Energy minimization method & comparison between global minima and bioactive coformationsajna.pdf

Energy minimization is a core computational chemistry technique used to find the most stable arrangement of atoms in a molecule by systematically reducing the system's potential energy. A molecule's total energy arises from bonded interactions (like bond stretching and torsional strain) and non-bonded interactions (van der Waals and electrostatics). Because molecules naturally seek the lowest energy state for maximum stability, minimization corrects unfavorable structural geometry, such as distorted bond lengths or angles that arise during model building, until it reaches an optimum structure where the net atomic force approaches zero.

Potential Energy Surface and Algorithms
The molecule's energy is mapped on the Potential Energy Surface (PES), where valleys represent stable conformations. The lowest point in any valley is an energy minimum (Local Minimum), and the lowest point overall is the Global Minimum (the most stable conformation). A Transition State or saddle point is a high-energy structure separating minima.

Minimization algorithms navigate the PES, categorized into first- and second-order methods. First-order methods use the gradient (force) to guide the search. Steepest Descent (SD) is simple and robust but slow, always moving in the direction of the negative gradient. Conjugate Gradient (CG) is faster, using past steps to choose a search direction that avoids zigzagging. Second-order methods use both the gradient and the Hessian matrix (curvature). Newton-Raphson (NR) is the fastest near the minimum but computationally expensive due to the Hessian calculation. Quasi-Newton Methods (QN), like BFGS, balance speed and cost by iteratively approximating the Hessian.

Comparison between global minimum and bioactive conformation
Energy minimization is a crucial preliminary step for molecular simulations and for systematic Conformational Search. A distinction is made between the most stable structure in isolation, the Global Minimum Conformation, and the Bioactive Conformation, which is the specific shape the molecule adopts when bound to its biological target (e.g., a receptor). While the global minimum is thermodynamically stable, the bioactive conformation's stability is heavily influenced by the energy gained from the binding interaction, and it defines the molecule's activity. These conformations are found through computational methods like Molecular Docking and experimental techniques.

1. Presented by,
Sajna K S
2
nd
Semester M.Pharm (2024 Batch)
Department of Pharmaceutical Chemistry
St. James College of Pharmaceutical Sciences, Chalakudy
1
ENERGY MINIMIZATION
METHOD

2. INTRODUCTION
Every molecule has potential energy arising from
1.bonded interactions (bond stretching, angle bending, torsional
strain) and
2.non-bonded interactions (van der Waals forces, electrostatics).
The spatial arrangement of atoms determines this potential energy
•The molecule naturally tends toward the arrangement
with the least energy.
Conformation is theshape adopted
by a molecule.
LOW
ENERGY
MORE
STABILITY
2
Molecular
modelling
Generates 3D
structure
Energy
minimization

3. Energy minimization is a process used in computational
chemistry and molecular modeling to find the most
stable arrangement of atoms in a molecule by reducing
the system's potential energy.
While building a molecule, process may produce
unfavorable bond length, bond angles, etc. The energy
minimization corrects this variation, obtain an optimum
structure with least energy
3
Find an arrangement net atomic force 0
Apomorphine

4. POTENTIAL
ENERGY SURFACE
•An energy surface (also
called potential energy
surface) is like a map that
shows how the energy of a
molecule changes when the
positions of its atoms
change.
• It connects geometry
(atomic positions) with
energy.
Goal of energy
minimization
4

5. TYPES OF ENERGY
MINIMIZATION
METHOD
ENERGY
MINIMIZATION
METHOD
NON
DERIVATIVE
SIMPLEX
METHOD
THE SEQUENTIAL
UNIVARIATE
METHOD
DERIVATIVE
METHODS
FIRST
DERIVATIVE
CONJUGATE
GRADIENT
METHOD
STEEPEST
DESCENTS
METHOD
LINEAR SEARCH
IN ONE
DIMENSION
ARBITRARY
SEARCH
METHOD
SECOND
DERIVATIVE
NEWTON
RAPHSON
METHOD
QUASI
NEWTON
METHOD
5

