Noncovalent Interactions in Biomolecules PDF

Noncovalent Interactions in Biomolecules Interparticle Forces of Attraction • Various forces of attraction that exist between the atoms and molecules of a substance • Hold particles together and determine many of the physical properties of a substance Electrostatic Energy and Coulomb’s Law • For...

Noncovalent Interactions in Biomolecules Interparticle Forces of Attraction • Various forces of attraction that exist between the atoms and molecules of a substance • Hold particles together and determine many of the physical properties of a substance Electrostatic Energy and Coulomb’s Law • Forces • Attractive or repulsive • Depends on properties of groups • Distances between interacting groups 𝒒𝟏 𝒒𝟐 𝑼 𝒓 = 𝟒𝝅𝜺𝒐 𝒓 q1 and q2 ≡ charges of particles in Coulombs r ≡ distance in meters 𝞮o ≡ permittivity constant = 8.854 x 10-12 C2 N-1 m-2 U(r) ≡ potential energy in Joules 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 Electrostatic Energy and Coulomb’s Law 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 (𝒆𝒏𝒆𝒓𝒈𝒚 𝒊𝒏 𝒌𝑱 𝒎𝒐𝒍:𝟏 ) 𝜺𝒓 𝞮 for hydrocarbons: 2 𝞮 for water: 80 • Interaction of two charges in a nonpolar membrane is much larger than when they are surrounded by water Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° -0.834 0.957 Å +0.417 W.L. Jorgensen et. al., 1983, J. Chem. Phys. 79:926 Find xy-coordinates for the atoms so that interatomic distances can be calculated. Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å Find xy-coordinates for the atoms so that interatomic distances can be calculated. -0.834 0.957 Å +0.417 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å -0.834 0.957 Å +0.417 The distances between each atom of the donor water molecule and each atom of the acceptor water molecule are calculated by 𝒓𝒊𝒋 = 𝒙𝒋 − 𝒙𝒊 𝟐 + 𝒚𝒋 − 𝒚𝒊 𝟐 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å -0.834 0.957 Å +0.417 The electrostatic energies are calculated using the following 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 𝑼𝑯:𝑯 𝒕𝒆𝒓𝒎𝒔 = 𝟏𝟑𝟖𝟗 𝟎. 𝟒𝟏𝟕 𝟐 × 𝑼𝑯:𝑯 𝒕𝒆𝒓𝒎𝒔 = +𝟑𝟓𝟏. 𝟏 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝟏 𝒓𝟏𝒂:𝟐𝒂 + 𝟏 𝒓𝟏𝒂:𝟐𝒃 + 𝟏 𝒓𝟏𝒃:𝟐𝒂 + 𝟏 𝒓𝟏𝒃:𝟐𝒃 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å -0.834 0.957 Å +0.417 The electrostatic energies are calculated using the following 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 𝑼𝑶:𝑯 𝒕𝒆𝒓𝒎𝒔 = 𝟏𝟑𝟖𝟗 𝟎. 𝟒𝟏𝟕 −𝟎. 𝟖𝟑𝟒 × 𝑼𝑶:𝑯 𝒕𝒆𝒓𝒎𝒔 = −𝟕𝟖𝟒. 𝟐 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝟏 𝒓𝑶𝟏:𝟐𝒂 + 𝟏 𝒓𝑶𝟏:𝟐𝒃 + 𝟏 𝒓𝑶𝟐:𝟏𝒂 + 𝟏 𝒓𝑶𝟐:𝟏𝒃 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å -0.834 0.957 Å +0.417 The electrostatic energies are calculated using the following 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 = 𝟏𝟑𝟖𝟗 −𝟎. 𝟖𝟑𝟒 𝟐 × 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 = +𝟑𝟖𝟔. 𝟓 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝑼𝒕𝒐𝒕𝒂𝒍 = −𝟒𝟔. 𝟔 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝟏 𝒓𝑶𝟏:𝑶𝟐 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 104.52° 2.500 Å -0.834 0.957 Å +0.417 The electrostatic energies are calculated using the following 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 𝟏 There is an attractive force 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 = 𝟏𝟑𝟖𝟗 −𝟎. 