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The binding of [2Fe–2S] and [3Fe–4S] in clMagRThree conserved cysteines (C60, C124, and C126) of clMagR in a CXnCGC sequence motif (n is 63–65 in most cases) play critical roles in iron–sulfur cluster binding18 (Fig. 1a), which has been further validated by alanines substitution mutant clMagR3M (C60A, C124A, and C126A mutation of clMagRWT). Strep-tagged clMagRWT and clMagR3M were freshly prepared (labeled as “as-isolated”) and purified to homogeneity under aerobic conditions. The clMagRWT protein showed brown color and clMagR3M appeared colorless in the solution, indicating the presence or absence of iron–sulfur cluster, respectively. Consistently, the Ultraviolet–visible (UV–Vis) spectrum of as-isolated clMagRWT showed absorption from 300-to-600-nm region, and with an absorption peak at 325 and 415 nm, and a shoulder at 470 nm, whereas these absorption peaks were abolished in clMagR3M (Fig. 1b). Circular dichroism (CD) spectroscopy was applied to characterize the types of iron–sulfur cluster and their protein environments during cluster maturation42,43,44. As shown in Fig. 1c, clMagRWT shows distinct positive peaks at 371 nm and 426 nm and three negative peaks at 324 nm, 396 nm, and 463 nm, respectively, suggesting the presence of [2Fe–2S] cluster45. However, it is worth pointing out that [4Fe–4S] or [3Fe–4S] clusters usually exhibit negligible CD intensity compared to [2Fe–2S] as shown previously in NifIscA45,46, thus CD spectroscopy cannot exclude the existence of [4Fe–4S] or [3Fe–4S]. Electron paramagnetic resonance (EPR) spectroscopy was then used to analyze different states of as-isolated clMagRWT. The oxidized clMagRWT was S = 1/2 species, characterized by a rhombic EPR signal with g values at g1 = 2.016, g2 = 2.002, and g3 = 1.997 (Fig. 1d) which disappeared at 45 K, suggesting the presence of [3Fe–4S]1+ cluster47,48. After reduced with sodium dithionite (Fig. 1e), EPR signal from [2Fe–2S] cluster can be observed until the temperature increased to 60K49,50,51. Thus, two distinct iron–sulfur clusters were assigned by EPR spectroscopy of clMagRWT. Figure 1Characterization of iron–sulfur clusters in as-isolated clMagR. (a) Sequence alignment of MagR in eight representative species. Predicted secondary structures are shown in the upper lines, with two alpha-helices (orange cylinders) and seven beta-strands (green arrows). Conserved residues with iron–sulfur cluster binding properties are shown in the red background (100% conserved), indicated by stars. Other conserved residues are shown in the gray background and bold fonts. Species’ common name, Latin name and sequence ID in NCBI are listed as follows: Pigeon (Columba livia), XP_005508102.1*; Zebra finch(Taeniopygia guttata), XP_002194930.1*; Fly(Drosophila melanogaster), NP_573062.1*; Monarch butterfly(Danaus plexippus), AVZ24723.1*; Salmon(Salmo salar), XP_013999046.1*; Octopus(Octopus bimaculoides), XP_014786756.1*; Little brown bat(Myotis lucifugus), XP_006102189.1*; Human(Homo sapiens), NP_112202.2*. (b) UV–Vis absorption spectrum of as-isolated pigeon MagR (clMagRWT, black) and C60AC124AC126A substitution mutant (clMagR3M, red), indicating three cysteines contribute to the iron–sulfur cluster binding. SDS-PAGEs of protein preparation are shown as inserts, theoretical mass of the clMagR monomer and clMagR3M monomer were 16.41 kDa, 16.31 kDa, respectively. (c) CD spectrum of as-isolated clMagRWT(black) and clMagR3M(red). (d, e) X-band EPR spectrum of as-isolated clMagRWT at oxidized (d) and reduced status (e). The samples were frozen in TBS buffer and the spectrums were recorded at various temperatures (10 K, 25 K, 45 K, 60 K). (f) Low-temperature resonance Raman spectra of as-isolated clMagRWT. Spectra were recorded at 17 K using 488 nm laser excitation.Full size imageConsidering some iron–sulfur clusters in proteins are diamagnetic and therefore EPR silent, low-temperature Resonance Raman (RR) spectroscopy was then utilized as a probe to characterize those clusters52. With 488 nm excitation, the RR spectra of clMagRWT in the iron–sulfur stretching region (240–450 cm−1) show the presence of [3Fe–4S]1+ cluster (represented by two bridging modes at 286 and 347 cm−1, and one terminal modes at 364 cm−1) and [2Fe–2S]2+ cluster (represented by three iron–sulfur bridging mode at 293, 308 and 330 cm−1 and two terminal modes at 407 and 422 cm−1, as shown in Fig. 1f)52,53,54,55,56. Taking together, we conclude that as-isolated clMagRWT contains both cystine-ligated [2Fe–2S] cluster and [3Fe–4S] cluster.The assembly and conversion of [2Fe–2S] and [3Fe–4S] in clMagRIron–sulfur cluster assembly of IscA, an clMagR homology protein in bacteria, is mediated by cysteine desulfurase IscS2. To elucidate how iron–sulfur cluster assembles in clMagR, time-course experiment was performed, and UV–Vis absorption and CD spectrum were used to monitor the IscS-catalyzed iron–sulfur cluster assembly in clMagR (Fig. 2). No signal of the iron–sulfur cluster was recorded when the reaction begins (0 min), and then the characteristic visible absorption peak and CD spectrum of clMagRWT appeared after 5 min, indicating [2Fe–2S] cluster assembled. As the reaction proceeds, the UV–Vis absorption intensity increased and after 180 min the signal was dominated by a broad shoulder centered at 415 nm (Fig. 2a). Concomitantly, the CD spectrum of the [2Fe–2S] center decreased and then almost disappeared after 180 min, indicating that [2Fe–2S] had been converted to [3Fe–4S] clusters and the reconstitution finished (Fig. 2b).Figure 2Iron–sulfur cluster assembly on clMagR. (a, b) IscS-mediated iron–sulfur cluster assembly on clMagR monitored as a function of time by UV–Vis absorption (a) and CD spectroscopy (b). The spectra shown were taken with samples of pretreated clMagR to remove iron–sulfur clusters before reconstitution (apo-clMagR, 0 min, light green), incubated with IscS after 5 min (green), and after 180 min (dark green). (c, d) chemical reconstitution-mediated iron–sulfur cluster assembly on clMagR monitored as a function of time by UV–Vis absorption (c) and CD spectroscopies (d). The spectra shown were taken with samples of pretreated clMagR to remove iron–sulfur clusters before reconstitution (apo-clMagR, light green) and chemically reconstituted clMagR (chem re clMagR, purple). (e) X-band EPR spectrum of chemically reconstituted clMagRWT. The spectrum was recorded at 10 K. (f) Low-temperature resonance Raman spectra of chemically reconstituted clMagR. Protein and reagent concentrations are described in the Methods. Spectra were recorded at 17 K using 488 nm laser excitation.Full size imageIron–sulfur cluster assembly can be achieved by chemical reconstitution as well, since iron–sulfur apo-proteins are able to spontaneously form iron–sulfur clusters in vitro when supplied with iron and sulfide under reducing conditions1,43,57. With this approach, started with apo-clMagRWT, we successfully reconstituted [3Fe–4S] cluster in clMagR protein, confirmed by UV–Vis absorption and CD spectrum result (Fig. 2c,d). To further validate if [3Fe–4S] is the sole type of iron–sulfur cluster in clMagR after chemical reconstitution, EPR and low-temperature Resonance Raman spectroscopy were applied (Fig. 2e,f). The chemically reconstituted clMagRWT was S = 1/2 species, characterized by a rhombic EPR signal with g values at g1 = 2.017, g2 = 2.002, and g3 = 1.994 (Fig. 2e). The signal is assigned to a S = 1/2 [3Fe–4S]1+ cluster. The Low-temperature Resonance Raman spectrum showed an intense band at 346 cm−1 and additional bands at 406 and 420 cm−1, which demonstrated that chemically reconstituted clMagRWT only contains [3Fe–4S]1+ cluster (Fig. 2f).We further investigated if clMagR could serve as an iron–sulfur carrier protein to accept [2Fe–2S] cluster from scaffold protein such as IscU58. Briefly, 400 µM holo-IscU was mixed with 400 µM strep-tagged apo-clMagRWT and incubated for 180 min under reduced condition, then, after desalting and strep-tactin affinity column separation, UV–Vis absorption and CD spectroscopy were applied the iron–sulfur cluster transfer process (Fig. 3a). The intensity of UV–Vis spectrum decreased in IscU (Fig. 3b) but significantly increased in clMagR after reaction (Fig. 3d), indicating [2Fe–2S] cluster was transferred from IscU to clMagR59. Consistently, CD spectrum of IscU and clMagR also confirmed that [2Fe–2S] transfer occurred between IscU and clMagR (Fig. 3c,e). The resulting spectrum is very similar to that of the [2Fe–2S] intermediate assembled on IscS mediated reconstituted apo-clMagR (Fig. 2b).Figure 3clMagR serve as carrier protein to accept [2Fe–2S] cluster from IscU in vitro. (a) A cartoon schematically illustrates the experimental procedures of in vitro iron–sulfur cluster transfer from IscU to clMagR. (b, c) The UV–Vis absorption (b) and CD spectra (c) of IscU. IscU protein samples were taken before mixing with apo-clMagR (holo-IscU, black lines) and after incubated with apo-clMagR for 180 min (pink lines). (d, e) The UV–Vis absorption (d) and CD spectra (e) of clMagR. clMagR samples were taken before mixing with holo-IscU (apo-clMagR, light green lines) and after incubated with holo-IscU for 