to another bead which is held in an optical trap,
00:02:40;09 as shown here.
00:02:42;04 So, what is going to happen is that when you add NTPs the RNA polymerase is going to
00:02:48;09 try to walk over the DNA.
00:02:50;13 But it cannot walk on the DNA because it's attached by its back to a surface.
00:02:54;24 So, what it's going to do is it's going to pull the DNA through itself, and in so...
00:02:59;22 in so doing it's going to pull the bead out of the center of the trap by an amount
00:03:04;16 delta x, right?
00:03:06;03 Okay.
00:03:07;03 So, what we want to calculate is what is that delta x that we see in this case.
00:03:10;20 See, the situation is shown here, mechanically, in this fashion.
00:03:15;25 Notice, here is the motor, the little guy here, who is about to make a step, delta x,
00:03:21;19 right?
00:03:22;19 So, we call that the motor step or delta x step.
00:03:26;12 But the guy is not grabbing the bead directly, but is grabbing the bead through this tether,
00:03:32;03 which in this case happens to be double-stranded DNA, right?
00:03:35;09 Between the motor and the bead, right?
00:03:39;12 So, this tether or spring of DNA has its own elasticity.
00:03:46;02 And we characterize that elasticity with a quantity, k-tether, which is the stiffness
00:03:51;05 of the tether, right?
00:03:53;02 Then you have the bead and then you have that the bead itself is attached to the
00:03:58;18 center of the trap, which is indicated here by the black line, here, by another spring,
00:04:04;04 which is the light spring.
00:04:05;04 Of course, this is the stiffness of the optical trap.
00:04:09;03 So essentially, what we have is two springs: one due to the tether and the other due
00:04:13;21 to the light, okay?
00:04:15;21 And it can be shown that in this case, in fact, what is going to happen is this.
00:04:20;27 You can see it intuitively.
00:04:22;00 The little guy, here, the motor, is going to try to make a step delta x.
00:04:29;10 But that's not going to produce a step delta x of the bead in the trap, because part...
00:04:34;26 once it applies force on the DNA, part of the movement of the motor is going to be absorbed
00:04:42;25 by the spring of the tether, right?
00:04:46;26 And so the result of that is going to be that the bead is going to move a lot less
00:04:51;20 than the actual motor.
00:04:52;20 So essentially, in a sense, in this type of experiment, you lose some resolution due
00:04:57;13 to the fact that your tether spring is not infinitely stiff, right?
00:05:02;21 Okay.
00:05:03;21 So, it can be shown that the delta x that is going to be experienced by the bead
00:05:10;10 in the trap -- namely the amount by which we are going to move the bead from the center
00:05:15;02 of the trap due to the movement of the motor by the amount delta x step --
00:05:20;22 is called delta x signal and is given by this expression, here.
00:05:24;25 Notice, it says that the signal is proportional to the... the signal is proportional to
00:05:31;20 the size of the step, but multiplied by this quantity, here, which is the ratio of stiffness.
00:05:39;07 Okay?
00:05:40;20 And because the stiffness of the tether is not very, very high, but rather the tether
00:05:45;16 is maybe flexible, you see that, in the end, this quantity -- that is, the ratio of the k's,
00:05:53;14 of these stiffnesses -- is going to be a smaller than one.
00:05:58;02 And therefore the signal that you measure is going to be smaller than the signal of
00:06:01;22 the actual step of the... of the motor, okay?
00:06:05;20 But let's take the case where... well, in the... in one extreme case, the tether
00:06:11;00 is so soft that k-tether is practically zero.
00:06:14;24 Then you have in the denominator just k-trap, and in the numerator you have zero.
00:06:18;11 Zero multiplied by delta x step gives you a very small signal.
00:06:22;05 Because, basically, every movement that the motor makes is absorbed basically by the tether,
00:06:27;25 right?
00:06:28;25 Of course, DNA is not that soft, right?
00:06:32;02 Double-stranded DNA is stiffer.
00:06:34;16 But let's take the other example, and the other example is when we make the...
00:06:39;00 imagine that if instead of having a piece of DNA we will have an absolutely rigid stick here,
00:06:43;22 right?
00:06:44;22 Then, of course, when the little motor moves by an amount, delta x, it's going to displace
00:06:49;24 the bead.
00:06:50;24 Since that inextensible, rigid tether is not going to extend, then the bead will move
00:06:58;06 its full amount, delta x.
00:06:59;20 And you'll see what happens, here.
00:07:01;04 If we take k-tether to be much, much larger than k-trap... k-tether is much, much larger
00:07:07;09 than k-trap, we can neglect k-trap in the denominator, and we have k-tether
00:07:12;02 divided by k-tether.
00:07:13;02 It cancels, and what we get is that the signal that we measure is identical to the size of
00:07:18;08 the trap.
00:07:20;04 So, the... the result, the conclusion of this analysis, is that, ultimately, the spring
00:07:30;07 that eventually connects our optical... our trapped bead to the motor actually determines
00:07:37;25 in... in a rather dramatic fashion the... the size of the signal that we can measure
00:07:45;22 in an experiment like this.
00:07:50;13 Let me give you an example of experiments done in that fashion.
00:07:53;23 This is a system that we study in our laboratory.
