Theory ofMolecular Modelling

Hardik MistryPharmaceutical ChemistryL. M. College of Pharmacy

Target Identification &Validation

HitIdentification

LeadIdentification

LeadOptimisa-tion

CDPrenomi-nation

ConceptTesting

Development for launch

LaunchPhase

FDA Submission

Launch

Finding Potential Drug Targets

Validating Therapeutic Targets

Finding Potential Drugs

Drug<>Target<>Therapeutic Effect Association Finalized

Testing in Man (toxicity and efficacy)

Drug Discovery is a goal of research. Methods and approaches from different science areas can be applied to achieve the goal.

The Drug Discovery pipeline

2

Hugo Kubinyi , www.kubinyi.de3

R&D cost per new drug is $500 to $700 millions

To sustain growth, each of top 20 pharma company should produce more new drugs

Currently, total industry produces only 32 new drugs per year.

Current rate of NDAs far below than required for sustained growth.

The Drug Discovery : Current Status

4

Atomistic Continuum

Finite Periodic

Quantum Mechanical Methodies Classical Methods

Semi-Empirical Ab Initio Deterministic Stochastic

QM/MMQuantum MC DFTMolecular Dynamics

Monte CarloHartree Fock

Computational Tools

5

Theoretical hierarchy

Model vs Scale> 10-4 m

10-6 m

10-8 m

10-10 m

Macroscale

Microscale

Nanoscale

Atomic Scale

Classical Mechanics

Organism

Cell

Protein, membrane

Small molecules, drug

Continuum Mechanics• Finite Element Method• Fluid Dynamics

Statistical Mechanics• Molecular Mechanics• Molecular Dynamics• Brownian Dynamics• Stochastic Dynamics

Quantum Mechanics•Density Functional Theory• Hartree-Fock Theory• Perturbation Theory

• Structural Mechanics

6

Receptor

Ligand

Unknown

Known

Unknown Known

Generate 3D structures,

HTS, Comb. ChemBuild the lock and then

find the key

Molecular Docking

Drug receptor interaction

2D/3D QSAR and PharmacophoreInfer the lock by

expecting key

De NOVO Design , Virtual screeningBuild or find the key

that fits the lock

Receptor based drug design

Rational drug designIndirect drug design

Homology modelling

Basic Modeling Strategies

7

• Molecular modelling allow scientists to use computers to visualize molecules means representing molecular structures numerically and simulating their behavior with the equations of quantum and classical physics , to discover new lead compounds for drugs or to refine existing drugs in silico.

• Goal : To develop a sufficient accurate

model of the system so that physical experiment may not be necessary .

Definition

8

• The term “ Molecular modeling “expanded over the last decades from a tool to visualize three-dimensional structures and to simulate , predict and analyze the properties and the behavior of the molecules on an atomic level to data mining and platform to organize many compounds and their properties into database and to perform virtual drug screening via 3D database screening for novel drug compounds .

9

Molecular modeling starts from structure determinationSelection of calculation methods in computational chemistry

Starting geometry from standard geometry, x-ray, etc.

Molecule

Molecularmechanics

Quantummechanics

Moleculardynamics or Monte Carlo

Is bond formation or breaking important?

Are many force fieldparameters missing ?

Is it smaller than 100 atoms?

Are chargesof interest ?

Are there many closelyspaced conformers?

Is plenty of computertime available?

Is the free energyNeeded ?

Is solvationImportant ?

10

Sticks

11

Ball and Stick

Space Filling (CPK)

Wire Frame

Molecular

Graphics

• Molecular modelling or more generally computational chemistry is the scientific field of simulation of molecular systems.

• Basically in the computational chemistry , the free energy of the system can be used to assess many interesting aspects of the system.

• In the drug design , the free energy may be used to assess whether a modification to a drug increase or decrease target binding.

• The energy of the system is a function of the type and number of atoms and their positions.

• Molecular modelling softwares are designed to calculate this efficiently.

Computational Chemistry approaches

12

• The energy of the molecules play important role in the computational chemistry. If an algorithm can estimate the energy of the system, then many important properties may be derived from it.

• On today's computer , however energy calculation takes days or months even for simple system. So in practice, various approximations must be introduced that reduce the calculations time while adding acceptably small effect on the result.