180 min (holo-clMagR, brown lines).Full size imageCys-60 is essential for clMagR to bind [3Fe–4S] cluster, not [2Fe–2S] clusterThree conserved cysteines (C60, C124, and C126) of clMagR play critical roles in iron–sulfur cluster binding, and the substitute mutation of these three residues abolished iron–sulfur binding (Fig. 1b,c)18. To elucidate if three cysteines bind [2Fe–2S] and [3Fe–4S] differently, single Cys-to-Ala substitutions (C60A, C124A, and C126A) were made and their iron–sulfur binding properties were characterized.Freshly purified as-isolated clMagRC60A showed light brown color, and [2Fe–2S] cluster binding was verified by UV–Vis absorption and CD spectrum (Fig. 4a,b). A typical protein-bound [2Fe–2S] cluster absorption peak at 325 nm and a shoulder at 415 nm are visible in UV–Vis absorption (Fig. 4a, light orange line). Consistently, the CD spectrum of as-isolated clMagRC60A mutant had a negative peak at 397 nm and a positive peak at 451 nm (Fig. 4b, light orange line), confirmed the [2Fe–2S] cluster binding, similar to clMagRWT. However, in contrast to clMagRWT, chemical reconstitution failed to convert [2Fe–2S] cluster to [3Fe–4S] cluster in clMagRC60A. As shown in Fig. 4a,b (orange line), chemically reconstituted clMagRC60A showed similar and characteristic [2Fe–2S] UV–Vis absorption peaks and CD spectrum, but not [3Fe–4S] (Fig. 4a,b, orange lines), suggesting that C60A mutation abolished [3Fe–4S] cluster binding ability in clMagR.Figure 4Three conserved cysteines play different roles in iron–sulfur binding in clMagR. (a, b) Chemical reconstitution-mediated iron–sulfur cluster assembly on apo-clMagRC60A monitored by UV–Vis absorption (a) and CD spectroscopies (b). The samples of spectra shown are as-isolated clMagRC60A (light orange) and chemically reconstituted clMagRC60A (chem re clMagRC60A, orange). (c, d) chemical reconstitution-mediated iron–sulfur cluster assembly on clMagRC124A monitored by UV–Vis absorption (c) and CD spectroscopies (d). The samples of spectra shown are as-isolated clMagRC124A (light purple) and chemically reconstituted clMagRC124A (chem re clMagRC124A, purple). (e, f) chemical reconstitution-mediated iron–sulfur cluster assembly on pigeon clMagRC126A monitored by UV–Vis absorption (e) and CD spectroscopies (f). The samples of spectra shown are as-isolated clMagRC126A (light blue) and chemically reconstituted clMagRC126A (chem re clMagRC126A, blue). SDS-PAGE results were shown in the right of corresponding UV–Vis spectra as inserts (a, c, e). The theoretical mass of the clMagRC60A monomer, clMagRC124A monomer and clMagRC126A monomer were 16.38 kDa. (g, h) The UV–Vis absorption (c) and CD spectra (d) of clMagRC60A obtained by mixing apo-clMagRC60A and holo-IscU which was recorded before the addition of apo-clMagRC60A (dotted orange lines) and after incubation with apo-clMagRC60A for 180 min (orange lines). Protein and reagent concentrations are described in the Experimental procedures.Full size imageIn contrast, purified as-isolated clMagRC124A and clMagRC126A were colorless, and the binding of iron–sulfur clusters was barely detectable by UV–Vis and CD spectrum (Fig. 4c–f, light purple, and light blue lines, respectively). However, chemical reconstitution successfully reconstituted [3Fe–4S] cluster binding in both clMagRC124A and clMagRC126A (Fig. 4c–f, purple and blue lines, respectively). After chemical reconstitution, the UV–Vis absorption of both clMagRC124A and clMagRC126A mutants showed the signal of iron–sulfur cluster binding (Fig. 4c,e). Parallel CD spectrum studies confirmed both chemically reconstituted clMagRC124A and clMagRC126A have [3Fe–4S] cluster binding (Fig. 4d,f), similar to chemically reconstituted clMagRWT. The results demonstrated that Cys-124 and Cys-126 in clMagR play important roles in [2Fe–2S] cluster binding, thus, mutating these two residues lead to clMagR favors [3Fe–4S] binding.Considering clMagR can act as a carrier protein to accept iron–sulfur cluster from IscU (Fig. 3), it is worth testing if three cysteines play a different role in this process as well. Holo-IscU was mixed with apo-clMagR single cysteine mutants in a reduced state for 180 min. The apo status of all three mutants (labeled as apo-clMagRC60A, apo-clMagRC124A, and apo-clMagRC126A) had no iron–sulfur cluster binding before mixing with holo-IscU, as shown by negligible UV absorption and CD intensities (Fig. 4g,h and Supplementary Fig. 1a–d, dotted lines). After incubation with holo-IscU and separation of IscU and clMagR mutants, clMagRC60A showed distinct changes in UV–Vis absorption and CD spectrum (Fig. 