Equations and assumptions

• Temperature factor (cells): Two models, user-selectable. Cardinal (default, Rosso, Lobry & Flandrois 1993, CTMI): f_T = (T−T_max)(T−T_min)² / [(T_opt−T_min) · ((T_opt−T_min)(T−T_opt) − (T_opt−T_max)(T_opt+T_min−2T))] for T_min < T < T_max, and exactly 0 outside that interval. Asymmetric (long shoulder below T_opt, sharp drop above), with biologically interpretable cardinal temperatures (input-field defaults T_min=5°C, T_opt=30°C, T_max=42°C; the Ethanol Red preset shifts T_opt to 32°C, the industrial fuel-ethanol value, on Load Preset). Gaussian (legacy): f_T = exp(−((T−T_opt)²/(2σ_T²))) — symmetric, never reaches zero. Useful for back-compatibility with older parameter sets.
• pH factor (cells): Two models, user-selectable. Cardinal pH Model (default, Rosso et al. 1995): f_pH = (pH−pH_max)(pH−pH_min)² / [(pH_opt−pH_min) · ((pH_opt−pH_min)(pH−pH_opt) − (pH_opt−pH_max)(pH_opt+pH_min−2pH))] — asymmetric (long acidic shoulder, sharp alkaline cliff), reflecting S. cerevisiae's real pH response. Gaussian (legacy): f_pH = exp(−((pH−pH_opt)²/(2σ_pH²))).
• Substrate kinetics (cells): Two models, user-selectable. Haldane (default, unified): mu_S = S / (K_s + S + S²/K_i,S). Combines Monod uptake and substrate-inhibition into one biologically-grounded expression. Monod × inhibition (legacy): Monod S/(K_s+S) multiplied by 1/(1 + S/K_i,S).
• Ethanol inhibition (cells): Three models, user-selectable. Luong (default, 1985): f_E = 1 below E_inh, ((E_max−E)/(E_max−E_inh))ⁿ in the inhibition range, 0 above E_max. Best empirical fit for S. cerevisiae. Hill (legacy): f_E = max(0, (1 − (max(E−E_inh,0))/E_max)ⁿ). Aiba (1968): f_E = exp(−k_E · E). Smooth exponential; never reaches zero.
• Ethanol inhibition (production, decoupled): Same three model choices, separate parameters (E_max,p, E_inh,p, n_p, k_E,p). Only the Luedeking-Piret β term is gated by f_E,p — this is the *specifically* ethanol-inhibited non-growth production contribution. Maintenance catabolism (m_S, regime-adjusted ~25× higher for anaerobic than aerobic) continues regardless of E, because live cells always need ATP for maintenance. What actually terminates fermentation is cell death (k_d,eff rising with E), which drives X → 0 and thus r_P = q_P·X → 0 — not a hard metabolic cliff.
• Smooth substrate depletion: A soft-switch factor f_avail = S/(S + K_s,min) (K_s,min = 0.01 g/L) replaces the hard S=0 cutoff, ensuring rates taper smoothly as substrate runs out rather than dropping discontinuously.
• Substrate consumption: q_S = μ/Y_X/S + m_S·f_avail + q_Gly/Y_S/Gly; maintenance scales with substrate availability.
• Cell death (v19.90): k_d,eff = k_d·(1 + k_d,E·E + k_d,T·[(T−T_opt)²_heat + f_cold·(T−T_opt)²_cold] + k_d,pH·(pH−pH_opt)²). dX/dt = (μ − k_d,eff) · X. Heat-stress (T > T_opt) drives most thermal death; cold-stress contributes a tunable fraction f_cold (default 0.3) of the same quadratic to reflect membrane/osmotic damage at low T. The pH term was added to capture real yeast mortality at extremes of acidity or alkalinity (default k_d,pH=0.05 per pH-unit²).
• Enzyme hydrolysis (two-step, Starch → Dextrins → Glucose; Pham, Sundstrom & Wright 2008): Real enzyme blends produce a sugar ladder, not pure glucose, so the suite tracks dextrins (DP 3–20) as a transient pool between starch (DP > 20) and free glucose. Three parallel reaction rates, each on the anhydroglucose mass basis (g/L/h):
  • v_liq = k_liq·(AA·1000)·f_env,E·Starch/(K_m,St+Starch) — α-amylase liquefaction, fast endo-attack on starch → dextrins. No mass gain.
  • v_sac = k_sac·(GA·1000)·f_env,E·Dx/(K_m,Dx+Dx) — glucoamylase saccharification, slow exo-attack on dextrin chain ends → glucose. The rate-limiter in SSF.
  • v_GA·St = k_GA·St·(GA·1000)·f_env,E·Starch/(K_m,St+Starch) — glucoamylase acting directly on starch (slow side-reaction; k_GA·St ≈ 0.1·k_sac).
  Glucose production rate r_hyd = (v_sac + v_GA·St)·Y_G/St, with Y_G/St = 1.11 g glucose / g anhydroglucose to account for the +H₂O added at hydrolysis (180/162). v_liq carries no Y_G/St factor because dextrins are still on the anhydroglucose basis — the water is added only at the saccharification step. Shared enzyme environment factor f_env,E = f_act·f_T,E·f_pH,E, with Gaussian f_T,E centred on T_opt,E=60°C and f_pH,E on pH_opt,E=4.5 — at typical fermentation T=32°C the enzymes operate at ~2% of their nominal rate constants, which is the kinetic price of SSF. f_act is the remaining active enzyme fraction (decays via first-order inactivation, see next bullet). The dextrin pool builds up early as v_liq > v_sac, then drains as GA catches up; residual dextrins at end-of-run are the brewer's attenuation limit.
• Mash pre-hydrolysis correction: When the Medium Calculator pushes a starch loading to the simulator via importFromMedium(), 0.9% of the starch (anhydroglucose mass) is moved into the initial free-glucose pool to model the small amount of saccharification already accomplished during mashing / liquefaction before fermentation starts. The two values together always equal the user-entered total — the correction avoids double-counting that fraction as both polymer and free sugar at t=0.
