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FermAxiom Yeast — Specific Growth Rate Calculator

Yeast Batch Growth Simulator

© 2026 FermAxiom LLC · Author: Peter Krasucki · peter.krasucki@fermaxiom.com  |  Batch Models S. cerevisiae  |  Time-course integrated growth models  |  v3.5

Six integrated models — Exponential, Logistic, Gompertz, Modified Gompertz, Baranyi-Roberts, Richards — with lag, carrying capacity, and side-by-side reference vs. novel strain comparison.

Reference Model Ref
Growth Model
Value Unit
N0M/mL
K (Xmax)M/mL
μmaxh⁻¹
λ (lag)h
ν (Richards)—
Novel Model Nov
Growth Model
Value Unit
N0M/mL
K (Xmax)M/mL
μmaxh⁻¹
λ (lag)h
ν (Richards)—
60

Growth Curve — Cell density vs. time

Strain · Model μmax (h⁻¹) td (h) λ (h) t½K (h) t95%K (h) N(tend)
Ref — Logistic — — — — — —
Novel — Mod Gompertz — — — — — —

Batch Growth Simulator — User Guide Guide · v3.5

Purpose

This simulator projects cell density N(t) over time for a batch S. cerevisiae culture, given an initial concentration N0, carrying capacity K, maximum specific growth rate μmax, lag time λ, and shape parameter ν. Six integrated growth models — Exponential, Logistic, Gompertz, Modified Gompertz, Baranyi-Roberts, and Richards — are available; assign one to the Reference strain and a different (or same) one to the Novel strain to compare side-by-side.

Intended users: researchers and bioprocess engineers who need rapid batch time-course projections, and educators demonstrating how lag-phase and carrying-capacity effects shape growth curves.

Intended uses: rapid what-if exploration, pre-experiment batch planning, teaching the differences between lag-phase encodings, benchmarking a new isolate's projected curve against a well-characterized reference strain.

How the simulator works

All six models are closed-form, time-explicit equations — no ODE integration required. Each expresses N(t) as a function of t with some subset of {N0, K, μmax, λ, ν} as parameters. The chart samples the curve at 240 points across the chosen time horizon and draws both Reference (teal) and Novel (dark red) traces on the same axes. The derived-kinetics table below compares doubling time, λ, t½K, t95%K, and N(tend) between the two strains.

Quick start — six-step workflow

  1. Pick the reference strain (S288c, CEN.PK, or Ethanol Red). Defaults for both Reference and Novel auto-load from the strain preset — override the Novel column to compare against a different parameter set.
  2. Name the novel strain in the text field (top-right).
  3. Choose models in the left (Reference) and right (Novel) cards. Default pairing is Logistic (Ref) vs Modified Gompertz (Novel) — a useful comparison that highlights how adding a lag term shifts the curve.
  4. Adjust parameters in the parameter tables. Each side has independent values. Unused parameters (λ for non-lag models, ν for non-Richards) are still stored but ignored by the compute.
  5. Set the time horizon via the slider at the bottom (10–200 h). The chart and derived-metrics table update live.
  6. Change the display unit (cells/mL, million cells/mL, billion cells/mL) and the y-axis scale (linear vs logarithmic) via the dropdowns. Use logarithmic scale to visually linearize the exponential phase and compare μmax across strains.

Choosing a model

Six models differ in how they handle lag, deceleration shape, and asymmetry:

  • Exponential — N0·exp(μ·t). Unbounded. Use only for early log phase or as a reference baseline to establish doubling time. Ignores K, λ, ν.
  • Logistic — classical Verhulst S-curve; symmetric about the inflection at N = K/2. No explicit lag. Pick as a first approximation when lag phase is short or not critical.
  • Gompertz — asymmetric S-curve with earlier inflection (around N ≈ K/e). Peak growth rate at t = 0. Pick when the culture shows an early-peak growth rate typical of yeast but lag isn't explicitly modeled.
  • Modified Gompertz (Zwietering 1990) — the most widely-used microbial growth model. Has an explicit lag parameter λ, but note that λ here is the x-intercept of the tangent through the inflection point, not a literal flat-growth phase. Pick when lag matters and a smooth (not sharply-kinked) curve is acceptable.
  • Baranyi-Roberts (Baranyi & Roberts 1994) — gold standard with a genuine flat lag phase. Uses an adjustment function α(t) that suppresses growth during lag, then releases it as α → 1. Pick when the flat-lag shape matters, or when fitting against data showing a clean lag-then-grow transition.
  • Richards (Richards 1959) — generalized logistic with shape parameter ν. At ν = 1 reduces to pure Logistic; ν → 0 approaches Gompertz; ν > 1 shifts inflection toward higher N (slower early growth, steeper approach to K). Pick when you need to tune asymmetry empirically without changing model family.

Interpreting the output

Growth Curve chart — both traces plotted on shared axes. X-axis: time (h). Y-axis: cell density in the selected unit. Teal = Reference, dark red = Novel. Linear y-scale reveals the sigmoidal shape; log y-scale linearizes the exponential phase so the slope directly equals μmax.

Derived Kinetics table — one row per strain, six metric columns:

  • μmax (h⁻¹) — echoed from input.
  • td (h) — doubling time, ln(2)/μmax. Analytical; same definition for all six models (exponential-phase doubling).
  • λ (h) — echoed input, shown only for Modified Gompertz and Baranyi-Roberts. Other models display "—".
  • t½K (h) — numerical bisection for N(t) = 0.5·K.
  • t95%K (h) — numerical bisection for N(t) = 0.95·K.
  • N(tend) — cell density at the end of the time horizon. For models approaching K asymptotically, this will be near K when tend is large enough.

Parameter reference — symbols, meanings, units

  • N0 — initial cell density at t = 0, in the selected display unit (default M cells/mL). Typical S. cerevisiae inoculation 1–10 M cells/mL.
  • K (Xmax) — carrying capacity; the asymptote where growth stops. 100–300 M cells/mL typical for yeast batch.
  • μmax — maximum specific growth rate, h⁻¹. Inherited from the SGR strain defaults: 0.48 (S288c), 0.42 (CEN.PK), 0.45 (Ethanol Red).
  • λ (lag) — lag time, h. Used only by Modified Gompertz and Baranyi-Roberts. Typical 2–10 h for lab strains in favorable media; longer for industrial strains under high-gravity conditions.
  • ν (Richards shape) — dimensionless. ν = 1 gives pure Logistic; ν → 0 approaches Gompertz; ν > 1 gives slower early growth and a steeper approach to K. Default 1.5 for lab strains, 2.0 for Ethanol Red.

Common pitfalls

  • μmax means subtly different things across models. In Exponential and Logistic it's the specific growth rate at N « K. In Gompertz it's the peak rate, occurring at t = 0. In Modified Gompertz and Baranyi-Roberts it's the max rate at the post-lag inflection point. In Richards the interpretation depends on ν. The same numerical input translates into slightly different realized peak rates across models.
  • The unit selector is a labeling toggle only. Switching the Cell Density Unit dropdown updates the y-axis label but does NOT auto-rescale N0 or K. If you entered "5" as M cells/mL and then switch to cells/mL, the chart still plots "5" — interpret it under the new label, or manually rescale your inputs.
  • λ is not a universal lag-time concept. In Modified Gompertz, λ is the x-intercept of the tangent through the inflection point (at t = λ, only ~1% of the log-growth has happened). In Baranyi-Roberts, λ enters through h0 = μmax·λ and produces a genuinely flat lag phase. The same numerical λ gives different-looking curves across the two models.
  • Exponential diverges. At default μmax = 0.48 with N0 = 5 M cells/mL and tend = 60 h, Exponential projects 5·exp(0.48·60) ≈ 2×1013 M cells/mL — nonsense because there's no carrying capacity. Use Exponential only for early-phase estimates over short horizons (under ~20 h).
  • Richards with ν = 1 is mathematically Logistic. If you pick Richards but leave ν = 1, the curve is indistinguishable from the Logistic model. Adjust ν to see Richards' distinctive behavior.
  • Changing the Reference strain dropdown resets both Ref and Novel parameters to that strain's defaults. If you've spent time tuning the Novel column, save your values before switching strains.

Relationship to the SGR Calculator

This simulator and the Specific Growth Rate Calculator (bottom tab) are designed to be used together:

  1. Estimate μmax using the SGR calculator: pick strain, set environmental conditions (S, P, T, pH), read off μ.
  2. Copy μmax into this batch simulator along with strain-appropriate N0, K, and λ estimates.
  3. Project the time course using whichever of the six integrated models best fits your system (use the Science tab for guidance on which model to pick).

The SGR calculator gives you an instantaneous rate under specified conditions; this batch simulator projects forward assuming those conditions persist (and that substrate depletion or product accumulation drives the deceleration captured by K). When environmental conditions change significantly during the run — temperature drift, pH ramp, feed pulses — neither tool captures the dynamics and you need a coupled-ODE simulator.

Aerobic vs anaerobic regime selection

The strain defaults assume aerobic operation. For anaerobic bioethanol work:

  • Reduce μmax by ~20–25% (anaerobic S. cerevisiae runs 0.25–0.40 h−1 depending on strain).
  • Adjust K based on the limiting factor: if substrate-limited keep the default; if ethanol-tolerance sets the ceiling (typical for Ethanol Red in high-gravity anaerobic fermentation) K often tracks close to the theoretical fermentation endpoint ~280 M cells/mL.
  • Expect longer λ for industrial strains — anaerobic lag is often 6–12 h vs 2–4 h aerobic because of the sterol and unsaturated-fatty-acid biosynthesis adaptation period.
  • Curve shape tends toward Modified Gompertz or Richards with ν > 1 (asymmetric, with later-phase deceleration) because progressive ethanol inhibition steepens the approach to K.

