Molecular Mechanics

a. Force fieldsb. Energy minimization / Geometry optimizationc. Molecular mechanics examples

Nonbonded energy

Van der Waals forceLenard-Jones type potentialShort-distance

Electrostatic forcesLong-distance

Bonded energy

Some popular force fields are:

AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) force field developed by Pengyu Ren (University of Texas at Austin) and Jay W. Ponder (Washington University).AMBER (Assisted Model Building and Energy Refinement) - widely used for proteins and DNA CHARMM (Chemistry at HARvard Molecular Mechanics) - originally developed at Harvard, widely used for both small molecules and macromoleculesOPLS (Optimized Potential for Liquid Simulations) (variations include OPLS-AA, OPLS-UA, OPLS-2001, OPLS-2005) - developed by William L. Jorgensen at the Yale University Department of Chemistry MMFF (Merck Molecular Force Field)- developed at Merck, for a broad range of molecules (MMFF94 for molecular dynamics, MM94s static for molecular mechanics).MM2, MM3, MM4 - developed by Norman Allinger, parametrized for a broad range of molecules.MM2 was developed by Norman Allinger primarily for conformational analysis of hydrocarbons and other small organic molecules. UFF - A general force field with parameters for the full periodic table up to and including the actinoids - developed at Colorado State University

Simplex method

Minimization Bounds Minimization Bounds Polygon of N+1 vertices Polygon of N+1 vertices Solution is a vertex of N+1-d polygonSolution is a vertex of N+1-d polygon

Procedure (Downhill Simplex Method)Procedure (Downhill Simplex Method)Begin with simplex for input coordinate valuesBegin with simplex for input coordinate valuesFind lowest point on simplex (best)Find lowest point on simplex (best)Find highest point on simplex (worst)Find highest point on simplex (worst)((Intuition: Move away from high point, towards low pointIntuition: Move away from high point, towards low point))

Reflect (x1=-xo)Reflect (x1=-xo) If E(x1)<E(xo) then expand (x=x+l)If E(x1)<E(xo) then expand (x=x+l)

ElseElse Try intermediate pointTry intermediate point

If E(xnew)<E(xo) expandIf E(xnew)<E(xo) expand If E(xnew)>E(xo) contractIf E(xnew)>E(xo) contract

Simplex MethodsSimplex Methods

Simplex downhill (amoeba) methodNelder, Meade 1965

Newton-Raphson method

An energy contour surface.

The minimizer must find the direction to the minimum and its distance from the initial guess

Improve efficiency by finding how the derivatives change

Most minimizers use line searches

line search - one-dimensional minimization along a direction vector determined at each iteration. (x´, y´) are coordinates along the line away from the current point (x0, y0) in the direction of the derivative at (x0, y0). For the path shown, it would be along derivative vector (2x,10y)

shows the minimization path followed by a steepest-descents approach for the simple quadratic function. As expected, each line search produces a new direction that is perpendicular to the previous gradient; however, the directions oscillate along the way to the minimum. This inefficient behavior is characteristic of steepest descent method, especially on energy surfaces having narrow valleys.

Conjugate Gradient method

It would be preferable to prevent the next direction vector from undoing earlier progress. This means using an algorithm that produces a complete basis set of mutually conjugate directions such that each successive step continually refines the direction toward the minimum. If these conjugate directions truly span the space of the energy surface, then minimization along each direction in turn must by definition end in arriving at a minimum. The conjugate gradient algorithm constructs and follows such a set of directions.

Model size and distance from the minimumThe choice of which algorithm to use depends on two factors--the size of the model and its current state of optimization. The conjugate gradient and steepest descents methods can be used with models of any size. Most Newton-Raphson methods cannot be used with very large models, because they need sufficient disk space to store a second-derivative matrix. (the storage requirements scale as approximately 3N x3N for N atoms)

Until the derivatives are well below 100 kcal/mol/1Å, it is likely that the point is sufficiently distant from a minimum that the energy surface is far from quadratic. Algorithms that assume the energy surface to be quadratic (Newton-Raphson, conjugate gradient) can be unstable when the model is far from the quadratic limit. The Newton-Raphson method is particularly sensitive because it must invert the Hessian matrix.

Therefore, as a general rule, steepest descents is often the best minimizer to use for the first 10-100 steps, after which the conjugate gradient and/or a Newton-Raphson minimizer can be used to complete the minimization to convergence.

Molecular docking simulation of aspirin with phospholipase A

Aspirin

Phospholipase cleavage sites

Phospholipase A2 is an enzyme present in the venom of bees and viper snakes,mammalian digestive juices and inflammatory exudates.

Optimization problems

 knapsack problem or rucksack problemwhich boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? A multiple constrained problem could consider both the weight and volume of the boxes.(Solution: if any number of each box is available, then three yellow boxes and three grey boxes; if only the shown boxes are available, then all but the green box.)

ant colony optimization algorithmDorigo 1992search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar

In the natural world, ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they

are likely not to keep travelling at random, but to instead follow the trail, returning and reinforcing it if they eventually find food.

Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at

all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be

constrained.Thus, when one ant finds a good (i.e., short) path from the colony to a food source,

other ants are more likely to follow that path, and positive feedback eventually leads to all the ants' following a single path. The idea of the ant colony algorithm is to mimic

this behavior with "simulated ants" walking around the graph representing the problem to solve.