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Energy Minimization
 -Non-Derivative Methods
 -First Derivative Methods



            Pinky Sheetal V
              1801110004
         M.Tech Bioinformatics




Background Image Courtesy: 3dciencia.com visual life sciences
Contents
•   Introduction
•   Criteria to start minimization
•   Energy Minimization –The Problem
•   Non Derivative Methods
    • Simplex Method
    • Sequential univariate method
• First Derivative Methods
    • Steepest Gradient
    • Conjugate Gradient Method
• References
Introduction

• The lowest energy conformation is the set of bond lengths and angles that
  gives the smallest steric energy.

• In other words, bonds find a compromise among competing forces to
  determine the lowest energy conformation.

• The goal of molecular mechanics is to determine the lowest energy
  conformation of a molecule. The process is called energy minimization
  [1]


• Knowing the stable conformers of a molecule is important because it
  allows us to understand its properties and behavior based on its structure
  [2]
After a number of
                                     steps, a local or
                                     global minimum on
                                     the potential energy
                                     surface is reached




General Formula:                     xnew - the value of the geometry at the next step
                                     Xold - refers to the geometry at the current step
                                     correction - some adjustment made to the
                                     geometry

In all these methods, a numerical test is applied to the new geometry to decide if
a minimum is reached. For example, the slope may be tested to see if it is zero
within some numerical tolerance. If the criterion is not met, then the formula is
applied again to make another change in the geometry.                     [2]
Criteria to start minimization
1. Starting set of atomic coordinates
2. Parameters for various terms of the potential energy function
3. Description of molecular topology [3]
                Energy Minimization –The Problem
– E = f(x)
– E - function of coordinates Cartesian /internal
– At minimum the first derivatives are zero and
  the second derivatives are all positive

– Derivatives of the energy with respect to the coordinates provide
  information about the shape of energy surface and also enhances the
  efficiency of the minimization. [5]
– A Hessian matrix is a square matrix of second derivatives of a
  function. [4]
Non Derivative Methods
– Require energy evaluation only and may require many energy evaluations
–– Simplex Method
    Simplex is a geometrical figure with M+1 interconnected vertices, where M is the dimensionality
     of the energy function
    It does not rely on the calculation of the gradients at all.
    As a result, it is the least expensive in CPU time per step.
    However, it also often requires the most steps.




                                                                                            [5]
• To implement the simplex algorithm it is first necessary to generate the vertices of the
  initial simplex.
• The initial configuration of the system corresponds to just one of these vertices. The
  remaining points can be obtained by adding a constant increment to each coordinate in
  turn. The energy of the system is calculated at the new point, giving the function value for
  the relevant vertex. [5]

• The point with the highest energy of the three is noted. Then, this point is reflected through
  the line segment connected the other two (to move away from the region of high energy).
• If the energy of point A is the highest out of the three points A, B and C, then A is reflected
  through line segment BC to produce point D.
• In the next step, the two original lowest energy points (B and C) along with the new point D
  are analyzed. The highest energy point of these is selected, and that point is reflected
  through the line segment connecting the other two. The process continues until a minimum
  is located. [2]
• The sequential univariate method is a non-derivative method that is
  considered more appropriate as it systematically cycles through the
  coordinates in turn
• For each coordinate, two new structures are generated by changing
  the current coordinate.
• The energies of these two structure are calculated.
• A parabola is then fitted through the 3 points corresponding to the
  two distorted structures and the original structure.
• The minimum point in this quadratic function is determined and the
  coordinate is then changed to the position of the minimum      [5]
Derivative Methods
• The direction of the first derivative of the energy (the gradient) indicates where
  the minimum lies, and the magnitude of the gradient indicates the steepness of
  the local slope.
• The energy of the system can be lowered by moving each atom in response to the
  force acting on it; the force is equal to minus the gradient.

