Chemosphere 117 (2014) 766–773
Contents lists available at ScienceDirect
Chemosphere journal homepage: www.elsevier.com/locate/chemosphere
Tracking sinks of atmospheric methane using small world networks Chih-Sheng Lee ⇑, Pei-Jen Su Department of Environmental Engineering, Kun Shan University, Tainan City 71003, Taiwan
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Use of small world network for
Major reactions: CH4 + OH ? CH3 + H2O; HO2 + NO ? NO2 + OH; CH3O2H ? CH3O + OH; and CH4 + Cl ? CH3 + HCl. Major elements: OH/HO2radicals, chlorine atoms and NO.
analyzing atmospheric CH4 sinks and sources. Analysis without considering rate constant as well as species concentration. OH and HO2 radicals, chlorine atoms and NO as key elements for CH4 sinks. Applied to other environmental issues, e.g., eutrophication.
Stratosphere
47
46
39 37
36 44
38
45
43 33 32
35
27
30 40
Troposphere
31
41
26
49
29
42 28
34
Soil
25
Sinks
48
24
20
Sources
6
4 3 2
8
5 1
7 16
17
19 18
15
i n f o
Article history: Received 19 June 2014 Received in revised form 30 September 2014 Accepted 2 October 2014 Available online 11 November 2014 Handling Editor: O. Hao Keywords: Methane Sinks Small world network Greenhouse gas
⇑ Corresponding author. Tel.: +886 6 2052532. E-mail address: [email protected] (C.-S. Lee). http://dx.doi.org/10.1016/j.chemosphere.2014.10.043 0045-6535/Ó 2014 Elsevier Ltd. All rights reserved.
22
23
10 11
13 14
Anthropogenic 1-10
a r t i c l e
21
9
12
Natural 11-19
a b s t r a c t The present study uses small world network to highlight the key hubs for CH4 atmospheric pathways without considering rate constant of each reaction and concentrations of each species. The atmospheric methane sources and sinks were formulated into a well-organized network of 49 nodes and 302 links. In the network, reactions (including substrates and products) are considered as nodes and their pathways as links. Using a small world model, we analyzed the weighted and directed network of methane sources and sinks. By analyzing the characteristic path length, clustering coefficient, and degree distribution, we obtained insights into the methane network. The results indicate that only a few key nodes serve as hubs or limiting reactions, such as CH4 + OH ? CH3 + H2O; HO2 + NO ? NO2 + OH; CH3O2H ? CH3O + OH; and CH4 + Cl ? CH3 + HCl. The network is highly efficient; when key hub reactions experience interruptions, pathways to other nodes can be accessed to complete the methane degradation process. Additionally, our directed network keeps sources and sinks independent of each other such that changes in the number or type of methane sources does not affect the findings related to sinks. Tracking the structure of methane sources and sinks not only provides valuable, and perhaps universal, information about the network structure, but also can lead to a better understanding of the dynamic processes that generate the network. Finally, this is the first attempt of the network model in analyzing environmental issues and may represent a common blueprint for the interconnected reactions (sources and sinks) of other greenhouse gases. Ó 2014 Elsevier Ltd. All rights reserved.
C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773
1. Introduction
2. Methodology
Atmospheric methane (CH4) accounts for about 15–20% of overall global warming, second only to carbon dioxide, and has a global warming potential of 20–30 times greater than that of CO2 (IPCC, 2007). As the most abundant reactive trace gas in the troposphere, CH4 is involved in both tropospheric and stratospheric chemistry (Wuebbles and Hayhoe, 2002) and has an atmospheric life of 8– 12 years (Hütsch, 2001; Houghton, 2005). Of the approximately 600 million metric tons (MMT) of CH4 emitted each year (Keppler and Röckmann, 2007), about 40% comes from natural sources and about 60% from anthropogenic sources (Wuebbles and Hayhoe, 2002; Lowe, 2006). Methane fate can be classified into three broad pathways: tropospheric oxidation, stratospheric loss, and soil microbial reactions. There are hundreds of photoreactions involving CH4 for its disappearance. Shallcross et al. (2007) have summarized the key reactions responsible for CH4 sinks with OH radical as the major sink for CH4, and chlorine atoms and possibly nitrate radicals as minor sinks in the troposphere. However, the rate constants of reactions and concentrations of species are uncertain considering their spatial and temporal variation. Moreover, reactions of the entire CH4 sinks are interconnected (some may have feedback reactions) that form a complete and structured network. To better understand overall atmospheric CH4 reactions and its eventual sink, many questions should be addressed. These include: Which reaction limits the overall CH4 sinks? Which reaction is the most subject to perturbation resulting in termination of certain reactions? Which is major hub of the network? Which reactions are most closely related? Which contributes to the accumulation of CH4? All questions essentially indicate one thing: which nodes/reactions are important and why? The current approach for estimating CH4 sinks may not fully address the above mentioned concerns. To examine parts of these questions and understand the behavior of the linked reactions, this study was undertaken by using a small world network (SWN) to provide a functional model. SWNs are originally associated with the phrase of ‘‘six degrees of separation,’’ which indicates how each individual is connected directly or indirectly to all other individuals by about six links (Milgram, 1967). Later, parameters such as path length and clustering were introduced and applied to diverse networks such as electric power grids (Sachtjen et al., 2000), neural networks (Latora and Marchiori, 2001), and connections among film actors (Watts and Strogatz, 1998). The ‘‘scale free network’’ concept was further introduced to refer to SWNs whose degree distribution typically follows a power law (Barabási and Albert, 1999). Not surprisingly, the SWN model with its complex connections and links has also been applied to biochemical reactions (Jeong et al., 2000, 2001; Wagner and Fell, 2001; Vendruscolo et al., 2002). Certain parameters identified in SWN analysis (e.g., path lengths, clustering coefficients, and degree distributions) can help rationalize and explain the complex reactions among and within methane sinks. Since SWN analysis has not been applied to environmental issues, this study aims to: (1) formulate the architecture of the atmospheric methane sources and sinks network; (2) understand the reaction pattern of methane sinks through a directed and weighted SWN analysis; (3) evaluate the important/relevant parameters of SWN analysis for methane network; (4) evaluate the roles of the complex reactions of which the links are perturbated for methane sinks; and (5) highlight the hubs and limiting reactions that govern the overall network.
