C-STABILITY an innovative modeling framework to leverage the continuous representation of organic matter

C-STABILITY description

Organic matter representation

The description of SOM in C-STABILITY consists of several subdivisions. First, organic matter is separated in two main pools, one for living microbes (noted C_mic) and one for the substrate (noted C_sub) (Fig. 1a). Several groups of living microbes can be considered simultaneously (e.g., bacteria, fungi, etc.) and C-STABILITY classes them into functional communities. Second, SOM is also separated between several biochemical classes, e.g., cellulose (or plant sugar), lignin, lipid, protein, and microbial sugar in this study (Fig. 1a). Third for each biochemical class, substrate accessible to its enzymes (noted ac) is separated from substrate which is inaccessible (noted in) due to specific physicochemical conditions, e.g., interaction between different molecules, inclusion in aggregates, sorption on mineral surfaces, etc.

Polymerization is a driver of interactions between substrate and living microbes and a continuous description of the degree of organic matter polymerization (noted p) is provided for each of these pools, as a distribution (Fig. 1b). The polymerization axis is oriented from the lowest to the highest degree of polymerization. A right-sided distribution corresponds to a highly polymerized substrate whereas a left-sided distribution corresponds to monomer or small oligomer forms. For each biochemical class ∗ (∗ = cellulose, lignin, lipid, etc.), the polymerization range is identical for both accessible and inaccessible pools. The total amount of C (in g_C) in the accessible and the inaccessible pools of any biochemical class is as follows:

where ({p}_{* }^{rm{min}}) and ({p}_{* }^{rm{max}}) are the minimum and maximum degrees of polymerization of the biochemical class ∗ and ({chi }_{* }^{rm{ac}}), ({chi }_{* }^{rm{in}}) (g_C.p⁻¹) are the polymerization distributions. Finally, the total substrate C pool is defined as the sum of all biochemical pools,

Accessibility to microbe uptake is described by the interval (also called domain) ({{mathcal{D}}}_{u}), which corresponds to small substrate compounds, monomers, dimers or trimers smaller than 600 Daltons¹, that microbes are able to take up (in red in Fig. 1b). Besides, accessibility to enzymes occurs in the ({{mathcal{D}}}_{rm{enz}}) domain (in blue in Fig. 1b). Over time the substrate accessible to enzymes is depolymerized and its distribution shifts toward ({{mathcal{D}}}_{u}) where it eventually becomes accessible to microbe uptake.

The numerical rules chosen to represent polymerization are as simple as possible in the context of theoretical simulations. Each pool is associated with a polymerization interval ([{p}_{* }^{rm{min}},{p}_{* }^{rm{max}}]) of length two. Initial substrate distributions are represented by Gaussian distributions centered at a relative distance of 25% from ({p}_{* }^{rm{max}}) (here 0.5), with the standard deviation set at 5% of polymerization interval length (here 0.1). In the accessible pool, the microbial uptake domains ({{mathcal{D}}}_{u}) are positioned at the left of the interval with a relative length of 20% (here 0.4), and enzymatic domains ({{mathcal{D}}}_{rm{enz}}) overlap the entire polymerization intervals.

Organic matter dynamics

As described in Fig. 1, three processes drive OM dynamics: (i) enzymatic activity, (ii) microbial uptake, biotransformation, and mortality, and (iii) changes in local physicochemical conditions. First, enzymes have a depolymerization role, which enables the transformation of highly polymerized substrate into fragments accessible to microbes. Second, microbial uptake of substrate is only possible for molecules having a very small degree of polymerization. When C is taken up, a fraction is respired and the remaining is metabolized, and biotransformed into microbial molecules that return to the substrate upon microbe death. Each microbial group has a specific signature that describes its composition in terms of biochemistry and polymerization. Third, changes in local substrate conditions drive exchanges between substrate accessible and inaccessible to enzymes (e.g., aggregate formation and break). All of these processes are considered with a daily time step (noted d).

