# The Evolutionary Mechanism of Formation of Biosphere Closure

^{*}

## Abstract

**:**

## 1. Introduction

_{2}in the atmosphere to 10% of the existing level in ~10 ÷ 15 years. This level corresponds to the compensation point of C3 plants, i.e., corresponds to the cessation of plant growth and then their death. Note that C3 plants account for 95% of all land plants.

^{−5}of the current level of atmospheric oxygen, reaching 0.1–1% of the current level by ~2 billion years ago. During the GOE, the primary biosphere arose, where the global change in the gas composition of the atmosphere affected all organisms in contact with the atmosphere, and instead of separate and practically non-interacting chemoautotrophic oases, a single, even though weakly connected, quasi-ecological system arose. If we compare it with the identification of four energy epochs of life on Earth proposed above, the appearance of the biosphere corresponds to the end of the second and the beginning of the third energy epochs—the transition to oxidative phosphorylation, when the oxygen and CO

_{2}cycle arose.

## 2. Materials and Methods

_{i}—biomasses of producers and consumers; X

_{m}—biomasses of decomposers; S

_{k}—concentrations or amounts of nutrients taking part in the cycling and taken into account in the CES description of a selected level of accuracy; and ${\alpha}_{ij}^{-}$—an analog of stoichiometric coefficient in ecology.

_{i}are velocities of flows entering the transformation node (metabolic species) and flows outgoing from it, which have different stoichiometry.

_{1}, X

_{2}) and one consumer (Y) (1). There are flows of biomass, carbon dioxide, oxygen, and water in the system.

_{1}is represented by a vector which components are stoichiometric coefficients of the form α

_{i}

_{1}.

_{1}, X

_{2}) and one consumer (Y) (1). There are flows of biomass, carbon dioxide, oxygen, and water in the system. For simplicity, we assume that dead organic matter is decomposed very quickly due to the high activity of the microbiota, whose biomass can be neglected.

_{2}, C—CO

_{2}, and H—H

_{2}O.

**A**is the matrix composed of column vectors of stoichiometric coefficients characterizing the processes carried out by the species included in the CES of stoichiometric coefficients of the system, and $\overrightarrow{V}={\left({V}_{1},{V}_{2},{V}_{3},{V}_{4}\right)}^{T}$. For the considered model to have steady state, Equation (2) linear in V

_{i}must have a solution. The system of linear homogeneous Equation (2) has a nonzero solution if and only if the rank of the matrix

**A**is less than the number of variables.

_{16}H

_{32}O

_{2}—will be used as a typical representative), lignin (cinnamic acid tetramer C

_{10}H

_{11}O

_{2}will be used as an average representative), glucose—C

_{6}H

_{12}O

_{6}, and cellulose—(C

_{6}H

_{10}O

_{5})

_{n}. These typical representatives were chosen to assess the potential level of closure of real CELSS [39].

_{16}H

_{32}O

_{2}: 16H

_{2}O + 16CO

_{2}↔ C

_{16}H

_{32}O

_{2}+ 23O

_{2}

_{10}H

_{11}O

_{2}: 22H

_{2}O + 40CO

_{2}↔ 4C

_{10}H

_{11}O

_{2}+ 47O

_{2}

_{6}H

_{12}O

_{6}: 6H

_{2}O + 6CO

_{2}↔ C

_{6}H

_{12}O

_{6}+ 6O

_{2}

_{6}H

_{10}O

_{5}: 5H

_{2}O + 6CO

_{2}↔ C

_{6}H

_{10}O

_{5}+ 6O

_{2}

_{1}corresponds to photosynthesis of biomass of Producer 1:

_{1}+ ν

_{2}= 1 and ν

_{3}+ ν

_{4}= 1.

_{1}= −0.746, ν

_{2}= −0.254, ν

_{3}= 0.385, and ν

_{4}= 0.615. As is customary, the sign in front of the stoichiometric coefficient indicates the loss or production of a substance in the reaction. We place the found stoichiometric coefficients in the matrix of Equation (2):

**A**is less than the number of variables. The rank of this matrix is 4, which means that this system has no solution.

_{i}exists, but there is no suitable ratio of steady-state variables included in the expressions for V

_{i}, for example, when they are negative. That is, the presence of a solution to this linear system does not guarantee the presence of a steady state for the model, but the absence of a solution to this system indicates the fatal absence of a steady state for the model.