• Yeast on dextrins (MAL pathway, glucose-repressed; Stewart 2017, Gancedo 1998): S. cerevisiae can consume short dextrins (maltose, maltotriose) via the MAL permease + maltase system, but the MAL operon is under strong glucose catabolite repression. The model gates the dextrin uptake rate by a repression factor that turns the pathway off when free glucose is abundant:
  q_Dx,yeast = q_Dx,max·(Dx/(K_Dx,yeast+Dx))·(K_glu,rep/(K_glu,rep+S))·f_E,p
  With K_glu,rep = 2 g/L (default), uptake is throttled to ~10% of maximum while S > 20 g/L and only switches on as glucose depletes below ~5 g/L. The f_E,p gate adds an ethanol-tolerance constraint — dying cells stop tapping dextrins regardless. Effective glucose-equivalent flux into the cell is q_Dx,yeast·Y_glu/Dx with Y_glu/Dx ≈ 1.05 (internal hydrolysis adds water, but less than full external hydrolysis since maltase processes only the terminal bond). This is folded into q_S upstream so the free-glucose balance dS/dt = r_hyd − q_S·X stays exact.
• Enzyme thermal inactivation: df_act/dt = −k_inact · Q10^{(T−T_ref)/10} · f_act. Activity decays faster at higher temperatures (Q10 = 2).
• Instant rates (cells): q_S = μ/Y_X/S + m_S·f_avail + q_Gly/Y_S/Gly, q_P = Y_P/S·(q_S − q_Gly/Y_S/Gly − (C_X/C_glu)·μ) + α·μ + β (mechanistic carbon balance — the glycerol draw and the carbon fixed into new biomass are removed before the Gay-Lussac split, so Y_P/S = 0.51142 is stoichiometric and the apparent yield, typically 88–90 % of theoretical, emerges from the carbon distribution rather than being imposed as a coefficient); r_P=q_P·X; q_Gly = [α_Gly·μ·(1 + k_VHG,Gly·f_osm) + q_G0 + q_g,osm + q_g,eth + q_g,T + q_g,pH + q_g,N]·f_S (seven additive components, substrate-gated; the earlier single-multiplier form is superseded).
• ODEs: dStarch/dt = −(v_liq + v_GA·St); dDx/dt = v_liq − v_sac − q_Dx,yeast·X; dS/dt = r_hyd − q_S·X; dX/dt=(μ − k_d,eff)·X; dEtOH/dt=q_P·X; dGly/dt=q_Gly·X; dCO₂/dt = q_CO₂·X; df_act/dt = −k_inact,eff·f_act; dLA/dt = Y_LA/S·q_S·X; dAA/dt = Y_AA/S·q_S·X; dZn/dt = (feed/dilution only); dP/dt, dCu/dt, dMn/dt, dFe/dt, dMo/dt, dCo/dt = consumption (−q_n·X) + feed/dilution; dBiotin/dt, dThi/dt, dRib/dt, dNia/dt, dPan/dt, dB6/dt, dFol/dt, dIno/dt = consumption + feed/dilution. 32 state variables, integrated with RK4 (full vector: st, s, x, e, gly, co2_aq, f_act, N, Mg, Erg_q, T, dx, V, la, aa, Zn, P, Cu, Mn, Fe, Mo, Co, Biotin, Thi, Rib, Nia, Pan, B6, Fol, Ino, Base_lys, y_{CO₂,headspace}). q_S here is the glucose-equivalent uptake rate — it absorbs the q_Dx,yeast·Y_glu/Dx flux from the MAL pathway, so dS only debits the residual draw on the free-glucose pool. Zn became dynamic in v20.07; 14 additional state variables (P + 5 trace metals + 8 vitamins) were added in v20.09 to support per-stream fed-batch composition. Consumption rates use q_n = μ/Y_X/n with published yield coefficients.
• Aqueous chemistry and pH: pH is solved at every step by bisection on the proton charge balance rather than carried as a fixed input. Five sources contribute, and the acid strengths themselves move as ethanol accumulates.
  • Baseline. The pH you enter is the pH of the finished medium, so the buffer and whatever lactic/acetic it already contains are referenced to that set point: a run set to 4.8 starts at exactly 4.8. Only acid produced during the run, H⁺ from NH₄⁺ assimilation, CO₂ taken up, and base from lysis perturb it.
  • Dissolved CO₂. CO₂(aq) is a state variable: dCO_2,aq/dt = q_CO₂·X/44.01 − k_La,eff·(CO_2,aq − k_H(T)·P_CO₂), the transfer signed so the liquid can absorb from a CO₂-rich headspace as well as strip into a lean one.
  • Headspace as a real reservoir. The headspace CO₂ mole fraction is itself a state: dy/dt = (J_strip/n_hs)(1 − y) − k_hs,exch(y − y_air), with n_hs = P_tot(V_gas/V_L)/RT. A fermenter purges its own headspace to essentially pure CO₂ while gassing (broth saturates near 1.24 g/L at 1 atm, more under hydrostatic head), and the blanket clears only by exchange with air afterwards — so dissolved CO₂ decays over days, not minutes. Interfacial area collapses when bubbling stops, so k_La,eff falls from its bubbled value to a surface-only one.
  • Late-fermentation pH recovery. CO₂ cannot drive it: at pH 4.1 only ~0.6 % of dissolved CO₂ is bicarbonate (pK_a1 = 6.35), so removing all of it is worth ~0.007 pH units. The recovery instead comes from base liberated as cells die and lyse — intracellular K⁺, amino acids, and the collapse of the H⁺-ATPase gradient — carried as a state driven by the death flux, dBase/dt = y_base,lysis·r_death. Because the driver is death rather than sugar exhaustion, the rise appears past the biomass peak, mid-run, as observed industrially (+0.3–0.5 units).
  • Ethanol pK_a shift. Ethanol lowers the medium dielectric constant, weakening every carboxylic acid present: ΔpK_a = k_pKa,EtOH·w_EtOH (~0.0145 per wt %, so acetic moves 4.76 → ~4.93 at 12 wt %). Applied to the buffer and organic acids; strong acids are unshifted.
  • Organic acids. Lactic and acetic are produced kinetically, dLA/dt = Y_LA/S·q_S·X (defaults 0.003 and 0.002 g/g), so they accumulate through the run and stop when sugar does.
  The solved pH feeds back into growth through f_pH and into death, so the recovery measurably improves completion at high gravity.