Enzymatic activity Enzymes are specific to biochemical classes. They are not individually reported, but rather as a family of enzymes contributing to the depolymerization of a biochemical substrate (e.g., combined action of endoglucanase, exoglucanase, betaglucosidase, etc., on cellulose will be reported as cellulolytic action). Figure 2 describes how substrate polymerization distributions are impacted by enzymes. The overall functioning of each enzyme family (noted enz) is described by two parameters: a depolymerization rate ({tau }_{rm{enz}}^{0}) providing the number of broken bonds per time unit and a factor accounting for the type of substrate cleavage α_enz. The term ({F}_{rm{enz}}^{rm{act}}) (g_C.p⁻¹.d⁻¹) represents the change in polymerization of ({chi }_{* }^{rm{ac}}) due to enzyme activity for all (pin {{mathcal{D}}}_{rm{enz}}),

The depolymerization rate, τ_enz (d⁻¹), is expressed as a linear function of microbial C biomass C_mic (g_C),

where ({tau }_{rm{enz}}^{0}) (g({,}_{C}^{-1}).d⁻¹) is the action rate of a given enzyme per amount of microbial C. If several microbial communities are associated with the same enzyme family, we replace the C_mic term by a weighted sum of the C mass of all communities involved in Eq. (5). The ({{mathcal{K}}}_{rm{enz}}) (p⁻¹) kernel provides the polymerization change from (p^{prime}) to p,

where ({{mathbb{1}}}_{ple p^{prime} }) equals 1 if (ple p^{prime}) and 0 otherwise. The α_enz cleavage factor denotes the enzyme efficiency to generate a large amount of small fragments and to shift the substrate polymerization distribution toward the microbe uptake domain ({{mathcal{D}}}_{u}) (Fig. 2). α_enz = 1 is typical of the action of endo-cleaving enzymes, which randomly disrupts any bond of its polymeric substrate and generates oligomers. The shift toward ({{mathcal{D}}}_{u}) is slower if α_enz increases. This is characteristic of exo-cleaving enzymes, which attack the end-members of their polymeric substrate, generate small fragments, and preserve highly polymerized compounds. To satisfy the mass balance, the kernel verifies (int {{mathcal{K}}}_{rm{enz}}(p,p^{prime} )dp=1). Then ({F}_{rm{enz}}^{rm{act}}) does not change the total C mass but only the polymerization distribution (i.e., (int {F}_{rm{enz}}^{rm{act}}(chi ,p,t)dp=0)).

Microbial biotransformation Each microbial group (denoted mic) produces new organic compounds from the assimilated C. After death, the composition of the necromass returning to each biochemical pool ∗ of SOM is assumed to be constant, accessible and is depicted with a set of distributions s_mic,∗, named signature. Each distribution s_mic,* (p^–1) describes the polymerization of the dead microbial compounds returning to the pool ∗. The signature is normalized and unitless to ensure mass conservation, i.e., if we note that,

then we have ∑_*S_mic,* = 1.

For each accessible pool of substrate, the term ({F}_{rm{mic},* }^{rm{upt}}) describes how the microbes utilize the substrate available in the microbial uptake ({{mathcal{D}}}_{u}) domain (Fig. 1b). For all (pin {{mathcal{D}}}_{u}),

where ({u}_{rm{mic},* }^{0}) (g({,}_{C}^{-1}).d⁻¹) is the uptake rate per amount of microbe C. The substrate uptake rate linearly depends on the microbial C quantity.

Depending on a carbon use efficiency parameter ({e}_{rm{mic},* }^{0}) (ratio between microbe assimilated C and taken up C), taken up C is respired or assimilated and biotransformed into microbial metabolites. This induces a change in the biochemistry and polymerization (Fig. 1a).

Finally, microbial necromass returns to the substrate pools with a specific mortality, which linearly depends on the microbial C quantity,

where ({m}_{rm{mic}}^{0}) (d⁻¹) is the mortality rate of the microbe (Fig. 1a).

