Molecular modelling

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Molecular modelling

  1. 1. TargetIdentification&ValidationHitIdentificationLeadIdentificationLeadOptimisa-tionCDPrenomi-nationConceptTestingDevelopmentfor launchLaunchPhaseFDA SubmissionLaunchFinding PotentialDrug TargetsValidating Therapeutic TargetsFinding Potential DrugsDrug<>Target<>TherapeuticEffect Association FinalizedTesting in Man(toxicity and efficacy)Drug Discovery is a goal of research. Methods and approaches from different science areascan be applied to achieve the goal.2
  2. 2. Hugo Kubinyi , www.kubinyi.de3
  3. 3.  R&D cost per new drug is $500 to $700 millions To sustain growth, each of top 20 pharma companyshould produce more new drugs Currently, total industry produces only 32 new drugsper year. Current rate of NDAs far below than required forsustained growth.4
  4. 4. Atomistic ContinuumFinite PeriodicQuantum Mechanical Methodies Classical MethodsSemi-Empirical Ab Initio Deterministic StochasticQM/MMQuantum MC DFTMolecularDynamicsMonte CarloHartree Fock5
  5. 5. > 10-4 m10-6 m10-8 m10-10 mMacroscaleMicroscaleNanoscaleAtomicScaleClassical MechanicsOrganismCellProtein, membraneSmall molecules, drugContinuum Mechanics• Finite Element Method• Fluid DynamicsStatistical Mechanics• Molecular Mechanics• Molecular Dynamics• Brownian Dynamics• Stochastic DynamicsQuantum Mechanics•Density Functional Theory• Hartree-Fock Theory• Perturbation Theory• Structural Mechanics6
  6. 6. UnknownKnownUnknown KnownGenerate 3Dstructures,HTS, Comb. ChemBuild the lock and thenfind the keyMolecularDockingDrug receptorinteraction2D/3D QSAR andPharmacophoreInfer the lock byexpecting keyDe NOVO Design ,Virtual screeningBuild or find the keythat fits the lockReceptor based drug designRational drug designIndirect drug designHomology modelling7
  7. 7. • Molecular modelling allow scientists to use computersto visualize molecules means representing molecularstructures numerically and simulating their behaviorwith the equations of quantum and classical physics , todiscover new lead compounds for drugs or to refineexisting drugs in silico.• Goal: To develop a sufficient accurate model of thesystem so that physical experiment may not benecessary .8
  8. 8. • The term “ Molecular modeling “expanded over the lastdecades from a tool to visualize three-dimensionalstructures and to simulate , predict and analyze theproperties and the behavior of the molecules on anatomic level to data mining and platform to organizemany compounds and their properties into database andto perform virtual drug screening via 3D databasescreening for novel drug compounds .9
  9. 9. Molecular modeling starts from structure determinationSelection of calculation methods in computational chemistryStarting geometry fromstandard geometry, x-ray, etc.MoleculeMolecularmechanicsQuantummechanicsMoleculardynamics orMonte CarloIs bond formation orbreaking important?Are many force fieldparameters missing ?Is it smaller than100 atoms?Are chargesof interest ?Are there many closelyspaced conformers?Is plenty of computertime available?Is the free energyNeeded ?Is solvationImportant ?10
  10. 10. 11
  11. 11. • Molecular modelling or more generally computational chemistryis the scientific field of simulation of molecular systems.• Basically in the computational chemistry , the free energy of thesystem can be used to assess many interesting aspects of thesystem.• In the drug design , the free energy may be used to assesswhether a modification to a drug increase or decrease targetbinding.• The energy of the system is a function of the type and numberof atoms and their positions.• Molecular modelling softwares are designed to calculate thisefficiently.12
  12. 12. • The energy of the molecules play important role in thecomputational chemistry. If an algorithm can estimatethe energy of the system, then many importantproperties may be derived from it.• On todays computer , however energy calculation takesdays or months even for simple system. So in practice,various approximations must be introduced thatreduce the calculations time while adding acceptablysmall effect on the result.13
  13. 13. • Example :• Familiar conformation of the Butane765432100 60 120 180 240 300 3600.30.250.20.150.100.10.050CBDEFPotentialenergyDihedral angleProbability14
  14. 14. Quantum mechanics Molecular mechanicsAb initio methods DFT method Semiimpirical methodsMolecular Modelling15