00:07:56;09 It's a bacteriophage, bacteriophage phi29, which is a virus that infects Bacillus subtilis.
00:08:06;22 And these capsids of the bacteriophage self-assemble inside the host cell.
00:08:13;24 And at the end, the DNA has to be actually put inside the capsid, okay?
00:08:18;25 And this is done by a tiny little motor that sits at the base of the... of the capsule,
00:08:24;18 and that grabs the DNA of the bacteriophage that has been replicated by the bacterium
00:08:29;27 and pulls it in and packages it inside the optical... inside the...
00:08:36;05 the bacteriophage prohead, okay?
00:08:39;11 And so, you notice that we have an experiment in which we have a bead in the optical trap,
00:08:45;01 indicated here by the tick marks, right?
00:08:47;28 We have the piece of DNA that hasn't been packaged yet, right, attached by a biotin-streptavidin
00:08:54;00 linker to that bead in the trap.
00:08:56;09 And we have the... the actual bacteriophage prohead, or capsid, attached to another bead
00:09:04;24 by antibodies against the protein in the capsid, and held there by suction through a micropipette
00:09:11;09 that is introduced in the optical tweezers chamber.
00:09:15;01 And now what you can do is this.
00:09:16;07 You add ATP to the... to the motor, and the motor will start pulling the DNA.
00:09:21;12 And as it pulls the DNA, it's going to pull the bead out of the trap, right?
00:09:25;01 It's going to start pulling the bead out of the trap.
00:09:27;06 And what is going to happen is that the length of the DNA that is outside, not yet packaged,
00:09:32;18 is going to be what we call the tether, the length of the... of the tether, the DNA tether length,
00:09:38;14 is going to be reducing or getting shortened in time.
00:09:41;20 And that's what you see here in this second plot, where you see that, in fact,
00:09:46;16 for four different phages, which are separated in time just so that you can see it...
00:09:51;24 see each one of them clearly, and also different colors, you see that the tether length decreases in time
00:09:56;28 as the bacteriophage motor packages the DNA inside the capsid.
00:10:03;03 We are very interested in this motor because it's a very nice example of a ring ATPase.
00:10:10;00 And as you know, cells are full of ring ATPases that carry out all kinds of mechanical
00:10:14;19 and transport functions.
00:10:16;00 And so we are studying this motor as a system to try to understand how is it that the motor
00:10:21;20 actually, in this case, translocates the DNA.
00:10:26;08 And this is one of the things that we observed.
00:10:28;12 We observed that we if we apply a small amount of force... of course, the stiffness of the DNA
00:10:33;18 depends on the amount of force applied.
00:10:36;28 In this case, we are using two optical traps.
00:10:39;07 Notice, one optical trap contains... has attached the... the bacteriophage capsid, and the other
00:10:46;10 optical trap, here, as I showed you before, via biotin-streptavidin attaches its bead
00:10:53;14 to the end of the DNA that hasn't been yet packaged.
00:10:57;13 And using this system, we can actually... we've been able to resolve the steps of
00:11:03;10 the motor.
00:11:04;10 And you can see that for different concentrations of ATP at these... each one of these curves
00:11:08;23 involves higher and higher concentrations of ATP... you see that the motor moves
00:11:14;08 in this dwell manner, followed by a burst, and then dwell, followed by a burst,
00:11:20;05 and so on and so forth.
00:11:21;05 So, it turns out that, actually, these dwells are the times when the motor is actually binding ATP,
00:11:25;16 and the bursts, of course, are the times where the motor is actually packaging the DNA,
00:11:30;15 because we are plotting here, in the y-axis, the length of the DNA
00:11:35;13 as a function of time, just as before.
00:11:37;01 It's just that, now, we have a novel solution, that we can actually see the individual steps
00:11:42;23 of the... of the motor.
00:11:44;16 Before, I showed you the same plot, here, but at low-resolution, and you see that
00:11:49;28 when you plot the length versus time all you see is a continuum curve.
00:11:54;22 But inside these curves, of course, is information about the step size of the motor.
00:11:59;12 And that's what you can see here, not at much higher resolution, where you see that
00:12:04;15 the motor pauses for a while during a dwell phase, and then translocates the DNA by ten base pairs,
00:12:10;05 and then pauses again, and then translocates the DNA by ten base pairs, and then pauses again,
00:12:15;06 and so on and so forth, okay?
00:12:17;20 Now, it turns out that ten base pairs is not the minimum size step of the motor.
00:12:25;07 This ten base pairs is made up of sub-steps of the individual ATPases in the ring.
00:12:30;11 The ring has five ATPases -- the motor has five ATPases, as shown here -- and so the
00:12:37;17 question...
00:12:38;17 what we are seeing presumably is the result of all of these ATPases making 10 base pairs
00:12:43;28 in one cycle, one full cycle, right, for the whole ATPase.
00:12:47;20 But now you may want to ask, what is the displacement caused by one ATPase at a time?
00:12:55;06 And we can do that by applying a lot of tension in our DNA, right?
00:12:59;22 We apply a lot of force, all the way to 40 or 50 picoNewtons, and now the DNA becomes
