In a recent study, it was shown by Arndt and Duret 22 that besides the positive correlation with recombination rate (RR), GC-content (GC%) is negatively correlated with the distance to telomere (DT). These results were mainly based on the analysis of noncoding sequence in a 1 Mb sized window that have high quality finished sequence available both in the chimpanzee and the macaque genome 22. We start our analysis by using both their data and our own dataset of the same 1 Mb windows for the human genome sequence, regardless of coding and noncoding status or the existence of quality sequence in other organisms. The GC% in these two datasets is not totally identical, but highly correlated (ρ = 0.98). Similarly, the HapMap estimate of RR 27 in the two datasets is correlated with ρ = 0.82. We discarded windows, if the number of HapMap single-nucleotide-polymorphism (SNP) is less than 20 or more than 30% of genomic sequence are missing. In total, 2647 and 2668 1 Mb windows are available with information on GC%, RR and DT for the two datasets. We performed log-transformation of distance to telomere (DT), because the scatter plot showed a non-linear correlation between DT with the other two variables, and then multiplied it by -1 to change the negative correlation with GC% to positive. The unconditional and conditional Pearson's correlation coefficients between GC%, RR and DT are shown in Table 1. All correlation coefficients are highly significant (p-value = 0) and results from both datasets are highly similar. Because an earlier study had observed that the correlation between RR and GC% is maximal when both variables are measured in the 50 kb window 15, we also looked at a dataset where GC%, RR, DT are measured by using the window size of 50 kb. Due to the smaller window size (1/20 of the 1 Mb window), RR is fluctuating in a much wider range as can be seen from the quantile values in Table 2. We also note that a square-root transformation of RR under 50 kb window leads to a slightly better linear correlation with GC%, and a larger correlation coefficient (result not shown).

The correlation and partial correlation between the three variables from 50 kb window is shown in Table 3. In contrast to 15, we found the correlation between GC% and RR to be higher using the 1 Mb sized window than the 50 kb window. This discrepancy may result from the threefold higher SNP density provided by the HapMap phase II 27. Importantly, the correlation between GC% and DT is less affected by the change of window size, although RR-DT correlation is far weaker in the 50 kb window than in the 1 Mb window. This change of the strength of the correlation of RR with GC% and DT from one window size to another may be related to the "domains within domains" phenomenon that had been found for GC-content variation and that may exist for fine-scale recombination rate variation too.

Because none of the pairwise correlations between GC%, RR and DT is rendered insignificant by conditioning on the third variable, it is not possible to remove any edge in the relationship graph for GC%, RR, and DT (Figure 1(A)).

In the next step, we checked the chromosome-specific correlations and partial correlations between the three variables. Table 4 shows these result in form of correlation and partial correlation coefficients (and p-value if it is larger than 0.01) for our main dataset (1 Mb window including all available human genome sequence independent from its coding status). There are several notable observations: (1) RR-log(1/DT) correlation is unchanged by conditioning on GC% for non-acrocentric chromosomes, indicating that the position of the window already explains RR, rendering GC% unlikely to be causal. (2) For acrocentric chromosomes (13, 14, 15, 21, 22), the position of the window (DT) is only marginally correlated with RR. In contrast, DT is correlated with GC% for all chromosomes including the acrocentric chromosomes. (3) For some (3, 7, 8, 9, 10, 11, 12, 18, 19), but not for all, chromosomes, the correlation between GC% and RR is weakened by conditioning on DT.(4) For chromosome 2 the positive correlation between RR and DT is not turned negative by conditioning on GC. This result is interesting, because chromosome 2 is known to result from a relatively recent fusion event of different chromosomes 3233

To examine the robustness of these chromosome specific correlations (Table 4), we carried out the same correlation analysis using the noncoding sequence 1 Mb windows 22 and the 50 kb window (Figure 2). Most of the correlations in Table 4 are confirmed in these two additional datasets. One interesting observation in Figure 2 is that the correlation between RR and DT is weaker for the 50 kb window, probably because finer details of recombination rate variation are revealed at this length scale and the dependence of RR on DT is no longer monotonic. Thus DT is primarily correlated with large scale recombination rate variation, which could relate to the proposed conservation of large-scale rates on longer time scales 1522.