Molecular behavior : Computing energy

13

• Example :• Familiar conformation of the Butane

76543210

0 60 120 180 240 300 360

0.30.250.20.150.100.10.050

C

B

D

E

F

Pot

entia

l ene

rgy

Dihedral angle

Probability

Need of molecular Modelling ?????

14

Quantum mechanicsMolecular mechanics

Ab initio methods DFT methodSemiimpirical methods

Molecular Modelling

15

• Quantum mechanics is basically the molecular orbital calculation and offers the most detailed description of a molecule’s chemical behavior.

• HOMO – highest energy occupied molecular orbital

• LUMO – lowest energy unoccupied molecular orbital

• Quantum methods utilize the principles of particle physics to examine structure as a function of electron distribution.

• Geometries and properties for transition state and excited state can only be calculated with Quantum mechanics.

• Their use can be extended to the analysis of molecules as yet unsynthesized and chemical species which are difficult (or impossible) to isolate.

Quantum Mechanics

16

• Quantum mechanics is based on Schrödinger equation

HΨ = EΨ = (U + K ) Ψ

E = energy of the system relative to one in which all atomic particles are separated to infinite distances

H = Hamiltonian for the system . It is an “operator” ,a mathematical

construct that operates on the molecular orbital , Ψ ,to determine the energy.

U = potential energy K = kinetic energy Ψ = wave function describes the electron

distribution around the molecule. 17

The Hamiltonian operator H is, in general,

Where Vi2 is the Laplacian operator acting on particle i. Particles are both electrons and nuclei. The symbols mi and qi are the mass and charge of particle I, and rij is the distance between particles.

The first term gives the kinetic energy of the particle within a wave formulation. The second term is the energy due to Coulombic attraction or repulsion of particles .

18

• In currently available software, the Hamiltonian above is nearly never used.

• The problem can be simplified by separating the nuclear and electron motions.

kinetic energy of electrons

Attraction of electrons to

nuclei

Repulsion between electrons

Born-Oppenheimer approximation

19

• Thus, each electronic structure calculation is performed for a fixed nuclear configuration, and therefore the positions of all atoms must be specified in an input file.

• The ab initio program like MOLPRO then computes the electronic energy by solving the electronic Schrödinger equation for this fixed nuclear configuration.

• The electronic energy as function of the 3N-6 internal nuclear degrees of freedom defines the potential energy surface (PES) which is in general very complicated and can have many minima and saddle points.

• The minima correspond to equilibrium structures of different isomers or molecules, and saddle points to transition states between them.

20

• The term ab initio is Latin for “from the beginning” premises of quantum theory.

• This is an approximate quantum mechanical calculation for a function or finding an approximate solution to a differential equation.

• In its purest form, quantum theory uses well known physical constants such as the velocity of light , values for the masses and charges of nuclear particles and differential equations to directly calculate molecular properties and geometries. This formalism is referred to as ab initio (from first principles) quantum mechanics.

Ab initio methods

21

HARTREE±FOCK APPROXIMATION

• The most common type of ab initio calculation in which the primary approximation is the central field approximation means Coulombic electron-electron repulsion is taken into account by integratinfg the repulsion term.

• This is a variational calculation, meaning that the approximate energies calculated are all equal to or greater than the exact energy.

• The energies are calculated in units called Hartrees (1 Hartree . 27.2116 eV) 22

• The steps in a Hartreefock calculation start with an initial guess for the orbital coefficients ,usually using a semiempirical method.

• This function is used to calculate an energy and a new set of orbital coefficients, which can then be used to obtain a new set ,and so on.

• This procedure continues iteratively untill the energies and orbital coefficient remains constant from one iteration to the next.

• This iterative procedure is called as a Self-consistent field procedure (SCF).

23

Advantage• Advantages of this method is that it breaks the

many-electron Schrodinger equation into many simpler one-electron equations.

• Each one electron equation is solved to yield a single-electron wave function, called an orbital, and an energy, called an orbital energy.

24

Method Advantages Disadvantages Best for

Ab initio Useful for a broad range of

systems

Computationally expensive

Small systems

Mathematically rigorous :no empirical parameters

Does not depend on experimental

data

Electronic transitions

System

Without experimental

data

Calculates transition states

and excited states

System requiring high

accuracy

Ab initio methods

25

• What is Density ?

