## Computer Modeling Of Biopolymers

The observable physical properties of any molecule are ultimately the result of the physical laws that determine the behavior of individual molecules and atoms. At the most fundamental level, of course, the properties of molecules are governed by quantum mechanics, and modeling the behavior of molecules numerically requires the direct solution of Schrodinger's equation for all of the nuclear and electronic degrees of freedom for all molecules in the system. Unfortunately, it is not practical to solve this time-dependent equation at the necessary level of accuracy for molecules containing more than a few heavy (nonhydro-gen) atoms. It is thus not possible to make a direct quantum mechanical simulation of large and complex systems such as the mixtures of proteins, fats, water, salts, and other components that make up a typical food. However, in many cases, part of the solution of the quantum mechanical problem is known by other means. Typically, important features of the molecular structure, such as equilibrium bond lengths and angles, are known approximately from studies of less complicated molecules, and the primary challenge is to understand the conformational behavior of polymers, how different molecules interact, and the connection between basic structure and macroscopic properties. The link between basic structure and bulk properties is statistical mechanics, and the most common theoretical studies of biopolymer physical properties are various attempts to model this statistical mechanical connection.

Theoretical calculations that exploit a knowledge of molecular energies as a function of nuclear coordinates are called molecular mechanics calculations (2—4). Because it is impossible to accurately solve Schrodinger's equation for large polymer systems, it is common in theoretical simulations to approximate the quantum mechanical energy with continuous energy functions that have approximately the same behavior. By invoking the Born-Oppenheimer approximation (3), which says that the electronic motions of a molecule are far faster than the nuclear motions, it is possible to separate the inherently quantal problem of the electronic motions from the much more classical problem of the motions of the nuclei, by treating the electronic energy for each value of the nuclear coordinates as a potential energy function that governs the motion of the nuclei. It is then possible to use continuous, analytic functions to approximate this nuclear potential energy function. These semiempirical energy equations have theoretically reasonable functional forms that are parameterized to the results of various experiments and theoretical calculations such that properties calculated using these functions are those found in the experiments. Energy functions of this type usually contain terms to represent bond stretching, bond angle bending, hindered torsional rotations about chemical bonds, and nonbonded and electrostatic interactions. A

typical example of such a function of the internal coordinates q might be

where the b's are the various bond lengths, which have equilibrium values b0, the O's are the bond angles with equilibrium values 80, the <p's are the torsional angles about each bond, and q, and q, are the partial atomic charges of each atom, separated by a distance rtj. This equation treats bond stretching and bending as harmonic oscillator terms with force constants kb and klh and also includes a periodic hindered torsional energy term in the angle <p with the force constant k,p, periodicity n, and phase 5. van der Waals interactions are treated as a typical 10-12 Lennard-Jones-type interaction, and the electrostatic interactions are represented by Coulomb's law. The various adjustable constants that appear in such equations (kb, ke, kv, Ay, By, qL) must be selected with great care in order for the results to be physically reasonable. This parameterization is accomplished by matching experimental properties of small molecules (equilibrium bond lengths and angles, IR and Raman spectral frequencies, etc) to the values calculated using the energy equation as a function of the adjustable parameters (2-4).

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