Change in local physicochemical conditions The polymerization of a substrate inaccessible to its enzymes remains unchanged over time. A specific event changing the accessibility to enzymes (e.g., aggregate disruption or desorption from mineral surfaces) is modeled with a flux from the inaccessible to the accessible pool. Transfer between these pools is described by the ({F}_{rm{ac},* }^{rm{loc}}) term for each biochemistry ∗,

where ({tau }_{rm{tr},* }^{rm{ac}}) (d⁻¹) is rate of local condition change toward accessibility.

Transfer in the opposite way (e.g., aggregate formation, association with mineral surfaces) is described by the ({F}_{rm{in},* }^{rm{loc}}) term,

where ({tau }_{rm{tr},* }^{rm{in}}) (d⁻¹) is rate of local condition change toward inaccessibility.

Organic matter input We defined time dependent distributions for carbon input fluxes. There are denoted ({i}_{* }^{rm{ac}}) and ({i}_{* }^{rm{in}}) (g_C.p⁻¹.d⁻¹) for both accessible and inaccessible pools of biochemical classes ∗. The total carbon input flux, expressed in g_C.d⁻¹, is:

General dynamics equations The distribution dynamics for each biochemical class * is obtained from Eqs. (4)–(6) and (8)–(11),

Then, the expended equations are,

where ({{mathbb{1}}}_{{{mathcal{D}}}_{u}}(p)) equals 1 if (pin {{mathcal{D}}}_{u}) and 0 otherwise.

The dynamics of the total substrate is ruled by,

The dynamics of microbial C_mic is obtained by,

and the CO₂ flux (g_C.d⁻¹) produced by the microbes is given by,

Model implementation

The model was implemented in the Julia^© language^63,64. An explicit finite difference scheme approximates the solutions of integro-differential equations with a Δt = 0.1d time step and a Δp = 0.01p polymerization step. Differential equations were solved with a Runge–Kutta method.

Scenarios

Scenario 1: cellulose decomposition kinetics and model sensitivity

A first simulation was run to depict cellulose depolymerization and uptake by a decomposer community over one year (see parameters in Table 1). A global sensitivity analysis focusing on the residual cellulose variable was made to determine (i) the relative influence of parameters, and (ii) how parameters influence varies over time (Fig. 3c). A specific attention was given on enzymatic parameters (especially α) to verify the pertinence of their introduction in the model.

We considered a specific method defined by Sobol for calculating sensitivity indices⁶⁵. It provides the relative contribution of the model parameters to the total model variance, here at different times of the simulation. The method relies on the same principle as the analysis of variance. It was designed to decompose the variance of a model output according to the various degrees of interaction between the n uncertain parameters ({({x}_{i})}_{iin {1,n}}). Formally, by assuming that the parameter uncertainties are independent, the model output, denoted y, could be expressed as a sum of functions that take parameter interactions into account,

Under independence assumptions between models parameters variability, model variance is:

This variance decomposition leads to the definition of several sensitivity indices. The first-order Sobol’s index of each parameter is,

and higher order indices are defined by:

and so on. These indices are unique, with a value of 0–1 and their sum equals 1. Here we focused on Sobol’s first-order indices as they are usually sufficient to give a straightforward interpretation of the actual influence of different parameters^66,67. We computed the sensitivity of the model outputs at several times of the simulation to highlight the role of model’s parameters at different phases. Figure 3c shows the normalized Sobol’s first-order indices to illustrate the relative influence of the model parameters on residual cellulose-C amount. Sobol’s indices were estimated using a Monte Carlo estimator of the variance⁶⁸. This was performed for a small variation in parameter values (±5% uniform variability), by running 12,000 model simulations for the Monte Carlo sampling.