For aerobic high-density biomass production, the defaults apply directly unless oxygen transfer (kL·a) becomes limiting. Above ~30–50 g DCW/L at 2-L scale with standard aeration, effective K drops to whatever biomass is O2-supportable regardless of substrate availability — you'll see saturation well below the theoretical substrate-limited K.

Regime applicability note

This simulator models batch culture only. For fed-batch, chemostat, and dynamic-environment runs, see the Science tab discussion of "Batch, fed-batch, and continuous systems". Brief summary: fed-batch needs an ODE simulator with an explicit feed term; chemostat needs steady-state analysis at varied dilution rate; dynamic-environment runs need time-varying parameters that closed-form models can't accommodate. The closed-form models here can serve as rough indicators for small perturbations around batch behavior but aren't quantitative for those regimes.

Scope and caveats

  • Cell density only. These models don't track substrate, ethanol, or byproducts. For coupled S–X–P dynamics, a full Monod-batch ODE simulator is needed — outside the scope of this tool.
  • Closed-form models only. No support for time-varying parameters (feed profiles, temperature cycling, pH ramps) — that's fed-batch or dynamic-batch territory.
  • No death or decline phase. All six models saturate at K asymptotically. Extended incubation (>72 h for most lab strains) typically shows cell decline as stationary-phase cells die, which isn't captured here.
  • Parameter defaults are strain-level averages. Real lag times and carrying capacities vary with inoculation history, medium composition, oxygen availability, and the specific sub-strain used. For quantitative prediction, calibrate parameters against your own growth-curve data.
  • No stochasticity. These are deterministic models; they don't capture the cell-to-cell variability or the inoculation-size dependence of observed lag times.

Batch Growth Models — Mathematical Formulations Science · v3.5

Six integrated (time-explicit) growth models predict cell density N(t) given initial density N0, carrying capacity K, maximum specific growth rate μmax, lag time λ, and shape parameter ν. All six are closed-form expressions — no ODE integration required.

Exponential

N(t) = N0 · exp(μmax · t)

Unbounded. Valid only during early log phase. Uses μmax; ignores K, λ, ν.

Logistic (Verhulst)

N(t) = K · N0 / [N0 + (K − N0) · exp(−μmax · t)]

Symmetric S-curve — derivative form dN/dt = μmax·N·(1−N/K). Inflection at N = K/2, reached at t* = ln((K−N0)/N0) / μmax. No explicit lag phase: growth starts immediately at t = 0.

Gompertz (classical)

N(t) = K · (N0/K)exp(−μr · t)

Asymmetric S-curve; inflection occurs earlier than logistic (at N ≈ K/e). The internal rate constant μr is derived from the user's μmax by μr = μmax / ln(K/N0), so that the specific growth rate at t = 0 equals the user's μmax. No explicit lag.

Modified Gompertz (Zwietering 1990)

N(t) = N0 · exp{A · exp[−exp((μmax·e/A)·(λ−t) + 1)]},   A = ln(K/N0)

Adds a lag term λ, interpreted as the x-axis intercept of the tangent through the inflection point — not a literal flat-lag period. Near t = 0 growth is slow but not zero, then accelerates smoothly. Inflection at ti = λ + A/(e · μmax), where N grows at its maximum rate μmax.

Baranyi-Roberts (1994)

The microbial-growth gold standard. A "physiological adjustment function" α(t) explicitly suppresses growth during lag. With curvature parameter m = 1 (logistic-style deceleration), the closed form is:

A(t) = t + (1/μmax) · ln[(e−μmax·t + e−h0) / (1 + e−h0)],   h0 = μmax · λ
ln(N(t)/N0) = μmax·A(t) − ln[1 + (exp(μmax·A(t)) − 1) · N0/K]

At t = 0, A(t) = 0 and α(0) = 1/(1 + h0)·h0, giving a near-flat initial phase for large h0. After t > λ, α → 1 and growth follows logistic-style deceleration toward K. More realistic than Modified Gompertz for actual lag phases.

Richards (generalized logistic)

N(t) = K / [1 + Q · exp(−μmax·t)]1/ν,   Q = (K/N0)ν − 1

Shape parameter ν tunes the position of the inflection point: ν = 1 recovers the pure Logistic model, ν → 0+ approaches Gompertz, ν > 1 shifts the inflection toward higher N (slower early growth, steeper approach to K). No lag handling.

When growth kinetics modeling applies

The closed-form integrated models in this tool (Exponential through Richards) rest on several implicit assumptions. When those assumptions hold, they fit time-course data well and give predictive power. When they break, the curves are still smooth and plausible-looking — which can be dangerously misleading.

Core assumptions:

  • Balanced growth. Cell composition is roughly constant during the window being modeled; μ reflects the coordinated increase of all cellular constituents. Holds in exponential phase; breaks down at the transitions into lag and stationary phase.
  • Well-mixed reactor. Every cell experiences the same substrate, product, temperature, and pH at any instant. Fails in stagnant zones of large bioreactors, in dough or solid-state fermentations, and in pellets or biofilms.
  • Single dominant limiting factor. One resource depletes (or one inhibitor accumulates) on a timescale that dominates the kinetics. If two resources co-limit or alternate (diauxic growth on glucose → galactose is the classic example), a single S-curve won't capture the dynamics — a two-phase curve with an intermediate plateau is characteristic.
  • Deterministic dynamics. Population is large enough that inoculation-size stochasticity doesn't dominate — typically N0 > 104 cells total. At lower inocula, lag-time variability between biological replicates can exceed 50%, and the deterministic λ in Modified Gompertz or Baranyi-Roberts becomes an ensemble average rather than a predictive quantity.
  • Pure culture. Mixed cultures have cross-feeding, competition, and shifting community structure that these models don't capture.
  • Slowly-changing environment. Temperature, pH, and shear remain roughly constant on the timescale of the growth phase. Cyclic or ramp changes require time-varying parameters, which closed-form models cannot accommodate.

Contraindicated regimes: stochastic / low-inoculum experiments (N0 < 104), diauxic or mixed-substrate systems, cultures with significant cell death or sporulation, dynamic-environment runs (temperature cycling, pH ramps, feed profiles), solid-state or heterogeneous systems, and any setup where substrate or product concentrations are being externally controlled rather than freely evolving.

Aerobic vs anaerobic systems

S. cerevisiae is facultative — it grows aerobically on glucose with respiration plus fermentation (Crabtree-positive) and anaerobically on glucose by pure fermentation. The same kinetic forms apply to both regimes, but the parameters differ substantially and the rate-limiting mechanisms differ.

Aerobic growth.

  • μmax ≈ 0.40–0.50 h−1 depending on strain (consistent with the SGR-calculator defaults).
  • Biomass yield YX/S ≈ 0.50 g DCW/g glucose under fully respiratory conditions (at low residual glucose, dilution rate D < Dcrit), dropping to 0.10–0.15 g/g when glucose overflow forces ethanol production (Crabtree effect, typically triggered above ~50 mg/L residual glucose even under ample O2).
  • Oxygen transfer (kL·a) often becomes the rate-limiting factor at high cell density. When oxygen uptake rate exceeds oxygen transfer rate, cells shift to fermentative metabolism regardless of glucose level and the effective μmax drops.
  • Carrying capacity K is typically set by accumulated ethanol (from Crabtree overflow) or by O2 limitation, not by substrate exhaustion.

Anaerobic growth.

  • μmax ≈ 0.25–0.40 h−1 — lower than aerobic because ATP yield per glucose is ~2 mol (substrate-level phosphorylation only) vs ~16 mol (respiration + SLP).
  • Biomass yield YX/S ≈ 0.08–0.12 g DCW/g glucose; most carbon goes to ethanol (~0.48 g/g theoretical, 0.42–0.45 g/g actual in bioethanol fermentations).
  • Growth is product-inhibited by ethanol (Pm ≈ 50–100 g/L depending on strain) rather than oxygen-limited. The g(P) factor in the SGR calculator captures this directly.
  • Requires sterol and unsaturated-fatty-acid supplementation for sustained growth, since those biosyntheses are oxygen-dependent steps.
  • Carrying capacity K is set by ethanol accumulation reaching Pm, not by substrate exhaustion.

Both regimes fit the same closed-form model families, but with different parameter sets. Aerobic batch curves typically show a relatively sharp saturation (substrate-limited, with a well-defined K at ethanol onset or O2 breakthrough), so pure Logistic or Baranyi-Roberts usually fit well. Anaerobic curves show a more gradual deceleration as ethanol accumulates — Modified Gompertz or Richards with ν > 1 often fits better than pure Logistic because the asymmetric approach to K reflects the progressively steeper product-inhibition effect. Workflow: use the strain/regime combination in the SGR calculator to estimate μmax, Pm, and T-optimum first, then transfer to this batch simulator for the time-course projection.

Batch, fed-batch, and continuous systems

The integrated models in this tool are strictly for batch culture — a closed system with all substrate added at t = 0, no feed, no harvest during the run. They assume monotonically decreasing substrate (or monotonically increasing inhibitor) as biomass grows, reaching K asymptotically.