  15. 15. • Quantum mechanics is basically the molecular orbital calculationand offers the most detailed description of a molecule’s chemicalbehavior.• HOMO – highest energy occupied molecular orbital• LUMO – lowest energy unoccupied molecular orbital• Quantum methods utilize the principles of particle physics toexamine structure as a function of electron distribution.• Geometries and properties for transition state and excited state can only becalculated with Quantum mechanics.• Their use can be extended to the analysis of molecules as yetunsynthesized and chemical species which are difficult (orimpossible) to isolate.16
  16. 16. • Quantum mechanics is based on Schrödinger equationHΨ = EΨ = (U + K ) ΨE = energy of the system relative to one in which all atomicparticles are separated to infinite distancesH = Hamiltonian for the system .It is an “operator” ,a mathematical construct that operateson the molecular orbital , Ψ ,to determine the energy.U = potential energyK = kinetic energyΨ = wave function describes the electron distribution around themolecule.17
  17. 17. The Hamiltonian operator H is, in general,Where Vi2 is the Laplacian operator acting on particle i. Particlesare both electrons and nuclei. The symbols mi and qi are the massand charge of particle I, and rij is the distance between particles.The first term gives the kinetic energy of the particle within awave formulation.The second term is the energy due to Coulombic attraction orrepulsion of particles . 18
  18. 18. • In currently available software, the Hamiltonian above is nearlynever used.• The problem can be simplified by separating the nuclear andelectron motions.kinetic energyof electronsAttraction ofelectrons tonucleiRepulsionbetweenelectronsBorn-Oppenheimer approximation19
  19. 19. • Thus, each electronic structure calculation is performed for afixed nuclear configuration, and therefore the positions of allatoms must be specified in an input file.• The ab initio program like MOLPRO then computes theelectronic energy by solving the electronic Schrödinger equationfor this fixed nuclear configuration.• The electronic energy as function of the 3N-6 internal nucleardegrees of freedom defines the potential energy surface (PES)which is in general very complicated and can have many minimaand saddle points.• The minima correspond to equilibrium structures of differentisomers or molecules, and saddle points to transition statesbetween them.20
  20. 20. • The term ab initio is Latin for “from the beginning” premises ofquantum theory.• This is an approximate quantum mechanical calculation for afunction or finding an approximate solution to a differentialequation.• In its purest form, quantum theory uses well known physicalconstants such as the velocity of light , values for the masses andcharges of nuclear particles and differential equations to directlycalculate molecular properties and geometries. This formalism isreferred to as ab initio (from first principles) quantummechanics.21
  21. 21. HARTREE±FOCK APPROXIMATION• The most common type of ab initio calculation in which theprimary approximation is the central field approximation meansCoulombic electron-electron repulsion is taken into account byintegratinfg the repulsion term.• This is a variational calculation, meaning that the approximateenergies calculated are all equal to or greater than the exactenergy.• The energies are calculated in units called Hartrees (1 Hartree .27.2116 eV)22
  22. 22. • The steps in a Hartreefock calculation start with an initial guessfor the orbital coefficients ,usually using a semiempirical method.• This function is used to calculate an energy and a new set oforbital coefficients, which can then be used to obtain a new set,and so on.• This procedure continues iteratively untill the energies andorbital coefficient remains constant from one iteration to thenext.• This iterative procedure is called as a Self-consistent fieldprocedure (SCF).23
  23. 23. Advantage• Advantages of this method is that it breaks the many-electronSchrodinger equation into many simpler one-electron equations.• Each one electron equation is solved to yield a single-electronwave function, called an orbital, and an energy, called an orbitalenergy.24
  24. 24. 25
  25. 25. • What is Density ?How something(s) is(are) distributed/spread about a given spaceElectron density tells us where the electrons are likely to exist.Allyl Cation:*26
  26. 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 dependson a set of variables.E = F [ f (x) ]E.g., energy depends on the wave function, which depends onelectron coordinates.12341234F(X)=Y27