4g,h). The UV–Vis absorption increased and showed better-resolved peaks at 322 nm, 410 nm, 504 nm (Fig. 4g, orange line), and parallel CD spectra had distinct positive peaks (319 nm, 355 nm, 445 nm, and 534 nm) and four negative peaks (333 nm, 392 nm, 477 nm, and 579 nm, Fig. 4h), indicating [2Fe–2S] cluster was transferred from IscU to clMagRC60A. Interestingly, clMagRC124A and clMagRC126A could also accept [2Fe–2S] cluster transferred from holo-IscU, though the binding efficiency is much lower than clMagRWT and clMagRC60A, as verified by UV–Vis and CD spectrum (Supplementary Fig. 1a–d). It seems that clMagRC60A accept [2Fe–2S] cluster from scaffold protein IscU more effectively compared with clMagRC124A and clMagRC126A. And after incubation with clMagR mutants, UV–Vis absorption of IscU significantly decreased, confirmed that iron–sulfur cluster transfer occurred in between holo-IscU and three clMagR mutants (Supplementary Fig. 1e).Again, our data demonstrated that three conserved cystines of clMagR played different roles on the iron–sulfur cluster binding, and especially Cys-60 is essential for clMagR to bind [3Fe–4S] cluster, not [2Fe–2S] cluster. Therefore, it is possible to obtain a [2Fe–2S] cluster binding only clMagR by mutating Cys-60. Thus, we labeled clMagR protein samples based on their iron–sulfur cluster in later experiments. For example, we labeled the chemically reconstituted clMagRWT as [3Fe–4S]-clMagRWT, and clMagRC60A that accepted [2Fe–2S] cluster from holo-IscU as [2Fe–2S]-clMagRC60A, to investigate the magnetic property of clMagR when it binds different iron–sulfur clusters.[3Fe–4S]-clMagR shows different magnetic properties from [2Fe–2S]-clMagRMagR has been reported as a putative magnetoreceptor and exhibits intrinsic magnetic moment experimentally and theoretically when forms complex with cryptochrome (Cry)18,20,21. To elucidate if different iron–sulfur clusters binding in clMagR have different magnetic features and respond to external magnetic fields differently, we obtained [3Fe–4S] and [2Fe–2S] bound only clMagR protein by chemical reconstitution of clMagRWT (as [3Fe–4S]-clMagRWT) and holo-IscU incubated and re-purified clMagRC60A (as [2Fe–2S]-clMagRC60A), respectively, and measured the magnetic moment of these proteins with Superconducting Quantum Interference Device (SQUID) magnetometry. SQUID is a highly sensitive magnetometry to measure extremely subtle magnetic fields and to study the magnetic properties of a range of samples, including extremely low magnetic moment biological samples. Therefore, it has been regularly used as a first test to identify the specific kind of magnetism of a given specimen, such as ferromagnetic, antiferromagnetic, paramagnetic or diamagnetic, by measuring at different temperatures and external magnetic field strength. For example, B-DNA was identified as paramagnetic under low temperature by SQUID60.Purified clMagR3M was utilized as a control since it had no iron–sulfur cluster binding due to lack of cysteine residues (Fig. 1b,c). The magnetic measurement was done at different temperatures (5 K and 300 K) and MH curves (magnetization (M) curves measured versus applied fields (H)) were generated for three proteins to reflect the protein magnetic anisotropy. The MH curves of clMagR3M clearly exhibited diamagnetic property at both 5 K and 300 K, suggesting that magnetism of clMagR is dependent on the iron–sulfur cluster (Fig. 5a,b, red lines). In contrast, [3Fe–4S]-clMagRWT showed superparamagnetic behavior at 5 K which has saturation magnetization (MS) at 2 T about 0.22771 emu/g protein (Fig. 5a, purple line), [2Fe–2S]-clMagRC60A is paramagnetic at 5 K (Fig. 5a, orange line). Interestingly, at higher temperature such as 300 K, [2Fe–2S]-clMagRC60A is diamagnetic while [3Fe–4S]-clMagRWT is paramagnetic (Fig. 5b, orange line and purple line). The different magnetism, as well as the different saturation magnetization of clMagR with different iron–sulfur binding, are clearly important features of this putative magnetoreceptor, and worth further investigation and validation in vivo in the future.Figure 5[3Fe–4S]-clMagRWT shows different magnetic properties from [2Fe–2S]-clMagRC60A. (a) Field-dependent magnetization curves (MH) at 5 K for [2Fe–2S]-clMagRC60A (orange), [3Fe–4S]-clMagRWT (chem re clMagRWT, purple), and clMagR3M (red). The magnetic susceptibility of [2Fe–2S]-clMagRC60A is 2.27749E−6 and the magnetic susceptibility of clMagR3M is − 4.0438E−7. (b) Field-dependent magnetization curves (MH) at 300 K for [2Fe–2S]-clMagRC60A (orange), [3Fe–4S]-clMagRWT (chem re clMagRWT, purple), and clMagR3M (red). And the magnetic susceptibility is − 1.83638E−7, 5.93483E−8, − 3.26432E−7, respectively.Full size image More

