Solutions

Solution 1 — RMS Height of a DOPC Patch

The real-space variance is the Fourier sum of the spectrum:

\[ \langle h^2(\mathbf{r}) \rangle = \dfrac{1}{(2\pi)^2} \int d^2q\, \dfrac{k_B T}{\kappa q^4 + \sigma q^2} = \dfrac{k_B T}{2\pi} \int_{q_{\min}}^{q_{\max}} \dfrac{dq}{\kappa q^3 + \sigma q} \]

Case (i): σ = 0

\[ \langle h^2 \rangle = \dfrac{k_B T}{4\pi\kappa}\left(\dfrac{1}{q_{\min}^2} - \dfrac{1}{q_{\max}^2}\right) \approx \dfrac{k_B T\, L^2}{16\pi^3 \kappa} \]

With L = 1 μm, κ = 20 k_BT = 8.55 × 10⁻²⁰ J:

\[ \langle h^2 \rangle = \dfrac{1}{16\pi^3 \cdot 20}\cdot L^2 \approx 1.01 \times 10^{-4}\, L^2 \]

So h_rms ≈ 10 nm— a free 1 μm bilayer ripples with ~10 nm amplitude at physiological temperature.

Case (ii): σ = 100 μN/m

Crossover wavevector: q* = √(σ/κ) = √(10⁻⁴/ 8.55 × 10⁻²⁰) ≈ 3.4 × 10⁷ m⁻¹, corresponding to λ* ≈ 185 nm. Since q_min = 2π/L ≈ 6 × 10⁶ m⁻¹ < q*, tension dominates the longest modes. In the tension-dominated regime:

\[ \langle h^2 \rangle \approx \dfrac{k_B T}{4\pi\sigma}\ln\!\left(\dfrac{q^*}{q_{\min}}\right) \approx \dfrac{k_B T}{4\pi\sigma}\ln\!\left(\dfrac{L}{\lambda^*/(2\pi)}\right) \]

Numerically: &langle;h²&rangle; ≈ (4.28 × 10⁻²¹)/(4π × 10⁻⁴) × ln(30) ≈ 1.2 × 10⁻¹⁷m², so h_rms ≈ 3.5 nm. The bending contribution above q* adds a further ~2 nm to give ~4 nm total.

Biological significance. The crossover λ* ~ 200 nm is comparable to typical transport-vesicle and organelle dimensions. Below it, the membrane undulates freely (dominated by bending); above it, tension pins it flat. A cell setting a non-zero tension — via the actin cortex, osmotic pressure, or membrane tethers — buys itself a flat membrane on cell-scale lengths while preserving short-wavelength flexibility. This is precisely the trick the plasma membrane uses.

Solution 2 — NPC Import Enrichment & Transit Time

FG-phase partitioning. Each bound importin-β adds a free energy ΔG_FG = −5 k_BTfor dissolution into the FG phase:

\[ \dfrac{[C]_{\mathrm{in}}}{[C]_{\mathrm{out}}} = \exp\!\left(-\dfrac{n \cdot \Delta G_{\mathrm{FG}}}{k_B T}\right) = e^{5n} \]

For n = 1: enrichment factor ≈ 148×. For n = 2: ~22 000×. For n = 3: ~3.3 × 10⁶×. Cargo decorated with a handful of importin-β receptors therefore enriches in the pore by 10³–10⁶× over cytosol, crowding the passive leakage of similarly-sized inert molecules to negligibility.

Occupancy of the cargo–importin complex. With K_d = 10 nM and free importin ≈ 1 μM, occupancy = [importin]/(K_d + [importin]) ≈ 0.99. The cargo is essentially always bound.

Transit time. For 1D diffusion across a 30 nm channel with local D = 10 μm²/s = 10⁻¹¹m²/s:

\[ \tau = \dfrac{L^2}{2D} = \dfrac{(30\times 10^{-9})^2}{2 \cdot 10^{-11}} = 4.5 \times 10^{-5}\,\mathrm{s} = 45\,\mu s \]

Check. ~2000 NPCs per nucleus, ~150× enrichment, ~45 μs transit → throughput ~150/(45 μs) per pore ≈ 3 × 10⁶ transits/s per pore × 2000 pores ≈ 6 × 10⁹/s nucleus-wide, consistent with the known ~10³translocation events/s per pore observed experimentally (FCS measurements) and the ~10⁴ ribosomes/min export rate.

Solution 3 — c-Ring Size vs H⁺/ATP

One full F₀ revolution translocates n_c protons (one per c-subunit) and drives three binding-change cycles on F₁(α₃β₃), producing 3 ATP. Therefore:

\[ \mathrm{H^+/ATP} = \dfrac{n_c}{3} \]

For ATP synthesis to be thermodynamically favourable, the free energy delivered by proton translocation must exceed the free energy of phosphorylation:

\[ n_c \cdot F\cdot |\mathrm{pmf}| \geq 3 \cdot \Delta G_{\mathrm{ATP}} \quad\Rightarrow\quad |\mathrm{pmf}|_{\min} = \dfrac{3\,\Delta G_{\mathrm{ATP}}}{n_c F} \]

With ΔG_ATP ≈ 50 kJ/mol, F = 96485 C/mol:

Why larger c-rings for low pmf. The alkaliphile Propionigenium modestum operates with a Na⁺-motive force of only ~150 mV and uses a gear reduction to survive on the available energy per ion. Chloroplasts, which experience a pmf limited by the light-harvesting cycle (ΔpH dominant, low Δψ), do the same thing. Both solutions demonstrate the c-ring as a mechanical gear ratio: larger rings → more protons per revolution → smaller proton free-energy required per ATP.