## 3. Results

#### 3.1. The Feasibility of Closing the Microbial Mat System by Organisms with Rigid Metabolism

_{i}) and be described by a formula close to the Mitscherlich formula [40]. Taking into account the discussion of an adequate form of the formula describing the extinction of organisms [41], we introduce a quadratic dependence of the rate of extinction on the concentration of biomass. The dying biomass turns into several forms of detritus D

_{k}corresponding to nutrients, which are then consumed by decomposers Y

_{i}. The proportion of nutrients for each species is given by the ratio a

_{i}/b

_{i}. The decomposer mineralizes detritus by first converting it into its own biomass and then lysing the biomass of dead heterotrophic bacteria. In addition, the laws of conservation of nutrients operate in the system (A

_{i}).

_{X}) and one of the heterotrophs (μ

_{1}) at different values of the stoichiometric composition of the other heterotroph.

_{1X}= 0.3; a

_{11}= 0.2. The ranges of variations in specific growth rates were: μ

_{X}= 0 ÷ 3; μ

_{1}= 0 ÷ 10. The parametric portraits (Figure 3) show that the region of existence of a steady/quasi-steady state is not very wide, but, nevertheless, it can ensure the long-term existence of the system with noticeable variations in the stoichiometric composition of organisms resulting from the mutation process.

_{X}> 1.

_{1X}= 0.3; a

_{2X}= 0.3; a

_{3X}= 0.4; a

_{11}= 0.1; a

_{21}= 0.2; a

_{31}= 0.7; a

_{13}= 0.1; a

_{23}= 0.7; and a

_{33}= 0.2. The ranges in variations in specific growth rates were μ

_{X}= 0÷3; μ

_{1}= 0÷10.

_{11}= 0.1, a

_{21}= 0.225, a

_{31}= 0.675, the rank of matrix (9) becomes equal to 2, and there may be a steady state in the system (Figure 6A).

#### 3.2. A Flexible Metabolism Model—“Catabolic Modifier”

_{h}—carbohydrates (in detritus); $C$—carbon (mineralized, suitable for use as food for the autotroph); $N$—nitrogen (the same as carbon); and a—average concentration of ATP as an indicator of vital functions in cells of the heterotroph.

_{x}, K

_{C}, K

_{N}are Mitscherlich’s coefficients for the function of autotroph’s reproduction, taking into account two limiting substances;

_{F}, λ

_{Ch}, K

_{P}, K

_{F}, and K

_{Ch}—coefficients of the function of heterotroph’s reproduction described below.

_{XP}, β

_{XF}, β

_{XCh}, β

_{YP}, β

_{YF}, and β

_{YCh}are stoichiometric coefficients (fractions) of proteins, fats, and carbohydrates in the autotroph (X) and heterotroph (Y); γ

_{PC}

_{,}γ

_{PN}

_{,}γ

_{FC}

_{,}γ

_{FN}

_{,}γ

_{ChC}

_{,}and γ

_{ChN}are fractions of carbon and nitrogen in proteins, fats, and carbohydrates.

_{P}, α

_{F}, and α

_{Ch}. Variations in these parameters change the final consumption stoichiometry

_{i}. This independence is so complete that it makes no sense to show all the options. Figure 8 shows a parametric portrait of the model for the case of selection according to the rate of substance oxidation during catabolism. All parametric portraits are characterized by a wide region of stable existence of a steady state. The dynamics of the initial transition process is shown in Figure 9.

#### 3.3. A flexible Metabolism Model—“Internal Regulatory Pool”

_{1}and X

_{2}) and one consumer (Y) and two nutrients (for example, nitrogen and phosphorus or carbon and nitrogen, the content of which in the pools is indicated by the letters a and b with the corresponding indices, and in free form by the symbols A and B). The model has the following form:

_{i}

^{m}and b

_{i}

^{m}denote capacity limits of the pools of the i-th organism; α

_{i}and β

_{i}are stoichiometric ratios of nutrients in the biomass of the i-th organism (α

_{i}+ β

_{i}= 1); P

_{i}() and F() are the rates of biomass synthesis from the nutrients contained in the pools; k

_{i}—constants of food uptake rate.

## 4. Discussion

- (1)
- The stochastic mechanism—random selection of closing organisms (decomposers) with the corresponding stoichiometric ratios;
- (2)
- Changing the consumption stoichiometry by switching catabolic pathways to different types of substances (proteins, fats, carbohydrates);
- (3)
- Changing the consumption stoichiometry by choosing food, depending on the state of internal nutrient pools.

## 5. Conclusions

- It is shown that models based on rigid metabolism (characterized by fixed values of stoichiometric coefficients of nutrient consumption) in general case do not have a non-trivial stationary state, and therefore cannot ensure the existence of a closed ecological system for an arbitrarily long time. However, the time of the transient process can be long enough to allow the appearance of mutants capable, in principle, of utilizing the dead ends of the organic matter and returning them back to the cycle.
- Models based on so-called flexible metabolism have a stable stationary state in a wide range of parameters, and therefore are capable of describing an arbitrarily long existence of a CES.
- Based on our research, we can conclude that the Vernadsky–Darwin paradox can be resolved in nature by combining the considered mechanisms that provide both the current competitive advantage and the ability to close trophic chains with a wide variation in the composition of substance flows.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Transition processes of the model at different values of the parameters. The lower graphs demonstrate the implementation of conservation laws for chemical elements. Note that such verification of the satisfiability of conservation laws was carried out for all cases in the course of computer simulations. To save space, such graphs are not shown again in the other figures.

