Suppose that there are m genes with transcription levels measured on recombinant offspring from an experimental genetic cross. (In the yeast experiment we consider, m = 6,216.) The goal is to use the data from such an experiment to estimate the probability that the transcription of gene i has a causal regulatory effect on the transcription of any other gene j, which we denote by P_ij, where 'causal regulatory effect' means that a change in the transcription level of gene i results in a predictable change in the level of gene j. This is not necessarily through a direct molecular interaction; however, if we directly modulate the transcriptional level of gene i, then this should result in a corresponding change in the transcriptional level of gene j. Trigger provides a conservative estimate of these probabilities, denoted by P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ for i = 1, ..., m and j = 1, ..., m.

These estimated regulatory probabilities can be used to build a regulatory network based on a directed graph. The probability that a directed edge exists from gene i to gene j in the network is estimated by P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@. One can directly threshold the entries, essentially setting those not meeting the threshold equal to zero. For example, one could remove all potential edges with P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ < 90% while including those with P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ 90%. Therefore, a directed edge would be drawn from gene i to gene j if and only if P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ 90% (Figure 2). The resulting network has an easily quantified and interpretable FDR, and each directed edge has an estimated probability that it is true (see Materials and methods [below] and Additional data file 1).

In addition to constructing a regulatory network from these estimated probabilities, each gene i can be examined as a putative regulator, and hence a quantitative trait gene or 'quantitative trait transcript' 34. Specifically, the probability that a specific gene i is a regulator for each other gene j is estimated as P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@. A threshold can be applied to these estimated probabilities to obtain the FDR of the significant genes (see Materials and methods [below] and Additional data file 1). This particular application of Trigger allows one to move beyond identifying QTL of expression traits to identifying a specific underlying causal quantitative trait transcript.

Trigger is based on a rigorous mathematical framework that we developed for utilizing randomized genetic backgrounds and genome-wide expression in order to test rigorously for causality among transcription levels. The approach starts with a pair of transcripts and a locus to which both are linked. Let L be the locus, T_i transcript i, and T_j transcript j.

The goal is to identify triplets (L, T_i, T_j) such that L → T_i → T_j, where the arrow '→' means causation. The definition of 'causal' has been a topic of much interest 1819. Although definitions of causality differ slightly among the many articles published on this topic, in essence T_i → T_j means that the ideal manipulation of T_i will change the distribution of T_j, whereas the ideal manipulation of T_j will not disturb the distribution of T_i. 'Ideal manipulation' of a variable means to change the variable in a manner that leaves every other variable unchanged, at the moment when the manipulation occurs 35. This framework also applies to causality among random variables.

With the genetic cross experimental design, the genotype at a fixed locus L is a random variable, whose random outcome occurs before and independently from the subsequently measured expression values. For example, in the yeast experiment analyzed below, two haploid parental strains (BY and RM) were crossed to produce 112 recombinant haploid segregant strains. Because of the random segregation of chromosomes during meiosis, the inheritance of L = BY or L = RM is random. Therefore, when measuring the alleles at a single locus L across 112 segregants, we observe 112 genotypes being generated from some probability distribution. (See Materials and methods [below] for explicit details on the assumptions we make about the randomized genotypes among the loci.)

Because the randomization of L takes place before the expression levels of T_i are measured, this implies that if T_i is linked to locus L then L → T_i. This property is due to the well established principles in statistics showing that an association between two variables when one of them is properly randomized implies causation 1920. Additionally, the randomization of L is carried through to the variation in T_i whenever L → T_i. If L → T_i, then segregants with L = BY have a different mean expression for T_i than segregants with L = RM. Therefore, the randomization of L provides a randomization of the mean level of expression for T_i. Figure 1a shows the transcriptional levels for a given gene, and Figure 1b shows a case in which it is linked to some locus L. Because the inherited allele L = BY or L = RM is random for each segregant, the mean level of expression for T_i is random when L → T_i.