2.1. SWNs
767
The SWN concept was developed to explain observed behaviors that failed to fit within the previous network models (Watts and Strogatz, 1998). A SWN lies between the two extreme categories of regular and random networks (see Fig. SM-1 in Supplementary material (SM)), having some characteristics of both networks, i.e., not completely random, but not having well-defined rules (Watts and Strogatz, 1998). Three important characteristics of SWNs are: (1) path lengths; (2) clustering coefficients; and (3) degree distribution among all nodes (Watts and Strogatz, 1998; Barabási and Albert, 1999). These parameters will be briefly discussed below. 2.1.1. Characteristic path length A network can have many paths (links) between any two nodes. The minimum path length (Lmin) is the path with the smallest number of links from any two nodes (Montoya and Sole, 2002). For example, in Fig. 1a one potential path from node i to node j is via node a (i–a–j); the path length, L, for this route is 2. Other paths between i and j nodes include i–b–k–j (L = 3) and i–c–l–m–j (L = 4). In this example, Lmin,i,j = 2. The Lmin from node i to any other node, m, can be determined in a similar manner. Once all the Lmin associated with node i are known, the equivalent path length (Leq) with respect to node i can be calculated by averaging all Lmin for the node i:
Leq;i ¼
N1 1 X Lmin;i;j N 1 j¼1
ð1Þ
Fig. 1. Example of path lengths and clustering: (a) paths from node i to j; (b) clustering coefficient for node i.
768
C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773
where N represents all nodes within a network except for those with no connections to other nodes. Unconnected nodes have a path equal to 1 and are discarded in the calculation (Arita, 2004). Leq is a micro-measure of center within the network, that is, the shorter the Leq of a given node, the closer this node is to the center of the network. The arithmetic average for all Leq of all network nodes is known as characteristic path length, Lcha (Watts and Strogatz, 1998), and is determined as follows:
Lcha ¼
N 1X Leq;i N i¼1
P (k)
(a)
ð2Þ
The magnitude of macro-Lcha determines how long it takes to transit the network. A lower Lcha indicates that nodes within the network are relatively close. Thus, in an example of network of co-authors, a low Lcha indicates close collaboration, whereas in a network of CH4 sinks, a low Lcha indicates closely related reactions within the network. 2.1.2. Clustering coefficient With a given node, by definition, ki refers to the number of connections to that node. If node i has ki paths that are directly connected to other nodes, then the maximum potential number of paths (Pi), referring to neighboring nodes all interconnected one another, for the neighboring nodes is given by (Wagner and Fell, 2001):
Pi ¼
ki ðki 1Þ 2
ð3Þ
The clustering coefficient of node i (Ci) is then defined as:
Ci ¼
Ei 2Ei ¼ P i k i ð k i 1Þ
ð4Þ
where Ei is similar to Pi, except that not all neighboring nodes are interconnected – it refers to the actual number of inter-connections. Thus, if Ci = 1, node i is connected to all other neighboring nodes and these neighboring nodes are also interconnected. In other words, Ci is a measure of closeness of the cluster, that is, the larger Ci in a given node indicates that this particular node exhibits more interconnected reactions among neighboring nodes. This is illustrated in Fig. 1b for node i, where ki = 7 (green lines) Pi = 21 (calculated from Eq. (4)), Ei = 6 (red lines) with the resultant Ci = 0.29. Typically, Ci is much less than 1, which indicates that the network nodes are not all interconnected. If Ci = 0, then node i is the only node connected to other nodes, whereas all other neighboring nodes remain unconnected. The overall clustering coefficient (C) of the entire network can be calculated as the average of the clustering coefficient of individual nodes Ci, such that
C¼
M 1X Ci M i¼1
ð5Þ
where M represents all nodes. Thus, C is a measure of the degree to which nodes in a network tend to cluster. 2.1.3. Degree and degree distribution P(k) Other groups of researchers use degree and P(k) to characterize SWN in terms of neighborhood and critical key nodes for network structure (Jeong et al., 2000). Degree distribution represents the probability that any randomly selected node has a certain number of links/paths. If node i has ki paths, it can be said that node i has ki degree. In other words, ki is a measure of number of links to the neighborhood, that is, the higher the ki of a given node, the more neighbors this node has. The overall degree of the entire network (K) is the average degree of all nodes (Jeong et al., 2000),
(b) P (k)
High K and lowȢ
Low K and steepȢ
Fig. 2. Degree distribution of scale-free power law: (a) normal plot; (b) log–log plot.
