Figure description: This figure shows the optimal flux distribution under autotrophic conditions (CO₂ + O₂).

Analysis: Under autotrophic growth with CO₂ as the carbon source and oxygen present, the model’s major metabolic fluxes are highlighted. The results indicate that the ATP maintenance reaction (R_ATPM) carries a significant flux, reflecting the essential energy demand required to sustain basic cellular metabolism. At the same time, the hydrogenase reaction (R_MBH_HYDROGENASE) and key reactions in the Calvin cycle (such as phosphoribulokinase) display non-zero fluxes, demonstrating that hydrogen oxidation and the Calvin cycle are the primary driving forces under autotrophic conditions.

Figure description: This figure shows the flux variability range of each reaction at the maximum growth rate.

Analysis: The urease reaction (R_UREASE_RXN) exhibits a wide flux range, indicating that under autotrophic conditions, nitrogen metabolism (urea degradation) is the pathway with regulatory flexibility. ATP maintenance (R_ATPM) appears as a rigid demand (with no variability), meaning that the cell’s energy maintenance cost must remain constant. The hydrogenase reaction (R_MBH_HYDROGENASE) shows no significant range, suggesting that under CO₂ + H₂ conditions, the flux of hydrogen oxidation is a fixed requirement.

Growth rate plot (left): As both carbon and nitrogen sources increase, the growth rate gradually rises. The contour lines appear close to a diagonal pattern, indicating that carbon and nitrogen contribute similarly to driving growth, and the lack of either will restrict it.

Lactoprotein flux plot (right): The overall trend is consistent with the growth rate: increases in both carbon and nitrogen lead to higher lactoprotein flux. Although the absolute values are much lower than those of the growth rate, the dependency shows a dual limitation by both sources rather than a single carbon limitation. In the low-carbon/low-nitrogen region, both act as bottlenecks; only in the high-carbon/high-nitrogen region can the yield be gradually released.

Under heterotrophic conditions, since lactoprotein synthesis has been coupled with biomass growth, the pFBA results show that the lactoprotein flux is no longer zero but instead appears alongside biomass and increases synchronously with the growth rate. This indicates that under this constraint, product formation has become an inevitable outcome of cellular metabolism rather than merely a byproduct.

At the same time, the ATP maintenance reaction remains at a high flux level, demonstrating that the energy maintenance requirement is still rigid under heterotrophic conditions. The urease reaction (R_UREASE_RXN) exhibits a significant negative flux, showing that nitrogen degradation continues to play a key role in fructose heterotrophic metabolism by providing nitrogen to support both growth and product synthesis.

Under heterotrophic conditions, the FVA results show that among the core reactions, only the urease reaction (R_UREASE_RXN) exhibits a clear flux variability range, while biomass, lactoprotein synthesis, ATP maintenance, and hydrogenase reactions are almost fixed at a single value.

Since the model has already coupled lactoprotein synthesis with growth, these two reactions have no degrees of freedom in the FVA, and their fluxes are completely locked at the fixed values determined by the optimal solution. ATP maintenance remains a strict energy demand with no flexibility, while the hydrogenase reaction is completely suppressed under heterotrophic conditions, keeping its flux range at zero.

The urease reaction, as the only flexible component, indicates that the system retains some redundancy in nitrogen metabolism, allowing for different levels of urea degradation to meet nitrogen requirements for both growth and product synthesis.

The left plot shows the growth rate (Biomass), while the right plot shows the lactoprotein flux. The x-axis represents the carbon source uptake rate (Fructose uptake), and the y-axis represents the nitrogen source uptake rate (Urea uptake).

From the plots, it can be observed that the growth rate and lactoprotein flux follow nearly identical trends: when both carbon and nitrogen sources increase, both values rise significantly and reach their maximum in the high-carbon, high-nitrogen region. This result is consistent with the coupling setup, where product formation is tied to growth, meaning that the increase in lactoprotein flux strictly depends on the increase in biomass. Furthermore, the limitation relationship between carbon and nitrogen sources shows a synergistic effect. Under low-carbon or low-nitrogen conditions, a shortage of either becomes a bottleneck, keeping both growth and product levels low; only when both carbon and nitrogen are sufficiently supplied can the metabolic system unleash higher potential.

Conclusion (feasibility analysis): Under autotrophic conditions, the trade-off plot shows that the model can sustain growth and produce a small amount of lactoprotein when both the carbon source (CO₂) and nitrogen source (urea) are present. As carbon and nitrogen inputs increase, the growth rate rises and lactoprotein synthesis follows, demonstrating that the model design can indeed drive the production of the target product. Although the overall yield is relatively low, this reflects that under autotrophy the system is constrained by energy supply and fixation efficiency, representing a “low-output but feasible” state. This result confirms that the model has basic

physiological validity under autotrophic conditions and is capable of generating the product.

Global map under autotrophic conditions:

Under autotrophic conditions, fluxes are concentrated in hydrogen oxidation and the Calvin cycle, with hydrogenase mainly providing electrons and energy to drive CO₂ fixation. The figure clearly shows that the Calvin cycle reaction pathway is strongly activated (with higher flux intensity in red), indicating that the model reasonably reproduces the mechanism of hydrogen-driven CO₂ fixation. The system can sustain a certain level of growth and metabolic flow.

Calvin cycle:

Feasibility analysis: The model can achieve target protein production under autotrophic conditions (mechanistically feasible).

Although under optimization for maximum growth, the cell prioritizes allocating resources to sustain biomass production and does not spontaneously channel flux toward the product synthesis reaction. As a result, the FVA shows zero flux for lactoprotein synthesis. In other words, the capability for product formation exists in the model but is not activated under the default optimization objective.

However, once metabolic engineering strategies are introduced—such as coupling lactoprotein synthesis with biomass—product formation increases in parallel with the growth rate, as confirmed by the pFBA and Escher results. Therefore, under autotrophic conditions, product synthesis is mechanistically feasible.