- Email: [email protected]
An elementary multistage discrete model of soil organic matter transformations with a continuous scale of stability
Ecological Modelling 393 (2019) 61–65
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
An elementary multistage discrete model of soil organic matter transformations with a continuous scale of stability Sergey I. Bartseva,b, Aleksei A. Pochekutova, a b
T
⁎
Institute of Biophysics SB RAS, Federal Research Center “Krasnoyarsk Science Center SB RAS”, Krasnoyarsk, Russia Institute of Fundamental Biology and Biotechnology of Siberian Federal University, Krasnoyarsk, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Soil organic matter Model of soil organic matter transformations Kinetics of soil organic matter transformations
The proposed elementary mathematical model of formation and decomposition of soil organic matter (SOM) is based on using equations of chemical kinetics to describe the multistage process of SOM transformation. The model both describes each step of transformation in accordance with the law of mass action and postulates the trend of increasing stability of the matter towards further transformation, which is common for all steps. Analysis of the model demonstrates that it is extremely difficult to construct a realistic model of SOM dynamics by assembling elementary models of the type presented in this study into the full description of SOM transformation processes.
1. Introduction We have previously developed a simple continuous model of SOM transformations (Bartsev and Pochekutov, 2015, 2016, 2017). To keep the model as simple as possible, the principle underlying the construction of that model was to reflect only the most general basic notions of these processes: a gradual increase in SOM stability towards transformation that occurs simultaneously with decomposition of some part of SOM. These notions were presented in the model phenomenologically, as the general trends of these processes, with no detailed description of their inherent microbiological and biochemical mechanisms. The increase in SOM stability and SOM transformation into more stable forms was described as continuous movement of the matter along the scale representing the stability of the matter towards further transformation. This scale was represented by the coefficient of the rate of SOM transformation into a more stable form — h. We also proposed a way to map this rather abstract scale one-to-one onto the depth of the matter's location in the soil profile, which was more convenient scale for observation and measurements. In the basic version of the model (Bartsev and Pochekutov, 2015), SOM movement along scale h is described by the transport equation. Hence, all SOM in the model characterized at this time point by a given value of h will move along the scale simultaneously and at the same velocity. This description appears too simplified, as it misses the fact that, according to the laws of chemical kinetics, at any instant of time, only part of the matter is undergoing the reaction of transformation to
⁎
the next form while another part of this matter has not been involved in this reaction yet. Moreover, some part of SOM may be physically inaccessible to decomposing microorganisms and their enzymes and may remain so for some time. To include this in the model, we had to complicate it by introducing additional terms into its basic equation and by adding another equation, which expressed dynamics of the part of the matter that had not been involved in the reaction. In that form, the model equations had an accurate analytical solution only in the stationary case. By using that complicated form of model equations, we managed to obtain a good quantitative fit between model calculations of stationary distributions of SOM and the distributions actually observed in the soil (Bartsev and Pochekutov, 2017). In this study, we have attempted to construct the simplest possible model of the gradual transformation of SOM to more stable forms occurring simultaneously with decomposition of some part of SOM. In this model, to describe the multistage transformation of SOM, we initially use explicit equations of chemical kinetics as basic equations of the model. To reveal advantages and disadvantages of this approach over the description of SOM transformation by the transport equation, which we used previously, the model of transformation is derived and discussed in the simplest possible schematic form. 2. Basic model equations However complex SOM composition is, it consists of definite
Corresponding author. E-mail addresses: [email protected] (S.I. Bartsev), [email protected] (A.A. Pochekutov).
https://doi.org/10.1016/j.ecolmodel.2018.12.012 Received 21 September 2018; Received in revised form 12 December 2018; Accepted 13 December 2018 0304-3800/ © 2018 Elsevier B.V. All rights reserved.