• Extended Liebig minimum: Eight of the 14 additional state variables enter the Liebig minimum, each with a literature-grounded Monod factor:
  Phosphate (P) — f_P = P/(K_P+P), K_P = 5 mg/L (Albers et al. 1996). Yeast can take up phosphate down to very low residual — limitation kicks in only at high biomass densities or with very low charge.
  Copper (Cu) — f_Cu = Cu/(K_Cu+Cu), K_Cu = 0.02 mg/L. Cytochrome cofactor; supplementation also suppresses H₂S formation.
  Manganese (Mn) — f_Mn = Mn/(K_Mn+Mn), K_Mn = 0.1 mg/L. Glucoamylase activator; matters for SSF productivity. Default initial Mn = 2 mg/L (top of the 0.2–2 mg/L range, reflecting that Mn is a minor-role nutrient present in most process waters), giving f_Mn ≈ 0.95 when replete — Mn limits growth only if a feedstock/water assay shows a genuinely low charge.
  Biotin — f_biotin = Bio/(K_bio+Bio), K_bio = 0.001 mg/L. The canonical limiting vitamin for industrial yeast, famously absent from beet molasses (Suomalainen 1971).
  Pantothenate — f_Pan = Pan/(K_Pan+Pan), K_Pan = 0.25 mg/L. Coenzyme-A precursor; its omission gave the largest single-vitamin growth defect after biotin (57% lower μ; Perli et al. 2020).
  Vitamin B₆ (pyridoxine) — f_B6 = B6/(K_B6+B6), K_B6 = 0.05 mg/L. Amino-acid metabolism cofactor (32% μ defect on omission; Perli et al. 2020).
  Thiamine — f_Thi = Thi/(K_Thi+Thi), K_Thi = 0.1 mg/L. Pyruvate-decarboxylase cofactor — directly on the fermentation pathway (22% μ defect on omission; Perli et al. 2020).
  Inositol — f_Ino = Ino/(K_Ino+Ino), K_Ino = 1.3 mg/L. Phosphatidylinositol precursor (19% μ defect on omission; Perli et al. 2020). Inositol also couples to ethanol toxicity — see Cell death below.
  The K values are calibrated so that well-supplied synthetic-medium defaults give f ≈ 0.95 (no false limitation when replete) while dropping steeply as the vitamin is consumed. The extended Liebig is f_nutrients = min(f_N, f_Mg, f_Zn, f_Erg, f_Tw, f_P, f_Cu, f_Mn, f_biotin, f_Pan, f_B6, f_Thi, f_Ino) — 13 active factors.
  The remaining six state variables (Fe, Mo, Co, riboflavin, niacin, folate) are tracked for mass balance and feed-stream delivery but do not limit growth: S. cerevisiae is effectively prototrophic for niacin, riboflavin, and folate (no significant μ reduction on omission after adaptation; Perli et al. 2020), and Fe/Mo/Co lack published half-saturation curves under industrial ethanol conditions, so adding them to the minimum with invented K values would produce poorly-grounded predictions.
• Inositol–ethanol toxicity coupling: Beyond its growth role, cellular inositol status modulates ethanol toxicity. A low-inositol cell synthesises less phosphatidylinositol (PI), which weakens the plasma-membrane H⁺-ATPase that maintains the ion barrier; under 12–20% ethanol such cells show a markedly higher death-rate constant and leak more intracellular K⁺, phosphate, and nucleotides (Furukawa et al. 2004; Krause et al. 2007). The model scales the ethanol term of the death rate by (1 + k_ino,tol·(1 − f_Ino)): a fully inositol-replete cell (f_Ino → 1) sees no penalty, while a depleted cell (f_Ino → 0) suffers up to (1 + k_ino,tol)× the baseline ethanol death rate, with k_ino,tol = 0.6 by default. This is distinct from the growth factor f_Ino above — inositol limitation slows growth and accelerates ethanol-driven death, the two mechanisms acting on different terms of dX/dt.
• Per-stream composition (v20.09): Every fed-batch stream now has independent composition fields for all 14 new species. The 5 streams × 14 species = 70 fed-batch composition fields, plus 14 fields for the semi-batch composite stream = 84 total. Most stay at industrial-realistic defaults; users typically only edit fb_c_feed_S (carbohydrate concentration in Stream 1) and fb_trace_feed_Zn (Zn in Stream 4). The unified mass-balance ODE collects feed contributions from all streams as Σ_k F_k·C_k,feed,i / V for each species i, then subtracts the cumulative dilution C_i·Σ_kF_k/V. This generalises the single-stream feed coupling of v20.07 to arbitrary numbers of streams without changing the form.
• Feed-flow chart (v20.09): New "Feed Flow" tab in the time-course chart area shows F_k(t) for each enabled stream (5 lines in fed-batch, 1 in semi-batch). The summary panel below reports total volume delivered per stream (∫F dt, trapezoidal) and per-stream peak flow rate, with a verdict line confirming the total flow against the final V trajectory. Useful for diagnosing controller behaviour: aggressive K_p shows up as a spiky F_C(t), exponential μ-targeted shows up as a smooth rising curve, stepped trace-element feed shows up as 3 distinct pulses.
• Mass balance check: Carbon in (glucose consumed + starch consumed) vs carbon out (ethanol + CO₂ + glycerol + biomass). Warns if balance deviates >5%.