K¼
M 1X ki M i¼1
ð6Þ
A large K indicates many links between nodes, resulting in a highly interconnected network. The degree distribution or frequency of ki for all nodes in the entire network [P(ki)] can be represented by a probability distribution as follows (Watts and Strogatz, 1998; Jeong et al., 2000):
ki Pðki Þ ¼ PM
i¼1 ki
100%
ð7Þ
The relation of P(ki) to ki can help distinguish the type of linkages that exists in a network. For example, a distribution in which only a few nodes have a large number of links indicates a network that follows the power law distribution; however, a distribution in which the network has nodes with a large number of connections follows the exponential power law. Most degree distributions in SWNs follow the scale-free power law (i.e., P(k) kc) (Barabási and Albert, 1999; Barabási and Oltvai, 2004), in which k and its probability, P(k), (i.e., the fraction of nodes with k links) follow a specific pattern (see Fig. 2a), and the log–log profile (Fig. 2b) has a power law distribution (P(k) kc). Thus, the probability is inversely proportional to k. A scale-free network is non-homogeneous because many nodes have few links (i.e., smaller k has higher P(k) – unimportant nodes) and a few nodes have many links (i.e., larger k has lower P(k) – critical key nodes). For example, in the case of CH4 network, the reaction 34 (HO2 + NO ? NO2 + OH, Table 1) has outgoing 6 links (high ki of 6) with lower P(k)) (Fig. 2); this node plays an important role in the overall CH4 sinks as will be discussed later. Any node with higher k and lower P(k) implies the smaller exponent (c). Typically c lies in the range of 2–3 in a SWN (Barabási and Albert, 1999). In short, average degree K and degree exponent c are useful indicators for identifying key nodes and their neighborhood in CH4 network. 2.2. Methane sources and sinks network Due to its complexity, it is impossible to list all relevant reactions for methane sources and sinks. Nonetheless, by searching existing literature (Higgins et al., 1981; Wahlen, 1993; Lelieveld et al., 1998; Bull et al., 2000; Hütsch, 2001; Lowe, 2006; IPCC,
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C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773 Table 1 Network nodes (methane sources and sinks) and links. Reaction
Reaction equation
Links (to other reactions) Outgoing
Incoming
Sources 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Animal wastes ? CH4 Lignite mining ? CH4 Fuel burning ? CH4 Landfills ? CH4 Biomass burning ? CH4 Rice paddies ? CH4 Ruminants ? CH4 Sewage/sludge fermentation ? CH4 Coal mining ? CH4 Industrial loss ? CH4 Tundra ? CH4 Termites ? CH4 Lakes ? CH4 Forest (plants) ? CH4 Ocean ? CH4 Wetlands ? CH4 Gas hydrates ? CH4 Volcanic mud ? CH4 Geothermal ? CH4
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20,
Sinks 20 21 22 23 24 25 26
CH4 + O2 + NADH + H+ ? CH3OH + H2O + NAD+ CH3OH ? CH2O CH2O ? HCOOH HCOOH + NAD+ ? CO2 + NADH + H+ CH4 + OH ? CH3 + H2O CH3 + O2 ? CH3O2 CH3O2 + NO ? CH3O + NO2
21 22 23 – 25 26, 42
1–19 20 21 22 1–19, 34, 40, 41, 43, 48 24, 46, 48
27,
35
25,
35 , 44
27 28 29 30 31 32 33 34
CH3O + O2 ? CH2O + HO2 CH2O + OH ? HCO + H2O HCO + O2 ? HO2 + CO CO + OH ? H + CO2 H + O2 ? HO2 CH2O ? HCO + H CH2O ? H2 + CO HO2 + NO ? NO2 + OH
28, 29 30, 31 34, 29, 30,
32, 33, 34, 42
26, 27, 28, 29, 30, 27, 27,
43 34, 40, 41, 43, 45, 48 32 33, 34, 40, 41, 43, 48 32, 41 45 45
35
NO2 ? NO + O (3P)
26 , 34 , 36 , 37
36
NO + O3 ? NO2 + O2
35
35
37 38 39 40 41 42
O (3P) + O2 ? O3 O3 ? O (1D) + O2 O2 ? O (3P) + O (3P) O (1D) + H2O ? 2OH O (1D) + H2 ? H + OH CH3O2 + HO2 ? CH3O2H + O2
38 40, 41, 48 37 24, 28, 30, 44, 49 24, 28, 30, 31, 44, 49
35, 39 37 – 38 33, 38
43,
25, 27, 29, 31,
43 44
CH3O2H ? CH3O + OH CH3O2H + OH ? CH3O2 + H2O
24, 27, 28, 30, 44, 49
45 46
CH3O2H ? CH2O + H2O CH4 + Cl ? CH3 + HCl
28, 32, 33
47 48 49
HCl + HOCl ? Cl2 + H2O CH4 + O (1D) ? CH3 + OH HCl + OH ? Cl + H2O
24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24,
46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46,
34, 42 42 31 41
24, 28, 30,
26,
48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
35 , 44, 49
44 , 45
27, 29, 31,
49
35
26 , 34 , 36
44
42 34, 40, 41,
42
25, 47,
– – – – – – – – – – – – – – – – – – –
42 , 45, 48
42 1–19,
49
– 24, 25, 28, 30, 44, 49
46 1–19, 38
46
34, 40, 41, 40,
46 , 48
# signifies reversible reactions. 2007), this study formulated atmospheric methane sources and sinks into a well-organized SWN of 49 nodes and 302 links (including 18 reversible links), as listed in Table 1 and shown in Fig. 3. Methane sources alone are not a complete network because the sources are parallel and not interconnected; thus, an integrated network of both sources and sinks was used. Methane sources are classified as anthropogenic (reactions 1–10) and natural (reactions 11–19). Major sources of methane include ruminants (reaction 7), fuel burning (reaction 3) and wetlands (reaction 16), all of which contribute more than 80 MMT CH4 yr1 (65% overall: 25% for wetlands, 20% each for fuel burning and ruminants) (Reay et al., 2007). Minor contributors ( 19). This long-tailed degree distribution of connections (cin = 1.1) indicates the robustness as well as the fragility of the network; if these specific reactions (nodes 20, 24, 46, and 48) are removed, the network can fragment into disconnected sub-networks (Montoya and Sole, 2002). In short, lower c values are associated with higher ki values (Barabási and Oltvai, 2004). The difference between incoming (cin = 1.1) and outgoing (cout = 3.8) exponential values indicates clearly different patterns in the two networks (e.g., long tail for incoming network). The implication is that the networks are directed, otherwise cin and cout would be similar. Also, for the incoming network, the four limiting reactions (nodes 20, 24, 46, and 48) serve as key nodes for efficient CH4 degradation. Any perturbation to these four nodes may result in the accumulation of methane; thus, smooth transit of these nodes appears to be of importance. By evaluating the incoming and outgoing network jointly, we can observe the total network structure in detail. This overview of the network is helpful for identifying and understanding the limiting reactions and hubs more completely than can be achieved through a stoichiometric approach. This is discussed in more detail in the following section. 3.4. Identification of hubs and limiting reactions Each parameter of two approaches (Table 2) cannot identify the key hub objectively, and may not represent the methane network
C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773