• Second derivatives indicate the curvature of the function, information that can be
  used to predict where the function will change direction ( pass through a
  minimum or some other stationary point)

• When discussing derivative methods it is useful to write the function[5] a Taylor
                                                                        as
  series expansion about the point Xk



• Therefore first order methods use the first derivatives (gradients), second order
  uses both first and second derivative. The simplex method can thus be considered
  as a zeroth-order method as it does not use any derivatives.
Steepest Descent
• In the steepest descents method, the Hessian is just approximated as a
  constant, γ


• γ as an effective force constant and is calculated at the beginning of
  the first step to give a specified step size. [1]

       γ is a constant


• In this method, the gradients at each point still must be calculated, but
  by not requiring second derivatives to be calculated.

• The method is much faster per step than the Newton-Raphson method.

• However, because of the approximation, it is not as efficient and so
  more steps are generally required to find the minimum. [2]
Conjugate Gradient Method
• Conjugate gradient is a variation of the steepest descents method. The calculation
  of the gradient is improved by using information from previous steps. [1]
• The gradients of the current geometry are first computed. The direction of the
  largest gradient is determined after the gradient of current geometry is
  computed.
• The geometry is minimized along this one direction (this is called a line search).
  Then, a direction orthogonal to the first one is selected (a “conjugate” direction).
  The geometry is minimized along this direction.
• This continues until the geometry is optimized in all the directions.
                                       • Pure steepest descents algorithms always take 90°
                                         turns after each stage, which may move the first
                                         minimization stage away from the optimal value.
                                       • If the initial steepest descents finds the minimum
                                         along the initial direction. The next best direction to
                                         look for the overall minimum is not necessarily
                                         perpendicular to the initial path.
                                       • Conjugate gradients is a good general purpose
                                         technique. [2]
Figure: potential energy surface for
the two variables
References
1.   https://www.uzh.ch/oci/efiles/OCPkb/EnergyMinNotes.pdf
2.   http://www.coursehero.com/file/2883095/38037emin/
3.   http://ocikbws.uzh.ch/education/molecular_science/mm/minimization.html
4.   http://www.gromacs.org/Documentation/Terminology/
5.   A R Leach, Molecular Modelling Principles and Applications, Pearson Education
     Second Edition
6.   http://i572-molecular-modeling.wikispaces.com/file/view/opt_lecture.pdf
Thank you