How something(s) is(are) distributed/spread about a given space

Electron density tells us where the electrons are likely to exist.Allyl Cation:

*

Density functional Theory

26

• A function depends on a set of variables. y = f (x) E.g., wave function depend on electron

coordinates.

What is a Functional?

• A functional depends on a functions, which in turn depends on a set of variables.

E = F [ f (x) ] E.g., energy depends on the wave function,

which depends on electron coordinates.

1234

1234

F(X)=Y

27

• The electron density is the square of wave function and integrated over electron coordinates.

• The complexity of a wave function increases as the number of electrons grows up, but the electron density still depends only on 3 coordinates.

x

x

y

r

Density functional Theory

• With this theory, the properties of a many-electron

system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent electron density .

28

• There are difficulties in using density functional theory to properly describe intermolecular interactions, especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other strongly correlated systems .

29

• Density functional theory has its conceptual roots in the Thomas-Fermi model .

• It is developed by Thomas and Fermi in 1927. • They used a statistical model to approximate the

distribution of electrons in an atom. • The mathematical basis postulated that electrons

are distributed uniformly in phase space with two electrons in every h3 of volume.

• For each element of coordinate space volume d3r we can fill out a sphere of momentum space up to the Fermi momentum pf

.

Thomas Fermi model

30

• Equating the number of electrons in coordinate space to that in phase space gives

• Solving for pf and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:

where

31

• As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density).

• The Thomas-Fermi equation's accuracy is limited because the resulting kinetic energy functional is only approximate, and because the method does not attempt to represent the exchange energy of an atom as a conclusion of the Pauli principle.

• An exchange energy functional was added by Dirac in 1928 called as the Thomas-Fermi-Dirac model

• However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect of electron correlation.

32

• DFT was originated with a theorem by Hoenburg and Kohn .

• The original H-K theorems held only for non-degenerate ground states in the absence of a magnetic field .

• The first H-K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates.

• It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to only 3 spatial coordinates, through the use of functionals of the electron density.

Hohenberg-Kohn theorems

33

• This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states.

• The second H-K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional.

34

• Within the framework of Kohn-Sham DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential.

• The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions.

kohn-Sham theory

EDFT[r] = T[r] + Ene[r] + J[r] + Exc[r]

Electronic Kinetic energy

Nuclei-electronsCoulombic energy

electrons-electrons Coulombic energy

electrons-electrons Exchange energy

35

• Modeling the latter two interactions becomes the difficulty within KS DFT.

• In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals.

• A determinant is then formed from these functions, called Kohn±Sham orbitals.

• It is the electron density from this determinant of orbitals that is used to compute the energy.

36

Semiempirical molecular orbital methods• So Semiempirical methods are very fast,

applicable to large molecules, and may give qualitative accurate results when applied to molecules that are similar to the molecules used for parameterization.

• Because Semiempirical quantum chemistry avoid two limitations,

namely slow speed and low accuracy, of the Hartree-Fock calculation by omitting or parameterzing certain integrals based on experimental data, such as ionization energies of atoms, or dipole moments of molecules.

• Rather than performing a full analysis on all electrons within the

molecule, some electron interactions are ignored .

37

• Modern semiempirical models are based on the Neglect of Diatomic Differential Overlap (NDDO) method in which the overlap matrix S is replaced by the unit matrix.

• This allows one to replace the Hartree-Fock secular equation |H-ES| = 0 with a simpler equation |H-E|=0.

• Existing semiempirical models differ by the further approximations that are made when evaluating one-and two-electron integrals and by the parameterization philosophy. 38

• Modified Neglect of Diatomic Overlap , MNDO ( by Michael Dewar and Walter Thiel, 1977)

• Austin Model 1, AM1 (by Dewar and co-workers)

• Parametric Method 3, PM3 (by James Stewart)

• PDDG/PM3 (by William Jorgensen and co-workers)

39

• Modified Neglect of Diatomic Overlap , by Michael Dewar and Walter Thiel, 1977

• It is the oldest NDDO-based model that parameterizes one-center two-electron integrals based on spectroscopic data for isolated atoms, and evaluates other two-electron integrals using the idea of multipole-multipole interactions from classical electrostatics.

• A classical MNDO model uses only s and p orbital basis sets while more recent MNDO/d adds d-orbitals that are especially important for the description of hypervalent sulphur species and transition metals.

MNDO

40

Deficiencies

• Inability to describe the hydrogen bond due to a strong intermolecular repulsion.