An example of chromosome specific patterns of recombination rate was recently discussed in the context of a putative gene that controls overall recombination rate 34. This study illustrates the effect of a SNP on increasing the female recombination rates by almost the same amount on all chromosomes with the exception of chromosome 21. Another SNP reduces the male recombination rates by variable degrees for different chromosomes 34.

Gene density constitutes a further variable that is known to be strongly correlated with GC% 45. To better understand this relationship, we counted the number of exons within a 1 Mb window, as it reflects both the number of genes and the intron count. The correlation and partial correlations between GC%, RR, and the number of exons (NE) are listed in Table 5. Unlike the previous situation, where we had looked at the three variables RR, DT and GC%, the consideration of NE instead DT is bringing up an observation that allows us to infer a causal relationship: although no significant direct correlation exists between RR-NE, a negative correlation between RR and NE emerges after conditioning on GC%.

This result (Table 5) suggests the causal model in Figure 1(B). In this causal model, RR and NE are two independent causes of GC%. The inference of this causal structure is based on the known fact that conditioning on a common child variable creates a correlation between two previously uncorrelated causes of this child variable 29. Or spoken more specifically, the relationship between NE and RR can be understood as follows: normally the two variables RR and NE do not contain any information about each other and are therefore uncorrelated. However, given the status of GC-content as third variable, this situation changes and RR and NE are now mutually informative. This mutual informativeness of NE and RR depending on GC% is explained by a model where both RR and NE are independent causes of GC%. When GC% in a region is high and RR is low, NE is more likely to be high. Vice versa, when NE is low, RR is more likely to be high. Thus, given the status of GC%, a previously invisible relationship between RR and NE emerges due to the causal influence of both variables on GC%.

Consistent with our present observation, a negative correlation between gene density and RR had been observed earlier in a multiple regression analysis when looking at 3 Mb windows, despite the fact that the unconditional RR/gene count correlation was weakly positive 35. Importantly, window size could be a factor that exerts some influence on the magnitude of observed correlations. Recombinations tend to occur more often in physical proximity to genes, when compared to intergenic regions; but on the other hand, they also tend to occur away from exons on a finer scale 17. It might be due to this subtle variation of RR at different length scales that the correlation between RR-NE is insignificant at the 1 Mb scale, but was weakly positive on the 3 Mb scale.

Nevertheless, when we repeated the chromosome-specific analysis using the variable NE (instead of DT), this confirmed the overall pattern of correlation between RR and NE. Unconditionally the correlation is not significant and can be both positive and negative. However, the partial correlations between NE and RR conditional on GC% are all negative with most of them being significant (results not shown). In principle, the absence of an unconditional correlation between RR and NE could also result from a phenomenon termed suppression 363738. Suppression refers to the situation, where different signs are obtained by following two paths with opposite effects from the same starting to the same ending node. However, the observed change of the correlation from insignificant to significant is inconsistent with suppression, because this conditional dependence indicates that both the NE and the RR link with GC% are pointing towards GC%.

In the final step, we extended our 3-variable analysis to a 4-variable analysis, which includes GC%, RR, -log(DT), and NE. Besides the previously calculated first-order partial correlation (conditional on one variable), we now also calculate the second-order partial correlations (conditional on two other variables). The result are shown in Table 6. When comparing the second-order partial correlations to the first-order partial correlations, we found that conditioning on GC% is mostly responsible for any change of correlation status. Conditioning on DT, RR or NE has only some quantitative effect, instead of introducing any qualitative changes into pairwise and first-order correlations. This implies a central position of GC% among these variables.

Figure 1(C) depicts a partially directed graph that is consistent with the results in Table 6. Importantly, the inclusion of DT does not alter causal relationships that were inferred above in the 3-variable analysis of RR, NE and GC%. Also, the above correlations between RR, DT and GC% remain largely unaltered by the inclusion of NE. As mentioned above, telomere distance is inversely correlated with GC% and RR. Additionally, we see in the unconditional pairwise analysis that telomere distance is inversely correlated with NE too, although this correlation is of smaller magnitude. This correlation between DT and NE does not change substantially when conditioning on RR. However, when conditioning on GC%, the correlation between DT and NE changes its direction. Following a similar line of reasoning as above, this suggests a model where DT and NE are two independent causes of GC%. On the contrary, this cannot be said for the influence of RR and DT on GC%, because the correlation RR and DT does not depend on conditioning on GC%.