PreliminariesA bipartite network captures connections between nodes of one type (agents) and nodes of a second type (artifacts). Throughout this section, we use the ecological case of Darwin’s Finches to provide a concrete example24,25. On his voyage to the Galapagos Islands on the H.M.S. Beagle, Darwin observed that only some species of finches lived on each island. These patterns can be represented as a bipartite network in which finch species (the agent nodes) are connected to the islands (the artifact nodes) where they are found26. A bipartite network can be represented as a binary matrix in which the agents are arrayed as rows, and the artifacts are arrayed as columns. We use ({mathbf {B}}) to denote a bipartite network’s representation as a matrix, where (B_{ik}=1) if agent i is connected to artifact k, and otherwise is 0. The sequence of row sums and the sequence of column sums of ({mathbf {B}}) are called the agent and artifact degrees sequences, respectively. These sequences are among the bipartite network’s most significant features and are known to have implications for bipartite projections and backbones15,27,28. In the ecological case, the agent degree sequence captures the number of islands where each species is found, while the artifact degree sequence captures the number of species found on each island.The projection of a bipartite network is a weighted unipartite co-occurrence network in which a pair of agents is connected by an edge with a weight equal to their number of shared artifacts. For example, the bipartite projection of Darwin’s finch network is a species co-occurrence network in which a pair of finch species is connected by an edge with a weight equal to the number of islands where they are both found. We use ({mathbf {P}}) to denote the matrix representation of a bipartite projection, which is computed as ({{mathbf {B}}}{{mathbf {B}}}^T), where ({mathbf {B}}^T) indicates the transpose of ({mathbf {B}}). In a projection ({mathbf {P}}), (P_{ij}) indicates the number of times agents i and j were connected to the same artifact k in ({mathbf {B}}). The diagonal entries of ({mathbf {P}}), (P_{ii}), are equal to the agent degrees, but in practice are ignored.The backbone of a bipartite projection is a binary representation of ({mathbf {P}}) that contains only the most ‘important’ or ‘significant’ edges. For example, the backbone of a species co-occurrence network connects pairs of species if they are found on a significant number of the same islands, which might be interpreted as evidence that the two species do not compete for resources and perhaps are symbiotic. We use ({mathbf {P}}’) to denote the matrix representation of the backbone of ({mathbf {P}}). Because multiple methods exist for deciding when an edge is significant and thus should be preserved in the backbone, we use (mathbf{P }^{‘{text {M}}}) denote a backbone extracted using method M. It is important to note that for a given bipartite projection, there is no ‘true’ backbone, but only backbones corresponding to specific backbone methods M. The backbone extracted using FDSM (i.e. (mathbf{P }^{‘{text{FDSM}}})) may be similar or different from a backbone extracted using another method such as SDSM (i.e. (mathbf{P }^{‘{text {SDSM}}})), and these similarities and differences depend on the information that is considered by the respective methods when determining whether edges’ weights are significant. It is these similarities and differences that we explore in the four studies below.Backbone extraction methods that were originally developed for non-projection weighted networks are often applied to weighted bipartite projections. One simple method preserves an edge in the backbone if its weight in the projection exceeds some global threshold T. However, when (T = 0), which is common, the backbone will be dense and have a high clustering coefficient because each artifact of degree d induces (d(d-1)/2) edges in the backbone29. Using (T > 0) can yield a sparser and less clustered backbone30,31,32, but still yields highly clustered networks in which low-degree nodes are excluded while high-degree nodes are preserved19. More sophisticated methods, including the disparity filter19 and likelihood filter20, aim to overcome these limitations of the global threshold method by using a different threshold for each edge based on a null model. However, all methods that were developed for non-projection weighted networks have the same shortcoming when applied to weighted