00:13:05;08 very taut, very stiff.
00:13:07;23 And remember, if you make the k-tether very stiff, right, then you can neglect k-trap,
00:13:16;26 here, and then it cancels, and basically the step that you measure is pretty much...
00:13:22;11 the step that you measure is pretty much the step of the motor.
00:13:24;21 So, you can improve your resolution.
00:13:26;15 And so, by increasing the force, we can actually improve the resolution of this...
00:13:32;00 of this method.
00:13:33;00 And by doing that, we were able to determine, using these optical tweezers, double-beam
00:13:37;19 optical tweezers, that the ten base pairs burst that I showed you here, in these curves,
00:13:44;15 in any one of these curves, are made up of four two-and-a-half base pairs steps, okay?
00:13:51;05 So, you see here, one, two-and-a-half, another two-and-a-half, another two-and-a-half,
00:13:58;17 and another two-and-a-half.
00:13:59;17 And that constitutes a full 10 base pair displacement of... or burst that we were seeing in this curve.
00:14:09;26 So, here is our... what we learned, for example, from this method.
00:14:14;13 By using that, we know that there are five ATPases, but for some reason, that now
00:14:18;24 we are beginning to understand, we know that only four of the ATPases that bind... that
00:14:23;25 bind to the dwell... in... during the dwell phase are used to translocate the DNA,
00:14:28;05 so that there are four substeps of translocation of 2.5 base pairs each during the burst phase,
00:14:34;18 right?
00:14:35;18 Now, what happens to the fifth subunit?
00:14:37;03 Well, now we know that the fifth subunit also binds ATP, also hydrolyzes ATP,
00:14:43;10 but it does it for regulation.
00:14:44;10 So, it regulates the cycle of the other four ATPases.
00:14:48;20 And that's data that actually is now just being published from the lab.
00:14:53;10 Okay, there is another experiment that we can do for studying motors, for example,
00:14:58;06 which is the so-called constant-force feedback experiment, right?
00:15:03;15 Constant-force feedback.
00:15:04;15 What we do is this.
00:15:05;15 We have, as before, our little motor, here, which is actually going to make a step,
00:15:10;21 delta x, right?... step.
00:15:14;12 We also have the tether spring here, which may be the piece of DNA that we saw
00:15:18;19 in the case of the polymerase, or it could be the DNA spring that we saw in the case of the
00:15:25;06 bacteriophage.
00:15:26;13 We have our bead in the trap, and we have that bead attached to the center of the trap
00:15:31;02 by another spring, which is the light spring.
00:15:34;27 In the constant-force feedback mode, what happens is this.
00:15:39;18 We want to make sure that the bead in the trap maintains its position in the trap and
00:15:45;03 is not displaced by the motor.
00:15:47;09 And so, what we tell the instrument is, when the motor tries to move the bead by pulling,
00:15:53;08 right, from here, by pulling... it tries to move the bead from the center of the trap,
00:15:58;15 what we want to do is we want to move the trap center by the same amount in order
00:16:04;17 to prevent the bead from being displaced from the center of the trap, or from the location
00:16:11;13 in the trap.
00:16:12;13 Because the bead is maintained in this location in the trap, the force it experience is constant
00:16:17;06 -- it's always F equals k (kappa, the stiffness of the trap) times delta x, whatever that
00:16:23;22 displacement... constant displacement we want to maintain.
00:16:27;01 And so that force is therefore constant, and that's why we call this method the constant-force feedback.
00:16:32;20 But essentially what it means is this.
00:16:34;20 As the... as the motor pulls, tries to pull the bead out of the trap, moving,
00:16:40;18 we move the trap by the same amount.
00:16:42;08 So that, actually, the bead inside the trap never moves, okay?
00:16:46;02 But what it means is this, that in order to accomplish that, what you have to be able
00:16:50;16 to do is to move the trap by an amount, delta x step, which is identical to the amount,
00:16:56;24 delta x step, moved by the trap.
00:16:59;01 You see... so essentially, the movement of your... of your... of your trap, or the movement
00:17:07;17 of your stage that controls the trap, is nothing but the movement of the motor, delta x.
00:17:12;27 So, in this case, it's equivalent to saying that the effective trap stiffness is essentially zero.
00:17:19;07 Because you essentially move the bead by the same amount... the trap by the same amount
00:17:24;08 that the motor moves.
00:17:26;08 So, I'm going to give you an example of this constant-force displacement, here, from experiments
00:17:33;13 that we did.
00:17:34;20 This is a movie that I'm going to show you, where we have a single ribosome.
00:17:39;26 Let me first give you a little bit of the background.
00:17:42;11 As you'll see here, we have a single ribosome that is about to translate this RNA hairpin, right?
00:17:50;05 And so what happens is we have an RNA hairpin, and we have grabbed the RNA hairpin
00:17:56;15 by both ends.
00:17:57;15 One end is attached to a bead in an optical trap, and the other end of the hairpin is
00:18:03;15 attached to be a bead atop micropipette.
00:18:06;26 And now we place the ribosome at the start site of translation, add all the components