Solution 4 — Marcus ETC Rate

Plug into the Marcus expression (λ = 0.7 eV, ΔG^°= −0.2 eV, |H_DA|² = 10⁻⁴ eV², k_BT = 0.0267 eV):

\[ \text{Exponent} = -\dfrac{(\Delta G^\circ+\lambda)^2}{4\lambda k_B T} = -\dfrac{(0.5)^2}{4(0.7)(0.0267)} = -3.34 \quad\Rightarrow\quad e^{-3.34} = 0.036 \]

\[ k_{ET} = \dfrac{2\pi}{\hbar}|H_{DA}|^2 \dfrac{1}{\sqrt{4\pi\lambda k_B T}} \cdot 0.036 \approx 2 \times 10^{12}\,\mathrm{s^{-1}} \cdot 0.036 \approx 7 \times 10^{10}\,\mathrm{s^{-1}} \]

One hop per ~15 ps. Since the whole ETC must pass an electron through ~7 hops, the net electron transit takes ~100 ps — vastly faster than the ~1 ms turnover of complex I. Therefore the rate-limiting step of the ETC is not electron transfer itself, but substrate binding, conformational changes, and proton-pumping cycles.

Optimal ΔG^°.The exponent is maximal at ΔG^° = −λ = −0.7 eV; this is the Marcus-activationless condition. At this point the pre-exponential factor alone sets the rate, k ≈ 2 × 10¹² s⁻¹.

Inverted region. At ΔG^° = −1.5 eV (deep inverted), the exponent is −(−1.5+0.7)²/(4×0.7×0.0267) = −8.56, giving exp(−8.56) ≈ 1.9 × 10⁻⁴. Rate drops by ~200× compared with the optimum. This is precisely the region the ETC avoids — successive redox centres are tuned so that ΔG^° stays near or below λ in magnitude.

Solution 5 — UPR Bifurcation Analysis

Fixed points of the UPR system satisfy simultaneously

\[ k_{\mathrm{syn}} = k_{\mathrm{fold}} U^* \dfrac{C^*}{C^* + K} + k_{\mathrm{deg}} U^*, \qquad C^* = g(U^*)/\gamma \]

with g(U) = g₀Uⁿ/(Uⁿ+ U_halfⁿ). Substituting the second into the first gives an implicit equation for U*. The Hopf condition (detJ > 0, trJ = 0) and the saddle-node condition (detJ = 0) partition parameter space.

Typical qualitative regimes as k_syn increases:

• (a) Resolved stress (moderate k_syn): the system settles to a new, higher steady state with elevated chaperones fully handling the load. Stable focus. Biological correlate: transient viral infection, secretory cell differentiation.
• (b) Oscillatory regime (intermediate k_syn with strong feedback n large): Hopf bifurcation gives rise to limit-cycle oscillations in U and C with period ~10τ_fold. Biological correlate: observed experimentally in pancreatic β-cells under mild ER stress (Zhang 2019).
• (c) Apoptotic bifurcation (large k_syn): the high-U branch either loses stability or exceeds a CHOP/PUMA threshold in a saddle-node-on-invariant-circle; the cell is committed to death. Biological correlate: chronic protein aggregation in neurodegeneration.

Relevance. In neurodegeneration (AD, PD, HD, ALS) the synthesis rate of misfolding-prone protein (e.g., Aβ, α-synuclein, mutant huntingtin, TDP-43) is slowly elevated over decades. The UPR can compensate transiently — explaining the decades-long prodromal phase — but eventually crosses the saddle-node bifurcation in a subset of neurons, initiating the cell-death cascade and producing the observed stochastic-loss topology of each disease. Pharmacological agents that shift bifurcations (ISRIB inhibits the ATF4 arm) have shown cognitive benefit in mouse models of neurodegeneration.

Solution 6 — Flory–Huggins Spinodal and Critical Point

Starting from f(φ)/k_BT = (φ/N) ln φ + (1−φ) ln(1−φ) + χ φ(1−φ):

\[ \dfrac{\partial^2 f}{\partial \phi^2} = \dfrac{1}{N\phi} + \dfrac{1}{1-\phi} - 2\chi \]

Spinodal: ∂²f/∂φ² = 0 gives

\[ \chi_{\mathrm{sp}}(\phi) = \dfrac{1}{2}\left[\dfrac{1}{N\phi} + \dfrac{1}{1-\phi}\right] \]

Critical point: ∂³f/∂φ³ = −1/(Nφ²) + 1/(1−φ)² = 0, so

\[ \phi_c = \dfrac{1}{1+\sqrt{N}}, \qquad \chi_c = \dfrac{(1+\sqrt{N})^2}{2N} \]

For N = 1 (monomer): φ_c = 1/2, χ_c = 2. For N = 100 (polymer): φ_c ≈ 0.091, χ_c≈ 0.605. In the long-polymer limit χ_c → 1/2 and φ_c → 0.

