Doing scientific research in post-Soviet Russia is challenging but there are solutions that could prevent the massive brain drain witnessed in recent decades

A scientist living in Russia is often asked two questions: “Why haven't you left?” and “Is it still possible to work there?” The best response to the first question is, “Why should I?”—which either terminates the conversation or leads to a stimulating discussion about the fate of the world. The second question, however, deserves a serious answer. In fact, this is the question that every one of us keeps asking ourselves.There is no simple answer. The biggest problems we face are brain drain, inadequate infrastructure, and lack of money (or perhaps, lack of money, lack of money, and lack of money). In the Soviet Union, fundamental science was supported to a great extent by military expenditure. Thus, it is not surprising that Soviet physics and mathematics were more successful than other fields, such as biology. In the 1990s, military spending on science declined sharply, although the exact numbers are hard to estimate. This year, the direct funding of science constitutes only 1.78% of Russia's national budget (an additional 0.46% is allocated to the space program), although the law stipulates that this figure should be at least 4%. Still, this funding amounts to 46.2 billion rubles (approximately US$1.6 billion), more than twice the amount spent in 2000. Although this figure looks negligible compared with spending on science in the United States and many European countries, it could still be sufficient to support existing actively working groups at a reasonable level.Funding There are several mechanisms for distributing funds for research. The major share comes via Russia's Department of Science and the Russian Academy of Sciences. The Academy, unlike its Western analogs, not only acts as a consulting body of experts, but also has the authority to distribute money ( Figure 1

 ). The funds come both as long-term support for scientific institutes and as National or Academy research programs. The former covers base salaries, which are small even by local standards (about US$200 per month for a laboratory chief), and basic infrastructure (water, electricity, etc.). This system of long-term support inherited all the old Soviet ills, such as the lack of correlation between scientific output and the level of funding. As a result, the available resources are spread thinly over hundreds of labs, most of which are just barely alive. The National or Academy research programs can provide funding at a higher level, sometimes even enough to do experimental research. However, the procedure of establishing such programs, though formally competitive, is often not transparent, and a major role is played by the so-called “administrative resource” ( Allakhverdov and Pokrovsky 2003 ). Figure 1 The Golden Brain (the Praesidium of the Russian Academy of Sciences) (Photograph, with permission, by Nataliya Sadovskaya) Figure 2 Papers Published in  Nature  and  Science  in Which at Least One of the Coauthors Lists an Address inside the Soviet Union or Russia The actual number of such papers is shown by the solid line. Their “effective number,” into which each paper contributes with the coefficient equal to the fraction of addresses inside the Soviet Union or Russia from all the listed addresses, is shown by the broken line. In recent years, the contribution of ethnic Russians to high-quality research increased, but their work is mostly performed outside Russia. Figure 3 Papers Published in  Nature  and  Science  by Researchers with the 15 Most Common Russian Surnames (Ivanov[a], Kuznetsov[a], Smirnov[a], Etc) The number of all papers in which at least one coauthor has a name from this list is shown by the solid line, and the number of such papers that list at least one address inside the Soviet Union or Russia is shown by the broken line. Whereas before 1992 nearly 100% of ethnic Russians doing toplevel science resided inside the Soviet Union and Russia (hardly surprising!), by now, this number has dropped to below 25%. Money is also distributed through the Russian Foundation for Basic Research (RFBR). The decisionmaking mechanism used by RFBR is closer to Western standards, and involves anonymous refereeing followed by board discussions. Although its grants are rather small (at most, several thousand dollars per year for a maximum of three years), they provide important additional support for many small and mediumsized groups that may receive several such grants for different projects. In addition, RFBR supports the publication and translation of books, travel to international conferences, the organization of conferences in Russia, and similar activities. Unfortunately, several programs (in particular, support for young scientists) have recently been transferred from RFBR to a newly established government office, and have thus become less independent.
Funding
There are several mechanisms for distributing funds for research. The major share comes via Russia's Department of Science and the Russian Academy of Sciences. The Academy, unlike its Western analogs, not only acts as a consulting body of experts, but also has the authority to distribute money ( Figure 1

 ). The funds come both as long-term support for scientific institutes and as National or Academy research programs. The former covers base salaries, which are small even by local standards (about US$200 per month for a laboratory chief), and basic infrastructure (water, electricity, etc.). This system of long-term support inherited all the old Soviet ills, such as the lack of correlation between scientific output and the level of funding. As a result, the available resources are spread thinly over hundreds of labs, most of which are just barely alive. The National or Academy research programs can provide funding at a higher level, sometimes even enough to do experimental research. However, the procedure of establishing such programs, though formally competitive, is often not transparent, and a major role is played by the so-called “administrative resource” ( Allakhverdov and Pokrovsky 2003 ).Money is also distributed through the Russian Foundation for Basic Research (RFBR). The decisionmaking mechanism used by RFBR is closer to Western standards, and involves anonymous refereeing followed by board discussions. Although its grants are rather small (at most, several thousand dollars per year for a maximum of three years), they provide important additional support for many small and mediumsized groups that may receive several such grants for different projects. In addition, RFBR supports the publication and translation of books, travel to international conferences, the organization of conferences in Russia, and similar activities. Unfortunately, several programs (in particular, support for young scientists) have recently been transferred from RFBR to a newly established government office, and have thus become less independent.Collaboration International collaboration and research grants are a major source of support for many active research groups ( Table 1 ). Several agencies and foundations have programs for Eastern Europe, Russia, and/or former Soviet republics. Some of them, such as the Howard Hughes Medical Institute (Chevy Chase, Maryland, United States), fund individual groups; others—for example the International Science and Technology Center (Moscow, Russia)—stipulate that projects should be submitted jointly by academic and military researchers; and some agencies—in particular, European INTAS (Brussels, Belgium) and the American John E. Fogarty International Center (Bethesda, Maryland, United States)—support collaboration between Russian and Western laboratories. Table 1 International Agencies Supporting Russian Science
 a The first three agencies are international organizations, followed by individual programs Another source of financial support is direct collaboration between Western and Russian laboratories. Even after a relatively short visit, the salary of a visiting researcher abroad can be stretched for several months back home in Russia; even more importantly, experimental biologists visiting foreign labs have access to modern instruments and chemicals, which allows them to do modern research. The hosts of such visits are often (although definitely not always) recent immigrants from Russia, and in such cases the collaborations may have roots in older days ( Box 1 ). As well as supporting Russian science directly, international collaboration plays an important indirect role because it is less influenced by local politics. In fact, one of the main positive impacts of the New York–based International Science Foundation set up by George Soros in early 1990s was that it demonstrated the possibility of open competition with clearly defined rules—something unheard of in Soviet times—and thus served as a model for the RFBR, which was organized at approximately the same time. Unfortunately, international ties, especially with the United States, have been adversely affected by recent changes in visa procedures, which have become lengthy (leading to many missed conferences) and, even worse, completely unpredictable (e.g.,  Brumfiel 2004 ). The grapevine distributes stories of “bad words” that should be avoided when describing one's research area during an interview at the consulate. Examples of such words include the adjective “nuclear” (even within a innocuous terms like “nuclear magnetic resonance”) or, more recently, anything that involves “bacteria.” The demand for fundamental and even for applied biological research from Russian industry is almost nonexistent. The pharmaceutical industry is content to produce generics, while Russian biotech companies are still exploiting old strains developed in the Soviet Union. However, some laboratories are conducting outsourced research, and there are now research outposts of Western and Japanese companies in Russia organized as standard industrial labs. On one hand, this work is a dead end for Russian scientists, because the results of such research normally cannot be published. This is a serious problem, especially for young scientists who want to establish themselves. On the other hand, royalties from patents or commercialization of the products can be used to support further research. One group that has followed this path successfully is Sergey Lukyanov's lab at the Shemyakin Institute of Bioorganic Chemistry in Moscow, Russia. They have developed the subtractive hybridization technique for enrichment of clone libraries by rare transcripts or specific genomic fragments ( Rebrikov et al. 2004 ), and are distributing it via a company called Evrogen ( http://www.evrogen.com/about.shtml ).