6. •A derivative tells you how fast
something is changing at a specific
moment.
•If you draw a curve showing something
changing over time:The derivative is
the slope of the curve at any point.
•Steep slope = fast change
•Flat slope = no change
DERIVATIVES
Fast
change
Slow
change
6

7. DERIVATIVES
The height of the hill
represents the energy of
the molecule.
High energy
Low
energy
The derivative tells
us the slope of the
hill at any point.
If the slope is
steep, the ball rolls
fast.
If the slope is flat
(derivative = 0), the
ball might be at the
bottom of a valley —
a minimum energy
point.
7

8. A.NON DERIVATIVE ENERGY MINIMIZATION
METHOD
These methods do not use
derivatives (slope
information).
Instead, they search for the
minimum by trial and error
– checking energies at
different positions.
Useful when it’s hard to
calculate derivatives (like in
complex systems).
Slower than derivative
methods but easier to
apply.
8

9. 1.THE SIMPLEX METHOD
SIMPLEX
A geometrical figure with M+1 points for M
variables
2 variables- Triangle
3 variables- Pyramid
Variables
describes the
position
Vertex = a possible
solution (a set of variable
values/atom positions)
represented as a corner of
the simplex.
9
y
x
(x,y)
y
x
(x,y,z)
z
The shape of the
simplex depends
on no of variables

10. 1.THE SIMPLEX
METHOD
•The Simplex Method is like an
amoeba-shaped figure (triangle,
pyramid, etc.) that crawls over
the energy surface.
•It flips, stretches, or shrinks
until it finds the lowest energy
point.
10

11. 1.THE SIMPLEX METHOD
REFLECTION
•Flip the worst
(highest energy)
point to the
opposite side → try
to find a lower
energy point.
REFLECTION AND
EXPANSION
•If reflection gives a
good point, stretch
further in that
direction to see if it
gets even lower.
CONTRACTION
•shrink (contract)
the simplex
towards better
points
CONTRACTION IN
ALL DIRECTION
•contractions
occur in all the
directions
towards the
lowest point.
TYPES OF MOVEMENTS
11

12. ADVANTAGES
•When the starting configuration of the
system is having high energy, it is best to
use simplex method
•Robust, good for early minimization
DISADVANTAGES
•requires large computational time for the
analysis of the high number of energy
instances.
•Usually combined with faster methods later
12

13. 2.The sequential univariate method
So, in this method, We try
to minimize energy by
adjusting one variable at a
time, while keeping all the
others fixed.
The Sequential Univariate
Method searches for the
minimum one direction at a
time, instead of moving in
all directions together.
one after another, step by step. one variable at a time.
13

14. Steps:
•First, change x
while keeping y
constant → checkif
energy decreases.
•Then, change y
while keeping x
constant → check
again.
•Repeat step by step
until you reach a
minimum energy.
14

15. B. DERIVATIVE
METHODS
•Derivative methods =
slope-based methods.
•They are faster and
more accurate than
non-derivative
methods.
15

16. (a) FIRST ORDER MINIMIZATION
METHOD
•It shows the direction in which energy decreases and how steep
the energy surface at the point
•In these methods coordinates of the atoms are altered step by
step with respect to their movement towards minimum point.
first derivative = slope = rate of change.
The direction of the first
derivative of the energy (the
gradient) points where the
minimum lies and the
magnitude of the gradient
tells about the steepness of
the local slope.
16

17. 1.THE STEEPEST DESCENTS METHOD
•These method is used to reduce the energy of a molecule by
moving in the direction of the steepest downhill path
17

18. 1.THE STEEPEST DESCENTS METHOD
2 steps:
1.The direction of the movement should be identified. The direction
is guided by net force given as sk
�??????=
−????????????
|????????????|
????????????=�??????��???????????? ??????�??????�??????�??????�
2. How much distance shou ld be moved along the gradient
Identification of
minimum point is by
Line search in one
dimension
Arbitrary search
method 18

