Energy Minimization MethodEnergy Minimization Method

Presented By:

Badgujar Pavan R.

DEPARTMENT OF PHARMACEUTICAL

CHEMISTRY

( M. Pharm II nd Sem )

R.C.Patel Institute of Pharmaceutical Education & Research, Shirpur.

Energy Minimization Energy Minimization MethodMethod

Presented By:Presented By:

Badgujar Pavan R.( M. Pharm II nd Sem )

DEPARTMENT OF PHARMACEUTICAL

CHEMISTRYR.C.Patel Institute of Pharmaceutical Education & Research, Shirpur.

Introduction Molecular mechanics Energy minimization Energy minimization method First order minimization : Steepest descent, Conjugate gradient minimization Second derivative methods : Newton Raphson method. Example References 2

Introduction:Introduction: Molecular mechanics:

It’s a approach of energy minimization that find stable, low energy conformation by changing the geometry of structure identifying a point in the configuration space at the force on each atom vanishes.Molecular mechanics depend on three parameter: I.Force field , II.Parameter set, III.Minimizing algorithm.3

I ) Force field: It is set of function & constant used to described potential energy of the molecule. General form of force field equation ;Epot = ∑Ebon+ ∑Eang+ ∑Etor + ∑Eoop+ ∑ Enb + ∑EelWhere; Epot : The total steric energy Ebon : The energy resulting from changing the bond length from it’s initial value calculated by Hook’s law for deformation spring E=1/2kb(b-b0)2 [ kb-force constant for bond, b0-equilibrium bond length ,b-current bond length] Eang: The energy resulting from deforming a bond angle from it’s original val. Etor : Deforming the torsinal or dihydral angle Eoop : Is the out of plane bending component of the steric energy Enb : Energy arising from non-bonded interaction Eel : Energy arising from coulombic forces4

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Energy Energy Minimization :Minimization : It is a systematic modification of the atomic coordinates of a model resulting in a 3-dimensional arrangement of atoms in the model representing an energy minimum (a stable molecular geometry to be found without crossing a conformational energy barrier) is called energy minimization and geometry optimization .EM used for : Locating a stable conformation Locating global & local energy minima Locating saddle point

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A potential energy surface (PES) is a plot of the mathematical relationship between the molecular structure and its energy.It can describe: Either a molecule or ensemble of molecules having constant atom composition, A system where a chemical reaction occurs, Relative energies for conformers

Potential Energy Surfaces:

Example; The conformations of n-butane as the global minimum is the anti conformer, local minima are the gauche conformers, and the saddle points are the eclipsed conformations.7

EM is an numerical procedure for find a minimum on the potential energy surface starting from a higher energy initial structure labelled "1" as illustrated in Figure .

During EM the geometry is change in a stepwise fashion; energy of molecule is reduced from step 2 to 3 to 4 shown in figure . 8

The numerical minimization technique then adjust the coordinate: - if slope is positive : it indicate the value of coordinate is reduced (point2) - if the slope is zero : a minimum has been reached , - if slope is still positive: coordinate reduced further (point 3) until a minimum is obtained .;

Most of EMM proceed by determining the energy & the slope of function at point 1. - if slope is positive : it indicate the coordinate is too large (point 1) - if slope is negative : the coordinate is too small &

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All the EM methods used to find a minimum on the potential energy surface of a molecule use an iterative formula to work in a step-wise fashion. These are all based on formulas of the type: Xnew = Xold + Correction Where; Xnew- The value of geometry at the next step ( moving from step 1 to 2 in figure ) Xold- The geometry at the current step & correction . In all these methods, a numerical test is applied to the new geometry (Xnew) to decide if a minimum is reached .

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I. First-order minimization:

Types of energy Minimization Types of energy Minimization Method :Method :

Steepest descent

Conjugate gradient

II. Second derivative methods :

Newton-Raphson

Steepest Descent Method : The second derivative is assumed to be constant , the equation to update the geometry becomes Xnew = Xold − Y E’ (Xold) Where ; Y is a constant In these method; - gradient at each point calculated - not required second derivative calculated - the method is much faster per step - relies on an approximation but not as efficient & more steps require to find minimum . The method is named Steepest Descent because the direction in which the geometry is first minimized in opposite to the direction in which the gradient is largest (i.e., steepest) at the initial point. 12

Steepest descent algorithm (thin line): The derivative vector from the initial point P0(x0,y0) defines the line search direction. The derivative vector does not point directly toward the minimum (O). The negative gradient of the potential energy (the force) points into the direction (P0→b,P1→c) of the steepest descent of the energy hyper surface and is always oriented perpendicular to energy isosurfaces. 13

Advantage of method is easy with which force field can be changed.

