Biology · Advanced · 30 min · Chapter 28 of 28

Enzymes, membranes, and scaling

Michaelis–Menten kinetics, the Nernst potential, diffusion time, and the allometric scaling that compares animals of different sizes.

Read first: Functions, exponentials, and logarithms, Acids, bases, and electrochemistry

The idea

Enzyme kinetics in the Michaelis–Menten model assumes a substrate S binding an enzyme to form a complex, which either falls apart or proceeds to product. At steady state the initial rate is v = Vmax [S] / (Km + [S]). At [S] = Km the rate is half of Vmax, which is the operational definition of Km and the reason it is read off a plot rather than derived from a microscope. At low substrate the reaction looks first-order in S. At high substrate the enzyme is saturated and the rate is zero-order in S, stuck at Vmax. A competitive inhibitor raises the apparent Km and leaves Vmax alone; it can be outcompeted by enough substrate.

Ions are not at the same concentration on both sides of a cell membrane, and the membrane potential is the electrical compromise. For a single ion species that can cross, the Nernst potential is the voltage at which the chemical gradient and the electrical gradient cancel. At mammalian body temperature the rule of thumb is E ≈ (61.5 mV / z) log₁₀([out]/[in]) for a cation. Potassium, high inside, has a negative Nernst potential. Sodium, high outside, has a positive one. The resting membrane sits near the potassium potential because the membrane at rest is much more permeable to potassium. The Goldman equation is the weighted average when several ions matter.

Diffusion is not a flow with a speed. The time to diffuse a distance x is on the order of x² / (2D), where D is the diffusion coefficient. Double the distance and you quadruple the time. A neurotransmitter crosses a synaptic cleft in microseconds because the cleft is tens of nanometers. The same molecule would take minutes to cross a centimeter. That square law is why circulatory systems and lungs exist: bulk flow covers distance, diffusion covers the last small gap. Allometric scaling compares whole animals. A quantity Y often follows Y = a Mᵇ. Metabolic rate scales closer to M^{3/4} than to M¹, so mass-specific metabolism falls as animals get larger. Surface-to-volume arguments suggest 2/3; the three-quarter law is empirical and still argued over, and the formula is what you fit.

Hold onto this

An enzyme saturates. At high substrate the rate approaches Vmax and stops depending on further substrate.
Km is the substrate concentration at half-maximum rate. It is not, by itself, a binding constant, though it is related.
The Nernst potential is the voltage that balances a concentration gradient for one ion.
Diffusion time grows with the square of distance, which is why cells are small.

Formulas

Michaelis–Menten

v = \frac{V _{m a x} [ S ]}{K _{m} + [ S ]}

Initial rate versus substrate concentration for a simple enzyme.

Vmax: Maximum rate, at saturating substrate
Km: Substrate concentration giving half of Vmax

Lineweaver–Burk

\frac{1}{v} = \frac{K _{m}}{V _{m a x}} \frac{1}{[ S ]} + \frac{1}{V _{m a x}}

The double-reciprocal form. Slope Km/Vmax, intercept 1/Vmax. Useful to see, noisy in practice.

1/v: Reciprocal of the measured rate

Nernst potential

E = \frac{R T}{z F} ln \frac{[ out ]}{[ in ]}

Equilibrium potential of one ion. At 37 °C, about (61.5 / z) log₁₀([out]/[in]) in millivolts.

z: Charge of the ion, +1 for K⁺ and Na⁺
[out], [in]: Concentrations on the two sides

Diffusion time

t \sim \frac{x ^{2}}{2 D}

Characteristic time to diffuse a distance x. Quadratic in distance.

D: Diffusion coefficient
x: Distance

Hill equation

Y = \frac{p ^{n}}{P _{50}^{n} + p ^{n}}

Fraction bound, for a cooperative binder such as hemoglobin. n > 1 is cooperativity.

P₅₀: Partial pressure, or concentration, at half saturation
n: Hill coefficient

Allometric scaling

Y = a M^{b}

How a biological quantity scales with body mass. Take logs: log Y = log a + b log M, a straight line.

M: Body mass
b: The scaling exponent

Worked examples

1. Half-maximum, by definition

An enzyme has Vmax = 100 μmol/min and Km = 2.0 mM. What is the rate at [S] = 2.0 mM, and at [S] = 18 mM?

At [S] = Km, v = Vmax / 2 = 50.
At 18 mM, v = 100 × 18 / (2 + 18) = 1800 / 20 = 90.
Nine times the substrate, from the Km point, only raises the rate from 50 to 90. Saturation is the whole story.

50 μmol/min at 2 mM, and 90 μmol/min at 18 mM.

2. Potassium’s equilibrium

A neuron has [K⁺]in = 140 mM and [K⁺]out = 5 mM. Estimate the Nernst potential at body temperature.

Use E ≈ 61.5 log₁₀([out]/[in]) mV, since z = 1.
[out]/[in] = 5/140 ≈ 0.0357. log₁₀ of that is about −1.45.
E ≈ 61.5 × (−1.45) ≈ −89 mV.

About −89 mV. Resting potentials near −70 mV sit close to this, pulled positive by a little sodium leak.

Try these

Oxygen diffuses with D about 2×10⁻⁹ m²/s in tissue. Estimate the time to diffuse 10 μm, then 1 mm.

If metabolic rate scales as M^{3/4}, by what factor does total metabolism rise when mass increases by 16?

Common mistakes

Km is the concentration at half Vmax. A small Km means the enzyme reaches half speed at low substrate; calling it “high affinity” is a shorthand that fails for multi-step mechanisms.
The Nernst potential of an ion is not the resting membrane potential. The resting potential is a permeability-weighted compromise.
Allometric exponents fitted on a log–log plot are sensitive to the range of masses and to whether you used species means or individuals. Quote the range.