Molecular Modeling of Proteins O. Michielin, SIB/LICR
Molecular Modeling of Proteins Lecture Plan: - Central role of partition function - Review molecular interactions - Modeling of molecular interactions: CHARMM force field - Recent techniques: implicit solvent models long range electrostatic treatment - Molecular dynamics simulations - elements of statistical mechanics - microcanonical sampling - canonical sampling - isothermal-isobaric sampling - Langevin dynamics - Other sampling techniques - Monte-Carlo - Simulated Annealing (SA) - Genetic Algorithms (GA) O. Michielin, SIB/LICR
Molecular Modeling: Introduction What is Molecular Modeling? Molecular Modeling is concerned with the description of the atomic and molecular interactions that govern microscopic and macroscopic behaviors of physical systems What is it good for? The essence of molecular modeling resides in the connection between the microscopic world and the macroscopic world provided by the theory of statistical mechanics Macroscopic Average of observable observable over selected microscopic (Solvation energy, states affinity between two proteins, H-H distance, conformation, ... ) O. Michielin, SIB/LICR
Connection micro/macroscopic: intuitive view E 1 , P 1 ~ e - b E1 Expectation value E 2 , P 2 ~ e - b E2 〈 O 〉= 1 Z ∑ i O i e − E i E 3 , P 3 ~ e - b E3 Z = ∑ i e − E i Where E 4 , P 4 ~ e - b E4 is the partition function E 5 , P 5 ~ e - b E5 O. Michielin, SIB/LICR
Central Role of the Partition function The determination of the macroscopic behavior of a system from a thermodynamical point of vue is tantamount to computing a quantity called the partition function , Z , from which all the properties can be derived. Z = ∑ i e − E i 〈 O 〉= 1 Z ∑ i O i e − E i . . . Expectation Value p = kT ∂ ln Z 〈 E 〉= ∂ A = -kT ln(Z) ∂ ln Z = U ∂ V N ,T Internal Energy Pressure Helmoltz free energy O. Michielin, SIB/LICR
Computation of the Partition function The partition function is a very complex function to compute, and, in most cases, only numerical approximations are possible Z = ∑ i e 1) − E i 2) Numerical approximations require: 1) the computation of the energy of the system for microstate i - performed using semi-empirical force fields CHARMM / Amber / Gromos / ... 2) a method to sample all (or a representative portion) of the microstates accessible to the system in a given macroscopic state, i.e: - microcanonical sampling for fixed N,V,E systems - canonical sampling for fixed N,V,T systems - isothermic-isobaric sampling for fixed N,P,T systems - ... O. Michielin, SIB/LICR
Introduction & historical note Theoretical milestones: Classical equations of motion: F(t)=m a(t) Newton (1643-1727): Schrödinger (1887-1961): Quantum mechanical equations of motion: - ih t (t)=H(t) (t) Boltzmann(1844-1906): Foundations of statistical mechanics Molecular dynamics milestones: Metropolis (1953): First Monte Carlo (MC) simulation of a liquid (hard spheres) Wood (1957): First MC simulation with Lennard-Jones potential Liquids Alder (1957): First Molecular Dynamics (MD) simulation of a liquid (hard spheres) Rahman (1964): First MD simulation with Lennard-Jones potential Karplus (1977) & First MD simulation of proteins McCammon (1977) Proteins Karplus (1983): The CHARMM general purpose FF & MD program Kollman(1984): The AMBER general purpose FF & MD program Car-Parrinello(1985): First full QM simulations Kollmann(1986): First QM-MM simulations O. Michielin, SIB/LICR
Molecular Interactions (I) Bonded Interactions: - Intramolecular energy terms associated with the deformation of the electronic structure of the molecule. 3 terms are introduced in all FF: 1) Bond stretch 2) Angle bend 3) Dihedral torsion and a fourth one is often used to maintain planarity 4) Improper torsion O. Michielin, SIB/LICR
Molecular Interactions (II) Non-Bonded Interactions: - Inter and intramolecular energy terms arising from electrostatics 1) Electrostatic interactions q i q j 1 V Ele = ∑ Coulomb law: 4 r i , j i j where e is the dielectric constant, 1 for vacuum, 4-20 for protein core, and 80 for water 2) van der Waals interactions Attractive part : due to induced-dipole/induced-dipole Repulsive part : due to Pauli exclusion principle Usually represented by the Lennard-Jones potential V vdW = ∑ 12 − ij / r ij 6 ] 4 ij [ ij / r ij i j e ij =( e i e j ) 1/2 , s ij = 1/2( s i + s j ) are obtained from the single atom param. e and s O. Michielin, SIB/LICR