Scenario 2: effect of substrate inaccessibility to enzyme on litter decomposition kinetics

A simulation of lignocellulose (76% cellulose, 24% lignin) degradation was performed by taking into account peroxidases, which deconstruct the lignin polymer, and cellulases, which hydrolyze cellulose. The cellulose was initially embedded in lignin and inaccessible to cellulase. The lignolytic activity (peroxidases) induces a disentanglement of the cellulose from the lignocellulosic complex. Therefore, the action of peroxidases was seen as a change of cellulose physicochemical local conditions resulting in a progressive transfer to the accessible pool. This transfer was assumed to be linearly related to the activity of lignolytic enzymes in Eq. (10),

where ({{mathcal{D}}}_{rm{lig.}}) is the domain of lignolytic activity and where the ({tau }_{rm{tr,cell.}}^{ac,0}) coefficient is set at 13 g({,}_{C}^{-1}) for the current illustration.

The simulation was performed over one year. Enzymatic and microbial parameters given in Table 1 were chosen to be closely in line with the litter decomposition and enzyme action observation^16,37,69.

Scenario 3: effect of community succession on C fluxes and substrate biochemistry

We simulated the succession of two microbial functional communities, on the same previous lignocellulose, considering microbial residue recycling. The parameters (Table 1) were chosen according to the microbial community succession observations^43,45,70. The first microbial community was specialized in plant substrate degradation, the second was specialized in the degradation of microbial residues. We referred to them as plant decomposers and microbial residue decomposers. Microbial residue decomposers were more competitive than plant decomposers because of their higher carbon use efficiency and lower mortality rate (Table 1). Both communities had the same biochemical signature, i.e., 50% polysaccharides, 30% lipids, and 20% proteins. We tested the impact of cheating as follows. Either uptake was impossible, i.e., u₀ equaled 0 for the community not involved in enzyme production, or uptake was possible but at a lower rate than the enzyme producers because the substrate fragments were released in the vicinity of the enzyme producers (Table 1).

Scenario 4: soil organic matter composition at steady state

We resolved the analytic formulation of the C stock and chemistry at steady state under several assumptions. We only considered one microbial community, a continuous constant plant input I at a rate of 2.74.10⁻⁴ g_C.cm⁻².d⁻¹ and microbial recycling⁴⁷. To be able to explicitly calculate the steady state, we only considered accessible pools, then cellulose substrate was not embedded in lignin but directly accessible to cellulolytic enzymes (Fig. 6). Finally, we considered that C use efficiency (({e}_{rm{mic}}^{0})) and uptake (({u}_{rm{mic}}^{0})) parameters were identical for all biochemical classes. A full mathematical proof is given in Supplementary Note 3.

At steady state, the amount of microbial carbon is,

For each biochemical class ∗, we define ({p}_{* }^{u}) which verifies ({p}_{* }^{rm{min}}<{p}_{* }^{u}<{p}_{* }^{rm{max}}) and ({{mathcal{D}}}_{u}=[{p}_{* }^{rm{min}},{p}_{* }^{u}]), and

By considering the scenario setup (Table 1) and Eq. (26), if ∗ = plant sugar and lignin, ({theta }_{* }(p)={m}_{rm{mic}}^{0}frac{(1-{e}_{rm{mic}}^{0}){i}_{* }(p)}{{e}_{rm{mic}}^{0}I}) and if ∗ = lipid, protein and microbial sugar, ({theta }_{* }(p)={m}_{rm{mic}}^{0}{s}_{* }(p)). The steady-state distribution of C is obtained for (pin [{p}_{* }^{u},{p}_{* }^{rm{max}}]),

and for (pin left[right.{p}_{* }^{rm{min}},{p}_{* }^{u}left[right.),

where (beta =frac{{tau }_{rm{enz}}^{0}-{alpha }_{rm{enz}}{u}_{rm{mic}}^{0}}{{tau }_{rm{enz}}^{0}+{u}_{rm{mic}}^{0}}).

Compared to previous scenarios related to laboratory experiments, enzymatic activities were divided by five to account for the impact of in situ climatic conditions. Other enzymatic and microbial parameters are given in Table 1. To explore how the catabolic traits of enzymes and anabolic traits of microbes affect the SOM composition, we modified individual parameters by 50% and documented their effects on the steady-state amount of C and its biochemistry.

Source: Ecology - nature.com