WEBVTT - autoGenerated
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Guten Tag und Wielkommen zur Rück. Gesten hater eschein Alptraum.
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Fier uns van sich Stunden befou der Klau zuwe und zemüssen für sein Wocken videos zu Schauen.
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Für zweifung zemnicht zie kannen immer die Videos mit doppelder Geschwindigkeit Abschpielen und viel er Fauck.
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Ich glauber heute ist ein English tag.
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And today I'm going to talk about motions inside proteins and frequencies and amplitudes.
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So how big are the motions in a protein and how fast are they?
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And I will have a few examples.
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And what will be very important is the idea of different models for motions.
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And they're just models. They're not really the truth.
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So motions without barriers and motions with barriers.
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And let me pose a question that I will answer later on.
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So imagine you look at a protein at two hundred and seventy two hundred and seventy three kelvins.
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That's pretty cold. And then you heat the protein up to three hundred Kelvin.
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Do you think the frequencies of the motions will change?
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That's a nasty question that's much more fun to do in a live lecture.
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It's usually fifty fifty the answers and we will have the answers soon.
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So why do we think that protein motions need a lecture to themselves?
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Protein motions or the mobility within a protein turn out to be necessary, not important, but essential to a protein functioning.
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So let me give you the first historic example.
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In the 60s, well before you were born, people solved the structure of myoglobin and it really did look like this.
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This is taken from a very old paper and myoglobin carries oxygen around.
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And even before people had a crystal structure, they knew that the oxygen had to bind to a heme group in the protein.
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Remember heme groups, right? The porphyrin group with an iron in the middle.
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The problem was after they solved the structure, nobody could see how an oxygen could get into the middle of the protein where the heme group is.
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So even in the 1960s, people said, well, a protein like hemoglobin or myoglobin can only function if atoms temporarily move up and down
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and let an oxygen travel into the middle of the protein.
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But there are lots of similar stories, lots of enzymes whose active site seems to be hidden by other atoms.
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And in general, the activity of a protein can't be explained just by looking at the crystal structure.
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So let's consider the kinds of motions we could have.
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First of all, at a fundamental level, everything has got kinetic energy.
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So we know that there are bond vibrations and angle vibrations here.
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There's rotation around torsion bonds.
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And when you think about it, it's actually these motions, vibrations and rotations, that are the basis of many kinds of spectroscopy.
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So think about infrared spectroscopy and bond vibrations, for example.
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Think about this is a bit more subtle.
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NMR would not work if there were no motions within proteins.
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But the explanation for that is a bit more complicated.
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Ask me in a Zoom session, if you are interested, why does NMR only work if you have motions?
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The trick is it's to do with relaxation.
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OK, we can have more fundamental arguments at absolute zero.
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Everything is either dead or very sleepy.
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It's not moving. But as soon as you have temperature, so at 300 Kelvin's, everything has kinetic energy.
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So kinetic energy is almost the definition of temperature.
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So remember, when we talk about temperature, we're talking about kinetic energy.
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But I always call e Kin.
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And kinetic energy is a half times the times the mass of a particle times its velocity squared.
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So everything that is not at absolute zero has some kind of atomic motion.
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An idea that I use a little bit this semester and much more next semester is the idea of an energy surface.
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So when I talk about an energy surface, it's conceptual.
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But I have a plot that looks like this.
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And on this axis, I have the energy of the system.
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And usually this is a potential energy.
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And on this axis here, this is what's not really physical.
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It's just meant to represent all of the possible confirmations of a molecule.
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So this might be the global optimum.
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This might be what you see in the protein data bank.
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Over here are unfolded confirmations.
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Up here are some misfolded confirmations.
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But it's everywhere that the molecule could possibly go.
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Now, if we knew the shape of the energy surface, we would actually know almost everything about motions.
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And this comes up or is explained much more in summer semester.
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But remember, the acceleration on a particle depends on the gradient here moving between different confirmations.
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So if I know the gradient, I can actually describe how much a particle will be accelerated.
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And you could also look at it.
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Maybe this is intuitively clear.