Ecological Modelling 393 (2019) 61–65
S.I. Bartsev, A.A. Pochekutov ∞
chemical compounds. Organic matter transformation in soil is a succession of transformations of these chemical compounds. Each reaction of transformation of one substance into another is described by the law of mass action. The model derived below is intended to provide an elementary demonstration of how the law of mass action works when SOM stability increases during the course of multistage chemical transformations. For this, we model the process of stability increase using an elementary example of one substance. The entire process of organic matter transformation is described as passing through a discrete succession of transformations. Each of the transformations is characterized by its own value of reaction rate coefficient, hj (index j = 0, 1, 2, … numbers transformations). The succession of transformations is generally directed towards increasing SOM stability. Although less stable SOM compounds can be transformed into more stable ones in different ways, for the elementary model we consider one chain of chemical transformations of one organic compound, which enters the soil as a component of plant litter. Then, each jth organic compound in soil will be matched by a definite value of hj of its transformation into the next compound. Thus, a discrete set of hj values placed along the common continuous scale, h, will describe the succession of SOM transformations. Let us remind you that h corresponds to stability of the matter towards further transformation. Then, the general process of SOM transformation will be described in the model as the succession of reactions of transformation of the jth organic compound into the (j + 1)th, with hj > hj+1 for each j. The model should also take into account that while part of the matter is being transformed into a more stable compound, another part of this matter is undergoing mineralization — decomposition to CO2, which leaves the soil. The assumption that SOM stability gradually increases also means that the coefficient of mineralization rate, kj, decreases with increasing j. Then SOM transformation dynamics will be described by a system of ordinary differential equations: dM0 (t ) dt dMj (t ) dt
S=
3. General pattern of the transformation rate decrease To make calculations using formulas (2), (3) or (4), one should determine the values of hj and kj for each number of the transformation stage j. As there are very many SOM transformation stages, and it is hard to determine even the exact chemical structure of SOM formed at each stage, it is impossible to measure kinetic coefficients for every stage. For this reason and striving to keep the model as simple as possible, we try to set common (for all SOM compounds and all transformation stages) empirical dependencies of changes in SOM stability, based on some general considerations. Each step of SOM transformation is one chemical reaction. It changes the value of h by some quantity, which cannot be infinitely small. That is why, the change of h from stage to stage must be discrete. The fundamental discreteness of the hj − hj+1 intervals, which must be considered as finite quantities, is determined by the differences between chemical-kinetic properties of the compounds in these stages. One chemical reaction changes only one bond. Supposing that as SOM stability increases, its structure becomes more complex, and a change in one bond in the more complex SOM molecule in the later transformation stage leads to a less considerable change in its kinetic properties than a change in one bond in the less complex molecule in an earlier stage, we reach the conclusion that the interval between hj and hj+1 must decrease with an increase in the number of j. The type of this decrease must ensure the convergence of series (4) and, at the same time, be mathematically simple, and its parameters are adjustable parameters in the model. As the simplest function that, as we will show below, satisfies these requirements, we use
1 ⎞ hj + 1 = hj ·⎛⎜1 − ⎟, j + ρ⎠ ⎝
j−1
j
∑ l=0
1 − exp(−Hl t ) , Hl ∏ij= 0; (Hl − Hi ) i≠l
(1)
hj = h 0
⎞ ⎛ Mj = D ⎜∏ hi ⎟ = i 0 ⎠ ⎝
ρ−1 . ρ+j−1
kj = bhpj ,
(7)
where b and p are adjustable parameters, b > 0, 0 < p < 1. Inserting (6) and (7) into (2) or (3), we can determine the amount of SOM at each jth stage of transformation and the total SOM from Eq. (4). Note that the ratio
(2)
Mj + 1 Mj
=
1 j + 1 −1 ρ−1
(1 + ) ⎛⎝ (1 + ) j ρ−1
j
⎞ ⎛ ⎜∏ Hi⎟. = i 0 ⎠ ⎝
(6)
Here ρ is an adjustable parameter, and ρ > 1. Note that for different substances of plant litter, which are characterized by different values of h0, the sequences of hj values for all subsequent transformation stages in the general case will not be the same, as is clear from (6). As the empirical dependence of the relationship between hj and kj, we use the function k(h) = bhp, which was previously used in the continuous model (Bartsev and Pochekutov, 2015, 2016, 2017):
where Hi ≡ hi + ki. Setting all derivatives in (1) equal to zero, we obtain a system of algebraic equations describing SOM in the stationary state. For an arbitrary number of j, it has the following solution: j−1
(5)
whence it follows that
where Mj(t) is the amount of SOM in the form of the jth compound, with the matter that directly enters the soil as litter taken as the 0th compound, t is time, and D is the litter input rate. All equations of system (1) have analytical solution. By solving them sequentially, beginning with the 0th, and inserting the solution into the following equation, we obtain a set of Mj(t) functions, which are generalized for the arbitrary j as
⎞ ⎛ Mj (t ) = (−1) jD ⎜∏ hi ⎟ ⎝ i=0 ⎠
(4)
j=0
= D − h 0 M0 (t ) − k 0 M0 (t ), = hj − 1 Mj − 1 (t ) − hj Mj (t ) − kj Mj (t ) for j ≥ 1,
∑ Mj.
+
b h 01 − p
j + 1 −p ⎞ ρ−1
(1 + )
⎠
(8)
at small j may be either above or below 1, but at j ≫ 1, it is always below 1. This suggests that Mj may either decrease or increase with an increase in j when j is small, but at large j, Mj necessarily decreases monotonically. Setting in (8) j + 1 ≈ j, at large j, the Mj+1/Mj ratio decreases monotonically with an increase in j.
(3)
Note that Mj(t) is not an alternating sequence: multiplier (− 1) in the odd degree compensates the sign of the odd number of negative cofactors in the denominator. All Mj(t) are positive, increase monotonically with increasing t, and Mj(t) → Mj at t→ ∞. The total amount of SOM formed from this substance will be expressed as the following series:
lim
j →∞
62
Mj + 1 Mj
= 0.
