Effects of Interactions Between the Segments

So far we did not talk about interactions between the different segments. For totally flexible and hydrophilic side chains, and for other completely soluble polymers, the non-interacting, or `phantom' polymer model that we have presented above is a good starting point for thinking about polymer shapes and elasticity.

But - we are supposed to be talking about biopolymers - RNAs, DNAs and proteins. In their active forms, these molecules are usually folded into particular 3d structures, precisely by strong interactions between the different segments.

For example, proteins have their active conformations determined to a large degree by the layout of hydrophobic and hydrogen-bonding residues, which stabilize a-helicies, b-sheets, and other 3d structures.

Nucleic acids have their folds defined by hydrogen bonding of bases, to form the stem-loop structures of RNAs and to form the DNA double helix.

So we need to think about interactions between segments.   The figure shows an interaction between vertex number i and j, with potential energy u which depends on the distance between the two vertices.    The interaction might look something like the following:

The interactions should generally have a strongly repulsive `core', and may or may not have an attractive well at distances of a few A or so.

Summing all pairs of interactions  we obtain an interaction free energy of the form (for one isolated polymer):

Einteraction = 
å 
i < j 		 ui,j(|ri-rj|) 
 
Note that this sum over pairs has N(N-1)/2 » N2/2 terms.

Although this formula does include the complication of having sequence dependence - the potential energy ui,j depends on the segment location along the chains - it is still a gross simplification of the `real' situation for biopolymers. The `real' interactions involve orientation and stretching degrees of freedom for all the flexible chemical bonds in the chemical groups making up each monomer, and at the monomer level cannot be considered to be simple pairwise interactions...

On the other hand, we have to start somewhere. To start with we will study the homopolymer case where all the monomers are the same, so that

ui,j(|ri-rj|) = u(|ri-rj|)
 
We'll be able to learn a lot about the basic physics driven by repulsive and attractive interactions from this simple model, plus a simple and remarkably effective statistical-mechanical treatment invented by Paul Flory in the 1940s.

Our aim will be to show first how repulsive interactions between monomers, even those of very short range associated with the fact that two segments can't be in the same place, will make a flexible polymer swell up to have a volume > N1/2 b. This will provide a slightly more sophisticated model for random coil polymers which could be applied to protein regions which are strings of hydrophilic amino acids.

Next we will show how attractive interactions cause polymers to collapse into compact `globules' reminiscent of folded, globular proteins, but without precisely defined 3d structure.   A rough overview of this is indicated in the next figure:

Then we will be in a good position to discuss the situation for `real' proteins and nucleic acids.

Flory's Calculation of the Effect of Segment-Segment Interactions:

Flory's idea was to reduce the complicated many-body problem of the conformational sum over all the segment orientations with interactions, to a simple approximate estimate of the net interaction energy, as a function of overall coil size R. Then he added together this interaction energy and the free energy of stretching that we figured out in the previous section:

F(R) = Fstretching(R) + Einteraction(R) 
 
The equilibrium value of R is then determined by minimization of this free energy, with respect to the free parameter R.

Some of you may recall that when there are free parameters in thermodynamics, you determine their equilibrium values by minimizing the free energy (F if you are working at fixed volume, G if you are working at fixed pressure).

We'll use the weak-stretching part of the stretching free energy:

F(R) =  3 kB T R2 
2 N b2		 + Einteraction(R) 
 
We'll make a rough, average estimate of the interaction energy,
Einteractions = < 
å 
i < j 		 u(|ri - rj|) > thermal, R
 
where the average is taken with respect to the usual thermal (Boltzmann) distribution, for segments on a polymer with overall coil size constrained to be R.

To make a rough estimate of this, we'll forget about the chemical connectivity of the polymer, and just suppose that we have an N-segment `gas' in a ball of radius R.

If this gas is dilute and if the interaction are weak, the simplest estimate of the interactions will be:

Einteractions »  N2 
2		 3 
4 pR3		 ó 
õ 

|r| < R 

 d3 r u(r) 
 
which is N2/2 (the number of distinct pairs of segments) times the interaction of one segment with another segment whose position is distributed randomly inside a sphere of radius R.

Slightly better is the slightly more complicated formula for the interaction of N segments with each other, inside a spherical `box' of radius R, from the theory of dilute gases:

Einteractions »  N2 
2		 3 kB T 
4 pR3		 ó 
õ 

|r| < R 

 d3 r (1 - exp[-u(r)/kB T]) 
 
For high temperature, or for weak interactions (u ® 0) you will see that this reduces to the simple averaged interaction. The latter formula has the great feature that is remains finite even for infinitely repulsive potentials (note that for u ® ¥, 1 - exp[-u(r)/kB T] ® 1).