19. 1.THE STEEPEST DESCENTS METHOD

Starting point- High energy state
Contour lines
showing levels of
constant energy
Lowest energy
point
A
B
The atoms moves with
direction of force –first
it reaches a minimum
and then increase
19

20. (a)LINE SEARCH IN ONE DIMENSION
•Line search finds the minimum energy along a specific direction
STEP 1
•BRACKETING THE MINIMUM
•This means Choosing 3 points along the line where the middle point has lower energy than the 2 ends
STEP 2
•NARROWING THE MINIMUM
•Reduce the distance between 3 points limits minimum to a smaller space
STEP 3
•USING A QUADRATIC FUNCTION
•The 3 points are set with a suitable quadratic function
STEP 4
•APPLY DIFFERENTIATION TO THE EQUATION AND FIND MINIMUM
20

21. 2 .LINE SEARCH IN
ONE DIMENSION
The algorithm searches along the
line and not the whole space and
hence called line search and
finding the minimum
DISADVANTAGES:
•Requires lot of calculations
•Computationally expensive
Search
direction
coordinates
A
C
B
Energy
position
A B C
21

22. (b)ARBITRARY SEARCH APPROACH
Random
This methods avoids expensive calculation of line search by taking random size in the direction of gradient(steepest
slope).
•A Step is taken in the direction of gradient with fixed size.
•If energy decreases, the next step is made larger.
•If energy increases, the step was too big , so the size is reduced.
The new set of coordinates after step k is given by the equation:
�
??????+1=�
??????+⋋
????????????
??????
Where ??????
??????=Gradient/ force unit vector
k= Step number
⋋
??????= Step size
22

23. (b)ARBITRARY SEARCH APPROACH
Each step is taken
and energy is
measured
The size of step is
decided according
to the nature of
energy
Advantages:
•Computational time is less
•Needs lesser functional analysis 23

24. 2.CONJUGATE GRADIENTS MINIMIZATION
Related
change
Improved version of
steepest descents method
This methods uses special direction called conjugate direction
instead of steepest slope
These directions don’t interfere with each other and they help
reach the minimum faster
Also the slopes at each step are perpendicular (orthogonal) to
previous ones
24

25. Follow the
direction of
gradient
•Step 1
Each new direction is
calculated using current
gradient and previous
direction
•Step 2
Reaches
minimum
faster
•Step 3
2.CONJUGATE GRADIENTS MINIMIZATION
Conjugate
gradient
Steepest
descents
method
25

26. (B) SECOND
DERIVATIVE
MINIMIZATION
METHOD
The second derivative tells us the
curvature of the energy surface.
It helps to find points where
the energy stops increasing
or decreasing, like minima,
maxima, or saddle points.
26

27. THE NEWTON RAPHSON METHOD
It is an iterative algorithm that updates atomic positions by considering the slope
(gradient) and curvature (second derivative) of the potential energy surface
allowing the system to reach minimum faster
Second derivative provides
curve informationMoves faster
and reaches
minimum
faster
Reaches minimum slowly
27

28. THE NEWTON RAPHSON METHOD
•This method is best for small molecules.
•For quadratic functions this finds minimum in just one step.
•This requires calculation and inversion of hessian matrix at every step.
??????x=��
2
+��+�
Hessian matrix is a square function that contains all the
second order partial derivatives of a function. It
describes curvature of a function
Negative value of Hessian indicates high
energy or saddle points
Hessian must have positive to have
lowest energy and stable
If position selected is far from minimum a more efficient method
should be used before application of this method
28

29. 2.QUASI NEWTON METHOD
calculating the Hessian for big molecules is too
expensive (needs lots of math).
The Quasi-Newton method is like Newton–Raphson, but instead of calculating the exact
Hessian (curvature), it guesses and updates an approximate Hessian at each step using the
gradient information.
In this method we keep updating a
matrix called Hk which is an
approximation of inverse Hessian
29