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And so if a system manages to get to the top of a mountain, it's probably moving very slowly.
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On average, if it gets to the bottom of a valley here on the energy surface, all of the potential energy has gone away.
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So it must have a lot of kinetic energy and moving fast down here.
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It's not really that simple, but that's not a bad way to think about it.
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The problem is we don't know what the energy surface really looks like, models for that next semester.
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The energy surface is just too complicated and our energy models are far from perfect.
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So what I'm going to do today is talk about two very, very simple models.
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And the two models are going to be what happens if you have an energy surface like this, which is basically just a valley.
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So it's one state and we have a particle which is moving inside this energy valley or on this energy surface.
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And I'm going to contrast that or compare that with a different picture of an energy surface where we've got a little energetic mountain
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between a minimum here and a minimum here.
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And these two extreme models give very different results for the kinds of motions that you will have and the frequencies of motions and their temperature dependence.
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So the first kind of energy model we will approximate as a harmonic energy surface.
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So this is not a word I invented.
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We talk about the harmonic oscillator.
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You could also, if you prefer, say it is moving on a quadratic energy surface.
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But the basic idea is that the potential energy depends on some constant, K, times the x-coordinate squared.
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So that corresponds exactly to this picture.
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Here is our x-coordinate and this here would be the energy.
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So this is a harmonic oscillator.
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And a result that I'm going to show you and explain, but not derive, is we can say that the x-coordinate at a certain time, T,
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depends on a constant here, the amplitude, times a periodic function, the cosine of omega, the frequency, times time, plus a phase offset that's called delta here.
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OK, this is a critical formula and I'm going to use it and say why it makes sense.
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And I'm not going to do the derivation today.
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So this is the prediction of the motion, this formula here.
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It tells us where the particle is as a function of time.
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Can I prove that this is a solution to the energy being quadratic?
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Yes.
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Otherwise, I would not have asked the question.
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Now is a good time for a Pinkelpouser and you should top up your coffee or red wine or whatever you drink during lectures.
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For the next three slides, we're going to have a few formula.
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There is a point to this.
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If I give you simple motion on a spring, can I then say that this here is a solution?
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So what I'm going to do is start with a spring, rewrite my formula and then show you the solution and show you why it is the truth.
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So harmonic oscillators, what do they really mean?
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This is my depiction of a spring.
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So spring and I have a mass attached to the spring.
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And I know that you have seen this before because I have said, do you know this?
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Do you remember Hooke's rule?
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And everyone goes, what?
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And I say, OK, the Hooke's shirt gazettes and people go, ah.
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So what did you learn in school about a spring?
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Here's a spring, here's a weight.
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And what I know is that if I pull on x in this direction, there's a force pulling me back in this direction.
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And the size of the force is just some constant times how far I pull you.
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So if I pull x, if x is pulled one centimeter and then it's pulled two centimeters, the size of the force doubles.
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Now, let's work our way down this side and then down that side.
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So we'll say the force is minus because it's the opposite of a constant times x.
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And remember Newton's law.
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So f, glite, m, r.
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F is mass times acceleration.
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So f is Massamol-Boschlöningen.
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So we can rewrite this f as an m, mol a.
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And I want to change nomenclature.
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So what is acceleration?
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What's the first time derivative of x, velocity?
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What's the second time derivative of x?
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It's acceleration.
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So I take a and I write it as d2x dt squared.
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So this is still the Hooke's law for a spring, just written down a little bit differently.
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So we take this version of Hooke's law and let's add kx to both sides.
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So we end up over here.
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And now remember that I can divide by a positive constant.
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So I'll divide this by m and I get this way of writing it down.
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And now humor me for a moment.
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And we could also say this is another nomenclature change.
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But if we were doing first year physics lectures, we would talk about it a bit differently.
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We'll say that the frequency squared depends on a constant divided by mass.
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I don't want to talk about this too much because I could call k on m the square root anything
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I want, but it has a physical meaning.
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So I'm going to say omega is the square root of some constant divided by the mass, which
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hopefully makes intuitive sense.
