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Hydro-pathy/phobicity/philicity • One of the most commonly used properties is the suitability of an amino acid for an aqueous environment • Hydropathy & Hydrophobicity – degree to which something is “water hating” or “water fearing” • Hydrophilicity – degree to which something is “water loving” Hydrophobicity/Hydrophilicity Tables • Describe the likelihood that each amino acid will be found in an aqueous environment - one value for each amino acid • Commonly used tables – Kyte-Doolittle hydropathy – Hopp-Woods hydrophilicity – Eisenberg et al. normalized consensus hydrophobicity Kyte-Doolittle hydropathy Amino Index Acid R -4.5 K -3.9 D -3.5 Q -3.5 N -3.5 E -3.5 H -3.2 P -1.6 Y -1.3 W -0.9 Amino Index Acid S -0.8 T -0.7 G -0.4 A 1.8 M 1.9 C 2.5 F 2.8 L 3.8 V 4.2 I 4.5 Example Hydrophilicity Plot This plot is for a tubulin, a soluble cytoplasmic protein. Regions with high hydrophilicity are likely to be exposed to the solvent (cytoplasm), while those with low hydrophilicity are likely to be internal or interacting with other proteins. Amphiphilicity/Amphipathicity • A structural domain of a protein (e.g., an helix) can be present at an interface between polar and non-polar environments – Example: Domain of a membrane-associated protein that anchors it to membrane • Such a domain will ideally be hydrophilic on one side and hydrophobic on the other • This is termed an amphiphilic or amphipathic sequence or domain Screenshot of a phospholipid bilayer in the process of its modeling. Shown is a computational cell consisting of 96 PhCh molecules and 2304 water molecules which on the whole make up 20544 atoms. Average number of hydrogen bonds within the first water shell around an ion Molecular Dynamics: Introduction Newton’s second law of motion Molecular Dynamics: Introduction We need to know The motion of the atoms in a molecule, x(t) and therefore, the potential energy, V(x) 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 Molecular Dynamics: Introduction Potential energy includes terms for (contd.) Electrostatic Interactions van der Waals Interactions 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, xi xi 1 vi 1t 1 dV ( x) vi vi 1 m xi1 t dx How a molecule changes during MD Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion Mixed terms • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • UvdW van der Waals • Ubend bend • Uel electrostatic • Utors torsion Mixed terms • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • UvdW van der Waals - + - + Attraction • Ubend bend • Uel electrostatic • Utors torsion Mixed terms • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • UvdW van der Waals + - + - Attraction • Ubend bend • Uel electrostatic • Utors torsion Mixed terms • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • UvdW van der Waals + - + - Attraction + - • Ubend bend • Uel electrostatic • Utors torsion Mixed terms • Upol polarization Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • Ubend bend • UvdW van der(2)Waals + - + - Attraction u u(2) ++ - • Uel electrostatic u(N) • Utors torsion Mixed terms • Upol polarization ++ - Contributions to Potential Energy • Total pair energy breaks into a sum U (r N ) U str U bend U tors U cross U vdW U el U pol of terms Intramolecular only Repulsion • Ustr stretch • Ubend bend • UvdW van der(2)Waals + - + - Attraction u u(2) ++ - • Uel electrostatic u(N) • Utors torsion Mixed terms • Upol polarization + - Modeling Potential energy dU 1 d 2U 2 U(r) (r req ) (r r ) U(req ) eq 2 dr r req 2 dr r r eq 1 d 3U 3 dr r req n 1 d U 3 (r req ) .... n! dr n (r req ) n r req Modeling