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MBW:Non-Dimensional Model of a Closed Ecological System for Space Life Support

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Life has existed for at least 4 billion years on Earth despite limited resources, due to the continuous material recycling of its biosphere. The biosphere is the region of Earth and its atmosphere that is occupied by living organisms. It is a self-regulating entity, with millions of organisms and hundreds of coordinated processes controlling the physical and chemical environment that supports life [1] . The biosphere is isolated in space, and thus its material cycle is essentially closed, self-maintaining, and sustainable. This idea dates back to the early ecological works of Liebig and Haeckal in the 19th century and Vernadsky in the early 20th century.[2] Vernadsky is known for his 1926 book The Biosphere in which he hypothesized that living organisms could reshape the planet, driving its material cycles. The human exploration (or even settlement) of space for long durations also necessitates balanced material cycling.

A life support system includes the environment, organisms, processes, and resources interacting to provide physical necessities of life, mainly food, energy, nutrients, air, and water [1]. An Environmental Control and Life Support System (ECLSS) for spacecraft must provide adequate oxygen to the human crew, remove excess carbon dioxide (CO2) from the air, maintain adequate temperature and relative humidity for metabolic processes and comfort, provide food for nutrition, provide potable water for drinking, remove (or store) crew metabolic wastes, and remove trace contaminants from the air, food, and water supply.

During the development of human space exploration in the 20th century, the idea of simulating the Earth’s biosphere to provide human life support led to the convergence of space biology and the field of ecology to develop closed manmade ecological systems [2]. The purpose of this effort was to model and understand Earth’s sustaining life support mechanisms (biospherics) in order to simulate them for sustaining human life in space. The benefit of a biological systems is that they can serve multiple functions (processing human waste while recycling it to generate oxygen, water, and food). They can operate at ‘normal’ cabin pressures and temperatures. And living systems can reproduce themselves, allowing continuous functioning and self-repair. Estimates of the true advantages and disadvantages of physicochemical and biological systems can only be made through comprehensive theoretical and experimental investigations, which will likely result in combined systems.

Biospheric research tools are limited due to the sheer size, mass, and process complexity of the biosphere. Our ability to observe global scale biogeochemical processes is improving with satellite remote sensing, but we still must rely on large scale mathematical models and smaller scale experiments to investigate the processes controlling biospheric homeostasis. We cannot experiment with our Earth to see how it will respond. Instead, we can use smaller scale closed ecosystem experiments to study the biosphere’s bounds and stability limits. Research with hermetically sealed micro-ecosystems began in the 1920’s, in which combinations of unicellular algae and bacteria were sealed in flasks, and under certain conditions, shown to be stable for several years.

In this paper, we review the basic principles of ecology and closed ecosystem and then describe the approaches that have been used to model and predict behavior of closed ecosystems for space life support. Given the expense and time required to conduct large full scale integrated system tests of closed systems, it becomes advantageous to validate ecosystem models with small scale prototypes. The creation of small prototype systems requires models of system dynamics that are invariant to system size. In this paper, a non-dimensional ‘invariant’ model of a space life support ecosystem is derived and presented based on the typical mass balance approach.

Basic Principles: Ecology and Closed Ecosystems

Biological systems for space life support are by definition ecological systems, or ecosystems. Webster’s dictionary defines an ecosystem as “the complex of a community of organisms and its environment functioning as an ecological unit”; and ecology as “a science that deals with the relationships between groups of living things and their environments”. A space habitat in of itself can be thought of an ecological system of human beings exchanging energy and material within their spacecraft environment. In order to study the material and energy exchange processes between biological entities and non-biological components of a spacecraft life support system, one must start with a basic understanding of the principles of ecology and ecosystem theory.

Ecological components combine to produce larger functional wholes with emergent properties that cannot be predicted by studying components in isolation. Ecology studies parts and wholes. Levels of ecological hierarchy are biosphere, biomes, landscape, ecosystem, community, population, and organisms. Ecosystem boundaries can be natural or arbitrary, with the sun as the primary energy input in addition to other energy sources like wind. Water, air, nutrients, and other materials constantly enter and leave. Biotic components are either autotrophic (producers), self-nourishing from inorganic material using chemical or solar energy, like plants; or heterotrophic (consumers or decomposers), getting nourishment from organic materials synthesized by autotrophs, including fungus, non-photosynthetic microorganisms and animals. Regardless of drastic difference in species, the same ecological functional components are always present (producers, consumers, decomposers). A habitat is the place where a species is found (address), while an ecological niche is the ecological role of an organism in the community (its profession). Diversity is the variety of species, allowing the system as a whole to adapt to changing conditions. Energy, essential for life, is governed by thermodynamics. It is never created or destroyed, but it does change. Energy is concentrated through trophic levels, decreasing in quantity but increasing in concentration. Primary production is the amount of organic matter fixed by autotrophs per area per time. Net community production is organic matter produced in excess of that used by producers and consumers. Abiotic processes that make an ecosystem operational are energy flow and material cycling. Energy flow is one way and reduces in quality as it changes form. However, chemical energy can be re-used. Materials cycle back and forth between abiotic and biotic components, called biogeochemical cycles. Inorganic compounds and elements essential for life are recycled within living systems, like carbon, hydrogen, nitrogen, phosphorus, and calcium, known as macronutrients. These elements might exist in different forms that may or may not be readily available for biological use. Materials exist in compartments or pools with varying exchange rates. Nutrients are recycled through decomposition of organic plant matter by consumers and microorganisms, indirect decomposition of animal excretion, direct return via symbiotic plants and microorganisms, or abiotic recycling with physical energy, or by fuel energy. Liebig’s law of the minimum says that growth is limited by the nutrient that is least available with respect to need. This results in steady state conditions where inflows balance outflows.