  27. 27. • The electron density is the square of wavefunction and integrated over electroncoordinates.• The complexity of a wave function increases asthe number of electrons grows up, but theelectron density still depends only on 3coordinates.xx• With this theory, the properties of a many-electronsystem can be determined by using functionals, i.e.functions of another function, which in this case isthe spatially dependent electron density .28
  28. 28. • There are difficulties in using density functional theory toproperly describe intermolecular interactions, especially van derWaals forces (dispersion); charge transfer excitations; transitionstates, global potential energy surfaces and some other stronglycorrelated systems .29
  29. 29. • Density functional theory has its conceptual roots in theThomas-Fermi model .• It is developed by Thomas and Fermi in 1927.• They used a statistical model to approximate the distribution ofelectrons in an atom.• The mathematical basis postulated that electrons are distributeduniformly in phase space with two electrons in every h3 ofvolume.• For each element of coordinate space volume d3r we can fill outa sphere of momentum space up to the Fermi momentum pf .Thomas Fermi model30
  30. 30. • Equating the number of electrons in coordinate space to that inphase space gives• Solving for pf and substituting into the classical kinetic energyformula then leads directly to a kinetic energy represented as afunctional of the electron density:where31
  31. 31. • As such, they were able to calculate the energy of an atom usingthis kinetic energy functional combined with the classicalexpressions for the nuclear-electron and electron-electroninteractions (which can both also be represented in terms of theelectron density).• The Thomas-Fermi equations accuracy is limited because theresulting kinetic energy functional is only approximate, andbecause the method does not attempt to represent the exchangeenergy of an atom as a conclusion of the Pauli principle.• An exchange energy functional was added by Dirac in 1928called as the Thomas-Fermi-Dirac model• However, the Thomas-Fermi-Dirac theory remained ratherinaccurate for most applications. The largest source of error wasin the representation of the kinetic energy, followed by the errorsin the exchange energy, and due to the complete neglect ofelectron correlation.32
  32. 32. • DFT was originated with a theorem by Hoenburg and Kohn .• The original H-K theorems held only for non-degenerateground states in the absence of a magnetic field .• The first H-K theorem demonstrates that the ground stateproperties of a many-electron system are uniquely determined byan electron density that depends on only 3 spatial coordinates.• It lays the groundwork for reducing the many-body problem ofN electrons with 3N spatial coordinates to only 3 spatialcoordinates, through the use of functionals of the electrondensity.Hohenberg-Kohn theorems33
  33. 33. • This theorem can be extended to the time-dependent domain todevelop time-dependent density functional theory (TDDFT),which can be used to describe excited states.• The second H-K theorem defines an energy functional for thesystem and proves that the correct ground state electron densityminimizes this energy functional.34
  34. 34. • Within the framework of Kohn-Sham DFT, the intractablemany-body problem of interacting electrons in a static externalpotential is reduced to a tractable problem of non-interactingelectrons moving in an effective potential.• The effective potential includes the external potential and theeffects of the Coulomb interactions between the electrons, e.g.,the exchange and correlation interactions.kohn-Sham theoryEDFT[ ] = T[ ] + Ene[ ] + J[ ] + Exc[ ]ElectronicKinetic energyNuclei-electronsCoulombic energyelectrons-electronsCoulombic energyelectrons-electronsExchange energy35
  35. 35. • Modeling the latter two interactions becomes the difficultywithin KS DFT.• In this formulation, the electron density is expressed as a linearcombination of basis functions similar in mathematical form toHF orbitals.• A determinant is then formed from these functions, calledKohn±Sham orbitals.• It is the electron density from this determinant of orbitals that isused to compute the energy.36
  36. 36. • So Semiempirical methods are very fast, applicable to largemolecules, and may give qualitative accurate results when applied tomolecules that are similar to the molecules used forparameterization.• Because Semiempirical quantum chemistry avoid two limitations,namely slow speed and low accuracy, of the Hartree-Fockcalculation by omitting or parameterzing certain integrals based onexperimental data, such as ionization energies of atoms, or dipolemoments of molecules.• Rather than performing a full analysis on all electrons within themolecule, some electron interactions are ignored .37