Derived Interactions Some interactions are often referred to as particular interactions, but they result from the two interactions previously described, i.e. the electrostatic and the van der Waals interactions. -d A f 1) Hydrogen bonds (Hb) -d +d - Interaction of the type D-H ··· A D H - The origin of this interaction is a dipole-dipole attraction - Typical ranges for distance and angle: d 2.4 < d < 4.5Å and 180º < f < 90º 2) Hydrophobic effect Water - Collective effect resulting from the energetically unfavorable surface of contact between the water and an apolar medium (loss of water-water Hb) - The apolar medium reorganizes to minimize the Oil water exposed surface O. Michielin, SIB/LICR
Semi-Empirical Force Fields 1) Goals of semi-empirical force fields - Definition of empirical potential energy functions V( r ) to model the molecular interactions described previously - These functions need to be differentiable in order to compute the forces acting on each atom: F =- V( r ) 2) Ways to compute semi-empirical potential energy functions - First, theoretical analytical functional forms of the interactions are derived - The system is divided into a number of atom types that differ by their atomic number and chemical environment, e.g. the carbons in C=O or C-C are not of the same type - Parameters are determined so as to reproduce the interactions between the various atom types by fitting procedures - experimental enthalpies (CHARMM) - experimental free energies (GROMOS, AMBER) Parametrization available for proteins, lipids, sugars, ADN, ... O. Michielin, SIB/LICR
The CHARMM Force Field 2 ∑ V r = ∑ 2 K b b − b 0 K − 0 Bonds Angles ∑ 2 K − 0 Impropers ∑ K [ 1 − cos n − ] Dihedrals q i q j 1 ∑ 4 r i , j i j ∑ 12 − ij / r ij 6 ] 4 ij [ ij / r ij i j O. Michielin, SIB/LICR
The CHARMM 22 Parameter Set (Sample) BONDS DIHEDRALS !V(bond) = Kb(b - b0)**2 !V(dihedral) = Kchi(1 - cos(n(chi) - delta)) !Kb: kcal/mole/A**2 !Kchi: kcal/mole !b0: A !n: multiplicity !atom type Kb b0 !delta: degrees C C 600.000 1.3350 ! ALLOW ARO HEM !atom types Kchi n delta ! Heme vinyl substituent (KK, from propene (JCS)) C CT1 NH1 C 0.2000 1 180.00 ! ALLOW PEP CA CA 305.000 1.3750 ! ALLOW ARO ! ala dipeptide update for new C VDW Rmin, adm jr., 3/3/93c ! benzene, JES 8/25/89 C CT2 NH1 C 0.2000 1 180.00 ! ALLOW PEP CE1 CE1 440.000 1.3400 ! ! ala dipeptide update for new C VDW Rmin, adm jr., 3/3/93c ! for butene; from propene, yin/adm jr., 12/95 C N CP1 C 0.8000 3 0.00 ! ALLOW PRO PEP CE1 CE2 500.000 1.3420 ! ! 6-31g* AcProN ! for propene, yin/adm jr., 12/95 CA CA CA CA 3.1000 2 180.00 ! ALLOW ARO ! JES 8/25/89 ANGLES CA CPT CPT CA 3.1000 2 180.00 ! ALLOW ARO !V(angle) = Ktheta(Theta - Theta0)**2 ! JWK 05/14/91 fit to indole !V(Urey-Bradley) = Kub(S - S0)**2 !Ktheta: kcal/mole/rad**2 IMPROPER !Theta0: degrees !V(improper) = Kpsi(psi - psi0)**2 !Kub: kcal/mole/A**2 (Urey-Bradley) !Kpsi: kcal/mole/rad**2 !S0: A !psi0: degrees !atom types Ktheta Theta0 Kub S0 !note that the second column of numbers (0) is ignored CA CA CA 40.000 120.00 35.00 2.41620 ! ALLOW ARO !atom types Kpsi psi0 ! JES 8/25/89 CPB CPA NPH CPA 20.8000 0 0.0000 ! ALLOW HEM CE1 CE1 CT3 48.00 123.50 ! ! Heme (6-liganded): porphyrin macrocycle (KK 05/13/91) ! for 2-butene, yin/adm jr., 12/95 CPB X X C 90.0000 0 0.0000 ! ALLOW HEM CE1 CT2 CT3 32.00 112.20 ! ! Heme (6-liganded): substituents (KK 05/13/91) ! for 1-butene; from propene, yin/adm jr., 12/95 CT2 X X CPB 90.0000 0 0.0000 ! ALLOW HEM CE2 CE1 CT2 48.00 126.00 ! ! Heme (6-liganded): substituents (KK 05/13/91) ! for 1-butene; from propene, yin/adm jr., 12/95 CT3 X X CPB 90.0000 0 0.0000 ! ALLOW HEM CE2 CE1 CT3 47.00 125.20 ! ! Heme (6-liganded): substituents (KK 05/13/91) ! for propene, yin/adm jr., 12/95 HA C C HA 20.0000 0 0.0000 ! ALLOW PEP POL ARO ! Heme vinyl substituent (KK, from propene (JCS)) HA CPA CPA CPM 29.4000 0 0.0000 ! ALLOW HEM O. Michielin, SIB/LICR
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