Importantly, some of the variation in T_i will not be explained by L, specifically the random fluctuations of the transcription levels within each genotype (Figure 1b). Therefore, it is not possible to conclude that T_i → T_j whenever T_i and T_j are significantly associated to L. This follows because there could be a common hidden variable affecting both T_i and T_j. (Note that if T_i were perfectly randomized, then there would be no causal hidden variable for T_i, which demonstrates the power of randomization.) Suppose that a hidden variable H is such that H → T_i and H → T_j. Because of this common hidden causal variable, any association between T_i and T_j would not allow us to conclude that T_i → T_j even though T_i has been partially randomized. In other words, the partial randomization of T_i caused by L is now confounded by the effect of H. The common causal hidden variable H does not prevent T_i → T_j from occurring; rather, we just are unable to draw any conclusion when this is the case, unless we are willing to model common hidden causal variables. Modeling common hidden causal variables has been shown to be particularly challenging in this high-dimensional setting 36, and doing so would require much additional work.

If there is a common causal hidden variable H that affects both T_i and T_j, then the Trigger method is designed to not make any conclusions about causality. However, if there is not a common hidden causal variable, then it is now possible, in a straightforward manner, to determine whether T_i → T_j. The following new theorem identifies three conditions that are equivalent to the case in which both L → T_i → T_j and no common causal hidden variable affects both T_i and T_j. (See Materials and methods [below] for a mathematical proof.)

We applied the Trigger algorithm to the yeast experiment (Materials and methods [below]) and found several interesting characteristics of the resulting regulatory probability matrix. Table 1 lists the overall significance results with different probability thresholds and Additional data file 2 contains the entire regulatory probability matrix. For example, at a probability threshold of 90%, we found 4,394 significant regulatory relationships among 2,145 genes where 127 are causal. Figure 2 shows a regulatory network drawn from the Trigger results at this threshold, where a directed edge is drawn from gene i to gene j if and only if P_ij ≥ 90%. It can be seen from Figure 2 that we have constructed a highly interconnected network where there is clearly a 'hub structure'.

We examined in detail four genes as putative regulators: CNS1 on chromosome 2, ILV6 on chromosome 3, SAL1 on chromosome 14, and NAM9 on chromosome 14. Each was highly significant for cis linkage, and the locus of each putative regulator had many significant secondary linking genes. At a 90% posterior probability cut-off (FDR = 6%), 144, 51 and 36 genes were significant for being regulated by CNS1, ILV6, and SAL1, respectively. At an 80% posterior probability cut-off (FDR = 11%), 14 genes were significant for being regulated by NAM9. The significant genes, posterior probabilities, and other relevant information for each putative regulator can be found in Additional data file 3. Note that each of these putative regulators is also a significant quantitative trait gene (or quantitative trait transcript) for each expression trait that it significantly regulates. Figure 3 shows heat maps of the four putative regulators and their corresponding significantly regulated genes. It can be seen that each significant gene is both linked to the locus of the putative regulator and has correlated expression with the regulator within each genotype, both of which are necessary but not sufficient for causality.

In order to determine whether the genes that are significant for each putative regulator show a coherent functional relationship, we employed the Gene Ontology (GO) database 37. For each putative regulator, we queried the database among all significant genes and the regulator itself. This approach takes independently performed experiments and synthesizes the information obtained from those. The GO searches allowed us to test specifically whether common processes, functions, and components are present among each set of genes. Indeed, we found an abundance of significance for enriched GO terms for each set of genes corresponding to a putative regulator.

Figure 4 shows the results of GO analysis for the putative regulator NAM9, which is a mitochondrial ribosomal component of the small subunit and inviable under deletion 38. It is a structural constituent of ribosome, involved in translation and mitochondrial small ribosome subunit 394041. For the 14 genes significant at an 80% posterior probability threshold (FDR = 11%), 13 are known to be in the same or similar pathway as NAM9. The other significant gene is heretofore uncharacterized. Translation, structural constituent of ribosome, and mitochondrial small ribosome subunit are all highly significant terms in the GO tree.