connectivity. However, by considering three parameters from both approaches it may obtain the true hub. Due to low Ci values, the clustering coefficient is unsuitable for the present directed network. Consequently, only center (approach 1) and neighborhood (approach 2) of the SWN were used to identify a few key nodes (lower portion of Table 2). These include nodes 34 and 43 for the outgoing network and nodes 24 and 46 for the incoming network as discussed before. 4. Conclusions In this study, a complete network of methane sources and sinks was first structured using SWN. Determination of relevant parameters is straightforward by identifying key nodes and important reactions based on uniqueness of SWN (characteristic path length and neighborhood). The results indicate that OH radicals play the most critical role in the CH4 sinks; other important elements include HO2 radical, chlorine atoms and NO. In particular, analysis is performed without consideration of rate constants of each reaction as well as species concentration. This certainly eliminates the uncertainty associated with these measurements in conventional approaches for analyzing atmospheric CH4 fate. Our first attempt using SWN analysis for understanding the fate of atmospheric methane can be easily applied to other complex biochemical reactions, e.g., other greenhouse gases and eutrophication. Our analysis only identifies major reactions for atmospheric CH4 sink. To lessen global warming effect, CH4 source emissions must be reduced. Acknowledgment The authors gratefully acknowledge the partial financial support from the National Science Council in Taiwan (NSC 1022221-E-168-003). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chemosphere. 2014.10.043. References Aftabuddin, M., Kundu, S., 2006. Weighted and unweighted network of amino acids within protein. Physica A 369, 895–904. Arita, M., 2004. The metabolic world of Escherichia coli is not small. Proc. Natl. Acad. Sci. U.S.A. 101, 1543–1547.
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Barabási, A.-L., Albert, R., 1999. Emergence of scaling in random networks. Science 286, 509–512. Barabási, A.-L., Oltvai, Z., 2004. Network biology: understanding the cell’s functional organization. Nat. Rev. Genet. 5, 101–113. Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. U.S.A. 101, 3747–3752. Bull, I.D., Parekh, N.R., Hall, G.H., Ineson, P., Evershed, R.P., 2000. Detection and classification of atmospheric methane oxidizing bacteria in soil. Nature 405, 175–178. Crutzen, P.J., 1974. Photochemical reactions initiated by and influencing ozone in unpolluted tropospheric air. Tellus 26, 47–57. Dunne, J.A., Williams, R.J., Martinez, N.D., 2002. Food-web structure and network theory: the role of connectance and size. Proc. Natl. Acad. Sci. U.S.A. 99, 12917– 12922. Higgins, I.J., Best, D.J., Hammond, R.C., Scott, D., 1981. Methane-oxidizing microorganisms. Microbiol. Rev. 45, 556–590. Houghton, J., 2005. Global warming. Rep. Prog. Phys. 68, 1343–1403. Hütsch, B.W., 2001. Methane oxidation in non-flooded soils as affected by crop production – invited paper. Eur. J. Agron. 14, 237–260. IPCC, 2007. Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, UK. Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.-L., 2000. The large-scale organization of metabolic networks. Nature 407, 651–654. Jeong, H., Mason, S.P., Barabási, A.-L., Oltvai, Z.N., 2001. Lethality and centrality in protein networks. Nature 411, 41–42. Keppler, F., Röckmann, T., 2007. Methane, plants and climate change. Sci. Am. 296, 40–45. Latora, V., Marchiori, M., 2001. Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701-1–198701-4. Lelieveld, J., Crutzen, P.J., Dentener, F.J., 1998. Changing concentration, lifetime and climate forcing of atmospheric methane. Tellus B 50, 128–150. Lowe, D.C., 2006. A green source of surprise. Nature 439, 148–149. Milgram, S., 1967. The small-world problem. Psychol. Today 1, 61–67. Montoya, J.M., Sole, R.V., 2002. Small world patterns in food webs. J. Theor. Biol. 214, 405–412. Reay, D.S., Smith, K.A., Hewitt, C.N., 2007. Methane: importance, sources and sinks. In: Reay, D.S., Hewitt, C.N., Smith, K.A., Grace, J. (Eds.), Greenhouse Gas Sinks. CABI Publishing, Oxford, UK, pp. 143–151. Sachtjen, M.L., Carreras, B.A., Lynch, V.E., 2000. Disturbances in a power transmission system. Phys. Rev. E 61, 4877–4882. Sander, S.P., Abbatt, J., Barker, J.R., Burkholder, J.B., Friedl, R.R., Golden, D.M., Huie, R.E., Kolb, C.E., Kurylo, M.J., Moortgat, G.K., Orkin, V.L., Wine, P.H., 2011. Chemical Kinetic and Photochemical Data for Use in Atmospheric Studies, Evaluation No. 17 JPL Publication 10-6. Jet Propulsion Laboratory. Pasadena, . Shallcross, D.E., Khalil, M.A.K., Butenhoff, C.L., 2007. The atmospheric methane sinks. In: Reay, D.S., Hewitt, C.N., Smith, K.A., Grace, J. (Eds.), Greenhouse Gas Sinks. CABI Publishing, Oxford, UK, pp. 171–183. Vendruscolo, M., Dokholyan, N.V., Paci, E., Karplus, M., 2002. Small-world view of the amino acids that play a key role in protein folding. Phys. Rev. E 65, 061910. Wagner, A., Fell, D.A., 2001. The small world inside large metabolic networks. Proc. R. Soc. London, Series B 268, 1803–1810. Wahlen, M., 1993. The global methane cycle. Annu. Rev. Earth Planet. Sci. 21, 407– 426. Watts, D.J., Strogatz, S.H., 1998. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. Wuebbles, D.J., Hayhoe, K., 2002. Atmospheric methane and global change. EarthSci. Rev. 57, 177–210.