(9)
Ecological Modelling 393 (2019) 61–65
S.I. Bartsev, A.A. Pochekutov
Theoretically, determination of k0 values of different chemical compounds could be performed as litterbag experiment, which involves chemical analysis of the composition of each litter component and measurement of the mass loss of each relevant chemical compound. In fact, however, some of the compounds in the litter sample may both decrease and increase — because they may be synthesized by microorganisms decomposing other compounds, which was experimentally found for cellulose (Talbot and Treseder, 2012). Hemicellulose may be expected to behave in a similar fashion, as it can be synthesized by bacteria as part of their cell walls. Thus, litterbag experiment is not a reliable way to obtain accurate k0 values of individual compounds. The model described above is knowingly oversimplified, as it is intended to be a starting point for constructing a more realistic model. For the model to be closer to the description of real transformations of organic matter in soil, it should take into account interactions between chemical compounds during transformations. These are primarily the well-known interactions between polysaccharides and lignin, in which polysaccharides are co-substrate for lignin decomposition, and lignin protects cellulose and hemicellulose against decomposition both physically and chemically (Talbot et al., 2012; Talbot and Treseder, 2012). Even better approximation could be achieved by taking into account that lignin may consist of various monomer types, which differ in both the effectiveness of the chemical protection of polysaccharides and their own stability (Talbot et al., 2012). Then, the model can be presented as a combination of transformation chains of various substances that intricately interact with each other. This model will consist of both firstorder and second-order equations of chemical kinetics. Taking into consideration the difficulty of determining parameters (even for the elementary model) mentioned above, it seems questionable that such a model could be constructed and used. Nevertheless, the model equations derived above demonstrate realism of the basic properties of model behavior. Namely, the model (with any values of the parameters that fall within the limits specified when it was derived) predicts accumulation of some stationary nonzero and finite amount of SOM caused by transformation of plant litter, asymptotic approaching of nonstationary values of the amounts of SOM components to their stationary values, and accumulation of SOM, which is more stable than the litter matter (Fig. 2). The presented model demonstrates only the minimum level of requirements of detail in litter description which arise when the model is based on an explicit description of the multistage litter transformation by the equations of chemical kinetics. But this minimum level is already too high, when we try to describe the real plant litter transformation processes. Since the model is a simplified idealization which reflects only one elementary fragment of the real SOM transformations process, then it is necessary to use also chemically simplified system for an experimental testing of the model. In these experiments, chemically homogeneous samples consisting of one organic chemical compound which can be decomposed but cannot be synthesized by soil micro- and mesoorganisms, should be exposed in the soil instead of samples of real plant litter. An observation of mass loss dynamics of the samples would make possible to evaluate the parameter h0 required for the model calculation. As a primary test of the model, a comparison between experimental and model calculated total amounts of new accumulated SOM can be proposed. A more advanced test of the model can be carried out by a mass spectrometric analysis detecting an appearance of new organic compounds in the soil and measuring their quantities and molecular masses. According the model, SOM compounds following each other in the order of increasing their numbers j analogously follow in order of increasing molecular mass. It would make possible to determine experimentally a sequence of Mj values and to compare them with the model calculations. An integrated approach, which takes into account all organic compounds in the litter, appears to be a better choice. For using this approach, it is enough to have experimental data on the litter, namely, the measured litter input rates, percentages of its major components (such
That is, series (4) converges according to D’Alembert's test. Note that if, instead of (5), we use a simpler relation, Mj + 1 hj+1 = hj · (1 − q), where q = const, 0 < q < 1, then lim M = ∞, j →∞
j
i.e. series (4) is divergent according to D’Alembert's test. Therefore, relation (5) can be considered as the simplest, ensuring series (4) convergence. To describe real soils, it makes sense to use only the first n terms of the sequence Mj, limiting n so as to make the remainder of the series ∞ Rn = ∑ j = n Mj negligibly small. Rn at n ≫ 1 can be quantified by using the inequality ensuing from the series convergence according to D’Alembert's test and the monotonicity of the decrease in Mj+1/Mj with the increase in j at j ≫ 1:
Rn ≤
Mn2 . Mn − Mn + 1
(10)
Then, evaluation of the total SOM with any required degree of accuracy depending on the choice of n can be calculated as n−1
Sn ≈
∑ Mj. j=0
(11)
Fig. 1 shows an example of the dynamics of accumulation of compounds formed in different stages of transformation of one plant litter substance, Mj(t), for several different j. Fig. 2 shows stocks of all products of transformation of one litter substance formed at different time points. 4. Discussion To be able to use the model to describe actual processes, one should know the values of its parameters. Determination of parameters b, p, and ρ, which are specific for the entire ecosystem, could be realized as an experiment with a sample of pure substance (e.g. cellulose) decomposed in soil. Having measured the curve of the sample mass loss with time, one can choose the values of b, p, and ρ to achieve the smallest possible discrepancy between the model curve, S(t) = ∑Mj(t), and the actual curve of the sample mass loss. To be able to use the model to describe transformations of real plant litter, one should know values of D and h0 for all significant chemical compounds constituting the litter. Taking into account (7), h0 can be determined using the value of the initial mineralization rate of chemical compound k0 found experimentally. However, the k0 value of even the same chemical compound must differ in different litter components (such as leaves, roots, or wood), as the availability of this compound to microorganisms, enzymes, and other factors causing SOM transformation is not the same.
Fig. 1. The dynamics of accumulation of transformation products of one litter chemical compound. Index j numbers compounds formed as a result of jth stage of transformation. D = 1.0 kgC/m2, k0 = 0.2 yr−1, b = 0.3, p = 0.8, ρ = 1.5. 63
Ecological Modelling 393 (2019) 61–65
S.I. Bartsev, A.A. Pochekutov
Fig. 2. Distributions of product amounts formed from one chemical compound at different durations of the transformation process. The heights of vertical lines designate Mj(t) the amounts of transformation products formed at jth stage of the process. Index j sequentially increases from right to left, beginning from j = 0. D = 1.0 kgC/m2, k0 = 0.2 yr−1, b = 0.3, p = 0.8, ρ = 1.5.