• The MNDO method is characterized by a generally poor reliability in predicting heats of formation.

• For example: highly substituted stereoisomers are predicted to be too unstable compared to linear isomers due to overestimation of repulsion is sterically crowded systems.

41

• By Michel Dewar and co-workers• Takes a similar approach to MNDO in

approximating two-electron integrals but uses a modified expression for nuclear-nuclear core repulsion.

• The modified expression results in non-physical attractive forces that mimic van der Waals interactions.

• AM1 predicts the heat of the energy more accurately than the MNDO.

• The results of AM1 calculations often are used as the starting points for parameterizations of the force fields in molecular dynamic simulation and CoMFA QSAR.

42

Austin Model 1 , AM1

Some known limitations to AM1 energies • Predicting rotational barriers to be one-third the

actual barrier and predicting five-membered rings to be too stable.

• The predicted heat of formation tends to be inaccurate for molecules with a large amount of charge localization.

• Geometries involving phosphorus are predicted poorly.

• There are systematic errors in alkyl group energies predicting them to be too stable.

• Nitro groups are too positive in energy.• The peroxide bond is too short by about 0.17 A0 .• Hydrogen bonds are predicted to have the

correct strength, but often the wrong orientation.

• So o n average, AM1 predicts energies and geometries better than MNDO, but not as well as PM3 .

43

• By James Stewart • Uses a Hamiltonian that uses nearly the same

equations as the AM1 method along with an improved set of parametersis.

• Limitations of PM3..• PM3 tends to predict that the barrier to rotation

around the C-N bond in peptides is too low. • Bonds between Si and the halide atoms are too

short Proton affinities are not accurate.• Some polycyclic rings are not flat. • The predicted charge on nitrogen is incorrect.• Nonbonded distances are too short..

44

Parametric Methods 3 , PM3

Strength• Overall heats of formation are more accurate

than with MNDO or AM1.• Hypervalent molecules are also predicted more

accurately• PM3 also tends to predict incorrect electronic

states for germanium compounds

• It tends to predict sp3 nitrogen as always being pyramidal.

• Hydrogen bonds are too short by about 0.1AÊ , but the orientation is usually correct .

• On average, PM3 predicts energies and bond lengths more accurately than AM1 or MNDO 45

• By William Jorgensen and co-workers• The Pairwize Distance Directed Gaussian (PDDG)• Use a functional group-specific modification of

the core repulsion function. • Its modification provides good description of the

van der Waals attraction between atoms .• PDDG/PM3 model very accurate for estimation of

heats of formation because of reparameterization .

• But some limitations common to NDDO methods remain in the PDDG/PM3 model: the conformational energies are unreliable, most activation barriers are significantly overestimated, and description of radicals is erratic.

• So far, only C, N, O, H, S, P, Si, and halogens have been parameterized for PDDG/PM3

46

PDDG/ PM3

• Some freely available computational chemistry programs that include many semiempirical models are MOPAC 6, MOPAC 7, and WinMopac .

47

• Computational modeling of structure-activity relationships

• Design of chemical synthesis or process scale-up

• Development and testing of new methodologies and algorithms

• Checking for gross errors in experimental thermochemical data

e.g. heat of formation

• Preliminary optimization of geomteries of unusual molecules and transition states that cannot be optimized with molecular mechanics methods

Usefulness

48

Method Advantages Disadvantages Best for

Semi empirical Less demanding computationally

than ab initio methods

Requires ab initio or

experimental data for

parameters.

Medium-sized systems

(hundreds of atoms)

• Use quantum physics• Uses experimental parameters• Uses extensive approximations

Calculates transition states

and excited states.

Less rigorous than ab initio

methods.

Electronic transitions

System

Semi empirical methods

49

• The Process of finding the minimum of an empirical potential energy function is called as the Molecular mechanics. (MM)

• The process produce a molecule of idealized geometry.

• Molecular mechanics is a mathematical formalism which attempts to reproduce molecular geometries, energies and other features by adjusting bond lengths, bond angles and torsion angles to equilibrium values that are dependent on the hybridization of an atom and its bonding scheme.