  37. 37. • Modern semiempirical models are based on the Neglect ofDiatomic Differential Overlap (NDDO) method in which theoverlap 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 furtherapproximations that are made when evaluating one-and two-electron integrals and by the parameterization philosophy.38
  38. 38. • Modified Neglect of Diatomic Overlap , MNDO ( byMichael 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
  39. 39. • Modified Neglect of Diatomic Overlap , by Michael Dewar andWalter Thiel, 1977• It is the oldest NDDO-based model that parameterizes one-center two-electron integrals based on spectroscopic data forisolated atoms, and evaluates other two-electron integrals usingthe idea of multipole-multipole interactions from classicalelectrostatics.• A classical MNDO model uses only s and p orbital basis setswhile more recent MNDO/d adds d-orbitals that are especiallyimportant for the description of hypervalent sulphur species andtransition metals.40
  40. 40. Deficiencies• Inability to describe the hydrogen bond due to a strongintermolecular repulsion.• The MNDO method is characterized by a generally poorreliability in predicting heats of formation.• For example: highly substituted stereoisomers are predicted tobe too unstable compared to linear isomers due tooverestimation of repulsion is sterically crowded systems.41
  41. 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 forcesthat mimic van der Waals interactions.• AM1 predicts the heat of the energy more accurately than theMNDO.• The results of AM1 calculations often are used as the startingpoints for parameterizations of the force fields in moleculardynamic simulation and CoMFA QSAR.42
  42. 42. Some known limitations to AM1 energies• Predicting rotational barriers to be one-third the actual barrierand predicting five-membered rings to be too stable.• The predicted heat of formation tends to be inaccurate formolecules with a large amount of charge localization.• Geometries involving phosphorus are predicted poorly.• There are systematic errors in alkyl group energies predictingthem 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, butoften the wrong orientation.• So o n average, AM1 predicts energies and geometries betterthan MNDO, but not as well as PM3 . 43
  43. 43. • By James Stewart• Uses a Hamiltonian that uses nearly the same equations as the AM1method along with an improved set of parametersis.• Limitations of PM3..• PM3 tends to predict that the barrier to rotation around the C-Nbond in peptides is too low.• Bonds between Si and the halide atoms are too short Protonaffinities are not accurate.• Some polycyclic rings are not flat.• The predicted charge on nitrogen is incorrect.• Nonbonded distances are too short..44
  44. 44. Strength• Overall heats of formation are more accurate than with MNDOor AM1.• Hypervalent molecules are also predicted more accurately• PM3 also tends to predict incorrect electronic states forgermanium compounds• It tends to predict sp3 nitrogen as always being pyramidal.• Hydrogen bonds are too short by about 0.1AÊ , but theorientation is usually correct .• On average, PM3 predicts energies and bond lengths moreaccurately than AM1 or MNDO45
  45. 45. • By William Jorgensen and co-workers• The Pairwize Distance Directed Gaussian (PDDG)• Use a functional group-specific modification of the corerepulsion function.• Its modification provides good description of the van der Waalsattraction between atoms .• PDDG/PM3 model very accurate for estimation of heats offormation because of reparameterization .• But some limitations common to NDDO methods remain inthe PDDG/PM3 model: the conformational energies areunreliable, most activation barriers are significantlyoverestimated, and description of radicals is erratic.• So far, only C, N, O, H, S, P, Si, and halogens have beenparameterized for PDDG/PM3 46
  46. 46. • Some freely available computational chemistry programs thatinclude many semiempirical models are MOPAC 6, MOPAC 7,and WinMopac .47
  47. 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 datae.g. heat of formation• Preliminary optimization of geomteries of unusual molecules andtransition states that cannot be optimized with molecularmechanics methods 48
  48. 48. 49