SOM, as all SOM components are transformed simultaneously and under continuously varying environmental conditions. (For instance, initial rates of transformation of the same substances must differ between each other depending on their availability for decomposing organisms, which in turn depends on how close the substance is to the surface of the plant parts in the litter. This dependence is continuous, as the distance from some point in the midst of plant litter to the surface contacting with the environment can be regarded as continuously changing value, depending on what point has been chosen. Continuousness of differences between initial transformation rates entails continuousness of differences in transformation rates in each following step.) Thus, the integrated process becomes actually continuous.
as leaves, roots, and wood), and their mass losses in the early phase of transformation (as the basis for estimating the average coefficients of mineralization rate for each component). Integrated approach, in combination with equations of chemical kinetics, is used to construct multi-compartment models. In these models, the entire SOM (as well as litter) is divided into a discrete set of compartments, each of them comprising a large class of compounds. Then, SOM transformation is described by kinetic equations of transmission of the substance between compartments with reaction rate constants being average for all substances in the compartment (e.g. Chertov et al., 2001). These models are rightly criticized for the rough description of SOM by using a small discrete set of compartments and for the arbitrary subdivision of the substances into the compartments. These drawbacks can be avoided by using the continuous approach, where the passing through all stages of multistage SOM transformation is described as SOM movement along the continuous scale, which, by some means or other, reflects decomposability or stability of the matter (Carpenter, 1981; Agren and Bosatta, 1998). Continuous models differ in the choice of the scale and formulation of the principle of the substance movement along the scale, which is common for the entire scale. This approach underlies the continuous model that we proposed previously (Bartsev and Pochekutov, 2017) if the unit of the transformed substance in it is represented not by an individual molecule but rather by a certain elementary portion (aliquot) of a litter component or SOM at a certain transformation stage. The movement of this aliquot along the stability scale must be thought of as a combination of all changes occurring in it: chemical transformations of some of the molecules of its constituent substances, changes in the proportions of the more stable and less stable substances in the sample, and changes in its availability for decomposing organisms and enzymes. The intrinsic discreteness of chemical transformations is not an obstacle to the continuous description of transformation of the entire
5. Conclusion This work is primarily a methodological study. We propose a model of the basic process in formation of SOM — an elementary transformation chain of one compound. The study shows that it gives qualitatively correct results, demonstrating accumulation of SOM, which is more stable than the litter matter but not infinitely stable, asymptotic approaching of SOM amounts in all transformation stages to stationary values with litter input being continuous, and accumulation of the nonzero finite total amount of SOM. Thus, the notions included in the model are accurate. Science generally uses elementary models as starting points to construct complex models, which describe real objects. Elementary models are usually extremely simple, and they describe nonexistent ideal objects such as material point, ideal gas, perfect black body, etc. In spite of their perfect simplicity, they provide an adequate approximate description of a number of physical systems (Rashevsky, 1938, pp. 1–3). Unfortunately, for even a simplified description of complex biochemical systems such as soil, a large number of elementary models 64
Ecological Modelling 393 (2019) 61–65
S.I. Bartsev, A.A. Pochekutov
should be used simultaneously. The contradiction between the need for a simple and transparent description of the SOM dynamics and the need to describe the transformation of a large number of compounds even in the elementary model can be resolved by a phenomenological description of the system dynamics as a whole. This description should empirically reflect the final result of the soil processes without detailed description of internal mechanisms. Thus, the present study is another argument in favor of using integrated continuous models to describe SOM dynamics.
transformations based on a scale of transformation rate. Ecol. Model. 302, 25–28. Bartsev, S.I., Pochekutov, A.A., 2016. The vertical distribution of soil organic matter predicted by a simple continuous model of soil organic matter transformations. Ecol. Model. 328, 95–98. Bartsev, S.I., Pochekutov, A.A., 2017. Quantitative description of vertical organic matter distribution in real soil profiles by means a simple continuous model. Ecol. Model. 360, 219–222. Carpenter, S.R., 1981. Decay of heterogenous detritus: A general model. J. Theor. Biol. 89, 539–547. Chertov, O.G., Komarov, A.S., Nadporozhskaya, M.A., Bykhovets, S.S., Zudin, S.L., 2001. Romul – a model of forest soil organic matter dynamics as a substantial tool for forest ecosystem modelling. Ecol. Model. 138, 289–308. Rashevsky, N., 1938. Mathematical Biophysics: Physicomathematical Foundations of Biology. University of Chicago Press, Chicago. Talbot, J.M., Treseder, K.K., 2012. Interactions among lignin, cellulose, and nitrogen drive litter chemistry-decay relationships. Ecology 93, 345–354. Talbot, J.M., Yelle, D.J., Nowick, J., Treseder, K.K., 2012. Litter decay rates are determined by lignin chemistry. Biogeochemistry 108, 279–295.