00:18:12;15 required for translation -- initiation factors, elongation factors, tRNAs... aminoacyl tRNAs,
00:18:20;17 everything, and EF-G, EF-Tu, everything -- and then we are going to see, essentially,
00:18:29;11 the ribosome walking and start translating the RNA hairpin.
00:18:32;24 So, as the RNA hairpin actually is being translated, it's going to start opening.
00:18:39;04 You know, the ribosome is going to start opening the RNA hairpin.
00:18:42;21 And what we do is we have the optical trap at constant tension, at constant force.
00:18:48;02 So, every time the ribosome, in this case, opens the hairpin... and the system is
00:18:53;01 going to become larger, and therefore the force is going to drop... the tension between this bead
00:18:58;10 and that bead, the tension applied to the hairpin, basically.... is going to drop,
00:19:02;28 the pipette actually is moved away by a given amount in order to maintain the tension constant.
00:19:08;27 And what we want to see is, essentially, how the ribosome opens the hairpin.
00:19:15;02 Or, in other words, how the ribosome moves during translation.
00:19:18;11 And what you will see in the movie is that these dashed lines in this graph are separated
00:19:24;03 by three nucleotides at the time.
00:19:26;12 So essentially, each one of them corresponds to a codon in the RNA.
00:19:30;24 And please... in the next movie you will see that the motor... there it is, right?
00:19:38;27 You'll see that... the RNA... the... the ribosome is in fact stepping in... in steps of
00:19:47;03 three nucleotides at a time -- codon by codon, right?
00:19:49;23 Sometimes we see two of them together.
00:19:52;04 And when the hairpin gets... gets very close to the end, notice that that hairpin is very
00:19:56;25 unstable, and then the hairpin opens completely, and we won't... we are not able to resolve
00:20:01;11 the individual codons in this experiment.
00:20:03;13 But early on, you can see very clearly that the ribosome in fact is moving codon by codon.
00:20:09;22 And of course, being able to actually resolve the individual steps of the ribosome is
00:20:15;13 very important, because all kinds of important chemical processes are occur... are occurring
00:20:20;04 here during these dwell times, and all kinds of important processes are occurring also
00:20:23;25 during the translocation itself.
00:20:25;14 And we want to characterize all the cycle of this motor.
00:20:30;17 Well, I showed you this geometry, which we call the hairpin geometry, right,
00:20:37;14 whereas the ribosome invades the RNA in order to translate, it makes the RNA longer, but the beads
00:20:46;28 are pulled away in order to keep the tension or the force constant in the optical trap.
00:20:51;14 And therefore we can actually follow the individual... individual steps of the ribosome.
00:20:59;09 Well, you may want to be able to play tug-of-war with a single ribosome, because notice that
00:21:04;11 in this experiment we never grabbed the ribosome directly.
00:21:07;23 In other words, we don't know how much force the ribosome can generate.
00:21:10;24 And it could be very interesting to know, to investigate the ribosome as a molecular motor.
00:21:16;24 Namely, how much of the energy of transpeptidation it is being used in order to actually,
00:21:23;12 for example, move against a force, or move against a barrier.
00:21:27;02 Like, it could be secondary structure of RNA.
00:21:30;05 So, that is the geometry that I showed you here, where you can see that, now,
00:21:36;17 we grab the ribosome itself via a biotin at a position in one of the small subunits, and... which
00:21:44;00 is attached... a tag in the small subunit and it's attached to streptavidin, which is
00:21:48;14 attached to a bead in the pipette.
00:21:50;22 And then the ribosome is attached to an mRNA that is going to translate.
00:21:55;19 And this mRNA is itself attached to a piece of DNA -- single-stranded DNA -- to form
00:22:05;13 a hybrid, here, that we can then connect via anti-digox -- digoxigenin... anti-digoxigenin
00:22:13;04 -- to a bead that is in the optical trap.
00:22:15;11 So, this experiment is a tour de force experiment, because now we are going to try to see...
00:22:22;00 observe how, through this very elastic spring, which is single-stranded RNA, whether we can
00:22:28;28 actually see the individual motion of the ribosome through the messenger RNA
00:22:35;05 as translation occurs.
00:22:37;08 Okay?
00:22:38;08 And so, in the next slide, we are going to see a movie of that situation, in fact,
00:22:43;13 here it is.
00:22:44;13 And now the dashed lines are separated by three nucleotides, as before.
00:22:49;08 And you will see that it will start here.
00:22:51;11 And you will see that in, fact, it is possible to follow very precisely... by double-clicking, here,
00:22:58;10 you see that the ribosome starts there and now makes the transition to the
00:23:05;01 first three nucleotides, then another three nucleotides, then another codon, here it is,
00:23:11;03 then yet another codon.
00:23:13;22 So, you can see that, in fact, it is possible to follow in real time a single ribosome
00:23:22;12 that is translating the messenger RNA against a force.
00:23:25;13 Because, remember, we are now applying some tension, here, across the mRNA.
00:23:29;17 And so we are actually loading, if you wish, the motor with some external force.