𝟖𝟑𝟒 𝟐 × 𝒓𝑶𝟏:𝑶𝟐 between the water :𝟏 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 = +𝟑𝟖𝟔. 𝟓 𝒌𝑱 𝒎𝒐𝒍 molecules in this orientation. 𝑼𝒕𝒐𝒕𝒂𝒍 = −𝟒𝟔. 𝟔 𝒌𝑱 𝒎𝒐𝒍:𝟏 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 0.957 Å 104.52° -0.834 +0.417 𝑼𝑯:𝑯 𝒕𝒆𝒓𝒎𝒔 =? 𝑼𝑶:𝑯 𝒕𝒆𝒓𝒎𝒔 =? 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 =? 𝑼𝒕𝒐𝒕𝒂𝒍 =? Using the same process for (b), the following terms are obtained and the total U. 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 Electrostatic Energy and Coulomb’s Law Calculate the electrostatic potential energy of interaction for two water molecules arranged (a) favorably for hydrogen bonding; (b) with the same distance between oxygen atoms but rotated so that a hydrogen is not between the O’s. 0.957 Å 104.52° -0.834 +0.417 Using the same process for (b), the following terms are obtained and the total U. 𝒒𝟏 𝒒𝟏 𝑼 𝒓 = 𝟏𝟑𝟖𝟗 𝒓 𝑼𝑯:𝑯 𝒕𝒆𝒓𝒎𝒔 = +𝟐𝟓𝟑. 𝟐 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝑼𝑶:𝑯 𝒕𝒆𝒓𝒎𝒔 = −𝟔𝟎𝟖. 𝟐 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝑼𝑶:𝑶 𝒕𝒆𝒓𝒎𝒔 = +𝟑𝟖𝟔. 𝟓 𝒌𝑱 𝒎𝒐𝒍:𝟏 𝑼𝒕𝒐𝒕𝒂𝒍 = +𝟑𝟏. 𝟓 𝒌𝑱 𝒎𝒐𝒍:𝟏 This arrangement produces a repulsive force on the water molecules. Net Atomic Charges and Dipole Moments • NET ATOMIC CHARGES • Qualitative picture of how molecules interact • Insights into the chemical properties of a molecule Although these calculations are approximate, the results are consistent with intuition. → Nitrogens attract positive species → Ring nitrogens are targets of electrophilic compounds Net Atomic Charges and Dipole Moments • DIPOLE MOMENT, 𝞵 • Measure of a molecule’s polarity If centers of both positive and negative charges are in the same position, there is no dipole moment. Positive Charge +q r -q Negative charge 𝝁 = q𝒓 Symmetry is very helpful in calculating dipole moments. 𝝁𝒙 = P 𝒒𝒊 𝒙𝒊 𝒊 𝝁𝒛 = P 𝒒𝒊 𝒛𝒊 𝒊 𝝁𝒚 = P 𝒒𝒊 𝒚𝒊 𝒊 Net Atomic Charges and Dipole Moments 𝝁 = 𝝁𝟐𝒙 + 𝝁𝟐𝒚 + 𝝁𝟐𝒛 The dimensions of dipole moments are charge times distance; chemists have traditionally used electrostatic units (esu). On 1 electron: -4.803 x 10-10 esu The magnitudes of dipole moments are about 10-18 esu cm. A special unit has been defined in honor of Peter Debye, who first explained the difference between polar and nonpolar molecules. 1 Debye (D) = 10-18 esu cm In SI units, electronic charge is in coulombs (C) and distance in meters. 1 D = 3.33564 x 10-30 C m Net Atomic Charges and Dipole Moments Net Atomic Charges and Dipole Moments Estimate the magnitude of the dipole moment of adenine using the calculated charge distribution shown in the figure. The molecule is assumed to be planar, and the atom position coordinates given in the table below are taken from analysis of X-ray diffraction studies of crystals of an adenine derivative. Dipole-Dipole Interactions When considering the interaction between 2 molecules, we can assign a dipole moment to each molecule. The charge distribution on a molecule is replaced by a molecular dipole – two equal and opposite charges that are very close together. Molecular dipole: vector Dipole-Dipole Interactions At a distance, 2 dipoles can attract each other, repel each other or have zero interaction energy. 