The Gaia hypothesis states that “the biosphere is a self-regulating entity with the capacity to keep our planet healthy by controlling the physical and chemical environment.” This implies self-regulating feedback mechanisms that provide balance. Numerous interacting functions and feedback loops enable moderation of temperature and chemical composition, keeping conditions relatively constant.

Ecosystem Theory (Jorgenson, 2012):[3]

Systems ecology focuses on the properties of ecosystems, using a systems approach to develop a theory explaining the characteristic processes and reactions of ecosystems, similar to how physics explains physical phenomena. Ecosystem theory allows us to predict how ecosystems would react to specific pollutants or well-defined changes. A system consists of cooperative components that result in new and emerging properties. The ecosystem is a cooperative unit of the organisms that are linked in a network. If we can understand the emergent system properties and underlying system processes, then we can develop an ecosystem theory that can be used to predict reactions to well-defined changes in forcing functions. Ecosystems, like all other systems, have to follow the 1st, 2nd and 3rd laws of thermodynamics. All biological components have surprising similar biochemistry with approximately the same need for about 20 to 25 elements. The reaction rates of biochemical processes set additional constraints and restrictions on ecosystems. Ecosystems evade thermodynamic and biochemical constraints by being open to energy flows and by being able to grow and develop (by increasing biomass, increasing network size, or increasing information). Ecosystems can get sufficient work energy by selecting pathways that move it farthest away from thermodynamic equilibrium, corresponding to the highest possible work capacity (i.e. power or ‘exergy’) to achieve the most growth under constraining conditions. This idea can be considered a translation of Darwin’s theory into thermodynamic terms, and is known as the ecological law of thermodynamics (ELT). Ecological thermodynamics is defined as “the study of energetic efficiencies of ecosystems and of their energetic pathways” [4]. The ability of an ecosystem to survive, grow, and develop despite thermodynamic and biochemical constraints stems from their basic characteristic properties. They are far from thermodynamic equilibrium, are organized into networks providing feedback regulation, and have with high diversity, information content and buffer capacity. These characteristics form the basis for what is has been called “ecosystem theory”.

Properties of Closed Ecological Systems (Gitelson et al., 2003):[2]

A life support system is an artificial ecosystem (AE) designed to support a specific species. An AE is subdivided into its constituents or ‘links’, which are species (or groups of species) that fulfill a specific function of energy and material exchange. The species being supported is the ‘governing link’ and the other links form its environment. AEs must have energy exchange with the outside world, and that exchange characterizes system efficiency. They may be open (with outside mass exchange) or sealed (functioning with no inflow or outflow of material from the outside). Sealed systems can be divided into those with stores of all needed substances, and those in which needed substances are regenerated in cycles. Life is limited in the former. The latter are closed ecological systems (CES). The links in a CES use or regenerate substances for biological exchange at certain rates. If coordinated, these material cycles can exist for infinitely long periods of time. Closed material cycles can be attained biologically, or by physicochemical means, or both. Ideally closure is full (100% of needed resources are cycled), but can be partial, in which link requirements are met by the material cycle and by stored substances.

Designing a CELSS

The purpose of a space life support system is to support life safely, reliably, comfortably, and sustainably (or autonomously). Such a system must be manufacturable (reproducible and designable to a plan); and it must have minimal mass, volume, and energy consumption [2]. The purpose of the closed ecosystem for human life support is to maintain an optimal ecological habitat for humans through material exchange, compensating for disturbances created by human metabolic activity. The system is built as a counterbalance or reciprocal to humans living in it, consuming all human excreta and restoring what humans consume. Human material exchange and energy have been well studied, and so there is a lot of data and documentation on human metabolic requirements.

