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

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 atoms.

Average number of hydrogen bonds within the first water shell around an ion

Molecular Dynamics: Introduction Newton’s second law of motion

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 x 1, v 1 and the potential energy V(x), the molecular trajectory x(t) can be calculated, using,

How a molecule changes during MD

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion Attraction

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion Attraction

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion Attraction

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion Attraction u (2) u (N)

Contributions to Potential Energy Total pair energy breaks into a sum of terms Intramolecular only U str stretch U bend bend U tors torsion U cross cross U vdW van der Waals U el electrostatic U pol polarization Mixed terms Repulsion Attraction u (2) u (N)

Modeling Potential energy

0 at minimum 0

Stretch Energy Expand energy about equilibrium position Model fails in strained geometries –better model is the Morse potential minimumdefine (neglect) harmonic dissociation energy force constant Morse

Bending Energy Expand energy about equilibrium position –improvements based on including higher-order terms Out-of-plane bending minimumdefine (neglect) harmonic   u (4)

Torsional Energy Two new features –periodic –weak (Taylor expansion in  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 

Van der Waals Attraction Correlation of electron fluctuations Stronger for larger, more polarizable molecules –CCl 4 > CH 4 ; Kr > Ar > He Theoretical formula for long-range behavior Only attraction present between nonpolar molecules –reason that Ar, He, CH 4, 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 Two popular treatments inverse powerexponential typically n ~ two parameters Combine with attraction term –Lennard-Jones model Exp-6 a.k.a. “Buckingham” or “Hill” Exp-6 repulsion is slightly softer Beware of anomalous Exp-6 short-range attraction

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

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

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

Polarization Approximation 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 –maybe anisotropic Model vdW forces via discontinuous potentials Ignore all attraction Model space as a lattice –especially useful for polymer molecules Qualitative models

Molecular Dynamics: Introduction Equation for covalent terms in P.E.

Molecular Dynamics: Introduction Equation for non-bonded terms in P.E.

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 Effetive dielectric constant