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Energetics of protein structures Energetics of protein structures • Molecular Mechanics force fields • Implicit solvent • Statistical potentials Energetics of protein structures • Molecular Mechanics force fields • Implicit solvent • Statistical potentials 1 What is an atom? • Classical mechanics: a solid object • Defined by its position (x,y,z), its shape (usually a ball) and its mass • May carry an electric charge (positive or negative), usually partial (less than an electron) Example of atom definitions: CHARMM MASS 20 C 12.01100 C ! carbonyl C, peptide backbone MASS 21 CA 12.01100 C ! aromatic C MASS 22 CT1 12.01100 C ! aliphatic sp3 C for CH MASS 23 CT2 12.01100 C ! aliphatic sp3 C for CH2 MASS 24 CT3 12.01100 C ! aliphatic sp3 C for CH3 MASS 25 CPH1 12.01100 C ! his CG and CD2 carbons MASS 26 CPH2 12.01100 C ! his CE1 carbon MASS 27 CPT 12.01100 C ! trp C between rings MASS 28 CY 12.01100 C ! TRP C in pyrrole ring Example of residue definition: CHARMM RESI ALA 0.00 GROUP ATOM N NH1 -0.47 ! | ATOM HN H 0.31 ! HN-N ATOM CA CT1 0.07 ! | HB1 ATOM HA HB 0.09 ! | / GROUP ! HA-CA--CB-HB2 ATOM CB CT3 -0.27 ! | \ ATOM HB1 HA 0.09 ! | HB3 ATOM HB2 HA 0.09 ! O=C ATOM HB3 HA 0.09 ! | GROUP ! ATOM C C 0.51 ATOM O O -0.51 BOND CB CA N HN N CA BOND C CA C +N CA HA CB HB1 CB HB2 CB HB3 DOUBLE O C 2 Atomic interactions Non-bonded Torsion angles pair Are 4-body Angles Bonds Are 3-body Are 2-body Forces between atoms Strong bonded interactions b U = K (b ! b0 ) 2 All chemical bonds θ U = K (! " ! 0 ) 2 Angle between chemical bonds Preferred conformations for φ Torsion angles: - ω angle of the main chain - χ angles of the sidechains U = K (1 " cos(n! )) (aromatic, …) Forces between atoms: vdW interactions r 1/r12 Lennard-Jones potential Rij 12 && R # 6 1/r6 &R # # ELJ ( r ) = ( ij $ $ ij ! ' 2$ ij ! ! $$ r ! $ r ! ! %% " % " " Ri + R j Rij = ; ! ij = ! i! j 2 3 Example: LJ parameters in CHARMM Forces between atoms: Electrostatics interactions r Coulomb potential 1 qi q j qi qj E (r) = 4"! 0! r Some Common force fields in Computational Biology ENCAD (Michael Levitt, Stanford) AMBER (Peter Kollman, UCSF; David Case, Scripps) CHARMM (Martin Karplus, Harvard) OPLS (Bill Jorgensen, Yale) MM2/MM3/MM4 (Norman Allinger, U. Georgia) ECEPP (Harold Scheraga, Cornell) GROMOS (Van Gunsteren, ETH, Zurich) Michael Levitt. The birth of computational structural biology. Nature Structural Biology, 8, 392-393 (2001) 4 Energetics of protein structures • Molecular Mechanics force fields • Implicit solvent • Statistical potentials Solvent Explicit or Implicit ? Potential of mean force A protein in solution occupies a conformation X with probability: U (X ,Y ) " X: coordinates of the atoms e kT of the protein P( X , Y ) = U (X ,Y ) " !!e kT dXdY Y: coordinates of the atoms of the solvent The potential energy U can be decomposed as: UP(X): protein-protein interactions U ( X , Y ) = U P ( X ) + U S (Y ) + U PS ( X , Y ) US(X): solvent-solvent interactions UPS(X,Y): protein-solvent interactions 5 Potential of mean force We study the protein’s behavior, not the solvent: PP ( X ) = ! P( X , Y )dY PP (X) is expressed as a function of X only through the definition: WT ( X ) " e kT PP ( X ) = WT ( X ) " !e kT dX WT(X) is called the potential of mean force. Potential of mean force The potential of mean force can be re-written as: WT ( X ) = U P ( X ) + Wsol ( X ) Wsol(X) accounts implicitly and exactly for the effect of the solvent on the protein. Implicit solvent models are designed to provide an accurate and fast estimate of W(X). Solvation Free Energy Wsol + + Sol Vac ! Wch Wch Wnp Wsol = Welec + Wnp = ( ch ! Wch )+ ( vdW + Wcav ) W sol vac W 6 The SA model Surface area potential N Wcav + WvdW = ! " k SAk k =1 Eisenberg and McLachlan, (1986) Nature, 319, 199-203 Surface area potentials Which surface? Accessible Molecular surface Surface Hydrophobic potential: Surface Area, or Volume? Surface effect (Adapted from Lum, Chandler, Weeks, J. Phys. Chem. B, 1999, 103, 4570.) Volume effect “Radius of the molecule” For proteins and other large bio-molecules, use surface 7 Protein Electrostatics • Elementary electrostatics • Electrostatics in vacuo • Uniform dielectric medium • Systems with boundaries • The Poisson Boltzmann equation • Numerical solutions • Electrostatic free energies • The Generalized Born model Elementary Electrostatics in vacuo Some basic notations: ! Fx ! Fy ! Fz Divergence " • F = div F = () !x + !y + !z ( !f !f !f % Gradient "f = grad ( f ) = & ' !x !y !z # $ !2 f !2 f !2 f Laplacian ( ) " • "f = div grad ( f ) = "2 f = + + !x 2 !y 2 !z 2 Elementary Electrostatics in vacuo Coulomb’s law: The electric force acting on a point charge q2 as the result of the presence of another charge q1 is given by Coulomb’s law: r q1q2 q1 q2 u F= u 4!" 