Physicochemical technology can be used to close the water and oxygen loop of a habitat, but not the carbon loop (food production). Currently the only means of achieving 100% material closure is with some combination of people, animals, plants, and micro-organisms with mechanical and physico-chemiccal support hardware [1]. Ideally the heterotrophic link decomposes all compounds synthesized by the autotrophic link. All energy released in decomposition converts to heat and is withdrawn from the system. Humans are the main heterotrophic agents, oxidizing organic substances, while primary synthesis of organic substances from inorganic is realized by phototrophs. Participation of other heterotrophs is less necessary, but they can perform auxiliary functions, like decomposition and mineralization of by-products that can’t be directly used by humans or phototrophs [2]. If food will be synthesized it must in turn be decomposed. But there is no known single autotroph that can fully utilize all human wastes and also produce biomass that is suitable as human food [2]. Thus a multi species system of both phototrophs and heterotrophs is necessary to support human material exchange requirements. Diverse combinations of micro-organisms, plants and animals can be proposed. “A closed human ecosystem must include components that provide a metabolic counterbalance in the form of an autotrophic link, which might include unicellular algae, higher plants, or chemotrophic bacteria, forming the basis of a closed material exchange.” [2]

Component selection should take into account food and nutritional requirements, compatibility with humans and other agents in the system, controllability of the components, energy efficiency, and other mission specific parameters.[2] The choice of LSS type for a given practical application might include many possible compositions, but since there is not enough experimental data on different LSS configurations, it is important to choose the most promising configurations a priori. Therefore quantitative objective criteria for component selection are needed. One such evaluation criteria might be the level of closure, which is related to the time the system can perform autonomously without external interference.[2] Closure also determines the total mass of the system. Many CES versions can be assemble with different degrees of closure.

Fractional closure is defined as:

K = (1-Δm/ΔM), where Δm is the mass input to the CES and ΔM is the mass recycled through the customer regeneration trophic level per unit time [2].

For stability, a steady state in mass balance must be achieved for each link, in terms of O2, CO2, H2O, etc. Non-zero mass in must equal mass out or else deadlocks will result, leading to system failure. Stability can be assessed from systems of differential equations describing mass flow dynamics between links. [2]

Another common criteria that can be used for comparing designs is Equivalent System Mass (ESM), which itself is a function of the level of closure and steady state. The equivalent system mass includes infrastructure plus all supplies needed over the duration of a mission. It might include additional mass of support systems like power and heat rejection and also the mass of spare parts. [2]

As mission distance from Earth increases, reliability and robustness may become more important design criteria than mass efficiency. To assess failure rates and consistency of performance under perturbation, component parameters are needed from experimental data under closed conditions (such as production and consumption rates). Often this data is time consuming and expensive to collect with full scale life support system prototypes. It would take a very long time to test all variations of environmental conditions that might be encountered over the life time of the system. Mathematical modeling can aid in predicting system behavior in response to changing conditions and outside influences, without a lot of experiments. Models can also reveal how we might force the system to behave the way we want. They allow us to size the components, figure out how much buffer or storage is needed, determine initial conditions, and illuminate issues related to scheduling and operation (like planting frequency). Many design options can be evaluation under a wide range of operating conditions. Small scale physical models that are comparable in function to the real system might also enable faster cycling times, reducing both size and cost of conducting multiple experiments.

Modeling of Closed Ecosystems for Space Life Support

Environmental models typically consist of the following elements:[5]

  • Forcing functions (i.e., external variables that influence the state of the system),
  • State variables, like concentration of O2, total edible biomass
  • Mathematical equations that represent the relationships between the forcing functions and state variables
  • Parameters (coefficients in the mathematical equations that are often considered constant)
  • Universal constants (like atomic weights and gas constants)

Because of the complexity of natural feedback mechanisms in biological systems and inherent non=linearity, parameter estimation is the most difficult and weakest aspect of environmental modeling. It would be impossible in our lifetimes to collect enough empirical data to accurately mathematically describe a complex ecosystem in detail. Therefore, it becomes advantageous to describe system level properties in holistic (rather than reductionistic) models [5].

Ecosystems are typically represented by network diagrams or diagraphs. Ecological networks have bins representing a component or stores, and connections representing flows between them, or external forcing functions. Since an ecosystem in space will be (ideally) closed to mass, a mass balance technique appears best suited to describe it.[6] “A model of an ecosystem can be considered as representation of a system of living and nonliving components occupying a defined space, through which energy, mass, and information flows.” [6]. At a very high level, a mass balance model for an ecosystem can be represented like that shown in Figure 1 below.

Figure 1. Flow of mass in a conceptualized small closed ecosystem (Averner, 1981) [6]"

In this approach the processes of polymer synthesis through photosynthesis, consumption and subsequent matter oxidation, and then decomposition back into inorganic minerals determine the flow and storage of elemental mass in an ecosystem. These processes are described with equations of mass transfer and transformation, guided by mass conservation constraints. The mass balance model tracks the flow of C, O, H, and N, through functional compartments. The mass flow rates are defined by a set of usually non-linear differential equations very similar in form to Lotka-Volterra equations, where the compartments are represented by mass instead of numbers of individuals. Rates are assumed to be dependent upon the amount of mass in the compartments, and often times upon the mass of both the source and sink compartment. In the simple 3 link example shown in Figure 2, we have inorganic nutrient mass storage, autotroph mass, and heterotroph mass represented by X1, X2, and X3.

