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Dinamica Molecular y el modelamiento de macromoleculas
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Historical Perspective
1946 MD calculation 1960 force fields 1969 Levinthal’s paradox on protein folding 1970 MD of biological molecules 1971 protein data bank 1998 ion channel protein crystal structure 1999 IBM announces blue gene project
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Proteins Polypeptide chains made up of amino acids or residues linked by peptide bonds 20 aminoacids residues, atoms Native structure believed to correspond to energy minimum, since proteins unfold when temperature is increased
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Proteins: Local Motions
AA, 1 fs -0.1s Atomic fluctuations Small displacements for substrate binding in enzymes Energy “source” for barrier crossing and other activated processes (e.g., ring flips) Sidechain motions Opening pathways for ligand (myoglobin) Closing active site Loop motions Disorder-to-order transition as part of virus formation
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Levinthal paradox Proteins simply can not fold on a reasonable time scale (Levinthal paradox; J. Chem. Phys., 1968, 65: 44-45) Each bond connecting amino acids can have several (e.g., three) possible states (conformations). A protein of, say, 101 AA could exist in 3100 = 5 x 1047 conformations. If the protein can sample these conformations at a rate of 1013/sec, 3 x 1020/year, it will take 1027 years to try them all. Nevertheless, proteins fold in a time scale of seconds.
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Proteins: Rigid-Body Motions
1-10 AA, 1 ns – 1 s Helix motions Transitions between substrates (myoglobin) Hinge-bending motions Gating of active-site region (liver alcohol dehydroginase) Increasing binding range of antigens (antibodies)
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Quantum Mechanical Origins
Fundamental to everything is the Schrödinger equation wave function H = Hamiltonian operator time independent form Born-Oppenheimer approximation electrons relax very quickly compared to nuclear motions nuclei move in presence of potential energy obtained by solving electron distribution for fixed nuclear configuration it is still very difficult to solve for this energy routinely usually nuclei are heavy enough to treat classically Nuclear coordinates Electronic coordinates
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Force Field Methods Too expensive to solve QM electronic energy for every nuclear configuration Instead define energy using simple empirical formulas “force fields” or “molecular mechanics” Decomposition of the total energy Force fields usually written in terms of pairwise additive interatomic potentials with some exceptions Neglect 3- and higher-order terms Single-atom energy (external field) Atom-pair contribution 3-atom contribution
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Conformation optimization for molecular interaction
Molecular Mechanics Approach:
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Energy minimisation Calculation of how atoms should move to minimise TOTAL potential energy At minimum, forces on every atom are zero. Optimising structure to remove strain & steric clashes However, in general finds local rather than global minimum. Energy barriers are not overcome even if much lower energy state is possible ie structures may be locked in. Hence not useful as a search strategy.
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Energy minimisation Steepest descents Conjugate gradients
Potential energy depends on many parameters Problem of finding minimum value of a function with >1 parameters. Know value of function at several points. Grid search is computationally not feasible Methods Steepest descents Conjugate gradients
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Molecular Dynamics: Introduction
Newton’s second law of motion
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Molecular dynamics F=ma
F is calculated from molecular mechanical potential. Model conformational changes. Calculate time-dependent properties (transport properties).
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Molecular Dynamics: Introduction
We need to know The motion of the atoms in a molecule, x(t) and therefore, the potential energy, V(x)
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Molecular Dynamics: Introduction
How do we describe the potential energy V(x) for a molecule? Potential Energy includes terms for Bond stretching Angle Bending Torsional rotation Improper dihedrals
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Molecular Dynamics: Introduction
Potential energy includes terms for (contd.) Electrostatic Interactions van der Waals
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Molecular Dynamics: Introduction
To do this, we should know at given time t, initial position of the atom x1 its velocity v1 = dx1/dt and the acceleration a1 = d2x1/dt2 = m-1F(x1)
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Molecular Dynamics: Introduction
The position x2 , of the atom after time interval t would be, and the velocity v2 would be,
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Molecular Dynamics: Introduction
In general, given the values x1, v1 and the potential energy V(x), the molecular trajectory x(t) can be calculated, using,
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How a molecule changes during MD
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The Necessary Ingredients
Description of the structure: atoms and connectivity Initial structure: geometry of the system Potential Energy Function: force field AMBER CVFF CFF95 Universal
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross - + Attraction - + Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross - + - + Attraction Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross - + - + Attraction + - + - Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross - + - + u(2) Attraction + - + + u(2) - + - + - u(N) + - Mixed terms
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Contributions to Potential Energy
Total pair energy breaks into a sum of terms Intramolecular only Repulsion UvdW van der Waals Uel electrostatic Upol polarization Ustr stretch Ubend bend Utors torsion Ucross cross - + - + u(2) Attraction + - + + u(2) - + - + - u(N) + - Mixed terms
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Modeling Potential energy
A Taylor series expansion of the full potential gives us an analytical form to which we apply mathematical techniques - eliminate terms - find extrema