• Feedstock mass balance (Medium Calculator): Total fermentable sugar required is computed as sugar = ethG / Y_E/S, then split between two paths. Dry feedstock needed = sugar / Y_{sugar/feedstock}, where Y_{sugar/feedstock} is computed live from the detailed composition table as (starch% × 1.11 + sugars%) ÷ 100 — the constant 1.11 = 180/162 is the stoichiometric glucose:starch mass ratio. Picking a feedstock from the dropdown populates the composition with literature defaults (e.g. corn: 74% starch + 1.5% sugars → 0.84 g/g) along with typical moisture (grains 12–14%, molasses ~20%, purified starch ~11%, pure sugars 0%); any cell of the composition table can be edited to match an assayed lot. As-received (wet) mass = dry mass ÷ (1 − moisture_fraction), so at 14% moisture, 20 kg dry corn weighs 23.3 kg as received. The starch/glucose split at t=0 is derived from composition as starchFrac = (starch% × 1.11) ÷ (starch% × 1.11 + sugars%); at t=0 the medium contains (1 − starchFrac) × sugar-eq as free glucose and (starchFrac × sugar-eq) ÷ 1.11 as solid starch; amylase enzymes (α-amylase 500 U/g starch, glucoamylase 200 U/g starch) hydrolyze the starch to glucose during fermentation.
• Instant-Predictions dual-axis charts: three tab groups in the right column of the Simulator sample rates across temperature (0–50 °C) and pH (2–8) at the current operating point, each with two series on a primary/secondary y-axis. Ethanol Product: r_P primary + r_Gly secondary (both volumetric, g/L/h). Yeast Product: μ primary + k_d,eff secondary (both specific, h⁻¹); crossover point = washout T. Glycerol Product: r_Gly primary (volumetric, g/L/h) + q_Gly secondary (specific, g/g/h), useful for osmotic / redox stress diagnosis. All three tabs re-render on every parameter change via the 180 ms debounce.
• Nutrient coupling (Phase 2): μ is multiplied by a Liebig's-minimum factor f_nutrients = min(f_N, f_Mg, f_Zn, f_Erg, f_Tween, f_P, f_Cu, f_Mn, f_biotin, f_Pan, f_B6, f_Thi, f_Ino), where each f is a Monod term on the corresponding nutrient. The full 13-factor set was activated in v20.54 when the four limiter-vitamins (pantothenate, B6, thiamine, inositol per Perli 2020) and the trace metals P, Cu, Mn, biotin were promoted from passive tracking to active growth limitation; see the detailed Liebig extension paragraph below. Growth slows sharply when any one nutrient falls below its half-saturation constant. The dominant limiter is reported in the "Limiting" pill above the time-course charts.
• FAN depletion: dN/dt = −(μ/Y_X/N)·X. With Y_X/N ≈ 10 g DCW/g N, 700 mg/L FAN supports ~7 g/L biomass growth — the boundary between normal-gravity and VHG fermentation.
• Cellular ergosterol dilution: d(Erg_q)/dt = −μ · Erg_q. Pure dilution — anaerobic yeast cannot synthesize sterols. The initial quota is erg_q_init + erg_init_broth / X0, so increasing either the per-cell reserve or the medium supplement shifts stuck-fermentation onset. This is the mechanism behind VHG stuck fermentations at 40–60% sugar utilisation.
• Adiabatic heat balance (optional): When temperature mode = adiabatic, T becomes a state variable with dT/dt = (Q_metab − Q_cool)/C_p, where Q_metab = r_P·ΔH_ferm/M_EtOH and Q_cool = k_cool·(T − T_set). Drop k_cool to see runaway heating.
• pH drift from NH₄⁺ assimilation (Phase 3): Each mole of ammonium taken up releases 1 mole of H⁺ as neutral N is incorporated into biomass. Modeled as a strong acid (pK_a = −2) with cumulative concentration (N_init − N_medium) · h_{per_N} / M_N, added to the existing aqueous-chemistry buffer solver. h_{per_N} is tunable (0 for urea-only, 1 for pure NH₄⁺, 0.5 for typical FAN mixtures).
• Inoculum sizing engineering identity (Phase 3): P = q_P·X_avg·t, so X_avg = P / (q_P·t). The sizing card solves this for target ethanol titer (auto-synced from Medium Calc) and fermentation time (read-only field auto-mirrored from Duration — the single source of truth for time), then converts X_avg → X_pitch via a tuneable ratio (default 0.6 for mild growth during lag + exponential phase).
• Live coupling to Medium Calculator (v20.04–v20.07): calc() in the Medium Calc updates window._calcCanonical with the unit-resolved totals (sugarG, ethVolL, volumeL, biomassG) on every input change, then calls importFromMedium() which is bio-process mode-aware. In Batch mode it pushes medium-target concentrations to the simulator's t=0 fields (s, st, n_init, mg_init, zn_init, erg_init_broth, tween80_init). In Semi-Batch mode it ALSO pushes sb_v_target, sb_v_heel (= batchFrac · V_target), and fill-mash composition (sb_feed_S, sb_feed_St, sb_feed_N, sb_feed_Mg). In Fed-Batch mode (v20.07) it pushes fb_v_max, fb_v_init (= batchFrac · V_max), and stream-specific composition fields: fb_c_feed_S and fb_c_feed_St (Stream 1, carbohydrate), fb_n_feed_N (Stream 2, nitrogen), and fb_trace_feed_Zn (Stream 4, trace — pre-populated even when Stream 4 is Off, so enabling the toggle picks up the target). Control strategies, timing windows, and F_max are NOT auto-populated — those are process choices, not concentration targets, and the user configures them per stream. All modes call syncTiterToInoculum() for the Inoculum Sizing card. A recursion guard prevents re-entry. The green "← Import from Medium Calculator" button remains as a manual refresh escape hatch.
