Among the theoretical methods available to predict the geometry of a molecule, the Molecular Mechanics method (often called Force Field method) is one of the most commonly used. Its advantages are that only limited resources are required, little experience is necessary to use the method and the results have almost the same quality with regard to geometry as those obtained by sophisticated quantum methods.
The molecule is viewed as a collection of points (atoms) connected by springs (bonds) with different elasticities (force constants). The forces holding the atoms together can be described by potential energy functions of structural features like bond lengths, bond angles, non-bonded interactions, and so on. The combination of these potential energy functions is the force field. There exist natural lengths and angles for bonds, and in the simplest cases a molecule assumes a geometry with those values. In general, steric, electrostatic and other strain forces must be included. Because of the intimate connection between structure and energy, molecular mechanics calculations always involve both and to find the structure, one necessarily has to examine the energy to find where the minima occur.
For a force field calculation to be performed, an equation needs to be used to calculate the energy as a function of the molecular geometry, and an algorithm has to be chosen to calculate new atomic coordinates.
The energy, E, of the molecule in the force field arises from deviations from ideal structures, and can be approximated by a sum of energy contributions.
Chapter 4 Design, synthesis and properties of conformationally-restricted PUFA analogues
E = E ^ s t r + E ^ b e n d ^ E ^ o o p + E ^ t o r s + E E y d w +
[ E ^ e l e + E ^ d i s t . c + E E a n g _ c + E ^ t o r s . c + E E r a n g e _ c ]
where the sums extend over all bonds, bond angles, torsion angles and non-bonded interactions between atoms not bound to each other or to a common atom (i.e. 1,4- interactions and higher).
E^tr energy of a bond stretched or compressed from its natural bod length. Ebend energy of bending bond angles from their natural values.
Eoop energy of bending planar atoms out of the plane. E(^,rs torsional energy due to twisting about bonds.
E^(j^ energy due to van der Waals non-bonded interactions. The optional energy terms are defined as follow :
Eg,g energy due to electrostatic interactions. Edist c energy associated with distance constraints. Eang_c energy associated with angle constraints.
Etors_c energy associated with torsion angle constraints. Era„ge c energy associated with range constraints.
E is only a measure of intramolecular strain relative to a hypothetical situation. By itself E has no physical meaning. The value of E is the difference in energy between the real molecule and a hypothetical molecule where all the structural features like bond length and bond angles are exactly at their ideal or natural values. Each energy term contains adjustable parameters that have been optimised empirically and that depend on the chosen force field.
Chapter 4 Design, synthesis and properties of conformationally-restricted PUFA analogues
Energy is a function of the atomic coordinates and the program attempts to generate the coordinates which correspond to a minimum of energy. This is accomplished by a minimisation procedure. All the minimisation methods currently used for this purpose are called descent series methods. They are iterative methods in which the atomic coordinates are modified from one iteration to the next in order to decrease the energy.
They are several methods to locate the minimum of a function. The methods can be classified as using no derivatives, using first derivatives only or using first and second derivatives. In the first case, a non derivative based procedure is used to optimise the variables on an atom by atom basis until the maximum force on any atom is below some specified value. In highly distorted structures the potential energy surface and its derivatives are often discontinuous and this method can handle these areas while a derivative based procedure cannot. The primary and secondary optimisation methods adjust the atomic coordinates of all the atoms simultaneously, based on the first and second derivatives of the energy equation with respect to the degrees of freedom. A series of line searches is used for finding the local minimum of this function. From the current position a direction in the n-dimensional space is chosen. A sequence of steps are taken in that direction until a minimum along the direction is bracketed. Then, a quadratic interpolation is done until the minimum is isolated to the required accuracy. The method of Steepest Descent uses as line search direction which is the derivative of the function at the current position and no information from previous iterations is used. The minimum is searched until the variance between the current minimum and the previous minimum is within a pre-defined value. The overall convergence properties of this method are poor. The Conjugate Gradient method accumulates information about the function from one iteration to the next. [229]. Its convergence properties are superior
Chapter 4 Design, synthesis and properties of conformationally-restricted PUFA analogues
to steepest descent. The Powell method belongs to the Conjugate Gradient family of minimisation methods, but uses more advanced rules to determine the descent direction. It is also more tolerant to inexact line searches, and as a result is faster than the Conjugate Gradient method and well suited for a wide variety of problems [211].
