Physics 414: Introduction to Biophysics Professor Henry Greenside September 21, 2017

Na ion channel as example of two-state system Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M. Mammals have ~10 kinds of Na channels, ~80 kinds of K channels, reason for diversity of channels remains puzzling

At the blackboard: background material about the chemical potential and the Gibbs distribution

Motivation for chemical potential from maximizing total entropy S of closed system in terms of entropies S1(N1) and S2(N2) of two subsystems Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Holds for ideal gas or dilute solutions Example I: more efficient and general way to derive Langmuir isotherm using chemical potentials and Gibbs factor Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M. Holds for ideal gas or dilute solutions

Compare simplicity of Gibbs distribution argument for ligand-receptor with lattice model, no combinatorics needed

First way to link chemical potential μ to concentration c: lattice model to estimate entropy of mixingSection 6.2.2, pages 262-263 of PBOC text Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

What is meant by a dilute solution? Concentrations much less that ~60 M What is concentration of water in water (of pure solvent)? Typical concentrations of ions in cell 0.01-0.3 M (10-300 mM) Typical protein concentrations in cell much less (fewer, heavier) See Chapter 2 of PBOC for cell census of components See website http://book.bionumbers.org/ “Cell Biology by the Numbers” Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Second way to link μ to c: Sackur-Tetrode formula for entropy of ideal gas Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Note how increasing concentration N/V generally increases μ Let’s look at actual numerical value of μ for He gas T = 300 K and p = 1 atm = 105 N/m2 Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M. Note how increasing concentration N/V generally increases μ

Chemical potentials and equilibrium constant Keq, pages 268-269 PBOC2

Example II: Receptor R that can bind two kinds of ligands 1 and 2 but only one at a time. What are the microstates? There are three: no particles bound to R, one particle of type 1 bound to R, and one particle of type 2 bound to R. What are the energies: binding energy for 1 is E_1, binding energy for 2 is E_2. New ingredient: need to take into account the number of particles of given type bound through the chemical potential for each particle, which is mu_1 and mu_2 respectively. Through the relation mu = mu_0 + k T ln(c/c0), valid for dilute solutions or for ideal gases, a large reservoir with constant chemical potentials mu_1, mu_2 requires mechanism to maintain constant concentrations throughout space Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Example II continued: receptor that can bind two kinds of ligands 1 and 2 Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M. What is probability for receptor to be unoccupied? Wha is probability for ligand 2 to be bound to receptor?

You try: Receptor R can now bind 1 with energy E_1, bind 2 with energy E_2, or bind 1 and 2 simultaneously with energy E_{12}. What is average amount of ligand 1 bound to the receptor (average occupancy <N1> Express your answer in terms of the concentrations c_1 and c_2 of the ligands Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Answer: Probability p1 is rational polynomial in concentrations c1, c2 Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M. Probability p1 is rational polynomial in concentrations c1, c2

What is average occupancy of receptor as function of concentrations c1, c2 of ligands? Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

Statistical physics of hemoglobin (Sec 7. 2 Statistical physics of hemoglobin (Sec 7.2.4) Cooperativity as interaction energy J Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

One-minute End-of-class Question