• Bio-Process operating modes (v20.03, extended v20.07): The simulator supports three modes through a dropdown in Strain & operating conditions, bidirectionally synced with a matching dropdown in the Medium Calc. Batch (default): all ingredients charged at t=0, no flow during fermentation — original behaviour preserved exactly. Semi-Batch: initial volume (V_init at t=0) plus a single composite fill mash F(t) added during the fill window [t_fill,start, t_fill,start+t_fill]. Three fill profiles (linear / exponential / stepped) and three inoculum-timing modes (pre_fill / in_feed / post_fill). Fed-Batch (v20.07): starts batch at V_init, then up to FIVE independent feed streams flow during fermentation, each with its own control strategy, timing, F_max, and composition:
  Stream 1 — Carbohydrate (C): always visible. Strategies: setpoint (P-controller on residual glucose, the industrial standard for fuel ethanol), exponential μ-targeted (F maintains μ_target by feeding substrate at the cellular consumption rate), constant flow, or off.
  Stream 2 — Nitrogen (N): always visible. Strategies: setpoint (P-controller on residual FAN), constant flow, one-shot pulse (smeared over ~3 min so RK4 catches it), or off.
  Stream 3 — Phosphorus (P): always visible, default Off. Strategies: constant flow, one-shot pulse, or off. Currently contributes only to volume balance (phosphate is not yet a state variable in the kinetic model — Stream 3 dilutes existing medium species but adds no kinetic effect of its own).
  Stream 4 — Trace elements (Tr): toggleable (default off). Carries dynamic Zn (the 16th state variable). Strategies: constant flow, stepped pulses (3 doses equally spaced over a duration window), or one-shot pulse. Useful for late-fermentation Zn supplementation to delay senescence in VHG ethanol.
  Stream 5 — Vitamins (V): toggleable (default off). Strategies: constant flow or one-shot pulse. Currently informational — the simulator does not track individual vitamins as state variables, so Stream 5 contributes only to volume balance (and adiabatic temperature mixing).
  Shared parameters V_init, V_max, t_feed,end apply globally: all streams stop when V ≥ V_max OR t ≥ t_feed,end, whichever fires first. Per-stream F_max hard-caps the flow rate of each individual stream.
• Feed coupling in the ODE (v20.03–v20.07): Semi-Batch and Fed-Batch share a unified multi-stream dilution math:
  dV/dt += Σ_k F_k (total volume gain)
  dC_i/dt += (1/V) · (Σ_k F_k · C_k,feed,i − C_i · Σ_k F_k) (per-species feed contribution minus dilution)
  dT/dt += (1/V) · Σ_k F_k·(T_k−T) (adiabatic mixed-stream heat balance)
  The sum Σ_k runs over 1 stream in Semi-Batch and up to 5 streams in Fed-Batch. Per-stream feed composition is tagged exactly to that stream — Stream 1 carries glucose+starch only (other species at 0 in C-stream), Stream 2 carries FAN only, Stream 4 carries Zn only. Most off-diagonal C_k,feed,i terms are zero by construction, so the algebra collapses to a small number of non-zero contributions per species. Species with non-zero feed composition: glucose (Stream 1), starch (Stream 1), FAN (Stream 2), Mg (Semi-Batch only — Mg is batch-only in Fed-Batch per the industrial convention), Zn (Stream 4). Pure-dilution species (feed=0 in all streams): ethanol, dextrins, glycerol, CO₂(aq), lactic acid, acetic acid, ergosterol, Tween-80. Intensive variables (Erg_q cellular quota, f_act normalized enzyme activity, T outside adiabatic) are unaffected by dilution. When neither bio-process mode is active, all F_k = 0 and the whole block contributes zero — batch behaviour preserved exactly.
• Regime multipliers are applied relative to the strain-preset base values and are indicative only; they do not compound on repeated changes. Calibrate with your own strain data.
• Environmental viability gate (v19.90): both the maintenance term and the non-growth-associated Luedeking–Piret production are now multiplied by f_env = f_T·f_pH. At cardinal-model extremes where f_T=0 or f_pH=0, m_S·f_S·f_E,p·f_env = 0 and β·f_E,p·f_env = 0, i.e. dormant cells don't consume sugar for maintenance or produce ethanol non-growth-associatedly. Previously a ~0.5 g/L/h floor persisted at T=0 or pH=2 (physically impossible). The fix makes the r_P vs T / pH charts drop cleanly to zero outside [T_min, T_max] and [pH_min, pH_max].
• VHG glycerol coupling: the growth-coupled glycerol term is now q_g,growth = α_Gly·μ·(1 + k_VHG,Gly·f_osm), coupling α_Gly to the same osmotic Hill that drives q_g,osm. This reflects biochemistry: under high total osmolyte, NADH disposal during biosynthesis becomes more glycerol-intensive because the cell is already stressed. At k_VHG,Gly=1 (default), full VHG (f_osm→1) doubles α_Gly. Setting k_VHG,Gly=0 recovers the legacy behaviour. q_gly is now decomposed into q_gly,stress (osm+eth+T+pH+N, U-shape in T/pH) and q_{gly,growthlike} (α·μ·(1+k_VHG·f_osm) + q_G0, Gaussian) so the two biochemically distinct mechanisms display as separate curves on the Glycerol Product instant chart.
• Instant-prediction chart semantics (v19.90): the three prediction tabs were rebuilt to show distinct information instead of three copies of the μ-Gaussian. Ethanol Product now shows r_P primary + Y_P/S,apparent = q_P/q_S secondary (the effective yield after maintenance and glycerol diversions). Yeast Product now shows μ primary, μ−k_d,eff (net growth, dashed purple), and k_d,eff (dotted red, right axis) — the zero-crossings of the net curve mark the WASHOUT boundaries. Glycerol Product plots the two decomposition buckets described above.