To find the missing orientations of the links between RR, DT and GC% in the 4-variable model, we next applied the TETRAD program 39 that implements the PC-algorithm to create a causal model by a systematic search strategy (see Methods for details) 4041. The graphical result that we obtained from running TETRAD is essentially the same as the one depicted in Figure 1(C) and confirmed the direction of the two arrows that we had inferred for causative influence of both RR and NE on GC%. However, the additionally proposed orientations of the links RR → -log(DT) and NE → -log(DT) are biologically counterintuitive, because telomere distance is unlikely to be an effect of any of the other variables. To explain the difficulty to infer the directions of these causal links between RR, DT and GC%, we hypothesize the causal model in Figure 1(D). This model includes as fifth hidden variable the proportion of recombination events that are resolved exclusively as gene conversion event without any crossing-over event (NCO/R), a variable that was recently suggested to be important 22. In this model in Figure 1(D), NCO/R is a cause of GC% that does not fully depend on RR, but is influenced in its magnitude by RR. A similar relationship might connect NCO/R with NE. On the other hand, distance-to-telomere, similar to other variables measuring position or time, might play the role of providing a common environment for several other variables. In other words, one can draw a directed arrow from DT to all other variables under discussion. A similar situation is seen for the linkage disequilibrium between two neighboring genetic markers, where the position can be considered is a "cause" of both markers. However, we could not test the validity of the model in Figure 1(D) because NCO/R data are not available.

A graph G = (V, E) contains vertex/node set V and edge/link set E ⊆ V × V. An edge (i, j) ∈ E is "directed" if (j, i) ∉ E; and is "undirected" if (j, i) ∈ E. If there is an edge between node i and j, either directed or undirected, we say there is a "direct association/relationship" between the two nodes. If there is no edge between node i and node j, the two are still connected through multiple-step edges, as all our nodes are in one single graph; then we say the two nodes are "indirectly associated".

If all edges are directed, the graph is said to be "directed graph" (e.g. Figure 1(B)). If all edges are undirected, the graph is an "undirected graph" (e.g. Figure 1(A)). If some edges are directed and other edges are undirected, the graph is a "partially directed graph" (e.g. Figure 1(C)).

For many situations, conditional correlation is equivalent to partial correlation 42 which is defined as follows (with one control variable z):

ρ x y . z = ρ x y − ρ x z ρ y z ( 1 − ρ x z 2 ) ( 1 − ρ y z 2 ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyWdi3aaSbaaSqaaiabdIha4jabdMha5jabc6caUiabdQha6bqabaGccqGH9aqpjuaGdaWcaaqaaiabeg8aYnaaBaaabaGaemiEaGNaemyEaKhabeaacqGHsislcqaHbpGCdaWgaaqaaiabdIha4jabdQha6bqabaGaeqyWdi3aaSbaaeaacqWG5bqEcqWG6bGEaeqaaaqaamaakaaabaGaeiikaGIaeGymaeJaeyOeI0IaeqyWdi3aa0baaeaacqWG4baEcqWG6bGEaeaacqaIYaGmaaGaeiykaKIaeiikaGIaeGymaeJaeyOeI0IaeqyWdi3aa0baaeaacqWG5bqEcqWG6bGEaeaacqaIYaGmaaGaeiykaKcabeaaaaGaeiOla4caaa@582C@

where ρxy=cov(x,y)/var(x)var(y) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOEaONaeiikaGIaeqyWdi3aaSbaaSqaaiabdIha4jabdMha5bqabaGccqGH9aqpcqWGJbWycqWGVbWBcqWG2bGDcqGGOaakcqWG4baEcqGGSaalcqWG5bqEcqGGPaqkcqGGVaWldaGcaaqaaiabdAha2jabdggaHjabdkhaYjabcIcaOiabdIha4jabcMcaPiabdAha2jabdggaHjabdkhaYjabcIcaOiabdMha5jabcMcaPaWcbeaaaaa@4D7E@ is the Pearson product-moment correlation coefficient. From the linear regression framework, partial correlation is the correlation after the main terms in regression over z are removed:

x = a x + b x z + ϵ x y = a y + b y z + ϵ y ρ x y . z = C o r ( ϵ x , ϵ y ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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v=aYoaaBaaaleaacqWG4baEaeqaaOGaeiilaWIae8x9di7aaSbaaSqaaiabdMha5bqabaGccqGGPaqkaaaaaa@69AE@

Partial correlation ρ_xy.z is often lower than ρ_xy, and a significantly lower partial correlation is an indication that the x - y correlation is indirect.