  49. 49. • The Process of finding the minimum of an empirical potentialenergy function is called as the Molecular mechanics. (MM)• The process produce a molecule of idealized geometry.• Molecular mechanics is a mathematical formalism which attemptsto reproduce molecular geometries, energies and other features byadjusting bond lengths, bond angles and torsion angles toequilibrium values that are dependent on the hybridization of anatom and its bonding scheme.50
  50. 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 componentThe general form of the force field equation isE P (X) = E bonded + E nonbonded51
  51. 51. •Bonded interactions• Used to better approximate the interaction of the adjacentatoms.• Calculations in the molecular mechanics is similar to theNewtonians law of classical mechanics and it will calculategeometry 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 isproportional to its elongation x ,and where k is the forceconstant or spring constant of the spring.52
  52. 52. • Ebonded = Ebond + Eangle + Edihedral• Bond termEbond = ½ kb (b – bo) 2• Angle termEAngle = ½ kθ (θ – θ0)• Energy of the dihedral anglesEdihedral = ½ kΦ(1 – cos (nΦ + δ)53
  53. 53. HCCHHGraphical representation of the bonded and non bonded interaction and thecorresponding energy terms.E coulombElectrostatic attractionE vdwVan der waalsYijθ0K θK bK ФE ФФ 0E θE bb0bBond stretchingDihedral rotationAngle bending54
  54. 54. • Nearly applied to all pairs of atoms• The nonbonded interaction terms usually include electrostaticinteractions and van der waals interaction , which are expressedas coloumbic interaction as well asLennard-Jones type potentials, respectively.• All of them are a function of the distance between atom pairs ,rij .55Non bonded interaction
  55. 55. • E Nonbonded = E van der waals + E electrostatic• E van der waals• E electrostatic56Lennard Jones potentialCoulombs Law
  56. 56. • The molecular mechanics energy expression consists of a simplealgebraic equation for the energy of the compound.• A set of the equations with their associated constants which arethe energy expression is called a force field.• Such equations describes the various aspects of the equation likestretching, bending, torsions, electronic interactions van derwaals forces and hydrogen bonding.57
  57. 57. • Valance term. Terms in the energy expression which describes asingle 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 howone motion of the molecule affect the motion of the another.Eg., Stretch-bend term which describes how equilibrium bondlength tend to shift as bond angles are changed.• Electrostatic term. force field may or may not include this term.Eg., Coulomb’s law.58
  58. 58. • Some force fields simplify the complexity of the calculations byomitting most of the hydrogen atoms.• The parameters describing the each backbone atom are thenmodified to describe the behavior of the atoms with the attachedhydrogens.• Thus the calculations uses a CH2 group rather than a Sp3carbon bonded to two hydrogens.• These are called united atom force field or intrinsic hydrogenmethods.• Some popular force fields are AMBER CHARMM CFF 59
  59. 59. • Assisted model building with energy refinement is the name ofboth a force field and a molecular mechanics program.• It was parameterized specifically for the protein and nucleicacids.• It uses only five bonding and nonbonding terms and no anycross term.60amber.scripps.edu
  60. 60. (Harvard University)• Chemistry at Harvard macromolecular mechanics is the name ofboth a force field and program incorporating the force field.• It was originally devised for the proteins and nucleic acids. Butnow it is applied to the range of the bimolecules , moleculardynamics, solvation , crystal packing , vibrational analysis andQM/MM studies.• It uses the five valance terms and one of them is an electrostaticterm.61www.charmm.org
  61. 61. • The consistent force field .• It was developed to yield consistent accuracy of results forconformations , vibrational spectras , strain energy andvibrational 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 iselectrostatic and four to six others are Cross terms.62
  62. 62. • Molecular mechanics energy minimization means to finds stable,low energy conformations by changing the geometry of astructure or identifying a point in the configuration space atwhich the net force on atom vanishes .• In other words , it is to find the coordinates where the firstderivative of the potential energy function equals zero.• Such a conformation represents one of the many differentconformations that a molecule might assume at a temperature of0 k0 .63