• Organic acid production & dynamic pH coupling (v19.95): Lactic acid (LA) and acetic acid (AA) are now dynamic ODE state variables: dLA/dt = Y_LA/S·q_S·X, dAA/dt = Y_AA/S·q_S·X, with production yields defaulting to 0 (no production unless user enables or optimizer fits). Both accumulate during fermentation and feed back into the pH calculation via their pK_a equilibria (lactic pK_a = 3.86, acetic pK_a = 4.76) through the buildDynamicAcids() function, which replaces the static initial organic acid concentrations in the pH solver at each RK4 timestep. The full pH model now includes four acid sources: CO₂ dissolution (H₂CO₃), NH₄⁺ assimilation H⁺ release, dynamic lactic acid, and dynamic acetic acid — all solved simultaneously in the bisection-based proton balance. Environment chart shows both species on a dedicated "Org. acids (g/L)" axis.
• Model Exp. Data tab (v19.95): Fourth main tab for fitting the model to observed fermentation data. Accepts all 12 standard ethanol fermentation HPLC columns: time, pH, temperature, DCW, DP4+, DP3, DP2, DP1, lactic acid, glycerol, acetic acid, ethanol. Column names are case-insensitive with multiple aliases supported (e.g. ethanol, EtOH, E all match). DP2 + DP3 are automatically summed to a dextrin pool for scoring against the model's dextrin trajectory. Replicate samples at the same time point are automatically detected and averaged to mean ± SD.
• Model Exp. Data — four actions (v19.95):
  • Statistical analysis — evaluates raw data quality: dataset overview (time points, duration, sampling density), per-species quality table (n valid, missing, min/max/range, CV% from replicates, trend detection ↑↓→↕, quality rating ✅⚠️❌), parameter estimability checklist (which parameters the data can support estimating), and data limitation warnings (too few points, missing species, no replicates, large gaps, short duration).
  • Evaluate fit — runs the simulator with current parameters, interpolates predictions at observation times, and reports R², RMSE, and mean bias per species. Overlays observed data as scatter points on the Metabolism chart.
  • Calculate parameters — derives kinetic parameters analytically from the time-course data: μ_max (corrected: slope of ln(X) + k_d,eff ÷ f_product at operating conditions), Y_X/S (apparent ΔX/ΔS), Y_P/S (apparent ΔE/ΔS), α/β (Luedeking–Piret regression of q_P vs μ), k_d (apparent k_d,eff from decline phase), m_S (apparent stationary-phase q_S), plus glycerol and organic acid yields. All parameters labeled "apparent" with explanations of entanglement with the full ODE model.
  • Train (optimise) — Nelder–Mead simplex on user-selected parameters, normalised [0,1] within bounds, async with Stop button. 14 fittable parameters across 3 groups: growth kinetics (μ_max, K_S, m_S, k_d, k_d,E), yields (Y_X/S, Y_P/S, α_Gly, Y_LA/S, Y_AA/S), ethanol production (α, β, E_max, E_inh).
  A "Next steps" panel appears after each action with contextual guidance and re-run buttons for iterative refinement.
• Medium volume tracking from CO₂ outgassing: ethanol fermentation loses mass as C₆H₁₂O₆ → 2 C₂H₅OH + 2 CO₂ drives 0.4886 g of CO₂ out of each gram of glucose consumed. On typical VHG mashes, 5–10% of medium volume exits as gas. Three modes are available (Fermentation Conditions → Strain & operating conditions → Volume tracking): Constant V (default, no tracking); Post-proc — ODE runs at constant V, then V_final is computed from cumulative CO₂ mass balance and all in-solution species are rescaled by V₀/V_final for display; In-ODE rigorous — V becomes a state variable in the 30-element vector (st, s, x, e, gly, co2_aq, f_act, N, Mg, Erg_q, T, dx, V, la, aa, Zn, P, Cu, Mn, Fe, Mo, Co, Biotin, Thi, Rib, Nia, Pan, B6, Fol, Ino — the 15 macro states plus the 15 nutrients made dynamic in v20.07–v20.09) with dV/dt = −r_CO₂·V/(ρ·1000); the companion dilution term +C·r_CO₂/(ρ·1000) is applied to every per-medium-volume species (starch, glucose, biomass, ethanol, glycerol, dissolved CO₂, FAN, Mg, dextrins, lactic acid, acetic acid), derived from the product-rule expansion d(C·V)/dt = r·V. Erg_q (per-biomass mg/g DCW), f_act (dimensionless), and T (intensive) do not receive the dilution term. In-ODE mode captures the nonlinear coupling where ethanol inhibition feels the rising concentration during the run — slightly lowering total ethanol mass but raising end-of-run g/L.
• DCW → cell count conversion: cells/mL = X (g DCW/L) × f_cells × 10⁷, where f_cells is derived from the ADY product spec on the Pitching & Inoculum tab and shown read-only in the Inoculum Sizing card (field: "Cells per g DCW", units ×10¹⁰). Derived 2.13 corresponds to ~47 pg single-cell dry weight (industrial S. cerevisiae exponential phase). Published values span 3–10 (60–200 pg/cell equivalent) depending on strain, growth phase (stationary cells carry glycogen/trehalose → heavier), ethanol stress (cells shrink in VHG runs), and measurement method. The factor is used by the Inoculum Sizing cell-density readout, the Yeast chart's cells-per-mL curve, and the Yeast summary panel's Pitch/Peak/End cell counts.