Molecular mechanics

50

• Molecular mechanics breaks down pair wise interaction into

√ Bonded interaction ( internal coordination ) - Atoms that are connected via one to three

bonds √ Non bonded interaction . - Electrostatic and Van der waals component

The general form of the force field equation is

E P (X) = E bonded + E nonbonded

51

• Bonded interactions• Used to better approximate the interaction of the

adjacent atoms.• Calculations in the molecular mechanics is

similar to the Newtonians law of classical mechanics and it will calculate geometry as a function of steric energy

• Hooke’s law is applied here• f = kx• f = force on the spring needed to stretch an

ideal spring is proportional to its elongation x ,and where k is the force constant or spring constant of the spring.

52

• Ebonded = Ebond + Eangle + Edihedral

• Bond term Ebond = ½ kb (b – bo) 2

• Angle term EAngle = ½ kθ (θ – θ0)• Energy of the dihedral angles Edihedral = ½ kΦ(1 – cos (nΦ + δ)

53

H

CC

H

H

Graphical representation of the bonded and non bonded interaction and the corresponding energy terms.

E coulomb

Electrostatic attraction

E vdw

Van der waals

Yij

θ0K θ

K b

K Ф

E Ф

Ф 0

E θE b

b0b

Bond stretching

Dihedral rotation

Angle bending

54

• Nearly applied to all pairs of atoms

• The nonbonded interaction terms usually include electrostatic interactions and van der waals interaction , which are expressed as coloumbic interaction as well as

Lennard-Jones type potentials, respectively.

• All of them are a function of the distance between atom pairs , rij .

55

Non bonded interaction

• E Nonbonded = E van der waals + E electrostatic

• E van der waals

• E electrostatic

56

Lennard Jones potential

Coulomb's Law

• The molecular mechanics energy expression consists of a simple algebraic equation for the energy of the compound.

• A set of the equations with their associated constants which are the energy expression is called a force field.

• Such equations describes the various aspects of the equation like stretching, bending, torsions, electronic interactions van der waals forces and hydrogen bonding.

Force Field

57

• Valance term. Terms in the energy expression which describes a single aspects of the molecular shape. Eg., such as bond stretching , angle bending , ring inversion or torsional motions.

• Cross term. Terms in the energy expression which describes how one motion of the molecule affect the motion of the another. Eg., Stretch-bend term which describes how equilibrium bond length tend to shift as bond angles are changed.

• Electrostatic term. force field may or may not include this term. Eg., Coulomb’s law.

58

• Some force fields simplify the complexity of the calculations by omitting most of the hydrogen atoms.

• The parameters describing the each backbone atom are then modified to describe the behavior of the atoms with the attached hydrogens.

• Thus the calculations uses a CH2 group rather than a Sp3

carbon bonded to two hydrogens.• These are called united atom force field or

intrinsic hydrogen methods.

• Some popular force fields are AMBER CHARMM CFF

59

AMBER

• Assisted model building with energy refinement is the name of both a force field and a molecular mechanics program.

• It was parameterized specifically for the protein and nucleic acids.

• It uses only five bonding and nonbonding terms and no any cross term.

60amber.scripps.edu

CHARMM (Harvard University)

• Chemistry at Harvard macromolecular mechanics is the name of both a force field and program incorporating the force field.

• It was originally devised for the proteins and nucleic acids. But now it is applied to the range of the bimolecules , molecular dynamics, solvation , crystal packing , vibrational analysis and QM/MM studies.

• It uses the five valance terms and one of them is an electrostatic term.

61www.charmm.org

• The consistent force field .• It was developed to yield consistent accuracy of

results for conformations , vibrational spectras , strain energy and vibrational enthalpy of proteins.

• There are several variations on this CVFF – consistent valence forcefield UBCFF – Urefi Bradley consistent forcefield LCFF – Lynghy consistent forcefield

• These forcefields use five to six valance terms . One of which is electrostatic and four to six others are Cross terms.

CFF

62

• Molecular mechanics energy minimization means to finds stable, low energy conformations by changing the geometry of a structure or identifying a point in the configuration space at which the net force on atom vanishes .

• In other words , it is to find the coordinates where the first derivative of the potential energy function equals zero.

• Such a conformation represents one of the many different conformations that a molecule might assume at a temperature of 0 k0 .