Figure 2 Flow of mass in a small closed ecosystem, regulated by compartments (Averner, 1981) [6]"

Because the system is closed, the total system mass is constant. The autotrophs take up inorganic nutrient at a rate of TX12. The synthesized plant biomass is then ingested by the heterotrophs at a rate of TX23. And then heterotrophic biomass is then remineralized back into inorganic nutrients at a rate of TX31. The rate of food consumption TX23 goes to zero if either the heterotrophs or autotroph population goes to zero, and becomes saturated in X2 as X2 increases to some level. In this example, the k parameters are rate constants and a values are saturation constants. The mass flow rates are assumed to be dependent upon the amount of mass in the compartments, and often times upon the mass of the source and sink compartment. These equations define the behavior in time of a simple closed element cycle model. The parameters are often treated as constants for a particular biological species though they themselves might be functions of other variables like temperature, pressure, or light intensity. The system dynamics can be described as follows.

X1+X2+X3 = M (constant)

dX1/dt = TX31 – TX12; dX2/dt = TX12 – TX23; dX3/dt = TX23 – TX31

∑(dXm)/dt=0, and

TX12 = k*X12*(X1/a1 + X1)*X2; TX23 = k*X23*(X2/a2 + X2)*X3; TX31 = k*X31*X3

With knowledge of the elemental composition of various process products, the major ecosystem elements (typically C, H, N, O, S, P) can be modeled separately. But, a realistic representation of element flow must allow for functional couplings. Linkages between the element cycles allow on cycle to be modulated by another. Figure 3 shows an example of functional coupling of C and P element cycles driving nutrient uptake flows.

Figure 3 Functional coupling of the C and P element cycles through inter-cycle rate modulations (Averner, 1981)[6]"

Autotrophic storage of carbon (C2) regulates the nutrient flow of phosphorus into the autotrophic compartment by means of a rate constant, and the state of the autotrophic storage of phosphorus (P2) regulates the flow of inorganic carbon into the autotroph compartment (C2) through another cross-coupling parameter. Real ecosystems are of course vastly more complex than the simple three-compartment, two-element cycle model depicted here.

Given stoichiometric composition of different materials flowing in and out of each compartment, the flow of each individual element (C,O,H,N,S,P) between compartments is then modeled at steady state with differential equations, and the instantaneous level of any element can be determined through integration.[6]

In a paper by Volk and Rummel (1987),[7] the basic biochemical stoichiometry was developed for the typical processes in a biological life support system, focusing on the elements (CHON) that constitute most of the system mass. Representative CHON fractions were used for different substances, and though they are generalizations of more variable molecules, they allow us to say something about average mass flow. Equations 1 through 6 below summarize the stoichiometric equations for production of plant matter, consumption of edible plant matter, waste processing of plant matter, and waste processing of human waste products.

Figure 4 Stoichiometric Mass Balance for CHON in a 3-link CES[7]"

Protein, carbohydrate, fat, fiber, and lignin are produced in the edible and inedible parts of plants. Humans consume food and produce organic solids in urine, feces, and wash water. The waste processor converts edible and inedible plant matter and human wastes back into CO2, H2O, and Nitrate. The specific technology of the waste processor to accomplish this is not specified. Adding this processes together and assuming mass conservation, we can form a system of differential equations to represent balanced matter transformations. The rate coefficients (n) are in moles/unit time. But Volk and Rummel (1987)[7] tell us that this results in 24 equations with 44 unknowns (the rate coefficients), necessitating more constraints be defined. The author propose that we can make assumptions about the proportions of substances that are produced. For example, we can assume a protein, carb, and fat ratio for wheat as well as a constant proportion of edible to inedible mass. This is known as the harvest index. The authors assume that urine, feces, and wash solids have constant ratios, and that the waste processor operates on materials in the same proportions that they are produced. So, the system of equations can be solved to determine steady state concentrations in each compartment, if we know the plant growth rate, harvest index, food consumption, and waste processing rates. The authors advocate that developing a stoichiometric method of mass balance allows the model to be readily changed for differences in these ratios and rates.

This approach has been commonly applied to compare design configurations and analyze mass balance in experimental systems. A-priori rate constants from published data are typically used, which may be constant or functions of environmental conditions (e.g. plant growth as a function of light flux or CO2).

Finn (1999)[8] describes a top level model similar in form to that described above, that was used to investigate design issues for the AlSSITB (Advanced Life Support Systems Integrated Test Bed) formerly named the Bioregenerative Planetary Life Support Systems Test Complex (BIO-Plex) at Johnson Space Center.