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Modeling Potential energy
0 at minimum A Taylor series expansion of the full potential gives us an analytical form to which we apply mathematical techniques - eliminate terms - find extrema
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Stretch Energy Expand energy about equilibrium position
Model fails in strained geometries better model is the Morse potential (neglect) define minimum harmonic Morse dissociation energy force constant
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Bending Energy q Expand energy about equilibrium position
improvements based on including higher-order terms Out-of-plane bending (neglect) define minimum harmonic u(4) c
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Torsional Energy f Two new features Fourier series periodic
weak (Taylor expansion in f not appropriate) Fourier series terms are included to capture appropriate minima/maxima depends on substituent atoms e.g., ethane has three mimum-energy conformations n = 3, 6, 9, etc. depends on type of bond e.g. ethane vs. ethylene usually at most n = 1, 2, and/or 3 terms are included
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Van der Waals Attraction
Correlation of electron fluctuations Stronger for larger, more polarizable molecules CCl4 > CH4 ; Kr > Ar > He Theoretical formula for long-range behavior Only attraction present between nonpolar molecules reason that Ar, He, CH4, etc. form liquid phases a.k.a. “London” or “dispersion” forces - + - + - + - +
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Van der Waals Repulsion
Overlap of electron clouds Theory provides little guidance on form of model Two popular treatments inverse power exponential typically n ~ two parameters Combine with attraction term Lennard-Jones model Exp-6 a.k.a. “Buckingham” or “Hill” Beware of anomalous Exp-6 short-range attraction Exp-6 repulsion is slightly softer
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Electrostatics 1. Interaction between charge inhomogeneities
Modeling approaches point charges point multipoles Point charges assign Coulombic charges to several points in the molecule total charge sums to charge on molecule (usually zero) Coulomb potential very long ranged
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Electrostatics 2. At larger separations, details of charge distribution are less important Multipole statistics capture basic features Dipole Quadrupole Octopole, etc. Point multipole models based on long-range behavior dipole-dipole dipole-quadrupole quadrupole-quadrupole Vector Tensor Axially symmetric quadrupole
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Polarization + - Charge redistribution due to influence of surrounding molecules dipole moment in bulk different from that in vacuum Modeled with polarizable charges or multipoles Involves an iterative calculation evaluate electric field acting on each charge due to other charges adjust charges according to polarizability and electric field re-compute electric field and repeat to convergence Re-iteration over all molecules required if even one is moved + - + - + - + - + -
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Polarization Approximation Electrostatic field does not include contributions from atom i
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Common Approximations in Molecular Models
Rigid intramolecular degrees of freedom fast intramolecular motions slow down MD calculations Ignore hydrogen atoms united atom representation Ignore polarization expensive n-body effect Ignore electrostatics Treat whole molecule as one big atom maybe anisotropic Model vdW forces via discontinuous potentials Ignore all attraction Model space as a lattice especially useful for polymer molecules Qualitative models
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Molecular Dynamics: Introduction
Equation for covalent terms in P.E.
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Molecular Dynamics: Introduction
Equation for non-bonded terms in P.E.
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An overview of various motions in proteins (1)
-14 to –13 0.5 to 1 Torsional vibration of buried groups -11 to –10 Rotation of side chains at surface -12 to –11 1 to 2 Elastic vibration of globular region 0.2 to 0.5 Relative vibration of bonded atoms Log10 of characteristic time (s) Spatial extent (nm) Motion
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An overview of various motions in proteins (2)
Log10 of characteristic time (s) Spatial Extent (nm) Motion -5 to 2 ??? Protein folding -5 to 1 0.5 to 1 Local denaturation -5 to 0 0.5 to 4 Allosteric transitions -4 to 0 0.5 Rotation of medium-sized side chains in interior -11 to –7 1 to 2 Relative motion of different globular regions (hinge bending)
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A typical MD simulation protocol
Initial random structure generation Initial energy minimization Equilibration Dynamics run – with capture of conformations at regular intervals Energy minimization of each captured conformation
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Essential Parameters for MD (to be set by user)
Temperature Pressure Time step Dielectric constant Force field Durations of equilibration and MD run pH effect (addition of ions)
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STARTING DNA MODEL
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DNA MODEL WITH IONS
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DNA in a box of water
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SNAPSHOTS
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Protein dynamics study
Ion channel / water channel Mechanical properties Protein stretching DNA bending Movie downloaded from theoreticla biophysics group, UIUC
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Molecular Interactions
water-water interaction van der Waals’ term Hydrogen bonding term ion-water interaction
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Average number of hydrogen bonds within the first water shell around an ion
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Solvent dielectric models
Effetive dielectric constant
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Introduction to Force Fields
Sophisticated (though imperfect!) mathematical function Returns energy as a function of conformation It looks something like this … U(conformation) = Ebond + Eangle + Etors + Evdw + Eelec+ …
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Why do we need force field?