References Ågren, G.I., Bosatta, E., 1998. Theoretical Ecosystem Ecology: Understanding Element Cycles. Cambridge University Press, Cambridge. Bartsev, S.I., Pochekutov, A.A., 2015. A continual model of soil organic matter
65
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
An elementary multistage discrete model of soil organic matter transformations with a continuous scale of stability Sergey I. Bartseva,b, Aleksei A. Pochekutova, a b
T
⁎
Institute of Biophysics SB RAS, Federal Research Center “Krasnoyarsk Science Center SB RAS”, Krasnoyarsk, Russia Institute of Fundamental Biology and Biotechnology of Siberian Federal University, Krasnoyarsk, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Soil organic matter Model of soil organic matter transformations Kinetics of soil organic matter transformations
The proposed elementary mathematical model of formation and decomposition of soil organic matter (SOM) is based on using equations of chemical kinetics to describe the multistage process of SOM transformation. The model both describes each step of transformation in accordance with the law of mass action and postulates the trend of increasing stability of the matter towards further transformation, which is common for all steps. Analysis of the model demonstrates that it is extremely difficult to construct a realistic model of SOM dynamics by assembling elementary models of the type presented in this study into the full description of SOM transformation processes.
1. Introduction We have previously developed a simple continuous model of SOM transformations (Bartsev and Pochekutov, 2015, 2016, 2017). To keep the model as simple as possible, the principle underlying the construction of that model was to reflect only the most general basic notions of these processes: a gradual increase in SOM stability towards transformation that occurs simultaneously with decomposition of some part of SOM. These notions were presented in the model phenomenologically, as the general trends of these processes, with no detailed description of their inherent microbiological and biochemical mechanisms. The increase in SOM stability and SOM transformation into more stable forms was described as continuous movement of the matter along the scale representing the stability of the matter towards further transformation. This scale was represented by the coefficient of the rate of SOM transformation into a more stable form — h. We also proposed a way to map this rather abstract scale one-to-one onto the depth of the matter's location in the soil profile, which was more convenient scale for observation and measurements. In the basic version of the model (Bartsev and Pochekutov, 2015), SOM movement along scale h is described by the transport equation. Hence, all SOM in the model characterized at this time point by a given value of h will move along the scale simultaneously and at the same velocity. This description appears too simplified, as it misses the fact that, according to the laws of chemical kinetics, at any instant of time, only part of the matter is undergoing the reaction of transformation to
⁎
the next form while another part of this matter has not been involved in this reaction yet. Moreover, some part of SOM may be physically inaccessible to decomposing microorganisms and their enzymes and may remain so for some time. To include this in the model, we had to complicate it by introducing additional terms into its basic equation and by adding another equation, which expressed dynamics of the part of the matter that had not been involved in the reaction. In that form, the model equations had an accurate analytical solution only in the stationary case. By using that complicated form of model equations, we managed to obtain a good quantitative fit between model calculations of stationary distributions of SOM and the distributions actually observed in the soil (Bartsev and Pochekutov, 2017). In this study, we have attempted to construct the simplest possible model of the gradual transformation of SOM to more stable forms occurring simultaneously with decomposition of some part of SOM. In this model, to describe the multistage transformation of SOM, we initially use explicit equations of chemical kinetics as basic equations of the model. To reveal advantages and disadvantages of this approach over the description of SOM transformation by the transport equation, which we used previously, the model of transformation is derived and discussed in the simplest possible schematic form. 2. Basic model equations However complex SOM composition is, it consists of definite
Corresponding author. E-mail addresses: [email protected] (S.I. Bartsev), [email protected] (A.A. Pochekutov).
https://doi.org/10.1016/j.ecolmodel.2018.12.012 Received 21 September 2018; Received in revised form 12 December 2018; Accepted 13 December 2018 0304-3800/ © 2018 Elsevier B.V. All rights reserved.
