Box 1: Ecosystem CDMZ model and temperature dependencies.
Based on (Allison et al. 2010) (Fig. 1a, Supplementary Fig. 1), the ecosystem model has four state variables measured in unit mass of carbon: soil (non decomposed) organic carbon (SOC), C ; soil decomposed soluble organic carbon (DOC), D ; microbial biomass, M ; and exoenzyme concentration, Z . Exoenzyme production drives the decomposition process of C into D , which is the only source of carbon for microbes. The model accounts for microbial production and death, exoenzyme decay, recycling of dead microbes and degraded exoenzymes, C input from plant litter, and leaching ofC and D .
Model equations. State variables C , D , M ,Z obey equations (1a-d):
(1a)\(\frac{\text{dC}}{\text{dt}}=I-\frac{v_{\max}^{D}C}{K_{m}^{D}+C}Z-e_{C}C\)
(1b)\(\frac{\text{dD}}{\text{dt}}=\frac{v_{\max}^{D}C}{K_{m}^{D}+C}Z+d_{M}M+d_{Z}Z-\frac{v_{\max}^{U}D}{K_{m}^{U}+D}M-e_{D}D\)
(1c)\(\frac{\text{dM}}{\text{dt}}=\left(1-\varphi\right)\gamma_{M}\frac{v_{\max}^{U}D}{K_{m}^{U}+D}M-d_{M}M\)
(1d)\(\frac{\text{dZ}}{\text{dt}}=\varphi\gamma_{Z}\frac{v_{\max}^{U}D}{K_{m}^{U}+D}M+d_{Z}Z\)
In equation (1a), decomposition follows from Michaelis-Menten kinetics of Z binding substrate C ; there is a constant input,I , of soil organic (non decomposed) carbon from aboveground litter, and a loss due to leaching at constant ratee C. In equation (1b), D is produced by decomposition and the recycling of dead microbial biomass and inactive enzymes; D is consumed by microbial uptake, and lost by leaching at constant rate e D. In equation (1c), growth of microbial biomass M is driven by the rate of D uptake (a Monod function of D ) times the fraction of uptaken Dturned into biomass, (1 – φ ) γ M, minus microbial mortality at constant rate d M. In equation (1d), enzyme variation is driven by the rate of D uptake times the fraction allocated to enzyme production, φ , and production efficiency, γ Z, minus enzyme deactivation at constant rate, d Z.
Ecosystem equilibria. The ecosystem model (equations (1a-d)) possesses either one globally stable equilibrium, or three equilibria (one of which is always unstable) (Supplementary Note 1, Supplementary Fig. 2). There are thresholds φ min andφ max such that the single globally stable equilibrium exists for φ < φ minor φ > φ max and is given byC = I /eC , D = 0, M = 0, Z = 0. Thus, at this equilibrium, the microbial population is extinct and no decomposition occurs. For φ min< φ < φ max, the microbial population can either go extinct (then the system stabilizes at the same equilibrium as before) or persists at or around a non-trivial equilibrium, which can be solved for analytically (Supplementary equation (1)). Note that φ min andφ max depend on all microbial and model parameters (Supplementary Note 1, Supplementary Fig. 3).
Effect of temperature on model parameters. Decomposition is predicted to respond to warming (Davidson & Janssens 2006) due to the temperature sensitivity of enzymatic activity (Wallensteinet al. 2009; German et al. 2012; Stone et al.2012). Microbial assimilation may also vary with temperature if the microbial membrane proteins involved in nutrient uptake are sensitive to warming. Following (Allisonet al. 2010), we assume that exoenzyme kinetics parameters (maximum decomposition rate \(v_{\max}^{D}\) and half-saturation constant \(K_{m}^{D}\)) and microbial uptake parameters (maximum uptake rate \(v_{\max}^{U}\) and half-saturation constant \(K_{m}^{U}\)) follow Arrhenius relations with temperature. This defines our baseline ‘kinetics-only’ scenario of temperature-dependent decomposition:
(2a)\(v_{\max}^{D}=\ v_{0}^{D}e^{-\frac{E_{v}^{D}}{R\left(T+273\right)}}\)
(2b)\(K_{m}^{D}=\ K_{0}^{D}e^{-\frac{E_{K}^{D}}{R\left(T+273\right)}}\)
(2c)\(v_{\max}^{U}=\ v_{0}^{U}e^{-\frac{E_{v}^{U}}{R\left(T+273\right)}}\)
(2d)\(K_{m}^{U}=\ K_{0}^{U}e^{-\frac{E_{K}^{U}}{R\left(T+273\right)}}\)
where T is temperature in Celsius, R is the ideal gas constant, and the E parameters denote the corresponding activation energies.
We consider two additional scenarios for the influence of temperature on decomposition. In the temperature-dependent microbial mortality scenario (Hagerty et al.2014), the microbial death rate d M depends on temperature according to
(3)\(d_{M}\left(T\right)=\ d_{M0}e^{-\frac{E_{\text{dM}}}{R\left(T+273\right)}}\)
as in (Hagerty et al.2014).
In the temperature-dependent microbial growth efficiency (MGE) scenario, the MGE decreases with temperature (Allisonet al. 2010; Wieder et al. 2013; Hagerty et al.2014), which is modeled by making the microbial growth efficiencyγ M vary linearly with temperature (Allisonet al. 2010; German et al. 2012; Wang et al. 2013; Li et al. 2014):
(4)\(\gamma_{M}\left(T\right)=\ \gamma_{M,\text{ref}}-m\left(T-T_{\text{ref}}\right)\)
with T ref = 20 °C.