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

2. 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

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

4. 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.

5. 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

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

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

8. 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 M ( mM)
Typical protein concentrations in cell much less (fewer, heavier)
See Chapter 2 of PBOC for cell census of components
See website “Cell Biology by the Numbers”
Assume V_box = (1 nm)^3, leads to effective concentration 10/6 ~ 1.7 M.

9. 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.

10. 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

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

12. 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.

13. 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?

14. 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.

15. 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

16. 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.

17. 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.

18. One-minute End-of-class Question