Tikhomirov et al (2003)[9] describe an experimental (physical) model of a biological life support system that they developed to investigate mass exchange and interactions between components responsible for utilization of inedible plant biomass through “biological combustion”. This model contained a higher plant component of wheat and radish, a heterotrophic components (of worms, mushrooms and microbial microflora in a soil-like substrate, and a human component).

Rummel and Volk (1987)[10] demonstrate this approach by evaluating modular BLSS designs (one of which is shown in Figure 5. There are 5 processors and 8 storage reservoirs, where gas separator and nutrient mixer were used as a means to maintain greater control over reservoirs.

Figure 5. A modular BLSS model (Volk & Rummel, 1987) [10]"

The reservoirs represent typical types of storage that will be found in a spacecraft life support system. The model was structured such that each processor could only interface with reservoirs. This paper goes into detailed description of the processes regulating flow between the reservoirs, but unfortunately, full system dynamics in a set of equation is not provided. They do provide a model of wheat growth that is a function of CO2 and NO3 concentrations and an intrinsic growth rate and edible biomass allocation proportion specific to wheat. Human diet and metabolic rates are kept constant, following a daily schedule. It is assumed that the waste processor recycles 100% of waste. Simulation results are described for various scenarios of planting schedules and system failures, and shows the utility in having storage buffers for gases to help maintain stability during these perturbations.[10]

Reduced Scale Prototypes for Model Validation

Once we have an ecosystem model structure defined and estimated parameters, at some point we need to validate the model with real world data. A lot can be measured at the component level in laboratory tests, like plant growth rates in a growth chamber. But, because of complex feedback mechanisms, ecosystems are greater than sum of their parts. Component level experiments may not tell you very much about how the links, the components will interact with one another. Therefore integrated tests of the whole system are needed to verify our dynamic models. There have only been 7 facilities built worldwide to do these kind of integrated full scale life support system tests though. That is because they are expensive and time consuming. Empirically measuring reliability and robustness requires replication, across a wide state space as well as long duration experiments to truly capture failure rates. With Integrated System Level Experimentation, we can answer questions like:

  • How do the components respond to one another?
  • Are the links biologically compatible?
  • How much buffering/storage is needed?
  • Robustness: What level of environmental perturbation can the system tolerate without disrupting performance?
  • What are the limits of self-regulation and repair?
  • What are failure conditions, rates, and recovery time?
Figure 6. One of C.E. Folsome’s Algal-microbial Materially Closed Ecosystems (in 2000 ml flask) With Sampling Ports (Folsome and Hanson, 1986)[4]

Model validation requires replication (across state space and for repeatability) and long duration experiments to assess stability over system life time. But again, these full scale integrated LSS facilities are large, expensive, take a lot of time and manpower to operate.

One way to test design configurations on a budget might be with small scale physical models that are comparable in function to the real system. Small prototypes enable faster cycling times, reducing both size and cost of conducting multiple experiments. Microcosms are small controllable, replicable, low cost experimental ecosystems. They can be open or closed to material exchange and usually contain many species. Typically there is a means of measuring metabolic rates inside. They are designed to be scalable in order to make inferences about larger similar systems. Their use dates back to 1851, and some from the 1970’s are still alive today, where alive means there is still active redox activity (regardless of species survival). Models can be verified on small scale systems like these, IF we trust our scaled model is a valid representation and that the scaling has not affected our system dynamics. Gitelson (2003)[2] alludes to the idea of ecological similarity. The idea is that there might be dimensionless properties for an ‘invariant system’ that can describe and predict system behavior no matter its size. This would be useful if we want to scale up a design for more crew, without having to build a larger test facility or mockup.

An Invariant Model for a Closed Ecological Life Support System

Recreating Stoichiometric Mass Balance Equations of Rummel and Volk

For this analysis, we recreate the dynamics of a 3-link Life Support System (Humans-Wheat-Decomposers), based on the modular BLSS model described by Rummel and Volk (1987).[10] In this model, it is assumed that humans and decomposers do not change in mass, and therefore they are treated only as a processor, converting edible plant mass into human waste, and waste to inorganic nutrients through oxidation during respiration. In this model, we will use stoichiometric relationships between canonical molecules of edible and inedible wheat, human waste solids, CO2, H2O, HNO3 and O2 to track their flow through the 3 system processors (Humans, Wheat, and Decomposers). Because the actual system of differential equations for this model were not provided in the paper, this analysis will reproduce a similar system in a simplified form.

To simplify the stoichiometric equations provided by Volk and Rummel (1987)[10], and assuming the constant mass proportions provided by the authors, we can estimate a canonical molecule of food (edible wheat), plant waste (inedible wheat), and human waste (urine solids, sweat solids, and feces solids).