The main problem with the steepest descent method is determining the appropriate step size for atom movement during the derivative estimation steps and the atom movement steps .

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Conjugate gradient minimization method:

It is a first-order minimization technique

It uses for both current gradient & the previous search direction

to drive the minimization.

The number of computing cycles required for a conjugated

gradient calculation is approximately proportional to the number of

atoms (N), and the time per cycle is proportional to N2. 15

Improves the step efficiency The method takes the next search direction to be a linear combination of the current gradient and the previous ones. Require fewer energy evaluations and gradient calculations. Convergence characterizations are better than with steepest descent

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The method is the most computationally expensive per step of all the methods utilized to perform EM. It is based on Taylor series expansion of the potential energy surface at the current geometry. The equation for updating the geometry is Xnew = Xold – E’(Xold)/ E”(Xold) - Is a powerful & convergent minimization procedure - Based on the assumption the energy is quadratically dependent like a classical spring.

Newton-Raphson minimization method:

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The advantage of the Newton-Raphson procedure that the minimization could converge in one or two steps. The major drawback is that this method requires the calculation of the second derivatives. The minimization can then become unstable when a structure is far from the minimum (or the energy surface is an harmonic).Advantage :• Only one iteration for quadratic functions• Efficient (relative to first -order methods)N/N-1 = (N-1/N-2)2 • Better energy estimate Disadvantages :N2 storage requirements (compared to N for conjugate gradient)N3 Involves calculating Hessian (~10 times time for gradient calculation)It used in transition-structure searches (saddle point locator)

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Example of the Use of Energy Minimization Methods: The geometry of lactic acid was optimized using the Newton-Raphson, Steepest Descent, and Conjugate Gradient methods. Lactic acid is a relatively small organic molecule

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Energy Minimization Using Conjugate Gradient Method : Molecular mechanics calculation was carried out for Glyburide & RepaglinideGlyburide:Glyburide:The molecule structure file contains 33 atoms, 35 bonds, and 244 connectors. Van der Waals interactions between atoms separated by greater than 9.00A are excluded. Optimization continues until the energy change is less than 0.00100000 kcal/mol, or until the molecule has been updated 300 times. The augmented force field is used for the bond stretch, bond angle, dihedral angle and improper torsion interactions. 3 organic ring(s) found in system, 2 ring(s) are found to be aromatic. The energy of the initial structure was 157.3633 kcal/mol. The energy of the final structure was 22.3486 kcal/mol.

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Repaglinide:The molecule structure file contains 33 atoms, 35 bonds, and 243 connectors Vander Waals interactions between atoms separated by greater than 9.00A will be excluded. Optimization continues until the energy change was less than 0.00100000kcal/mol, or until the molecule has been updated 300 times. The augmented force field was used for the bond stretch, bond angle, dihedral angle and improper torsion interactions. 3 organic ring(s) found in system, 1 ring(s) are found to be aromatic The energy of the initial structure was 75.9242 kcal/mol. The energy of the final structure was 16.0877 kcal/mol. 21

Energy Minimization Algorithms Displaying Energy States of Five Molecules before and after Minimization Steps Using Conjugate Gradient Method :

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Why minimization important :I. Comparison of structures/properties II. Template forcing III. Systematic mapping of E spaceIV. Binding energiesV. Docking VI. Harmonic analysis VII. Comparing/Fitting force fields.

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References:I. Thomas L. Lemke , David A. Williams , Victoria F. Roche , S. William Iito, Foye’s Principles of MEDICINAL CHEMISTRY , 6th ed., Wolters kluwer (India) Pvt. Ltd., New Delhi ., 2008 , pp.62-63.II. Andrew R. Leach , Molecular Modelling Principles & Application, Second edition ., Pearson Education Limited, 2001, pp. 253-279.III. John h. Block & John M. Beale , Wilson & Gisvold’s Textbook of Organic Medicinal & Pharmaceutical Chemistry , 11th ed. , Lippincott Williams & Wilkins , 2004 , pp.929-930.IV. Thomas J. Perun, C.L. Propst , Computer Aided Drug Design method & Application , Marcel Dekker , Inc ., New York , 1983 , pp.70-72.V. B. J. Jaidhn , P. Srinivasa Rao & Allam Apparo , Energy Minimization & Conformation Analysis of Molecule using Conjugate Gradient Method , IJRET , Vol.2, May 2014, pp.111-116 .VI. C.Stan Tsai , Introduction to Computational Biochemistry, Wiley- Liss , 2002, pp.285-288 ,291,292.

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