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