Collaboration
International collaboration and research grants are a major source of support for many active research groups ( Table 1 ). Several agencies and foundations have programs for Eastern Europe, Russia, and/or former Soviet republics. Some of them, such as the Howard Hughes Medical Institute (Chevy Chase, Maryland, United States), fund individual groups; others—for example the International Science and Technology Center (Moscow, Russia)—stipulate that projects should be submitted jointly by academic and military researchers; and some agencies—in particular, European INTAS (Brussels, Belgium) and the American John E. Fogarty International Center (Bethesda, Maryland, United States)—support collaboration between Russian and Western laboratories.International Agencies Supporting Russian Science
a The first three agencies are international organizations, followed by individual programsAnother source of financial support is direct collaboration between Western and Russian laboratories. Even after a relatively short visit, the salary of a visiting researcher abroad can be stretched for several months back home in Russia; even more importantly, experimental biologists visiting foreign labs have access to modern instruments and chemicals, which allows them to do modern research. The hosts of such visits are often (although definitely not always) recent immigrants from Russia, and in such cases the collaborations may have roots in older days ( Box 1 ).As well as supporting Russian science directly, international collaboration plays an important indirect role because it is less influenced by local politics. In fact, one of the main positive impacts of the New York–based International Science Foundation set up by George Soros in early 1990s was that it demonstrated the possibility of open competition with clearly defined rules—something unheard of in Soviet times—and thus served as a model for the RFBR, which was organized at approximately the same time.Unfortunately, international ties, especially with the United States, have been adversely affected by recent changes in visa procedures, which have become lengthy (leading to many missed conferences) and, even worse, completely unpredictable (e.g.,  Brumfiel 2004 ). The grapevine distributes stories of “bad words” that should be avoided when describing one's research area during an interview at the consulate. Examples of such words include the adjective “nuclear” (even within a innocuous terms like “nuclear magnetic resonance”) or, more recently, anything that involves “bacteria.”The demand for fundamental and even for applied biological research from Russian industry is almost nonexistent. The pharmaceutical industry is content to produce generics, while Russian biotech companies are still exploiting old strains developed in the Soviet Union. However, some laboratories are conducting outsourced research, and there are now research outposts of Western and Japanese companies in Russia organized as standard industrial labs. On one hand, this work is a dead end for Russian scientists, because the results of such research normally cannot be published. This is a serious problem, especially for young scientists who want to establish themselves. On the other hand, royalties from patents or commercialization of the products can be used to support further research. One group that has followed this path successfully is Sergey Lukyanov's lab at the Shemyakin Institute of Bioorganic Chemistry in Moscow, Russia. They have developed the subtractive hybridization technique for enrichment of clone libraries by rare transcripts or specific genomic fragments ( Rebrikov et al. 2004 ), and are distributing it via a company called Evrogen ( http://www.evrogen.com/about.shtml ).Infrastructure and Bureaucracy Another major problem is the degradation of infrastructure. Only a few labs can afford modern equipment and instruments, and for many others, even standard chemicals are too expensive. This leads to a vicious circle: without equipment, a lab cannot conduct experiments at the level demanded by highimpact journals—and without such publications, it cannot compete for large grants. Smaller RFBR grants, while simpler to obtain, are insufficient to purchase large pieces of equipment, and funds from several grants or several years cannot be combined due to bureaucratic restrictions. Thus, the only hope for these labs, apart from international collaboration, is a personal connection with senior bureaucrats that might result in an (un)expected windfall. Having the funds to purchase modern equipment abroad is only the first hurdle; the many conflicting rules and restrictions, inefficiency, and corruption within the system can subsequently hold up the process. Some items, such as tissue samples or animals, are virtually impossible to import legally. The process of clearing the shipments through customs is a difficult, timeconsuming job. Grigory Kopelevich, the Howard Hughes Medical Institute's Russian representative, recalls a story of a grantee whose microscope was stopped at customs because the box contained two screwdrivers not specified in the order. Fortunately, to resolve the issue, it was sufficient to present one of the screwdrivers to a customs officer as a gift. Even basic access to journals is a problem, especially outside the main research centers. Indeed, out of a random sample of ten major universities where electronic library catalogs were available via the Internet, only six had subscribed to  Nature , and only two to  Science . More specialized journals are available only in Moscow and perhaps St. Petersburg. This is partially offset by the proliferating open-access journals from the Public Library of Science and BioMed Central, free electronic versions of older issues provided by some journals, free subscriptions for Russian academic institutes granted by some publishers or purchased by international foundations (e.g., the  e-library.ru  project organized by the RFBR and supported by the Open Society Institute [the Soros Foundation, based in New York] and the Department of Education) ( Table 2 ), reprints at authors' Web pages, and last but not least, colleagues abroad who break copyright laws by e-mailing PDF files; there is even a popular bulletin board coordinating this activity. However, these are only partial solutions. Russia is not considered a developing country, and thus is excluded from many international efforts that provide free access to journals (such as HINARI). Moreover, many journals have page charges, but no Russian grants cover these, and the cost of publication may be prohibitively high for many groups. Table 2 Journals and Other Resources, Available to Russian Academic Institutes under the elibrary.ru Project ( e-library.ru )
Infrastructure and Bureaucracy
Another major problem is the degradation of infrastructure. Only a few labs can afford modern equipment and instruments, and for many others, even standard chemicals are too expensive. This leads to a vicious circle: without equipment, a lab cannot conduct experiments at the level demanded by highimpact journals—and without such publications, it cannot compete for large grants. Smaller RFBR grants, while simpler to obtain, are insufficient to purchase large pieces of equipment, and funds from several grants or several years cannot be combined due to bureaucratic restrictions. Thus, the only hope for these labs, apart from international collaboration, is a personal connection with senior bureaucrats that might result in an (un)expected windfall.Having the funds to purchase modern equipment abroad is only the first hurdle; the many conflicting rules and restrictions, inefficiency, and corruption within the system can subsequently hold up the process. Some items, such as tissue samples or animals, are virtually impossible to import legally. The process of clearing the shipments through customs is a difficult, timeconsuming job. Grigory Kopelevich, the Howard Hughes Medical Institute's Russian representative, recalls a story of a grantee whose microscope was stopped at customs because the box contained two screwdrivers not specified in the order. Fortunately, to resolve the issue, it was sufficient to present one of the screwdrivers to a customs officer as a gift.Even basic access to journals is a problem, especially outside the main research centers. Indeed, out of a random sample of ten major universities where electronic library catalogs were available via the Internet, only six had subscribed to  Nature , and only two to  Science . More specialized journals are available only in Moscow and perhaps St. Petersburg. This is partially offset by the proliferating open-access journals from the Public Library of Science and BioMed Central, free electronic versions of older issues provided by some journals, free subscriptions for Russian academic institutes granted by some publishers or purchased by international foundations (e.g., the  e-library.ru  project organized by the RFBR and supported by the Open Society Institute [the Soros Foundation, based in New York] and the Department of Education) ( Table 2 ), reprints at authors' Web pages, and last but not least, colleagues abroad who break copyright laws by e-mailing PDF files; there is even a popular bulletin board coordinating this activity. However, these are only partial solutions. Russia is not considered a developing country, and thus is excluded from many international efforts that provide free access to journals (such as HINARI). Moreover, many journals have page charges, but no Russian grants cover these, and the cost of publication may be prohibitively high for many groups.NatureScienceJournals and Other Resources, Available to Russian Academic Institutes under the elibrary.ru Project ( e-library.ru )
Brain Drain These problems, along with low salaries, have naturally led to a huge brain drain. Entire generations have been decimated ( Box 1 ); the dearth of researchers at the postdoc level, has caused a gap in the teaching and maintenance of scientific traditions. Many labs now consist of older chiefs and senior researchers, and graduate students who plan to leave immediately after getting the candidate degree (the equivalent of a Western doctorate). “Leaving” does not necessarily mean leaving the country; many capable young people go into business. While that might be good for the country in general, it is bad for science, at least in the short term. However, even emigration is not a completely negative thing; it creates a network of collaborators, and in many cases enhances ties with the international community. Despite all this, science in Russia is very much alive. Not-yet-Nobel-prizewinner Alexei Abrikosov's repeated exhorations to the scientific community “to help all the talented scientists leave Russia and to ignore the rest” were met by universal disgust ( Hoffman 1993 ;  Leskov 1993 ;  Migdal 1993 ). There are several competitive Russian labs doing first-rate research and publishing in the top-tier journals. Old habits die hard; even in these days, very decent results are often published in Russianlanguage journals, the best of which have impact factors that are around 1. Each year, many intelligent and capable students enroll in universities, and competition for admission is steadily increasing from the lows of the mid-1990s. There are also well-attended international conferences in several Russian cities.
Brain Drain
These problems, along with low salaries, have naturally led to a huge brain drain. Entire generations have been decimated ( Box 1 ); the dearth of researchers at the postdoc level, has caused a gap in the teaching and maintenance of scientific traditions. Many labs now consist of older chiefs and senior researchers, and graduate students who plan to leave immediately after getting the candidate degree (the equivalent of a Western doctorate). “Leaving” does not necessarily mean leaving the country; many capable young people go into business. While that might be good for the country in general, it is bad for science, at least in the short term. However, even emigration is not a completely negative thing; it creates a network of collaborators, and in many cases enhances ties with the international community.Despite all this, science in Russia is very much alive. Not-yet-Nobel-prizewinner Alexei Abrikosov's repeated exhorations to the scientific community “to help all the talented scientists leave Russia and to ignore the rest” were met by universal disgust ( Hoffman 1993 ;  Leskov 1993 ;  Migdal 1993 ). There are several competitive Russian labs doing first-rate research and publishing in the top-tier journals. Old habits die hard; even in these days, very decent results are often published in Russianlanguage journals, the best of which have impact factors that are around 1. Each year, many intelligent and capable students enroll in universities, and competition for admission is steadily increasing from the lows of the mid-1990s. There are also well-attended international conferences in several Russian cities.Prospects What can be done by the international community to support what is left of Russian science? Of course, direct support in the form of competitive grants is important, especially if there are few restrictions on spending; even the most carefully considered procedure cannot foresee all possible situations. But even more useful is the creation of joint research centers, such as the one opened by the international Ludwig Institute for Cancer Research (LICR) based jointly in Zurich, New York, and London, and the Belozersky Institute of Physico-Chemical Biology of Moscow State University in Moscow, Russia. This research center began with limited support for several stronger groups, and is gradually moving toward integration of a research program in Moscow with other LICR projects, and real collaboration between Moscow groups and LICR labs elsewhere. One of the most essential elements of successful research is access to up-to-date information. Consequently, any initiative that provides open access to scientific literature and databases is extremely useful. Seminars, lecture courses (such as the Moscow University [Moscow, Russia] cycle on oncology and immunology sponsored by LICR;  www.oncoimmunology.ru/index_e.htm ), and the participation of Western scientists in scientific conferences in Russia are important not only because they provide a fresh understanding of emerging trends, but also because they create personal contacts between Russian and Western scientists that often lead to fruitful collaboration. By contrast, some other types of joint project may be less successful. Artificial programs aimed at creating various participant “networks” usually do not work as expected, and training programs in Western universities often attract potential emigrants rather that those willing to continue active research inside Russia. The contribution of the international community cannot be the sole decisive factor in the future growth of Russian science. Important as it is in this transition period, it is no substitute for a systemic change. The ills of Russian science are not unique; the same issues have been raised by scientists from other Eastern European countries (e.g.,  Wojcik 2004 ). Even the existing funds could go much further if scientific policies were more open, better structured, and more competitive. Large grants should be provided, on the basis of well-defined criteria, to only the strongest labs doing the best research. An often-heard opinion that no independent review is possible in a small, well-entrenched community is irrelevant, since international boards of experts can be formed—the example of the Soros foundation clearly demonstrates that this is feasible. However, smaller pilot grants are also needed to support young scientists and labs contemplating new projects. This would create competition at all levels and provide doctoral students and postdocs with an incentive to stay in Russia and enroll in a strong lab. But again, the procedure for awarding such grants should be well defined, transparent, and independent from administrative influences. Thus, the traditional model of top-down distribution of funds must be changed, and this may be difficult. The current system of decision making by Russian funding agencies is clearly inadequate. Moreover, the problems of Russian science mirror the problems of Russian society in general, and it would be naive to expect that they will be solved overnight, even given the political will. Still, if successful, this combination should provide both high-level research in established fields and sufficient flexibility to find new directions. Box 1. Top-Level Publications by Russian Scientists The vast majority of papers published in recent years in the best journals by scientists working in Russia have foreign coauthors (who are often Russian émigrés), indicating that international collaboration is the most reliable source of support for top-level research. (Text and figures in this box courtesy of Alexey Kondrashov.)