30. 1. Start with
initial guess
+
approximate
Hessian.
2. Compute
gradient
(slope).
3. Update
approximate
Hessian from
gradient
changes.
4. Compute
new search
direction.
5. Update
coordinates.
6. Repeat
until
minimum is
found.
2.QUASI NEWTON METHOD
STEPS INVOLVED
ADVANTAGES
1.Faster than steepest descent
& conjugate gradient.
2.Much cheaper than full
Newton–Raphson.
3.Works well for large
systems.
DISADVANTAGES
1.Approximation may be
imperfect.
2.Can still be costly compared
to steepest descent.
3.Slower to start but excellent
near minimum.
30

31. EXAMPLE
❖Energy minimization by using steepest
descents method in Glyburide,
Repaglinide,Pioglitazone,Nateglinide
•Jaidhan et.al., conducted study on 5 compounds
using a CaChe 6.1.12 software.
•In order to perform analysis on a set of molecules,
rotatable bond counts were made for each
molecule.
•Number of freely rotatable bonds are counted
using CaChe .
31
Energy terms for the
following interactions
are included:
1. Bond Stretch
2. Bond Angle
3.Dihedral Angle
4.Torsion Stretch
5.Torsion Bend
6.Van Der Waals
7. Electrostatics
8. Hydrogen Bond

32. Pioglitazone: Number of rotatable bonds – 6
EXAMPLE
32
PIOGLITAZONE
•Steepest descent search was used to locate the energy minimum.
•All atoms are moved at once during minimization.
•Optimization continues until the energy change was less than 0.00100000 kcal/mol
Number of rotatable bonds = 6
Potential energy map showing local minimum with its
structure
Potential energy map showing saddle point with its
structure

33. EXAMPLE
33
Potential energy map showing local minimum with its
structure
Potential energy map showing saddle point with its
structure
NATEGLINIDE

34. COMPARISON BETWEEN
GLOBAL MINIMUM AND
BIOACTIVE
CONFORMATION
34

35. GLOBAL MINIMUM
LOCAL MINIMUM
GLOBAL MINIMUM AND LOCAL MINIMUM
•The global minimum on the PES corresponds with
the lowest possible energy for the molecule
•Represents thermodynamically most stable
conformation
•These are points where the molecule in a relatively
stable conformation
•Energy is not as low as in global minimum
•Important in drug design- represent meta stable
conformation that may adopt in biological
systems.
35

36. BIOACTIVE CONFORMATION
•It refers to 3D shape that
aligand(drug) assumes when
interacting with target protein.
•The structure mostly differs from
conformation of global minimum
36

37. COMPARISON BETWEEN GLOBAL MINIMUM AND
BIOACTIVE CONFORMATION
Feature Global minimum conformation Bioactive conformation
Definition The lowest energy conformation The conformation adopted by a
ligand while binding to a biological
target
Energy level Absolute lowest potential energyNot lowest. Higher than global
minimum
Determination methods Conformational search, Energy
minimization using force fields or
quantum mechanics
Experimental: Xray crystallography,
NMR Spectroscopy.
Computational: Molecular docking
37

38. COMPARISON BETWEEN GLOBAL MINIMUM AND BIOACTIVE
CONFORMATION
Feature Global minimum conformation Bioactive conformation
Relevance Represents most stable
conformation
Most relevant for drug- target
interaction.
Biological activity May not be biologically activeDefines biological activity.
Challenges Hard to find Computationally Depends on flexibility and target
interactions.
Stability High stability Stability influenced by target
binding
38

39. REFERENCES
1.Gautam B. Energy Minimization [Internet]. Homology Molecular Modeling - Perspectives and
Applications. IntechOpen.2021;1-14
2.Jaidhan BJ. Srinivas RP. Apparao A. Energy minimization and conformation analysis of molecules
using steepest descent method. International Journal of computer science information
technologies.2014,Vol 15[3] ;3525- 3528
3.Leach A R.Energy minimization and related methods for exploring the energy surface. Molecular
modelling: Principles and applications, 2
nd
edition,Harlow,England, Pearson prentice hall:
2001.Chapter 5,253-301.
4.Baker J.An algorithm for the location of transition states. Journal of computational chemistry;
1986 ,385-395
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40. THANKYOU
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