00:23:36;10 And the ribosome, which is attached by the back, has to actually pull the RNA in order
00:23:42;04 to actually... through itself... in order to actually be able to carry out translation.
00:23:48;06 And so doing this type of experiment, we have recently characterized the... the...
00:23:54;07 the kind of chemical conversion in the ribosome motor.
00:24:00;17 Okay.
00:24:02;02 So finally, I want to actually discuss the issue of...
00:24:10;01 I want to discuss the issue of the resolution in optical tweezers in a slightly more formal way.
00:24:15;18 And the resolution has to do with the fact that when you have a bead in an optical trap
00:24:20;19 -- the optical trap is here, represented by this harmonic potential that you see here,
00:24:25;17 this parabola, right? -- you see that the harmonic potential is my optical trap,
00:24:30;18 and holds my bead in the center of the trap.
00:24:33;07 And this bead has two springs attached to it, as we discussed before, the light spring,
00:24:39;03 which is the spring that tends to maintain the center of the bead in the trap, right,
00:24:44;02 so it's... this is related to the stiffness of the trap, and the tether spring with which
00:24:50;24 I add actually grab that bead and connect it, maybe, to a molecular motor, as indicated
00:24:55;10 here, right?
00:24:57;00 The problem that I have is the following, and it's that this bead that is in the center of...
00:25:01;26 of the optical trap is subjected, as we saw, to a lot of fluctuations and noise,
00:25:07;22 because we are doing these experiments in water and at room temperature.
00:25:11;10 And so there are a lot of fluctuations going on.
00:25:14;24 And so what I want to estimate is what is the fluctuations that I may expect at
00:25:20;23 room temperature due to the movement of the bead.
00:25:24;08 Well, for a system such as this, it can be shown that the bead behaves as if it
00:25:34;22 would be under the effect combined of the two stiffnesses: the stiffness due to the light spring and
00:25:42;23 the stiffness due to the tether spring.
00:25:44;15 So, the k-effective that actually is determining how much noise do I actually experience
00:25:50;28 in this bead in the center of the trap due to the thermal noise, the thermal fluctuations,
00:25:56;24 is ultimately controlled by the sum of the stiffness of the light and the stiffness of the...
00:26:02;02 of the tether, okay?
00:26:04;13 An effective stiffness.
00:26:07;13 Well, I showed you the spectrum of fluctuations, the mean quadratic fluctuations as a function
00:26:14;24 of frequency.
00:26:15;24 If you multiply it by the square of the stiffness, you can obtain the mean quadratic fluctuation
00:26:22;01 in the force.
00:26:23;19 And again, you obtain... since that's a constant, you obtain again a Lorentzian curve.
00:26:28;10 Notice, flat at the beginning, then the corner frequency, and then a decay that we saw before.
00:26:34;15 Okay?
00:26:35;16 So, here is the mean quadratic fluctuation in force experienced by the... by the bead
00:26:43;02 at zero frequency, when the frequency, here... when... when the frequency is zero, right?
00:26:48;28 It's given by 4*kB*T*gamma, gamma being the diffraction... the friction coefficient
00:26:56;04 of the bead.
00:26:57;04 Now, notice, this curve here is completely flat, right?
00:27:01;09 It's supposed to be flat here because it's a Lorentzian.
00:27:04;08 And that means that you can ask what is the mean quadratic total fluctuation that I
00:27:10;13 should experience all the way to a particular frequency, say, a hundred Hertz.
00:27:15;07 That's what we call our bandwidth of measurement, okay?
00:27:18;02 So, we can measure that by simply, it's just the area under this curve.
00:27:22;17 It's this spectrum, in picoNewtons squared per unit Hertz, multiplied by a hundred Hertz.
00:27:30;06 That should give me the total amount of noise in terms of quadratic noise in force.
00:27:36;09 So, that is expressed here, mathematically, as the mean quadratic noise in force
00:27:44;03 that I expect due to the thermal motion for a bandwidth that is smaller than the corner frequency.
00:27:50;13 Remember, the corner frequency is the place where the curve starts decaying, right?
00:27:55;15 It's just nothing but what we care about -- 4*kB*T*gamma*B, B being this frequency,
00:28:03;12 which is my bandwidth, right?
00:28:04;23 Alright.
00:28:05;23 So, this is the bandwidth of my measurement, okay, in Hertz.
00:28:14;08 So, here is the noise.
00:28:17;04 The noise that I'm going to measure is nothing... of a bead in the trap, that we're going to
00:28:22;13 observe of the bead in the trap, is nothing but the root-mean-square displacement of the bead
00:28:29;01 due to thermal noise, which is equal to the root-mean displacement in force
00:28:34;26 due to thermal noise divided by the combined kappas -- the combined stiffness of the tether and
00:28:41;05 the trap -- as I indicated here, right?
00:28:42;25 Because I told you that this is an effective kappa, or stiffness, that comes to...
00:28:48;07 that is equal to the sum of these two.
00:28:50;01 I know this quantity in the numerator because I just showed it to you here.
00:28:54;04 It's 4*kB*T*gamma*B. So, I can put it there, in the denominator, take the square root,
00:28:59;26 because it's root-mean-square, and the 4 becomes a 2 and then the rest is just inside this
