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The refGene list of human genes (last updated in April 2018) is obtained from Pseudogenes are removed from the list by requiring the gene ending position to be larger than the starting position. Transcripts with overlapping coordinates are merged using the bedtools program |
Quinlan and Hall, 2010 , with the command bedtools merge -i file -d 100 -c 4 -o collapse This processing leads to 18,757 gene sequences. |
The non-transposon/non-repetitive unique sequence can be read out from the reference genome directly: unique sequences are in uppercase, whereas RepeatMasker |
) identified repetitive sequences are in lowercase. The telomere and centromere region is first based on the cytoband information from |
The first and the last bands of a chromosome is considered to be the telomere regions. We first use the same file to bracket the centromere region (when the band is labeled as “acen”). Then we further fine-tune the boundary by an observation made in Thanos et al., |
2018 that windowed statistical qualities (e.g. entropy) have extremely low variations in the centromere region. 3.2 Distribution of Erdös sequences in the human genome at the chromosome level |
For R/Y binarization, out of 2.911 billions overlapping 10-mers in the human genome (chromosomes 1-22,X, excluding any 10-mers which contain unsequenced bases) there are 6,161,338 counts of R/Y [MATH] , or 0.21% of all 10-mer counts. The ratio between R/Y [MATH] and non-R/Y- [MATH] sequence counts is 1:471. This freque... |
For W/S [MATH] , there are 10,073,985 copies of them in the human genome (0.35%). For each W/S [MATH] , there are 288 10-mers that are not [MATH] compatible. This frequency is slightly higher than the expected value of 0.32% (O/E=1.05), if we make a simple assumption that the GC-content is 40% (see Appendix). The 40% v... |
2013 Even if we use the observed GC-content at the individual chromosome, the conclusion remains true that the frequency is slightly higher than expected (except for chromosomes 16 and 19) (see Appendix). |
Finally, there are 8,782,253 copies of M/K based [MATH] (0.3%), slightly lower than expected (0.39%, assuming strand symmetry). There are more W/S or M/K [MATH] than R/Y [MATH] even though R/Y [MATH] is expected to appear more often than W/S [MATH] and equally likely as M/K [MATH] Combining the three types of [MATH] , ... |
There is yet another R/Y based 10-mer Trifonov, 2010 Li et al., 2013 RRRRRYYYYY /YYYYYRRRRR (R5Y5), proposed as a nucleosome positioning sequence or motif |
Trifonov and Sussman, 1980 Drew and Travers, 1985 Peckham et al., 2007 Segal et al., 2006 Jiang and Pugh, 2009 There are other nucleosome positioning sequence patterns proposed, in particular, the periodicity 10-11 of AA/TT steps Calladine et al., |
2004 Unlike the underrepresented R/Y-based Erdös sequences, the R5Y5 motif is overrepresented: observed frequency is 0.36% vs. the expected 0.195% (O/E=1.87). The overrepresentation of R5Y5 is also consistent with the abundance of R- and Y-tracts Behe, |
1995 and their connection to coding/noncoding regions has been reported Almirantis and Provata, 1997 Table 1(a) shows the correlation among various [MATH] ’s. The R/Y based Erdös frequency is negatively correlated with both W/S based and K/M based Erdös frequencies, at the chromosome level, with Spearman rank correlati... |
1999 , sec. 8.6) of [MATH] and [MATH] . The W/S based and K/M based Erdös sequences are positively correlated. The overall Erdös frequency is closely tied to that of the W/S based Erdös sequence, probably because there are more W/S based Erdös motifs than either R/Y based or K/M based ones. |
Table 1(b) shows the correlation between Erdös sequence frequencies with those of other sequence features. The R/Y based Erdös sequence is always opposite to W/S based or to K/M based, as well as overall Erdös sequence, in terms of its correlation with other features. Notably, the rate of W/S based Erdös sequence is po... |
For completeness, we also show the correlation among the frequencies of various sequence features in Table 1(c). Interestingly (and perhaps counter-intuitively), both the polyA/polyT density and polyC/polyC density (which is very low) are positively correlated with the GC-content. |
Fig. highlights a few strong correlations: negative correlation between two types of Erdös [MATH] ’s, positive correlation between R/Y based [MATH] |
and sequence length, negative correlation between R/Y based [MATH] and GC-content, and positive correlation between polyA/polyT density and GC-content. Other large (in absolute value) and significant correlations are marked as bold in Table 1. |
