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Δ N = − x t N o u t (2) N o u t = N D η F η H (3)

Maximum N depletion occurs when no recycling or N₂ fixation is performed. N loss is therefore assumed to be equal to potential losses incurred during fertilization and harvest, as well as the total N consumed by crewmembers (N _D ). Maximum depletion is modeled over time as

Where η _F is equal to the efficiency of fertilization (the fraction of N converted to crop biomass) and η_H is equal to the efficiency of harvest (the fraction of edible crop biomass).

The + NF Depletion model represents a system in which N losses are still equal to the losses incurred during fertilization, harvest, and consumption by crewmembers. However, ΔN for the colony is increased by the addition of N to the system by an NF fermenter. Attempting to replenish all lost N using the +NF system would likely demand very large reactor (>2500 L CM⁻¹) and energy requirements, so depletion is still expected to occur over time and will be reduced by the addition of fixed N. NF depletion is modeled over time as Δ N = x t N i n − N o u t (4) N i n = r V η B η W C Z (5) η x = η A E x + η A N 1 − x (6)

where N _out is as defined in Eq. 3, r is the volumetric rate of N₂ fixation (g L⁻¹ d⁻¹), V is the reactor volume per crewmember (L CM⁻¹), η _B is the fraction of N harvestable from the fermenter, η[x] is a function (Eq. 6) for calculating the weighted average of aerobic (AE) and anaerobic (AN) digestion efficiencies (η _AE andη _AN , respectively, representing the fraction of N retained) based on flow fraction x, and W _C is the fraction of microbial biomass fed to AE (with remainder to AN). Z is equal to 1 if biomass is treated in digesters prior to agriculture integration, or 0 otherwise.

The + R Depletion model represents a system in which N losses are reduced through the recycling of various waste sources including inedible crop biomass (ICB) and human waste in urinary and fecal forms. AE and AN digestion processes are used to recycle ICB and feces, while N from urine in the form of urea is recycled directly back to crops. Inefficiencies stemming primarily from AE and AN result in gradual depletion over time. These losses may also be compounded by any N loss during fertilization. +R depletion is modeled over time as Δ N = x t N r e c − N o u t (7) N r e c = N D η W C 1 η H − 1 + W F   η W W + η U 1 − W F (8) where N _out is as defined in Eq. 3, W _C is the fraction of ICB fed to AE (with remainder to AN), W _W is the fraction of feces fed to AE (with remainder to AN), W _F is the fraction of human waste N converted to feces (with remainder to urine), and η _U is the fraction of urine N recyclable. The function η[x] is as defined in Eq. 6. This assumes a complete conversion of consumed N to waste products in humans and complete feed of all waste to either AE or AN in the colony.

The Separate NFR Model represents the first of the two proposed systems in which NF and R occur simultaneously and separately. This system forms a closed loop using R processes implementing AE and AN, with an NF fermenter acting to supply N from outside of the loop. The transfer of urea directly to agriculture (post-salts removal) serves to simplify the loop closure and to produce effective crop growth (Rathnaweera et al., 2019). The losses incurred because of recycling processes can be made up for through NF and drastically reduce the fermenter size required to break even. The Separate NFR system is modeled as Δ N = x t N i n + N r e c − N o u t (8a) where N _in is as defined in Eq. 5, N _rec is as defined in Eq. 8, and N _out is as defined in Eq. 3.

The Combined NFR Model makes the NF fermenter a part of the loop instead of an external N supplier. R processes are a part of the loop in this model as well. By cycling urine through the NF fermenter, heterotrophic microbial growth can be accelerated as fixed N and carbon sources are provided. Though N₂-fixing microbes typically stop expressing nitrogenase in the presence of fixed N, the photoheterotrophic Rhodopseudomonas palustris (R. palustris) nifA* mutant expresses nitrogenase constitutively and may be able to fix N₂ while consuming fixed N (Adessi et al., 2012; Heiniger et al., 2012). For organisms that do not constitutively express nitrogenase, limited amounts of fixed N may be used to accelerate growths at the beginning of fermenter cycles, which may also result in a greater rate of NF in addition to accomplishing N recycling (Lee et al., 1995). The increase in NF in both cases is assumed to be representable by a scalar multiplier of the standard fixation rate r. The Combined NFR Model is modeled as Δ N = x t N i n + N r e c − N o u t (8b) N i n = a r V η B η W C Z (9) N o u t = N D η F η H + Z N D η U 1 − W F   1 − η W c (10) where a is a scalar multiplier to account for an altered rate when fed with urea and other waste components, the η[x] function is as defined in Eq. 6, and N _rec is as defined in Eq. 8. As in +NF -R (B), Z is equal to 1 if biomass is treated in digesters prior to agriculture integration, or 0 otherwise. The second term in Eq. 10 represents the additional N loss incurred should reactor N be treated with digestor processes.