Prospects
What can be done by the international community to support what is left of Russian science? Of course, direct support in the form of competitive grants is important, especially if there are few restrictions on spending; even the most carefully considered procedure cannot foresee all possible situations. But even more useful is the creation of joint research centers, such as the one opened by the international Ludwig Institute for Cancer Research (LICR) based jointly in Zurich, New York, and London, and the Belozersky Institute of Physico-Chemical Biology of Moscow State University in Moscow, Russia. This research center began with limited support for several stronger groups, and is gradually moving toward integration of a research program in Moscow with other LICR projects, and real collaboration between Moscow groups and LICR labs elsewhere.One of the most essential elements of successful research is access to up-to-date information. Consequently, any initiative that provides open access to scientific literature and databases is extremely useful. Seminars, lecture courses (such as the Moscow University [Moscow, Russia] cycle on oncology and immunology sponsored by LICR;  www.oncoimmunology.ru/index_e.htm ), and the participation of Western scientists in scientific conferences in Russia are important not only because they provide a fresh understanding of emerging trends, but also because they create personal contacts between Russian and Western scientists that often lead to fruitful collaboration.By contrast, some other types of joint project may be less successful. Artificial programs aimed at creating various participant “networks” usually do not work as expected, and training programs in Western universities often attract potential emigrants rather that those willing to continue active research inside Russia.The contribution of the international community cannot be the sole decisive factor in the future growth of Russian science. Important as it is in this transition period, it is no substitute for a systemic change. The ills of Russian science are not unique; the same issues have been raised by scientists from other Eastern European countries (e.g.,  Wojcik 2004 ). Even the existing funds could go much further if scientific policies were more open, better structured, and more competitive. Large grants should be provided, on the basis of well-defined criteria, to only the strongest labs doing the best research. An often-heard opinion that no independent review is possible in a small, well-entrenched community is irrelevant, since international boards of experts can be formed—the example of the Soros foundation clearly demonstrates that this is feasible. However, smaller pilot grants are also needed to support young scientists and labs contemplating new projects. This would create competition at all levels and provide doctoral students and postdocs with an incentive to stay in Russia and enroll in a strong lab. But again, the procedure for awarding such grants should be well defined, transparent, and independent from administrative influences.Thus, the traditional model of top-down distribution of funds must be changed, and this may be difficult. The current system of decision making by Russian funding agencies is clearly inadequate. Moreover, the problems of Russian science mirror the problems of Russian society in general, and it would be naive to expect that they will be solved overnight, even given the political will. Still, if successful, this combination should provide both high-level research in established fields and sufficient flexibility to find new directions.Box 1. Top-Level Publications by Russian Scientists
The vast majority of papers published in recent years in the best journals by scientists working in Russia have foreign coauthors (who are often Russian émigrés), indicating that international collaboration is the most reliable source of support for top-level research. (Text and figures in this box courtesy of Alexey Kondrashov.)


Background Correlation between the expression levels of genes which are located close to each other on the genome has been found in various organisms, including yeast, drosophila and humans. Since such a correlation could be explained by several biochemical, evolutionary, genetic and technological factors, there is a need for statistical models that correspond to specific biological models for the correlation structure.
Background
Correlation between the expression levels of genes which are located close to each other on the genome has been found in various organisms, including yeast, drosophila and humans. Since such a correlation could be explained by several biochemical, evolutionary, genetic and technological factors, there is a need for statistical models that correspond to specific biological models for the correlation structure.Results We modelled the pairwise correlation between the expressions of the genes in a Drosophila microarray experiment as a normal mixture under Fisher's z-transform, and fitted the model to the correlations of expressions of adjacent as well as non-adjacent genes. We also analyzed simulated data for comparison. The model provided a good fit to the data. Further, correlation between the activities of two genes could, in most cases, be attributed to either of two factors: the two genes both being active in the same age group (adult or embryo), or the two genes being in proximity of each other on the chromosome. The interaction between these two factors was weak.
Results
We modelled the pairwise correlation between the expressions of the genes in a Drosophila microarray experiment as a normal mixture under Fisher's z-transform, and fitted the model to the correlations of expressions of adjacent as well as non-adjacent genes. We also analyzed simulated data for comparison. The model provided a good fit to the data. Further, correlation between the activities of two genes could, in most cases, be attributed to either of two factors: the two genes both being active in the same age group (adult or embryo), or the two genes being in proximity of each other on the chromosome. The interaction between these two factors was weak.Conclusions Correlation between the activities of adjacent genes is higher than between non-adjacent genes. In the data we analyzed, this appeared, for the most part, to be a constant effect that applied to all pairs of adjacent genes.
Conclusions
Correlation between the activities of adjacent genes is higher than between non-adjacent genes. In the data we analyzed, this appeared, for the most part, to be a constant effect that applied to all pairs of adjacent genes.

Background Several studies (Hamilton [ 1 ], Fukuoka[ 2 ]) have found stronger correlation between the expression levels of genes that are located close to each other on the genome than between those of distant genes: when gene expressions of many genes are measured for multiple tissue samples, for example using microarray technology, adjacent genes are sometimes found to be consistently up- or downregulated in a subset of the tissue samples. Gene expression is influenced by many factors (for a review, see Orphanides[ 3 ]), many of which could influence the correlation between the expression of two genes in general, and that between two adjacent genes in particular. Of particular interest are  chromatin domains . DNA can exist in either one of two states: a condensed state, termed heterochromatin, which is broadly inaccessible to transcription (although there are exceptions (Orphanides[ 3 ])), and an active state, termed euchromatin. A  chromatin domain  (a segment of DNA which, in a given cell at a given moment, is either entirely euchromatin or entirely heterochromatin) typically spans several genes (Roy[ 4 ]). Therefore, one would expect the expressions of two adjacent genes to tend to be positively correlated, at least if it was possible to measure transcription in individual cells. If the chromatin state was completely random (Jackson[ 5 ]) suggested a dynamic equilibrium, where chromatin fluctuates, to some extent randomly, between the two states), the effect of chromatin domains would vanish when gene expression is measured in pools of many cells, as with microarray technology. However, there is ample evidence for non-randomness. For example, chromatin states tend to be preserved after cell division (Orphanides[ 3 ]). And Cho[ 6 ] demonstrated that the states of chromatin domains in yeast are related to the cell cycle. In addition to the chromatin theory, several other explanations have been suggested for the apparent correlation between the expressions of adjacent genes. Several authors (Cohen[ 7 ], Kruglyak[ 8 ]) have noted that divergent gene pairs show stronger correlation than tandem and convergent pairs, possibly because divergent pairs share an Upstream Activation Sequence. Lercher[ 9 ] found that many of the co-expressed adjacent genes in  Caenorhabditis elegans  are either operons or homologues (see also Llorente[ 10 ] and Rossfll]), and it has been suggested that evolution has arranged for functionally related genes to be located close to each other, either in order to promote consistent inheritance (Bleiweiss[ 12 ]), or in order to benefit from the correlation accounted for by the chromatin domains (Cohen[ 7 ]). Parisi[ 13 ] found a nonrandom distribution of the chromosomal location of genes with high expression level in testis and ovaria in Drosophila. Jackson[ 5 ] suggested that the location of a gene in the nucleus plays a role for its transcription, in relation to gradients of the concentration of transcription factors. Finally, since the action of a transcription factor on a promotor gets weaker with distance, genes belonging to the same pathway should show stronger correlation if they are located close to each other (Dorsett [ 14 ]). Due to this abundance of alternative theories, a study of gene-expression correlations should be designed in a way that makes it possible to distinguish correlation structures predicted by one model from those predicted by other models. The same applies to the statistical analysis techniques used. An important consequence of the evolution-based theories is that they predict a  consistent  coregulation structure. Suppose that two genes (in this case, two adjacent genes) are co-regulated because, for example, they participate in the same pathway. They would, then, show a strong correlation because they would be co-regulated in all tissue samples. This need not necessarily be the case with the chromatin domain model: the segments of euchromatin in one tissue sample may overlap with those in another tissue sample. This latter scenario, we call an  inconsistent  coregulation structure. With consistent coregulation, adjacent gene pairs will show either strong positive correlation, or they will be uncorrelated. With inconsistent coregulation, all adjacent gene pairs will show a modest positive correlation. In a microarray analysis of gene expressions in 35 pools of drosophila embryos and 54 adult drosophilae (Spellman and Rubin[ 15 ], reviewed by Oliver[ 16 ]), it was shown that adjacent genes with correlated expression levels tend to cluster. The method they used to demonstrate this was the following: let  w  be a fixed window size, e.g. 10. For each window of  w  adjacent genes, the average pairwise Pearson correlation coefficient within the window was computed. If that measure was found to be significant at, say, 1 -  α  = 0.999 (the p-value was estimated in a permutation experiment), all the genes in the sequence were tagged. Doing this for all windows (they were allowed to