bipartite projections: they ignore information about the artifacts, which is lost when generating the projection18. In the ecological case, the global threshold, disparity filter, and likelihood filter methods all decide whether two species should be connected in the backbone only by examining how many islands these two species are both found on, but do not consider the characteristics of those islands, including how many other species are found there, or even how many islands there are. Therefore, although these methods are promising for extracting the backbone from non-projection weighted networks, different methods are required for extracting the backbone from a bipartite projection.Bipartite ensemble backbone modelsBipartite ensemble backbone models decide whether an edge’s observed weight (P_{ij}) is significantly large, and thus whether a corresponding edge should be included in the backbone by comparing it to an ensemble of random bipartite networks. Let ({mathscr {B}}) be the set of all bipartite networks (mathbf {B^*}) having the same number of agents and artifacts as ({mathbf {B}}). In the ecological case, (mathbf {B^*}) might be viewed as representing a possible world containing the same species and islands, but in which locations of species on islands is different, and likewise ({mathscr {B}}) is the set of all such possible worlds. The bipartite ensembles used in backbone models take a subset ({mathscr{B}}^{text{M}}) of ({mathscr {B}}), subject to certain constraints M, and impose a probability distribution on it. In all models except the SDSM, the uniform probability distribution is imposed on ({mathscr{B}}^{text{M}}), that is, each element of the ensemble is equally likely. The backbone is then extracted from the projection of ({mathbf {B}}) by using the distribution of edge weights arising from projections of members of the ensemble to evaluate their statistical significance.We use (P^*_{ij}) to denote a random variable equal to ((mathbf {B^*}mathbf {B^*}^T)_{ij}) for (mathbf {B^*}~in ~{mathscr {B}}^{text {M}}). That is, (P^*_{ij}) is the number of artifacts shared by i and j in a bipartite network randomly drawn from ({mathscr {B}}^{text {M}}). In the ecological case, (P^*_{ij}) represents the number of islands that are home to both species i and j in a possible world, while the distribution of (P^*_{ij}) is the distribution of the number of islands shared by species i and j in all possible worlds.Decisions about which edges should appear in a backbone extracted at the statistical significance level (alpha) are made by comparing (P_{ij}) to (P^*_{ij})$$begin{aligned} P_{ij}’= {left{ begin{array}{ll} 1 &{} quad {text { if }} Pr (P^*_{ij} ge P_{ij}) < frac{alpha }{2},\ 0 &{} quad {text {otherwise.}} end{array}right. } end{aligned}$$This test includes edge (P'_{ij}) in the backbone if its weight in the observed projection (P_{ij}) is uncommonly large compared to its weight in projections of members of the ensemble (P^*_{ij}). We use a two-tailed significance test in the studies below because, in principle, an edge’s weight in the observed projection could be uncommonly larger or uncommonly smaller than its weight in projections of members of the ensemble, however a one-tailed test may also be used. In the ecological case, two species are connected in the backbone if their number of shared islands in the observed world is uncommonly large compared to their number of shared islands in all possible worlds.There are many ways that ({mathscr {B}}) can be constrained33, with each set of constraints describing a particular ensemble ({mathscr {B}}^{text {M}}), which is used in a particular ensemble backbone model M to yield a particular backbone ({mathbf {P}}^{'M}). In the case of ensembles used to extract the backbone of bipartite projections, our focus in this paper, two broad types of constraints are common23. First, ensembles can be distinguished by what they constrain: only the number of edges, the degrees of the agent nodes, the degrees of the artifact nodes, or the degrees of both the agent and artifact nodes. Second, ensembles can be distinguished by how they impose these constraints: the constraints can be satisfied exactly, or only on average. In statistical physics, ensembles that impose exact or ‘hard’ constraints are known as microcanonical, while ensembles that satisfy constraints on average or impose ‘soft’ constraints are known as canonical9.Prior work on these ensembles generally adopts either a theoretical focus on the ensembles themselves, or an applied focus on the consequences of ensemble choice. In the theoretical