3.3 Distribution of Erdös sequences in the human genome at the window level Quantities calculated at the chromosome level may not be fine scaled enough. In order to check whether Erdös sequence frequencies are correlated with other sequence features at a more local scale, we partition the human genome into non-overlapp... |
Interestingly, all highlighted strong correlations in Table 1 and Fig. at the chromosome level remain true at the 1Mb window level (Table 2, Fig. ), including the negative correlation between R/Y based and W/S (and K/M based) Erdös sequence frequency, the negative (positive) correlation between R/Y (W/S and overall) Er... |
The overrepresentation of W/S based [MATH] sequence observed at chromosome level is still true at 1Mb window level. We use the local GC-content to estimate the expected frequency of W/S based [MATH] in each window. The distribution of all O/E is single-peaked, with median 1.097, mean 1.091, mode around 1.12-1.13. Of al... |
In Li et al., 2013 it was observed that the R5Y5 density is negatively correlated with that of the transposon and other repetitive sequences at 64kb window level. Table 2 shows that this conclusion remains true at the 1Mb window level (positive correlation between R5Y5 density and unique sequence density). |
3.4 Frequency of Erdös motifs in various functional, regional, sequence-feature classes Besides counting Erdös motifs in the whole genome, we also count them in sequences belonging to specially defined sub-categories. The first subcategory consists of the genome free of transposons and repeats. This sequence has been f... |
Fig. summarizes the observed vs expected Erdös motifs in the following situations: (1) non-transposon, non-repetitive “unique” sequences; (2) messenger RNA sequences; (3) gene sequences; (4) centromeres; and (5) telomeres. The expected frequencies derived by the formulae in Appendix use the exact S%, R%, K% frequencies... |
sequence, and GC%=0.4 assumption for W/S based one. Fig. shows that the underrepresentation of R/Y-based Erdös motifs is true for all categories examined, in particular the centromere region. However, the extreme lower R/Y based Erdös motif frequency in centromere might be an artifact. Centromeres are dominated by a la... |
For W/S based Erdös motifs, most observed frequencies are consistent or slightly higher than the expected (Fig. ), except for mRNA sequences. We hypothesize that it might be related to a “hidden” periodicity in the [MATH] sequence. If we examine the [MATH] sequence closely in Fig. , the positions 1, 4, 7, 10 are all po... |
Finally, adding the three types of Erdös frequency together, for both expected and observed sequences, we see in Fig. a pattern of underrepresentation (with the exception of mRNA). This underrepresentation, generally speaking, supports the idea that the human DNA sequences are not locally random. |
Discussion and conclusions There is a long history in characterizing statistical patterns in DNA sequences: from recognizing the nearest neighbor correlations Swartz et al., |
1962 to the detection of long-range correlations Li and Kaneko, 1992 Peng et al., 1992 Voss, 1992 . The debate on whether the human genome sequences are homogeneous, iid (independent and identically distributed), random IHGSC, |
2001 Clay and Bernardi, 2001 Li et al., 2002 is often based on concepts defined on infinitely long sequences. There is always a lack of adequate concepts related to randomness on finite scales. In this study, our aim is to consider a type of low-discrepancy sequences, called Erdös sequences [MATH] and perhaps their gen... |
The underrepresentation of overall [MATH] compatible motifs in the human genome (Fig. ) can be interpreted as a lack of locally random sequences. This is particularly true for R/Y based [MATH] 10-mers. Interestingly, it was observed that long-range correlation scaling is best observed in the R/Y binarization Peng et al... |
1992 . The surprising overrepresentation of W/S based [MATH] 10-mers in mRNA sequences might reflect, among others, a potentially periodicity-3 tendency in certain reading frames in this sequence. Further investigation is required to pinpoint the cause of the overrepresentation. |
The underrepresentation of R/Y based [MATH] is not a consequence of lower or higher R content, or equivalently, a violation of Chargaff’s second parity. Indeed, if R% is too high or too low, other R/Y-based 10-mers may become the most frequent motifs, such as R-track or Y-track |
Almirantis and Provata, 1997 . However, Chargaff’s rule is well preserved both at the chromosomal and at the 1Mb window levels (result not shown). Furthermore, we observed that among R/Y based 10-mers with R%=Y%, the Erdös motifs are still underrepresented (result not shown). |