  • Batch: directly applicable. The closed-form solutions capture the full S-curve from inoculation through stationary phase. All six models fit.
  • Fed-batch: not directly applicable. Adding substrate during the run breaks the assumption of monotone substrate depletion — you can maintain cells in exponential phase for extended periods by matching feed rate to cellular demand. Fed-batch simulation requires integrating an ODE system (dX/dt, dS/dt, dV/dt) with a user-specified feed profile F(t); the closed-form models here don't apply. As a rough approximation: if the feed is small relative to the initial batch (under ~20% volume increase) and starts only after exponential phase, the batch-simulator Logistic or Baranyi-Roberts prediction remains roughly indicative of the biomass trajectory. For accurate fed-batch design, use a coupled-ODE simulator with the SGR factor decomposition μ(S, P, T, pH) as the kinetic core.
  • Continuous (chemostat): also not directly applicable. At steady state μ = D (dilution rate) and the system sits on a single operating point — there's no time-course S-curve to model. Chemostat data is used instead to extract Monod parameters (μmax, KS) by varying D and measuring steady-state biomass and residual substrate; those parameters then feed back into batch or fed-batch simulators for time-course work.

Exponential fed-batch is a useful special case. If F(t) is set so that dX/dt = μset·X, the culture runs at a user-specified μset below μmax for an extended period with balanced growth preserved throughout. This is the workhorse for baker's-yeast propagation (typically μset = 0.15–0.25 h−1 to avoid Crabtree overflow) and for high-density microbial production. The batch simulator here doesn't model exponential fed-batch directly, but the doubling-time output (td = ln(2)/μmax) is directly useful for scoping fed-batch design: choose μset to hit a target td, then calculate the exponential feed profile F(t) = μset·X(t)·V(t) / (YX/S·Sfeed).

Choosing a model

  • Exponential: early log phase only.
  • Logistic: quick estimate; symmetric around N = K/2.
  • Gompertz: asymmetric early-peak growth; no lag.
  • Modified Gompertz: typical choice when lag matters; curve is still smooth through the lag region.
  • Baranyi-Roberts: most realistic lag representation; has a distinct flat lag phase.
  • Richards: flexibility to tune inflection shape; ν is an extra fitting parameter.

Derived metrics

Doubling time td = ln(2) / μmax — analytical, same definition for all six models (exponential-phase doubling).

Time to reach half-K (t½K) and 95% K (t95%K) are found numerically by bisection: for each target N-value, the interval [0, 10·tend] is successively halved until N(t) brackets the target within 10−3 hours. These metrics are meaningless for pure Exponential (no carrying capacity) and are reported as "—" in that case.

Notes on numerical stability

Modified Gompertz clamps its inner exponent at 50 to prevent overflow at very negative t values; Baranyi-Roberts returns K directly when μmax·A(t) > 700 to avoid exp overflow at large times. These edge-case guards don't affect the curves in any physical regime.

Batch Growth Simulator — References Refs · v3.5

Primary sources for the six integrated growth models implemented in this simulator, plus broader predictive-microbiology reviews and strain-specific benchmarks for S. cerevisiae batch culture.

Reading order for new users. Start with Verhulst 1838 and Gompertz 1825 as the historical foundations. For modern microbial growth modeling, Zwietering 1990 is the most-cited Modified Gompertz paper and a good entry point. For mechanistic lag-phase treatment, Baranyi & Roberts 1994 is essential. Richards 1959 for the generalized logistic. Swinnen 2004 provides a comparative review of all the lag-phase formulations.

Classical growth-curve foundations

  1. Malthus TR (1798). An Essay on the Principle of Population. J. Johnson, London. — Origin of exponential-growth formalization; establishes N(t) = N0·exp(r·t) as the simplest population model. Conceptual precursor to every model in this simulator.
  2. Verhulst PF (1838). Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique 10:113–121. — Introduced the logistic equation dN/dt = r·N·(1 − N/K); the foundation for every S-curve growth model that followed.
  3. Gompertz B (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London 115:513–583. — Original Gompertz formulation for human mortality curves; later adapted to microbial growth. The asymmetric early-inflection behavior is characteristic of the model family.

Modified Gompertz — lag-phase extension

  1. Zwietering MH, Jongenburger I, Rombouts FM, van 't Riet K (1990). Modeling of the bacterial growth curve. Applied and Environmental Microbiology 56(6):1875–1881. — The canonical modern paper; introduces the re-parametrized Gompertz form with λ (lag time), μm (max growth rate), and A (asymptotic log-increase). Widely used in predictive food microbiology and yeast fermentation research. This is the form implemented in the simulator.
  2. Gibson AM, Bratchell N, Roberts TA (1988). Predicting microbial growth: Growth responses of salmonellae in a laboratory medium as affected by pH, sodium chloride and storage temperature. International Journal of Food Microbiology 6(2):155–178. — Earlier application of Gompertz-style curves to microbial lag-phase modeling; predates Zwietering's cleaner re-parametrization.

Baranyi-Roberts — mechanistic lag model

  1. Baranyi J, Roberts TA (1994). A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23(3–4):277–294. — Introduced the adjustment-function α(t) formalism for mechanistic lag-phase modeling. Considered the gold standard of predictive microbial growth curves. This is the form implemented in the simulator (with curvature parameter m = 1).
  2. Baranyi J, Roberts TA, McClure P (1993). A non-autonomous differential equation to model bacterial growth. Food Microbiology 10:43–59. — Theoretical foundation paper preceding the 1994 full treatment; develops the mathematical structure of the adjustment function.
  3. Baranyi J (1998). Comparison of stochastic and deterministic concepts of bacterial lag. Journal of Theoretical Biology 192(3):403–408. — Discusses the biological interpretation of λ and the conditions under which the deterministic lag formulation is appropriate.

Richards — generalized logistic

  1. Richards FJ (1959). A flexible growth function for empirical use. Journal of Experimental Botany 10(2):290–300. — Original introduction of the generalized logistic with shape parameter ν. Developed for plant growth but widely applied to microbial populations for its flexibility in tuning inflection shape.
  2. Tsoularis A, Wallace J (2002). Analysis of logistic growth models. Mathematical Biosciences 179(1):21–55. — Modern synthesis of logistic-family models including Richards; useful for understanding the unified parameter space and the limiting behaviors at ν → 0 (Gompertz) and ν → ∞.

Predictive microbiology reviews

  1. McMeekin TA, Olley JN, Ross T, Ratkowsky DA (1993). Predictive Microbiology: Theory and Application. Research Studies Press, Taunton. — Foundational textbook for predictive microbiology; establishes the framework for applying these growth models to food safety and fermentation design.
  2. Swinnen IAM, Bernaerts K, Dens EJJ, Geeraerd AH, Van Impe JF (2004). Predictive modelling of the microbial lag phase: a review. International Journal of Food Microbiology 94(2):137–159. — Comprehensive review of lag-phase models including all formulations used here; systematically compares Modified Gompertz, Baranyi-Roberts, and empirical alternatives on the same datasets.
  3. López S, Prieto M, Dijkstra J, Dhanoa MS, France J (2004). Statistical evaluation of mathematical models for microbial growth. International Journal of Food Microbiology 96(3):289–300. — Comparative evaluation of growth-model fitting quality across many formulations. Useful for understanding which model fits which type of data best.
  4. Peleg M, Corradini MG (2011). Microbial growth curves: what the models tell us and what they cannot. Critical Reviews in Food Science and Nutrition 51(10):917–945. — Critical review of growth-model interpretation; discusses the biological meaning of λ and μmax across formulations and common misuses.

S. cerevisiae-specific and batch-fermentation references

  1. Walker GM (1998). Yeast Physiology and Biotechnology. Wiley, Chichester. — Comprehensive reference for S. cerevisiae batch-culture kinetics; source for typical lag times, carrying capacities, and strain-specific kinetic benchmarks behind the defaults.
  2. Verduyn C, Postma E, Scheffers WA, van Dijken JP (1990). Physiology of Saccharomyces cerevisiae in anaerobic glucose-limited chemostat cultures. Journal of General Microbiology 136(3):395–403. — Anchor reference for S288c / CEN.PK μmax values (0.40–0.50 h⁻¹) inherited by the simulator from the SGR defaults.
  3. Bai FW, Anderson WA, Moo-Young M (2008). Ethanol fermentation technologies from sugar and starch feedstocks. Biotechnology Advances 26(1):89–105. — Industrial S. cerevisiae benchmarks for commercial ethanologenic strains; source for the higher K, longer λ, and μmax = 0.45 defaults used in the Ethanol Red preset.
  4. Warringer J, Blomberg A (2003). Automated screening in environmental arrays allows analysis of quantitative phenotypic profiles in Saccharomyces cerevisiae. Yeast 20(1):53–67. — High-throughput growth-curve measurement methodology; important for understanding how the classical kinetic parameters are extracted from optical-density data, which feeds into the defaults used here.

Operating-mode design: batch, fed-batch, chemostat

  1. Bailey JE, Ollis DF (1986). Biochemical Engineering Fundamentals, 2nd ed. McGraw-Hill, New York. — Foundational textbook treatment of batch, fed-batch, and continuous bioreactor design; source for the closed-form-vs-ODE scope distinction that underlies this simulator's applicability boundaries.
  2. Shuler ML, Kargi F (2002). Bioprocess Engineering: Basic Concepts, 2nd ed. Prentice Hall. — Parallel textbook coverage emphasizing the kinetic modeling workflow from Monod parameters through batch projection to fed-batch design.
  3. Doran PM (2013). Bioprocess Engineering Principles, 2nd ed. Academic Press. — Modern textbook with worked examples for fed-batch exponential feed design and chemostat parameter extraction.
  4. van Hoek P, Van Dijken JP, Pronk JT (1998). Effect of specific growth rate on fermentative capacity of baker's yeast. Applied and Environmental Microbiology 64:4226–4233. — Canonical reference for the μset = 0.15–0.25 h−1 window that keeps aerobic fed-batch S. cerevisiae respiratory (avoiding Crabtree overflow).
  5. Postma E, Verduyn C, Scheffers WA, van Dijken JP (1989). Enzymic analysis of the Crabtree effect in glucose-limited chemostat cultures of Saccharomyces cerevisiae. Applied and Environmental Microbiology 55:468–477. — Chemostat methodology for extracting Monod parameters (μmax, KS, YX/S); parameter-feedback path between chemostat characterization and batch/fed-batch simulation.