Contents lists available at ScienceDirect
Chemosphere journal homepage: www.elsevier.com/locate/chemosphere
Tracking sinks of atmospheric methane using small world networks Chih-Sheng Lee ⇑, Pei-Jen Su Department of Environmental Engineering, Kun Shan University, Tainan City 71003, Taiwan
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Use of small world network for
Major reactions: CH4 + OH ? CH3 + H2O; HO2 + NO ? NO2 + OH; CH3O2H ? CH3O + OH; and CH4 + Cl ? CH3 + HCl. Major elements: OH/HO2radicals, chlorine atoms and NO.
analyzing atmospheric CH4 sinks and sources. Analysis without considering rate constant as well as species concentration. OH and HO2 radicals, chlorine atoms and NO as key elements for CH4 sinks. Applied to other environmental issues, e.g., eutrophication.
Stratosphere
47
46
39 37
36 44
38
45
43 33 32
35
27
30 40
Troposphere
31
41
26
49
29
42 28
34
Soil
25
Sinks
48
24
20
Sources
6
4 3 2
8
5 1
7 16
17
19 18
15
i n f o
Article history: Received 19 June 2014 Received in revised form 30 September 2014 Accepted 2 October 2014 Available online 11 November 2014 Handling Editor: O. Hao Keywords: Methane Sinks Small world network Greenhouse gas
⇑ Corresponding author. Tel.: +886 6 2052532. E-mail address: [email protected] (C.-S. Lee). http://dx.doi.org/10.1016/j.chemosphere.2014.10.043 0045-6535/Ó 2014 Elsevier Ltd. All rights reserved.
22
23
10 11
13 14
Anthropogenic 1-10
a r t i c l e
21
9
12
Natural 11-19
a b s t r a c t The present study uses small world network to highlight the key hubs for CH4 atmospheric pathways without considering rate constant of each reaction and concentrations of each species. The atmospheric methane sources and sinks were formulated into a well-organized network of 49 nodes and 302 links. In the network, reactions (including substrates and products) are considered as nodes and their pathways as links. Using a small world model, we analyzed the weighted and directed network of methane sources and sinks. By analyzing the characteristic path length, clustering coefficient, and degree distribution, we obtained insights into the methane network. The results indicate that only a few key nodes serve as hubs or limiting reactions, such as CH4 + OH ? CH3 + H2O; HO2 + NO ? NO2 + OH; CH3O2H ? CH3O + OH; and CH4 + Cl ? CH3 + HCl. The network is highly efficient; when key hub reactions experience interruptions, pathways to other nodes can be accessed to complete the methane degradation process. Additionally, our directed network keeps sources and sinks independent of each other such that changes in the number or type of methane sources does not affect the findings related to sinks. Tracking the structure of methane sources and sinks not only provides valuable, and perhaps universal, information about the network structure, but also can lead to a better understanding of the dynamic processes that generate the network. Finally, this is the first attempt of the network model in analyzing environmental issues and may represent a common blueprint for the interconnected reactions (sources and sinks) of other greenhouse gases. Ó 2014 Elsevier Ltd. All rights reserved.
C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773
1. Introduction
2. Methodology
Atmospheric methane (CH4) accounts for about 15–20% of overall global warming, second only to carbon dioxide, and has a global warming potential of 20–30 times greater than that of CO2 (IPCC, 2007). As the most abundant reactive trace gas in the troposphere, CH4 is involved in both tropospheric and stratospheric chemistry (Wuebbles and Hayhoe, 2002) and has an atmospheric life of 8– 12 years (Hütsch, 2001; Houghton, 2005). Of the approximately 600 million metric tons (MMT) of CH4 emitted each year (Keppler and Röckmann, 2007), about 40% comes from natural sources and about 60% from anthropogenic sources (Wuebbles and Hayhoe, 2002; Lowe, 2006). Methane fate can be classified into three broad pathways: tropospheric oxidation, stratospheric loss, and soil microbial reactions. There are hundreds of photoreactions involving CH4 for its disappearance. Shallcross et al. (2007) have summarized the key reactions responsible for CH4 sinks with OH radical as the major sink for CH4, and chlorine atoms and possibly nitrate radicals as minor sinks in the troposphere. However, the rate constants of reactions and concentrations of species are uncertain considering their spatial and temporal variation. Moreover, reactions of the entire CH4 sinks are interconnected (some may have feedback reactions) that form a complete and structured network. To better understand overall atmospheric CH4 reactions and its eventual sink, many questions should be addressed. These include: Which reaction limits the overall CH4 sinks? Which reaction is the most subject to perturbation resulting in termination of certain reactions? Which is major hub of the network? Which reactions are most closely related? Which contributes to the accumulation of CH4? All questions essentially indicate one thing: which nodes/reactions are important and why? The current approach for estimating CH4 sinks may not fully address the above mentioned concerns. To examine parts of these questions and understand the behavior of the linked reactions, this study was undertaken by using a small world network (SWN) to provide a functional model. SWNs are originally associated with the phrase of ‘‘six degrees of separation,’’ which indicates how each individual is connected directly or indirectly to all other individuals by about six links (Milgram, 1967). Later, parameters such as path length and clustering were introduced and applied to diverse networks such as electric power grids (Sachtjen et al., 2000), neural networks (Latora and Marchiori, 2001), and connections among film actors (Watts and Strogatz, 1998). The ‘‘scale free network’’ concept was further introduced to refer to SWNs whose degree distribution typically follows a power law (Barabási and Albert, 1999). Not surprisingly, the SWN model with its complex connections and links has also been applied to biochemical reactions (Jeong et al., 2000, 2001; Wagner and Fell, 2001; Vendruscolo et al., 2002). Certain parameters identified in SWN analysis (e.g., path lengths, clustering coefficients, and degree distributions) can help rationalize and explain the complex reactions among and within methane sinks. Since SWN analysis has not been applied to environmental issues, this study aims to: (1) formulate the architecture of the atmospheric methane sources and sinks network; (2) understand the reaction pattern of methane sinks through a directed and weighted SWN analysis; (3) evaluate the important/relevant parameters of SWN analysis for methane network; (4) evaluate the roles of the complex reactions of which the links are perturbated for methane sinks; and (5) highlight the hubs and limiting reactions that govern the overall network.