Strain presets are indicative; calibrate with your strain data. Regime modifies yields and glycerol base to mimic redox shifts.

References

Key primary sources for the kinetic, stoichiometric, and engineering choices encoded in the suite. All citations are illustrative — the implementation often blends or adapts published forms; see the bullets above for exact functional forms.

• Luedeking, R. & Piret, E. L. (1959). A kinetic study of the lactic acid fermentation. J. Biochem. Microbiol. Technol. Eng. 1: 393–412. — Original growth-associated + non-growth product term q_P = α·μ + β.
• Luong, J. H. T. (1985). Kinetics of ethanol inhibition in alcohol fermentation. Biotechnol. Bioeng. 27: 280–285. — f_E with E_inh threshold and E_max; default ethanol-inhibition model in this suite.
• Aiba, S., Shoda, M. & Nagatani, M. (1968). Kinetics of product inhibition in alcohol fermentation. Biotechnol. Bioeng. 10: 845–864. — Exponential f_E = exp(−k_E·E); legacy alternative.
• Andrews, J. F. (1968). A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10: 707–723. — Haldane substrate-inhibition term S/(K_s+S+S²/K_i,S).
• Bai, F. W., Anderson, W. A. & Moo-Young, M. (2008). Ethanol fermentation technologies from sugar and starch feedstocks. Biotechnol. Adv. 26: 89–105. — VHG fermentation, industrial yield benchmarks (Y_E/S=0.45–0.49), feedstock yield coefficients.
• Walker, G. M. (2011). Pichia and Saccharomyces yeast biology. The Yeasts: A Taxonomic Study, 5th ed. — Magnesium / zinc / vitamin requirements, sterol & unsaturated-fatty-acid auxotrophy under anaerobiosis.
• Casey, G. P. & Ingledew, W. M. (1986). Ethanol tolerance in yeasts. CRC Crit. Rev. Microbiol. 13: 219–280. — Membrane / Mg²⁺ / lipid mechanisms behind ethanol toxicity; basis for VHG sterol & oleate supplementation.
• Pham, T. K. & Wright, P. C. (2008). The proteomic response of Saccharomyces cerevisiae in very high glucose conditions. J. Proteome Res. 7: 4766–4774. — Osmotic-stress glycerol overproduction; basis for the Hill term q_g,osm.
• Atala, D. I. P., Costa, A. C., Maciel, R. & Maciel Filho, R. (2001). Kinetics of ethanol fermentation with high biomass concentration. Appl. Biochem. Biotechnol. 91–93: 353–365. — Cell-death rate dependence on ethanol & temperature.
• Verduyn, C. et al. (1990). Physiology of Saccharomyces cerevisiae in anaerobic glucose-limited chemostat cultures. J. Gen. Microbiol. 136: 395–403. — Anaerobic Y_X/S ≈ 0.10 g/g, ergosterol & UFA quotas, maintenance coefficient m_S.
• Cot, M., Loret, M.-O., François, J. & Benbadis, L. (2007). Physiological behaviour of Saccharomyces cerevisiae in aerated fed-batch fermentation for high-level production of bioethanol. FEMS Yeast Res. 7: 22–32. — Industrial fed-batch & ergosterol dilution behaviour.
• Stewart, G. G. (2017). Brewing and Distilling Yeasts. Springer. — α-Amylase / glucoamylase loading conventions (≈500 U/g starch / 200 U/g starch), nitrogen targets (200–700 mg FAN/L for normal-gravity to VHG).
• Pham, H. T. B., Sundstrom, E. R. & Wright, A. R. (2008). Kinetic modeling of ethanol fermentation from wheat flour under simultaneous saccharification and fermentation. Biotechnol. Prog. 24: 118–126. — Two-step starch hydrolysis kinetics (Starch → Dextrins → Glucose) with separate K_m for each substrate; basis for the v_liq, v_sac, v_GA·St rate decomposition.
• Gancedo, J. M. (1998). Yeast carbon catabolite repression. Microbiol. Mol. Biol. Rev. 62: 334–361. — Mechanism of glucose repression of MAL gene expression; basis for the K_glu,rep/(K_glu,rep+S) gate on q_Dx,yeast in the dextrin uptake model.
• Pirt, S. J. (1965). The maintenance energy of bacteria in growing cultures. Proc. R. Soc. B 163: 224–231. — Maintenance-energy framework underpinning the m_S·X term.
• Kosaric, N. & Vardar-Sukan, F. (2001). Potential source of energy and chemical products. In The Biotechnology of Ethanol, Wiley-VCH. — Feedstock-yield reference values for corn / wheat / cassava / molasses.

Model Notes — BibliographyRefs

Master bibliography for the full suite — an organised superset of the topic-specific reference lists on the Simulator, Medium Calculator, and Model Exp. Data tabs. Grouped following Part VIII of the v20.85 technical report: Primary literature, Background reading, Ergosterol dilution mechanism, and Kinetic modelling choices.

What changed in v21.21. The suite now has six top tabs: Yield Calculator, Medium Calculator, Pitching & Inoculum (new), Ethanol Simulator, Model Data and Model Notes.

• The calculators are chained. Yield → Medium → Pitching → Simulator. Each receiving tab carries an import button that pulls from the one before it: Import ethanol target in the Medium Calculator, Import fermenter from Medium Calculator on the Pitching tab, and Import from Pitching Calculator in the simulator's Inoculum Sizing card.
• Calculation basis. The Medium Calculator can be driven three ways: ethanol output + titer → volume (the original), fermenter volume + titer → ethanol, or fermenter volume + mash sugar → titer. An optional fill fraction converts a nameplate vessel volume to working volume.
• One theoretical constant. Gay-Lussac is 0.51142 g/g in all three tabs. Y_E/S is no longer typed in: it is derived as theoretical × fermentation efficiency (default 90% → 0.4603). It sizes the sugar charge, so it must be the apparent yield achieved, never the ceiling.
• One biomass conversion. Cells per g DCW is derived from the ADY product spec (20 ×10⁹ viable cells/g ÷ 0.94 dry matter = 2.13 ×10¹⁰ cells/g, ≈ 47 pg/cell) rather than entered separately in each tool.
• Metric by default throughout, and the file is fully offline — Chart.js and the fonts are embedded, so no internet connection is required.
• Semi-batch selects a 10% heel automatically instead of inheriting the batch 100%, and the heel fraction can now go below 30%.

Medium Calculator — Quick Start

The calculator is organized as a two-column layout with five top-level collapsible cards: Target Production & Medium Target Concentrations on the left (inputs); Summary, Carbon Mass Balance, and Results & Tabs on the right (outputs). Each card with sub-categories opens further into Macronutrients / Trace Metals / Vitamins / Anaerobic Lipid Factors sub-collapsibles, all collapsed by default to keep the view tidy.