Additional data file 1 (Figure S1) shows the results for the putative regulator CNS1, which is an essential tetratricopeptide repeat (TPR)-containing co-chaperone, deletion of which is inviable 42. It binds both heat shock protein 82p (Hsp82p) and Ssa1p (Hsp70), and stimulates the ATPase activity of SSA1. CNS1 is involved in the protein binding process, and its cellular component is associated with cytoplasm 42434445. Of the 144 genes significant at the 90% joint posterior probability cut-off (FDR = 6%), a substantial subset is involved in transferase activity and ribosome biogenesis and assembly, which coincides with the key role played by CNS1 in yeast. Many of the 144 genes were also found to be in the same pathway as CNS1; for example, TRM8 and CNS1 are both involved in a pathway for protein binding 4647.

Additional data file 1 (Figure S2) shows the significant GO results for ILV6 and its 51 genes under statistically significant regulation. ILV6 is a regulatory subunit of acetolactate synthase, which catalyzes the first step of branched-chain amino acid biosynthesis 4849. Amino acid biosynthesis and its associated pathways are significantly enriched GO terms with P values below 10^-10. Cyclohydrolase activity and lyase activity are some other significant pathways identified by GO analysis.

The putative regulator SAL1 is a probable transporter and a member of the calcium-binding subfamily of the mitochondrial carrier family, with two EF-hand motifs. It works in transporter activity and calcium ion binding 50, with its corresponding cellular component involved in the mitochondrial inner membrane 51. From the GO analysis (Additional data file 1 [Figure S3]), we can see that a number of the 36 genes significantly regulated by SAL1 are associated with the mitochondrian and membrane GO terms. Six of the 36 significantly regulated genes are involved in mitochondrial inner membrane with high statistical significance (P < 10^-8), a trend that is consistent with previous findings 5051.

It should be noted that in the case of SAL1 no polymorphism exists in the immediate 500 base regions upstream or downstream of the SAL1 open reading frame. The linkage peaks occur approximately 13 kilobases and 21 kilobases on either side. This illustrates that linkage does not have to be due to an unequivocally cis-acting regulatory polymorphism in order for Trigger to work. On the contrary, there must simply be some locus to which both expression traits T_i and T_j are linked. We justified limiting the locus L to be in the 50 kilobases region of T_i based on computational and statistical increases in efficiency (Additional data file 1).

In addition to these four well characterized putative regulators, we noticed that expression levels of a number of genes with relatively unknown function (for instance, YSW1, PHM7, and so on), were predicted to regulate a number of genes, with significant GO terms appearing for each set. Therefore, our results can potentially be used to predict properties of relatively unknown genes as well. Furthermore, several transcription factors significantly regulated a number of genes, including HAP1 5253 and RAD16 5455. In previous work it was found that mutations in GPA1 and AMN1 lead to expression changes in genes whose expression exhibits linkage to each respective locus 14. Missense mutations (leading to amino acid changes in the protein product) were identified in both GPA1 and AMN1 that appear to be the cause of the expression changes in the linking genes. In work to be reported in the future we examine the GPA1 and AMN1 cases in detail, showing that there appears to be common causal hidden variables involved. The Trigger approach is extended to take into account these common causal hidden variables, allowing us to recapitulate the previous findings regarding GPA1 and AMN1.

We have presented Trigger within the context of inferring regulatory relationships based on gene expression data from organisms with randomized genetic backgrounds. However, this method may actually be applied to a much broader class of data types. Because the estimation is done in a nonparametric and scale-free manner (Materials and methods [below] and Additional data file 1), it is possible to combine any combination of expression, proteomic, metabolomic, and phenotypic data as the variables among which causal relationships are inferred. These may be considered separately or simultaneously, allowing one to discover regulatory relationships, say, among protein levels and transcriptions levels. The general requirement is that one must acquire organisms with random genetic backgrounds that are essentially stable as the expression levels and other potential traits are measured. The computational approach and statistical principles underlying the method remain the same for all of these data types.

