Pivoting Polymer

A pivoting polymer is a rigid set of links where the points joining the links can rotate. When a point is rotated, the whole polymer pivots rigidly about that point. Pivots always happen in increments of 90 degrees. A red dot indicates the pivot point at each step in the animation. This simple system was used to model protein folding at the 2011 NKS Summer School.


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In this simple pivoting polymer, pivot points are chosen at random at each step, and the polymer segments are allowed to overlap. While pivots are only allowed in increments of 90 degrees, each pivot movement is animated to allow the eye to track the way the polymer is rigidly rotating about the pivot point.
Although extremely simple, this model shows interesting similarities to single-molecule protein-folding dynamics. If one measures the rate at which various substructures occur in the polymer (analogous to measuring the varying conformations of an active site in an enzyme), the distribution of such rates is exponential. For real proteins, this distribution appears to be peaked with an exponential tail.
One possible extension of this model is to make the probability of pivot events dependent on the local environment of the pivot point. For example, each pivot point could be given a hydrophobicity (attracting or repelling the surrounding medium), giving rise to bending rules that would favor bringing together points of the same type. Would such rules produce folded structures with hydrophobic cores, as in real proteins?
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