Molecular Mechanics Energy Minimization

63

• The potential energy function is evaluated by a certain algorithm or minimizer that moves the atoms in the molecule to a nearest local minimum

• Examples ;o Steepest Decent o Conjugate Gradiento Newton-Raphson procedure

64

• There are three main approaches to finding a minimum of a function of many variables. infalliable

! Search Methods :

- Utilize only values of function

- Slow and inefficient - Search algorithms

infalliable and always find minimum Example :SIMPLEX

! Gradient Methods :

- Utilizes values of a function and its

gradients.- Currently most popular Example : The conjugated gradient algorithm

! Newton Methods :

- Require value of function and its 1st

and 2nd derivatives.- Hessian matrix Example : BFGS algorithm 65

• Geometry optimization is an iterative procedure of computing the energy of a structure and then making incremental increase changes to reduce the energy.

• Minimization involves two steps 1 – an equation describes the energy of the

system as a function of its coordinates must be defined and evaluated for a given conformation

2 – the conformation is adjusted to lower the value of the potential function .

66

VL

G

X

L

X

X (1) X (2) X (min)

L = Local minimumG = Global minimum

Local and global minima for a function of one variable and an example of a sequence of solution.

Algorithm for decent series minimization.

67

! In Cartesian presentation of potential energy surface , the picture would like the lots of narrow tortuous valleys of similar depth.

→ This is because low energy paths for individual atoms are very narrow due to the presence of hard bond stretching and angle bending terms.

→ The low energy paths corresponds only to the rotation of groups or large portions of the molecule as a result of varying torsional angles.

• In the Cartesian space the minimizer walks along the bottom of a narrow winding channel which is frequently a dead-end .

Cartesian space

68

• In internal coordinates presentation , the potential energy surface looks like a valley surrounded by high mountains.

• → The high peaks corresponds to stretching and bending terms and close Vander Waals contacts while the bottom of the valley represents the torsional degree of freedom.

• → If you happen to start at the mountain tops in the internal coordinates space , the minimizer sees the bottom of the valley clearly from the above .

Internal coordinates

69

• Using the internal coordinates there is a clear separation of variables into the hard ones ( those whose small changes produces large changes in the function values ) and soft ones ( those whose changes do not affect the function value substantially).

• During the function optimization in the internal coordinates, the minimizer first minimizes the hard variables and in the subsequent iterations cleans up the details by optimizing the soft variables.

• While in the Cartesian spaces all variables are of the same type. 70

• The atoms and molecules are in the constant motion and especially in the case of biological macromolecules , these movement are concerted and may be essential for biological function.

• And so such thermodynamic properties cannot be derived from the harmonic approximations and molecular mechanics because they inherently assumes the simulation methods around a systemic minimum.

• So we use molecular Dynamic simulations.

Molecular dynamics

71

• Used to compute the dynamics of the molecular system, including time-averaged structural and energetic properties, structural fluctuations and conformational transitions.

• The dynamics of a system may be simplified as the movements of each of its atoms. if the velocities and the forces acting on atoms can be quantified, then their movement may be simulated.

72

• There are two approaches in molecular dynamics for the

simulations . Stochastic ! Called Monte Carlo simulation ! Based on exploring the energy surface by

randomly probing the geometry of the molecular

system. Deterministic ! Called Molecular dynamics ! It actually simulates the time evolution of

the molecular system and provides us with the actual

trajectory of the system

73

• Based on exploring the energy surface by randomly probing the geometry of the molecular system.

• Steps 1 - Specify the initial coordinates of atoms 2 - Generate new coordinates by changing the

initial coordinates at random. 3 - Compute the transition probability W(0,a) 4 - Generate a uniform random number R in the

range [0,1] 5 - If W(0,a) < R then take the old coordinates as

the new coordinates and go to step 2 6 – Otherwise accept the new coordinates and go

to step 2.

Stochastic approachMonte Carlo Simulation

74

The most popular of the Monte Carlo method for the molecular system

See the pamplet for description75

Deterministic approachMolecular Dynamic Simulation

• Actually time evaluation of the molecular system and the information generated from simulation methods can be used to fully characterized the thermodynamic state of the system.

• Here the molecular system is studied as the series of the snapshots taken at the close time intervals. ( femtoseconds usually) .

76

• Based on the potential energy function we can find components Fi of the force F acting on atom as

Fi = - dV/ dxi

This force in an acceleration according to Newton’s equation of motion

F = m a• By knowing the acceleration we can calculate the

velocity of an atom in the next time step. From atom position , velocities and acceleration at any moment in time, we can calculate atom positions and velocities at the next time step.