Aerobic vs anaerobic operation for batch kinetics

  1. de Deken RH (1966). The Crabtree effect: A regulatory system in yeast. Journal of General Microbiology 44:149–156. — Original Crabtree-effect paper; explains why aerobic S. cerevisiae ferments even under full O2 when glucose is plentiful, which constrains effective K and YX/S in aerobic batch runs.
  2. van Dijken JP, Scheffers WA (1986). Redox balances in the metabolism of sugars by yeasts. FEMS Microbiology Reviews 32:199–224. — Quantitative aerobic vs anaerobic ATP and biomass yield ratios underlying the regime-dependent μmax ranges used when switching strain defaults between aerobic and anaerobic operation.
  3. Andreasen AA, Stier TJB (1953). Anaerobic nutrition of Saccharomyces cerevisiae. I. Ergosterol requirement for growth in a defined medium. Journal of Cellular and Comparative Physiology 41:23–36. — Ergosterol and UFA supplementation requirement for sustained anaerobic growth; explains why anaerobic lag (λ) is typically 2–3× longer than aerobic.
  4. Visser W, Scheffers WA, Batenburg-van der Vegte WH, van Dijken JP (1990). Oxygen requirements of yeasts. Applied and Environmental Microbiology 56:3785–3792. — Classification of yeast O2 requirements; useful when benchmarking a novel isolate against the S288c / CEN.PK / Ethanol Red strain presets.
Default strain parameters for the simulator: μmax inherits from the SGR calculator defaults (0.48 / 0.42 / 0.45 h⁻¹ for S288c / CEN.PK / Ethanol Red). Batch-specific defaults: N0 = 5 M cells/mL for all strains (typical inoculation density). K = 200 / 220 / 280 M cells/mL (carrying capacity rising with strain robustness). λ = 4 / 4 / 6 h (industrial Ethanol Red under stress has longer lag). ν = 1.5 / 1.5 / 2.0 (Richards shape; higher for the more-asymmetric industrial strain curve).
© 2026 FermAxiom LLC · Author: Peter Krasucki · peter.krasucki@fermaxiom.com  |  Conceptual tool for rapid what-if analysis. Not a substitute for strain-specific calibration.

Yeast Specific Growth Rate Calculator

© 2026 FermAxiom LLC · Author: Peter Krasucki · peter.krasucki@fermaxiom.com  |  Kinetic Model S. cerevisiae  |  Conceptual tool for rapid what-if analysis  |  v3.5

μ = μmax · f(S) · g(P) · h(T) · i(pH) — multiplicative decomposition with strain presets (S288c, CEN.PK, Ethanol Red) and seven substrate / five product / five temperature / two pH sub-models.

Substrate Model f(S)
Ref. Strain
Novel
Ref. Novel
μmax
KS
Sm or KI,S
n (L)
f(S) · Ref —
f(S) · Novel —
Ethanol Model g(P)
Ref. Strain
Novel
Ref. Novel
Pm
n or k (HO) or PI
k (HIN)
g(P) · Ref —
g(P) · Novel —
Temperature Model h(T)
Ref. Strain
Novel
Ref. Novel
Tmin
Topt
Tmax
b or R
c (factor)
h(T) · Ref —
h(T) · Novel —
pH Model i(pH)
Ref. Strain
Novel
Ref. Novel
pHopt
σ
i(pH) · Ref —
i(pH) · Novel —
Instantaneous Specific Growth Rate (μ): Calculating...

Specific Growth Rate Calculator — User Guide Guide · v3.5

Purpose

This calculator predicts the instantaneous specific growth rate μ (h⁻¹) of Saccharomyces cerevisiae under a given combination of environmental conditions — glucose, ethanol, temperature, and pH. It's built for rapid what-if analysis: pick a strain preset, choose kinetic sub-models for each environmental factor, adjust parameters, and watch μ respond. Side-by-side Reference vs Novel comparison lets you benchmark a new isolate against a well-characterized strain without leaving the tool.

Intended users: yeast researchers doing rapid comparative analysis, bioprocess engineers exploring sensitivity to operating conditions, and students learning microbial growth kinetics.

Intended uses: educational exploration, pre-screening strain hypotheses before wet-lab work, teaching, building intuition. It's not a substitute for time-course data fitting or validated bioreactor simulations.

How the calculation works

The tool uses a multiplicative decomposition into four independent environmental factors:

μ = μmax · f(S) · g(P) · h(T) · i(pH)

Each factor f, g, h, i is dimensionless and bounded in [0, 1]; μmax is the maximum rate under simultaneously-optimal conditions. Seven f(S), five g(P), five h(T), and two i(pH) sub-models are available; pick independently for Reference and Novel strains. Full formulas live in the Science sub-tab.

Quick start — six-step workflow

  1. Pick the reference strain from the top-left dropdown (S288c, CEN.PK, or Ethanol Red). Each preset loads literature-plausible defaults for every parameter.
  2. Name the novel strain in the top-right text field. Defaults to "Custom Strain".
  3. Choose sub-models in the four model-cards — Substrate f(S), Ethanol g(P), Temperature h(T), pH i(pH). Each card exposes a Ref dropdown and a Novel dropdown. The parameter rows update automatically to match.
  4. Override defaults by editing any value in the parameter table. Non-physical entries (Ks ≤ 0, Tmin ≥ Tmax, σ ≤ 0, pHopt outside [2, 9]) light up red with a warning banner; calculation proceeds with safe fallbacks.
  5. Use the sliders — Glucose (S, 0–300 g/L), Ethanol (P, 0–120 g/L), Temperature (T, 10–45 °C), pH (3.0–6.0) — to explore the response surface. μ and the four factor readouts update in real time.
  6. Click "Show Optimal Parameters" to auto-position the sliders at the reference strain's numerical optimum and display a 2-row × 6-column comparison table (Strain | S | P | T | pH | μ).

Choosing a sub-model

The right sub-model depends on what phenomenon you're trying to capture. Short decision guide:

Substrate f(S) — glucose uptake kinetics

  • Monod — default saturation kinetics. Pick when there's no substrate inhibition and one dominant transporter system. Most common default for lab yeast in well-controlled conditions.
  • Moser — sigmoidal Hill-type Sn/(Ks+Sn). Pick when growth shows cooperative uptake (multiple transporters with shared regulation, allosteric effects).
  • Tessier — exponential saturation. Approaches saturation faster than Monod at matched Ks. Rarely the first choice; useful for fitting historical datasets that originally used this form.
  • Luong — Monod × (1 − S/Sm)n. Pick when high glucose causes complete growth arrest at some cutoff Sm (e.g., very-high-gravity fermentations approaching osmotic limits).
  • Haldane / Andrews — reversible substrate inhibition via S²/KI,S. Pick when high substrate depresses growth but doesn't cause complete arrest. Analytical peak at S* = √(Ks·KI,S). Haldane and Andrews are identical in functional form here.
  • Edwards — Monod with exponential substrate inhibition. Sharper inhibition onset than Haldane; useful for strains that tolerate substrate up to a point and then abruptly fail.

Ethanol g(P) — product inhibition

  • Generalized (Levenspiel) — default. (1 − P/Pm)n. Smooth decline with a true cutoff at Pm. Pick this unless you have a specific reason to deviate.
  • Linear — simplest decline. Same cutoff as Generalized but constant-slope loss of activity.
  • Hopkins — exp(−k·P). Exponential decay, no hard cutoff. Pick when inhibition is gradual and the population retains small residual activity even at very high ethanol.
  • Aiba — exp(−P/PI). Same functional form as Hopkins but PI is the 1/e-decay concentration, a more interpretable parameter. Canonical ethanol-fermentation form from Aiba 1968.
  • Hinshelwood — 1 − (P/Pm)·exp(k·P). Sharper transition; PmHIN is a scale parameter, not a literal cutoff (zeros below nominal Pm).

Temperature h(T) — thermal response

  • Cardinal (Rosso) — default 3-parameter model. Cleanest interpretation: Tmin and Tmax are the biological limits; Topt is the peak; h(Topt) = 1 exactly.
  • Arrhenius — activation energy term plus quadratic thermal-inactivation. Pick when Ea is known from literature or when you want thermodynamic interpretation of the rising branch.
  • Ratkowsky / Zwietering / Square-root Ratkowsky — empirical square-root family. Zwietering is the standard 4-parameter Ratkowsky form squared throughout. Use when fitting against published data that used a square-root formulation.

pH i(pH) — pH response

  • Gaussian — default bell shape exp(−½((pH−pHopt)/σ)²). Smooth with infinite tails.
  • Quadratic — 1−((pH−pHopt)/σ)², clamped to ≥ 0. Same bell but with hard zeros outside [pHopt±σ] and sharper curvature near the peak.

Interpreting the output

Instantaneous μ display (below the button) shows the computed specific growth rate at the current slider position for both strains. Teal for Reference, dark red for Novel. Units are h⁻¹.

