Homework 2 — Diffusion, Mass Action & Membrane Potential

Problem 1 — A + B → X + Y (irreversible, mass action)

With a = a₀−x and b = b₀−x (conservation of mass), the correct model is dx/dt = k(a₀−x)(b₀−x) = 2(10.5−x)(6.5−x) — statement (a) is TRUE, statement (b) is FALSE (y is a product, not a reactant), and “A depletes first” is FALSE: B is limiting (6.5 < 10.5 µg/ml), so B → 0 while A levels off at a₀−b₀ = 4 µg/ml. Curve shown is the exact analytic solution; markers are the numerical (Euler) solution as built in Simulink.

a₀ (µg/ml)

b₀ (µg/ml)

k (1/(µg/ml)/s)

Problem 2 — Diffusion + X + Y → Z with product feedback inhibition

Dynamical system: dx/dt = k_x(x₀^e − x) − k₁xy, dy/dt = k_y(y₀^e − y) − k₁xy, dz/dt = k₁xy − δz, with k₁ = max(0, α − γz). X and Y diffuse in from the ECF (k_x=2, k_y=3 /s; x₀^e=10, y₀^e=6 µg/mg); Z is consumed (δ=2/s) and inhibits its own production (α=10, γ=2).

α (base rate constant)

γ (feedback strength)

δ (Z consumption /s)

Run time (s)

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z at steady state (µg/mg)

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final k₁ (does NOT reach 0)

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x steady state

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y steady state

Answer to the assignment’s question: the reaction does not stop — it reaches a steady state where production of Z exactly balances its consumption, and k₁ settles at a positive value (α − γz_ss > 0).

Problem 3 — Resting membrane potential vs extracellular K⁺ (chord conductance)

V_m = (g_Na/g_tot)E_Na + (g_K/g_tot)E_K with g_Na:g_tot = 0.1, g_K:g_tot = 0.9, and E_ion = 61.5 log₁₀([out]/[in]) mV at 37 °C. Baseline: Na⁺ 14 in / 140 out, K⁺ 140 in / 4 out → V_m = −79.3 mV (matches the measured value given). Drag the slider to add KCl to the ECF.

Extracellular [K⁺] (mM) — baseline 4, assignment asks +25 → 29

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E_K (mV)

+61.5

E_Na (mV)

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V_m (mV)

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excitability state

Why IV KCl is lethal: raising [K⁺]_o to 29 mM depolarizes V_m from −79 mV to about −32 mV — far above the ≈−55 mV threshold. Neurons and cardiac muscle fire once, then their Na⁺ channels remain inactivated (h gates closed) at the depolarized potential, so no further action potentials are possible: the heart stops in diastole. This connects directly to the hyperkalemia experiment in Homework 4.