A unified theory for organic matter accumulation

Significance Organic matter in the global ocean, soils, and sediments stores about five times more carbon than the atmosphere. Thus, the controls on the accumulation of organic matter are critical to global carbon cycling. However, we lack a quantitative understanding of these controls. This prevents meaningful descriptions of organic matter cycling in global climate models, which are required for understanding how changes in organic matter reservoirs provide feedbacks to past and present changes in climate. Currently, explanations for organic matter accumulation remain under debate, characterized by seemingly competing hypotheses. Here, we develop a quantitative framework for organic matter accumulation that unifies these hypotheses. The framework derives from the ecological dynamics of microorganisms, the dominant consumers of organic matter.

the uncertainty from the other potential impacts. 141 For the pools that can equilibrate at subsistence concentrations, the impact of temperature is ambiguous. 142 An increase in ρ max alone should decrease the equilibrium concentrations of organic matter. However, if 143 uptake affinity ρ max k −1 remains constant and k also increases proportionally, then there will be a negligible 144 effect on concentrations. 145 For some of the functionally recalcitrant pools (Q ≤ 1), an increase in ρ max with temperature may result 146 in a relatively small (though still nonlinear) decrease in organic matter concentration. Specifically, the rate 147 of consumption ρ max B may increase quadratically with temperature. To explain, below (Eqn. S24), we 148 show that the processing-limited biomass B * proc is directly proportional to ρ max . If ρ max itself increases 149 linearly over small increases in temperature, then both v and B may increase linear, together giving 150 a quadratic response in the consumption rate. We can express this explicitly by considering how ρ max 151 changes with temperature as ρ max = ρ max γ T , where γ T is the modification of the rate due to temperature. 152 Thus, ρ max B * proc can be expressed as ρ max γ T B * proc γ T = (ρ max B * proc )γ 2 T . However, since these recalcitrant 153 consumption rates are generally low, the total decrease in concentration will likely be low. Furthermore, 154 ρ max may not change with temperature for some pools, such as those with organic compounds physically 155 protected by mineral structures. 156 However, for other functionally recalcitrant pools for which recalcitrance is near the threshold Q = 1, 157 small changes in temperature may induce a large response. Even small changes in rates can be enough 158 to change the value of Q for a pool from less than one (or approximately equal to one) to greater than 159 one. We show, for a subset of recalcitrant pools, that warming may result in a 'tipping point:' a dramatic 160 decrease in concentration when the threshold Q = 1 is crossed (Fig. S10c). This suggests the potential for 161 the standing stock of organic matter in the oceans and soils to decrease nonlinearly with warming, with 162 much of the newly consumed organic carbon respired and released as CO 2 . Of course, the degree of this 163 impact depends also on how temperature affects other ecosystem processes that set the population loss 164 rates and other factors in Q. 165 Emily J. Zakem, B. B. Cael, and Naomi M. Levine 5 of 33 Subsistence concentrations The subsistence concentration C * ij for OM pool i and population j is derived as follows. We derive it for a generalist that can consume multiple pools, and show that it reduces to the specialist's subsistence concentration (Eqn. 2). For conciseness and clarity, we neglect the probability of presence P j of the populations in the following derivation, and then add it to the final expressions. For population j = 1 consuming multiple pools, for the uptake of n up OM pools by population 1, where for clarity, L 1 = m q 1 B 1 − m l 1 (Eqn. 6). Any pool consumed by B 1 may also be consumed by other populations. For pool i = A, ρ Aj B j uptake by pops other than B 1 [S2] for the time-averaged supply rate s A and uptake of pool A by n cons populations. Assuming a steady state and rearranging gives the subsistence concentration C * A1 of pool A by microbial population 1 as Now we can define q ij as: Thus q ij is a population-and OM pool-specific value. (Below (Eqn. S19), we explain how Q i is the maximum across the values of q ij for all populations consuming pool i.) Substituting q A1 into the expression gives the subsistence concentration as: Next we derive more general expressions for q ij and C * ij that also do not depend on the biomass concentration. Again neglecting P j in Eqn. S1 for the concentration of pool i, and again pulling out the uptake of population 1 from the summation gives for uptake of pool i by n cons populations. Assuming steady state, and multiplying through by the yield y i1 equates the production of biomass of population 1 from pool i to the supply and the uptake by other consumers as We can now substitute the above expression into Eqn. S1: where n up is the number of OM pools consumed by population 1. This can be arranged assuming steady state to give the steady state biomass concentration B * 1 as Substituting this into Eqn. S4 for q A1 gives The production of biomass of population 1 on pool A is [S11] We can substitute this into the denominator of Eqn. S10, and also substitute in the growth balance of Eqn. S7 into the numerator of Eqn. S10 to give We now pull the ratio of uptake for pool i = A out of the summation, and sum over the remaining pools n up − 1, where now k is any other pool taken up by population 1. This gives Generalized to pool i and population j,and adding back in the overall probability of presence P j , the population-and OM pool-specific values q ij , of which Q i is the maximum across the populations, is Thus the subsistence concentration for pool i and population j is [S18] line. For a specialist taking up no other pools, k y kj ρ kj ρ ij = 0, and Eqn. S18 reduces to Eqn. 2 in the main 168 text, and is identical to that used in a previous study for chemoautotrophic nitrifying microorganisms (11).