Factor readouts at the bottom of each model card show f(S), g(P), h(T), i(pH) as fractions in [0, 1]. Readout near 1 means that factor is not limiting; near 0 means that factor is strongly suppressing growth. μ = μmax · (product of readouts).

Charts show μ as a function of each slider variable with the other three factors fixed at their current slider values. So the Substrate chart shows μ vs S at the current P, T, pH — move those sliders and the Substrate chart scales accordingly. Y-axis is clamped at 0–0.6 h⁻¹ so strain comparisons are visually consistent regardless of model choice.

Show Optimal Parameters button runs a numerical solver (see Science tab for details), repositions all four sliders at the reference strain's optimum, and displays a 2×6 comparison table. For inhibition models the reported optimal μ lands below μmax by a factor equal to the model's intrinsic peak (e.g., for Haldane: 1/(1 + 2√(Ks/KI,S))).

Warning banner (amber text above the model cards) appears when any input fails validation. The offending inputs get a red border. Calculation still proceeds with safe fallbacks — you can keep adjusting without first fixing all warnings.

Parameter reference — symbols, meanings, units

Quick lookup for every parameter that can appear:

  • μmax — maximum specific growth rate, h⁻¹. 0.40–0.50 typical for S. cerevisiae under joint-optimal conditions.
  • KS — half-saturation constant for glucose, g/L. 0.1–0.3 in defined-medium chemostat; 1–10 batch-effective under industrial conditions.
  • KI,S — substrate inhibition constant, g/L. 200–500 typical.
  • Sm (Luong) — substrate cutoff above which growth = 0, g/L. 200–360.
  • n (Moser, Luong, Generalized) — dimensionless shape parameter. 1.0–3.0.
  • Pm — ethanol concentration at which growth vanishes, g/L. 75–100 typical.
  • k (Hopkins, Hinshelwood) — inhibition rate constant, 1/(g/L). 0.02–0.05.
  • PI (Aiba) — ethanol 1/e-decay concentration, g/L. 55–85.
  • Tmin, Topt, Tmax — cardinal temperatures, °C. 8/30/39 for S288c & CEN.PK; 12/33/42 for Ethanol Red.
  • Ea (Arrhenius) — activation energy, J/mol. 50 000–60 000 typical for yeast growth.
  • R — universal gas constant, 8.314 J/(mol·K). Fixed.
  • b, c (Ratkowsky family) — empirical shape parameters. b in 1/((°C)·h½), c in 1/°C. 0.035–0.045 and 0.3–0.4 typical.
  • pHopt — optimal pH. 5.0 for all three strain presets.
  • σ — pH response bandwidth, pH units. 0.5–0.6 typical.

Common pitfalls

  • KS values aren't transferable between model families. The Monod KS measured in glucose-limited chemostat (~0.18 g/L for S288c) is NOT the KS you should use in Haldane for the same strain — the inhibition fit needs batch-effective values (~1–2 g/L). The calculator uses different KS defaults for saturating vs inhibition models to reflect this. Overriding one side without understanding this distinction can produce misleading comparisons.
  • The multiplicative form assumes factor independence. At extreme conditions — very high glucose, very high ethanol, temperatures far from Topt — cross-terms become significant and the factored form over-predicts μ. For example, osmotic stress at high glucose also reduces h(T) tolerance, not just f(S).
  • "Optimal glucose" from monotone models is capped physiologically. Monod, Moser, and Tessier are monotone in S and would otherwise return the slider maximum (300 g/L) as optimal. The solver caps S at strain-specific upper bounds (30 g/L for S288c, 40 for CEN.PK, 70 for Ethanol Red) because above those concentrations osmotic stress and Crabtree overflow depress growth in ways not captured by f(S). See the Science tab for the full rationale.
  • Inhibition models have intrinsic peaks below μmax. For Haldane with KS = 2, KI,S = 400, the peak f(S*) value is about 0.87, so reported optimal μ ≈ 0.87·μmax. This isn't a bug or solver artifact — it's the model's mathematical ceiling.
  • The four charts aren't truly independent plots. Each chart fixes the other three sliders at their current positions. Moving a slider redraws the other three charts too. To see a pure factor response, zero out the other variables (set P = 0, T = Topt, pH = pHopt) before reading a given chart.
Parameter label convention: when Reference and Novel use different sub-models with the same symbol (e.g. Ks in Monod vs Luong), the row header shows both with two-letter superscripts (KsMD | KsLG) so each side's number is unambiguous. The same applies to Pm (GN = Generalized, LN = Linear, HIN = Hinshelwood), making clear that Pm is a true cutoff for Generalized/Linear but only a scale parameter for Hinshelwood.

Aerobic vs anaerobic operation

The strain defaults in this tool reflect typical aerobic respirofermentative conditions — μmax ≈ 0.42–0.48 h−1, KS in the 0.1–1.5 g/L range, ethanol-tolerance parameters tuned for standard lab and industrial regimes. Translating to other regimes:

  • Fully anaerobic operation (bioethanol, controlled fermentation): reduce μmax by ~20–25% (anaerobic S. cerevisiae typically 0.25–0.40 h−1 because ATP yield from pure fermentation is ~2 mol per mol glucose vs ~16 mol respirative). Set g(P) tight — Pm 50–110 g/L depending on strain. Note that sterol / unsaturated-fatty-acid supplementation is required for sustained anaerobic growth, independent of anything captured by the factor decomposition.
  • Microaerobic / oxygen-limited: metabolism slides toward fermentation when kL·a limits oxygen uptake rate. Expect parameters to sit between the aerobic and anaerobic extremes; the multiplicative form doesn't capture the kL·a coupling directly, so use values interpolated between the two regime estimates.
  • Crabtree overflow (respirofermentative): at residual glucose above ~50 mg/L, S. cerevisiae ferments even under full aerobiosis. The f(S) factor captures substrate saturation, but the overflow-to-fermentation switch is a yield effect (biomass vs ethanol) not a rate effect — it doesn't show up in μ directly but drops the effective YX/S from ~0.50 g/g respiratory to 0.10–0.15 g/g fermentative.

When to use SGR vs time-course modeling

The two top-level tools in this suite answer different questions and are designed to be used together:

  • SGR Calculator (this tool) answers "what's the instantaneous growth rate μ under these conditions?" — useful for sensitivity analysis, strain benchmarking, environmental parameter sweeps, and screening hypothesis space before committing wet-lab runs.
  • Growth Kinetics Batch Simulator (top tab) answers "when does the culture hit N = 200 M cells/mL?" — useful for batch timing, harvest planning, lag-phase comparison, and projecting time course given a μmax estimate from this calculator plus strain / medium-specific K and λ.
  • Fed-batch and chemostat regimes are out of scope for both tools. For fed-batch design, use a coupled-ODE simulator with the SGR factor decomposition μ(S, P, T, pH) as the kinetic core. For chemostat characterization, extract parameters (μmax, KS, YX/S) by steady-state runs at varied dilution rate and feed them back into batch projection.

Scope and caveats

This calculator is a conceptual / educational tool for rapid what-if analysis and strain benchmarking. Known limitations:

  • No cross-terms. Multiplicative factor decomposition ignores temperature-dependent Ks, pH-dependent ethanol tolerance, and similar interactions. Real bioprocesses have these.
  • Instantaneous μ only. No time-course integration. For cell-density vs. time, use the Growth Kinetics Models top tab.
  • Defaults are literature-anchored, not strain-specific. Regulatory, scale-up, or investment-grade predictions require calibrating constants against your own batch or chemostat time-course data.
  • Parameters for S. cerevisiae only. Other yeasts or bacteria would need different cardinal temperatures, different KS, different ethanol tolerances. The sub-model forms themselves are general but the defaults and ranges are yeast-specific.
  • Glucose only. Substrate kinetics assume glucose as the limiting carbon source. Mixed-sugar feeds (sucrose, fructose, maltose blends) have their own KS values and sometimes sequential uptake, which this form doesn't capture.

Specific Growth Rate — Mathematical Formulations Science · v3.5

Multiplicative decomposition

Environmental factors are assumed to act independently on the specific growth rate:

μ = μmax · f(S) · g(P) · h(T) · i(pH)

Each factor is dimensionless and bounded in [0, 1]; μmax is the maximum achievable rate under simultaneously-optimal conditions. The independence assumption is reasonable away from extremes but loses fidelity when cross-terms dominate — for example, temperature-dependent Ks, pH-dependent ethanol tolerance, or osmotic stress that reduces both f(S) and h(T) together. Use this as a first-order approximation, not as a rigorous bioreactor model.

Substrate kinetics — f(S)

Seven models, split into two families by whether they include substrate inhibition:

Monod:    f(S) = S / (KS + S)
Moser:    f(S) = Sn / (KS + Sn)
Tessier:  f(S) = 1 − exp(−S / KS)
Luong:    f(S) = [S / (KS + S)] · (1 − S/Sm)n
Haldane:  f(S) = S / (KS + S + S²/KI,S)
Andrews:  f(S) = S / (KS + S + S²/KI,S)   (same functional form as Haldane)
Edwards:  f(S) = [S · exp(−S/KI,S)] / (KS + S)

Monod, Moser, Tessier are monotone in S (saturate to 1). Luong, Haldane, Andrews, Edwards are interior-max models with substrate inhibition. For Haldane and Andrews, the analytical optimum is:

S* = √(KS · KI,S),   f(S*) = 1 / (1 + 2·√(KS / KI,S))

The peak of Haldane is never 1 — for the default parameters (KS ≈ 1–8, KI,S ≈ 400–450) the peak value is 0.85–0.96. This mathematical ceiling is why the calculator's "optimal μ" for inhibition models falls below μmax.

