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Generalized adjacency and the conservation of gene clusters in genetic networks defined by synthetic lethals
BMC Bioinformatics volume 13, Article number: S8 (2012)
Abstract
Background
Given genetic networks derived from two genomes, it may be difficult to decide if their local structures are similar enough in both genomes to infer some ancestral configuration or some conserved functional relationships. Current methods all depend on searching for identical substructures.
Methods
We explore a generalized vertex proximity criterion, and present analytic and probability results for the comparison of random lattice networks.
Results
We apply this criterion to the comparison of the genetic networks of two evolutionarily divergent yeasts, Saccharomyces cerevisiae and Schizosaccharomyces pombe, derived using the Synthetic Genetic Array screen. We show that the overlapping parts of the networks of the two yeasts share a common structure beyond the shared edges. This may be due to their conservation of redundant pathways containing many synthetic lethal pairs of genes.
Conclusions
Detecting the shared generalized adjacency clusters in the genetic networks of the two yeasts show that this analytical construct can be a useful tool in probing conserved network structure across divergent genomes.
Introduction
As two related organisms diverge through evolutionary time, functional relationships among genes may alter. Some relationships may weaken, others strengthen, some may disappear while new ones appear. New genes or variants of genes may take on specific functions, while other genes may be inactivated or lost. And these changes proceed independently in the two evolving species. Even if most changes are local, affecting one or two relationships and two or three genes, after a long enough period of time the inventory of relationships in each of the species may reflect relatively little of the original pattern in the common ancestor, and may be quite different from each other.
Given two graphs representing functional genetic networks of two organisms, then, it may be difficult to decide if the local structures are similar enough in both graphs to infer some ancestral configuration or some conserved functional relationships. Current methods all depend on searching for identical substructures [1]. We have recently explored the notion of generalized adjacency to compare chromosomal gene ordering in two or more genomes [2–4] as way of parametrizing the relative importance of conserved gene order versus total gene content within a cluster. However, this concept is not tied to the physical nature of chromosomes; it has a graphtheoretical definition based solely on the adjacency of pairs of genes as a consequence their linear order along the chromosome. As such it is applicable to more general graphs. In this paper we will use generalized adjacency to compare the genetic networks of two species, representing the functional interaction between their genes.
Our work falls in the tradition of situating small world networks between regular lattice structures, with their dense local connections throughout, and completely random graphs with their short characteristic path lengths. Small world networks tend to have both properties, as discussed by Goldberg and Roth [5]. In the next section we define generalized vertex adjacency in a graph, and generalized adjacency clusters. Since these definitions involve a parameter, we invoke our previous work on finding a "natural" value for this parameter, and discuss its application to networks. We then sketch some analytic results on the distribution of the number of generalized adjacencies in the comparison of two randomly labelled regular lattices, and propose a general result for the comparison of two arbitrary graphs on the same set of vertices.
We apply our concepts to the comparison of genetic networks of Saccharomyces cerevisiae and Schizosaccharomyces pombe. The networks were obtained using Synthetic Genetic Array screens for "synthetic lethals" among virtually all pairs of genes whose individual inactivation is not lethal [6–8]. Typically, these pairs are organized in two parallel pathways that converge on a common endpoint, as illustrated in Figure 1. These pathways buffer each other so that the inactivation of one or more genes on a single one of the pathways will not affect survival, but inactivating at least one gene on both pathways is lethal.
We discover a pattern of local clustering in the edges common to both networks beyond what is defined by vertex adjacency alone. We suggest this is a consequence of the synthetic lethals methodology for building the networks.
Methods
Generalized vertex adjacency
Let S be a gene network with a gene set V = {1,..., n}. Two genes g and h are iadjacent, and the pair (g, h) is an iadjacency, in the gene network S, written in $g\stackrel{i}{~}h$ in S, if there are i  1 genes between them in S along a shortest path from one gene to the other. We define genes g and h to be (i, j)adjacent, and the pair (g, h) is called an (i, j)adjacency, in two gene networks S and T, if they are iadjacent in either one of the gene networks and jadjacent in the other. We say g is an iadjacent neighbor of the gene h in a gene network S, if g and h are iadjacent in S.