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Thus Eqn. S18 for C * ij is a general expression appropriate for diverse metabolisms and substrates. We note 170 that the term representing the impact of another pool k may also represent the impact of inorganic nutrient 171 consumption and chemolithotrophy (12).

Recalcitrance indicator
The expression for Q i in the main text is the maximum across many populationspecific subcomponents for OM pool i. Here in the Supplement, we explicitly label each of these populationspecific subcomponents as q ij , for population j consuming pool i (Eqn. S17). The values of q ij were used to calculate the "community recalcitrance" in Fig. 3 in the main text. For example, if 40% of the values of q ij for pool i are less than or equal to 1, then we can consider 40% of the populations to experience pool i as recalcitrant. However, even if the "community recalcitrance" is high, it only takes one value of q ij > 1 for the pool to be considered functionally labile overall at that time and place. Thus proximal community recalcitrance of 100% equates to functional recalcitrance. Thus the recalcitrance indicator is the maximum among the population-specific values as [S19] Though this formulation is applied diagnostically in the model, it was derived with the goal of being useful 173 for application to real systems because it requires an estimation of relative rates of uptake of multiple 174 substrates, which may be more consistent across environments, rather than the absolute magnitude of the 175 uptake, which depends on the local concentration.

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Although Q is related to fitness, investigation of competition and cooperation among populations should 177 use C * ij as the metric of fitness. Recalcitrance indicator Q is always appropriate for determining whether 178 or not a pool will accumulate. However, when multiple populations compete for the same resource, the 179 population with the lowest subsistence concentration wins. Thus, the minimum of C * ij determines the 180 equilibrated concentration for pool i (Fig. S8). concentration with respect to uptake, Eqn. 4 is approximated as [S20]

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With this approximation, q ij >> 1. This is equivalent to the approximation made in ref. (13). With this, 188 the population-specific supply-limited equilibrium concentration of each OM pool C * supp ij is: Plugging in q for a specialist gives the direct comparison to Eqn. 2 in the main text:  does not depend on concentration, we can make an approximation to Eqn. 6 for processing-limited biomass B proc as and thus uptake of pool i does not depend on its concentration. We can then take the steady state approximation of Eqn. S23. Even though the associated recalcitrant pool i is not at steady state (i.e. the steady state of Eqn. 5 does not approximate the solutions), this approximation ends up in a close match to the simulated biomass since the change in the biomass concentration is small relative to the rate of growth and loss of the population (Fig. S15f). From Eqn. S23 at steady state, the steady state processing-limited biomass B * proc is Thus the biomass associated with recalcitrant pools reaches quasi-equilibrium, suggesting that natural  Similarly, we can approximate the supply-limited biomass B * supp . Using the above approximation for supply-limited consumption, C << k, and again assuming consumption of one OM pool for simplicity, At steady state, Thus in contrast to B * proc , B * supp must be related diagnostically to the concentration of the pool (rather uptake rate. Rather, the initial abundance of each heterotrophic population determined its prevalence in 228 the newly fueled biomass production, and this abundance was set by the combination of model parameters. 229 We also note that because the experimental set-up alters the ecological context, increasing or decreasing 230 the loss rates experienced by each population (which we simulated by changing the mortality parameters),

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both Q and C * Supp should also increase or decrease in such experiments relative to the in situ conditions.

