Flexibility and Rigidity in Molecules



Flexibility and Rigidity in Molecules

Submitted by:

Pramod AbrahamKurian

111-38-243

Computation In Structural Biology

The University of Texas at Dallas

Dallas, Texas

Flexibility and Rigidity in Molecules

Abstract

This paper describes the Flexibility and Rigidity in molecules based on mathematical models and the related algorithms. These Mathematical models are used to describe the snap-shot flexibility in molecules from a single snap-shot of a molecule. We begin with discussion about the bar and joint framework and the reason for modeling the molecules into a body and bar joint. Finally based on these results the flexibility of secondary structures like Alpha-helices, Beta-sheets is illustrated.

Introduction

From 1960s people have been counting vertices and edges to determine the rigidity and flexibility of plane frameworks. This counting technique was proved and confirmed by Laman’s theorem in 1970 for plane frameworks. Different counting algorithms can also predict the rigidity and flexibility of the framework by calculating the rank of the rigidity matrix associated with the framework. It is being calculated that a framework with n vertices is rigid in m space if and only if the rigid matrix has a rank of mn-(m(m+1))/2.

Unfortunately an attempt to generalize these counting algorithms to frameworks in higher dimensions to predict rigidity was a failure. An alternate framework was the framework model of bodies and bars that efficiently uses these counting algorithms for finding the rigidity, flexibility and related decomposition. A molecular structure of atom and bonds in 3D can be represented into a geometric body and bar model and its flexibility can be easily determined by these counting algorithms. These counting techniques and algorithms are being used in FIRST and other software to predict flexibility and rigidity in molecules that are present in proteins.

Snap-Shot Flexes and Flexibility

A Flex P’ of a structure is a vector Pi’ assigned to each vertex Pi of the structure such that (Pi – Pj)(Pi’ – Pj’) = 0,where Pi’ represents the unknown virtual velocity of the point Pi. For a bar and joint framework represented as a graph G = (V,E) and a configuration P E R^3V, gives a system of |E| equations in 3|V| variables P’, which can be recorded with the rigidity matrix. If the solution to the rigidity matrix is not trivial, then P’ is snap-shot flex. So it can be said that if a framework has a snap-shot flex then it is snap-shot flexible, else it is shap-shot rigid.

Modeling Molecules as Bar and Joint in 3D

In this section we model a molecule into a Bar and Joint Framework. Each molecule is assumed to be a set of atoms and covalent bonds and can be modeled into a Graph M , as shown in fig: 1

[pic]

Since the angle between the covalent bonds are fixed we need to add additional bond bending edges between the neighboring atoms to fix these angles. This forms a Graph M^2 (fig: 2)

[pic]

If a double or peptide bond exists between two atoms then an addition edge is added between the further next two neighbors. This prevents the rotation on this edge. (fig 3).

[pic]

Additional constrains like the Hydrogen bond or the Hydrophobic interactions can be modeled by adding additional fixed distance between points.

This model of Bar and joint is too slow to solve by linear algebra techniques , when there are thousands of atoms and bonds.

Theorem

Laman theorem for a generic configuration P E R^2|V| states that non-empty subset of edges E is independent in the rigidity matrix if and only if for all non-empty subsets E’ subset of E, |E’| ................
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