Multivalency effect. Increasing effective N (number of “stickers” per chain) lowers both χ_cand φ_c. A multivalent IDP condenses at orders of magnitude lower concentration than a monomeric stickily-interacting protein. The cell exploits this: nucleolar and stress-granule scaffolds all have > 10 sticker residues each.

ALS relevance. FUS has ~27 tyrosines in its LC domain, TDP-43 has ~10 aromatic stickers. Disease mutations (FUS P525L, R521C; TDP-43 A315T, Q343R) either add stickers (increasing effective valence) or disrupt post-translational modifications (methylation, phosphorylation) that normally modulate χ. The cell shifts from a dilute to a dense phase at physiological concentration and never returns — the condensate ages into a gel. Pharmacologically reducing N or χ (e.g., disrupting the aromatic network) is being investigated as a therapeutic approach.

Solution 7 — Mitochondrial Capacitance

(a) Capacitance. Inner-membrane area A = 200 μm² = 2 × 10⁻⁶ cm². With specific capacitance 1 μF/cm²:

\[ C = 1\,\mu F/cm^2 \times 2\times 10^{-6}\,cm^2 = 2\,pF \]

(b) Charge at |pmf| = 210 mV.

\[ Q = CV = 2\times 10^{-12}\,\mathrm{F} \cdot 0.21\,\mathrm{V} = 4.2\times 10^{-13}\,\mathrm{C} \]

In electron units: N = Q/e = 4.2 × 10⁻¹³ / 1.6 × 10⁻¹⁹ ≈ 2.6 × 10⁶ charges. This is the net excess positive charge separated across the IMM. Proton-equivalents: ~few × 10⁶, negligible compared with the total matrix H⁺ buffer pool.

(c) Dissipation time at full ATP-synthase turnover. 10⁵ complexes × 100 H⁺/s (low-activity estimate; full turnover is higher):

\[ \tau = \dfrac{N}{J} = \dfrac{2.6\times 10^6}{10^7\,s^{-1}} \approx 260\,\mathrm{ms} \]

Interpretation. The IMM stores only ~250 ms worth of proton current at its full potential. Respiration must match ATP-synthase consumption on sub-second timescales to maintain Δψ. This is why mitochondrial uncoupling (e.g., 2,4-DNP) so rapidly collapses Δψ, and why respiration is so tightly coupled to ATP demand (respiratory control). It is also why cellular quiescence does not stop respiration: even a resting cell must continuously pump protons to hold Δψ at −180 mV. The energetic “battery” is remarkably small; the engine must never quite stop running.

Solution 8 — LLPS in Module 2, 3, and 7

Three concrete points where the Flory–Huggins framework applies inside “membrane-bound” biology:

1. Module 2 — FG phase in the NPC.The selective channel of the nuclear pore is a Flory–Huggins polymer condensate of FG-nucleoporins. Cargo partitioning follows exp(−ΔG/k_BT). The NPC is a membrane-bound compartment (the nuclear envelope) within which a membraneless condensate performs selectivity. The nucleolus, Cajal bodies, nuclear speckles, and paraspeckles are all LLPS condensates inside the nucleus.
2. Module 3 — ER-phagy and ribosome quality control. Recent work (Liang 2020; Jomaa 2022) shows that ER-phagy receptor FAM134B forms condensates on the ER membrane that drive curvature-dependent membrane fragmentation. Ribosome-associated quality control (RQC) similarly uses NEMF/Rqc2-nucleated condensates to dock stalled ribosomes. The ER thus hosts membraneless compartments on its cytoplasmic face for specialised proteostasis.
3. Module 7 — condensates at contact sites. The MICOS complex at mitochondrial cristae junctions and the VAP–OSBP/FFAT-motif tether at ER–Golgi contacts display condensate behaviour (Sou 2022; Kumar 2023). Phase separation at contact sites is proposed to enhance lipid transfer by concentrating lipid-transfer proteins into a dense phase.

Is the dichotomy natural? Probably not. The cell’s compartments span a spectrum: some are pure bilayer (lysosome, outer mitochondrial membrane), some are pure condensate (stress granules), and many are hybrid — a bilayer-bounded organelle with membraneless subcompartments coupled to its surface. The historical classification reflects the accessibility of techniques (electron microscopy sees membranes, fluorescence recovery sees condensates) rather than any deep dichotomy in biology. A more modern taxonomy would treat every compartment as a non-equilibrium steady state defined by its boundary type and its internal phase structure; “membrane-bound” and “membraneless” are the two limiting cases of a continuum.