Potential energy 0 0 at minimum 2 1dU dU 2 U(r) U(req ) (r r ) (r req ) eq 2 2 dr r r dr r req eq 1 d 2U 1 2 2 U(r) (r req ) kAB (r req ) 2 2 dr r r 2 eq Stretch Energy • Expand energy about equilibrium position o U (r12 ) U (r12 ) define dU dr r ro o (r12 r12 ) d 2U dr 2 (neglect) o 2 (r12 r12 ) r r o minimum o 2 U (r12 ) k (r12 r12 ) harmonic Morse Energy (kcal/mole) 250 200 150 U (r12 ) in D strained 1 e geometries • Model fails r12 2 100 50 – dissociation better energy model is 0 theforce Morse constantpotential -0.4 -0.2 0.0 0.2 0.4 Stretch (Angstroms) 0.6 0.8 Bending Energy • Expand energy about2equilibrium position U ( ) U ( o ) define dU d U ( o ) d o d 2 ( o ) 2 (neglect) o minimum U ( ) k ( o )2 harmonic u(4) – improvements based on including higher-order o 2 U (c ) k (c c ) terms c • Out-of-plane bending Torsional Energy • Two new features f – periodic – weak (Taylor expansion in f not appropriate) • Fourier series U (f ) n1Un cos(nf ) – 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 Van der Waals Attraction +- +• Correlation of electron fluctuations • Stronger for larger, more polarizable molecules – CCl4 > CH4 ; Kr > Ar > He att U vdW -+ -+ C 8 O ( r ) 6 r • 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 Van der Waals Repulsion • Overlap of electron clouds • Theory provides little guidance on form of model rep A rep UvdW U vdW rn • Two popular treatments inverse power exponential • typically A C n ~ 910- 12 U 20 12 r6 LJ Exp-6 • Combine with attraction term r 8 3 x10 10 – Lennard-Jones model Beware of anomalous Exp-6 short-range attraction 4 C Br parameters two U Ae 6 r a.k.a. “Buckingham” or “Hill” 6 15 Exp-6 Exp-6 repulsion is slightly softer 2 5 0 0 2 4 6 8 Ae Br 1.0 1.2 1.4 1.6 1.8 2.0 Electrostatics 1. • Interaction between charge inhomogeneities • Modeling approaches – point charges – point multipoles • Point charges – assign Coulombic charges to several points in the qi q j molecule U (r ) 4 r 0 – total charge sums to charge on molecule (usually zero) Lennard-Jones Coulomb 1.5 1.0 0.5 0.0 -0.5 -1.0 1 2 3 4 Electrostatics 2. • At larger separations, details of charge distribution are less important 0, Q 0 Vector features i qiri capture • Multipole statistics basic Q i qiriri – Dipole – Quadrupole – Octopole, etc. Tensor 1 3 2 3(ˆ1 rˆ )(ˆ 2 rˆ ) ( ˆ1 ˆ 2 ) r 0, Q 0 Q Q • Pointuddmultipole models based on long-range behavior 3 1Q2 – dipole-dipole u ( ˆ dQ uQQ 2 r4 ˆ) 1 r 5(Qˆ2 rˆ )2 1 2(ˆ1 ˆ2 )(Qˆ2 rˆ ) 3 Q1Q2 2 2 2 2 2 1 5 c 5 c 2 c 35 c c2 20c1c2c12 1 2 12 1 5 4 r – dipole-quadrupole Axially symmetric quadrupole 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 Polarization ind E Approximation ind ,i E i Ei ji q j rij rij3 ij 3 3rij 1 r r ij ji ij rij Electrostatic field does not include contributions from atom i 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 Qualitative – maybe anisotropic • Model vdW forces via discontinuous potentials models • Ignore all attraction • Model space as a lattice – especially useful for polymer molecules Molecular Dynamics: Introduction Equation for covalent terms in P.E. k (l l ) k Vbonded ( R) l 0 bonds 2 2 0 angles k ( 0 ) 2 impropers A [1 cos(nf f )] n torsions 0 Molecular Dynamics: Introduction Equation for non-bonded terms in P.E. Vnonbonded ( R) i j rijmin 12 rijmin 6 qi q j ( ij [( ) 2( ) ] rij rij 4 r 0rij DNA in a box of water SNAPSHOTS Protein dynamics study • Ion channel / water channel • Mechanical properties – Protein stretching – DNA bending Movie downloaded from theoreticla biophysics group, UIUC Solvent dielectric models QiQ j V rij Effetive dielectric constant eff r r r 1 rS rS 2rS 2 e 2 S 0.15Å1 ~ 0.3Å1 2