overlap), the total number of tagged genes was counted. Then the experiment was repeated with shuffled genes (i.e., as it would behave in the absence of positionally related correlation), and the number of tagged genes in the shuffled experiment was subtracted from the number of tagged genes in the original experiment. This difference (called "net genes") grows with window size and starts plateauing for a window size of approximately 10. Spellman and Rubin interpreted this as evidence for gene interaction within regions of approximately that size. One problem with the above method of analysis is that the increasing number of "net genes" would occur even without direct interactions between genes separated by up to ten positions. As shown in figure  1 , the analysis gives similar results when applied to simulated data from a normal distribution, in which an autocorrelation of AC = 0.10 or 0.05 was imposed artificially. So we cannot, on the basis of the analysis described above, reject the hypothesis that the data arose from a simple first-order autocorrelation process, in which no clustering of correlated genes exists. It is true that gene-pairs with high correlation form clusters: the autocorrelation of Pearson's R for adjacent genes is 0.1, with a standard error 0.01. However, this can be explained by the fact that genes that tend to correlate strongly with other genes in general (for example because of low measurement noise) tend to correlate with both their neighbors. If one eliminates that confounder by looking at non-overlapping gene pairs only, the autocorrelation vanishes (0.01, standard error = 0.01). Another way of showing this is by means of cross-tables. We divided the adjacent gene-pairs into three groups: positively correlated pairs(R>0.7), negatively correlated (R< -0.7) and non-correlated. (The threshold of 0.7 was suggested by Cohen[ 7 ]). If the correlated gene-pairs were clustered, one would expect that a gene-pair belonged to the same group as the next gene-pair more often than would happen by chance. This is indeed the case when overlapping gene-pairs are considered: 627 gene pairs out of 12949 (4.8%) had an R > 0.7 while the next (overlapping) gene pair also had an R > 0.7. This is 2.22 times more than what we expected due to chance alone. However, the same was observed when only one of the two overlapping gene-pairs was was a neighbor pair and the other was a random pair (if the genes were labelled ABCD...Z, a strong correlation between A and B predicted a strong correlation between B and C but also between B and X, where X is a random gene). But when non-overlapping adjacent gene pairs were considered (say, AB versus CD), the contingency was 332 out of 12878(2.6%) which is only 1.18 times more than expected due to chance. So the apparent clustering of correlated gene pairs is mainly due to overlap rather than to adjacency. On the other hand, it is clear that there is some higher order correlation structure in Spellman and Rubin's data. This can be seen by computing the average correlation coefficient for subgroups of the gene pairs, based on their physical distance (table  1 ) – it decreases much slower with distance than would a first-order process. Hence, the question remains how the correlation structure should be modelled and analyzed. In this paper, we present a method to separate A) Correlation of gene expression that can be attributed to consistent coregulation, from B) The uniform correlation expected under an hypothesis of inconsistent corregulation.
Background
Several studies (Hamilton [ 1 ], Fukuoka[ 2 ]) have found stronger correlation between the expression levels of genes that are located close to each other on the genome than between those of distant genes: when gene expressions of many genes are measured for multiple tissue samples, for example using microarray technology, adjacent genes are sometimes found to be consistently up- or downregulated in a subset of the tissue samples.Gene expression is influenced by many factors (for a review, see Orphanides[ 3 ]), many of which could influence the correlation between the expression of two genes in general, and that between two adjacent genes in particular. Of particular interest are  chromatin domains . DNA can exist in either one of two states: a condensed state, termed heterochromatin, which is broadly inaccessible to transcription (although there are exceptions (Orphanides[ 3 ])), and an active state, termed euchromatin. A  chromatin domain  (a segment of DNA which, in a given cell at a given moment, is either entirely euchromatin or entirely heterochromatin) typically spans several genes (Roy[ 4 ]). Therefore, one would expect the expressions of two adjacent genes to tend to be positively correlated, at least if it was possible to measure transcription in individual cells. If the chromatin state was completely random (Jackson[ 5 ]) suggested a dynamic equilibrium, where chromatin fluctuates, to some extent randomly, between the two states), the effect of chromatin domains would vanish when gene expression is measured in pools of many cells, as with microarray technology. However, there is ample evidence for non-randomness. For example, chromatin states tend to be preserved after cell division (Orphanides[ 3 ]). And Cho[ 6 ] demonstrated that the states of chromatin domains in yeast are related to the cell cycle.chromatin domainschromatin domainIn addition to the chromatin theory, several other explanations have been suggested for the apparent correlation between the expressions of adjacent genes. Several authors (Cohen[ 7 ], Kruglyak[ 8 ]) have noted that divergent gene pairs show stronger correlation than tandem and convergent pairs, possibly because divergent pairs share an Upstream Activation Sequence. Lercher[ 9 ] found that many of the co-expressed adjacent genes in  Caenorhabditis elegans  are either operons or homologues (see also Llorente[ 10 ] and Rossfll]), and it has been suggested that evolution has arranged for functionally related genes to be located close to each other, either in order to promote consistent inheritance (Bleiweiss[ 12 ]), or in order to benefit from the correlation accounted for by the chromatin domains (Cohen[ 7 ]). Parisi[ 13 ] found a nonrandom distribution of the chromosomal location of genes with high expression level in testis and ovaria in Drosophila. Jackson[ 5 ] suggested that the location of a gene in the nucleus plays a role for its transcription, in relation to gradients of the concentration of transcription factors.Caenorhabditis elegansFinally, since the action of a transcription factor on a promotor gets weaker with distance, genes belonging to the same pathway should show stronger correlation if they are located close to each other (Dorsett [ 14 ]). Due to this abundance of alternative theories, a study of gene-expression correlations should be designed in a way that makes it possible to distinguish correlation structures predicted by one model from those predicted by other models. The same applies to the statistical analysis techniques used.An important consequence of the evolution-based theories is that they predict a  consistent  coregulation structure. Suppose that two genes (in this case, two adjacent genes) are co-regulated because, for example, they participate in the same pathway. They would, then, show a strong correlation because they would be co-regulated in all tissue samples. This need not necessarily be the case with the chromatin domain model: the segments of euchromatin in one tissue sample may overlap with those in another tissue sample. This latter scenario, we call an  inconsistent  coregulation structure. With consistent coregulation, adjacent gene pairs will show either strong positive correlation, or they will be uncorrelated. With inconsistent coregulation, all adjacent gene pairs will show a modest positive correlation.consistentinconsistentIn a microarray analysis of gene expressions in 35 pools of drosophila embryos and 54 adult drosophilae (Spellman and Rubin[ 15 ], reviewed by Oliver[ 16 ]), it was shown that adjacent genes with correlated expression levels tend to cluster. The method they used to demonstrate this was the following: let  w  be a fixed window size, e.g. 10. For each window of  w  adjacent genes, the average pairwise Pearson correlation coefficient within the window was computed. If that measure was found to be significant at, say, 1 -  α  = 0.999 (the p-value was estimated in a permutation experiment), all the genes in the sequence were tagged. Doing this for all windows (they were allowed to overlap), the total number of tagged genes was counted. Then the experiment was repeated with shuffled genes (i.e., as it would behave in the absence of positionally related correlation), and the number of tagged genes in the shuffled experiment was subtracted from the number of tagged genes in the original experiment. This difference (called "net genes") grows with window size and starts plateauing for a window size of approximately 10. Spellman and Rubin interpreted this as evidence for gene interaction within regions of approximately that size.wwαOne problem with the above method of analysis is that the increasing number of "net genes" would occur even without direct interactions between genes separated by up to ten positions. As shown in figure  1 , the analysis gives similar results when applied to simulated data from a normal distribution, in which an autocorrelation of AC = 0.10 or 0.05 was imposed artificially. So we cannot, on the basis of the analysis described above, reject the hypothesis that the data arose from a simple first-order autocorrelation process, in which no clustering of correlated genes exists. It is true that gene-pairs with high correlation form clusters: the autocorrelation of Pearson's R for adjacent genes is 0.1, with a standard error 0.01. However, this can be explained by the fact that genes that tend to correlate strongly with other genes in general (for example because of low measurement noise) tend to correlate with both their neighbors. If one eliminates that confounder by looking at non-overlapping gene pairs only, the autocorrelation vanishes (0.01, standard error = 0.01). Another way of showing this is by means of cross-tables. We divided the adjacent gene-pairs into three groups: positively correlated pairs(R>0.7), negatively correlated (R< -0.7) and non-correlated. (The threshold of 0.7 was suggested by Cohen[ 7 ]). If the correlated gene-pairs were clustered, one would expect that a gene-pair belonged to the same group as the next gene-pair more often than would happen by chance. This is indeed the case when overlapping gene-pairs are considered: 627 gene pairs out of 12949 (4.8%) had an R > 0.7 while the next (overlapping) gene pair also had an R > 0.7. This is 2.22 times more than what we expected due to chance alone. However, the same was observed when only one of the two overlapping gene-pairs was was a neighbor pair and the other was a random pair (if the genes were labelled ABCD...Z, a strong correlation between A and B predicted a strong correlation between B and C but also between B and X, where X is a random gene). But when non-overlapping adjacent gene pairs were considered (say, AB versus CD), the contingency was 332 out of 12878(2.6%) which is only 1.18 times more than expected due to chance. So the apparent clustering of correlated gene pairs is mainly due to overlap rather than to adjacency.On the other hand, it is clear that there is some higher order correlation structure in Spellman and Rubin's data. This can be seen by computing the average correlation coefficient for subgroups of the gene pairs, based on their physical distance (table  1 ) – it decreases much slower with distance than would a first-order process. Hence, the question remains how the correlation structure should be modelled and analyzed. In this paper, we present a method to separateA) Correlation of gene expression that can be attributed to consistent coregulation, fromB) The uniform correlation expected under an hypothesis of inconsistent corregulation.Results Data We used the Drosophila data published by Spellman and Rubin[ 15 ]. This data set consisted of normalized expression levels of 13090 genes in 89 flies (35 pools of embryos and 54 pools of adults), obtained with Affymetrix GeneChip microarrays. For the purpose of analysis based on physical distance between genes, the data set was linked to Flybase[ 17 ] on the basis of the CG-identifiers provided by Spellman and Rubin. 