0 r 2 Electric field due to a charge: By definition: F q1 q1 E= = u q2 4!" 0 r 2 E “radiates” 8 Elementary Electrostatics in vacuo Gauss’s law: The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. Integral form: Differential form: q "(X ) ! E • dA = " div ( E ( X )) = 0 !0 Notes: - for a point charge q at position X0 , ρ(X)=q δ (X-X0) - Coulomb’s law for a charge can be retrieved from Gauss’s law Elementary Electrostatics in vacuo Energy and potential: - The force derives from a potential energy U: F = ! grad ( ) U - By analogy, the electric field derives from an electrostatic potential φ: E = " grad (! ) For two point charges in vacuo: Potential produced by q1 at at a distance r: q1q2 q1 U= #= 4!" 0 r 4!" 0 r Elementary Electrostatics in vacuo The cases of multiple charges: the superposition principle: Potentials, fields and energy are additive For n charges: qN N qi qi % (X ) = ! i =1 4#$ 0 X " X i X N qi q1 E( X ) = ! u 2 i q2 i =1 4#$ 0 X " X i qi q j U =! i< j 4#$ 0 X " X i 9 Elementary Electrostatics in vacuo Poisson equation: " div E =() !0 " ( ) div grad (# ) = % • %(# ) = %2 (# ) = $ !0 Laplace equation: !2" = 0 (charge density = 0) Uniform Dielectric Medium Physical basis of dielectric screening An atom or molecule in an externally imposed electric field develops a non zero net dipole moment: - + (The magnitude of a dipole is a measure of charge separation) The field generated by these induced dipoles runs against the inducing field the overall field is weakened (Screening effect) The negative charge is screened by a shell of positive charges. Uniform Dielectric Medium Electronic polarization: Under external - - - - - field - - - - - - + - - - - - - + - - - - - - - - - Resulting dipole moment Orientation polarization: Under external field Resulting dipole moment 10 Uniform Dielectric Medium Polarization: The dipole moment per unit volume is a vector field known as the polarization vector P(X). " $1 In many materials: P( X ) = # E ( X ) = E( X ) 4! χ is the electric susceptibility, and ε is the electric permittivity, or dielectric constant The field from a uniform dipole density is -4πP, therefore the total field is E = E applied # 4" P E applied E= ! Uniform Dielectric Medium Some typical dielectric constants: Molecule Dipole moment Dielectric (Debyes) of a constant ε of the single molecule liquid at 20°C Water 1.9 80 Ethanol 1.7 24 Acetic acid 1.7 4 Chloroform 0.86 4.8 Uniform Dielectric Medium Modified Poisson equation: " ( ) div grad (# ) = %2 (# ) = $ ! 0! Energies are scaled by the same factor. For two charges: q1q2 U= 4"! 0!r 11 Uniform Dielectric Medium The work of polarization: It takes work to shift electrons or orient dipoles. A single particle with charge q polarizes the dielectric medium; there is a reaction potential φ that is proportional to q for a linear response. !R = Cq The work needed to charge the particle from q i =0 to q i =q: q q 1 2 1 W = ! "R (qi )dqi = C ! qi dqi = Cq = q"R 0 0 2 2 For N charges: 1 N W= ! qi"iR Free energy 2 i =1 System with dielectric boundaries The dielectric is no more uniform: ε varies, the Poisson equation becomes: " (X ) ( ) div ! (X )grad (# ( X ) ) = % • ! (X )%# ( X ) = $ !0 If we can solve this equation, we have the potential, from which we can derive most electrostatics properties of the system (Electric field, energy, free energy…) BUT This equation is difficult to solve for a system like a macromolecule!! The Poisson Boltzmann Equation ρ(X) is the density of charges. For a biological system, it includes the charges of the “solute” (biomolecules), and the charges of free ions in the solvent: ! ( X ) = ! solute ( X ) + ! ions ( X ) The ions distribute themselves in the solvent according to the electrostatic potential (Debye-Huckel theory): " qi! ( X ) ni ( X ) ni : number of ions of type i per unit volume =e kT qi : charge on type i ion ni0 N " ions ( X ) = ! qi ni ( X ) i =1 The potential f is itself influenced by the redistribution of ion charges, so the potential and concentrations must be solved for self consistency! 