Haldane Substrate Inhibition and Growth Kinetics Model
Compare non-monotonic Haldane (Andrews) substrate-inhibition kinetics against classical saturating Monod uptake in the interactive bioreactor simulator. The workspace opens the Pichia pastoris methanol preset, whose cited kinetics use the Haldane model (Ks 1.5, Ki 8.86 g/L), so growth rises then falls with substrate; the Monod presets have no Ki. The Haldane term peaks at Sopt = √(Ks Ki) and depresses specific growth at high substrate.
Uses the shipped Pichia pastoris methanol preset (25 °C, exponential feed in a 10 L vessel), whose growth kinetics are the Haldane (uncompetitive-inhibition) fit of Zhang et al. 2000 quoted by Cos et al. 2006: μmax 0.146 per hour, Ks 1.5 g/L and Ki 8.86 g/L, giving Sopt 3.65 g/L where specific growth tops out near 0.08 per hour.
Contrast saturating Monod kinetics against non-monotonic Haldane inhibition
In the classical Monod model, specific substrate uptake qS (and derived specific growth rate μ) is a strictly increasing, concave-down function of extracellular substrate concentration S: once S exceeds roughly 10 × Ks, the factor S/(Ks + S) exceeds 0.91 and further increases in substrate concentration leave the specific rate virtually unchanged. While convenient for glucose-limited teaching balances, a purely saturating curve cannot represent the rate depression observed when microorganisms encounter high concentrations of glucose, methanol, phenol, or ammonia.
Borrowing from J. B. S. Haldane’s enzyme kinetics for inactive ES2 complexes, Andrews (1968) introduced the quadratic denominator term S²/Ki into microbial growth modeling. Under Haldane kinetics, the rate factor rises sharply at low substrate (governed by Ks), reaches a unique maximum at the geometric mean Sopt = √(Ks Ki), and then declines hyperbolically as S approaches and exceeds Ki. For the shipped Pichia preset, Ks = 1.5 g L⁻¹ and Ki = 8.86 g L⁻¹ give Sopt = 3.65 g L⁻¹, so charging methanol far above a few g L⁻¹ depresses the specific rate rather than raising it; growth tops out near 0.08 per hour at Sopt.
Understand why substrate inhibition motivates fed-batch operation over high-sugar batch
If a fermentation target requires accumulating 50 g L⁻¹ of dry cell weight with a biomass yield YX/S = 0.5 g g⁻¹, a total of 100 g L⁻¹ of glucose equivalent must be supplied over the run. Charging all 100 g L⁻¹ into the initial batch medium under Haldane kinetics (or in an overflow-prone E. coli strain) subjects the inoculum to severe high-substrate inhibition and osmotic stress right when biomass X is smallest.
By starting with a modest initial batch substrate charge near or slightly above Sopt and transitioning after batch depletion to a controlled fed-batch feed, the operator keeps residual liquid-phase substrate S in the low milligram-per-liter to gram-per-liter window where S²/Ki is negligible while still delivering hundreds of grams of total carbon. Inspect the run view with the Pichia preset, and an E. coli or yeast preset for contrast, to compare how Haldane and Monod kinetics respond to initial substrate and feed-rate changes.
Equations and variables
Haldane (Andrews) substrate-inhibited uptake or growth factor
fHaldane(S) = S / (Ks + S + S² / Ki)
- fHaldane(S)
- Dimensionless specific rate factor accounting for both affinity and high-S inhibition
- S
- Extracellular limiting carbon substrate concentration in the liquid phase (g L⁻¹)
- Ks
- Half-saturation affinity constant at low substrate concentrations (g L⁻¹)
- Ki
- Substrate inhibition constant governing high-concentration rate depression (g L⁻¹)
Optimal substrate concentration maximizing the Haldane factor
Sopt = sqrt(Ks × Ki); fmax = 1 / (1 + 2 sqrt(Ks / Ki))
- Sopt
- Substrate concentration where dfHaldane/dS = 0 (geometric mean of Ks and Ki) (g L⁻¹)
- fmax
- Maximum attainable fraction of the uninhibited asymptotic rate at S = Sopt
Scope and limitations
- Haldane kinetics apply only where a substrate inhibition constant Ki is set; the Pichia methanol preset carries one, while the E. coli and yeast glucose presets are Monod (no Ki).
- The Haldane equation lumps osmotic stress, overflow metabolism (such as acetate excretion via the Crabtree/bacterial Crabtree effect), and direct enzyme inhibition into a single empirical Ki constant without modeling intracellular acetate pools.
- Model outputs are mathematical simulations of the entered differential equations and parameters, not experimental measurements or process predictions for a specific manufacturing fermentation.
Sources and provenance
- Bioreactor model methods and independent verificationOriginal reference calculations; see source notices
Model outputs are simulations, not observations from a reactor.
- Andrews 1968 a mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesArticle citation; Haldane-type microbial substrate inhibition model
- Kovárová et al. 1996 glucose-limited E. coli ML30 growth kineticsCited parameter reference; low-substrate Monod affinity context
Technical questions
How does the Haldane equation reduce to the classical Monod equation?
In the limit as the inhibition constant Ki approaches infinity (Ki → ∞), the quadratic term S²/Ki in the denominator vanishes, reducing S / (Ks + S + S²/Ki) identically to the classical Monod saturation factor S / (Ks + S).
Why does the maximum specific rate occur at the geometric mean Sopt = √(Ks · Ki)?
Differentiating f(S) = S / (Ks + S + S²/Ki) with respect to S and setting the numerator (Ks + S + S²/Ki) − S(1 + 2S/Ki) = Ks − S²/Ki to zero yields S² = Ks · Ki, or Sopt = √(Ks · Ki). Below Sopt, nutrient limitation dominates; above Sopt, substrate inhibition dominates.
Why can a Haldane-inhibited culture exhibit bistability or sudden washout when overfed?
Because the specific uptake curve slopes downward for S > Sopt, an abrupt surge in residual substrate concentration lowers the specific consumption rate rather than accelerating it. Left uncorrected, unconsumed feed accumulates further, pushing S deeper into the inhibitory regime in a self-reinforcing runaway toward growth arrest.
Connected workflows
- Monod fed-batch growth simulation
Explore the baseline E. coli Monod fed-batch growth and maintenance model.
- Oxygen transfer and uptake modelling
Check how volumetric oxygen transfer (OTR) constrains high-density growth.
- Bell-shaped dose-response analysis
Compare bell-shaped non-monotonic inhibition curves in biochemical assays.