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.