12 The Poisson Boltzmann Equation q $( X ) & (X ) 1 N " i # • % (X )#$ ( X ) = " " ! qi ni0e kT %0 % 0 i =1 Linearized form: $ (X ) & • ! (X )&# ( X ) = % % ! ( X )" 2 ( X )# ( X ) !0 N 1 2 "2 = 'n q 0 2 i i = I I: ionic strength ! 0!kT i =1 ! 0!kT Solving the Poisson Boltzmann Equation • Analytical solution • Only available for a few special simplification of the molecular shape and charge distribution • Numerical Solution • Mesh generation -- Decompose the physical domain to small elements; • Approximate the solution with the potential value at the sampled mesh vertices -- Solve a linear system formed by numerical methods like finite difference and finite element method • Mesh size and quality determine the speed and accuracy of the approximation Linear Poisson Boltzmann equation: Numerical solution • Space discretized into a cubic lattice. • Charges and potentials are defined on grid points. εw • Dielectric defined on grid lines • Condition at each grid point: εP j : indices of the six direct neighbors of i 6 qi !# $ j =1 ij j + # 0h $i = 6 Solve as a large system of linear !# ij 2 + # ij" ij h 2 equations j =1 13 Electrostatic solvation energy The electrostatic solvation energy can be computed as an energy change when solvent is added to the system: 1 1 Welec = ! qi# RF (i ) = 2 ! qi (#S (i ) " #NS (i )) 2 i i The sum is over all nodes of the lattice S and NS imply potentials computed in the presence and absence of solvent. Approximate electrostatic solvation energy: The Generalized Born Model 1 N Remember: "Gelec = ! qi#iRF 2 i =1 For a single ion of charge q and radius R: q2 & 1 # (GBorn = $ ' 1! Born energy 8*) 0 R % ) " For a “molecule” containing N charges, q1 ,…qN, embedded into spheres or radii R1 , …, RN such that the separation between the charges is large compared to the radii, the solvation energy can be approximated by the sum of the Born energy and Coulomb energy: N qi2 ( 1 % 1 N N qi q j ( 1 % *Gelec = ! & ) 1# + !! & ) 1# i =1 8,+ 0 Ri ' + $ 2 i =1 j "i 4,+ 0 rij ' + $ Approximate electrostatic solvation energy: The Generalized Born Model The GB theory is an effort to find an equation similar to the equation above, that is a good approximation to the solution to the Poisson equation. The most common model is: 1 (1 % N N qi q j )GGB = & " 1#!! 8+* 0 ' * $ i =1 j =1 " rij2 4 ai a j rij2 + ai a j e ΔG GB is correct when rij 0 and r ij ∞ ai : Born radius of charge i: Assuming that the charge i produces a Coulomb potential: 1 1 dV = ! ai 4" r > Ri r4 14 Approximate electrostatic solvation energy: The Generalized Born Model &1 # 1 $ ' 1! %( " rij ΔG GB Further reading • Michael Gilson. Introduction to continuum electrostatics. http://gilsonlab.umbi.umd.edu • M Schaefer, H van Vlijmen, M Karplus (1998) Adv. Prot. Chem., 51:1-57 (electrostatics free energy) • B. Roux, T. Simonson (1999) Biophys. Chem., 1-278 (implicit solvent models) • D. Bashford, D Case (2000) Ann. Rev. Phys. Chem., 51:129-152 (Generalized Born models) • K. Sharp, B. Honig (1990) Ann. Rev. Biophys. Biophys. Chem., 19:301-352 (Electrostatics in molecule; PBE) • N. Baker (2004) Methods in Enzymology 383:94-118 (PBE) Energetics of protein structures • Molecular Mechanics force fields • Implicit solvent • Statistical potentials 15 Statistical Potentials a f(r) r b r(Ǻ) & P (r) # E ( a, b, r ) = ' ln$ ( a ,b ) ! $ ! % P( r ) " Counts Energy Ile-Asp Ile-Asp r(Ǻ) r(Ǻ) Ile-Leu Ile-Leu r(Ǻ) r(Ǻ) The Decoy Game Finding near native conformations 1CTF Score cRMS (Ǻ) ' P( ai , a j , rij ) $ E = ! E (i, j ) = ( ! ln% " i< j i< j % P( r ) " & ij # 16

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protein structure, protein folding, protein structures, secondary structure, amino acids, protein structure prediction, free energy, protein-protein interactions, protein sequence, j. mol. biol, tertiary structure, ab initio, protein interactions, structure prediction, secondary structures

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posted: | 6/9/2010 |

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