𝝁𝟏 𝑎𝑛𝑑 𝝁𝟐 𝑹𝟏𝟐 𝑼 𝑹 = 𝟏𝟑𝟖𝟗𝝁𝟏 𝝁𝟐 𝑹𝟑𝟏𝟐 𝒄𝒐𝒔𝜽𝟏𝟐 − 𝟑𝒄𝒐𝒔𝜽𝟏 𝒄𝒐𝒔𝜽𝟐 London Interactions An instantaneous dipole occurs in one atom that induces an oppositely directed dipole in a neighboring atom. This universal attraction is named after Fritz London, who derived the equation explaining the effect, often called London dispersion forces. 𝑼 𝒓 =− 𝑨𝒊𝒋 𝒓𝟔𝒊𝒋 Polarizability: measure of the mobility or ease of delocalization of the electrons in a molecule London Interactions The London fluctuation dipoleinduced dipole attraction between 2 oxygen atoms in two water molecules. Van der Waals repulsion You cannot put more than two electrons in one electronic orbital. The repulsion can be calculated using quantum mechanics. Hard-sphere model for atoms: Two atoms are assumed to have zero interaction energy until they come in contact at the hard-sphere distance; the interaction energy then goes suddenly to infinity. A better model is a repulsion that depends on the inverse twelfth power of the distance. 𝑩𝒊𝒋 𝑼 𝒓 = 𝟏𝟐 𝑩𝒊𝒋 𝒊𝒔 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒓𝒊𝒋 Van der Waals repulsion Repulsive potential energy of interaction between 2 oxygen atoms in two water molecules. Van der Waals repulsion Van der Waals repulsion: universal repulsion of atoms and molecules at short distances; named after the Dutch scientists who studied it experimentally, Joannes Diderik van der Waals Van der Waals radius: of an atom is determined from the distance between two atoms where the repulsion energy starts rising rapidly. London-van der Waals Interaction 6-12 potential: the sum of the attractive and repulsive energies of interaction 𝑼 𝒓 = 𝑩𝒊𝒋 𝒓𝟏𝟐 𝒊𝒋 − 𝑨𝒊𝒋 𝒓𝟔𝒊𝒋 Distance Dependence of Noncovalent Interactions Relative plots of energy versus distance for charge-charge, dipole-dipole, and London attraction, which show the long-range nature of chargecharge interactions and the short nature of London attraction. Distance Dependence of Noncovalent Interactions Stacking interaction The Lowest Energy Conformation Molecular interactions • Bonded Interactions: • Bond stretching • Bond angle bending • Torsion angle rotation • Nonbonded interactions • London-van der Waals • Coulomb interactions The Lowest Energy Conformation Total energy: 𝑼 = +𝒌 𝒓 − 𝒓𝒆𝒒 𝟐 +𝒌𝒃 𝜽 − 𝜽𝒆𝒒 Bond stretching 𝟐 𝑽𝒏 + 𝟏 + 𝒄𝒐𝒔 𝒏𝝓 − 𝝓𝟎 𝟐 𝑩𝒊𝒋 + 𝟏𝟐 𝒓𝒊𝒋 − 𝑨𝒊𝒋 𝒓𝟔𝒊𝒋 𝒒𝒊 𝒒𝒋 +𝑪 𝒓𝒊𝒋 Bond angle stretching Torsion angle rotation; n=2 or 3 Van der Waals repulsion; Bij is positive London attraction; Aij is positive Coulomb interaction; C depends on the units The Lowest Energy Conformation Total energy: 𝑼 = +𝒌 𝒓 − 𝒓𝒆𝒒 𝟐 +𝒌𝒃 𝜽 − 𝜽𝒆𝒒 𝟐 𝑽𝒏 + 𝟏 + 𝒄𝒐𝒔 𝒏𝝓 − 𝝓𝟎 𝟐 𝑩𝒊𝒋 + 𝟏𝟐 𝒓𝒊𝒋 − 𝑨𝒊𝒋 𝒓𝟔𝒊𝒋 𝒒𝒊 𝒒𝒋 +𝑪 𝒓𝒊𝒋 Energy calculations Docking: prediction of how proteins or nucleic acids will bind a drug or, in general, how two molecules will interact. Interaction energy: difference between the energy when 2 molecules are separated from each other and when they are docked together Energy calculations Hydrogen bonds: an interaction mediated by a hydrogen atom between two electronegative