The authors assume 16.9:80.5:2.6 grams of protein to carbohydrate to fat in the edible portion of wheat, so we can assume the following molar proportions: 16.9 g / 83 g/mole protein to 80.5/180 g/mole carbohydrate to 2.6 g / 256 g/mole fat, or 0.2036:0.4472:0.0102 moles of protein, carbohydrate, and fat.

We can estimate a new canonical ‘edible wheat’ molecule as: 0.2036 C4H5ON + 0.4472 C6H12O6 + 0.0102 C16H32O2 ==> 3.6608 moles C + 6.7108 moles H + 0.2036 moles N + 2.902 moles O which is approximately C18H33NO14 at 487 g/mole.

Similarly, for inedible plant mass, the authors assume 0.35:0.50:0.15 grams protein to fiber to lignin, so we can assume the following molar proportions: 0.35 g / 83 g/mole protein to 0.50 g / 162 grams/mole fiber to 0.15 g / 163 g/mole lignin, or 0.0042:0.0031:9.2025E-4 moles of protein, fiber, and lignin.

We can estimate a new canonical ‘inedible wheat’ molecule as: 0.0042 C4H5ON + 0.0031 C6H10O5 + 9.2025E-4 C10H11O2 ==> 0.0446 moles C + 0.0621 H + 0.0215 O + 0.0042 N, which is approximately C11H15NO5 at 241 g/mole.

Edible wheat (C18H33NO14) is consumed with O2 and broken down into human waste solids, water and CO2. The authors break down 1.74 moles-N in protein to 0.55:0.12:0.02 moles of urine, feces, and sweat solids, respectively, so dividing by 1.74 moles, gives a proportion of 0.3161:0.0690:0.0115 moles of each, respectively, per mole of Nitrogen consumed in protein. This gives 3.6797 moles C + 6.9796 moles H + 1.6787 moles O + 1.0002 moles N which is approximately C11H21O5N3 for ‘human solid waste’.

The molar weight in grams of all of the molecules tracked in this analysis is listed for reference in Table 1, below.

Table 1: Atomic Weight Estimates
Grams/mole Element
12 Carbon
1 Hydrogen
14 Nitrogen
16 Oxygen
32 O2
62 NO3
18 H2O
83 Protein(C4H5ON)
180 Carbohydrate (C6H12O6)
256 Fat (C16H32O2)
162 Fiber (C6H10O5)
163 Lignin (C10H11O2)
90 Urine (C2H6O2N2
851 Feces (C42H69O13N5)
420 Sweat Solids (C13H28O13N2)
487 Edible Wheat (C18H33NO14)
241 Inedible Wheat (C11H15NO5)
275 Solid Human Waste (C11H21O5N3)

Food Production (growth of edible plant matter): 18CO2 + 16H2O + HNO3 ==> C18H33NO14 (food) + 20.5O2, such that for every mole of edible plant matter produced, there are 1 moles of nitrate (HNO3), 16 moles of water (H2O), and 18 moles of CO2 consumed, with 20.5 moles of oxygen (O2) produced.

Inedible Wheat Production: 11CO2 + 7H2O + HNO3 ==> C11H15NO5 (inedible wheat) + 13.5O2, such that for every mole of inedible plant matter produced, there are 1 moles of nitrate (HNO3), 7 moles of water (H2O), and 11 moles of CO2 consumed, with 13.5 moles of oxygen (O2) produced.

Food Consumption: Humans metabolize each mole of edible plant matter as 3C18H33NO14 (food) + 44O2 ==> C11H21O5N3 (human waste) + 43CO2 + 39H2O

As Rummel and Volk (1987)[7] did, we will also assume that whatever is produced by the plants is eaten by the people, and so, there is no waste processing of edible plant matter, only of inedible plant matter and human waste. Stoichiometrically, this becomes the reverse of inedible wheat production.

Inedible Wheat Consumption: C11H15NO5 (inedible wheat) + 13.5O2 ==> 11CO2 + 7H2O + HNO3 and

Human Waste Decomposition: C11H21O5N3 (human waste) + 17.5O2 ==> 11CO2 + 9H2O + 3HNO3

Given a plant growth rate (G), a harvest index (H), an inedible wheat consumption rate (C), and a human waste decomposition rate (D), we can then form a system of differential equations to track the flux of the following substances through the 3-link system in moles per day: Plant (P), Food (F), inedible wheat (I), human waste (W), water (H2O), nitrate (NHO3), oxygen (O2), and carbon dioxide (CO2), in moles per day. Any plant mass growth is assumed to a combination of edible matter (food) and inedible matter. The food is immediately consumed, and inedible matter goes directly into decomposition. Plant growth is a function of inedible plant material (represented here in moles), CO2, and HNO3. Though this is an oversimplification of a true plant growth model it will suffice for the purposes of this analysis. We are also assuming that plants do not die before being harvested. Also in this simplified example, we are not limiting the size of crop growth.