1871 genes had to be omitted from that analysis because of unmatched CG-identifiers. We checked that these omitted genes were not a biased sample with respect to correlation in expression with their neighbor genes. Non-adjacent gene pairs The distribution of the correlation coefficient between gene pairs in general, (i.e., genes that are not necessarily adjacent) was fitted with a three-component normal mixture, based on the Arcus Tangens Hyperbolic-transform (Fisher's z) of the correlation coefficients, which was validated with a Kolmogorov-Smirnov test (p = 0.6). As shown in table  2 , the mean of the middle component, tanh( μ 0 ), was 0.014 with a standard error of 0.002. (In all tables,  μ  has been transformed back with tanh so that it can be interpreted as a correlation). The fact that the standard deviation of the third component,  σ + , is greater than that of the other components, suggests that the co-regulation of some gene pairs was stronger than that of others. Adjacent gene pairs Table  3  shows the fitted parameters in the same model, but for adjacent gene pairs. The results are very similar to those for the random gene pairs, although  μ 0  and  μ +  are substantially higher. This suggests that the effect of adjacency is, for the most part, a mechanism that applies to adjacent-gene pairs in general, not just to specific adjacent-gene pairs. One way to illustrate this is by means of a qq-plot (figure  2 ), which shows that the correlation for random gene pairs and adjacent gene pairs have a similar structure, although the overall correlation is stronger in adjacent gene pairs. This contrasts with the qq-plot in figure  3 , in which the gene-expression correlations in the subset of adult flies is compared to those in a mixed fly group. On average, the correlation is zero in both the adult group and the mixed group. However, the standard deviation of the correlation is much higher in the mixed group, presumably because genes that are expressed specifically in the adults (or specifically in the embryos) correlate strongly with each other in the mixed group. Table  4  shows the fitted values of the parameters  μ 0 ,  μ +  and the size of the third component (that is, the fraction of the genes that are positively co-regulated), for random and adjacent gene pairs in four subsets of the flies: one containing all 54 adult pools, one containing all 35 embryo pools, and two random subsets of 35 and 54 pools, respectively. One can observe that  μ 0  (which can be attributed to corregulation of gene pairs in general, i.e., inconsistent co-regulation) is always high for adjacent gene pairs and low for random gene pairs. On the other hand  μ +  and the "+"-fraction (which can be attributed to corregulation of specific gene pairs, i.e., consistent co-regulation) are always high in heterogenous fly groups and low in homogenous one. That difference between heterogenous and homogenous groups is what we expected: In a homogenous fly group, less gene pairs will come up as co-regulated, because some pathways may either be active in all flies, or inactive in all flies. That age groups does not explain all correlation, is hardly surprising: although the adult group and the embryo group are less heterogenous than the mixed group, neither is homogenous. Gene pairs of intermediate distance To get an impression of the size of the segments involved in co-regulation, we fitted the three-component mixture to the correlation coefficient between the expressions of more distant gene pairs.  μ 0  was 0.114 for the correlation between the expression of a gene and that of its direct neighbor, and 0.08 for the correlation with its second neighbor. From the fifth neighbor and until at least the tenth, a stable level of 0.03 is reached, which is still higher than for distant gene pairs. One possible interpretation of this is that two different co-regulation effects exist: a short-segment effect accounting for a correlation of approximately 0.114-0.03 = 0.081, and a long-segment effect accounting for a correlation of approximately 0.03-0.014 = 0.016. At first we had the suspicion that this stable level of 0.03 was a whole-chromosome effect, but that was not the case. We found no significant difference between the overall average correlation of random gene pairs from the same chromosome and random gene pairs from different chromosomes. In a first-order autocorrelation process, tanh( μ 0 ) for the second neighbor would be the square of that for the first neighbor. However, the unexpectedly small difference between the two values could be due to the fact that some second neighbors are physically quite close. To confirm or reject the hypothesis of a first-order process, however, the analysis must be based on physical distance rather than simple adjacency. As shown in table  1 , the decrease in average correlation as a function of physical distance is still too slow for a first-order process. Table  5  shows the fitted parameters for gene pairs within a physical distance of between 100 and 1000 bases. Unlike the fitted parameters for neighbor genes,  μ +  is not significantly higher than for random gene pairs. Simulated data To validate our approach, we computed the correlation coefficients of adjacent and random gene pairs in simulated data, based on either consistent or inconsistent co-regulation. As expected, the simulated data with inconsistent co-regulation resulted in a correlation structure with only one component, whereas those with consistent coregulation showed two components. (Not three, since the  segment effect  (see Methods) was always positive in the simulated models).
Results
Data We used the Drosophila data published by Spellman and Rubin[ 15 ]. This data set consisted of normalized expression levels of 13090 genes in 89 flies (35 pools of embryos and 54 pools of adults), obtained with Affymetrix GeneChip microarrays. For the purpose of analysis based on physical distance between genes, the data set was linked to Flybase[ 17 ] on the basis of the CG-identifiers provided by Spellman and Rubin. 1871 genes had to be omitted from that analysis because of unmatched CG-identifiers. We checked that these omitted genes were not a biased sample with respect to correlation in expression with their neighbor genes.
Data
We used the Drosophila data published by Spellman and Rubin[ 15 ]. This data set consisted of normalized expression levels of 13090 genes in 89 flies (35 pools of embryos and 54 pools of adults), obtained with Affymetrix GeneChip microarrays. For the purpose of analysis based on physical distance between genes, the data set was linked to Flybase[ 17 ] on the basis of the CG-identifiers provided by Spellman and Rubin. 1871 genes had to be omitted from that analysis because of unmatched CG-identifiers. We checked that these omitted genes were not a biased sample with respect to correlation in expression with their neighbor genes.Non-adjacent gene pairs The distribution of the correlation coefficient between gene pairs in general, (i.e., genes that are not necessarily adjacent) was fitted with a three-component normal mixture, based on the Arcus Tangens Hyperbolic-transform (Fisher's z) of the correlation coefficients, which was validated with a Kolmogorov-Smirnov test (p = 0.6). As shown in table  2 , the mean of the middle component, tanh( μ 0 ), was 0.014 with a standard error of 0.002. (In all tables,  μ  has been transformed back with tanh so that it can be interpreted as a correlation). The fact that the standard deviation of the third component,  σ + , is greater than that of the other components, suggests that the co-regulation of some gene pairs was stronger than that of others.
Non-adjacent gene pairs
The distribution of the correlation coefficient between gene pairs in general, (i.e., genes that are not necessarily adjacent) was fitted with a three-component normal mixture, based on the Arcus Tangens Hyperbolic-transform (Fisher's z) of the correlation coefficients, which was validated with a Kolmogorov-Smirnov test (p = 0.6). As shown in table  2 , the mean of the middle component, tanh( μ 0 ), was 0.014 with a standard error of 0.002. (In all tables,  μ  has been transformed back with tanh so that it can be interpreted as a correlation). The fact that the standard deviation of the third component,  σ + , is greater than that of the other components, suggests that the co-regulation of some gene pairs was stronger than that of others.μμσAdjacent gene pairs Table  3  shows the fitted parameters in the same model, but for adjacent gene pairs. The results are very similar to those for the random gene pairs, although  μ 0  and  μ +  are substantially higher. This suggests that the effect of adjacency is, for the most part, a mechanism that applies to adjacent-gene pairs in general, not just to specific adjacent-gene pairs. One way to illustrate this is by means of a qq-plot (figure  2 ), which shows that the correlation for random gene pairs and adjacent gene pairs have a similar structure, although the overall correlation is stronger in adjacent gene pairs. This contrasts with the qq-plot in figure  3 , in which the gene-expression correlations in the subset of adult flies is compared to those in a mixed fly group. On average, the correlation is zero in both the adult group and the mixed group. However, the standard deviation of the correlation is much higher in the mixed group, presumably because genes that are expressed specifically in the adults (or specifically in the embryos) correlate strongly with each other in the mixed group. Table  4  shows the fitted values of the parameters  μ 0 ,  μ +  and the size of the third component (that is, the fraction of the genes that are positively co-regulated), for random and adjacent gene pairs in four subsets of the flies: one containing all 54 adult pools, one containing all 35 embryo pools, and two random subsets of 35 and 54 pools, respectively. One can observe that  μ 0  (which can be attributed to corregulation of gene pairs in general, i.e., inconsistent co-regulation) is always high for adjacent gene pairs and low for random gene pairs. On the other hand  μ +  and the "+"-fraction (which can be attributed to corregulation of specific gene pairs, i.e., consistent co-regulation) are always high in heterogenous fly groups and low in homogenous one. That difference between heterogenous and homogenous groups is what we expected: In a homogenous fly group, less gene pairs will come up as co-regulated, because some pathways may either be active in all flies, or inactive in all flies. That age groups does not explain all correlation, is hardly surprising: although the adult group and the embryo group are less heterogenous than the mixed group, neither is homogenous.