  63. 63. • The potential energy function is evaluated by a certain algorithmor minimizer that moves the atoms in the molecule to a nearestlocal minimum• Examples ;o Steepest Decento Conjugate Gradiento Newton-Raphson procedure64
  64. 64. • There are three main approaches to finding a minimum of afunction of many variables. infalliable! Search Methods :- Utilize only values of function- Slow and inefficient- Search algorithms infalliable andalways find minimumExample :SIMPLEX! Gradient Methods :- Utilizes values of a function and itsgradients.- Currently most popularExample : The conjugated gradientalgorithm! Newton Methods :- Require value of function and its 1stand 2nd derivatives.- Hessian matrixExample : BFGS algorithm65
  65. 65. • Geometry optimization is an iterative procedure of computingthe energy of a structure and then making incremental increasechanges to reduce the energy.• Minimization involves two steps1 – an equation describes the energy of the system as a functionof its coordinates must be defined and evaluated for a givenconformation2 – the conformation is adjusted to lower the value of thepotential function .66
  66. 66. VLGXLXX (1) X (2) X (min)L = Local minimumG = Global minimumLocal and global minima for a function of one variable and an example of a sequence ofsolution.Algorithm for decent series minimization.67
  67. 67. ! In Cartesian presentation of potential energy surface , the picturewould like the lots of narrow tortuous valleys of similardepth.→ This is because low energy paths for individual atoms are verynarrow due to the presence of hard bond stretching and anglebending terms.→ The low energy paths corresponds only to the rotation of groupsor large portions of the molecule as a result of varying torsionalangles.• In the Cartesian space the minimizer walks along the bottom of anarrow winding channel which is frequently a dead-end .68
  68. 68. • In internal coordinates presentation , the potential energy surfacelooks like a valley surrounded by high mountains.• → The high peaks corresponds to stretching and bending termsand close Vander Waals contacts while the bottom of the valleyrepresents the torsional degree of freedom.• → If you happen to start at the mountain tops in the internalcoordinates space , the minimizer sees the bottom of the valleyclearly from the above .69
  69. 69. • Using the internal coordinates there is a clear separation ofvariables into the hard ones ( those whose small changesproduces large changes in the function values ) and soft ones (those whose changes do not affect the function valuesubstantially).• During the function optimization in the internal coordinates, theminimizer first minimizes the hard variables and in thesubsequent iterations cleans up the details by optimizing the softvariables.• While in the Cartesian spaces all variables are of the same type.70
  70. 70. • The atoms and molecules are in the constant motion andespecially in the case of biological macromolecules , thesemovement are concerted and may be essential for biologicalfunction.• And so such thermodynamic properties cannot be derived fromthe harmonic approximations and molecular mechanics becausethey inherently assumes the simulation methods around asystemic minimum.• So we use molecular Dynamic simulations.71
  71. 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 movementsof each of its atoms. if the velocities and the forces acting onatoms can be quantified, then their movement may be simulated.72
  72. 72. • There are two approaches in molecular dynamics for thesimulations .Stochastic! Called Monte Carlo simulation! Based on exploring the energy surface by randomlyprobing the geometry of the molecular system.Deterministic! Called Molecular dynamics! It actually simulates the time evolution of the molecularsystem and provides us with the actual trajectory of thesystem73
  73. 73. • Based on exploring the energy surface by randomly probing thegeometry of the molecular system.• Steps1 - Specify the initial coordinates of atoms2 - Generate new coordinates by changing the initial coordinates atrandom.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 newcoordinates and go to step 26 – Otherwise accept the new coordinates and go to step 2. 74
  74. 74. The most popular of the Monte Carlo method for the molecular systemSee the pamplet for description 75
  75. 75. • Actually time evaluation of the molecular system and theinformation generated from simulation methods can be used tofully characterized the thermodynamic state of the system.• Here the molecular system is studied as the series of the snapshotstaken at the close time intervals. ( femtoseconds usually) .76