Although this looks complicated, the last formula has a pretty simple structure:

Einteractions »  kB T N2 
R3		 w 
 
where w is a quantity with dimensions of a volume, which is
w »  ó 
õ 

|r| < R 

 d3 r (1 - exp[-u(r)/kB T])
 
where some overall numerical factors have been dropped. This number depends only the inter-segment potential and not on the total number of segments, and characterizes the overall attraction vs. repulsion of segments. w is sometimes called the excluded volume, since for interactions which are simple hard-core interactions, that is what it is.

If w > 0, the interactions are predominantly repulsive, and the segments will tend to stay away from one another. For repulsive interactions, w will be a volume of the order of a segment volume.

If w < 0, the interactions are predominantly attractive, and the segments will tend to aggregate.

So, we obtain Flory's estimate of the net free energy of a polymer coil, as a function of the coil size R, in units of kB T:

F(R) 
kB T		 »  3 R2 
2 N b2		 +  w N2 
R3
 
What is particularly nice here is how a lot of complexity of intermolecular interaction is dumped into the parameter w, which may have some complicated dependence on temperature, salinity, pH, etc. However, in terms of w, the theory is kind of simple. This approach is rather typical of modern condensed matter physics, most notably Landau's theories of magnetic phase transitions, superconductivity, and of interacting conduction electrons in solids. All of these, plus Flory's theory, are mean-field theories, where an averaging procedure is used to decouple the degrees of freedom in a many-body problem.

Net Repulsive Interactions - Flory Swelling

If w is positive, the net free energy blows up as R-3 as R® 0, thanks to the strong repulsion that occurs between segments forced into a small volume. On the other hand, for R® ¥, the polymer connectivity and elasticity come into play and the free energy again blows up, as R2.

So, the equilibrium (free-energy-minimizing) radius Req is somewhere in between 0 and ¥, and can be determined by looking for zero slope of F(R):

æ 
ç 
è 		 d F 
d R 		 ö 
÷ 
ø 

R = Req 

 = 0 =  3 Req 
N b2		 -  3 w N2 
Req4
 
Solving this simple equation tells us
Req = (w b2)1/5 N3/5 
 
i.e. that if w > 0, for large enough N, we will always have swelling of a polymer due to its self-avoidance.

Of particular interest is the scaling behavior R µ N3/5 which has been verified for a huge range of polymers in good solvent. So, we say that isolated polymers in `good solvent' or which are `self-avoiding', swell, to be larger than simple random walks. The best numerical estimates and real experiments agree that the exponent of N is close to 0.60, making Flory's simple-minded theory remarkably accurate on this point. de Gennes' book has some nice discussion of this.

Problem: For a given value of N, how small does w have to be to not affect the random-walk coil size?

The result Req » (w b2)1/5 N3/5 has the property that Req ® 0 when w ® 0. But we know that Req should not get smaller than the random-walk size R0 = b N1/2 when the interactions are turned off. So, we can figure when w becomes unable to perturb the size of a polymer by figuring out when

Req < R0
 
or when
b N1/2 < (w b2)1/5 N3/5 
 
This gives the condition
w <  b3 
N1/2 
 
for the segment-segment interactions to be so weak that they do not change the coil size from its random-walk value R0 = b N1/2.

Isolated polymers in solution where w < b3 / N1/2 are often called ideal polymers or theta-solvent polymers. Because w can be tuned to some degree by changing temperature, salinity, or some other parameter(s), one can often find the theta point where polymers are essentially taking on random-walk conformations because their segment-segment interactions are very weak.

Theta conditions occur in two ways:

(a) solution conditions where there are balancing attractive and repulsive contributions to the segment-segment interaction

(b) for polymers with segments which are long, thin rods

Case (b) will be important for thinking about the flexible polymer behavior of long double-helix DNAs.

Summary:

When w < b3 / N1/2, the size of a flexible polymer is about R0 = N1/2 b.

When w > b3 / N1/2, the size of a flexible polymer is about RFlory = (w b2)1/5 N3/5.

Problem: Find the size Req of a self-avoiding polymer in 1, 2, 4 and 5 dimensions using Flory's theory.
Hint: in any space dimensions, Req cannot be less than R0 = N1/2 b if there are net repulsive interactions.

