Problem Set
Problems & Further Study
Eight problems at increasing difficulty. Stars indicate rough effort level: ☆ a short calculation; ☆☆ requires combining two concepts from the course; ☆☆☆ open-ended or numerical.
Problem 1 Ā ☆
Starting from the Helfrich spectrum \( \langle|h_q|^2\rangle = k_B T / (\kappa q^4 + \sigma q^2) \) , compute the root-mean-square height of thermal undulations of a 1 Ī¼m Ć 1 Ī¼m patch of pure DOPC bilayer (Īŗ = 20Ā kBT) under zero tension. Compare with the same patch under Ļ = 100Ā Ī¼N/m. Discuss the biological significance of the crossover wavelength.
Problem 2 Ā ☆☆
A nuclear import cargo binds importin-Ī² with Kd = 10 nM and the importinācargo complex partitions into the FG-phase with free energy ā5Ā kBT. Estimate the concentration enrichment of cargo in the NPC over the cytosol, and the diffusion time across a 30 nm channel given a local diffusion coefficient of 10Ā Ī¼mĀ²/s.
Problem 3 Ā ☆☆
Derive the stoichiometry relation between F1F0 c-ring size and the effective H+/ATP ratio. Why might organisms adapted to low proton-motive force (e.g., alkaliphiles, Propionigenium modestum, spinach chloroplast) be expected to have larger c-rings?
Problem 4 Ā ☆☆☆
Using the Marcus rate expression, estimate the rate-limiting step of the ETC given typical values |HDA|Ā² = 10ā4 eVĀ², Ī» = 0.7 eV, and ĪGĀ° = ā0.2 eV for each hop. At what driving force does the rate maximise? What is the cost of operating in the inverted region?
Problem 5 Ā ☆☆☆
Numerically solve the UPR ODEs of ModuleĀ 3 for a chronic stress input ksyn = 2āksyn0. Identify the parameter regimes in which (a) the UPR resolves stress, (b) the system oscillates, (c) a bifurcation to cell-death attractor occurs. Comment on the relevance to neurodegeneration.
Problem 6 Ā ☆☆☆
For a two-component FloryāHuggins system, derive the spinodal from the free energy expression of ModuleĀ 6 and sketch the phase diagram as a function of Ļ and Ļ. Discuss how multivalency (effective N) shifts the critical concentration, and why this matters for ALS-linked mutations in FUS and TDP-43.
Problem 7 ā Integrative Ā ☆☆☆
A small cell (10 Ī¼m diameter) sustains a mitochondrial pmf of ā210 mV across an inner-membrane area of 200Ā Ī¼mĀ². Estimate (a) the electrical capacitance of the IMM (assume 1 Ī¼F/cmĀ²), (b) the charge separation corresponding to the pmf, (c) the time to dissipate this charge at full ATP-synthase current (~100 electrons/s per complex, 105 complexes). What does this timescale imply about the minimum sustainable rate of respiration?
Problem 8 ā Conceptual Ā ☆
In one page: identify three specific points at which the biophysics of ModuleĀ 2 (the nucleus), ModuleĀ 3 (the ER), and ModuleĀ 7 (membrane contact sites) depend on the FloryāHuggins framework of ModuleĀ 6. Does this suggest that āmembraneboundā and āmembranelessā are a natural dichotomy, or an artefact of historical classification?
Further Reading
- Alberts et al., Molecular Biology of the Cell, 7th ed. ā canonical reference; chapters 12ā14 cover intracellular compartments and trafficking.
- Phillips, Kondev, Theriot & Garcia, Physical Biology of the Cell, 2nd ed. ā for the biophysical formalism used throughout this course.
- Boal, Mechanics of the Cell, 2nd ed. ā the most complete treatment of the Helfrich Hamiltonian and its experimental implications.
- Lane, Power, Sex, Suicide: Mitochondria and the Meaning of Life ā readable counterpart to the bioenergetics in ModuleĀ 4.
- Hyman, Weber & JĆ¼licher (2014); Shin & Brangwynne (2017); Mittag & Pappu (2022) ā successive review generations on LLPS biophysics.
- Scorrano et al. (2019); Prinz (2014); Phillips & Voeltz (2016) ā MCS field reviews.