Introduction to Biophysics Lecture 2 Molecular forces in Biological Structures

The Concept of Free Energy: -“useful” energy of a system - the part of total energy that can be harnessed to do “useful” work F=E-TS (total energy – randomness (or disorder)) If  F<0 – process is spontaneous, T=constant F can decrease if E decreases (exmp. - heat loss) S increases (disorder tents to increase) Life doesn’t create order from nowhere. Life captures order, ultimately from the Sun. Prosesses of free energy transduction then transmit order through the biosphere.

Biological molecules are polymers : The free energy associated with a covalent bond is ~ 100 – 150 k B T. These bonds are therefore not disrupted by thermal fluctuations. lipids

The Schrödinger equation is the theoretical basis for calculation of the wave functions of electrons and the probability of their presence at a particular point in space. Quantum numbers: n, l, m l, m s Pauli exclusion principle - It is impossible for two electrons with identical quantum numbers to occur in the same atom.

Atomic orbitals - illustration of the statistical nature of the electron distribution: Note that spin quantum number does not influence either the shape or the size of the orbitals.

How chemical bond is made? What determines its length? Energy diagram for formation of the hydrogen molecule E = Ee,kin + Ee,e +Ee,n

Molecular orbitals / Molecular geometries Carbon: Peculiar property of carbon atom which enabled the emergences of life. H2OH2O  (H-O-H) = 90   (H-O-H) =  Electrostatic repulsion of the valence electrons sp-orbital sp 3 -orbital (hybrid orbital)  - bond (  -electrons) have rotational symmetry, can be subjected to thermal rotation sp 3 sp 2 CH 4

Name non-covalent interactions:

non-covalent interactions: Electrostatic interaction (special case of ionic bond) Charge – dipole interaction Induced Dipoles Dispersion forces Hydrophobic forces Hydration forces Hydrogen bond Steric repulsion

non-covalent interactions: govern how protein folds determine structure of nucleic acids and lipid bilayers drive association between macromolecule and ligand They are generally well understood to the extent that good approximate mathematical expressions are available. Much is known about their relative strength under various conditions. Structure and dynamics of biological macromolecules are determined by interplay of many forces.

Question biophysicists ask: How are Genes Turned On and Off? The discovery of the double helical structure of the DNA was a breakthrough for many branches of biology, because it explained how the structure of the molecule allows it to function as a template for copying genetic information. However, it was not clear how DNA works in three dimensions – what sorts of three-dimensional structures are superimposed on the double helix and how these influence the interactions of DNA with other molecules in its environment. Binding of proteins to DNA determines when and how genes are turned on and off and in turn regulates both normal and abnormal development.

Electrostatic interaction Transport of ions across the biological membrane. Some macromolecules are electrically charged (DNA) Electrical charge on macromolecules (mostly negative) prevents them from aggregation Detailed pattern of the negatively and positively charged residues on protein surface can be responsible for protein-protein or protein- substrate interaction and binding

Electrostatic interaction Two charges q 1 and q 2, separated by a distance r. The Coulomb force is given by k=1/4  0, where  0 = 8.85x C 2 /Nm 2 is the permittivity of free space (vacuum). (k = 9x10 9 Nm 2 /C 2 ). The potential energy U of two charges is defined as the work required to bring the two charges to a distance r apart if they are initially infinitely far away: The potential energy is negative when the force is attractive (q 1 and q 2 have opposite signs), and positive when the force is repulsive. The potential energy for the simple Coulomb interaction falls off rather gradually, as 1/r, and hence it is also referred to as a long-range interaction. 1=19992

Dielectric constant (  ) depends on how easy the molecules in the environment are polarized. In completely unresponsive medium (vacuum  =1), U ~ 30 k B T for two charges q 1 = q 2 = e - (1.6x C) separated by r ~ 2nm. In strongly responsive media (In water,  =80) polarization of molecules counteracts the electric field. The Coulomb interactions are reduced by a factor which is equal to the . Water molecules have a permanent dipole; they align in the direction of the local electric field and effectively screen the charges. Therefore, in water, U ~ 0.4 k B T for two ions carrying unit charges separated by ~ 2 nm. Dielectric Constant Ion in water does a considerable amount of work on the surrounding water molecules by forcing them to rotate and orient their dipoles.

In hydrocarbon,  =2, this makes electrostatic interactions within proteins and membranes very strong. Complications: spatial variation in  inside biomolecules prevent simple use of Coulomb potential. (solution – employ Poisson or Laplace equations with boundary conditions given by geometry of system) We will deal with simple planar, spherical or cylindrical systems. Dielectric Constant

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.

Home work 1.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. Using  G number you found estimate a partition coefficient of Na + ion in water versus hydrocarbon (model for membrane) media. Partition coefficient is the ratio of concentrations of a compound in the two phases of a mixture of two immiscible solvents at equilibrium. 2. Estimate the value of the bonding energy between the Na + and Cl - in NaCl. Consider interaction as pure ionic (Coulomb law). 3. Nelson page 31 problem 1.3 Metabolism Suggested reading: Chapter 2.1 (till 2.1.4) Roland Glaser “ Biophysics”