**Figure 2.**Dynamics of variables of a microbial mat closed system: (

**A**)—a mat with a stable steady state; (

**B**)—rapid destruction of the system after the equilibrium of nutrients and detritus is established. Here and in all subsequent figures, the lower graphs represent visual control of the conformity to the laws of nutrient conservation.

**Figure 3.**Parametric portraits of the model in coordinates of the specific growth rate of an autotroph (μ

**) and one of the heterotrophs (μ**

_{X}**) at different values of the stoichiometric composition of the second heterotroph (in the order of the portraits): a**

_{1}_{12}= 0.35, 0.50, 0.80. The integration time was 5000 relative units. That is, the state of the system was assessed upon reaching this time. Coordinates of selected points for viewing system dynamics: (

**A**) μ

_{X}= 1.5, μ

_{1}= 1.9: (

**B**) μ

_{X}= 1.5, μ

_{1}= 1.8; (

**C**) μ

_{X}= 0.2, μ

_{1}= 3.5.

**Figure 4.**Dynamics of the system corresponding to different points in the parametric portrait. The designations of the points are the same as the designations in Figure 3: (

**A**) true steady state; (

**B**) very fast destruction of the system when crossing the stability boundary; (

**C**) a long transition period.

**Figure 5.**Parametric portraits of the three-nutrient system with different stoichiometric relationships of the second decomposer: (

**A**) a

_{12}= 0.25, a

_{22}= 0.1, a

_{32}= 0.65; (

**B**) a

_{12}= 0.4, a

_{22}= 0.1, a

_{32}= 0.5; (

**C**) a

_{12}= 0.7, a

_{22}= 0.1, a

_{32}= 0.2.

**Figure 6.**Dynamics of the transition process in the three-nutrient system with a degenerate matrix (9) (non-rough case) for small deviations in the stoichiometric ratio from the singular point. The values of the variable parameter in the presence of a steady state a

_{21}= 0.225 (

**A**) and in the case of the extinction of the system a

_{21}= 0.2253 (

**B**), a

_{21}= 0.2246 (

**C**).

**Figure 7.**The parametric portrait of the system for the non-rough case of stoichiometric ratios of the second decomposer (the values correspond to Figure 5): (

**A**) a

_{12}= 0.25, a

_{22}= 0.1, a

_{32}= 0.65; (

**B**) a

_{12}= 0.4, a

_{22}= 0.1, a

_{32}= 0.5; (

**C**) a

_{12}= 0.7, a

_{22}= 0.1, a

_{32}= 0.2. The red arrow denotes the regions of the parameters corresponding to the steady state.

**Figure 8.**Parametric portraits of the system for the case of selection according to the rate of substance oxidation during catabolism. Stoichiometric ratios: (

**A**) β

_{XP}= 0.1, β

_{YP}= 0.1; (

**B**) β

_{XP}= 0.1, β

_{YP}= 0.2; (

**C**) β

_{XP}= 0.1, β

_{YP}= 0.4. Red dot on sub picture (

**B**) represents the parameter values V

_{X}= 2 and V

_{S}= 3 which are chosen to illustrate typical dynamics of the system shown on Figure 9.

**Figure 9.**The dynamics of the transition process in the “Switching Paths” model. The values of the model parameters correspond to a point on the parametric portrait in Figure 8B with the parameter values V

_{X}= 2 and V

_{S}= 3.

**Figure 10.**A parametric portrait of a model with switching pools with different stoichiometric proportions of the biomass of one of the producers (α

_{1}= 0.5, 0.7, 0.9).

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Bartsev, S.; Degermendzhi, A.
The Evolutionary Mechanism of Formation of Biosphere Closure. *Mathematics* **2023**, *11*, 3218.
https://doi.org/10.3390/math11143218

**AMA Style**

Bartsev S, Degermendzhi A.
The Evolutionary Mechanism of Formation of Biosphere Closure. *Mathematics*. 2023; 11(14):3218.
https://doi.org/10.3390/math11143218

**Chicago/Turabian Style**

Bartsev, Sergey, and Andrey Degermendzhi.
2023. "The Evolutionary Mechanism of Formation of Biosphere Closure" *Mathematics* 11, no. 14: 3218.
https://doi.org/10.3390/math11143218