Adjacent gene pairs
Table  3  shows the fitted parameters in the same model, but for adjacent gene pairs. The results are very similar to those for the random gene pairs, although  μ 0  and  μ +  are substantially higher. This suggests that the effect of adjacency is, for the most part, a mechanism that applies to adjacent-gene pairs in general, not just to specific adjacent-gene pairs. One way to illustrate this is by means of a qq-plot (figure  2 ), which shows that the correlation for random gene pairs and adjacent gene pairs have a similar structure, although the overall correlation is stronger in adjacent gene pairs. This contrasts with the qq-plot in figure  3 , in which the gene-expression correlations in the subset of adult flies is compared to those in a mixed fly group. On average, the correlation is zero in both the adult group and the mixed group. However, the standard deviation of the correlation is much higher in the mixed group, presumably because genes that are expressed specifically in the adults (or specifically in the embryos) correlate strongly with each other in the mixed group.μμTable  4  shows the fitted values of the parameters  μ 0 ,  μ +  and the size of the third component (that is, the fraction of the genes that are positively co-regulated), for random and adjacent gene pairs in four subsets of the flies: one containing all 54 adult pools, one containing all 35 embryo pools, and two random subsets of 35 and 54 pools, respectively. One can observe that  μ 0  (which can be attributed to corregulation of gene pairs in general, i.e., inconsistent co-regulation) is always high for adjacent gene pairs and low for random gene pairs. On the other hand  μ +  and the "+"-fraction (which can be attributed to corregulation of specific gene pairs, i.e., consistent co-regulation) are always high in heterogenous fly groups and low in homogenous one. That difference between heterogenous and homogenous groups is what we expected: In a homogenous fly group, less gene pairs will come up as co-regulated, because some pathways may either be active in all flies, or inactive in all flies. That age groups does not explain all correlation, is hardly surprising: although the adult group and the embryo group are less heterogenous than the mixed group, neither is homogenous.μμμμGene pairs of intermediate distance To get an impression of the size of the segments involved in co-regulation, we fitted the three-component mixture to the correlation coefficient between the expressions of more distant gene pairs.  μ 0  was 0.114 for the correlation between the expression of a gene and that of its direct neighbor, and 0.08 for the correlation with its second neighbor. From the fifth neighbor and until at least the tenth, a stable level of 0.03 is reached, which is still higher than for distant gene pairs. One possible interpretation of this is that two different co-regulation effects exist: a short-segment effect accounting for a correlation of approximately 0.114-0.03 = 0.081, and a long-segment effect accounting for a correlation of approximately 0.03-0.014 = 0.016. At first we had the suspicion that this stable level of 0.03 was a whole-chromosome effect, but that was not the case. We found no significant difference between the overall average correlation of random gene pairs from the same chromosome and random gene pairs from different chromosomes. In a first-order autocorrelation process, tanh( μ 0 ) for the second neighbor would be the square of that for the first neighbor. However, the unexpectedly small difference between the two values could be due to the fact that some second neighbors are physically quite close. To confirm or reject the hypothesis of a first-order process, however, the analysis must be based on physical distance rather than simple adjacency. As shown in table  1 , the decrease in average correlation as a function of physical distance is still too slow for a first-order process. Table  5  shows the fitted parameters for gene pairs within a physical distance of between 100 and 1000 bases. Unlike the fitted parameters for neighbor genes,  μ +  is not significantly higher than for random gene pairs.
Gene pairs of intermediate distance
To get an impression of the size of the segments involved in co-regulation, we fitted the three-component mixture to the correlation coefficient between the expressions of more distant gene pairs.  μ 0  was 0.114 for the correlation between the expression of a gene and that of its direct neighbor, and 0.08 for the correlation with its second neighbor. From the fifth neighbor and until at least the tenth, a stable level of 0.03 is reached, which is still higher than for distant gene pairs. One possible interpretation of this is that two different co-regulation effects exist: a short-segment effect accounting for a correlation of approximately 0.114-0.03 = 0.081, and a long-segment effect accounting for a correlation of approximately 0.03-0.014 = 0.016. At first we had the suspicion that this stable level of 0.03 was a whole-chromosome effect, but that was not the case. We found no significant difference between the overall average correlation of random gene pairs from the same chromosome and random gene pairs from different chromosomes.μIn a first-order autocorrelation process, tanh( μ 0 ) for the second neighbor would be the square of that for the first neighbor. However, the unexpectedly small difference between the two values could be due to the fact that some second neighbors are physically quite close. To confirm or reject the hypothesis of a first-order process, however, the analysis must be based on physical distance rather than simple adjacency. As shown in table  1 , the decrease in average correlation as a function of physical distance is still too slow for a first-order process.μTable  5  shows the fitted parameters for gene pairs within a physical distance of between 100 and 1000 bases. Unlike the fitted parameters for neighbor genes,  μ +  is not significantly higher than for random gene pairs.μSimulated data To validate our approach, we computed the correlation coefficients of adjacent and random gene pairs in simulated data, based on either consistent or inconsistent co-regulation. As expected, the simulated data with inconsistent co-regulation resulted in a correlation structure with only one component, whereas those with consistent coregulation showed two components. (Not three, since the  segment effect  (see Methods) was always positive in the simulated models).
Simulated data
To validate our approach, we computed the correlation coefficients of adjacent and random gene pairs in simulated data, based on either consistent or inconsistent co-regulation. As expected, the simulated data with inconsistent co-regulation resulted in a correlation structure with only one component, whereas those with consistent coregulation showed two components. (Not three, since the  segment effect  (see Methods) was always positive in the simulated models).segment effectDiscussion There are many theories that could explain correlation between the expressions of two genes in general, and that of two adjacent genes in particular. For the verification of the specific correlation structures predicted by each theory, Spelmann and Rubin's data turned out to be very useful. Their experimental design was based on two distinct classes of experimental conditions (in this case, embryos and adults), which makes a relatively simple correlation structure plausible. Also, the data from each of their microarrays could be fit nicely with a normal mixture, which makes it possible to analyze their data on the basis of the Pearson R. As expected, most of the consistent correlation was found to be related to age group and unrelated to adjacency. Maybe more surprisingly, most of the correlation that was related to adjacency could not be accounted for by consistent co-regulation. This suggest that, at the statistical level, co-regulation of adjacent genes should be understood in terms of the mechanics of the transcription process, rather than in terms of the evolutionary origin of specific gene groups. In other words, in this particular data set, consistent correlation was very widespread, applying to 40% of all gene pairs (table  2 ), half of this could be attributed to the two age groups (table  3 ), but this consistent correlation was not the most important contributor to the elevated correlation coefficients for adjacent gene pairs. Correlation between the expression of adjacent genes is evident, but it is possible, as suggested by Spellman and Rubin, that such a correlation between expression levels for two adjacent genes does not generally imply a relationship with respect to biological function. Spellman and Rubin concluded that the adjacent genes with correlated expressions were confined to certain domains, which spanned 20% of all genes. Our findings are different, in that we distinguished between two components of the correlation structure. We found a baseline correlation of 0.1 (the difference between  μ 0  in table  2  and table  3 ) applying to all adjacent gene pairs. In addition to that, we found that 8% of all adjacent gene pairs (the difference between the "+" -fractions in table  2  and table  3 ) were strongly positively correlated. Finally, unlike Spellman and Rubin, we found no evidence for clustering of the strongly correlated gene pairs. Since this data set contained data from whole animals, we cannot say anything about the role of chromatin domains (or other mechanisms causing correlation of the expression of adjacent genes) in tissue differentiation. Parisi et al. [ 13 ] found clustering of genes that were upregulated in drosophila germ cells, which suggests that one would find elevated consistent coregulation of adjacent genes when applying our model to data sets with different tissue classes. Even for random (non-adjacent) gene pairs,  μ 0  was slightly positive. It is hard to imagine a biological effect that could cause  μ 0  to be significantly greater than zero, but it could be an artifact of the normalization: if gene  g  is expressed at the same level in all tissue samples, and gene  h  as well, they should have a correlation coefficient of zero, but since they will both yield elevated measurements in flies in which the normalization bias is positive, the observed correlation will be positive. The fact that  μ 0  is almost zero suggests that normalization bias is not a major problem with this data set. The fact that  μ 0  for random gene pairs is not generally higher in mixed age groups than in pure adult or embryo groups (table  4 ) is consistent with the assumption that  μ 0  for random gene pairs does not have a biological interpretation. Although looking at adjacent genes is computationally convenient, it would probably be more correct to use physical distance, rather than adjacency, as a covariate. Actually, Fukuoka[ 2 ] found that physical distance is stronger related to correlation of expression levels than is adjacency. Table  5  shows that when a subgroup of the gene pairs is selected on the basis of physical distance, it becomes even more clear that the elevated correlation is a general property of all near-by gene pairs. This is not surprising since the group of adjacent gene pairs is heterogenous, containing gene pairs with vastly varying distances. Unfortunately, there are some problems related to the use of physical distance. First, it is not clear with respect to which anchor the physical distance should be defined. We have chosen to use the minimal distance (using the end nearest to the neighbor as an anchor), because it is not confounded by the length of the gene. However, depending on which co-regulation mechanism one has in mind, other distance measures might be more natural. This question becomes crucial if one wants to make subgroup analysis based on convergent, divergent and tandem gene pairs. Second, physical distance is a continuous variable, which means that the model must be augmented with an assumption regarding the relationship between distance and correlation. The question remains to what biological phenomena the baseline correlation between adjacent genes should be attributed. Several of the theories that have been proposed could account for it, and it is likely that several mechanisms play a role. In addition to the baseline correlation, we also found more consistent correlation between adjacent genes and inconsistent co-regulation, but this does not necessarily imply a combination of two mechanisms: one could imagine a kind of quasi-consistent mechanism, in which groups of adjacent, co-regulated genes could have boundaries everywhere, but with some boundary locations being more likely than others. Finally, it has been suggested that the correlation between the expressions of adjacent genes should be understood in terms of enhancer-promotor interaction, which weakens with distance(Dorsett[ 14 ]). It is possible that such a model would predict a pattern similar to what we have found.