00:29:04;25 square root, right, 1/2, divided by k-tether + k-trap.
00:29:09;26 Where B, remember, is the bandwidth of the measurement in Hertz.
00:29:15;28 And now I want to know what is the signal-to-noise ratio of my experiment.
00:29:21;06 Okay, what I want is a signal-to-noise ratio greater than 1, which means that the delta
00:29:26;11 signal that I measure at the level of my optical tweezers experiment is greater than the noise
00:29:33;08 due to thermal... to thermal fluctuations, right?
00:29:35;08 In other words, delta-signal greater than delta x root-mean-square thermal, right?
00:29:41;17 Okay.
00:29:42;17 So, I know this quantity because we determined it before for the case of the... of the little
00:29:49;28 motor moving and pulling the string, remember?
00:29:53;11 You don't get the full delta x, the step, but you get delta x-signal is equal to
00:29:58;27 delta x-step multiplied by k-tether divided by k-tether plus k-trap, remember?
00:30:03;17 We...
00:30:04;17 I told you that this quantity usually is smaller than 1, so what you measure as a signal is
00:30:11;08 multiplied... is the... is the size of the... of the motor... the step of the motor
00:30:15;08 multiplied by a quantity smaller than 1.
00:30:16;24 So, you end up actually losing some resolution in that experiment.
00:30:20;02 Nonetheless, we want that delta x-signal to still be larger than the noise in the experiment
00:30:27;26 due to thermal fluctuations.
00:30:30;12 But the noise is given by this quantity.
00:30:32;04 So, I want this quantity to be larger than that quantity.
00:30:36;27 And then I... that's the same thing as saying that I want the step size of the motor
00:30:42;18 greater than this quantity.
00:30:44;03 In other words, the motor has to be at least this size in order for me to resolve it above
00:30:50;09 the noise with a signal-to-noise ratio greater than 1, right?
00:30:55;03 And notice what we obtain here.
00:30:56;23 We obtain a very simple relationship, and we are going to see it again, here.
00:31:00;04 Here it is.
00:31:02;17 The size of the step that we can resolve with the optical tweezers machine is given by
00:31:07;22 2 times the square root of kT -- that's the Boltzmann constant and the temperature of
00:31:12;00 the... of the experiment -- gamma -- which is the friction coefficient of the bead, alright,
00:31:18;06 in water, in the viscous medium water -- times the bandwidth, square root of that,
00:31:23;22 divided by k-tether.
00:31:24;28 The first thing that you notice is that the resolution of the optical trap does not depend
00:31:30;07 on the stiffness of the trap, but it depends only on the stiffness of the tether.
00:31:36;00 That's the first thing to realize, okay?
00:31:38;15 So, there is no point in actually trying to make stiffer traps just to get better resolution,
00:31:45;13 because the resolution doesn't depend on the... on the resolution... on the stiffness of the...
00:31:50;00 of the trap, only on the stiffness of the tether.
00:31:52;24 The other thing that you notice is that the bandwidth appears here.
00:31:56;13 The bandwidth is how long are you willing to average your measurement in order to actually
00:32:01;09 get a signal, right?
00:32:03;17 But of course you make the bandwidth 0, then of course this quantity will be 0.
00:32:10;20 But if you bring your bandwidth 0, it means that actually you are ave... very...
00:32:16;12 averaging the... your signal for an infinite amount of time.
00:32:19;15 And then you lose time resolution.
00:32:21;24 Presumably your motor is stepping at some frequency, and you want to be able to
00:32:26;21 see this process, and so you cannot average to infinite time.
00:32:30;24 You have to average to times shorter than the stepping time of the motor.
00:32:35;03 So, that limits the... what can you do with the bandwidth, because you make the bandwidth 0,
00:32:40;05 of course, you... this quantity will eventually be 0, and this will be fulfilled for any value
00:32:46;13 of delta x, but you will have no time resolution in your experiment.
00:32:51;12 Okay.
00:32:52;12 And the other thing to realize is that a quantity that is important is the friction coefficient
00:32:58;16 of the... of the bead.
00:33:00;13 The friction coefficient of the bead... if you make the friction coefficient of the bit
00:33:04;13 smaller, then this... this inequality will be fulfilled, and you can see it smaller and
00:33:10;03 smaller steps.
00:33:11;03 So, the idea is that you can make the friction coefficient smaller but using the smaller beads.
00:33:16;26 And so in general, high resolution experiments using optical tweezers utilize small beads
00:33:21;20 in order to actually minimize this gamma, or the drag of friction coefficient of the bead.
00:33:28;11 Now, we are pretty close to the end of this lecture, and I just want to remind you that,
00:33:32;21 in fact, reducing the bandwidth is great because it averages... it means increasing the amount
00:33:40;03 of time that you average your noise, so of course if you have actually some kind of signal,
00:33:46;27 as indicated here by this staircase feature here, but that is superimposed with the noise
00:33:55;07 of your trap, as you actually reduce the bandwidth you will actually average out quite a bit
00:34:02;04 of that noise.
00:34:03;05 Reduce your bandwidth and eventually you may be able to actually begin to see, in fact,