Ethanol (product) inhibition — g(P)

Generalized (Levenspiel): g(P) = (1 − P/Pm)n,   clamped to 0 for P ≥ Pm
Linear: g(P) = max(0,  1 − P/Pm)
Hopkins: g(P) = exp(−k · P)
Aiba: g(P) = exp(−P/PI)
Hinshelwood: g(P) = max(0,  1 − (P/Pm) · exp(k · P))

Pm is a true ethanol cutoff for Generalized and Linear (g reaches 0 there). In Hinshelwood, Pm is a scale parameter and the function zeros at a value below it — hence the PmHIN superscript in the parameter labels. All five g(P) models are maximized (= 1) at P = 0, so the optimal-finder sets P* = 0 analytically.

Temperature response — h(T)

Cardinal (Rosso 1993):
h(T) = [(T−Tmin)² · (T−Tmax)] / [(Topt−Tmin) · ((Topt−Tmin)·(T−Topt) − (Topt−Tmax)·(Topt+Tmin−2T))]

Zero outside [Tmin, Tmax]; equals 1 at T = Topt. Three-parameter model with clean cardinal-temperature interpretation.

Arrhenius + thermal deactivation:
h(T) = exp(−(Ea/R) · (1/TK − 1/Topt,K)) · [1 − ((T−Topt)/(Tmax−Topt))²]+

Temperatures TK, Topt,K in Kelvin. The deactivation bracket applies only for T > Topt and is clamped to zero at T = Tmax. Below Topt, only the Arrhenius activation term operates. This extension is needed because pure Arrhenius has no maximum and cannot represent thermal death.

Ratkowsky: h(T) = [b · (T−Tmin)]² · [1 − exp(c · (T−Tmax))]
Zwietering: h(T) = {b · (T−Tmin) · [1 − exp(c · (T−Tmax))]}²
Sqrt Ratkowsky: h(T) = b · (T−Tmin) · [1 − exp(c · (T−Tmax))]

All three go to 0 outside [Tmin, Tmax]. Zwietering is the standard 4-parameter Ratkowsky μ-form (squared throughout); the "Ratkowsky" and "Sqrt Ratkowsky" options here expose variant forms using the same shape parameters b, c. Unlike Cardinal, these models do not guarantee h = 1 at a specific T — the output is further clamped to [0, 1] so it fits into the multiplicative decomposition.

pH response — i(pH)

Gaussian: i(pH) = exp(−(pH − pHopt)² / (2σ²))
Quadratic: i(pH) = max(0,  1 − ((pH − pHopt) / σ)²)

Both peak at pH = pHopt, so the optimal-finder sets pH* = pHopt analytically (clamped to the [3, 6] slider range). Gaussian has infinite support; Quadratic is zero outside [pHopt − σ, pHopt + σ].

Biological interpretation of parameters

What the kinetic constants physically represent — useful for translating between model-fitting values and observable biology:

  • μmax (h⁻¹) is the maximum rate at which a cell population can double under simultaneously-optimal S, P, T, pH. Biologically rate-limited by ribosome biogenesis in exponential phase; for S. cerevisiae, this caps out near 0.45–0.50 h⁻¹ even under ideal conditions because protein-synthesis machinery has physical turnover limits.
  • KS (g/L) is the glucose concentration at which f(S) = 0.5 (Monod). Biologically reflects the affinity of active glucose transporters — the 17-member Hxt hexose-transporter family in yeast, with Hxt6/7 being high-affinity (Km ≈ 1 mM ≈ 0.18 g/L) and Hxt1/3 being low-affinity. The sub-1 g/L chemostat KS reflects induction of high-affinity transporters under glucose limitation.
  • KI,S (g/L) parameterizes substrate-inhibition kinetics. For S. cerevisiae, this is not a single mechanism but a composite of osmotic stress (high solute → reduced water activity), catabolite repression (glucose represses respiration genes), and Crabtree overflow (aerobic ethanol fermentation diverts carbon from biomass). No single molecular parameter captures KI,S.
  • Pm (g/L) is the ethanol concentration at which growth ceases. Reflects membrane integrity loss and cytoplasmic protein denaturation. Commercial strains like Ethanol Red evolved higher Pm via membrane ergosterol content, unsaturated fatty acid composition, and trehalose/chaperone expression — hence their 90–100 g/L tolerance vs. 70–80 g/L for lab strains.
  • Tmin, Topt, Tmax are the cardinal temperatures. Tmin (~5–12 °C) is set by membrane fluidity loss; Topt (~28–34 °C) is the enzymatic-rate × protein-stability tradeoff peak; Tmax (~38–42 °C) is the protein-denaturation threshold. Industrial strains typically tolerate higher Tmax through trehalose accumulation and heat-shock protein expression.
  • Ea (J/mol) is the Arrhenius activation energy of the rate-limiting enzymatic step in growth. For yeast, 50 000–60 000 J/mol (50–60 kJ/mol) matches typical enzymatic activation energies measured in vitro.
  • pHopt is the pH at which proton motive force across the plasma membrane is optimal. For yeast, pHopt ≈ 5.0 reflects the balance between cytoplasmic pH regulation (~7.0–7.2) and environmental pH gradient requirements for nutrient uptake.

Key derivations

Haldane peak location. For f(S) = S / (KS + S + S²/KI,S), set df/dS = 0:

df/dS = [(KS + S + S²/KI,S) · 1 − S · (1 + 2S/KI,S)] / (KS + S + S²/KI,S)² = 0

The numerator simplifies to KS − S²/KI,S = 0, giving S* = √(KS·KI,S). Substituting back:

f(S*) = √(KS·KI,S) / [KS + √(KS·KI,S) + KS] = 1 / [1 + 2·√(KS/KI,S)]

For S288c defaults KS = 1, KI,S = 400: S* = 20 g/L, f(S*) = 1/(1 + 2·0.05) = 0.909. This is why the "optimal μ" for Haldane lands 9–10% below μmax — the model itself imposes this ceiling.

Monod saturation. For f(S) = S/(KS+S), f approaches 1 asymptotically as S increases. Half-maximum at S = KS, 90% at S = 9·KS, 99% at S = 99·KS, 99.9% at S = 999·KS. For KS = 0.18 g/L (S288c chemostat value), 99% is reached at just 18 g/L glucose — so above ~20 g/L, Monod effectively saturates and "optimal S" becomes ill-defined without a physiological cap.

Arrhenius normalization at Topt. The pure Arrhenius form k = A·exp(−Ea/RT) has no fitting constant A in our calculator — instead, we normalize so h(Topt) = 1:

hactivation(T) = exp(−(Ea/R) · (1/T − 1/Topt)),   with T, Topt in Kelvin

This makes Ea the only free parameter of the rising branch. The deactivation branch is a separate empirical quadratic that brings h down to 0 at Tmax.

Model assumptions and limitations

Each sub-model carries implicit assumptions about the underlying biology:

Substrate kinetics.

  • Monod assumes steady-state Briggs-Haldane enzyme kinetics with one rate-limiting transporter. No substrate inhibition. Breaks down when: (a) multiple transporters contribute with different Km, (b) substrate inhibition matters, (c) growth rate is not substrate-limited.
  • Moser assumes cooperative substrate uptake or allosteric regulation with Hill coefficient n. Reduces to Monod at n = 1. Harder to justify mechanistically for glucose in yeast, since Hxt transporters are largely non-cooperative.
  • Tessier is purely empirical; no mechanistic derivation. Included for compatibility with older datasets.
  • Luong is Monod multiplied by an empirical product-style inhibition term. The Sm cutoff has no thermodynamic basis — it's a convenient fitting parameter.
  • Haldane / Andrews derives from quasi-steady-state enzyme kinetics with two substrate molecules binding the enzyme (one productive, one inhibitory). Physically meaningful for enzymes with discrete substrate-binding sites. For microbial growth, it's applied phenomenologically even when the underlying mechanism is different (e.g., osmotic stress).
  • Edwards replaces Haldane's rational-function inhibition with an exponential form; empirical, sharper onset.

Product inhibition.

  • Generalized / Linear assume a threshold-like cutoff at Pm. Useful for fitting; Pm has direct interpretation.
  • Hopkins / Aiba exponential decay reflects thermodynamic equilibrium with protein denaturation. Aiba's PI is the 1/e-decay concentration — more interpretable than Hopkins' k.
  • Hinshelwood is empirical; the form comes from early chemical-kinetics textbooks without microbial-specific justification.

Temperature response.

  • Cardinal (Rosso) is empirical but produces a clean bell shape with Tmin, Topt, Tmax directly interpretable. No thermodynamic basis.
  • Arrhenius + deactivation combines Arrhenius theory (activation-energy barrier) with a purely empirical quadratic deactivation. The activation branch has mechanistic grounding; the deactivation branch is curve-fitting.
  • Ratkowsky family is empirical square-root relation. Widely used in predictive food microbiology but has no mechanistic derivation. Zwietering (Ratkowsky squared) is the most common form in the literature.

pH response.

  • Gaussian assumes a single normally-distributed activity window. Smooth, widely tolerated by fitting, infinite tails.
  • Quadratic is a computationally simpler alternative with hard cutoffs — sharper but less forgiving at the edges of the allowed pH range.

The overall framework assumes independent factor contributions (the multiplicative form). This fails most noticeably when: (i) temperature and glucose combine to produce osmotic stress at high-gravity + elevated T, (ii) pH modulates membrane permeability to ethanol (so g(P) depends on pH), (iii) substrate inhibition and temperature interact via Crabtree-effect intensification at warm temperatures. These cross-terms are absent from this tool by design — it's a decomposition, not a dynamical simulator.