These methods are generally unable to find the global energ> minimum, i.e. the set of atomic coordinates corresponding to the lowest value of the energy. Most of the time, only a local minimum is found, the one closest to the starting set of coordinates. The only way to find the global minimum is to systematically explore different sets of starting coordinates. These can be generated by rigid geometry calculations or postulated on the basis of other considerations and data (e.g. NMR studies or crystallographic results)
4.1.2.2 Distance Geometry
Distance Geometry is a method for producing three-dimensional structures consistent with a set of distances bounds [230, 231]. It automatically generates many of the possible interatomic distances from the starting structure given as input: it assumes that all bond lengths and angles are fixed, and generates the interatomic distances corresponding to all covalent bonds and all bond angles. Appropriate ranges are assigned for all distances. For other atom pairs. Distance Geometry uses the sum of their Van der Waals radii as a lower bound for the interatomic distance. Using only these distance constraints, this method produces structures which represent a random sampling of the possible conformations of the molecule. These may be useful in further analyses, such as starting conformations for molecular dynamics simulations. The best known application of Distance Geometry to structure determination is in the area of
Chapter 4 Design, synthesis and properties of conformationally-restricted PUFA analogues
incorporating interatomic distances obtained from NMR NOE experiments [232, 233].
4.1.2.3 Molecular Dynamics
Molecular Dynamics (MD) is a method of studying the motions and the conformational space of molecular systems by integration of the classical Newtonian equations of motion given a potential energy function and its associated force field. Earlier applications of MD techniques were concerned with the calculations of ensemble averages on models of simple liquids [234-236].The basic assumption that justified thc^e calculations was the use of the ergodic hypothesis that states that (infinite) time averages are equal to ensemble averages or integrals over conformational space. Standard Monte Carlo methods try to estimate those integrals more directly [237]. The initial success of MD and Monte Carlo simulations in reproducing values of thermodynamic properties and average geometrical features indicated that the computational techniques were valid and that the potential energy functions could be trusted. These developments set the stage for the first MD calculations of large bio molecules [238, 239], and later, for long simulations of peptides [240, 241] and proteins in vacuo [242], proteins in a crystalline environment and in solution [243], and oligosaccharides [244].
Operationally, and at the simplest possible level, two entities are required in an MD program : a force field and a way of integrating the equations of motion. The Verlet method [245], also known as the Leapfrog method, is generally used for this integration. As in any MD method implemented in a digital computer, the calculations of motion are done at discrete intervals: the length of these intervals defines the time step. A common type of simulation is a run at constant temperature. In this type of simulation the velocities of the atoms are scaled at each step so that the kinetic energy of the system
Chapter 4 Design, synthesis and properties of conformationally-restricted PUFA analogues
corresponds to a set of temperatures. The desired simulation temperature is commonly achieved by slowly raising the temperature from near absolute zero up to the target value. This is done by specifying a series of temperatures and the times to remain at these temperatures. A common use of the constant temperature simulation is to explore the conformational space available at the given temperature. This is done by simulating the motions at a very high temperature, 2000 K for example, where nearly all conformations are energetically accessible, then slowly cooling down to room temperature or below. The molecules settles into a natural conformation at that temperature. This sequence can be repeated many times, heating and cooling, to explore the possible families of conformations energetically attainable at a given temperature. This methodology, also known as simulated annealing has been introduced to molecular modelling of proteins using NMR [246] or crystallographic data [247].
4.1.3 Arachidonic acid and other polyunsaturated fatty acids in biochemical