Up till now I have spoken about motion of a harmonic oscillator, a system or a particle
moving inside a quadratic energy well. Sounds reasonable for some atoms, but let's consider
something not more complicated. Imagine you look at a side chain on the surface of a protein.
The side chain can exist like this, or in the next rotamer, or in the next rotamer. And
now a harmonic oscillator doesn't look appropriate. What's happening is this is an energy minimum,
and occasionally the side chain moves and it finds the next energy minimum. So a better
picture would be something like this one up here. We have an energy surface, but it has
more than one energy minimum. So the particle can wobble around in here, but every so often
it is able to jump over the mountain and land on this side. And when I talk about the frequency
of emotion, what I'm probably talking about is how many times per second can the system jump from
this energy minimum over to this energy minimum here. Now we can talk about kinetic energy as a
half mv squared, and say this is roughly the temperature with a few constants there, and
velocity is the square root of the temperature, or proportional to the square root of the
temperature. A better way to think about this system is do not forget I have a particle, but
it's always being hit by other particles, other parts of the protein, other solvent atoms and so
on. So it has an inertial component of velocity that stays with it, but it also has a random
influence. And this is important because we are going to talk about probabilities. So what we're
looking at is a system with two energy minima here and over on this side, and a barrier with a
certain height in between them, and a particle with a velocity which is not at all fixed, and is
subject to random forces. So it's a system where maybe you can reach the top, and then you're moving
very slowly, and maybe you fall back down to this side, or you fall over to the other side, but now
there's a consequence. You can only move from left to right, or vice versa, if you had enough energy
to reach the top of the energy mountain. So the rate of movement in this system is proportional to
your probability of reaching the top of the energy barrier. If you cannot reach the top of the energy
mountain, you're stuck on one side. The probability of jumping between states is directly proportional to
the probability of reaching the top of the energy mountain. This probability we can quantify relatively
exactly in a certain sense. I'm going to write down a rule which is familiar to people who've
done physical chemistry, and if not, do not worry. I'm going to talk endlessly about this rule next
semester. But the Boltzmann rule, so this thing here, you can write it down in different ways,
but what it says is that the probability of reaching a certain state i
is proportional to an exponential function to the power of the opposite of the energy level
minus Ei divided by Boltzmann's constant and the temperature.
Oh, we have to stop and think about this for a second.
Here is a little plot. This is the energy of some state, and this is probability on this axis here,
not normalized, so don't take this too seriously. At a low temperature, we have this red line here,
so if we have a small value of T that makes this fraction bigger, which is putting E up to a more
negative number, so the probabilities are relatively small. If we make T bigger,
right, it's always a positive number, temperature, if we make T bigger, that makes this fraction
smaller, so E up to a smaller negative number is coming closer to 1, so by heating the system up,
we heat up the probability, we increase the probabilities of having a certain energy level
in this direction. So the probability, now why is this relevant? We're talking about
the probability of reaching a certain energy level to get over a mountain.
If I have a very small temperature here, then I've got E up to a big negative number,
which is close to zero, so for a small temperature, the probability of getting
over this energy mountain is very small. If I have a great big temperature here,
imagine I have infinite, a gigantic positive temperature, this fraction goes towards zero,
so with infinite positive temperature, then I have a very high probability of being able to reach
any energy level, so the implications are very clear. If I have small barriers with small energy,
those are easy and far, and I cross them all the time, but when I have more interesting energy
barriers, as the temperature goes up and I have more energy, my probability of crossing
those energy barriers goes up, so as I increase the temperature, the rate or the frequency that
I see also increases, so now we have a very different situation to a harmonic oscillator.
Harmonic oscillator frequency does not change with temperature. In a two-state system,
as we increase the temperature, then the rate does change because of the
energy barrier between the two states.
So finally, the original question near the start of the lecture was, as I increase the temperature
of a system, do the frequencies in the system change? If we have a harmonic oscillator, no.
If we are moving between energy minima, then yes. Now you should ask, which model is better?
There is no better model, there are different models. You might say which model applies to
which kinds of motions? That's a better question. Well, empirically, you can heat the system up
and maybe you can measure the frequencies and see if they change, and then you've answered the
question, did you have more than one state? Is your kind of spectroscopy dependent on movement
between states or just simple vibrations within one energy well? When we're talking about a real
protein and a physical model, you can decide what you believe in. The truth will be that both of the
models that I have described are really too simple, of course. If we have small emotions,
so bond-stretching angles, those are actually nearly harmonic, and then the harmonic oscillator
is a good approximation. If we have something like the system here, rotation around this bond,
then we've got separate energy wells. So you would say that for these two systems,
I do not expect temperature-dependent frequencies. For rotation around a torsion angle,
I think the frequencies will be temperature-dependent.
And remember, nothing is really harmonic. Nothing at this level. So the harmonic oscillator is just
a convenient approximation because we can do all the maths for it. So if you look inside the middle
of a protein, I have a bunch of atoms here, and I'm considering what is this atom doing here.
In the middle of a packed protein, you could argue that you expect something like
the motions of a harmonic oscillator. What happens when we talk about big emotions?
Then I would say everything changes and becomes more complicated. Let's consider an example.
Big emotions. Let's consider the classic protein example of calmodulin. It's awfully famous because
this protein has been solved in more than one conformation. It has conformations that are so
distinct you can crystallize it this way or that way. You have to cheat a little bit
by changing the chemistry, but it's pretty clear that calmodulin can be squashed up like this
or it can be a bit extended like that. These motions going from here over to here are big.