2.1. SWNs
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The SWN concept was developed to explain observed behaviors that failed to fit within the previous network models (Watts and Strogatz, 1998). A SWN lies between the two extreme categories of regular and random networks (see Fig. SM-1 in Supplementary material (SM)), having some characteristics of both networks, i.e., not completely random, but not having well-defined rules (Watts and Strogatz, 1998). Three important characteristics of SWNs are: (1) path lengths; (2) clustering coefficients; and (3) degree distribution among all nodes (Watts and Strogatz, 1998; Barabási and Albert, 1999). These parameters will be briefly discussed below. 2.1.1. Characteristic path length A network can have many paths (links) between any two nodes. The minimum path length (Lmin) is the path with the smallest number of links from any two nodes (Montoya and Sole, 2002). For example, in Fig. 1a one potential path from node i to node j is via node a (i–a–j); the path length, L, for this route is 2. Other paths between i and j nodes include i–b–k–j (L = 3) and i–c–l–m–j (L = 4). In this example, Lmin,i,j = 2. The Lmin from node i to any other node, m, can be determined in a similar manner. Once all the Lmin associated with node i are known, the equivalent path length (Leq) with respect to node i can be calculated by averaging all Lmin for the node i:
Leq;i ¼
N1 1 X Lmin;i;j N 1 j¼1
ð1Þ
Fig. 1. Example of path lengths and clustering: (a) paths from node i to j; (b) clustering coefficient for node i.
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where N represents all nodes within a network except for those with no connections to other nodes. Unconnected nodes have a path equal to 1 and are discarded in the calculation (Arita, 2004). Leq is a micro-measure of center within the network, that is, the shorter the Leq of a given node, the closer this node is to the center of the network. The arithmetic average for all Leq of all network nodes is known as characteristic path length, Lcha (Watts and Strogatz, 1998), and is determined as follows:
Lcha ¼
N 1X Leq;i N i¼1
P (k)
(a)
ð2Þ
The magnitude of macro-Lcha determines how long it takes to transit the network. A lower Lcha indicates that nodes within the network are relatively close. Thus, in an example of network of co-authors, a low Lcha indicates close collaboration, whereas in a network of CH4 sinks, a low Lcha indicates closely related reactions within the network. 2.1.2. Clustering coefficient With a given node, by definition, ki refers to the number of connections to that node. If node i has ki paths that are directly connected to other nodes, then the maximum potential number of paths (Pi), referring to neighboring nodes all interconnected one another, for the neighboring nodes is given by (Wagner and Fell, 2001):
Pi ¼
ki ðki 1Þ 2
ð3Þ
The clustering coefficient of node i (Ci) is then defined as:
Ci ¼
Ei 2Ei ¼ P i k i ð k i 1Þ
ð4Þ
where Ei is similar to Pi, except that not all neighboring nodes are interconnected – it refers to the actual number of inter-connections. Thus, if Ci = 1, node i is connected to all other neighboring nodes and these neighboring nodes are also interconnected. In other words, Ci is a measure of closeness of the cluster, that is, the larger Ci in a given node indicates that this particular node exhibits more interconnected reactions among neighboring nodes. This is illustrated in Fig. 1b for node i, where ki = 7 (green lines) Pi = 21 (calculated from Eq. (4)), Ei = 6 (red lines) with the resultant Ci = 0.29. Typically, Ci is much less than 1, which indicates that the network nodes are not all interconnected. If Ci = 0, then node i is the only node connected to other nodes, whereas all other neighboring nodes remain unconnected. The overall clustering coefficient (C) of the entire network can be calculated as the average of the clustering coefficient of individual nodes Ci, such that
C¼
M 1X Ci M i¼1
ð5Þ
where M represents all nodes. Thus, C is a measure of the degree to which nodes in a network tend to cluster. 2.1.3. Degree and degree distribution P(k) Other groups of researchers use degree and P(k) to characterize SWN in terms of neighborhood and critical key nodes for network structure (Jeong et al., 2000). Degree distribution represents the probability that any randomly selected node has a certain number of links/paths. If node i has ki paths, it can be said that node i has ki degree. In other words, ki is a measure of number of links to the neighborhood, that is, the higher the ki of a given node, the more neighbors this node has. The overall degree of the entire network (K) is the average degree of all nodes (Jeong et al., 2000),
(b) P (k)
High K and lowȢ
Low K and steepȢ
Fig. 2. Degree distribution of scale-free power law: (a) normal plot; (b) log–log plot.
