Protein-like AB-copolymers

The general idea just outlined above will be first illustrated taking as an example a globular conformation of a polymer chain.

The primary structure in real globular proteins is known to be highly specific. Globular proteins/enzymes functioning in living systems are the products of molecular evolution.

Their primary structures involve 20 possible types of monomeric units (20 amino-acid

Polymer physics: from basic concepts to modern developments 71

residues), therefore globular proteins are much more complicated objects than typical AB-copolymers. However, the most essential distinction between different monomeric unit of proteins is that some of these units are hydrophobic, while others are hydrophilic or charged (Dickerson et al. 1969, Grosberg 1997). Thus in a very rough approximation it is possible to represent a globular protein as a kind of AB-copolymer. The spatial (ternary) structure of such a copolymer in the native state would then normally correspond to the structure in which hydrophilic units (A-type) cover the globular surface and prevent different globules from aggregation, while hydrophobic units (B-type) constitute the dense globular core.

Because the hydrophobic links should be in the core and hydrophilic ones should lie on the surface, there should be some long-range correlations within the primary struc- ture, since these correlations are related to the spatial conformation of the globule. It is therefore interesting to ask whether it is possible to have a synthetic AB-copolymer with long-range correlations in primary structure analogous to the correlations in evolved proteins.

So, we are looking for preparation of such an AB-sequence that in the most dense globular conformation, all the B-units form a dense core while all the A-units are on the surface, i.e. such a sequence mimics one of the properties of a real protein what allows us to call the desired copolymer chain a protein-like AB-copolymer. Such copolymers should have interesting physical properties, for example, they should not precipitate when the dense globular conformation is formed. This is not the case for ordinary macromolecules (Lifshitz 1978, Grosberg et al. 1994; see also Section 2.4).

4.2.1

It is very easy to prepare such protein-like sequences in computer simulation (Khokhlov et al. 1998a,l998b,1998~,1999); the main steps of the real laboratory experiments should probably be the same. A computer realisation of the procedure is illustrated in Figure 22.

Preparation scheme for protein-like AB-copolymers

Figure 22. Colouring procedure f o r preparation of protein-like copolymer We take some particular conformation of a usual homopolymer coil with excluded volume, and switch on a strong attraction of monomeric links to let the chain collapse into the conformation of a homopolymer globule (Figure 22, left part). Next we take an ‘instant snapshot’ of the globule and assign the colour index A to those units that are on the surface of the globule and call these units hydrophilic, and assign the index B to the units in the core of the globule and call these units hydrophobic. Then we fix this primary structure (Figure 22, middle part; the structure shown in this figure will be further referred to as the parent globule). Finally, the last step is to remove the uniform strong attraction of monomeric units, and to add different interaction potentials for A- and B-units (Figure 22, right part).

72 Alexei Khokhlov

4.2.2 Coil-globule transition for AB-copolymers with different structures Let us consider the coil-globule transition for the protein-like copolymer thus formed, which occurs when one introduces attraction between hydrophobic B-units (hydrophilic A-units remaining repulsive to each other and to hydrophobic units). The Monte Carlo computer simulation was performed using the bond fluctuation algorithm (Carmesin 1988;

see also Kremer, this volume). The coil-globule transition was realised upon the increase in attraction interaction between B-links (which is equivalent to a decrease of temperature).

We studied primary sequences with an A/B composition ratio of 50/50.

We compared the coil-globule transition for three types of sequence: (i) our protein- like sequences; (ii) random AB-copolymers of the same A/B ratio but an entirely random sequence; (iii) random-block AB-copolymers, with the same A/B ratio and also the same degree of blockiness, i.e. with the same mean length of the blocks of A- and B-links as for protein-like copolymers. This comparison was done to distinguish the effects connected with the block lengths (visible in Figure 22) from the effects coming from the presence of long-range correlations between blocks, within the primary sequence. (Such correlations are, by construction, absent in case (iii).)

The data for temperature dependences of the mean energy per monomeric unit and of the specific heat are presented in Figure 23. We have found the transition for protein-like copolymers to occur at higher temperatures and to be more abrupt than that for either random or random-block copolymers (the peak of specific heat is narrower and higher for the protein-like copolymer). The kinetics of the coil-globule transition is also faster for protein-like copolymers (i.e. protein-like copolymers form the dense globule faster than random and random-block ones, under similar

O

-I . I . 4 . I . I

0 1 2 3 4

T T

Figure 23. The temperature dependences of the mean energy per monomeric unit (a) and of the specific heat (b) for copolymers with chain length N = 512 and primay sequences as indicated.

Polymer physics: from basic concepts to modern developments 73

Figure 24. Typical snapshots of the globular structures for ^(a) protein-like, (b) random, and (c) random-block copolymers.

What is the reason for such effects? To answer this question let us look at the mor- phology of different globules. In Figure ^24, typical snapshots are shown for globular conformations obtained for the three types of copolymers at equilibrium at low tempera- ture. (The hydrophilic A-links are shown in a darker colour than the hydrophobic ones.) It can be seen that for protein-like AB-copolymers, practically all B-units are concentrated in the dense core of the globule which is stabilised by long dangling loops of hydrophilic A-links. On the other hand, the core of the globules formed by random and random-block AB-copolymers is much looser, and approximately 30% larger in average size than that for our designed AB-copolymers. A fraction of the hydrophilic A-links are now inside the core, and those of them which belong to the surface form very short dangling loops which apparently are not sufficient to prevent the aggregation of such globules in the solution.

It is reasonable to assume that the formation of the dense core shown in Figure ²⁴ for protein-like copolymer globules is facilitated by the fact that the dense globule preexisted in the parent conditions shown in Figure ²² (middle part). Since all the B-links in this parent core are fitting next to each other, there is no connectivity obstacle to reassembly of most of this core when the effective attraction between the B-units is switched on. In other words, we can say that the protein-like copolymer inherited some important features of the parent globule which were then reproduced in the other conditions.