We denote ${E}_{M}^{\mathrm{\Theta}}$ the set of all i adjacencies in a network M , where 1 ≤ i ≤ Θ For two networks S and T with the same vertex set V = {1,..., n}, we define a subset of C ⊆ V to be a (θ, ψ) generalized adjacency cluster, or (θ, ψ) cluster, if all vertices in the subset C are also the whole vertices of a connected component of the graph ${G}_{ST}^{\theta \psi}=\left(V,\left({E}_{S}^{\theta}\cap {E}_{T}^{\psi}\right)\cup \left({E}_{S}^{\psi}\cap {E}_{T}^{\theta}\right)\right).$
To obtain (θ, ψ) clusters of two gene networks, S and T, the new network ${G}_{ST}^{\theta \psi}$ need to be created first. The network ${G}_{ST}^{\theta \psi}$ can be constructed by connecting two genes of gene networks S and T if they are iadjacent in S and jadjacent in T, where max(i, j) ≤ max (θ, ψ) and min(i, j) ≤ min (θ, ψ). Figure 2 illustrates how the grid networks S and T determine the (1, 2) clusters {2,3,4,5,7,9,10,12,13,14,15,19,20},{11,17,18,22} and {16,21,23,24}. Figures 3 and 4 depict the same process for triangular graphs and hexagonal graphs, respectively.
Weight function
The definition of generalized adjacency cluster in the previous section does not discriminate among pairs of (i, j)adjacent genes as long as i and j are less than some cutoff values. However, it seems reasonable to think that (i, j) with smaller i and j should be weighted more heavily in defining clusters. To explore this, consider two networks S and T with the same vertices. Let w_{ ij } be the weight on two vertices that are (i, j)adjacent, i.e., iadjacent in one of the networks and jadjacent in the other, such that

1.
0 ≤ ω_{ ij }= ω_{ ji }, i, j ∈ {1, 2,..., n1}

2.
${\sum}_{i=1}^{n1}{\sum}_{j=1}^{n1}{\omega}_{ij}=1$

3.
ω_{ i, j } ≥ ω_{ k, l } if

(a)
max(i, j) < max(k, l) or

(b)
max(i, j) = max(k, l) and min(i, j) < min(k, l)

(a)
This is a very general class of weights with reasonable monotonicity and total weight conditions. We define the dissimilarity between two gene networks S and T as
where P is the number of pairs (x, y) that are (1, 1)adjacent in two identical gene networks. n_{ ij } is the total number of pairs (x, y) that are iadjacent in S and jadjacent in T. l is the diameter of the network. We have argued elsewhere [4] that the "natural" way of finding weights is to minimize d and we proved the following surprising
Theorem 1. Let ${\alpha}_{k}=\u230a\frac{\sqrt{1+8\left(k1\right)}+1}{2}\u230b.$The weight ω that minimizes d(S, T) has
where k* is an integer and maximizes the function
where n_{ ij } is the number of gene pairs iadjacent on S and jadjacent on T.
This suggests that uniform weights are appropriate for all (i, j) adjacencies up to a certain cutoff. Empirical work indicates that k* is of the order of $\sqrt{n},$where n is the number of vertices in the network and so the cutoff would be for i and j to be less than some value $\alpha \approx {n}^{\frac{1}{4}}.$ E.g., for a network with 100 vertices, it should suffice to consider 2 and 3adjacencies, but 4adjacencies need not be considered.
The expected number of (i, j) adjacencies in two random networks
An essential step in studying gene clusters is to verify their significance. Random networks are often used to estimate the significance of clusters. In this section, we represent some characteristics of the expected number of (i, j) adjacencies in two random networks, which can then be used in evaluating cluster significance.