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The bigger the mass, the lower the frequency.
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So now I've really just renamed my formula from this form here to writing it in terms
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of something called omega, which if you think in terms of physics lectures is the rotational
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frequency.
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But if you prefer, you can just pretend it's a name.
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OK, this here is important.
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It's Hooke's law written down rather differently, but it will come back in a slide or two.
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The point of these overheads, remember, was to say if I have a simple mass on a spring,
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which is a harmonic oscillator, I made the claim that this expression here is a solution,
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that we can have the coordinate as a function of time is given by the amplitude of the motions
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times co-sinus of the angular frequency times time plus some offset delta.
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So what let me show you not the handle item that can also be done.
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That takes 15 minutes.
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But what I want to show you is this expression here is a valid solution for Hooke's law.
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It's not the same as saying it's the only solution.
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I'm just saying this expression is a valid solution.
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So look at this expression here and take the first time derivative.
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So dx dt, I bring the a down, derivative of co-sinus is meanus sinus.
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Take the derivative, so bring that down here, take the derivative of the inside, keton regal.
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So the chain rule, I bring omega down here and then I just move omega and the minus sign
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out the front.
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So the first time derivative, dx dt, is minus a omega sinus of the original insides here.
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Then let's take the next time derivative.
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So from dx dt, the next ubliton, imbutsu dzeit, so the next time derivative is d2 x dt squared.
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Bring the constants straight down, derivative of sinus is co-sinus.
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Bring the insides down, take the derivative of what's in the middle.
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Remember the chain rule.
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So we get this omega here, then tidy up the terms.
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So bring omega here, so it's minus a omega squared, co-sinus of omega mild tape.
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And let's rewrite this expression a little bit differently.
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So we'll move this a in this side and write it down like this.
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Now look at this expression here, from here over to there.
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That's what I have at the start.
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So amplitude times the cosine function.
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So let's take this x here and substitute it in here.
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So now we have the second time derivative is minus omega x squared times x.
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So we've reached this point here.
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Does this look familiar?
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Two slides ago, and you can go back and check, I said this is one way to write Hooke's law.
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But now let's take this term here, and remember from the previous slide, the second time derivative
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is actually just minus omega squared times x.
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So let's substitute that in here, and then I end up with the claim that minus omega x
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squared minus omega squared x plus omega squared x is zero, which is not an unreasonable
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thing to say.
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So I reached this point here by testing this as a solution to my form of Hooke's law.
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So what it's saying is that this was not a hair alightal, not a derivation, but it's
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saying that this way of expressing coordinates as a function of time is a valid solution
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to Hooke's law.
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So we have an expression here for periodic motion, for a harmonic oscillator.
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And that was what we were aiming for.
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So now you can have another little rest, hit pause, and another pinkelpouser.
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And if you want a moment of entertainment, next time a Hamburg professor says to you,
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oh, you have to learn tech to typeset formula, show him or her the previous slide, haha.
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Why did I spend so much time getting to this kind of expression here to do with the second
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derivative of the energy or the, in this case, the coordinates with respect to time?
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Think about the meaning of this second derivative here.
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Right, remember the first derivative of an expression is the gradient.
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The second derivative is the curvature, right, or auf Deutsch, boygung.
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But that means there's no velocity term coming into this at all.
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And it's second derivative, acceleration, berschleinigle, not velocity.
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And what's the connection to temperature?
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Temperature is die Hälfte and fau hulkswe.
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So temperature depends on velocity, or temperature is velocity squared.
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So the angular frequency has no dependence on temperature.
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It has a dependence on the energy surface, absolutely no dependence on temperature.
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All right, so does this make intuitive sense?
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Now it's fun to have a video because now we can think of a practical demonstration.
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What is a harmonic oscillator that most people have met?
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You might say a spring.
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Or think about a pendulum, right, so a swinging pendulum as in an old-fashioned clock.
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Why did people put a pendulum in a clock?
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Because the frequency of the pendulum does not depend on how hard you push it.
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If you push it very slowly, the frequency does not change.
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It just doesn't move as far.