• And so integrating these infiniteimal steps yields the trajectories of the system for any desired time range.

Principle

77

Verlet algorithm

The Verlet algorithm uses positions and accelerations at time t and the positions from time t-δt to calculate new positions at time t+δt.

r(t+δt) = 2r(t) - r(t-δt)+a(t) δt2

78

• Advantages:• – Position integration is accurate (errors on order

of Δt4).• – Single force evaluation per time step.• – The forward/backward expansion guarantees

that the path is reversible.

• Disadvantages:• – Velocity has large errors (order of Δt2).• – It is hard to scale the temperature (kinetic

energy of molecule).

79

1. the velocities are first calculated at time t+1/2δt (the velocities leapover the positions)

2. these are used to calculate the positions, r, at time t+δt. (then the positions leapover the velocities)

Leapfrog algorithm

r(t+δt) = r(t) + v( t + ½ δt) δt

v( t + ½ δt) = v( t - ½ δt) +a(t) δt

80

• Advantages: – High quality velocity calculation, which is

important in temperature control.

• Disadvantages: – Velocities are known accurately at half time

steps away from when the position is known accurately.

– Estimate of velocity at integral time step: v(t) = [v(t-Δt)+v(t+Δt)]/2

81

1) We need an initial set of atom positions (geometry) and atom velocities.

• The initial positions of atoms are most often accepted from the prior geometry optimization with molecular mechanics.

• • Formally such positions corresponds to the

absolute zero temperature.

Procedure

82

2) The velocities are assigned to each atom from the Maxwell distribution for the temperature 20 oK .

• Random assignment does not allocate correct velocities and the system is not at thermodynamic equilibrium.

• To approach the equilibrium the “equilibration” run is performed and the total kinetic energy of the system is monitored until it is constant.

• The velocities are then rescaled to correspond to some higher temperature. i.e heating is performed.

• Then the next equilibration run follows.83

• The absolute temperature T, and atom velocities are related through the mean kinetic energy of the system.

N = number of the atoms in the system m = mass of the i-th atom k = Boltzman constant. • And by multiplying the velocities by

we can effectively “heat “ the system and that accelerate the atoms of the molecular system.

• These cycles are repeated until the desired temperature is achieved and at this point a “production’ run can commence.

T =2

3 N ki=1

Nmi Vi

2

2

Tdesired / Tcurrent

84

• Molecular dynamics for larger molecules or systems in which solvent molecules are explicitly taken into account is a computationally intensive task even for supercomputers.

• For such a conditions we have two approximations

Periodic boundary conditions Stochastic boundary conditions

85

Periodic boundary conditions

Here we are actually simulating a crystal comprised of boxes with ideally correlated atom movements.

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Stochastic boundary conditions

Reaction zone :

Portion of the system which

we want to study

Reservoir zone Portion of the system

which Is inert and

uninteresting

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Method Advantages Disadvantages Best for

Molecular Mechanics

Computationally cheap ,fast and

useful with limited computer

resources

Does not calculate electronic properties

Large systems(Thousands of

atoms)

• Use Classical physics• Relies on force field with embeded empirical parameters

Can be used for large molecules

like enzymes

Requires ab initio or

experimental data for

parameters

Systems or processes that do not involve bond breaking

Commercial software

applicable to a limited range of

molecules

Molecular Mechanics

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• So molecular dynamics and molecular mechanics are often used together to achieve the target conformer with lowest energy configuration

• Visualise the 3D shape of a molecule• Carry out a complete analysis of all possible

conformations and their relative energies• Obtain a detailed electronic structure and the

polarisibility with take account of solvent molecules.• Predict the binding energy for docking a small

molecule i.e. a drug candidate, with a receptor or enzyme target.

• Producing Block busting drug • Nevertheless, molecular modelling, if used with

caution, can provide very useful information to the chemist and biologist involved in medicinal research.

Conclusion

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References

1) Cohen N. C. “Guide book on Molecular Modelling on Drug Design” Academic press limited publication, London, 1996.

2) Young D. C. “Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems”. John Wiley & Sons Inc., 2001.

3) Abraham D. J. “Burger’s Medicinal Chemistry and Drug Discovery” sixth edition, A John Wiley and Sons, Inc. Publication,1998.

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THANK YOU

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