In general form, the seven ‘reservoirs’ can be modeled as:

dP/dt = Growth – Food Consumption – Inedible Wheat Consumption = G*I*CO2*H2O*HNO3 – H*G*I*CO2*H2O*HNO3 – C*I*O2, assuming growth of edible biomass occurs from inedible biomass only. This reduces to (1-H)*G*I*CO2*H2O*HNO3 – C*I*O2.

dF/dt = 0, assuming food is immediately consumed by humans and H is the harvest index (proportion of plant mass that is edible).

dI/dt = Growth – Food Consumption – Inedible Wheat Consumption = (1-H)* G*I*CO2*H2O*HNO3 – C*I*O2 (same as dP/dt)

dW/dt = 1/3(Food Consumption) – Human Waste Decomposition = 1/3 * H*G*I*CO2*H2O*HNO3 – D*W*O2

dH2O/dt = Food Consumption + Inedible Wheat Consumption + Human Waste Decomposition - Food Production – Inedible Wheat Production = 39/3* H*G*I*CO2*H2O*HNO3 + 7*C*I*O2 + 9*D*W*O2 – 16*H*G*I*CO2*H2O*HNO3 – 7*(1-H)*G*I*CO2*H2O*HNO3, which simplifies to ((39/3-16+7)H – 7)*G*I*CO2*H2O*HNO3 + 7*C*I*O2 + 9*D*W*O2

dHNO3/dt = Inedible Wheat Consumption + Human Waste Decomposition - Food Production – Inedible Wheat Production = 3*D*W*O2 + C*I*O2 - H*G*I*CO2*H2O*HNO3 - (1-H)*G*I*CO2*H2O*HNO3, which reduces to 3*D*W*O2 + C*I*O2 - G*I*CO2*H2O*HNO3

dO2/dt = Food Production + Inedible Wheat Production - Food Consumption - Inedible Wheat Consumption - Human Waste Decomposition = 20.5*H*G*I*CO2*H2O*HNO3 + 13.5*(1-H)*G*I*CO2*H2O*HNO3 – 44/3* H*G*I*CO2*H2O*HNO3 – 13.5C*I*O2 – 17.5D*W*O2,which simplifies to ((20.5-13.5-44/3)H + 13.5)G*I*CO2*H2O*HNO3 – 13.5C*I*O2 – 17.5D*W*O2

dCO2 = Inedible Wheat Consumption + Human Waste Decomposition + Food Consumption - Food Production – Inedible Wheat Production = 11*C*I*O2 + 11*D*W*O2 + 43/3H*G*I*CO2*H2O*HNO3 – 18*H*G*I*CO2*H2O*HNO3 – 11*(1-H)*G*I*CO2*H2O*HNO3, which then reduces to 11*C*I*O2 + 11*D*W*O2 + ((43/3-18+11)H – 11)G*I*CO2*H2O*HNO3

P = F+I and

Fmf+Imi+Wmw+HNO3mHNO3+O2mO2+CO2mCO2+H2OmH2O = M, the total mass of C, O, H, and N in the system, where mx is the molar mass of molecule x.

Using the Michaelis–Menten kinetic form, as used by Averner (1981)[6] and Rummel and Volk (1987)[10], and setting the saturation constants equal to ½ the molar value required for the reaction, we can then develop the following set of differential equations:

dI/dt = (1-H)*G*I*CO2/(5.5+CO2)*H2O/(3.5+H2O))*HNO3/(0.5+HNO3) –C*I*O2/(6.75-O2)

dW/dt = 1/3HG*I*CO2/(5.5+CO2)*H2O/(3.5+H2O)*HNO3/(0.5+HNO3) – D*W*O2/(8.75+O2)

dH2O/dt = ((39/3-16+7)H – 7)*G*I*CO2/(5.5+CO2)*H2O/(3.5+H2O)*HNO3/(0.5+HNO3) + 7*C*I*O2/(6.75-O2) + 9*D*W*O2/(8.75+O2)

dHNO3/dt = 3*D*W*O2/(8.75+O2) + C*I*O2/(6.75-O2) - G*I*CO2/(5.5+CO2)*H2O/(3.5+H2O)*HNO3/(0.5+HNO3)

dO2/dt = ((20.5-13.5-44/3)H + 13.5)G*I*CO2/(5.5+CO2)*H2O/(3.5+H2O)*HNO3/(0.5+HNO3) – 13.5C*I*O2/(6.75-O2) –17.5D*W*O2/(8.75+O2)

dCO2 = 11*C*I*O2/(6.75-O2) + 11*D*W*O2/(8.75+O2) + ((43/3-18+11)H – 11)G*I*CO2/(5.5+CO2)*H2O /(3.5+H2O)*HNO3/(0.5+HNO3)

Non-Dimensionalizing the System

Next, we can transform the system of equations into a non-dimensional system.