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For example, reducing the mortality rate in the experimental set-up increases Q, which means that some 233 pools that were functionally recalcitrant could cross the threshold and become functionally labile. This 234 could spur even more microbial production.  Table S1.

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The water column vertical domain is 2,000 m with 5 m vertical resolution. Each tracer X is vertically 250 diffused throughout the water column as function of the diffusive coefficient κ as: where S X are additional sources and sinks. Sources and sinks (as well as the advection of POM according 253 to vertical velocity w s ) for organic pool i and population j with carbon-based biomass B j are as follows: where µ j is the growth rate of population j (detail below for phytoplankton vs. heterotrophs), ρ i,j is the 255 uptake of pool i by population j, and L j = m q B j . Total DOC is produced as a fraction of mortality     Phytoplankton growth. Phytoplankton growth is limited by maximum growth rate µ max (d −1 ), the concen-308 tration of DIN, and light, and modified by temperature (γ T ). Light limitation was parameterized using an 309 exponential form as a function of the instantaneous photosynthetic rate Γ (d −1 ) and the Chl a to carbon 310 ratio θ (g/g) (23, 24) as: Michaelis-Menten form may not appropriately capture the process. 350 We can consider a simple description of a solid particle-resolving model for particle (or polymer) P and monomer M , neglecting physical transport, as: where β is the specific rate of particle degradation to monomer. Solving for the net change in polymer and 351 monomer gives: We note that Eqn. S39 is identical in form to our Eqn.    The decrease in ρ max is 1/ √ n up . Stronger penalty for generalist ability: The decrease in ρ max is 1/n 2 up . One generalist (no penalty): Only one population consumes all pools, with no decrease in ρ max . In other words, the magnitudes of the rates remain the same as for the model with pure specialists, except that they all are associated with uptake by one population of biomass. Illustrated are the individual OM concentrations C (grey dots), the individual model diagnostic C * supp (the supply-limited subsistence concentrations of the microbial consumers, identifying the most labile of the functionally labile pools; Eqn. S21; light blue dots), the binned mean of C (maroon line), and the 16th and 84th percentiles (equivalent to one standard deviation for a Gaussian distribution) for C (maroon error bar) against recalcitrance indicator Q (Eqn. 3). Q = 1 (grey dashed line) separates the functionally recalcitrant OM pools from the functionally labile pools. Each simulation has 1,000 OM pools.    S16. Simulation of the experiment measuring bulk sediment decomposition rate after additions of organic matter (31). In their experiment, Westrich and Berner (1984) observed that the rate of organic carbon decomposition increased linearly with the amount of organic carbon added after ten days (their Fig. 4). From this, they confirmed the utility of first-order decomposition models (or, "multi-G" models) for describing the decomposition. Here, we show that our model, with nonlinear (Michaelis-Menten) uptake by microbial populations, also results in this bulk linear behavior. We replicated the experiment by adding different amounts of organic matter (OM) to the resulting solutions from the default model (n = 1,000), and integrating the model for ten days. The additions were accomplished by initiating the model with the original solutions concentrated by a range of factors, as indicated by the x-axis. (a) The bulk rate of decomposition (calculated as the sum of the consumption by all populations of all pools; Eqn. 5) increases linearly with the additions (black dots in panel a). To test whether additions of microbial biomass impacted the results, we also conducted the experiment in which biomass concentrations were increased by the same factor as OM (red dots in panel a). Best fit linear regressions demonstrate the linearity (black and red lines in panel a, R 2 = 0.99 for both). (b) The decomposition rates of the individual OM pools for the experiment with only the OM additions (calculated as the sum of the consumption by all populations for each pool; Eqn. 5).