Force field and potential energy surface. Changes in the energy of a system can be considered as movements on a multidimentional surface call the “energy surface”. Force is the first derivative of the energy. In molecular mechanics approach, the dimension of potential surface is 3N, N is number of particles. The probability of the molecular system stay in certain conformations can be calculated if the underlying potential is known.
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Three types of force field
Quantum mechanics (Schrodinger equation for electrons), usually deal with systems with less than 100 atoms. Empirical force field: molecular mechanics (for atoms), can be used for systems up to millions of atoms. Statistical potential (flexible), no restriction.
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Source of FF components
Geometrical terms: bond, angle, torsion & vdw parameters come from empirical data. Electrostatic charges: two problems arise no exp.data for charges basis underlying molec. model
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Molecular mechanical force field
Potential is the summation of the following terms : Bond stretching, Angle bending, Torsion rotation, Non-bonded interactions Vdw interaction, Electrostatic interaction. (Hydrogen bonds). (Implicit solvent). …
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Figures are taken from NIH guide of molecular modeling
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non-bonded terms
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Common empirical force fields
Class I CHARMM CHARMm (Accelrys) AMBER OPLS/AMBER/Schrödinger ECEPP (free energy force field) GROMOS Class II CFF95 (Biosym/Accelrys) MM3 MMFF94 (CHARMM, Macromodel, elsewhere) UFF, DREIDING
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Assumptions Hydrogens often not explicitly included (intrinsic hydrogen methods) “Methyl carbon” equated with 1 C and 3 Hs System not far from equilibrium geometry (harmonic) Solvent is vacuum or simple dielectric
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Assumptions: Harmonic Approximation
Energy is in Joules/molecule Distance is in Angstroms
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Assumptions: Harmonic Approximation
Energy is in Joules/molecule Distance is in Angstroms
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Brief History of FF
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Force Field classification
1.- with rigid/partially rigid geometries ECEPP, … 2.- without electrostatics SYBYL, … 3.- simple diagonal FF Weiner, GROMOS, CHARMm, OPLS/AMBER, … 4.- more complex FF MM2, MM3, MMFF, … OVERALLLLLLLLLLLLLLLLLLLLLLLLLLL …. IN SUMMARY ……..
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Comparison of the simple diagonal FF
Force Field Electrostatics van der Waals CHARMm empirical fit to quantum mechanics dimers empirical (x-ray, crystals) GROMOS empirical OPLS/AMBER empirical (Monte Carlo on liquids) empirical (liquids) Weiner ESP fit (STO-3G) Cornell RESP fit (6-31G*)
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Transferability AMBER (Assisted Model Building Energy Refinement)
Specific to proteins and nucleic acids CHARMM (Chemistry at Harvard Macromolecular Mechanics) Widely used to model solvent effects Molecular dynamics integrator
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Transferability MM? – (Allinger et. al.)
Organic molecules MMFF (Merck Molecular Force Field) Molecular Dynamics Tripos/SYBYL Organic and bio-organic molecules
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Transferability UFF (Universal Force Field) YETI
Parameters for all elements Inorganic systems YETI Parameterized to model non-bonded interactions Docking (AmberYETI)
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MMFF Energy Electrostatics (ionic compounds) D – Dielectric Constant
d - electrostatic buffering constant
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MMFF Energy Analogous to Lennard-Jones 6-12 potential
London Dispersion Forces Van der Waals Repulsions The form for the repulsive part has no physical basis and is for computational convenience when working with large macromolecules. K. Gilbert: Force fields like MM2 which is used for smaller organic systems will use a Buckingham potential (or expontential) which accurately reflects the chemistry/physics.
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Pros and Cons N >> 1000 atoms Easily constructed Accuracy
Not robust enough to describe subtle chemical effects Hydrophobicity Excited States Radicals Does not reproduce quantal nature
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Simple Statistics on MD Simulation
Atoms in a typical protein and water simulation Approximate number of interactions in force calculation Machine instructions per force calculation 1000 Total number of machine instructions Typical time-step size –15 s Number of MD time steps steps Physical time for simulation –4 s Total calculation time (CPU: P4-3.0G ) days 10,000
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Hardware Strategies Parallel computation
PC cluster IBM (The blue gene), 106 CPU Massive distributive computing Grid computing (formal and in the future) Server to individual client (now in inexpensive) Examples: SETI,
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# publications/year mentioning FF used to model proteins
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