Discussion
There are many theories that could explain correlation between the expressions of two genes in general, and that of two adjacent genes in particular. For the verification of the specific correlation structures predicted by each theory, Spelmann and Rubin's data turned out to be very useful. Their experimental design was based on two distinct classes of experimental conditions (in this case, embryos and adults), which makes a relatively simple correlation structure plausible. Also, the data from each of their microarrays could be fit nicely with a normal mixture, which makes it possible to analyze their data on the basis of the Pearson R. As expected, most of the consistent correlation was found to be related to age group and unrelated to adjacency. Maybe more surprisingly, most of the correlation that was related to adjacency could not be accounted for by consistent co-regulation. This suggest that, at the statistical level, co-regulation of adjacent genes should be understood in terms of the mechanics of the transcription process, rather than in terms of the evolutionary origin of specific gene groups. In other words, in this particular data set, consistent correlation was very widespread, applying to 40% of all gene pairs (table  2 ), half of this could be attributed to the two age groups (table  3 ), but this consistent correlation was not the most important contributor to the elevated correlation coefficients for adjacent gene pairs. Correlation between the expression of adjacent genes is evident, but it is possible, as suggested by Spellman and Rubin, that such a correlation between expression levels for two adjacent genes does not generally imply a relationship with respect to biological function.Spellman and Rubin concluded that the adjacent genes with correlated expressions were confined to certain domains, which spanned 20% of all genes. Our findings are different, in that we distinguished between two components of the correlation structure. We found a baseline correlation of 0.1 (the difference between  μ 0  in table  2  and table  3 ) applying to all adjacent gene pairs. In addition to that, we found that 8% of all adjacent gene pairs (the difference between the "+" -fractions in table  2  and table  3 ) were strongly positively correlated. Finally, unlike Spellman and Rubin, we found no evidence for clustering of the strongly correlated gene pairs.μSince this data set contained data from whole animals, we cannot say anything about the role of chromatin domains (or other mechanisms causing correlation of the expression of adjacent genes) in tissue differentiation. Parisi et al. [ 13 ] found clustering of genes that were upregulated in drosophila germ cells, which suggests that one would find elevated consistent coregulation of adjacent genes when applying our model to data sets with different tissue classes.Even for random (non-adjacent) gene pairs,  μ 0  was slightly positive. It is hard to imagine a biological effect that could cause  μ 0  to be significantly greater than zero, but it could be an artifact of the normalization: if gene  g  is expressed at the same level in all tissue samples, and gene  h  as well, they should have a correlation coefficient of zero, but since they will both yield elevated measurements in flies in which the normalization bias is positive, the observed correlation will be positive. The fact that  μ 0  is almost zero suggests that normalization bias is not a major problem with this data set. The fact that  μ 0  for random gene pairs is not generally higher in mixed age groups than in pure adult or embryo groups (table  4 ) is consistent with the assumption that  μ 0  for random gene pairs does not have a biological interpretation. Although looking at adjacent genes is computationally convenient, it would probably be more correct to use physical distance, rather than adjacency, as a covariate. Actually, Fukuoka[ 2 ] found that physical distance is stronger related to correlation of expression levels than is adjacency. Table  5  shows that when a subgroup of the gene pairs is selected on the basis of physical distance, it becomes even more clear that the elevated correlation is a general property of all near-by gene pairs. This is not surprising since the group of adjacent gene pairs is heterogenous, containing gene pairs with vastly varying distances. Unfortunately, there are some problems related to the use of physical distance.μμghμμμFirst, it is not clear with respect to which anchor the physical distance should be defined. We have chosen to use the minimal distance (using the end nearest to the neighbor as an anchor), because it is not confounded by the length of the gene. However, depending on which co-regulation mechanism one has in mind, other distance measures might be more natural. This question becomes crucial if one wants to make subgroup analysis based on convergent, divergent and tandem gene pairs.Second, physical distance is a continuous variable, which means that the model must be augmented with an assumption regarding the relationship between distance and correlation.The question remains to what biological phenomena the baseline correlation between adjacent genes should be attributed. Several of the theories that have been proposed could account for it, and it is likely that several mechanisms play a role.In addition to the baseline correlation, we also found more consistent correlation between adjacent genes and inconsistent co-regulation, but this does not necessarily imply a combination of two mechanisms: one could imagine a kind of quasi-consistent mechanism, in which groups of adjacent, co-regulated genes could have boundaries everywhere, but with some boundary locations being more likely than others. Finally, it has been suggested that the correlation between the expressions of adjacent genes should be understood in terms of enhancer-promotor interaction, which weakens with distance(Dorsett[ 14 ]). It is possible that such a model would predict a pattern similar to what we have found.Conclusions It is possible to analyze correlation structures in gene expression data on the basis of simple, parametric models with known mathematical properties and parameters with biological interpretations, or at least some candidate interpretations. When applied to the data from Spellman and Rubin, it appeared that the expressions of two genes can show positive correlation for either of two largely distinct reasons: because they share some confounder unrelated to adjacency (this is often the age group, but could also be some other biological parameter, or a technical artefact), or because they are located close to each other on the chromosome. The underlying biological mechanisms remain unknown, but there appears to be a component of the correlation that depends on distance only and not of the biological function of the genes. If this is true, gene clustering algorithms might benefit from making distinction between the adjacency-related and not-adjacency-related co-regulation, for example by subtracting the effect of adjacency from the correlation of adjacent genes. Doing that would, according to our findings, lead to more reliable identifications of gene pairs with related biological functions.
Conclusions
It is possible to analyze correlation structures in gene expression data on the basis of simple, parametric models with known mathematical properties and parameters with biological interpretations, or at least some candidate interpretations. When applied to the data from Spellman and Rubin, it appeared that the expressions of two genes can show positive correlation for either of two largely distinct reasons: because they share some confounder unrelated to adjacency (this is often the age group, but could also be some other biological parameter, or a technical artefact), or because they are located close to each other on the chromosome. The underlying biological mechanisms remain unknown, but there appears to be a component of the correlation that depends on distance only and not of the biological function of the genes. If this is true, gene clustering algorithms might benefit from making distinction between the adjacency-related and not-adjacency-related co-regulation, for example by subtracting the effect of adjacency from the correlation of adjacent genes. Doing that would, according to our findings, lead to more reliable identifications of gene pairs with related biological functions.Methods Non-adjacent gene pairs First, we constructed a model for those co-regulation effects that were unrelated to the relative location of the two genes. Suppose that a gene can be active either in adults, in embryos, in both, or in neither. Let  Y gi  be the log-scale measured activity of gene  g  in fly  i , fly  i  having development stage  s  (either "adult" or "embryo"). A natural model for  Y gi  would be a two-component normal mixture: Since the data were normalized so that the average logscale activity across all 89 flies was zero, the activity of gene  g  in the other development stage should be taken into account. We therefore assumed a three-component normal mixture: Actually, of the 89 flies, the gene expressions in 87 of them could be nicely fitted with a three-component, normal mixture, while in two of the adult flies we identified only two components (the component of genes that were passive in adults while active in embryos vanished in those two cases). If the three-component model is correct, Fisher's z-transform, which is the hyperbolic arcus tangens of the Pearson correlation coefficient  R gh  for two genes,  g  and  h , is We expected  μ 0  to be close to zero. This model was validated by implementing it on the neighbor correlations in a  shuffled  data set, that is, for each gene its Pearson R with a random gene was computed. This procedure was repeated with 10 shuffled data sets, each generated by shuffling the genes in the original data set and subsequently computing the Pearson R for each pair of adjacent genes. For each shuffled data set, the parameters in the model were estimated with the EM algorithm (Dempster et.al[ 18 ]). Adjacent genes Second, the distribution of the correlation coefficients of the pairs of adjacent genes in the unshuffied data set were compared to the same distribution for the shuffled data sets. Doing that, we could distinguish two different contributions to the correlation; A) co-regulation effects that are unrelated to the fact that two genes are adjacent, and B) co-regulation effects related to the fact that two genes are adjacent. We also analyzed each gene's correlation with its second, third etc. (up to the tenth) neighbor, in order to get an idea of the lengths of the chromatin domains in question. Third, we applied the above two procedures to two subsets of the data, namely the data for adult flies and the data for embryos. If our assumption, that all (or at least most) correlation could be attributed to the development stage, one would expect much less correlation in those subsets. However, those subsets differ from the entire data set by the mere fact that they are smaller. Therefore, we also applied the same procedure to two random subsets of 35 and 54 flies, respectively, stratified by development stage. Physical distance For those gene pairs where the start and the end of the open reading frame (ORF) was available, we defined the physical distance as the distance between the end of the ORF of the first gene and the start of the ORF of the second gene. We chose that definition because other distance definitions (defined on the basis of the direction of transcription, or on the basis of mid-points) would be confounded by the length of the ORFs. Simulated data Finally, in order to validate our method, we applied it to 6 simulated data sets, which we simulated on the basis of the two alternative hypothesis of consistent and inconsistent co-regulation. The simulated data sets contained 100 tissue samples and 1000 genes. The genes were randomly divided into either 67, 200 or 500 segments, corresponding to average segment sizes of 15, 5 and 2. This means that each gene belonged to exactly one segment. For each average segment size, we simulated one data set with  consistent co-regulation , in which the same segments were used for all 100 tissue samples, and one data set with  inconsistent co-regulation , in which a separate segmentation was sampled for each tissue sample. If gene  g  belongs to segment  s  in sample  i , the expression  Y gi  was given by Y gi  =  x si  +  ε gi      (4) where  x si  + and  ε gi  were both sampled from the standard normal distribution. Notice that this model does not account for different tissue classes, so it should be compared to the results from the adults-only and embryos-only subsets. The error term  ε gi  should be interpreted as a combination of measurement errors and biological effects that are unrelated to the segmentation.
