04-Bio-A8 · May 2014
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, May 2014 — 04-Bio-A8 Biophysical Measurements. Three hours, open book, any non-communicating calculator. Seven questions of equal value (20 marks each); five constitute a complete paper and only the first five appearing in the answer book are marked. All seven are solved here, because this set is a study resource rather than an examination script. Every question is qualitative/descriptive — there is no numerical data to compute — so each answer follows flowing prose with instrumentation block diagrams where the question explicitly asks for one.
Reference texts (the books an open-book candidate should have on the desk for this subject):
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
The excitable membrane is modelled electrically as a set of parallel branches sandwiched between an outside and an inside node, one branch per permeant ion species — sodium, potassium, and a lumped "leak" (chiefly chloride and other ions) — plus a membrane capacitance $C_m$ formed by the lipid bilayer itself. Each ionic branch consists of a conductance $g_i$ in series with a battery equal to that ion's Nernst equilibrium potential $E_i = (RT/z_iF)\ln([\text{ion}]_o/[\text{ion}]_i)$, the voltage at which the electrical and diffusional (concentration-gradient) driving forces on that ion exactly balance so its net current is zero. Sodium is concentrated outside the cell and potassium inside, maintained against their gradients by the metabolically active Na–K pump, so for a mammalian excitable cell at 37 °C (typical $[K^+]_o/[K^+]_i=4/140$ mM, $[Na^+]_o/[Na^+]_i=145/12$ mM, $[Cl^-]_o/[Cl^-]_i=110/4$ mM) $E_{Na}\approx+67$ mV and $E_K\approx-95$ mV, while the leak battery (dominated by chloride) sits at $E_L\approx-89$ mV, close to $E_K$. (The often-quoted $E_K\approx-72$ mV and $E_{Na}\approx+55$ mV are squid-axon values; they cannot produce a $-85$ to $-90$ mV resting potential, since a conductance-weighted average can never be more negative than its most negative battery.)
At rest none of the branch currents is individually zero, but the membrane settles at the potential $V_m$ at which the sum of the three branch currents is zero, i.e. the circuit's Thévenin (steady-state) potential:
$$V_m=\frac{g_{Na}E_{Na}+g_K E_K+g_L E_L}{g_{Na}+g_K+g_L}$$This is the weighted average of the individual equilibrium potentials, weighted by each branch's conductance. The crucial physiological fact is that the resting membrane's potassium conductance greatly exceeds its sodium conductance ($g_K\gg g_{Na}$ at rest, roughly 15–30:1), because far more potassium "leak" channels are open at rest than sodium channels. The weighted average is therefore pulled strongly toward $E_K$, giving the observed resting potential of $-85$ to $-90$ mV — close to, but slightly less negative than, pure $E_K$ ($-95$ mV) because the small resting $g_{Na}$ pulls it positive. For example, with $g_K:g_{Na}:g_L=1:1/15:0.2$ the formula gives $V_m\approx-85.5$ mV, and with $g_K:g_{Na}=30:1$ it gives $V_m\approx-89.6$ mV, bracketing the stated range. This is the chord-conductance equation — the electrical-circuit counterpart of the Goldman–Hodgkin–Katz equation — applied to a resistive-battery network, and it makes explicit why the resting potential is not simply the Nernst potential of one ion: it is a conductance-weighted compromise set by whichever channels happen to be open.
For small, sub-threshold perturbations the membrane behaves as a fixed, linear RC network: the conductances $g_{Na}$, $g_K$, $g_L$ are treated as constants (their rest values), so an injected current step produces an exponential charging/discharging response of $V_m$ toward a new steady state with time constant $\tau=R_mC_m$, exactly as an ordinary resistor–capacitor circuit would respond — this is the electrotonic (passive, decremental) response seen for hyperpolarizing or small depolarizing stimuli. The action potential arises because this picture breaks down once depolarization becomes large enough to open voltage- and time-dependent sodium and potassium conductances: the branches are no longer fixed resistors but conductances that are themselves functions of $V_m$ and time, $g_{Na}(V_m,t)$ and $g_K(V_m,t)$, which is the nonlinear, regenerative element Hodgkin and Huxley identified.
As depolarization crosses threshold, the sodium conductance activates rapidly: $g_{Na}$ rises steeply, driving $V_m$ toward $E_{Na}$ ($\approx+67$ mV) because the weighted-average formula above is now dominated by the sodium branch — this is the regenerative upstroke, since depolarization itself increases $g_{Na}$ further in a positive-feedback loop until the membrane approaches $E_{Na}$. Sodium conductance then inactivates (a separate, slower voltage-dependent gate closes the channel even though $V_m$ is still depolarized), while the potassium conductance, which activates more slowly than sodium, continues to rise, pulling $V_m$ back toward $E_K$ — the repolarization phase. Because $g_K$ remains elevated briefly after $V_m$ returns to rest, the membrane transiently overshoots toward $E_K$, producing the after-hyperpolarization, before conductances relax to their resting values and the linear passive behaviour resumes. The whole cycle is thus the nonlinear, voltage-gated conductances temporarily overwhelming the fixed-conductance linear model that governs sub-threshold behaviour.
This conductance model is the basis for every subsequent question on this paper: pacemaker sensing circuits detect the depolarization (R-wave) this mechanism produces, impedance plethysmography and evoked-potential instrumentation record the aggregate extracellular field it generates, and the amplifier front-ends throughout biomedical instrumentation are designed around the millivolt amplitudes and millisecond time constants this membrane circuit implies.