00:34:08;08 the individual signal that you want.
00:34:10;26 But of course, if you overdo it, maybe here, still you can see the individual steps
00:34:16;25 that are underneath, corresponding to that dynamics of the motor, for example, if you imagine
00:34:22;12 that that's what you are observing, but if you actually reduce further bandwidth,
00:34:26;28 notice that eventually you start averaging out the signal itself, and so you lose your resolution again.
00:34:34;21 Because you are now losing time resolution rather than spatial resolution.
00:34:38;08 So, let me give you a final example so that you see the kind of quantities that we
00:34:44;23 can expect to measure with optical tweezers.
00:34:46;23 Suppose that you have a bead that has a two-micron diameter.
00:34:50;02 It's immersed in water.
00:34:52;02 The bead has a tether.... attached to a tether of DNA, double-stranded DNA that is 10 kilobase pairs.
00:35:00;10 And you can either apply a tension of 2 picoNewtons or a tension of 20 picoNewtons to that tether.
00:35:08;01 The signal that you want to measure is maybe some stepping of a motor that occurs at about
00:35:13;18 1 Hertz, so that much bandwidth is required.
00:35:16;26 You'll need least 1 Hertz or more in order to see the stepping of the motor.
00:35:24;05 The tether stiffness of the DNA can be calculated from the so-called worm-like chain theory
00:35:32;28 of... of polymer elasticity, of DNA elasticity.
00:35:37;01 And you can assume that... for that, you can assume that your DNA has a persistence length
00:35:42;17 of about 50 nanometers.
00:35:44;10 And then you can say, well, what is the stiffness of my tether at 2 picoNewtons of force or
00:35:49;12 at 20 picoNewtons of force.
00:35:51;19 And using the worm-like chain result, you can actually obtain that at 2 picoNewtons
00:35:55;23 of force your tether is 12 picoNewtons per micrometer -- that's the stiffness of your DNA,
00:36:00;13 of your 10 kilobase pairs of DNA.
00:36:03;24 And at 20 picoNewtons, notice your stiffness is 170 picoNewtons per micrometer.
00:36:11;15 So obviously, you... it's convenient to work at higher forces.
00:36:15;02 You can see that your k-tether is much stiffer, now, right?
00:36:18;23 And so you want that generally.
00:36:22;26 The smallest resolvable step, we decided, was actually given by an inequality.
00:36:28;09 But then we make it an equality, because we are going to assume that we want just signal-to-noise
00:36:32;08 equal to 1.
00:36:33;11 So, the smallest step that we can resolve is given by this amount, right?
00:36:39;00 And notice that k-tether is in the denominator, so having a larger k-tether -- the stiffness
00:36:45;02 of your tether, namely, using higher forces -- means that you will have a much larger
00:36:49;24 quantity here, which means that you can resolve a much smaller value, here, of the step...
00:36:55;26 of the step... of distance, of displacement.
00:36:58;16 So, we can plot all these numbers, here.
00:37:01;22 We know them.
00:37:03;00 And it turns out that you can actually, at 2 picoNewtons of tension with a 10,000 base pair tether...
00:37:09;10 which is very long, you don't need to use a 10,000 base pairs... base pairs...
00:37:14;22 you can actually just a 1000 base pairs and then it will be 10 times better than that,
00:37:19;13 but let's use 10,000 as in this example.
00:37:21;28 At 2 picoNewtons of tension you should be able to resolve displacement of about
00:37:30;03 1.5 nanometers with a bandwidth of 1 Hertz -- occurring every...
00:37:34;07 every cycle...
00:37:35;09 every second, right?
00:37:36;28 But if you actually work, now, at 20 picoNewtons, notice that you can do the calculation again,
00:37:41;26 and what you obtain is that you are able to resolve essentially 1 Angstrom at...
00:37:47;00 at that 1 Hertz of... of bandwidth.
00:37:50;15 So in fact, optical tweezers are really wonderful instruments in the sense that they allow you
00:37:56;05 to study molecular dynamic processes at high resolution, at reasonable temporal resolution,
00:38:04;22 and that... are of the order of displacement and resolution that... that are... that are
00:38:11;21 important in biological process: conformational changes, movements of one part of the molecule
00:38:17;20 relative to another, or just changes in the position of the motor in a track.
00:38:24;02 Of course, this is somewhat of an ideal result, because it assumes that the instrument
00:38:28;26 is only limited by Brownian...
00:38:31;25 Brownian noise, thermal noise.
00:38:35;28 And that is an important point of consolation to which we should end.
00:38:41;13 And it is that, in as much as possible, wonderful things can be learned as long as you have
00:38:46;23 instruments that actually are only limited by Brownian noise.
00:38:51;13 And that's one of the big efforts in this field, to actually have instruments that are
00:38:56;02 only Brownian-limited and not limited by other problems like, for example, drift or temperature gradients
00:39:02;08 in the laboratory, or... or mechanical noise due to the building, vibrations,
00:39:08;05 or acoustic noise, etc etc.
00:39:10;20 So, let me finish here, and with... with this example, and remind you again that
00:39:18;26 optical tweezers are powerful tech... it's a powerful technique, but you really need to know
00:39:22;22 what you are doing in order to make sure that you don't make a big mistake.