Ecological Modelling 393 (2019) 61–65
S.I. Bartsev, A.A. Pochekutov ∞
chemical compounds. Organic matter transformation in soil is a succession of transformations of these chemical compounds. Each reaction of transformation of one substance into another is described by the law of mass action. The model derived below is intended to provide an elementary demonstration of how the law of mass action works when SOM stability increases during the course of multistage chemical transformations. For this, we model the process of stability increase using an elementary example of one substance. The entire process of organic matter transformation is described as passing through a discrete succession of transformations. Each of the transformations is characterized by its own value of reaction rate coefficient, hj (index j = 0, 1, 2, … numbers transformations). The succession of transformations is generally directed towards increasing SOM stability. Although less stable SOM compounds can be transformed into more stable ones in different ways, for the elementary model we consider one chain of chemical transformations of one organic compound, which enters the soil as a component of plant litter. Then, each jth organic compound in soil will be matched by a definite value of hj of its transformation into the next compound. Thus, a discrete set of hj values placed along the common continuous scale, h, will describe the succession of SOM transformations. Let us remind you that h corresponds to stability of the matter towards further transformation. Then, the general process of SOM transformation will be described in the model as the succession of reactions of transformation of the jth organic compound into the (j + 1)th, with hj > hj+1 for each j. The model should also take into account that while part of the matter is being transformed into a more stable compound, another part of this matter is undergoing mineralization — decomposition to CO2, which leaves the soil. The assumption that SOM stability gradually increases also means that the coefficient of mineralization rate, kj, decreases with increasing j. Then SOM transformation dynamics will be described by a system of ordinary differential equations: dM0 (t ) dt dMj (t ) dt
S=
3. General pattern of the transformation rate decrease To make calculations using formulas (2), (3) or (4), one should determine the values of hj and kj for each number of the transformation stage j. As there are very many SOM transformation stages, and it is hard to determine even the exact chemical structure of SOM formed at each stage, it is impossible to measure kinetic coefficients for every stage. For this reason and striving to keep the model as simple as possible, we try to set common (for all SOM compounds and all transformation stages) empirical dependencies of changes in SOM stability, based on some general considerations. Each step of SOM transformation is one chemical reaction. It changes the value of h by some quantity, which cannot be infinitely small. That is why, the change of h from stage to stage must be discrete. The fundamental discreteness of the hj − hj+1 intervals, which must be considered as finite quantities, is determined by the differences between chemical-kinetic properties of the compounds in these stages. One chemical reaction changes only one bond. Supposing that as SOM stability increases, its structure becomes more complex, and a change in one bond in the more complex SOM molecule in the later transformation stage leads to a less considerable change in its kinetic properties than a change in one bond in the less complex molecule in an earlier stage, we reach the conclusion that the interval between hj and hj+1 must decrease with an increase in the number of j. The type of this decrease must ensure the convergence of series (4) and, at the same time, be mathematically simple, and its parameters are adjustable parameters in the model. As the simplest function that, as we will show below, satisfies these requirements, we use
1 ⎞ hj + 1 = hj ·⎛⎜1 − ⎟, j + ρ⎠ ⎝
j−1
j
∑ l=0
1 − exp(−Hl t ) , Hl ∏ij= 0; (Hl − Hi ) i≠l
(1)
hj = h 0
⎞ ⎛ Mj = D ⎜∏ hi ⎟ = i 0 ⎠ ⎝
ρ−1 . ρ+j−1
kj = bhpj ,
(7)
where b and p are adjustable parameters, b > 0, 0 < p < 1. Inserting (6) and (7) into (2) or (3), we can determine the amount of SOM at each jth stage of transformation and the total SOM from Eq. (4). Note that the ratio
(2)
Mj + 1 Mj
=
1 j + 1 −1 ρ−1
(1 + ) ⎛⎝ (1 + ) j ρ−1
j
⎞ ⎛ ⎜∏ Hi⎟. = i 0 ⎠ ⎝
(6)
Here ρ is an adjustable parameter, and ρ > 1. Note that for different substances of plant litter, which are characterized by different values of h0, the sequences of hj values for all subsequent transformation stages in the general case will not be the same, as is clear from (6). As the empirical dependence of the relationship between hj and kj, we use the function k(h) = bhp, which was previously used in the continuous model (Bartsev and Pochekutov, 2015, 2016, 2017):
where Hi ≡ hi + ki. Setting all derivatives in (1) equal to zero, we obtain a system of algebraic equations describing SOM in the stationary state. For an arbitrary number of j, it has the following solution: j−1
(5)
whence it follows that
where Mj(t) is the amount of SOM in the form of the jth compound, with the matter that directly enters the soil as litter taken as the 0th compound, t is time, and D is the litter input rate. All equations of system (1) have analytical solution. By solving them sequentially, beginning with the 0th, and inserting the solution into the following equation, we obtain a set of Mj(t) functions, which are generalized for the arbitrary j as
⎞ ⎛ Mj (t ) = (−1) jD ⎜∏ hi ⎟ ⎝ i=0 ⎠
(4)
j=0
= D − h 0 M0 (t ) − k 0 M0 (t ), = hj − 1 Mj − 1 (t ) − hj Mj (t ) − kj Mj (t ) for j ≥ 1,
∑ Mj.
+
b h 01 − p
j + 1 −p ⎞ ρ−1
(1 + )
⎠
(8)
at small j may be either above or below 1, but at j ≫ 1, it is always below 1. This suggests that Mj may either decrease or increase with an increase in j when j is small, but at large j, Mj necessarily decreases monotonically. Setting in (8) j + 1 ≈ j, at large j, the Mj+1/Mj ratio decreases monotonically with an increase in j.
(3)
Note that Mj(t) is not an alternating sequence: multiplier (− 1) in the odd degree compensates the sign of the odd number of negative cofactors in the denominator. All Mj(t) are positive, increase monotonically with increasing t, and Mj(t) → Mj at t→ ∞. The total amount of SOM formed from this substance will be expressed as the following series:
lim
j →∞
62
Mj + 1 Mj
= 0.