atoms such as O and N. Its contribution to the energy of a structure can be calculated as a sum of nonbonded interactions, but with parameters different from non-hydrogen bonded atoms. Hydrophobic effects: helps determine how proteins fold, how they fit into membranes, and how lipids interact in water. Treatments of Entropy and Condensed Phases A complete treatment of the entropy of a system and surroundings is very difficult to implement on a computer. An even more difficult calculation is involved in trying to understand the stability of condensed-phase structures that are not covalently linked, such as phospholipid bulayers, micells, and even the liquid phase itself. Hydrogen Bonds The H atom in a molecule (HA) must be somewhat acidic. This acidic hydrogen will interact with an electron donor species, B, to form a hydrogen bond. Hydrogen bonds donors: strong acids (HCl), weak acids (H2O) and even molecules containing C−H bonds (CHCl3). Hydrogen bond acceptors: atoms of the most electronegative elements (F, O, N) Hydrogen Bonds Hydrogen Bonds Hydrogen bonds are directional. Hydrogen Bonds The experimental criterion that establishes that a hydrogen bond exists between A and B (A, B = N or O) is that the A • • • B distance in A − H • • • B is about 2.8 to 3.0 Å. Quantum mechanical description: involves the electrostatic attraction of the positively charged hydrogen nucleus of the donor to a negatively charged, nonbonding orbital on the acceptor. Hydrogen bonds provide a path for moving charges. Amphiphilic Biomolecules HYDROPATHY INDEX Hydropathy profile of lactate dehydrogenase from dogfish. Amphiphilic Biomolecules Charged residues within hydrophobic regions Molecular Dynamics Simulation By simulating the motions of large molecules on a computer, the free energy of a system can be estimated. The free energy characterizes which reactions occur and which species or conformations are present at equilibrium. Molecular Dynamics Simulation Two methods for calculating entropy and free energy for a system: • Monte Carlo method • Molecular dynamics Free energy depends on the average of 𝑒 :d/fg , averaged over the different arrangements of the systems: 𝑮 = −𝑹𝑻 ln 𝑒 :d/fg Monte Carlo Method Molecular Dynamics Method Directly simulates the motions of all molecules in the system; Newton’s Law is used for each molecule The force on each atom of a molecule is calculated from the potential energy, U(r). 𝑭𝒐𝒓𝒄𝒆 = (𝒎𝒂𝒔𝒔)(𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏) Molecular Dynamics Method Simulated Annealing • MD calculations are initiated at very high temperatures to try to avoid local minima and then the motions are gradually cooled to yield a candidate conformation. Free energy change of unfolding Binding of a substrate to an enzyme Summary A large number of covalent and noncovalent interactions determine the conformations of macromolecules. The sheer number of weak interactions leads to a specific structure of reasonable thermodynamic stability. There have been great successes in the use of physical principles to understand biological structure. E.g. Structure-based drug design