With Formulas 1–10, simulations for each of the 5 N regiments were run. A literature review was conducted to determine likely values for each efficiency variable included. N₂ fixation rates were estimated experimentally (Supplementary Table S1; Supplementary Table S2; Supplementary Figure S1). Maxima and minima were also tested to determine the maximum possible effect of each variable on N management. Table 1 shows the default values used in standard models, other feasible values tested, maxima, and minima. N depletion or accumulation (kg) over the course of 1 year (365 days) was compared using the Combined NFR model.

Equations 1–10 do not account for likely interdependencies between system variables. Digestion was assumed to positively impact fertilization efficiency by allocating greater bioavailable N to plants. The effect of digester efficiencies (η_AE and η_AN) on biomass fertilization efficiency (η_F) was accounted for by setting fertilization efficiency equal to 100% in digestive systems, and 90% otherwise.

Minimum Reactor Volumes (MRVs) were defined as the minimum reactor volume required to keep N supplies constant over time under a set of given conditions (such that N _in + N _rec = N _out ). MRVs were useful in quantifying the influence of individual variables and regiments on N management and summarizing the results shown in sensitivity analyses. The influence (I) of a variable (X) on the system volume was quantified by taking the difference in MRVs at the maximum and minimum values (Eq. 11). The MRVs used in this work were calculated using the Combined NFR model. I X = M R V MAX − M R V MIN (11)

Daily protein intake required to sustain a healthy human diet varies anywhere from 0.8 to 1.6–2.0 g per kg bodyweight, depending on the level of activity and fitness of the individual (Forslund et al., 1999; Friedman and Lemon, 1989; Organization and others, 1985). Assuming N makes up 16% of protein mass and a mean crew body mass of 80 kg, this offers a range between 10.2 and 20.5 g CM⁻¹ d⁻¹. Moderately trained endurance athletes have been shown to break even on an N dose of 12 g d⁻¹ and have a recommended intake around 16 g d⁻¹ (Meredith et al., 1989; Tarnopolsky, 2004). A conservative estimate to maintain the health of physically active crewmembers was considered to be 14 g CM⁻¹ d⁻¹ and is a foundational assumption for the data presented in Figures 3, 4.

The selected N consumption rate of 14 g CM⁻¹ d⁻¹ assumes no N production besides dietary usage. Pharmaceutical and other types of production would add to the daily N demand of the colony. This value also assumes complete consumption of incoming fixed N supplies, and neither accounts for a surplus of nutrients nor its storage. With these factors considered, the N consumption rate should be treated as a bare minimum to support a healthy human diet, with zero additional demands or accumulation.

The N₂ fixation rate r is very influential in the presented NF regiments, acting as the only significant means by which additional N may enter the closed-loop system. The rate itself is dependent on many variables in chemical engineering design, including N₂ solubility in water, the N₂-fixing organism (including its growth rate and metabolism), media composition, and reactor conditions (temperature, pH, pressure, mixing, and lighting where applicable). The reactor type (continuous flow stirred tank reactor (CFSTR), batch, plug flow, etc.) also will play a significant role in the rate of production. Future studies on scale-ups are essential to narrow down required operating values for microbiological N₂-fixing systems.

To simplify the model, a constant daily output of reactor biomass was assumed based on observed laboratory data without accounting for growth kinetics. A more sophisticated reaction model may be developed in which a specific reactor type is selected and incorporated into the N management model. For a batch kinetic model, influxes of N would occur according to an estimated harvest interval, rather than a constant influx of N assumed in the presented model. On the other hand, for a CFSTR kinetic model, a constant influx of N may indeed be valid.

Regardless of the type of reactor modeled, downtimes should also be accounted for to estimate the effects of maintenance on N management. Reactor maintenance may result in several days or weeks in which no additional N is being added to the system, which would need to be accounted for in the design of any such system. The effects of random equipment failure, possibly resulting in even longer downtimes, may be easily modeled. The exact downtimes for maintenance or failure are dependent on the specific components and systems used.