(9)
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Theoretically, determination of k0 values of different chemical compounds could be performed as litterbag experiment, which involves chemical analysis of the composition of each litter component and measurement of the mass loss of each relevant chemical compound. In fact, however, some of the compounds in the litter sample may both decrease and increase — because they may be synthesized by microorganisms decomposing other compounds, which was experimentally found for cellulose (Talbot and Treseder, 2012). Hemicellulose may be expected to behave in a similar fashion, as it can be synthesized by bacteria as part of their cell walls. Thus, litterbag experiment is not a reliable way to obtain accurate k0 values of individual compounds. The model described above is knowingly oversimplified, as it is intended to be a starting point for constructing a more realistic model. For the model to be closer to the description of real transformations of organic matter in soil, it should take into account interactions between chemical compounds during transformations. These are primarily the well-known interactions between polysaccharides and lignin, in which polysaccharides are co-substrate for lignin decomposition, and lignin protects cellulose and hemicellulose against decomposition both physically and chemically (Talbot et al., 2012; Talbot and Treseder, 2012). Even better approximation could be achieved by taking into account that lignin may consist of various monomer types, which differ in both the effectiveness of the chemical protection of polysaccharides and their own stability (Talbot et al., 2012). Then, the model can be presented as a combination of transformation chains of various substances that intricately interact with each other. This model will consist of both firstorder and second-order equations of chemical kinetics. Taking into consideration the difficulty of determining parameters (even for the elementary model) mentioned above, it seems questionable that such a model could be constructed and used. Nevertheless, the model equations derived above demonstrate realism of the basic properties of model behavior. Namely, the model (with any values of the parameters that fall within the limits specified when it was derived) predicts accumulation of some stationary nonzero and finite amount of SOM caused by transformation of plant litter, asymptotic approaching of nonstationary values of the amounts of SOM components to their stationary values, and accumulation of SOM, which is more stable than the litter matter (Fig. 2). The presented model demonstrates only the minimum level of requirements of detail in litter description which arise when the model is based on an explicit description of the multistage litter transformation by the equations of chemical kinetics. But this minimum level is already too high, when we try to describe the real plant litter transformation processes. Since the model is a simplified idealization which reflects only one elementary fragment of the real SOM transformations process, then it is necessary to use also chemically simplified system for an experimental testing of the model. In these experiments, chemically homogeneous samples consisting of one organic chemical compound which can be decomposed but cannot be synthesized by soil micro- and mesoorganisms, should be exposed in the soil instead of samples of real plant litter. An observation of mass loss dynamics of the samples would make possible to evaluate the parameter h0 required for the model calculation. As a primary test of the model, a comparison between experimental and model calculated total amounts of new accumulated SOM can be proposed. A more advanced test of the model can be carried out by a mass spectrometric analysis detecting an appearance of new organic compounds in the soil and measuring their quantities and molecular masses. According the model, SOM compounds following each other in the order of increasing their numbers j analogously follow in order of increasing molecular mass. It would make possible to determine experimentally a sequence of Mj values and to compare them with the model calculations. An integrated approach, which takes into account all organic compounds in the litter, appears to be a better choice. For using this approach, it is enough to have experimental data on the litter, namely, the measured litter input rates, percentages of its major components (such
That is, series (4) converges according to D’Alembert's test. Note that if, instead of (5), we use a simpler relation, Mj + 1 hj+1 = hj · (1 − q), where q = const, 0 < q < 1, then lim M = ∞, j →∞
j
i.e. series (4) is divergent according to D’Alembert's test. Therefore, relation (5) can be considered as the simplest, ensuring series (4) convergence. To describe real soils, it makes sense to use only the first n terms of the sequence Mj, limiting n so as to make the remainder of the series ∞ Rn = ∑ j = n Mj negligibly small. Rn at n ≫ 1 can be quantified by using the inequality ensuing from the series convergence according to D’Alembert's test and the monotonicity of the decrease in Mj+1/Mj with the increase in j at j ≫ 1:
Rn ≤
Mn2 . Mn − Mn + 1
(10)
Then, evaluation of the total SOM with any required degree of accuracy depending on the choice of n can be calculated as n−1
Sn ≈
∑ Mj. j=0
(11)
Fig. 1 shows an example of the dynamics of accumulation of compounds formed in different stages of transformation of one plant litter substance, Mj(t), for several different j. Fig. 2 shows stocks of all products of transformation of one litter substance formed at different time points. 4. Discussion To be able to use the model to describe actual processes, one should know the values of its parameters. Determination of parameters b, p, and ρ, which are specific for the entire ecosystem, could be realized as an experiment with a sample of pure substance (e.g. cellulose) decomposed in soil. Having measured the curve of the sample mass loss with time, one can choose the values of b, p, and ρ to achieve the smallest possible discrepancy between the model curve, S(t) = ∑Mj(t), and the actual curve of the sample mass loss. To be able to use the model to describe transformations of real plant litter, one should know values of D and h0 for all significant chemical compounds constituting the litter. Taking into account (7), h0 can be determined using the value of the initial mineralization rate of chemical compound k0 found experimentally. However, the k0 value of even the same chemical compound must differ in different litter components (such as leaves, roots, or wood), as the availability of this compound to microorganisms, enzymes, and other factors causing SOM transformation is not the same.
Fig. 1. The dynamics of accumulation of transformation products of one litter chemical compound. Index j numbers compounds formed as a result of jth stage of transformation. D = 1.0 kgC/m2, k0 = 0.2 yr−1, b = 0.3, p = 0.8, ρ = 1.5. 63
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Fig. 2. Distributions of product amounts formed from one chemical compound at different durations of the transformation process. The heights of vertical lines designate Mj(t) the amounts of transformation products formed at jth stage of the process. Index j sequentially increases from right to left, beginning from j = 0. D = 1.0 kgC/m2, k0 = 0.2 yr−1, b = 0.3, p = 0.8, ρ = 1.5.
