ENV-35

Monod specific growth

μ = μm S /(Ks + S). Hyperbolic microbial growth.

Reading speed

BiokineticsMonod 1949

Governing equation

\mu=\mu_m\dfrac{S}{K_S+S}

$S \in \left[0.1, 500\right]$

where

\mu_m: Max specific growth (1/d)
S: Substrate (mg/L)
K_S: Half-saturation (mg/L)
\mu: Specific growth (1/d)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\mu=\mu_m\dfrac{S}{K_S+S}

$S \in \left[0.1, 500\right]$

$S = 0.1$ $\mu = 0.02$

Along the curve

②

Domain of the variable — blue

S_{\min} = 0.1,\quad S_{\max} = 500

$S \in \left[0.1, 500\right]$

$S = 0.1$ $S = 0.1$

Along the curve

③

Response curve — green

\mu=\mu_m\dfrac{S}{K_S+S}

$S \in \left[0.1, 500\right]$

$S = 0.1$ $\mu = 0.02$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mu}{\mathrm{d}S}

$S \in \left[0.1, 500\right]$

$S = 3.244$ $\mu^{\prime} = 0.151$

Along the curve

Lecture brief

Historical brief

Streeter–Phelps (1925) oxygen sag, settling theory and Guldberg–Waage kinetics made water and air quality a rate problem. The lab computes sag, overflow and a snapshot of reactor mass balance. This sheet (ENV-35 — Monod specific growth) is the form associated with Monod 1949. Working symbols: $\mu_m$ , $S$ , $K_S$ $\rightarrow$ $\mu$ . Monod adapted Michaelis–Menten to whole-cell growth. Half-saturation Ks is the S at μm/2.

Purpose

Purpose: compute

\mu

from

\mu_m

S

K_S

in Environmental via

\mu=\mu_m\dfrac{S}{K_S+S}

μ = μm S /(Ks + S). Hyperbolic microbial growth. Use it when a real environmental question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\mu_m = 4.000\,\mathrm{1/d}

S = 50.000\,\mathrm{mg/L}

K_S = 20.000\,\mathrm{mg/L}

, the governing relation

\mu=\mu_m\dfrac{S}{K_S+S}

yields

\mu = 2.857\,\mathrm{1/d}

. A μ vs S hyperbola, a Ks tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Max specific growth \mu_m4 1/dSubstrate S50 mg/LHalf-saturation K_S20 mg/L

Outputs

Specific growth \mu2.857 1/d

Reading speed

Watch on YouTube

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Narration of this film

A μ vs S hyperbola, a Ks tick.

Monod adapted Michaelis–Menten to whole-cell growth. Half-saturation Ks is the S at μm/2.

Reading speed

Watch on YouTube