The expression and genotype data were recently reported elsewhere 1233. In that work, 112 segregants (one from each tetrad) were grown from a cross involving parental strains BY4716 (isogenic to the laboratory strain S288C) and the wild isolate RM11-1a. RNA was isolated and cDNA was hybridized to microarrays in the presence of the same BY reference material. Each array assayed 6216 yeast open reading frames. GeneChip Yeast Genome S98 microarrays were purchased from Affymetrix (Santa Clara, CA, USA). Genotyping was performed using GeneChip Yeast Genome S98 microarrays (Affymetrix) on all 112 F₁ segregants. The resulting genetic map of 3,312 markers covered more than 99% of the genome.

We simply point out here that the main assumption regarding random genotypes is that the L_i are random variables occurring before and independently from the subsequently measured expression values. We also assume that the alleles inherited by different individuals at a fixed locus occurs independently; in other words, we assume that the crosses have been carried out independently. (If related segregants or offspring are collected, then Trigger can be adjusted to account for this.) However, we do not assume that the inheritance at several loci on a given chromosome occurs independently, and we make no other assumptions about independence of inheritance among loci. Segregation distortion, selection, and other traditionally problematic issues arising when performing genetic crosses for the purpose of genetic mapping do not invalidate Trigger.

As in all genetic crosses, the more independent the inheritance of the loci is, the more information there is in the experiment. For example, suppose that loci L_i and L_k are dependent (for instance, they are located on the same chromosome, or their segregation is dependent because of selection). Suppose also that L_i → T_i → T_j and L_k → T_j, but it is not the case that L_k → T_i. Because L_i and L_k are dependent, it will not be the case that L_i ⊥ T_j | T_i, as not all linkage information for T_j is captured by T_i. Specifically, L_i contains some information about L_k because of their dependence, so T_j | T_i is not independent from L_i. This is an example of how dependence of inheritance of different loci can reduce the power of Trigger. However, Trigger does not produce false positives because of this, so it is robust to linkage among loci on the same chromosome or other forms of dependence among loci.

The proof of the theorem follows from well-established theory in graphical and causal modeling 356163. Several basic assumptions are typically made in causal modeling to avoid nonsensical situations. The 'causal Markov assumption' states that in a causal model, each variable is independent of all of its non-descendants given information about all of its direct causes. The 'faithfulness assumption' states that any conditional independence relationships in the population exist in the presence of the causal Markov assumption. Under the faithfulness assumption, conditional independence of two variables implies there is no direct edge between the two. Our proof also relies on the known result that if a hidden variable is causal for both X and Y, then the directed graph associated with X and Y can be represented by X → Y 63.

We first show that if L → T_i → T_j with no hidden variables causal for both T_i and T_j, then L → T_i, L → T_j, and L ⊥ T_j | T_i. Under these assumptions, the first two properties (L → T_i and L → T_j) are trivially true. Because there are no hidden variables involved, T_i is the only direct cause of T_j, and L is a non-descendant of T_j, it follows by the causal Markov assumption that the third property (L ⊥ T_j | T_i) holds.

We now show the more important direction of this equivalence: if L → T_i, L → T_j, and L ⊥ T_j | T_i, then L → T_i → T_j and there are no hidden variables causal for both T_i and T_j. The third property (L ⊥ T_j | T_i) implies that there is no direct edge between L and T_j by the faithfulness assumption.