SOM, as all SOM components are transformed simultaneously and under continuously varying environmental conditions. (For instance, initial rates of transformation of the same substances must differ between each other depending on their availability for decomposing organisms, which in turn depends on how close the substance is to the surface of the plant parts in the litter. This dependence is continuous, as the distance from some point in the midst of plant litter to the surface contacting with the environment can be regarded as continuously changing value, depending on what point has been chosen. Continuousness of differences between initial transformation rates entails continuousness of differences in transformation rates in each following step.) Thus, the integrated process becomes actually continuous.
as leaves, roots, and wood), and their mass losses in the early phase of transformation (as the basis for estimating the average coefficients of mineralization rate for each component). Integrated approach, in combination with equations of chemical kinetics, is used to construct multi-compartment models. In these models, the entire SOM (as well as litter) is divided into a discrete set of compartments, each of them comprising a large class of compounds. Then, SOM transformation is described by kinetic equations of transmission of the substance between compartments with reaction rate constants being average for all substances in the compartment (e.g. Chertov et al., 2001). These models are rightly criticized for the rough description of SOM by using a small discrete set of compartments and for the arbitrary subdivision of the substances into the compartments. These drawbacks can be avoided by using the continuous approach, where the passing through all stages of multistage SOM transformation is described as SOM movement along the continuous scale, which, by some means or other, reflects decomposability or stability of the matter (Carpenter, 1981; Agren and Bosatta, 1998). Continuous models differ in the choice of the scale and formulation of the principle of the substance movement along the scale, which is common for the entire scale. This approach underlies the continuous model that we proposed previously (Bartsev and Pochekutov, 2017) if the unit of the transformed substance in it is represented not by an individual molecule but rather by a certain elementary portion (aliquot) of a litter component or SOM at a certain transformation stage. The movement of this aliquot along the stability scale must be thought of as a combination of all changes occurring in it: chemical transformations of some of the molecules of its constituent substances, changes in the proportions of the more stable and less stable substances in the sample, and changes in its availability for decomposing organisms and enzymes. The intrinsic discreteness of chemical transformations is not an obstacle to the continuous description of transformation of the entire
5. Conclusion This work is primarily a methodological study. We propose a model of the basic process in formation of SOM — an elementary transformation chain of one compound. The study shows that it gives qualitatively correct results, demonstrating accumulation of SOM, which is more stable than the litter matter but not infinitely stable, asymptotic approaching of SOM amounts in all transformation stages to stationary values with litter input being continuous, and accumulation of the nonzero finite total amount of SOM. Thus, the notions included in the model are accurate. Science generally uses elementary models as starting points to construct complex models, which describe real objects. Elementary models are usually extremely simple, and they describe nonexistent ideal objects such as material point, ideal gas, perfect black body, etc. In spite of their perfect simplicity, they provide an adequate approximate description of a number of physical systems (Rashevsky, 1938, pp. 1–3). Unfortunately, for even a simplified description of complex biochemical systems such as soil, a large number of elementary models 64
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should be used simultaneously. The contradiction between the need for a simple and transparent description of the SOM dynamics and the need to describe the transformation of a large number of compounds even in the elementary model can be resolved by a phenomenological description of the system dynamics as a whole. This description should empirically reflect the final result of the soil processes without detailed description of internal mechanisms. Thus, the present study is another argument in favor of using integrated continuous models to describe SOM dynamics.
transformations based on a scale of transformation rate. Ecol. Model. 302, 25–28. Bartsev, S.I., Pochekutov, A.A., 2016. The vertical distribution of soil organic matter predicted by a simple continuous model of soil organic matter transformations. Ecol. Model. 328, 95–98. Bartsev, S.I., Pochekutov, A.A., 2017. Quantitative description of vertical organic matter distribution in real soil profiles by means a simple continuous model. Ecol. Model. 360, 219–222. Carpenter, S.R., 1981. Decay of heterogenous detritus: A general model. J. Theor. Biol. 89, 539–547. Chertov, O.G., Komarov, A.S., Nadporozhskaya, M.A., Bykhovets, S.S., Zudin, S.L., 2001. Romul – a model of forest soil organic matter dynamics as a substantial tool for forest ecosystem modelling. Ecol. Model. 138, 289–308. Rashevsky, N., 1938. Mathematical Biophysics: Physicomathematical Foundations of Biology. University of Chicago Press, Chicago. Talbot, J.M., Treseder, K.K., 2012. Interactions among lignin, cellulose, and nitrogen drive litter chemistry-decay relationships. Ecology 93, 345–354. Talbot, J.M., Yelle, D.J., Nowick, J., Treseder, K.K., 2012. Litter decay rates are determined by lignin chemistry. Biogeochemistry 108, 279–295.
References Ågren, G.I., Bosatta, E., 1998. Theoretical Ecosystem Ecology: Understanding Element Cycles. Cambridge University Press, Cambridge. Bartsev, S.I., Pochekutov, A.A., 2015. A continual model of soil organic matter
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