Videos in this Talk

Part 1: Optical Traps: An Introduction
Audience:
- Researcher
Part 2: Optical Traps: Building and Calibrating
Audience:
- Researcher
- Educators of Adv. Undergrad / Grad
Part 3: Optical Traps: Measuring Steps by Molecular Motors
Audience:
- Researcher
- Educators of Adv. Undergrad / Grad

Speaker: Carlos Bustamante

Total Duration: 1:33:21

Recorded: September 2012

Full Playlist: Microscopy Series

Talk Overview

Light can exert force on physical objects. In optical trapping this is used to immobilize objects (often beads) in a focussed laser beam. These videos describe the principles of optical traps, show how to build and calibrate optical traps and show how optical traps are used in the study of molecular motors.

Questions

Which of the following is false for optical trapping of particles that are greater than the wavelength of light?
1. Can be better described by refraction of light by the particle than by an induced dipole of particle by interacting with light.
2. Requires focused light from a high numerical aperture objective
3. The particle is trapped at the center of the beam and slightly above the focus (between the focus and coverslip)
4. Conservation of momentum can be used to explain the forces acting upon a particle in an optical trap.
True or false. A highly reflective particle makes it easier to trap.
For particles that are considerably smaller than the wavelength of light, which of the following is true:
1. The light induces a dipole momentum in the particle,
2. The dipole will move if the electromagnetic field of the light beam is non-homogeneous; ie towards a more intense region of illumination at center of a focused beam of light.
3. The particle will gain momentum if it absorbs light.
4. Small particles do not have appreciable scattering.
Filling the back of the objective with light is important for trapping because
1. It allows for optical trapping in the x-y plane
2. It reduces light absorption by the particle.
3. The momentum change of lateral light rays interacting through the particle generates forces that counter the scattering forces to allow stable optical trapping
4. It provides a better means of increasing laser intensity than turning up the laser power

Answers

View Answers

Speaker Bio

Carlos Bustamante

Dr. Bustamante received his B.S. in biology from Cayetano Heredia Universtiy in Lima, Peru, and his M.S. in biochemistry from San Marcos University, also in Lima. He then moved to the University of California, Berkeley where here completed his Ph.D. in biophysics. Currently, Dr. Bustamante is a Howard Hughes Medical Institute Investigator and a Professor… Continue Reading

Part 1: Optical Traps: An Introduction

Part 2: Optical Traps: Building and Calibrating

Part 3: Optical Traps: Measuring Steps by Molecular Motors