literature, some (primarily mathematicians) have aimed to characterize the properties of ensembles, such as estimating the cardinality of the ensemble of matrices with fixed rows and columns (below, we call this ensemble ({mathscr{B}}^{{text{FDSM}}}))34. Others (primarily physicists) have aimed to identify conditions under which ensembles are equivalent or non-equivalent, typically interpreting ensembles as representing thermodynamic systems35,36,37. In the applied literature, the focus is not on identifying fundamental properties of ensembles, but instead on understanding the implications of choosing a particular ensemble when detecting a particular pattern, such as nestedness38 or community structure23,27. The present work falls into this latter group: we are not directly concerned with identifying fundamental properties of ensembles, but instead on identifying the consequences of ensemble choice, with the ultimate goal of offering practical guidance to applied researchers wishing to extract the backbone of a bipartite projection.In the remaining subsections below, we first describe the FDSM in terms of its ensemble. We then present four potential alternative backbone models whose ensembles differ only slightly from FDSM, in terms of either what they constrain or how they impose constraints. We then turn to exploring the consequences of choosing one of these alternatives over FDSM when extracting a backbone.Fixed degree sequence model (FDSM)In the fixed degree sequence model (FDSM), (mathbf {B^*}~in ~{mathscr {B}}^{{text{FDSM}}}) are constrained to have the same agent and artifact degree sequences as ({mathbf {B}}). That is, FDSM constrains the degrees of both the agent and artifact nodes, and requires that these constraints are satisfied exactly, making it a tightly-constrained microcanonical ensemble. Adopting the FDSM implies, for example, that in all possible worlds a given species is found on exactly the same number of islands, and a given island is home to exactly the same number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text{FDSM}}}) is unknown, but can be approximated by uniformly sampling (mathbf {B^*}) from ({mathscr {B}}^{text{FDSM}}), constructing (mathbf {P^*}), and saving the values (P^*_{ij}). In the studies below, we use 1000 samples of (mathbf {B^*}) generated using the ‘curveball’ algorithm, which is among the fastest methods to sample ({mathscr {B}}^{text{FDSM}}) uniformly at random39,40. The FDSM has been used to extract the backbone of bipartite projections of, for example, movies co-liked by viewers21 and conference panel co-participation by scholars41,42.The FDSM offers an intuitively appealing approach to extracting the backbone of bipartite projections because it fully controls for both bipartite degree sequences, which are known to be responsible for many of the projection’s structural characteristics15,16. However, because the distribution of (P^*_{ij}) must be computed via Monte Carlo sampling, it is computationally costly, making it impractical for all but relatively small bipartite projections. There are at least three distinct computational challenges. First, although the curveball algorithm is the fastest among existing methods for randomly sampling a bipartite graph with fixed degree sequences (i.e. for sampling (mathbf {B^*}) from ({mathscr {B}}^{text{FDSM}})), it still can require several seconds per sample for large graphs. Second, once a (mathbf {B^*}) has been sampled, constructing each (mathbf {P^*}) requires matrix multiplication, which must be performed repeatedly and has complexity of at least ({mathscr {O}}(n^{2.37}))43. Finally, computing an edge’s p value (i.e. (Pr (P^*_{ij} ge P_{ij}))) with sufficient precision to achieve a specified familywise error rate that controls for Type-I error inflation due to multiple testing22 can require these sampling and multiplication steps to be performed a very large number of times (see Supplementary Text S2).These computational challenges have led researchers to develop other backbone models3,9,18. Many such models exist, however here we are focused on identifying methods that yield backbones similar to what would be obtained using FDSM, and thus which may serve as computationally-feasible alternatives to FDSM. Therefore, we consider only those models whose ensembles involve at least one of the two types of constraints imposed by FDSM. That is, we consider models that either (1) impose exact constraints, or (2) impose constraints on both the agent and artifact degrees.Fixed fill model (FFM)In the fixed fill model (FFM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FFM}}}) are