  76. 76. • Based on the potential energy function we can find componentsFi of the force F acting on atom asFi = - dV/ dxiThis force in an acceleration according to Newton’s equation ofmotionF = m a• By knowing the acceleration we can calculate the velocity of anatom in the next time step. From atom position , velocities andacceleration at any moment in time, we can calculate atompositions and velocities at the next time step.• And so integrating these infiniteimal steps yields the trajectoriesof the system for any desired time range.77
  77. 77. The Verlet algorithm uses positions and accelerations at time t andthe positions from time t-δt to calculate new positions at time t+δt.r(t+δt) = 2r(t) - r(t-δt)+a(t) δt278
  78. 78. :• – Position integration is accurate (errors on order of Δt4).• – Single force evaluation per time step.• – The forward/backward expansion guarantees that the path isreversible.:• – Velocity has large errors (order of Δt2).• – It is hard to scale the temperature (kinetic energy of molecule).79
  79. 79. 1. the velocities are first calculated at time t+1/2δt (the velocitiesleapover the positions)2. these are used to calculate the positions, r, at time t+δt. (thenthe positions leapover the velocities)r(t+δt) = r(t) + v( t + ½ δt) δtv( t + ½ δt) = v( t - ½ δt) +a(t) δt80
  80. 80. :– High quality velocity calculation, which is important intemperature control.:– Velocities are known accurately at half time steps away fromwhen the position is known accurately.– Estimate of velocity at integral time step:v(t) = [v(t-Δt)+v(t+Δt)]/281
  81. 81. 1) We need an initial set of atom positions (geometry) and atomvelocities.• The initial positions of atoms are most often accepted from theprior geometry optimization with molecular mechanics.•• Formally such positions corresponds to the absolute zerotemperature.Procedure82
  82. 82. 2) The velocities are assigned to each atom from the Maxwelldistribution for the temperature 20 oK .• Random assignment does not allocate correct velocities and thesystem is not at thermodynamic equilibrium.• To approach the equilibrium the “equilibration” run is performedand the total kinetic energy of the system is monitored until it isconstant.• The velocities are then rescaled to correspond to some highertemperature. i.e heating is performed.• Then the next equilibration run follows.83
  83. 83. • The absolute temperature T, and atom velocities are relatedthrough the mean kinetic energy of the system.N = number of the atoms in the systemm = mass of the i-th atomk = Boltzman constant.• And by multiplying the velocities by we caneffectively “heat “ the system and that accelerate the atoms ofthe molecular system.• These cycles are repeated until the desired temperature isachieved and at this point a “production’ run can commence.T =23 N k i=1Nmi Vi22Tdesired / Tcurrent84
  84. 84. • Molecular dynamics for larger molecules or systems in whichsolvent molecules are explicitly taken into account is acomputationally intensive task even for supercomputers.• For such a conditions we have two approximations Periodic boundary conditions Stochastic boundary conditions85
  85. 85. Here we are actually simulating a crystal comprised of boxes with ideallycorrelated atom movements.86
  86. 86. Reaction zone :Portion of thesystem which wewant to studyReservoir zonePortion of thesystem which Isinert anduninteresting87
  87. 87. 88
  88. 88. • So molecular dynamics and molecular mechanics are often usedtogether to achieve the target conformer with lowest energyconfiguration• Visualise the 3D shape of a molecule• Carry out a complete analysis of all possible conformations andtheir relative energies• Obtain a detailed electronic structure and the polarisibility withtake account of solvent molecules.• Predict the binding energy for docking a small molecule i.e. a drugcandidate, with a receptor or enzyme target.• Producing Block busting drug• Nevertheless, molecular modelling, if used with caution, can providevery useful information to the chemist and biologist involved inmedicinal research. 89
  89. 89. References1) 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 ApplyingTechniques to Real-World Problems”. John Wiley & Sons Inc., 2001.3) Abraham D. J. “Burger’s Medicinal Chemistry and Drug Discovery” sixthedition, A John Wiley and Sons, Inc. Publication,1998.90
  90. 90. 91

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