For organisms capable of constitutive nitrogenase expression in the presence of fixed N, wastewater components may be added to create combined NFR systems. Both inhibition and facilitation of growth may occur depending on the composition of this waste stream. Determining inhibitory and facilitative components of wastewater will be essential in the design of wastewater treatment systems intending to recycle waste streams upstream to agriculture and/or fermenters. We consider a 65% increase in growth to be possible in streams containing reasonably high amounts of urea as well as trace amounts of acetic acid, propionic acid, glutamic acid, and other organic compounds supportive of growth (Verostko et al., 2004).

An in-depth analysis of reactor kinetics is beyond the scope of this work. However, such analyses will be an essential component of ongoing modeling and design of N management systems. Subsequent studies focusing on modeling reactor kinetics and output may be able to implement a similar finite differences approach as presented here. Additionally, a variety of software and libraries exist to support the modeling of chemical systems (Charlton & Parkhurst, 2011; Müller et al., 2011).

Future research on N₂ fixation within the scope of this model must seek determination of 1) performance in upscaled systems; 2) ideal organisms across system types; 3) ideal reactor types and designs; 4) operational profiles with a variety of substrates and growth conditions; and 5) alternatives to biological N₂ fixation systems. Assessing N₂ fixation systems not only in terms of volume, but also mass, power, reliability, and resource demands will allow for a more direct assessment and determination of N₂ fixation capabilities in life support systems.

Crop harvest index η_H is very influential and appears to be one of the most controllable factors in N management, determined chiefly by the types of produce grown and species selected. Maximizing the percentage of high-harvest index produce such as potatoes, lettuce, beets, spinach, and onions would minimize waste between fertilization and consumption by humans. This is, of course, limited by the nutritional demands and dietary requirements of the colony. A crop harvest index of 50% has generally been a realistic average for a system that offers a greater variety of produce, including those with lower harvest indices, such as tomatoes and wheat. Different species of the same crop type appear to exhibit some variation in harvest indices as well: U.K. winter wheat has exhibited a range of 0.43–0.54 (Gent & Kiyomoto, 1989), while U.S. spring wheat may vary from 0.31 to 0.51 (McLaren, 1981). A selection of crops to meet the nutritional demands of a colony while maintaining a high harvest index will have a significant effect on colonial N management.

The bioreactor harvest index η _B is very influential in NF regiments, directly limiting the amount of N that can be added to the system. N “losses” described by this variable do not consist of N physically leaving the system, but rather represent a fraction of precipitated N that is effectively locked into the reactor system, unavailable to organisms and/or harvesting mechanisms (Marti et al., 2008; Feyissa et al., 2020; Achilleos et al., 2022). These inefficiencies result from processes such as struvite precipitation, recalcitration, and biofilm formation.

In laboratory experiments with R. palustris, biofilm formation was observed to occur on the sides of a batch reactor (Supplementary Figure S1). When biomass is harvested from media via centrifugation, the accumulation of biofilm ultimately has a negative effect on production, as biofilm blocks light entry and cannot be accessed via pumping. In batch systems, however, biofilm formation may become advantageous. Instead of liquid media harvests via pumping and centrifugation, biofilm reactors could allow larger amounts of biomass to simply be scraped off a scaffolding, similar to the operation of rotating algal biofilm reactors (Christenson & Sims, 2012). Successful biomass harvests without centrifugation would save a significant amount of energy over time (Young, 2011). N loss during bioreactor harvests has been estimated at 20%, with higher losses devastating N management in the presented model.

As growth kinetics are more accurately accounted for, the bioreactor harvest index will become more significant to account for N which may not be harvestable. The bioreactor harvest index may be reactor-dependent, with some designs and types allowing for more thorough harvests than others. Batch reactor systems may facilitate the removal of biofilm layers during harvest periods better than CFSTRs. Given that R. palustris depends on light to perform N fixation, and that biofilm accumulation inhibits light entry into the media, the reactor may require thorough biofilm harvests.

Fertilization efficiency η _F is a measure of the fraction of N that is integrated into crop biomass in the agricultural stage and is very influential in all the regiments in this model. On Earth, η _F is predicted to be approximately 50% or less for large farms due to factors such as leaching and runoff (Smil, 1997; Ruiz et al., 2020). Nitrogen use efficiency is typically between 30% and 80% in terrestrial agriculture (Dobermann, 2005).