K¼
M 1X ki M i¼1
ð6Þ
A large K indicates many links between nodes, resulting in a highly interconnected network. The degree distribution or frequency of ki for all nodes in the entire network [P(ki)] can be represented by a probability distribution as follows (Watts and Strogatz, 1998; Jeong et al., 2000):
ki Pðki Þ ¼ PM
i¼1 ki
100%
ð7Þ
The relation of P(ki) to ki can help distinguish the type of linkages that exists in a network. For example, a distribution in which only a few nodes have a large number of links indicates a network that follows the power law distribution; however, a distribution in which the network has nodes with a large number of connections follows the exponential power law. Most degree distributions in SWNs follow the scale-free power law (i.e., P(k) kc) (Barabási and Albert, 1999; Barabási and Oltvai, 2004), in which k and its probability, P(k), (i.e., the fraction of nodes with k links) follow a specific pattern (see Fig. 2a), and the log–log profile (Fig. 2b) has a power law distribution (P(k) kc). Thus, the probability is inversely proportional to k. A scale-free network is non-homogeneous because many nodes have few links (i.e., smaller k has higher P(k) – unimportant nodes) and a few nodes have many links (i.e., larger k has lower P(k) – critical key nodes). For example, in the case of CH4 network, the reaction 34 (HO2 + NO ? NO2 + OH, Table 1) has outgoing 6 links (high ki of 6) with lower P(k)) (Fig. 2); this node plays an important role in the overall CH4 sinks as will be discussed later. Any node with higher k and lower P(k) implies the smaller exponent (c). Typically c lies in the range of 2–3 in a SWN (Barabási and Albert, 1999). In short, average degree K and degree exponent c are useful indicators for identifying key nodes and their neighborhood in CH4 network. 2.2. Methane sources and sinks network Due to its complexity, it is impossible to list all relevant reactions for methane sources and sinks. Nonetheless, by searching existing literature (Higgins et al., 1981; Wahlen, 1993; Lelieveld et al., 1998; Bull et al., 2000; Hütsch, 2001; Lowe, 2006; IPCC,
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C.-S. Lee, P.-J. Su / Chemosphere 117 (2014) 766–773 Table 1 Network nodes (methane sources and sinks) and links. Reaction
Reaction equation
Links (to other reactions) Outgoing
Incoming
Sources 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Animal wastes ? CH4 Lignite mining ? CH4 Fuel burning ? CH4 Landfills ? CH4 Biomass burning ? CH4 Rice paddies ? CH4 Ruminants ? CH4 Sewage/sludge fermentation ? CH4 Coal mining ? CH4 Industrial loss ? CH4 Tundra ? CH4 Termites ? CH4 Lakes ? CH4 Forest (plants) ? CH4 Ocean ? CH4 Wetlands ? CH4 Gas hydrates ? CH4 Volcanic mud ? CH4 Geothermal ? CH4
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20,
Sinks 20 21 22 23 24 25 26
CH4 + O2 + NADH + H+ ? CH3OH + H2O + NAD+ CH3OH ? CH2O CH2O ? HCOOH HCOOH + NAD+ ? CO2 + NADH + H+ CH4 + OH ? CH3 + H2O CH3 + O2 ? CH3O2 CH3O2 + NO ? CH3O + NO2
21 22 23 – 25 26, 42
1–19 20 21 22 1–19, 34, 40, 41, 43, 48 24, 46, 48
27,
35
25,
35 , 44
27 28 29 30 31 32 33 34
CH3O + O2 ? CH2O + HO2 CH2O + OH ? HCO + H2O HCO + O2 ? HO2 + CO CO + OH ? H + CO2 H + O2 ? HO2 CH2O ? HCO + H CH2O ? H2 + CO HO2 + NO ? NO2 + OH
28, 29 30, 31 34, 29, 30,
32, 33, 34, 42
26, 27, 28, 29, 30, 27, 27,
43 34, 40, 41, 43, 45, 48 32 33, 34, 40, 41, 43, 48 32, 41 45 45
35
NO2 ? NO + O (3P)
26 , 34 , 36 , 37
36
NO + O3 ? NO2 + O2
35
35
37 38 39 40 41 42
O (3P) + O2 ? O3 O3 ? O (1D) + O2 O2 ? O (3P) + O (3P) O (1D) + H2O ? 2OH O (1D) + H2 ? H + OH CH3O2 + HO2 ? CH3O2H + O2
38 40, 41, 48 37 24, 28, 30, 44, 49 24, 28, 30, 31, 44, 49
35, 39 37 – 38 33, 38
43,
25, 27, 29, 31,
43 44
CH3O2H ? CH3O + OH CH3O2H + OH ? CH3O2 + H2O
24, 27, 28, 30, 44, 49
45 46
CH3O2H ? CH2O + H2O CH4 + Cl ? CH3 + HCl
28, 32, 33
47 48 49
HCl + HOCl ? Cl2 + H2O CH4 + O (1D) ? CH3 + OH HCl + OH ? Cl + H2O
24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24,
46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46,
34, 42 42 31 41
24, 28, 30,
26,
48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
35 , 44, 49
44 , 45
27, 29, 31,
49
35
26 , 34 , 36
44
42 34, 40, 41,
42
25, 47,
– – – – – – – – – – – – – – – – – – –
42 , 45, 48
42 1–19,
49
– 24, 25, 28, 30, 44, 49
46 1–19, 38
46
34, 40, 41, 40,
46 , 48
# signifies reversible reactions. 2007), this study formulated atmospheric methane sources and sinks into a well-organized SWN of 49 nodes and 302 links (including 18 reversible links), as listed in Table 1 and shown in Fig. 3. Methane sources alone are not a complete network because the sources are parallel and not interconnected; thus, an integrated network of both sources and sinks was used. Methane sources are classified as anthropogenic (reactions 1–10) and natural (reactions 11–19). Major sources of methane include ruminants (reaction 7), fuel burning (reaction 3) and wetlands (reaction 16), all of which contribute more than 80 MMT CH4 yr1 (65% overall: 25% for wetlands, 20% each for fuel burning and ruminants) (Reay et al., 2007). Minor contributors ( 19). This long-tailed degree distribution of connections (cin = 1.1) indicates the robustness as well as the fragility of the network; if these specific reactions (nodes 20, 24, 46, and 48) are removed, the network can fragment into disconnected sub-networks (Montoya and Sole, 2002). In short, lower c values are associated with higher ki values (Barabási and Oltvai, 2004). The difference between incoming (cin = 1.1) and outgoing (cout = 3.8) exponential values indicates clearly different patterns in the two networks (e.g., long tail for incoming network). The implication is that the networks are directed, otherwise cin and cout would be similar. Also, for the incoming network, the four limiting reactions (nodes 20, 24, 46, and 48) serve as key nodes for efficient CH4 degradation. Any perturbation to these four nodes may result in the accumulation of methane; thus, smooth transit of these nodes appears to be of importance. By evaluating the incoming and outgoing network jointly, we can observe the total network structure in detail. This overview of the network is helpful for identifying and understanding the limiting reactions and hubs more completely than can be achieved through a stoichiometric approach. This is discussed in more detail in the following section. 