With more than 3 variables (x, y, z, w), the partial correlation can be defined by conditional on one variable (e.g. z, first order), or two variables (z, w, second order). Both Eq.(1) and Eq.(2) can be extended for calculating second-order partial correlation:

ρ x y . z w = ρ x y . z − ρ x w . z ρ y w . z ( 1 − ρ x w . z 2 ) ( 1 − ρ y w . z 2 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@648C@

and

x = a x + b x z + c x w + ϵ x y = a y + b y z + c y w + ϵ y ρ x y . z w = C o r ( ϵ x , ϵ y ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabmqaaaqaaiabdIha4jabg2da9iabdggaHnaaBaaaleaacqWG4baEaeqaaOGaey4kaSIaemOyai2aaSbaaSqaaiabdIha4bqabaGccqWG6bGEcqGHRaWkcqWGJbWydaWgaaWcbaGaemiEaGhabeaakiabdEha3jabgUcaRmbvLH1qn1uy0Hws0fgBPngaryWyT1wAXadaiqGacqWF1pGSdaWgaaWcbaGaemiEaGhabeaaaOqaaiabdMha5jabg2da9iabdggaHnaaBaaaleaacqWG5bqEaeqaaOGaey4kaSIaemOyai2aaSbaaSqaaiabdMha5bqabaGccqWG6bGEcqGHRaWkcqWGJbWydaWgaaWcbaGaemyEaKhabeaakiabdEha3jabgUcaRiab=v=aYoaaBaaaleaacqWG5bqEaeqaaaGcbaGaeqyWdi3aaSbaaSqaaiabdIha4jabdMha5jabc6caUiabdQha6jabdEha3bqabaGccqGH9aqpcqWGdbWqcqWGVbWBcqWGYbGCcqGGOaakcqWF1pGSdaWgaaWcbaGaemiEaGhabeaakiabcYcaSiab=v=aYoaaBaaaleaacqWG5bqEaeqaaOGaeiykaKIaeiOla4caaaaa@76B3@

Higher order partial correlation can be defined in an analog fashion.

Figure 3 illustrates an example for inferring relationship and causal graph from data for three variables x, y, z. In Figure 3(A), we assume all pairwise correlations are significant, so all nodes are linked to other nodes. If the correlation between two of the variables is not due to a direct cause-effect relationship, but mediated via a third variable, then the correlation between the two conditional on that third variable will be greatly reduced. Accordingly, we would end up Figure 3(B), if assuming that the partial correlation Cor(x, y|z) becomes insignificant, while the other two partial correlations remain significant. In that case, partial correlation cannot determine the orientation of causal arrows. Except the first causal model on the top of Figure 3(C), all the three other causal models were possible. However, in a special situation, a directed causal model can be inferred uniquely. Suppose Cor(x, z) and Cor(y, z) are both significant, but Cor(x, y) is insignificant, then we start with the undirected graph in Figure 3(B) from the unconditional analysis. Further suppose Cor(x, y|z), Cor(x, z|y), Cor(y, y|x) are all significant. Then by the rule of d-separation 41 only the top model in Figure 3(C) is consistent with these assumptions.

The TETRAD program 39 implements the PC-algorithm to automatically infer causal relationships from partial correlation analysis 40. This algorithm can be broken into two phases: an adjacency phase and an orientation phase. In the adjacency phase, a complete undirected graph over the variables is constructed and then edges X - Y are removed, if some set S among either the adjacents of X or the adjacents of Y can be found such that I(X, Y|S). Then the orientation phase is begun. The first step examines unshielded triples and considers to orient them as colliders. An unshielded triple is a triple (X, Y, Z) where X is adjacent to Y, Y is adjacent to Z, but X is not adjacent to Z. Since X is not adjacent to Z, the edge X - Z must have been removed during the adjacency search by conditioning on some set S_xz; (X, Y, Z) is oriented as a collider X → Y ← Z just in case Y is not in this S_xz. Once all such unshielded triples have been oriented as colliders, a series of rules orients any edge whose orientation is implied by previous orientations.