ENTROPY IN BIOLOGY
by Charles S. Peskin

Handwritten lecture notes of a course taught at the Courant Institute of Mathematical Sciences, New York University

Copyright, Charles S. Peskin, 2020

Lecture 1: ideal gas, entropy of mixing, ideal solution, incompressible ideal solution, osmotic pressure.

Lecture 2: Gibbs free energy, Stirling's formula, entropy = k log(# microstates), finite osmotic system, ensemble of finite osmotic systems, macroscopic osmotic flow.

Lecture 3, Onsager Relation Example and Proof:
Part 1: single-site channel with coupled solvent and solute fluxes, steady-state, equilibrium, near-equilibrium Onsager relation.
Part 2: proof of the Onsager relation for the steady state of a continuous-time Markov chain near equilibrium.
Appendix: uniqueness and positivity of the steady state of a continuous-time Markov chain.

Lecture 4, the Poisson-Boltzmann Equation: derivation from entropy and electrostatic energy with volume constraint, standard and linearized Poisson-Boltzmann equations, balance of pressure and electrostatic forces.

Lecture 5, Control of Cell Volume: osmotic flow and electroneutrality, ionic fluxes and the Einstein relation, space-charge layers, dynamical equations, dynamics on the slow manifold, steady-state.

Lecture 6, Rotary Motors Driven by Transmembrane Ionic Currents:
mobility matrix for the infinite-length case, Onsager relation satisfied.

Lecture 7, Rotary Motors (continued): positive-definiteness of the mobilitiy matrix, boundary conditions, mobility matrix for the finite-length case, motor characteristics and efficiency, ion-pump characteristics and efficiency, Brownian dynamics.

Lecture 7, continued: the multivariable diffusion equation of Brownian dynamics, and the generalized Einstein relation.

Lecture 8, Entropic Spring: force-extension relation of a freely-jointed chain, thermodynamics of the freely jointed chain, entropy as a function of the applied force.

Lecture 9, Introduction to Crossbridge Dynamics in Skeletal Muscle: The version of crossbridge dynamics described in this lecture does not account for the thermal aspect of muscle contraction. This will be discusssed in Lecture 10. The homework problems in Lecture 9 are optional, but you might find the one on pages 15-16 interesting!

Lecture 10, Heat of Shortening and Crossbridge Dynamics in Skeletal Muscle: discoveries of A.V. Hill on the force-velocity curve and the heat of shortening, solutions of the direct and inverse problems of steady-state crossbridge dynamnics, brief comparison with some 21st century experimental data, and a project suggestion.
Papers cited in Lecture 10: Hill 1938; Hill 1964; Lacker & Peskin 1986; Piazzesi 2007.

Lecture 11, Bimolecular Reaction: microscopic equilibrium, Markov chain consistent with microscopic equilibrium and mass action, application of microscopic equilibrium to event-driven simulation with a mixture of fast and slow reactions, macroscopic chemical kinetics, and diffusion-limited microscopic reaction rate.

Lecture 12, Biomolecular Reaction Rates with Diffusion and Detailed Balance: implication of detailed balance for a spatial model with binding/unbinding, equilibrium constant of the reaction, evaluation of k_on, evaluation of p_escape and k_off, project suggestion. Appendix: analysis of the Laplace transform of the probability density function for the binding time.

Homework Review, by Zan Ahmad, Alain Boldini, Tristan Goodwill, Ondrej Maxian, and Guanhua Sun.