3.4. Identification of hubs and limiting reactions Each parameter of two approaches (Table 2) cannot identify the key hub objectively, and may not represent the methane network
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connectivity. However, by considering three parameters from both approaches it may obtain the true hub. Due to low Ci values, the clustering coefficient is unsuitable for the present directed network. Consequently, only center (approach 1) and neighborhood (approach 2) of the SWN were used to identify a few key nodes (lower portion of Table 2). These include nodes 34 and 43 for the outgoing network and nodes 24 and 46 for the incoming network as discussed before. 4. Conclusions In this study, a complete network of methane sources and sinks was first structured using SWN. Determination of relevant parameters is straightforward by identifying key nodes and important reactions based on uniqueness of SWN (characteristic path length and neighborhood). The results indicate that OH radicals play the most critical role in the CH4 sinks; other important elements include HO2 radical, chlorine atoms and NO. In particular, analysis is performed without consideration of rate constants of each reaction as well as species concentration. This certainly eliminates the uncertainty associated with these measurements in conventional approaches for analyzing atmospheric CH4 fate. Our first attempt using SWN analysis for understanding the fate of atmospheric methane can be easily applied to other complex biochemical reactions, e.g., other greenhouse gases and eutrophication. Our analysis only identifies major reactions for atmospheric CH4 sink. To lessen global warming effect, CH4 source emissions must be reduced. Acknowledgment The authors gratefully acknowledge the partial financial support from the National Science Council in Taiwan (NSC 1022221-E-168-003). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chemosphere. 2014.10.043. References Aftabuddin, M., Kundu, S., 2006. Weighted and unweighted network of amino acids within protein. Physica A 369, 895–904. Arita, M., 2004. The metabolic world of Escherichia coli is not small. Proc. Natl. Acad. Sci. U.S.A. 101, 1543–1547.
773
Barabási, A.-L., Albert, R., 1999. Emergence of scaling in random networks. Science 286, 509–512. Barabási, A.-L., Oltvai, Z., 2004. Network biology: understanding the cell’s functional organization. Nat. Rev. Genet. 5, 101–113. Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. U.S.A. 101, 3747–3752. Bull, I.D., Parekh, N.R., Hall, G.H., Ineson, P., Evershed, R.P., 2000. Detection and classification of atmospheric methane oxidizing bacteria in soil. Nature 405, 175–178. Crutzen, P.J., 1974. Photochemical reactions initiated by and influencing ozone in unpolluted tropospheric air. Tellus 26, 47–57. Dunne, J.A., Williams, R.J., Martinez, N.D., 2002. Food-web structure and network theory: the role of connectance and size. Proc. Natl. Acad. Sci. U.S.A. 99, 12917– 12922. Higgins, I.J., Best, D.J., Hammond, R.C., Scott, D., 1981. Methane-oxidizing microorganisms. Microbiol. Rev. 45, 556–590. Houghton, J., 2005. Global warming. Rep. Prog. Phys. 68, 1343–1403. Hütsch, B.W., 2001. Methane oxidation in non-flooded soils as affected by crop production – invited paper. Eur. J. Agron. 14, 237–260. IPCC, 2007. Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, UK. Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.-L., 2000. The large-scale organization of metabolic networks. Nature 407, 651–654. Jeong, H., Mason, S.P., Barabási, A.-L., Oltvai, Z.N., 2001. Lethality and centrality in protein networks. Nature 411, 41–42. Keppler, F., Röckmann, T., 2007. Methane, plants and climate change. Sci. Am. 296, 40–45. Latora, V., Marchiori, M., 2001. Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701-1–198701-4. Lelieveld, J., Crutzen, P.J., Dentener, F.J., 1998. Changing concentration, lifetime and climate forcing of atmospheric methane. Tellus B 50, 128–150. Lowe, D.C., 2006. A green source of surprise. Nature 439, 148–149. Milgram, S., 1967. The small-world problem. Psychol. Today 1, 61–67. Montoya, J.M., Sole, R.V., 2002. Small world patterns in food webs. J. Theor. Biol. 214, 405–412. Reay, D.S., Smith, K.A., Hewitt, C.N., 2007. Methane: importance, sources and sinks. In: Reay, D.S., Hewitt, C.N., Smith, K.A., Grace, J. (Eds.), Greenhouse Gas Sinks. CABI Publishing, Oxford, UK, pp. 143–151. Sachtjen, M.L., Carreras, B.A., Lynch, V.E., 2000. Disturbances in a power transmission system. Phys. Rev. E 61, 4877–4882. Sander, S.P., Abbatt, J., Barker, J.R., Burkholder, J.B., Friedl, R.R., Golden, D.M., Huie, R.E., Kolb, C.E., Kurylo, M.J., Moortgat, G.K., Orkin, V.L., Wine, P.H., 2011. Chemical Kinetic and Photochemical Data for Use in Atmospheric Studies, Evaluation No. 17 JPL Publication 10-6. Jet Propulsion Laboratory. Pasadena, . Shallcross, D.E., Khalil, M.A.K., Butenhoff, C.L., 2007. The atmospheric methane sinks. In: Reay, D.S., Hewitt, C.N., Smith, K.A., Grace, J. (Eds.), Greenhouse Gas Sinks. CABI Publishing, Oxford, UK, pp. 171–183. Vendruscolo, M., Dokholyan, N.V., Paci, E., Karplus, M., 2002. Small-world view of the amino acids that play a key role in protein folding. Phys. Rev. E 65, 061910. Wagner, A., Fell, D.A., 2001. The small world inside large metabolic networks. Proc. R. Soc. London, Series B 268, 1803–1810. Wahlen, M., 1993. The global methane cycle. Annu. Rev. Earth Planet. Sci. 21, 407– 426. Watts, D.J., Strogatz, S.H., 1998. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. Wuebbles, D.J., Hayhoe, K., 2002. Atmospheric methane and global change. EarthSci. Rev. 57, 177–210.