Aerobic vs anaerobic parameter regimes

S. cerevisiae is facultative and uses distinctly different metabolic modes under aerobic vs anaerobic conditions. The multiplicative SGR factor decomposition handles both regimes with the same mathematical forms but with different numerical parameters:

  • μmax differs by ATP-yield ratio. Full respiration generates ~16 mol ATP per mol glucose (substrate-level phosphorylation + oxidative phosphorylation through the electron transport chain); pure fermentation generates ~2 mol (SLP only). Measured μmax values scale accordingly: 0.40–0.50 h−1 aerobic, 0.25–0.40 h−1 anaerobic.
  • Biomass yield YX/S differs by 4–5×: ~0.50 g DCW/g glucose under fully respiratory conditions, ~0.08–0.12 g/g anaerobic. Under Crabtree overflow, aerobic YX/S falls to 0.10–0.15 g/g even with adequate O2. YX/S is not a direct SGR parameter — it enters the batch simulator's kinetic coupling through substrate depletion dynamics — but the regime dependence is important context for interpreting the f(S) factor's meaning.
  • Product (ethanol) inhibition g(P): aerobic respiratory strains are not ethanol-tolerant (Pm ~30–50 g/L effective) because sustained growth on ethanol as a C-source is slow. Anaerobic industrial strains are bred for tolerance (Pm ~80–110 g/L). The Ethanol Red preset defaults reflect the industrial anaerobic regime.
  • Temperature response h(T) shifts modestly between regimes: anaerobic Topt tends to be 1–2 °C lower than aerobic for the same strain because fermentation heat output and heat-stress interact with ethanol toxicity at warm temperatures.
  • pH response i(pH) is largely regime-independent; membrane proton-motive-force requirements don't depend strongly on whether the cell is fermenting or respiring.

Crabtree effect. In S. cerevisiae and other Crabtree-positive yeasts, residual glucose above ~50 mg/L triggers fermentative (ethanol-producing) metabolism even under full aerobiosis. This is a substrate-concentration threshold, not an O2-availability response: the glucose-signaling machinery (Snf3/Rgt2 sensors, Hxk2, glucose repression of TCA-cycle genes) shuts down respiratory pathways when sugar is plentiful. Practically, in aerobic batch culture with initial glucose above ~50 mg/L, yeast ferments glucose to ethanol regardless of oxygen until the substrate is depleted, then enters a diauxic shift and respires the accumulated ethanol. The SGR f(S) factor models substrate saturation but not this overflow switch — expect biomass yield to drop to fermentative values above the threshold.

Pasteur effect. The reverse — under oxygen limitation, glucose flux shifts from respiration to fermentation. Physiologically distinct from the Crabtree effect (O2-driven vs glucose-driven) but visually similar (fermentation dominates in both). Pasteur shift becomes relevant at the high-cell-density end of aerobic batch runs, where kL·a starts limiting and oxygen uptake rate saturates.

Workflow recommendation. Use this SGR calculator to compare strain μ under named conditions; when projecting batch time course, pass μmax forward into the Batch Simulator top tab. For fed-batch or chemostat design, neither tool in this suite applies directly — see the Batch Simulator's Science sub-tab discussion of "Batch, fed-batch, and continuous systems" for the rationale and recommended workflow.

Numerical optimal-parameter finder

The Show Optimal Parameters button runs the following algorithm:

  • S* — Log-spaced scan over [10−3, Sopt,max] g/L with 400 points to locate the argmax of f(S), followed by golden-section refinement (φ = (√5−1)/2, 40 iterations) within ±2 log-spacing units of the scan maximum. Sopt,max is the strain-specific physiological cap (30 / 40 / 70 g/L for S288c / CEN.PK / Ethanol Red). The scan uses f ≥ fbest (not strict >) so saturating models like Tessier walk their numerical-saturation plateau up to the cap rather than stopping at the smallest S where f first reaches 1 within machine precision.
  • T* — Linear scan over [10, 45] °C with 200 points, then golden-section refinement.
  • P* = 0 — analytical; every g(P) model peaks at P = 0.
  • pH* = clamp(pHopt, 3, 6) — analytical; Gaussian and Quadratic both peak at pHopt.

The reported optimal μ is evaluated at (S*, P*, T*, pH*) using the actual model, so for inhibition f(S) models it lands below the user-entered μmax — the difference is the model's inherent peak-reduction factor, not a solver artifact.

Physiological glucose cap

Why constrain the S optimizer at Sopt,max rather than the slider maximum (300 g/L)?

The multiplicative form μ = μmax·f(S)·g(P)·h(T)·i(pH) assumes the four factors are independent. In reality, at glucose concentrations above ~50–100 g/L S. cerevisiae experiences osmotic stress, Crabtree-effect carbon overflow into ethanol, and catabolite repression — all of which depress growth but aren't captured by the f(S) factor alone. Without a cap, saturating models (Monod, Moser, Tessier) would report the 300 g/L slider max as "optimal" even though real yeast growth is strongly inhibited at that concentration. The caps (30 / 40 / 70 g/L) sit well below the onset of these non-substrate effects, giving physiologically defensible optima across all model choices.

Input validation

Before each μ computation, parameters are validated for sign and ordering: KS > 0, σ > 0, Ea > 0, Pm > 0, Tmin < Topt < Tmax, pHopt ∈ [2, 9]. Violations are flagged in red around the offending input with a warning banner at the top of the panel; calculation proceeds using safe fallback values so the user can continue exploring. This is a deliberate design choice — validation informs without interrupting.

Specific Growth Rate Calculator — References Refs · v3.5

Primary sources for the kinetic sub-models, temperature-response formulations, and default parameter values used in this calculator. Citations are grouped by factor — f(S), g(P), h(T), i(pH) — plus strain presets and broader review literature.

Reading order for new users. Start with Monod 1949 and the Bailey-Ollis or Shuler-Kargi textbook chapter on microbial growth (these frame the multiplicative decomposition). For substrate-inhibition theory, Andrews 1968 followed by Luong 1987. For temperature, Rosso 1993 is the foundational Cardinal-model paper; Ratkowsky 1982/1983 and Zwietering 1991 for square-root formulations. For ethanol inhibition, Aiba 1968 and Levenspiel 1980 are the classical sources. For strain-specific S. cerevisiae kinetics, Verduyn 1990 is the anchor for chemostat μmax and KS; Bai 2008 for commercial fuel-ethanol strains.

Substrate kinetics — f(S)

  1. Monod J (1949). The growth of bacterial cultures. Annual Review of Microbiology 3:371–394. — Foundational saturation kinetic μ = μmax · S/(Ks + S).
  2. Moser H (1958). The Dynamics of Bacterial Populations Maintained in the Chemostat. Carnegie Institution of Washington Publication 614. — Hill-type sigmoidal generalization, Sn/(Ks + Sn), accommodating cooperative or allosteric substrate uptake.
  3. Luong JHT (1987). Generalization of Monod kinetics for analysis of growth data with substrate inhibition. Biotechnology and Bioengineering 29:242–248. — Monod × (1 − S/Sm)n; the SmLG parameter.
  4. Andrews JF (1968). A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnology and Bioengineering 10:707–723. — Substrate self-inhibition S / (Ks + S + S²/KI,S); analytical optimum at S* = √(Ks · KI,S).
  5. Haldane JBS (1930). Enzymes. Longmans, Green & Co., London. — Original enzyme-kinetic derivation of the substrate-inhibition form later applied to microbial growth by Andrews.
  6. Tessier G (1942). Croissance des populations bactériennes et quantité d'aliment disponible. Revue Scientifique 3208:209–216. — Exponential saturation 1 − exp(−S/Ks); approaches 1 faster than Monod at comparable Ks.
  7. Edwards VH (1970). The influence of high substrate concentrations on microbial kinetics. Biotechnology and Bioengineering 12:679–712. — Variant of Andrews with exponential rather than quadratic inhibition; internal maximum shifts slightly below the Haldane/Andrews optimum.

Ethanol (product) inhibition — g(P)

  1. Levenspiel O (1980). The Monod equation: A revisit and a generalization to product inhibition situations. Biotechnology and Bioengineering 22:1671–1687. — The (1 − P/Pm)n generalized nonlinear form; PmGN is the true ethanol cutoff.
  2. Aiba S, Shoda M, Nagatani M (1968). Kinetics of product inhibition in alcohol fermentation. Biotechnology and Bioengineering 10:845–864. — Exponential form exp(−P/PI) from continuous ethanol fermentation experiments; the canonical ethanol-inhibition citation.
  3. Hinshelwood CN (1946). The Chemical Kinetics of the Bacterial Cell. Clarendon Press, Oxford. — Origin of the 1 − (P/Pm) · exp(k · P) form used here; note that in this model PmHIN is a scale parameter and the function zeros out at a value below it.
  4. Ghose TK, Tyagi RD (1979). Rapid ethanol fermentation of cellulose hydrolysate. II. Product and substrate inhibition and optimization of fermentor design. Biotechnology and Bioengineering 21:1401–1420. — Linear-decline form (1 − P/Pm) applied to S. cerevisiae; source of industrial Pm reference values.