1. Open Target Production → Ethanol Fermentation Targets. Enter Target Ethanol Production (default 10 L; switchable to US gallons) and Target Ethanol Titer (default 17% v/v — VHG territory; also % w/v or g/L). The calculator converts internally before running the mass balance.
2. Open Target Production → Feedstock & Substrate. Pick a Feedstock from the 10-option dropdown (default: Corn — whole kernel). Selecting one auto-populates Moisture % and two collapsible tables (▶ both collapsed by default): Detailed composition (7 components on a dry basis — starch, free sugars, protein, oil, fiber, ash, other) and Elemental composition (21 elements grouped into Macro / Trace / Vitamins, each with Total ppm, an Availability factor, and a computed Available = Total × Avail). The fermentable-sugar yield (g sugar / g dry feedstock) is computed live from composition as (starch% × 1.11 + sugars%) ÷ 100 and shown next to the dropdown; both yield and the t=0 starch/glucose split recompute when you edit any composition row. Optional: enable "Credit nutrients provided by feedstock" to subtract Available element content from the salt recipe.
3. Open Target Production → Bio-Process & Fermentation Technology. Two selectors stacked here. Bio-Process Technology picks the front-end mash-prep pathway (Dry Grind — mill → cook → SSF, the predominant US fuel-ethanol process; or Wet Grind — steep → fractionate → fermentation, used by integrated corn refineries producing oil / gluten / fiber co-products). Fermentation Technology selects the run mode: Batch (all ingredients at t=0), Semi-Batch (initial volume + fill mash F(t)), or Fed-Batch (up to 5 independent feed streams). For Semi-Batch and Fed-Batch, additional parameter cards appear below.
4. Set the Stoichiometric Yields and Batch % in the Summary card. Three sliders sit below the Summary table: Y_E/S (0.38–0.51142, default 0.4603 — no longer entered directly, but derived as theoretical × fermentation efficiency), Y_X/S (0.02–0.10, default 0.05), Y_Gly/S (0.01–0.08, default 0.04). The % Nutrients in Initial Batch / Initial Volume slider lives in the Bio-Process & Fermentation Technology card (30–100, default 100 = full batch).
5. Click "Load Standard" in the Medium Target Concentrations card to populate all 23 nutrient rows (7 macronutrients, 6 trace metals, 8 vitamins, 2 lipid factors) with fuel-ethanol defaults. Expand any of the four nutrient sub-cards (▶ Macronutrients / Trace Metals / Vitamins / Anaerobic Lipid Factors) to edit individual targets.
6. Read the Summary card — three snapshot-style rows of computed values:
   • Production overview: Total Sugar Required (kg / lbs / ton), Ethanol Produced (L abs. / gal / kg / lbs), Medium Volume (L / gal / hL) with biomass-produced as descriptor. Switch units via the compact Units strip at the bottom.
   • Substrate split (visible when feedstock contains starch): Starch Needed (starchFrac × sugar ÷ 1.11), Direct Glucose ((1 − starchFrac) × sugar), Feedstock as-received (dry mass ÷ (1 − moisture)). starchFrac is derived from composition.
   • Enzyme loading (same trigger): α-Amylase @ 500 U/g starch, Glucoamylase @ 200 U/g starch, plus combined total. Auto-formatted as U / kU / MU based on magnitude.
   For glucose-only feedstocks (cane / beet molasses, pure sugars), the substrate + enzyme rows hide and a small italic notice replaces them.
7. Check the Carbon Mass Balance card — the stacked horizontal bar should show 95–100% closure across Ethanol / CO₂ / Glycerol / Biomass / Other. Below 95% means missing products; above 100% means yield coefficients exceed stoichiometry and need reducing.
8. Open Results & Tabs. Five tabs across the top:
   • Total Required — every nutrient grouped into the same 4 sub-cards (Macro / Trace / Vitamin / Lipid), each row showing the chosen salt, mass to weigh per batch, and a ✓ "covered" indicator when an element is auto-supplied as a co-element from another salt (e.g., S from MgSO₄).
   • Batch Medium & VHG Feed — splits the Total Required between the initial batch and the fed-batch supplement based on the Batch % slider.
   • Reference — typical concentrations and molar masses; also includes an Ethanol Fermentation — Process Notes collapsible with 12 topical guidance items (yield expectations, FAN targets, Mg / Zn / vitamin roles, VHG operation, pH & temperature ranges, sterility tips).
   • Test Medium — enter actual salt masses you weighed out and see coverage % per nutrient (green ≥100%, amber 50–99%, red <50%) plus the most-limiting row identified in the verdict bar.
9. Customize individual nutrients by editing target concentrations in Medium Target Concentrations or picking different salts from the dropdowns in Total Required. Recipe updates live across all tabs.
10. Switch to the Time-Course Simulator tab. Initial glucose, starch, FAN, Mg, Zn, ergosterol, oleate, medium volume, and target titer are already live-synced from the Medium Calc — no Import click needed for routine edits. Click Run Simulation or rely on the 180 ms auto-debounce.

Live sync: Every input change in the Medium Calculator (target ethanol, titer, feedstock selection, detailed-composition and elemental-composition table edits, moisture, Y_E/S / Y_X/S / Y_Gly/S sliders, batch %, and every medium-target concentration) propagates automatically to the Simulator's Substrate Pools, Init Conditions, Nutrient Coupling, and Inoculum Sizing cards. The "← Import from Medium Calculator" button in the Simulator is still there as a manual refresh if you ever need to force a re-sync. All headers (outer h2-style and inner sub-collapsibles) are click-to-toggle — useful once you've configured a section.

Medium Calculator — ReferencesRefs

Primary sources for the nutrient targets, FAN tiers, element stoichiometry, feedstock yields, and complex-material compositions used by the medium calculator. Where the calculator's defaults represent a range in the literature, the citation lists both endpoints and the implemented midpoint.