**8. Conclusion**

*Homology Molecular Modeling - Perspectives and Applications*

saddle point one or more eigenvalues are negative.

**6. What should be the convergence criteria?**

matrices (if appropriate) [22].

require the means to inter-convert between the internal coordinate representation and the Cartesian coordinates that are often used as input and desired as output. Of particular importance is the need to transform energy derivatives and the Hessian

A configuration at which all the first derivatives are zero need not necessarily be a minimum point; this condition holds at both maxima and saddle points as well. From simple calculus we know that the second derivative of a function of one variable, f'(x) is positive at a mini- minimum and negative at a maximum. It is necessary to calculate the eigenvalues of the Hessian matrix to distinguish between minima, maxima and saddle points. At a minimum point there will be six zero and 3 N — 6 positive eigenvalues if 3 N Cartesian coordinates are used. The six zero eigenvalues correspond to the translational and rotational degrees of free- freedom of the molecule (thus these six zero eigenvalues are not obtained when internal coordinates are used). At a maximum point all eigenvalues are negative and at a

In contrast to the simple analytical functions that we have used to illustrate the operation of the various Minimization methods, in 'real' molecular modeling applications it is rarely possible to identify the 'exact' location of minima and saddle points. We can only ever hope to find an approximation to the true minimum or saddle point. Unless instructed otherwise, most Minimization methods would keep going forever, moving ever closer to the minimum. It is therefore necessary to have some means to decide when the Minimization calculation is sufficiently close to the minimum and so can be terminated. Any calculation is of course limited by the precision with which numbers can be stored on the computer, but in most instances it is usual to stop well before this limit is reached. A simple strategy is to monitor the energy from one iteration to the next and to stop when the difference in energy between successive steps falls below a specified threshold. An alternative is to monitor the change in coordinates and to stop when the difference between successive configurations is sufficiently small. A third method is to calculate the root-meansquare gradient. This is obtained by adding the squares of the gradients of the energy with respect to the coordinates, dividing by the number of coordinates and taking the square root. It is also useful to monitor the maximum value of the gradient to ensure that the Minimization has properly relaxed all the degrees of freedom

and has not left a large amount of strain in one or two coordinates [23].

complex systems such as macromolecules or large molecular assemblies.

Energy Minimization is very widely used in molecular modeling and is an integral part of techniques such as conformational search procedures. Energy Minimization is also used to prepare a system for other types of calculation. For example, energy mini- Minimization may be used prior to a molecular dynamics or Monte Carlo simulation in order to relieve any unfavorable interactions in the initial configuration of the system [24]. This is especially recommended for simulations of

**7. Applications of energy minimization**

**5. Differentiating between minima, maxima and saddle points**

**130**

The energetic state of a protein is one of the most important representative parameter of its stability. The energy of a protein (E) can be defined as a function of its atomic coordinates, thus providing a quantitative criterion for model selection and refinement. This energy function consists of several components e.g. (1) Bond energy and angle energy, representative of the covalent bonds, bond angles. (2) Dihedral energy, due to the dihedral angles. (3) A van der Waals term (also called Leonard-Jones potential) to ensure that atoms do not have steric clashes. (4) Electrostatic energy accounting for the Coulomb's Law m protein structure, i.e. the long-range forces between charged and partially charged atoms. All these quantitative terms have been parameterized and are collectively referred to as the 'forcefield'. The goal of energy Minimization is to find a set of coordinates representing the minimum energy conformation for the given structure. Various algorithms have been formulated by varying the use of derivatives. The common algorithm used for this optimization is steepest descent, conjugate gradient and Newton–Raphson etc. These methods complement each other in search of the local minima. Therefore, a reasonable energy Minimization protocol involves few initial steps of steepest descent, followed by a larger number of conjugate gradient iterations. Although energy Minimization is a tool to achieve the nearest local minima, it is also an indispensable tool in correcting structural anomalies, viz. bad stereo-chemistry and short contacts. An efficient optimization protocol could be devised from these methods in conjunction with a larger space exploration algorithm, e.g. molecular dynamics.