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If you give it a big push, it moves faster.
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The instantaneous velocity is bigger, but the frequency doesn't change.
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So this answers the question, what does change in this system when you heat it up?
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When I change the temperature of a system, I'm changing the kinetic energy.
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Kinetic energy is a half m fau hulkswe.
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This V here is the first time derivative of the coordinates.
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Remember dx dt is V. So we can substitute this part here for this V over here.
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So we could say kinetic energy is this expression here depending on mass and amplitude and frequency.
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The frequency is not changing.
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And this means we can finally answer the question that I posed earlier in the lecture.
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If I double the temperature of a system, I go from 100 Kelvin to 200 Kelvin, the amplitude
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squared will go up by 2.
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Everything else will stay constant in here.
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So in this model of motions, frequency is temperature independent.
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It's just the amplitude of the motions that changes when I heat the system up.
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So that is the important result.
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There is one more interesting consequence.
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If you ever hear a seminar from somebody who does molecular dynamics simulations, they
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will probably say, oh, we've got the small high frequency motions and the lower frequency
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large motions.
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And you might look at it and say, is this necessarily the case?
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Are small motions higher frequency?
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It turns out in this model, it is the case.
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Small emotions tend to have higher frequencies.
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Let me explain why.
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For some temperature, energy distributes amongst the modes of a system.
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And it has to, not just because there is something called the equi-partition theorem, but if
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you've got particles moving around, they will always be exchanging energy with each other.
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If you have a system which can move in certain different manners with certain different modes,
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the energy can flow from one mode into another one.
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The practical consequence is that the energy within each mode tends to be the same.
00:28:20.000 --> 00:28:29.000
Now, if we can say the energy within a mode, let's say it's E kinetic energy, so it's
00:28:29.000 --> 00:28:37.000
a half mv squared, so we can expand the velocity turnout to this thing here.
00:28:37.000 --> 00:28:48.000
If I have two modes within a protein, one is a slow, low frequency mode, so the bending
00:28:48.000 --> 00:28:52.000
of a big hinge inside a protein.
00:28:52.000 --> 00:28:59.000
The other is fast, a high frequency mode, so a side chain moving around very quickly.
00:28:59.000 --> 00:29:07.000
The kinetic energy in both of these modes is on average going to be about the same.
00:29:07.000 --> 00:29:15.000
But that means that as the frequency goes up, the amplitude has to go down, or as it's
00:29:15.000 --> 00:29:23.000
written here, as the square of the frequency goes up, the square of the amplitudes goes
00:29:23.000 --> 00:29:24.000
down.
00:29:24.000 --> 00:29:33.000
So if I have a big amplitude motion, it's also going to be a low frequency motion.
00:29:33.000 --> 00:29:41.000
Or conversely, a very small amplitude motion will be a high frequency motion.
00:29:41.000 --> 00:29:49.000
So low frequency motions in proteins are the ones with large amplitudes.
00:29:49.000 --> 00:29:54.000
And that comes almost directly from the physics of the situation.
00:29:54.000 --> 00:30:05.000
So it is true when you hear a seminar and somebody refers to small, high frequency motions.
00:30:05.000 --> 00:30:09.000
Let us summarise so far.
00:30:09.000 --> 00:30:14.000
Something I've said has been about a harmonic oscillator.
00:30:14.000 --> 00:30:24.000
And maybe that's not such a bad model when I have motion within an energy valve.
00:30:24.000 --> 00:30:27.000
But it's only an approximation.
00:30:27.000 --> 00:30:30.000
It's not the truth.
00:30:30.000 --> 00:30:38.000
But it does have the interesting consequence that the frequencies of motions are independent
00:30:38.000 --> 00:30:41.000
of temperature.
00:30:41.000 --> 00:30:45.000
Let's not forget that we're talking about real molecules.
00:30:45.000 --> 00:30:50.000
So a real protein or nucleotide has many such oscillators.
00:30:50.000 --> 00:30:52.000
Some are fast and some are slow.
00:30:52.000 --> 00:30:58.000
But the bigger motions will be associated with lower frequencies.