[Insert Figure 4 here]
We then applied the empirical model to estimate the CH4sink strength of global grasslands (Fig. S8 ), under four treatments: ambient (i.e. pre-industrial) levels of N and P depositions, elevated N deposition (contemporary N deposition and pre-industrial P deposition), elevated P deposition (contemporary P deposition and pre-industrial N deposition), and concurrently elevated N + P depositions (contemporary N and P depositions) (Fig. 4 ). Simulated global grassland CH4 sinks amounted to 4.43 ± 0.20 Tg C-CH4 y-1 for the ambient scenario, 3.92 ± 0.16 Tg C-CH4 y-1 for the elevated N scenario, 4.60 ± 0.22 Tg C-CH4y-1 for the elevated P scenario, and 4.18 ± 0.18 Tg C-CH4 y-1, for the N + P scenario (Fig. 4 ). Addition of N-only thus suppressed the global grassland CH4 sink by ~0.50 Tg C (~11.4%), while concurrent P deposition alleviated this suppression by more than half (~5.8%).

Conceptual Model for P Alleviation of N-suppressed CH4Sink

Based on our field experiments, the global meta-analysis, and the empirically-derived insights of N and P impacts on CH4uptake, we here propose a conceptual framework that summarizes the possible mechanisms underlying the interactive impacts of N and P additions on CH4 uptake (Fig. 5 ). In this conceptual model, MMO represents the whole group of enzymes responsible for CH4 oxidation under aerobic conditions (Dunfield & Knowles 1995) (Fig. 5a ). Under ambient conditions (e.g. pre-industrial N and P deposition), grassland productivity is generally limited by low soil N availability (Ladwig et al. 2012) and by low soil P availability in more than half of the grasslands (Fayet al. 2015) and grassland plant species have therefore optimized their N uptake and allocation processes during ecological succession (Bai et al. 2004). Soil N does not leach under these low-N conditions and available N is either assimilated by plants or immobilized by soil microbes. Because soil mineral N (particularly NH4+) is maintained at a low level, competition with CH4 for the MMO enzyme is weak (Fig. 5a ). In contrast, sustained or high N addition will push the system out of N limitation and result in NH4+ accumulation in the soil (Fig.5b ). This in turn strengthens the competition with CH4 for the MMO enzyme, thereby suppressing the oxidation of atmospheric CH4 in grassland soils (Fig. 5b ). If N and P are concurrently added, the added P stimulates vegetation growth and uptake of mineral N, especially NH4+, and thus alleviates the N-induced suppression of CH4 oxidation. (Fig. 5c ). This theoretical framework emphasizes the substrate competition theory when explaining the P alleviation of N-suppression on CH4 uptake.