In open-loop N management systems, which encompasses most industrial terrestrial agriculture, fertilization losses result from a variety of inefficient practices. Nitrate leaching occurs when excess nitrate is applied. This anion is repelled by negatively charged soil particles and leaches into the surrounding environment. In poorly aerated root-zones, anaerobic conditions can lead to denitrification, which releases nitrous oxide and dinitrogen gases. Nitrous oxide is especially potent due to its greenhouse warming potential 300 times that of carbon dioxide and its destruction of atmospheric ozone (Griffis et al., 2017). When ammonium is applied to alkaline soils, equilibrium can promote the production of ammonia gas leading to increased losses. In well-aerated root-zones with pH control these losses should be minimized. A fertilization efficiency near 100% appears to be possible in precision liquid hydroponic systems using a mass balance guided approach to match nutrient delivery with plant requirements (Langenfeld et al., 2022). Runoff and leaching would represent major losses for other valuable resources such as water, but agricultural systems on Mars will use closed root-zones to eliminate these losses.

Continued research on fertilization efficiency is required to validate the high efficiencies assumed in this model. This would include measuring and optimizing N efficiency using biomass in a wider variety of agricultural system types. A soil-based approach to biomass fertilization may incur additional loss from foregoing biomass digestion (Foereid et al., 2021). Accurately accounting for N in soil-based, hydroponic, and aeroponic agricultural systems will be essential in the design of an N-efficient life support system. The results of this model suggest that high fertilization efficiencies, independent of digestion, would best support N management in a closed loop.

Little research has been done on N-efficient AE and AN systems. It has been shown in existing work that AE and AN have efficiencies around 75% and 50%, respectively (Novak et al., 2011). These were the approximate efficiency values assumed in the previous life support N-recycling models presented (Langenfeld et al., 2021). The surrounding research and literature around earth-based AE and AN processes differ in metrics, with the primary goal of such systems being the removal of N from waste streams regardless of form. Thus, reported efficiencies of 50% or 75% may refer to fixed N harvested from water and converted to usable forms (i.e., microbial biomass), concentrated in fixed forms (i.e., ammonium and nitrates), or lost to the environment as N₂. In space life support systems, the majority of N must be kept in bioavailable forms for digestive recycling systems to work properly. The design and optimization of digestive and other recycling systems to achieve this goal represents an important pathway for future research.

Barring losses during bioreactor harvests and fertilization, AE and AN processes represent the only source of loss in the regiments incorporating these processes. N losses through AE and AN are proportional to the total N flow imposed on them. Reducing flow through these systems as much as possible while maintaining recyclability on each N stream will reduce losses and sensitivity to digester inefficiencies.

Exploring alternatives to biomass digestion is of particular importance to the combined NFR model. Otherwise, waste N from urine would be subjected to digestive losses that are avoided in the separated NFR model, significantly reducing the predicted gains that come from combining NF with recycling. The treatment of microbial biomass with sulfuric acid appears to be a viable alternative to digestion and can preserve N content (Valgardson, 2020).

N forms fed to plants are an important consideration in choosing AE or AN. Plants can take up amino acids, urea, nitrate, or ammonium as forms of N. Amino acids and urea can be taken up by plants in small amounts, but urea is largely hydrolyzed to ammonium prior to root uptake. Nitrate uptake is slower than ammonium uptake, but it may be stored in the vacuole and redistributed to cells as needed. Many studies show that plants given some ammonium in addition to nitrate grow better than those fed nitrate only (Zhang et al., 2019; Daiane et al., 2021). Ammonium is taken up much faster than the other three combined and can lead to potential toxicities because it cannot be stored by the plant like nitrate (Marschner, 2011). Excessive ammonium can lead to plant toxicity by inducing cationic nutrient deficiencies and decreasing rhizosphere pH.

AE remains a significant process in N management even without treating additional N from the fermenter. Maximizing the efficiency of AE will be important in minimizing the losses from crop harvest inefficiencies and the digestion of human feces. AN, however, appears to have a somewhat-limited influence in these circumstances, treating a smaller fraction of crop biomass (20%) and 50% of feces, which makes up only 10% of waste that ultimately undergoes digestion in the combined NFR model. AN maintains relevancy as a means of converting waste N to ammonia, which is an efficient N source for plants when supplied in limited amounts alongside nitrate (Zhu et al., 2021; Wang et al., 2022). AE differs from AN digestion further in terms of resource demands, energy, and time: AE requires O₂ and high energy input while accomplishing digestion in a short time. Conversely, AN can be performed without O₂ and with low energy input with longer digestion times. Although AE appears to be far more significant in N management, the extent to which it can be implemented may be limited by the energy demands and O₂ production of the colony.

