Homework 2 (due We, Feb. 5): Reading: Van Holde, Chapter 1 Van Holde Chapter 3.1 to 3.3 Van Holde Chapter 2 (we’ll go through Chapters 1 and 3 first. 1.Van Holde 3.1(Coulomb potential) 2.Van Holde 3.2(Dipole-diople interaction) 3.Van Holde 3.3(Lennard-Jones potential) 4.Van Holde 3.4(force is gradient of potential) 5.Van Holde 3.6(Manning theory of counterion condensation) 6.Van Holde 3.9(Hydrogen-bond potential) Paper list (for presentations) is posted on web site Introduction-2 Important molecular interactions in Biomolecules

Type of Interaction Distance Relationship Charge-charge1/r Charge-dipole1/r 2 Dipole-dipole1/r 3 Charge-induced dipole 1/r 4 Dispersion1/r 6 Figue 1.6 Energies of molecular interactions. The interactions that define the structure of a molecule range from the strong interactions of covalent bonds (200 to 800 kJ/mol) to the weak ion-ion, dipole-dipole, dispersion and hydrogen-bonding interactions (0 to 60 kJ/mol)

The notion of potential energy The force, with which molecules attract or repel each other is: Force is negative of gradient (slope) of potential well.  Molecules get “pushed” to the lowest point of potential.  Molecules sit in minimum of potential function. V x Molecular potentials: (Temperature makes them jiggle around Once you know the potential, you can calculate the force fields and simulate dynamics (molecular dynamics simulations): For example: Fibrinogen stretching, Drug binding to proteinFibrinogen stretchingDrug binding to protein

Bonding potential (covalent bonds) V bonding is big, but  V bonding between (protein) conformations is not so big.  V nonbonding is what counts most for (protein) folding and binding. The bond potentials are often approximated by harmonic potentials (good for small deflection): ( kJ/mole) Figure 3.3 The potential energies for deforming a covalent bond V B and the bond angle between three bonded atoms V , as treated by molecular mechanics force fields. The deformations are modeled as harmonic springs, with a spring constant for stretching +  r or compressing –  r a bond from equilibrium bond length r 0. The spring model overestimates the potential energy for stretching at large  r. Deformations to the bond angle  0 are similarly treated as a spring between 1-3 atoms of the three bonded atoms.

Charge-charge interaction A lot of biological molecules are charged. Amino acids: Asp -, Glu -, Lys + Arg +, His + ; DNA: phosphates - in backbone, salt ions, etc. Like charges repel, opposite charges attract. The potential energy of two point charges is: q 1,2 … point charges (charge of electron e = 1.6* C) r … separation of charges D … Dielectric constant; D =  r  0 (  r  0 in SI units)  0 … permittivity constant  0 = 8.85* C 2 /Nm 2  r … relative permittivity (depends on material) The tricky part 1: D not constant) D water = 78.5·  0 (easy). D inside a protein varies (1·  0 to 20·  0, average ~3.5  0 ) and depends on the local environment Various approx. to deal with this Tricky part 2: (counterion screening) Really also related to D not being constant) Counterions (salt) condense/surround fixed charges on protein/DNA (~ 60 kJ/mole, long-range)

Counterion shielding (screening) N fixed point charges in vacuum have potential energy: The ions in solution are then forming a “cloud” around those charges, which effectively screens the fixed charges from each other: Debye-Hückel screening length: (For an aqueous electrolyte at T = 25 o C) I … ionic strength: c … ion concentration (moles/liter) Z… valency of ion species Solution ions form a “cloud’ around the fixed charges on protein surfaces or DNA.  The electric potential of these fixed charges is weakened (damped). This damping is called Screening. General: with I in mole/m 3

Counterion condensation (Manning theory) How large is the net charge of the phosphates then? Amount of charge neutralized: Thus, for B-DNA, 76% is neutralized in aqueous Na + environment and 24% is not compensated. 88% is neutralized in aqueous Mg 2+ environment! Charges also condense (bind longer) onto fixed charges (mostly applies to polyions, like DNA), and partly neutralize them. This effect does not depend on ion concentration (like screening). For condensation of counterions onto DNA: If a charge is actually “condensed”, depends on the parameter  For  > 1, condensation occurs. For an electrolyte in water,  = 0.71/b, with b in nm. At 25°C,  = 4.2 for B DNA.  So, counterions do condense on DNA. b … distance between charges (make polyion a line charge ). For DNA: b = h/Z = 0.34/2 = 0.17nm Manning, G.S. (1969). "Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions I. Colligative Properties". J. Chem. Phys. 51, 924

Dipole-dipole Interactions Dipole moment (note: vectors!) +- General dipole-dipole interaction: If  1 and  2 are side by side: If  1 and  2 are parallel: (~ -2 to 2 kJ/mole, shorter range, can act in series)

Induced dipole-induced dipole interactions (van der Waals interactions) Attractive London Dispersion potential: I … ionization energy of atoms  1,  2 polarizabilities of atoms Repulsive potential from electron cloud: Strong force when r is small, m = 5 to 12 Combining them we get the van der Waals potential: And for m = 12 the Lennard-Jones potential: A, B are constants that depend on the type of interacting atoms in table 3.4 (~ kJ/mole, very short range)

Hydrogen-bonds Important for protein/DNA stability Easy make-easy break, directional  requires very accurate alignment. 11 22 D-H …………. A This potential is added to the dipole-dipole interaction and only contributes ~ 2 KJ/mole of energy. H-bonds – it is still debated how to treat H-bond potential. Treat like dipole-dipole interaction (4-48 kJ/mole). Best alignment (diagram) gives: However, some aspects of H-bond are similar to covalent bonds – e.g. the optimal distance between donor and acceptor is very short (see next page)! So, model that by a van der Waals potential: C, D depend on the particular donor and acceptor In real life, we need to consider H-bonding to water too!! So unless solvent is accounted for, H-bonding affect is overestimated. Hydrogen bonds are important in the inside of proteins and DNA base-pairing. (0-48 kJ/mole, very short range, directional  provide specificity)

H-bond examples Watson-Crick base-pairing Hydrogen bonds in water © Sinaur and Associates, Inc.

 -helix (© by Irvine Geis) Biochemistry Voet & Voet Red – oxygen Black – carbon Blue – nitrogen Purple – R-group White – C  Hydrogen-bonds between C-O of n th and N-H group of n+4 th residue. Non-Watson-Crick base-pairing Watson-Crick base- pairing

… A word about pH, pK and problem 3.1 (on white board) Then assume  S is zero, and  H is equal to the Coulomb potential.