simply constrained to contain the same number of 1s as ({mathbf {B}}). That is, the FFM constrains only the number of edges, but requires that this constraint is satisfied exactly. Adopting the FFM implies, for example, that in all possible worlds only the total number of species-island pairs is fixed, but any given species may be found on a different number of islands and any given island may be home to a different number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FFM}}}) has not been described before, but is derived in Supplementary Text S1.1. We call it a Jacobi distribution because it is related to Jacobi polynomials.Fixed row model (FRM)In the fixed row model (FRM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FRM}}}) are constrained to have the same agent degree sequence as ({mathbf {B}}), but have unconstrained artifact degree sequences. That is, the FRM constrains the degrees of the agent nodes, and requires that this constraint is satisfied exactly. A canonical variant of the FRM, the (hbox {BiPCM}_r), also constrains the degrees of the agent nodes, but only requires this constraint to be satisfied on average; we do not consider it here because it involves neither of FDSM’s constraints9. Adopting the FRM for backbone extraction implies, for example, that in all possible worlds a given species is found on the same number of islands, but a given island may be home to a different number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FRM}}}) is hypergeometric (see Supplementary Text S1.2), and for this reason it is sometimes referred to as the hypergeometric model22,23,44. The FRM has been used to extract the backbone of bipartite projections of, for example, movies co-starring actors22, papers co-written by authors22, parties co-attended by women44, majority opinions joined by Supreme Court justices44, and microRNAs co-associated with diseases45.Fixed column model (FCM)In the fixed column model (FCM), (mathbf {B^*}~in ~{mathscr {B}}^{{text {FCM}}}) are constrained to have the same artifact degree sequence as ({mathbf {B}}), but have unconstrained agent degree sequences. That is, the FCM constrains the degrees of the artifact nodes, and requires that this constraint is satisfied exactly. A canonical variant of the FCM, the (hbox {BiPCM}_c), also constrains the degrees of the artifact nodes, but only requires this constraint to be satisfied on average; we do not consider it here because it involves neither of FDSM’s constraints9. Adopting the FCM for backbone extraction implies, for example, that in all possible worlds a given species may be found on a different number of islands, but a given island is home to the same number of species. The distribution of (P^*_{ij}) arising from ({mathscr {B}}^{{text {FCM}}}) has not been described before, but is derived in Supplementary Text S1.3, where we show it is Poisson-binomial.Stochastic degree sequence model (SDSM)Finally, the stochastic degree sequence model (SDSM) takes ({mathscr {B}}^{{text {SDSM}}}) to be all binary (m times n) matrices, but also gives a process for generating these matrices with different probabilities. Each (mathbf {B^*}) is generated by filling the cells (B^*_{ik}) with a 0 or 1 depending on the outcome of an independent Bernoulli trial with probability (p^*_{ik}). The distribution of the random variable (P^*_{ij}) arising from ({mathscr {B}}^{{text {SDSM}}}) is Poisson-binomial with parameters which can be computed using the (p^*_{ik}) (see Supplementary Text S1.4)27,46. There are many ways to choose (p^*_{ik}), but in the studies below we choose (p^*_{ik}) so that it approximates (Pr (B^*_{ik} = 1)) for (mathbf {B^*}~in ~{mathscr {B}}^{{text{FDSM}}}). This choice of (p^*_{ik}) ensures that the SDSM constrains the degrees of both the agent and artifact nodes, but only requires these constraints to be satisfied on average. Adopting such a version of SDSM implies, for example, that in each possible world a given species may be found on many or few islands and a given island may be home to many or few species, but the average number of islands on which a given species lives in all possible worlds and the average number of species that live on an given island in all possible worlds matches these values the observed world. The SDSM has been used to extract the backbone of bipartite projections of, for example, legislators co-sponsoring bills1,18,47,48,49, zebrafish (Danio rerio) sharing operational taxonomic units50, countries sharing exports3, and genes expressed in genesets51. More

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