They are many, many angstroms. And when I look at those motions, I'm looking at many energetic
barriers. So I would say that the frequency of exchange going from compact to less compact
will definitely be temperature dependent because of all of the barriers along the way,
even if my two-state model was too simple.
So let's summarize the various models that I've had. They are definitely too simple.
And in a real protein, energy terms are not going to be simply a constant times
coordinate squared, but locally that may not be a bad approximation. We definitely do not have
many simple energy barriers. And usually what we've got are many more than two states.
So people tend to talk about a spectrum of motions, a big collection of motions
with different energy barriers between different states. And people also talk about
what are called concerted motions. And that means to go from this state to that state
requires several things to happen at the same time, to be concerted with each other.
So the real world is more complicated than my toy models.
Let's continue to try to relate what I've said back to what you might be able to measure
and talk about detecting states. We've got different states of a protein. And states is
a really general word. It could mean unfolded compared to folded, or we could be talking about
nearly the same states just differing by side chain rotations. So it's a very general term.
But imagine I've got two states. I can talk about the probability of being in state A
compared to this one here, the probability of being in state B. And using the relation
that I had on the previous slide, the Boltzmann rule, we can talk about the ratio of these two
probabilities. And it depends on the energy difference of these two states. So if state
A has a lower energy than state B, you are more likely to be in state A.
So maybe we've got these alternative states. But if the energy difference between them
is much bigger than the typical kinetic energy that you have, so if state A is much better than
state B, all of the population will be in state A. So yes, maybe state B exists in principle.
But if the probability of being there is 1 in 1,000, it means you may not be able to detect it.
If you're talking about NMR, maybe you see a main confirmation, and you can't even see
that there is a second state with one-tenth the probability. So if energy differences are much
bigger than the kinetic energy that the system has, then you are only going to see one state.
So all of this could be a little bit abstract. And don't forget the importance of the size of
energy barriers. So if I have an energy surface that looks like this, I have a state here, and
a state here, and here, and here, and here, each one of these little NMR. But the barriers between
them are not very big. So the energy barrier to jump from here over to there is this barrier here,
and that's pretty small. But that means that all particles have got plenty of kinetic energy to
jump over these barriers. So it's as if the barrier is not really there. It's so small compared to
kinetic energy. So it's effectively, let's exaggerate, Ubert Riven a little bit,
we could say the barrier is effectively not present. And that's definitely the case for
some rotations. So sometimes a rotation is so fast that you don't know that there's more than one
state, even if we think in principle there should be. And I hesitate to talk about methyl groups,
because the physics there can get even more complicated. So if you do Prof Hammond's lectures,
you will probably learn that methyl groups are not totally classic entities. But a methyl group
rotates so fast that you may not know that there is more than one confirmation for the protons on
a methyl group as an extreme example. Let's put this even more into physical context and think
about the amplitudes and time scales. When we're talking about bond vibrations, we're talking about
very small amplitude motions, a tenth or a hundredth of an angstrom. We can only talk
about orders of magnitude for the times. So let's consider logarithm base 10 of the time.
For bond vibrations, we're talking about minus 14 to minus 13. All right, so it's the same
Hall meanest delights zone of a second. We're talking about side chain rotation on the surface
of a protein. The motions might be five to ten angstroms. And we're talking about time scales,
let's say a tenth of a nanosecond down to a hundredth of a nanosecond.
When we talk about big emotions of a protein, I think all bets are off. They could be still not
so big or they might be many tens of angstroms. And the time scales are similar. They could be
from a fraction of a microsecond down to a hundredth of a nanosecond, 10 to the minus 11.
In the middle of a protein, strangely, here there's actually experimental data.
In the middle of a protein, how fast does a phenylalanine ring go from this conformation
to that conformation as an example? So motions are sort of five, six, seven, eight angstroms.
The time scale here might be 10 to the zero, so that's one, might be once a second down to fractions
of a millisecond. And the reason I know about this is some of these you can see with NMR.
Now we get to things that are less concrete. If you've got a helix or a strand, these things
actually open up, right? So an alpha helix is not absolutely fixed. There are good experimental
ways to, or at least there is good experimental evidence that a helix occasionally opens up and
the hydrogen bonds break. So you might be talking about largish motions, several angstroms,
and the time scale here can be over several seconds down to maybe 10 microseconds.
So these are all broad ranges, but the idea is to say that we go from motions which are
very small and very fast through to things like opening of protein secondary structure,
which can happen on the time scale of fraction of a second over to many seconds. This opening
of secondary structure you can probably see by things like isotope exchange inside a protein,
exchanging normal protons for deuterated protons or deuterons.
Good, so almost finished. You really do need motions to explain biochemistry.
If you don't have motions, proteins don't work. What you see from NMR and X-ray crystallography
are time averaged structures. You see a set of coordinates and these are the average positions
of atoms over really long time scales. So for X-ray crystallography we're talking about hours
and for nuclear magnetic resonance seconds, but still that is a very long time. Nearly always,
small motions are fast and big movements are slow, and don't forget what for me is entertaining the
temperature dependence of these different kinds of motions. So we're going to have different kinds
of temperature dependence depending on the kinds of motions that we're talking about,
but the fun thing is that using temperature dependence, if we have some, we can estimate
energy barriers. So, ich glauber das Reicht führer Heuter.