What is the size of a self-avoiding polymer in 9 (or 10) space dimensions?

Problem: Using an appropriate modification of the Flory theory, estimate the size (and shape) of an isolated polymer whose segments interact by a unscreened repulsive Coulomb interaction u(r) = k e2 / er in water (e = 80).

Hint: you will need to think carefully about how to estimate Einteraction.

Problem: Suppose we have a denatured protein of N = 300 amino acids. Assuming reasonable values for b and w for excluded-volume interaction of the segments, how different are the random-walk and Flory swollen-coil estimates? How long a protein would you need for the random-walk size estimate to be 1/2 of the swollen-coil estimate?

Net Attractive Interactions - Polymer Collapse

If the segments have attractive interactions with one another, Flory's free energy curve is totally different.

At small R, the attractive interactions lead to a free energy which becomes smaller and smaller, µ -1/R3. So, Flory's theory indicates that when w < 0, a polymer will collapse to a point, i.e. Req = 0 and Feq = -¥.

To make more sense of this, we have to remember that there are strong repulsive interactions that keep atoms from overlapping, which will therefore make this collapse stop at the point where the monomers are compactly packed together. If we suppose that the volume per segment is v, then we should have Req3 » Rmin = N v for this collapsed globule, or

Req = Rmin » v1/3 N1/3
 
This will be not too different from b N1/3 if the segments are reasonably sphere-shaped (e.g. amino acids).  Again, the situation if the segments are thin rods is a little special and we will discuss it later.

We can ask two questions about this picture of a collapsed polymer:

(a) is there really a collapse to a compact globule for all w < 0?

(b) is this a reasonable model for a compactly folded (globular) protein?

The answer to both questions is no.

The first question can be simply answered by looking at how big the free energy of interaction is for the random-walk coil. If this free energy is less than kB T, then we know that the random-walk coil will not be strongly perturbed by the attractive interactions:

| Einteraction 
kB T		 | »  |w| N2 
R3
 
Plug in R = b N1/2 to estimate the size of this term for a random walk:
| Einteraction 
kB T		 | »  |w| N2 
b3 N3/2 		 =  |w| N1/2 
b3
 
When this is less than 1, the attractive interactions can be ignored. The condition for this is therefore
|w| < b3 
N1/2		 < 1
 
This is the same condition encountered when we considered how small w should be for no swelling to occur.

Another way to think about this is that along a region of chain of size

N <  b6 
w2
 
the interactions are too weak to perturb that region of chain to be very different from just a random walk. As w is made more and more negative, this `collapse length' gets smaller and smaller.

Finally when w » -b3, the polymer will be folded into a compact globule. The meaning of this threshold is simply that when w » -b3, the attractive interactions exceed kB T per monomer, enough to crush out almost all of the random-walk entropy.

We'll answer the second question in the next lecture.

Interaction Effects - Summary:

Flory Self-Avoiding Polymer
w > b3 / N1/2
Repulsive interactions are strong enough to swell the polymer up to a size » (w b2)1/5 N3/5

Ideal Polymer (or Gaussian Polymer, Theta Point Polymer)
b3 / N1/2 > w > 0
Repulsive interactions are too weak to swell the chain to appreciably larger than its random-walk size » b N1/2

Ideal Polymer
0 > w > -b3 / N1/2
Attractive interactions are too weak to condense the polymer to be appreciably smaller than its random-walk size » b N1/2

Partially Collapsed Polymer
-b3 / N1/2 > w > -b3
Attractive interactions are strong enough to make polymer quite a bit smaller than its random-walk size

Molten Globule Polymer (or Fully Collapsed Polymer)
w < -b3
Attractive interactions are so strong that polymer is a globule of monomers, of size µ N1/3. The solvent is excluded from the interior of the globule.

Problem: Find the approximate diameter of a 100-amino-acid protein if it folds tightly into a globular structure. You will have to make a realistic estimate of the volume per amino acid.

How much larger is the diameter of a 1000-amino-acid globular protein?

Problem: Estimate the conformational entropy of a condensed homopolymer, using compact walks on a lattice as a model. How do you think the entropy will go up with N for N large?

This problem is hard, and can only be semi-quantitatively answered without a big numerical analysis. However, it will start you thinking about protein (heteropolymer) folding.

File translated from TEX by TTH, version 2.53.
On 25 Feb 2001, 21:11.