Let us first consider the case when there are no hidden variables causal for both T_i and T_j, so that the only variables involved in this causal graph are L, T_i, and T_j. Because of the second property (L → T_j), and there is no direct edge between L and T_j, it must follow that there is a direct edge between T_i and T_j. Otherwise, T_j is completely independent of L, which violates the second property. Thus, L → T_i - T_j, where an edge without arrowheads implies dependence. If any two variables are dependent, then one is a cause of the other or there must be a third variable causal for both 63. Thus, either T_i is causal for T_j, or T_j is causal for T_i, or both cases are true. L cannot be the common direct cause for both T_i and T_j, because no direct edge exists between L and T_j. If L is an indirect cause of T_j, then T_i as the only other variable in the graph must be a direct cause of T_j, implying that T_i → T_j. If T_j → T_i and the first property (L → T_i) holds, then it cannot be the case that the third property (L ⊥ T_j | T_i) holds. Thus, T_j is not causal for T_i but it is true that T_i → T_j, implying that L → T_i → T_j.

Now consider the second case in which there might be causal hidden variables in the graph. Because L is an independently randomized, static variable, there cannot be any hidden variables causal for both L and T_i or both L and T_j. The only possible existence of hidden causal variable in this graph is one affecting both T_i and T_j. However, if there is a common hidden cause for T_i and T_j, then T_i → T_j 63. If this is true, then T_j | T_i is dependent with L, contradicting the third property (L T_j | T_i). Therefore, L → T_i → T_j with no hidden variables affecting either of the two.

Note that it can be shown that the second and third properties (L → T_j and L ⊥ T_j | T_i, respectively) imply the first property (L → T_i). However, we have designed Trigger to test for all three properties because conditioning on the first property increases the power to detect the state of the second and third properties.

Text

The following method was developed to estimate the regulatory probabilities. Recall that by the causality equivalence theorem:

P_ij = Pr(L_i → T_i → T_j)

= Pr(L_i → T_i) × Pr(L_i → T_j | L_i → T_i) × Pr(L_i ⊥ T_j | T_i | L_i → T_i and L_i → T_j)

To compute the joint posterior probability, the probabilities on the right hand side of the equation are estimated from left to right in that respective order. The basic algorithm works as follows (with specific details following) (Note that further details about steps 1 to 6 can be found in Additional data file 1.)

A significance threshold can be applied to the probabilities for either the entire regulatory probability matrix or for a specific putative regulator. For the entire probability matrix, this would entail applying a threshold λ to the P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@, where we call L_i → T_i → T_j significant if and only if P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ λ. For a given putative regulator, the exact same thresholding would take place, except only the P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ for a fixed putative regulator, gene i, would be considered. The estimate of the FDR corresponding to λ, FDR(λ), is as follows:

F D ^ R ( λ ) = ∑ i , j ( 1 − P ^ i j ) 1 ( P ^ i j ≥ λ ) # { P ^ i j ≥ λ } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8GiVeY=Pipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacaWGgbGabmirayaajaGaamOuaiaacIcaiiGacqWF7oaBcaGGPaGaeyypa0tcfa4aaSaaaeaadaaeqaqaaiaacIcacaaIXaGaeyOeI0IabmiuayaajaWaaSbaaeaacaWGPbGaamOAaaqabaaabaGaamyAaiaacYcacaWGQbaabeGaeyyeIuoacaGGPaGaaGymaiaacIcaceWGqbGbaKaadaWgaaqaaiaadMgacaWGQbaabeaacqGHLjYScqWF7oaBcaGGPaaabaGaai4iaiaacUhaceWGqbGbaKaadaWgaaqaaiaadMgacaWGQbaabeaacqGHLjYScqWF7oaBcaGG9baaaaaa@531D@

Where 1(P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ λ) is 1 or 0 according to whether P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ λ or not, respectively, and # {P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ λ} is the total number of P^ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqee0evGueE0jxyaibaiKI8=vI8viVeY=Nipec8Eeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGabmiuayaajaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@328F@ ≥ λ 1765. Further details and justification can be found in Additional data file 1.