Theorem 2. Let M be a randomly labelled square grid network with N vertices. Then the number of i adjacencies, n_{ i }, in the network M converges in distribution to the Poisson with parameter
the expected number of i adjacencies in the network M.
Proof. Because M is a random square grid network, we can use a coordinate system to represent it. Vertices in the network correspond to the points in the plane with integer coordinates, xcoordinates being in the range 1,..., m, ycoordinates being in the range 1,..., n, where N = mn. Without loss of generality, we set m ≤ n. Two vertices in the network are iadjacent if the L_{1} distance between them in the integer coordinates is i.
Let ${Y}_{M}^{i}\left(u,v\right)$be 1 if vertices u, v are iadjacent in the network M and 0 for otherwise. Then n_{ i } = ${\sum}_{\left(u,v\right)}{Y}_{M}^{i}\left(u,v\right).$Since most vertices have 4i iadjacent neighbors, we can show that
where the error term is due to edge effects [9]. Since N = mn,
where the error term includes the edge effects detailed in equation(5). Then
Therefore, based on the proof of Theorem 2 in [10], we can conclude that n_{ i } converges in distribution to the Poisson with parameter E(n_{ i }) the expected number of i adjacencies in the network M.
Theorem 3. For S and T two random square grid networks with the same N vertices, the number of pairs of vertices n_{ ij } that are iadjacent in S and jadjacent in T converges in distribution to the Poisson with parameter
the expected number of (i, j) adjacencies in networks S and T.
Proof. Let ${Y}_{S}^{i}\left(g,h\right)$be 1 if vertices g, h are iadjacent in the random square grid network S and 0 otherwise. Similarly, define${Y}_{T}^{j}\left(g,h\right)$to be 1 if vertices g, h are jadjacent in the random square grid network T and 0 otherwise. Let ${Y}_{\left(S,T\right)}^{\left(i,j\right)}\left(g,h\right)$be 1 if vertices g, h are iadjacent in S and jadjacent in T. Otherwise ${Y}_{\left(S,T\right)}^{\left(i,j\right)}\left(g,h\right)=0.$ Because of the independence of g, h being iadjacent in S and jadjacent in T, the probability that g and h are (i, j)adjacent in S and T is
So the expected number of (i, j)adjacencies in the two networks S and T is
The term ${\sum}_{\left(g,h\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{in}}\phantom{\rule{0.3em}{0ex}}S,T}1$in equation (10) represents the total number of (g, h) combinations in two networks S and T based on pairs of location of (g, h) in S and T. There are $\frac{1}{2}N\left(N1\right)$ pairs of location possible for (g, h) in each of two networks and 2 alternatives for each gene pair (g, h) in S and T. So ${\sum}_{\left(g,h\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{in}}\phantom{\rule{0.3em}{0ex}}S,T}1=\frac{1}{2}{N}^{2}{\left(N1\right)}^{2}.$ Hence, the expected number of (i, j)adjacencies in the two networks S and T is
Therefore, based on the proof of Theorem 2 in [10], we can conclude that n_{ ij } converges in distribution to the Poisson with parameter E(n_{ ij }), the expected number of (i, j) adjacencies in networks S and T.
More generally we can use the same techniques to prove Theorems 4 and 5:
Theorem 4. Let D be the degree of a gene in the random genetic grid network, i.e. the number of 1adjacent neighbors of this gene in the network. For two random genetic lattice networks S and T with same genes, the number of pairs of genes n_{ ij } that are iadjacent in S and jadjacent in T converges in distribution to the Poisson with parameter
Even for networks as small as 400, simulations indicate that the distribution of n_{ ij } is close to the Poisson in Theorem 4, for square (D = 4), hexagonal (D = 3), triangular (D = 6) grids as well as linear networks (D = 2). Looking beyond regular networks:
Theorem 5. Let D_{ k }(M) be the number of kadjacent gene pairs in the random network M. For two random networks S and T with same vertices, the number of pairs of vertices n_{ ij } that are iadjacent in S and jadjacent in T converges in distribution to the Poisson with parameter
Results: genetic networks in S. cerevisiae and S. pombe
Dixon et al. [8] presented an extraordinary comparison of the genetic networks of Saccharomyces cerevisiae and Schizosaccharomyces pombe, two rather distant yeast genomes. Their results are summarized in their Figure 2, which we reproduce here as Figure 5. We separated the two overlapping networks based on the colours in this diagram, as depicted in Figure 6.