Methods
Non-adjacent gene pairs First, we constructed a model for those co-regulation effects that were unrelated to the relative location of the two genes. Suppose that a gene can be active either in adults, in embryos, in both, or in neither. Let  Y gi  be the log-scale measured activity of gene  g  in fly  i , fly  i  having development stage  s  (either "adult" or "embryo"). A natural model for  Y gi  would be a two-component normal mixture: Since the data were normalized so that the average logscale activity across all 89 flies was zero, the activity of gene  g  in the other development stage should be taken into account. We therefore assumed a three-component normal mixture: Actually, of the 89 flies, the gene expressions in 87 of them could be nicely fitted with a three-component, normal mixture, while in two of the adult flies we identified only two components (the component of genes that were passive in adults while active in embryos vanished in those two cases). If the three-component model is correct, Fisher's z-transform, which is the hyperbolic arcus tangens of the Pearson correlation coefficient  R gh  for two genes,  g  and  h , is We expected  μ 0  to be close to zero. This model was validated by implementing it on the neighbor correlations in a  shuffled  data set, that is, for each gene its Pearson R with a random gene was computed. This procedure was repeated with 10 shuffled data sets, each generated by shuffling the genes in the original data set and subsequently computing the Pearson R for each pair of adjacent genes. For each shuffled data set, the parameters in the model were estimated with the EM algorithm (Dempster et.al[ 18 ]).
Non-adjacent gene pairs
First, we constructed a model for those co-regulation effects that were unrelated to the relative location of the two genes. Suppose that a gene can be active either in adults, in embryos, in both, or in neither. Let  Y gi  be the log-scale measured activity of gene  g  in fly  i , fly  i  having development stage  s  (either "adult" or "embryo"). A natural model for  Y gi  would be a two-component normal mixture:YgigiisYgiSince the data were normalized so that the average logscale activity across all 89 flies was zero, the activity of gene  g  in the other development stage should be taken into account. We therefore assumed a three-component normal mixture:gActually, of the 89 flies, the gene expressions in 87 of them could be nicely fitted with a three-component, normal mixture, while in two of the adult flies we identified only two components (the component of genes that were passive in adults while active in embryos vanished in those two cases).If the three-component model is correct, Fisher's z-transform, which is the hyperbolic arcus tangens of the Pearson correlation coefficient  R gh  for two genes,  g  and  h , isRghghWe expected  μ 0  to be close to zero. This model was validated by implementing it on the neighbor correlations in a  shuffled  data set, that is, for each gene its Pearson R with a random gene was computed. This procedure was repeated with 10 shuffled data sets, each generated by shuffling the genes in the original data set and subsequently computing the Pearson R for each pair of adjacent genes. For each shuffled data set, the parameters in the model were estimated with the EM algorithm (Dempster et.al[ 18 ]).μshuffledAdjacent genes Second, the distribution of the correlation coefficients of the pairs of adjacent genes in the unshuffied data set were compared to the same distribution for the shuffled data sets. Doing that, we could distinguish two different contributions to the correlation; A) co-regulation effects that are unrelated to the fact that two genes are adjacent, and B) co-regulation effects related to the fact that two genes are adjacent. We also analyzed each gene's correlation with its second, third etc. (up to the tenth) neighbor, in order to get an idea of the lengths of the chromatin domains in question. Third, we applied the above two procedures to two subsets of the data, namely the data for adult flies and the data for embryos. If our assumption, that all (or at least most) correlation could be attributed to the development stage, one would expect much less correlation in those subsets. However, those subsets differ from the entire data set by the mere fact that they are smaller. Therefore, we also applied the same procedure to two random subsets of 35 and 54 flies, respectively, stratified by development stage.
Adjacent genes
Second, the distribution of the correlation coefficients of the pairs of adjacent genes in the unshuffied data set were compared to the same distribution for the shuffled data sets. Doing that, we could distinguish two different contributions to the correlation; A) co-regulation effects that are unrelated to the fact that two genes are adjacent, and B) co-regulation effects related to the fact that two genes are adjacent. We also analyzed each gene's correlation with its second, third etc. (up to the tenth) neighbor, in order to get an idea of the lengths of the chromatin domains in question.Third, we applied the above two procedures to two subsets of the data, namely the data for adult flies and the data for embryos. If our assumption, that all (or at least most) correlation could be attributed to the development stage, one would expect much less correlation in those subsets. However, those subsets differ from the entire data set by the mere fact that they are smaller. Therefore, we also applied the same procedure to two random subsets of 35 and 54 flies, respectively, stratified by development stage.Physical distance For those gene pairs where the start and the end of the open reading frame (ORF) was available, we defined the physical distance as the distance between the end of the ORF of the first gene and the start of the ORF of the second gene. We chose that definition because other distance definitions (defined on the basis of the direction of transcription, or on the basis of mid-points) would be confounded by the length of the ORFs.
Physical distance
For those gene pairs where the start and the end of the open reading frame (ORF) was available, we defined the physical distance as the distance between the end of the ORF of the first gene and the start of the ORF of the second gene. We chose that definition because other distance definitions (defined on the basis of the direction of transcription, or on the basis of mid-points) would be confounded by the length of the ORFs.Simulated data Finally, in order to validate our method, we applied it to 6 simulated data sets, which we simulated on the basis of the two alternative hypothesis of consistent and inconsistent co-regulation. The simulated data sets contained 100 tissue samples and 1000 genes. The genes were randomly divided into either 67, 200 or 500 segments, corresponding to average segment sizes of 15, 5 and 2. This means that each gene belonged to exactly one segment. For each average segment size, we simulated one data set with  consistent co-regulation , in which the same segments were used for all 100 tissue samples, and one data set with  inconsistent co-regulation , in which a separate segmentation was sampled for each tissue sample. If gene  g  belongs to segment  s  in sample  i , the expression  Y gi  was given by Y gi  =  x si  +  ε gi      (4) where  x si  + and  ε gi  were both sampled from the standard normal distribution. Notice that this model does not account for different tissue classes, so it should be compared to the results from the adults-only and embryos-only subsets. The error term  ε gi  should be interpreted as a combination of measurement errors and biological effects that are unrelated to the segmentation.
Simulated data
Finally, in order to validate our method, we applied it to 6 simulated data sets, which we simulated on the basis of the two alternative hypothesis of consistent and inconsistent co-regulation. The simulated data sets contained 100 tissue samples and 1000 genes. The genes were randomly divided into either 67, 200 or 500 segments, corresponding to average segment sizes of 15, 5 and 2. This means that each gene belonged to exactly one segment. For each average segment size, we simulated one data set with  consistent co-regulation , in which the same segments were used for all 100 tissue samples, and one data set with  inconsistent co-regulation , in which a separate segmentation was sampled for each tissue sample. If gene  g  belongs to segment  s  in sample  i , the expression  Y gi  was given byconsistent co-regulationinconsistent co-regulationgsiYgiY gi  =  x si  +  ε gi      (4)Ygixsiεgiwhere  x si  + and  ε gi  were both sampled from the standard normal distribution. Notice that this model does not account for different tissue classes, so it should be compared to the results from the adults-only and embryos-only subsets. The error term  ε gi  should be interpreted as a combination of measurement errors and biological effects that are unrelated to the segmentation.xsiεgiεgiAuthors' contributions HT did the explorative data analysis and suggested the concept of consistent co-regulation and using three-components mixtures for the correlation structure. AZ suggested using Fisher's z-transform for the correlations. HT did the literature review. Both authors contributed to the manuscript.
Authors' contributions
HT did the explorative data analysis and suggested the concept of consistent co-regulation and using three-components mixtures for the correlation structure. AZ suggested using Fisher's z-transform for the correlations. HT did the literature review. Both authors contributed to the manuscript.