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Energy minimization

  • 1. Energy Minimization -Non-Derivative Methods -First Derivative Methods Pinky Sheetal V 1801110004 M.Tech Bioinformatics Background Image Courtesy: 3dciencia.com visual life sciences
  • 2. Contents • Introduction • Criteria to start minimization • Energy Minimization –The Problem • Non Derivative Methods • Simplex Method • Sequential univariate method • First Derivative Methods • Steepest Gradient • Conjugate Gradient Method • References
  • 3. Introduction • The lowest energy conformation is the set of bond lengths and angles that gives the smallest steric energy. • In other words, bonds find a compromise among competing forces to determine the lowest energy conformation. • The goal of molecular mechanics is to determine the lowest energy conformation of a molecule. The process is called energy minimization [1] • Knowing the stable conformers of a molecule is important because it allows us to understand its properties and behavior based on its structure [2]
  • 4. After a number of steps, a local or global minimum on the potential energy surface is reached General Formula: xnew - the value of the geometry at the next step Xold - refers to the geometry at the current step correction - some adjustment made to the geometry In all these methods, a numerical test is applied to the new geometry to decide if a minimum is reached. For example, the slope may be tested to see if it is zero within some numerical tolerance. If the criterion is not met, then the formula is applied again to make another change in the geometry. [2]
  • 5. Criteria to start minimization 1. Starting set of atomic coordinates 2. Parameters for various terms of the potential energy function 3. Description of molecular topology [3] Energy Minimization –The Problem – E = f(x) – E - function of coordinates Cartesian /internal – At minimum the first derivatives are zero and the second derivatives are all positive – Derivatives of the energy with respect to the coordinates provide information about the shape of energy surface and also enhances the efficiency of the minimization. [5] – A Hessian matrix is a square matrix of second derivatives of a function. [4]
  • 6. Non Derivative Methods – Require energy evaluation only and may require many energy evaluations –– Simplex Method  Simplex is a geometrical figure with M+1 interconnected vertices, where M is the dimensionality of the energy function  It does not rely on the calculation of the gradients at all.  As a result, it is the least expensive in CPU time per step.  However, it also often requires the most steps. [5]
  • 7. • To implement the simplex algorithm it is first necessary to generate the vertices of the initial simplex. • The initial configuration of the system corresponds to just one of these vertices. The remaining points can be obtained by adding a constant increment to each coordinate in turn. The energy of the system is calculated at the new point, giving the function value for the relevant vertex. [5] • The point with the highest energy of the three is noted. Then, this point is reflected through the line segment connected the other two (to move away from the region of high energy). • If the energy of point A is the highest out of the three points A, B and C, then A is reflected through line segment BC to produce point D. • In the next step, the two original lowest energy points (B and C) along with the new point D are analyzed. The highest energy point of these is selected, and that point is reflected through the line segment connecting the other two. The process continues until a minimum is located. [2]
  • 8. • The sequential univariate method is a non-derivative method that is considered more appropriate as it systematically cycles through the coordinates in turn • For each coordinate, two new structures are generated by changing the current coordinate. • The energies of these two structure are calculated. • A parabola is then fitted through the 3 points corresponding to the two distorted structures and the original structure. • The minimum point in this quadratic function is determined and the coordinate is then changed to the position of the minimum [5]
  • 9. Derivative Methods • The direction of the first derivative of the energy (the gradient) indicates where the minimum lies, and the magnitude of the gradient indicates the steepness of the local slope. • The energy of the system can be lowered by moving each atom in response to the force acting on it; the force is equal to minus the gradient. • Second derivatives indicate the curvature of the function, information that can be used to predict where the function will change direction ( pass through a minimum or some other stationary point) • When discussing derivative methods it is useful to write the function[5] a Taylor as series expansion about the point Xk • Therefore first order methods use the first derivatives (gradients), second order uses both first and second derivative. The simplex method can thus be considered as a zeroth-order method as it does not use any derivatives.
  • 10. Steepest Descent • In the steepest descents method, the Hessian is just approximated as a constant, γ • γ as an effective force constant and is calculated at the beginning of the first step to give a specified step size. [1] γ is a constant • In this method, the gradients at each point still must be calculated, but by not requiring second derivatives to be calculated. • The method is much faster per step than the Newton-Raphson method. • However, because of the approximation, it is not as efficient and so more steps are generally required to find the minimum. [2]
  • 11. Conjugate Gradient Method • Conjugate gradient is a variation of the steepest descents method. The calculation of the gradient is improved by using information from previous steps. [1] • The gradients of the current geometry are first computed. The direction of the largest gradient is determined after the gradient of current geometry is computed. • The geometry is minimized along this one direction (this is called a line search). Then, a direction orthogonal to the first one is selected (a “conjugate” direction). The geometry is minimized along this direction. • This continues until the geometry is optimized in all the directions. • Pure steepest descents algorithms always take 90° turns after each stage, which may move the first minimization stage away from the optimal value. • If the initial steepest descents finds the minimum along the initial direction. The next best direction to look for the overall minimum is not necessarily perpendicular to the initial path. • Conjugate gradients is a good general purpose technique. [2] Figure: potential energy surface for the two variables
  • 12. References 1. https://www.uzh.ch/oci/efiles/OCPkb/EnergyMinNotes.pdf 2. http://www.coursehero.com/file/2883095/38037emin/ 3. http://ocikbws.uzh.ch/education/molecular_science/mm/minimization.html 4. http://www.gromacs.org/Documentation/Terminology/ 5. A R Leach, Molecular Modelling Principles and Applications, Pearson Education Second Edition 6. http://i572-molecular-modeling.wikispaces.com/file/view/opt_lecture.pdf