Continuous input variables and uncertainty of FT

For reactions a to h and k, concentrations were 1 mM hexose, 60 mM acetate, 20 mM propionate, 12.5 mM butyrate and 40 mM bicarbonate, 0.25 bar partial pressure of CH₄ and pH was equal to 6.45; for reaction j, Fd_RED^2-/Fd_OX was equal to 9. Values for ΔG^o of fermentation pathways and standard redox potentials of cofactors were taken from refs. [3, 10]. Values of ΔG^o of metabolite formation were adjusted to rumen temperature using the Van’t Hoff equation (e.g., [26]).

The uncertainty of F_T to variation in inputs other than P_H2 was assessed for the five glucose fermentation pathways, the three VFA interconversions and methanogenesis (Table 1). Ten thousand different samples were drawn randomly from uniform distributions for glucose, acetate, propionate, butyrate concentrations, pH, P_CO2, P_CH4 and ΔG_P ranging from 0.1 to 2.0 mM, 35 to 90 mM, 7 to 30 mM, 5 to 21 mM, 5.7 to 7.2, 0.35 to 0.80 bar, 0.15 to 0.35 bar and 35 to 50 kJ⋅mol^-1, respectively. For completeness, proton concentrations were calculated from pH and HCO3− concentrations were calculated using the Henderson-Hasselbalch equation (e.g., [27]).

Uncertainty of F_T approaches zero when F_T approaches unity. If F_T of a specific reaction deviated from unity for the range of P_H2 considered, a 95% confidence interval of F_T was calculated for 10 values of P_H2 for which F_T was close to zero at the previously mentioned fixed concentrations. Values of P_H2 increased exponentially in steps according to P_{H2, n} = a⋅bⁿ⁻¹, where a is the start value, b is the factor by which P_{H2, n} increases per step, and n runs from 1 to 10 for the number of steps. The exact values of a and b were chosen based on the visual representation of the uncertainty by the error bar. Applying this, the uncertainty of F_T was assessed for P_H2 at {2.00⋅10⁻⁵, 2.60⋅10⁻⁵, …, 2.12⋅10⁻⁴} bar for methanogenesis yielding 0.5 ATP, {6.00⋅10⁻⁴, 8.10⋅10⁻⁴, …, 8.94⋅10⁻³} bar for methanogenesis yielding 1.5 ATP, {7.70⋅10⁻⁵, 9.63⋅10⁻⁵, …, 5.74⋅10⁻⁴} bar for acetate to propionate conversion, {5.00⋅10⁻⁶, 6.50⋅10⁻⁶, …, 5.30⋅10⁻⁵} bar for propionate to acetate conversion, and {1.95⋅10⁻⁶, 2.93⋅10⁻⁶, …, 7.50⋅10⁻⁵} bar for butyrate to acetate conversion. The actual ranges of the 95% confidence intervals of F_T depends on metabolite concentrations and values of ΔG^o, ΔG_C and χ, explicitly shown for the particular conversion of glucose into two equivalents of acetate (Eq 6). Eq 6 also shows the nonlinearity of F_T to its input, which makes the 95% confidence intervals asymmetric.

Calculation of the 95% confidence intervals of F_T at discrete values of P_H2 and plotting of F_T as a function of P_H2 was performed in R statistical software. Code is provided as supporting information (S1–S3 Files).