About 90% of the N in urine is contained in urea (Kirchmann and Pettersson, 1994). Urea is hydrolyzed to ammonium via urease. This enzyme is ubiquitous among microorganisms and is also found in plants (Kryštofová et al., 2007). If urine contamination occurs after leaving the body, rapid hydrolysis may occur. Hydrolysis increases the pH of the urine, which can lead to ammonia volatilization (Ray et al., 2018). Acid addition is required to optimize recovery efficiency. A near-100% recovery of urea N from urine is achievable if contamination is minimized and acidic conditions are maintained (Langenfeld et al., 2021).

The remaining N in urine can be attributed to a variety of low-concentration ammonium salts, which are considerably more difficult to recycle. A variety of systems have been developed that can recover N in these forms. Bioelectrochemical (BEC) systems have been shown to be highly effective at recovering ammonium from wastewater and energy from organics. BEC systems convert ammonium to volatile ammonia, which may be recovered in liquid form. Recoveries of ammonium in BEC systems have been estimated between 94% and 97% (Wu & Modin, 2013). Recovery up to 80% has also been achieved through electrodialysis (Shi, Li). Membrane systems such as forward osmosis systems have been able to recover between 50% (Volpin et al., 2018) and 66% (Singh et al., 2019) of total urinary ammonia.

A combination of sterilization and acidification (for urea recovery) followed by BEC, electrodialysis, and/or membrane filtration methods (for ammonium recovery) would potentially allow for maximal N recovery from urine (Shi et al., 2018; Li et al., 2021). Effective capture of volatile ammonia will be necessary for systems such as BEC and will help ensure minimal losses during other steps.

Of these W_X values, W_C is the only variable that appears capable of significantly altering N management. This becomes the case when fermenter biomass is treated with digestion, in which a greater amount of N will be entering the digestion processes. Assuming that AE remains more efficient than AN, maximizing W_C is valuable in minimizing digestive losses. This, however, must be balanced by any ammonia demand that has to be met via AN. The higher energy demand of AE must also be considered with increased inputs.

When fermenter biomass bypasses digestion processes, W_C and W_F appear to diminish drastically in their impacts on the system. The primary factors governing the proportion of biomass or waste to enter AE versus AN will likely be how much power is available to support AE, combined with the needs of nitrate or ammonia for crop fertilization and soil health. Assuming ample power and sufficient ammonia availability for fertilization, maximizing W_C and W_W would minimize N losses.

The fraction of N in human waste going to feces W _F seems to be mostly dependent on diet, but rarely exceeds 20%, with the remainder exiting in urine (LAPID, 2004). This represents the fraction of N in human waste that must undergo digestion to be recycled, as urinary N can directly reenter the agricultural or NF steps (post-salts removal). The amount of N going to feces thus corresponds to higher losses, dependent on the efficiency of AE and AN. W_F is not anticipated to exceed 20% by a significant margin, however, and is therefore unlikely to significantly alter N management.

The significance of boundary estimations for natural and man-made closed-loop systems has been well-established as a means of determining unsustainable practices and establishing sustainability milestones (Rockström et al., 2009; Steffen et al., 2015; Willett et al., 2019; Cousins et al., 2022). Additionally, poor N management has resulted in catastrophic effects on ecosystems across the U.S. and world at large (Glibert et al., 2014; Yarandi et al., 2021; Demertzioglou et al., 2022). The National Academy of Engineering has listed effective management of the N cycle as one of the grand challenges of the 21st century (Katsouleas et al., 2013). The application of similar boundary systems to Earth N management may help to identify key variables that can be changed to better control the flow of N into crops and the environment.

Applying the model presented here to N management in closed-loop systems on Earth naturally brings with it a host of challenges (Eid, 2000). While N management on an early Mars colony will be small-scale and precisely controlled, Earth N management encompasses global ammonia production from industrial and biotic processes amounting to over 200 million tons per year (Yarandi et al., 2021), its imprecise application to croplands, its application in other uses (i.e., pharmaceuticals, refrigeration, and textiles), and its entry into the environment as runoff and through volatilization. Earth’s N cycle occurs vastly through an established network linking industrial and natural N activity together, summarized by Figure 5.

Applying N boundaries to N management on Earth can be simplified through a variety of means. The scope of the model may be restricted to specific regions and emphasize the practices and necessary changes in smaller communities. Additionally, some N uses may be ignored if they are negligible in comparison to agricultural N use. In-depth discussion on the application of this system to Earth N management is beyond the scope of this work. However, this model represents a prototype for a future analyses of N boundaries in space and on Earth.