Temperature dependence — h(T)

  1. Rosso L, Lobry JR, Flandrois JP (1993). An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model. Journal of Theoretical Biology 162:447–463. — The Cardinal (CTMI) 3-parameter model using Tmin, Topt, Tmax; default implementation in this calculator.
  2. Arrhenius S (1889). Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Zeitschrift für Physikalische Chemie 4:226–248. — Original activation-energy formulation. This calculator implements Arrhenius activation normalized at Topt, multiplied by a quadratic thermal-inactivation term that decays to 0 at Tmax (pure Arrhenius has no maximum and cannot represent thermal death alone).
  3. Ratkowsky DA, Olley J, McMeekin TA, Ball A (1982). Relationship between temperature and growth rate of bacterial cultures. Journal of Bacteriology 149:1–5. — Square-root model √μ = b(T − Tmin).
  4. Ratkowsky DA, Lowry RK, McMeekin TA, Stokes AN, Chandler RE (1983). Model for bacterial culture growth rate throughout the entire biokinetic temperature range. Journal of Bacteriology 154:1222–1226. — Four-parameter extension covering the full Tmin–Tmax range.
  5. Zwietering MH, de Koos JT, Hasenack BE, de Wit JC, van 't Riet K (1991). Modeling of bacterial growth as a function of temperature. Applied and Environmental Microbiology 57:1094–1101. — Squared-Ratkowsky variant; smoother curvature near Topt than the linear-squared form.

pH dependence — i(pH)

  1. Rosso L, Lobry JR, Bajard S, Flandrois JP (1995). Convenient model to describe the combined effects of temperature and pH on microbial growth. Applied and Environmental Microbiology 61:610–616. — CTMI-style pH model; supports the bell-shaped formulations implemented here.
  2. Presser KA, Ratkowsky DA, Ross T (1997). Modelling the growth rate of Escherichia coli as a function of pH and lactic acid concentration. Applied and Environmental Microbiology 63:2355–2360. — Quadratic pH-response form used in the Quadratic option.
  3. Narendranath NV, Power R (2005). Relationship between pH and medium dissolved solids in terms of growth and metabolism of lactobacilli and Saccharomyces cerevisiae during ethanol production. Applied and Environmental Microbiology 71:2239–2243. — Reference for S. cerevisiae pH tolerance (pHopt ≈ 5.0, tolerable range 3.5–6.0) used in default σ values.

Reviews and syntheses

  1. Kovárová-Kovar K, Egli T (1998). Growth kinetics of suspended microbial cells: from single-substrate-controlled growth to mixed-substrate kinetics. Microbiology and Molecular Biology Reviews 62(3):646–666. — Comprehensive review of microbial growth kinetics beyond pure Monod; discusses substrate inhibition, mixed substrates, and when classical models fail. Essential reading for understanding the limits of the multiplicative decomposition.
  2. Roels JA (1983). Energetics and Kinetics in Biotechnology. Elsevier Biomedical Press, Amsterdam. — Rigorous thermodynamic and stoichiometric treatment of microbial growth; classical reference for the relationship between kinetic parameters and cellular energetics.
  3. Panikov NS (1995). Microbial Growth Kinetics. Chapman & Hall, London. — Book-length synthesis of growth-kinetic models including substrate inhibition, product inhibition, and multi-substrate systems. Covers most of the f(S) and g(P) sub-models in this calculator.
  4. Ross T, Dalgaard P (2004). Secondary models. In: McKellar RC, Lu X (eds.), Modeling Microbial Responses in Food. CRC Press. — Modern review of secondary models (T, pH, aw) with emphasis on Ratkowsky-family formulations widely used in predictive microbiology.
  5. Maier RM (2009). Bacterial growth. In: Maier RM, Pepper IL, Gerba CP (eds.), Environmental Microbiology, 2nd ed. Academic Press. — Textbook chapter giving a concise integrated overview of growth kinetics suitable as a quick-reference introduction.

Strain defaults and textbook references

  1. Verduyn C, Postma E, Scheffers WA, van Dijken JP (1990). Physiology of Saccharomyces cerevisiae in anaerobic glucose-limited chemostat cultures. Journal of General Microbiology 136:395–403. — S288c and CEN.PK reference μmax ≈ 0.45–0.50 h⁻¹ under anaerobic glucose-limited chemostat conditions; Ks on the order of 0.1–1 mg/L glucose.
  2. van Hoek P, Van Dijken JP, Pronk JT (1998). Effect of specific growth rate on fermentative capacity of baker's yeast. Applied and Environmental Microbiology 64:4226–4233. — Kinetic constants for S288c under defined-medium chemostat, including the Ks = 0.183 g/L glucose used as the S288c default.
  3. Bai FW, Anderson WA, Moo-Young M (2008). Ethanol fermentation technologies from sugar and starch feedstocks. Biotechnology Advances 26:89–105. — Industrial S. cerevisiae (Ethanol Red, Superstart, etc.) benchmark kinetics; Pm ≈ 90–100 g/L ethanol tolerance for VHG strains.
  4. Bailey JE, Ollis DF (1986). Biochemical Engineering Fundamentals, 2nd ed. McGraw-Hill, New York. — Standard textbook treatment of Monod, Haldane, Andrews, and related microbial growth kinetics.
  5. Shuler ML, Kargi F (2002). Bioprocess Engineering: Basic Concepts, 2nd ed. Prentice Hall. — Textbook presentation of multiplicative environmental-factor decomposition for μ; pedagogic reference for the overall framework used here.
  6. Doran PM (2013). Bioprocess Engineering Principles, 2nd ed. Academic Press. — Contemporary textbook reference for temperature, pH, and product-inhibition kinetics in bioreactor design.

Aerobic / anaerobic metabolism and Crabtree effect

  1. de Deken RH (1966). The Crabtree effect: A regulatory system in yeast. Journal of General Microbiology 44:149–156. — Original description of glucose-triggered respiratory suppression in S. cerevisiae. Foundational reference for why aerobic yeast ferments when glucose is plentiful.
  2. Pronk JT, Steensma HY, van Dijken JP (1996). Pyruvate metabolism in Saccharomyces cerevisiae. Yeast 12:1607–1633. — Detailed treatment of the pyruvate branch point that decides respiration vs fermentation; includes the mechanistic basis for Crabtree overflow.
  3. van Dijken JP, Scheffers WA (1986). Redox balances in the metabolism of sugars by yeasts. FEMS Microbiology Reviews 32:199–224. — Quantitative treatment of aerobic vs anaerobic energy metabolism; source for the ~16 vs ~2 mol ATP/mol glucose ratios underlying regime-dependent μmax.
  4. Andreasen AA, Stier TJB (1953). Anaerobic nutrition of Saccharomyces cerevisiae. I. Ergosterol requirement for growth in a defined medium. Journal of Cellular and Comparative Physiology 41:23–36. — Original ergosterol and UFA supplementation requirements for sustained anaerobic growth; still the standard reference for why anaerobic lag (λ) is longer than aerobic.
  5. Visser W, Scheffers WA, Batenburg-van der Vegte WH, van Dijken JP (1990). Oxygen requirements of yeasts. Applied and Environmental Microbiology 56:3785–3792. — Classification framework for yeast O2 requirements; useful for deciding whether a novel isolate is facultative, strictly aerobic, or Crabtree-positive.
  6. van Hoek P, Van Dijken JP, Pronk JT (1998). Effect of specific growth rate on fermentative capacity of baker's yeast. Applied and Environmental Microbiology 64:4226–4233. — Quantitative measurement of how μ affects respiratory vs fermentative capacity; directly relevant to fed-batch μset selection (0.15–0.25 h−1 to stay respiratory).
  7. Postma E, Verduyn C, Scheffers WA, van Dijken JP (1989). Enzymic analysis of the Crabtree effect in glucose-limited chemostat cultures of Saccharomyces cerevisiae. Applied and Environmental Microbiology 55:468–477. — Chemostat methodology for extracting Monod parameters under controlled dilution rate; source for the parameter feedback loop between chemostat characterization and batch/fed-batch simulation.
Default parameter anchors: S288c μmax = 0.48 h⁻¹, Ks = 0.18 g/L (Verduyn 1990, van Hoek 1998); CEN.PK μmax = 0.42 h⁻¹, Ks = 0.22 g/L (Verduyn 1990, defined-medium chemostat); Ethanol Red μmax = 0.45 h⁻¹, Ks = 1.5 g/L, Pm = 100 g/L (Bai 2008, industrial-strain benchmarks). Cardinal Tmin/Topt/Tmax = 8/30/39 °C for S288c and CEN.PK, 12/33/42 °C for Ethanol Red (composite of Rosso 1993 framework with industrial fermentation data). pHopt = 5.0 with σ = 0.5–0.6 across strains (Narendranath 2005). Inhibition-model Ks (used by Haldane, Andrews, Luong, Edwards) is held separately at 2–8 g/L — batch-effective rather than chemostat values — so model peaks land in physiologically realistic ranges (Haldane peak S* = √(Ks·KI,S) ≈ 28–60 g/L across the three strains). Luong Sm = 200–280 g/L, KI,S = 400–450 g/L. Physiological glucose cap: The Show-Optimal solver constrains the S search to a strain-specific upper bound (S288c: 30, CEN.PK: 40, Ethanol Red: 70 g/L) set well below the concentrations at which osmotic stress, catabolite overflow, and Crabtree-effect carbon diversion begin to suppress growth — effects not captured by the pure multiplicative factor decomposition. Without this cap, saturating models (Monod / Moser / Tessier) would return the 300 g/L slider maximum as "optimal", which contradicts observed S. cerevisiae physiology.
© 2026 FermAxiom LLC · Author: Peter Krasucki · peter.krasucki@fermaxiom.com  |  Conceptual tool for rapid what-if analysis. Not a substitute for strain-specific calibration.