Introduction to Biophysics Lecture 3 Molecular forces in Biological Structures

Discrete distribution Probability of observing x i N i /N =P(x i ) for large N  i P(x i ) = 1 normalization condition

Continuous distributions Bins of width dx dN (x 0 )/N =P(x 0 )dx for large N If P(x) approaches a smooth limiting function we call P(x) probability distribution. Always nonnegative. Has dimension inverse to these of x.  dxP(x) = 1 normalization condition. Example: Uniform distribution: P(x) = 1/a if 0≤x≤a P(x) = 0 otherwise Gaussian distribution (normal distribution) Variance =  2 (= k B T/m for ideal gas) Boltzmann distribution: where E is total energy

Average (or mean or expectation value):  x  =  i x i  P(x i ) for discrete distribution  x  =  dx  x  P(x) for continuous distribution Example: for symmetric distribution, such as Gaussian distribution, the mean in the center point. Root-mean-square deviation (or RMS deviation or standard deviation):

Electrostatic self energy: The energy of placing ion in a dielectric medium. Consider the work done to bring a small increment of charge  q’ to the surface of a sphere with radius r, already carrying a charge, q’ This charging process can be integrated to get the total work done, starting with charge = 0 and final charge = q. Find a difference between the electrostatic self-energy for ion (Na +, r=0.95 Å) in two media (water and hydrocarbon(membrane)) and estimate the free energy of transfer of Na + ions between two media. Give an answer in Joule, k B T (at Room Temperature) and kcal/mole.

Can we predict partition- (P) or distribution coefficient (D) (the ratio of concentrations of a compound in the two phases of a mixture of two immiscible solvents at equilibrium.) of charged particle (ion) in water/hydrocarbon system by knowing its electrostatic self-energy?

Ion in the membrane 50 Å putting an ion into the middle of a lipid bilayer is almost as hard as transferring it to a bulk hydrocarbon medium. How it can be stabilized? Ion in the globular protein 1.Small pocket of water can stabilize an ion within non-polarizable medium. 2.High relevance for ion channels - water cavity about 5Å stabilizes charge inside membrane.

Channels – allow the passage of specific molecules under specific conditions. Pumps – actively pull ions across membrane.

In the K + channel four alfa-helices forming a right-handed bundle Nature 2002 The membrane electric potential across the pore changes in opening

non-covalent interactions Electrostatic interaction Charge – dipole interaction Induced Dipoles Cation -  interaction Dispersion forces Hydrophobic forces Hydration forces Hydrogen bounds Steric repulsion

Charge – Dipole Interaction (fixed geometry): When r >> a Dipole moment Rotation weakens interaction Here and further on  =4  0  How rotation of molecules affect this interaction?

The dipoles with fixed orientations (U  1/r 3 ), as would be the case within a folded protein with a fairly rigid structure. For example, the carbonyl groups of polypeptides have substantial dipole moments and their electrostatic interactions make an important energetic contribution to the stability of an a-helix Dipole – Dipole Interaction in rigid structure: O C D= 3.7 Debye

The van der Waals force (or van der Waals interaction), is the attractive or repulsive forces between molecules (or between parts of the same molecule) other than those due to covalent bonds or to the electrostatic interaction of ions with one another. Example: force between a permanent dipole and a corresponding induced dipole (Debye force) force between two instantaneously induced dipoles (London dispersion force). It is also sometimes used loosely as a synonym for the totality of intermolecular forces. Van der Waals forces are relatively weak compared to covalent bonds, but play a fundamental role in fields as diverse as supramolecular chemistry, structural biology, polymer science, nanotechnology, surface science, condensed matter physics.

Induced Dipoles: Molecules with neither a net charge nor a permanent dipole moment can be influenced by electrical forces (polarization) Charge – Induced dipole: Permanent dipole – Induced dipole: Home work: derive this d-induced dipole moment  - polarizability

Interaction energy of argon dimer. The long range part is due to London dispersion forces. Without London forces, there would be no attractive force between noble gas atoms, and they wouldn't exist in liquid form. Dispersion forces (London fource)

The charge distribution of a molecule fluctuates rapidly with time. At any instant there will be a transient dipole moment. Since the attractive configurations have a lower potential energy than the repulsive configurations, they will have larger weights in a Boltzmann average, leading to a net attraction. There are many theoretical approaches to the calculation of this interaction energy, all of which are quite complicated. where  1 and  2 denote the polarizabilities of the two interacting molecules, I 1 and I 2 denote their ionization energies, and n denotes the refractive index of the medium. The fluctuations in the electronic structure responsible for transient dipole moments are much faster than molecular rotations in a liquid. In water media dispersion force between molecules is much stronger than interactions involving rotating permanent dipoles. Two rotating permanent dipoles

Quantities that vary with frequency are said to exhibit dispersion, and that is why this force is commonly called the dispersion force. Dispersion forces (London fource) ‘‘like dissolves like.’’ An important consequence of dispersion is that two molecules may not attract one another strongly if the frequency ranges in which their polarizabilities are large do not overlap. For interactions between similar molecules the frequency ranges will be well matched, and the dispersion force will be stronger.

Home work: 1.Find the value A required to normalize Gaussian distribution. 2.Derive Energy of the Charge – Induced dipole interaction. Suggested reading: Glaser, Chapter 2.1 and Nelson