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Brian Hayes

Foursquare Folding

The specific model I've been toying with was devised 10 years ago by Ken A. Dill of the University of California at San Francisco, who has continued to explore it since then with the help of several colleagues. Almost all of my experiments merely replicate their earlier work.

Dill's molecules would not be recognized as proteins by a biochemist (or by a ribosome, for that matter). They are radically simplified in three ways.

First, whereas real proteins are constructed from 20 kinds of amino acids (which differ in size, shape, electric charge, affinity for water and other properties), the building blocks of prototeins come in just two flavors. Dill designates them H and P, for hydrophobic and polar; the H units repel water while the P units attract it.

Second, the various forces acting between amino acids in proteins (electrostatic attractions and repulsions, hydrogen bonds, solvent interactions) are reduced in prototeins to a single rule: H's like to stick together. The P units in prototeins are inert, neither attracting nor repelling.

Third, prototeins do their folding on a lattice, as if the molecules were laid out on graph paper. Think of the H's and P's as colored dots placed at the grid points of the lattice; the chemical bonds in the backbone of the prototein are lines drawn on the grid to connect the dots. Confining the molecules to a lattice is a major computational convenience. It keeps the number of configurations finite. If the chain could bend and twist in continuous space, there would be no clear way of counting the arrangements, and you could never be sure you had tried them all. Dill and others have explored several lattice geometries in both two and three dimensions. My own experiments all inhabit the two-dimensional square lattice, which is the simplest.

Dots and lines on graph paper: That's really all there is to a prototein. Or else the model could be described in terms of colored beads, laid down on a board with a gridlike pattern of dimples to hold the beads in place. To build a sequence of amino acids, you string together H-beads and P-beads in whatever order you choose. To fold the molecule, you arrange the string of beads on the lattice board. The string is not allowed to stretch or break, and so successive beads in the sequence have to occupy nearest-neighbor sites on the lattice. No two beads can be piled up at the same site, and so the chain cannot cross itself. If two H-beads that are not adjacent within the linear sequence wind up on adjacent sites after the chain is folded, their attraction creates a cross-link, or contact, that helps to stabilize the molecule. Foldings that give rise to many such contacts are favored over those with few contacts.

Simple and abstract the model surely is—so much so that you can't help wondering if the process of abstraction hasn't sucked all the life out of it. The squared-off, flattened molecules certainly don't look very biological. But the proof of the prototein is in the folding.

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