We compiled the graphtheoretical characteristics of these networks: number of vertices, average vertex degree, number of edges, and present them in Table 1. The details of the vertex degree distributions are given in Figure 7.
We then carried out a number of simulations. First, we simulated random networks having the same statistical characteristics as in Table 1 and Figure 7. This showed the random networks to be deficient in (2,2), (3,3) and (4,4)clusters of genes compared to the yeast networks, under all of the (1,1), (2,2) or (3,3)adjacency criteria (see Table 2). In passing, we mention that the analysis of regular grid networks 7 earlier in this paper predicts very much smaller numbers of clusters than the random networks. Second, we fixed the common edges in both yeasts to initialize the random networks, and then generated the rest of the edges in conformity with Table 1 and Figure 7. This assured the (1,1)adjacency results would be the same or close to the yeast results (see Table 2), but again the yeast networks showed a significant excess of clusters under (2,2)adjacency. (The significance can be verified in Figure 8.)
One of the factors responsible for the increase in clusters under 2adjacency is the incidence of parallel buffering pathways in the genetic organization of these yeasts. Figure 9 illustrates how such pathways determine subgraphs in the network that are essentially bipartite. There are no 1adjacencies among the genes in a single pathway, but the backandforth pattern of edges between the two sides of the bipartite structure ensures that under 2adjacency, the genes in both pathways participate in clusters of various sizes.
As for the observation that 3adjacency does not increase the number of clusters over random networks more than is achieved by fixing the common edges, this is partly explained by the fact that the yeast show only about 50% more clusters of each size than the random network, compared to the 250% 600% under 2adjacency. Increasing the adjacency parameter in these networks simply results in large numbers of random clusters that swamp any subtle distinction between the fixed edge simulation and the yeast network.
Conclusions
Generalized adjacency is a flexible but rigorous concept in the search for patterns of similarity among genetic networks. Although we analytically calculate properties of regular grid networks, e.g., linear, triangular, square and hexagonal grids, and though the average vertex degree of the empirically derived networks is in the same range as the hexagonal and square grids, the predicted number of clusters is much higher in the real data. This can be attributed in large part to the dispersion of the degree distribution, which is nonexistent for the grids.
Of greater interest is the inability of random networks with the same characteristics as the real network to generate the same number of clusters. This is largely due to the small number of common adjacencies in the random networks, but even when this is forced to be the same, the yeast data showed an unexpected pattern of increased clustering under (2, 2)adjacency, for all sizes of cluster (see Table 2). This was partly explicable in the way the networks were constructed using the synthetic lethals screen.
In conclusion, generalized adjacency is potentially a useful tool in exploring the special combinatorial structure of genetic networks.
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Acknowledgements
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 9, 2012: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2011: Bioinformatics. The full contents of the supplement are available online at http://0www.biomedcentral.com.brum.beds.ac.uk/bmcbioinformatics/supplements/13/S9.
Research supported in part by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC). DS holds the Canada Research Chair in Mathematical Genomics.
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ZY and DS formulated the problem, carried out the calculations and simulations, and wrote the paper. Both authors read and approved the final manuscript.
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Yang, Z., Sankoff, D. Generalized adjacency and the conservation of gene clusters in genetic networks defined by synthetic lethals. BMC Bioinformatics 13, S8 (2012). https://0doiorg.brum.beds.ac.uk/10.1186/1471210513S9S8
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DOI: https://0doiorg.brum.beds.ac.uk/10.1186/1471210513S9S8
Keywords
 Random Network
 Small World Network
 Genetic Network
 Generalize Adjacency
 Synthetic Lethal