Recall that Fmf+Imi+Wmw+HNO3mHNO3+O2mO2+CO2mCO2+H2OmH2O = M, the total mass of C, O, H, and N in the system, where mx is the molar mass of molecule x, as listed in Table 1, and M is the total mass of C, H, O, and N at any given time in the system (100% mass closure).

Let I = iI’, W = wW’, H2O = hH’, HNO3 = nN’, O2 = oO’, CO2 = cC’, and t = τT’

Choose T’ = 1/G such that G*iT’ = i is now dimensionless, giving:

di/dτ = (1-H)*i*cC’/(5.5+cC’)*hH’/(3.5+hH’))*nN’/(0.5+nN’) – C/G*i*oO’/(6.75-oO’)

dw/dτ = 1/W’ * 1/3 * H*iI’*cC’/(5.5+cC’)*hH’/(3.5+hH’)*nN’/(0.5+nN’) – D/G*w*oO’/(8.75+oO’)

dh/dτ = 1/H’((39/3-16+7)H – 7)*iI’*cC’/(5.5+cC’)*hH’/(3.5+hH’)*nN’/(0.5+nN’) + 1/H’7*C/G*iI’*oO’/(6.75-oO’) + 1/H’9*D/G*wW’*oO’/(8.75+oO’)

dn/dτ = 1/N’*3*D/G*wW’*oO’/(8.75+oO’) + 1/N’C/G*iI’*oO’/(6.75oO’)-1/N’*iI’*cC’/(5.5+cC’)*hH’/(3.5+hH’)*nN’/(0.5+nN’)

do/dτ = 1/O’((20.5-13.5-44/3)H + 13.5)*iI’*cC’/(5.5+cC’)*hH’/(3.5+hH’)*nN’/(0.5+nN’)– 13.5C/GiI’*oO’/(6.75-O2)/O’ – 17.5D/G*wW’*oO’/(8.75+oO’)/O’

dc/dτ = 11*C/G*iI’*oO’/(6.75-oO’)/C’ + 11*D/G*wW’*oO’/(8.75+oO’)/C’ + ((43/3-18+11)H –11)*iI’*cC’ /(5.5+cC’)*hH’/(3.5+hH’)*nN’/(0.5+nN’)/C’

To translate into dimensionless units for each molecule, we can convert to mass (using molecular weight for each molecule), and divide by the total mass (M) of the closed system.

Let i = Imi/M, w = Wmw/M, h = H2OmH2O/M, n = HNO3mHNO3/M, o = O2mO2/M, and c = CO2mCO2/M ==>

I = iM/mi, W = wM/mw, H2O = hM/mH2O, HNO3 = nM/mHNO3, O2 = oM/mo2, and CO2 = cM/mCO2

The non-dimensional system of equations becomes (after simplification):

di/dτ = (1-H)M4/(mi)*(ichn)/[(mCO25.5+ cM)(mH2O3.5+ hM)(mHNO30.5+ nM)] – MC/G*io/(mO26.75- oM)

dw/dτ = 1/3H*M3mwmi*(ichn)/[(mCO25.5+ cM)(mH2O3.5+ hM/(mHNO30.5+ nM)]–MD/G*wo/(mo28.75+ oM)

dh/dτ = (4H – 7) M3 mH2O/mi * ichn/[(mCO25.5+cM)(mH2O3.5+hM)(mNO30.5+nM)] + 7MmH2O/mi C/G* io/(mO26.75-oM) + 9MmH2O/mw D/G*wo/(mO28.75+oM)

dn/dτ = 3MmHNO3/mwD/G*wo/(mo28.75+ oM) + MmHNO3/mi *C/G*io(mo26.75- oM) – M3mHNO3/mi*ichn/[(mCO25.5+ cM)(mH2O3.5+ hM)(mHNO30.5+ nM)]

do/dτ = M3mo2/mi (-7.77H + 13.5)* ichn/[(mCO25.5+ cM)(mH2O3.5+ hM)(mHNO30.5+ nM)] – 13.5M2/miC/G*io/(mo26.75- oM) – 17.5M2/mw D/G* wo/(mo28.75+ oM)

dc/dτ = 11mCO2/mi *C/G*io/(mo26.75- oM) + 11mCO2/mw*D/G*wo/(mo28.75+ oM) + M3mCO2/mi (7.33H – 11)* ichn/[(mCO25.5+ cM)(mH2O3.5+ hM)(mHNO30.5+ nM)]

In this new system of equations, the system behavior is not directly dependent upon the growth rate of the autotrophic link, but upon the ratio of the decomposition rate (of inedible wheat and human waste) to the growth rate, making C/G and D/G scale independent parameters. The harvest index H is already scale independent. The only parameter in the model that will change with system size is the total system mass (M), the scaling parameter of the model.


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