The key component in an Erdös sequence is its low discrepancy, either in the direct cumulative plot or in many of its equally spaced subsequences. The cumulative plot or (random) walk representation has been frequently used in DNA sequences Berthelsen et al., |
1992 Peng et al., 1992 Zhang and Zhang, 1994 , but the inclusion of equally spaced subsequences is an ingenious device to exclude periodic sequences. An open question is how low discrepancy property fits other hallmarks of local randomness. |
One feature of finite random sequences is their lack of periodic patterns. Indeed, [MATH] is identical to the first 10 digits of the Thue-Morse sequence, a well known aperiodic or quasiperiodic infinite sequence |
Allouche and Shallit, 1999 Riklund et al., 1987 . Interestingly, Thue-Morse sequence has bounded discrepancy for specific spacing values ( [MATH] even in the infinite sequence length limit Leong, |
2011 Leong and Shallit, 2013 Using a semi-periodic pattern to reduce discrepancy while at the same time to avoid exact periodicity may have other applications in sequence analysis. For example, it is well known that in the promoter region of housekeeping genes, CpG dinucleotide is common, forming CpG islands Vinson and... |
2012 However, they can not be arranged in a periodic fashion. When they appear in a periodic arrangement of CGG repeats with more than 200 copies in gene FMR1, it leads to a form of intellectual disability Park et al., |
2015 Another property of local randomness is the difficulty in recreating the sequence. This property makes randomness equivalent to a measure of complexity |
Li, 1991 . The use of Kolmogorov complexity as a way to measure local randomness is an involved topic Allouche, 1999 Vitányi, 2001 Li and Vitányi, |
2009 Soler-Toscano et al., 2014 and will not be addressed here. However, compressibility based calculation can be easily carried out |
Ziv and Lempel, 1977 Benedetto et al., 2002 Estevez-Rams et al., 2013 . Preliminary calculations indicate that [MATH] belong to a harder-to-compress group, but not the hardest-to-compress (results not shown). |
In conclusion, the search for low discrepancy sequences in the human DNA has shown that these motifs are overall underrepresented (except for some particular cases). This results is consistent with nonrandomness at the local scale. In future studies, it would be interesting to extend this search into the full 4-letter ... |
Acknowledgement: WL thanks the Robert S Boas Center for Genomics and Human Genetics for support. Appendix: estimation of the [MATH] frequencies in random sequences |
If each symbol appears with equal probability, the expected [MATH] frequency can be estimated by counting the number of 4-symbol sequences matching [MATH] |
of the total number of possible 10-mers. There are 4 motifs in Eq.( 2.2 ), each containing [MATH] 4-symbol sequences. Combining Eqs.( 2.2 2.2 2.2 ), there are roughly |
[MATH] =12288 4-symbol 10-mers associated with [MATH] (the actual number is 12240 because one 4-symbol sequence can simultaneously belong to (e.g.) Eq.( 2.2 ) or Eq.( 2.2 ) ). The total number of 4-symbol 10-mers is [MATH] Therefore the [MATH] associated 10-mer frequency is 12240/1048576 [MATH] |
For individual type of [MATH] sequence, R/Y based Erdös sequence is expected to appear with the frequency of [EQUATION] If the strand symmetry Fickett et al., |
1992 Li, 1997 Forsdyke and Mortimer, 2000 holds true, i.e., A% [MATH] T%, G% [MATH] C%, we have R% [MATH] Y% [MATH] Then the expected frequency is [MATH] The same argument is also applied to K/M based Erdös motifs. For W/S based Erdös sequence, the expected frequency is |
[EQUATION] If the strand symmetry is true, S% [MATH] 2 G% [MATH] 2C%, W% [MATH] 2 A% [MATH] 2T%. In the human genome, it is observed that G% [MATH] C% [MATH] 0.2, and A% [MATH] T% [MATH] 0.3 Li, |
2013 . The expected W/S based [MATH] frequency is [MATH] We also ran a simulation to generate artificial chromosomes with the same base composition as the real human chromosomes, but bases are scrambled. The numbers of R/Y and W/S based Erdös sequence [MATH] |
in the simulated artificial chromosomes are shown in Table A1 (column “exp(simu)”. These can be compared to the estimated from the formula (“exp(form)”). These two columns match very well. In comparison, the observed numbers of R/Y based Erdös sequences in the human genome are greatly underrepresented, whereas W/S base... |
531402 Table A1: The number of R/Y and W/S-based Erdös sequences in 1-22 & X chromosomes. The counts are based on the formulae in Appendix (left columns), artificially constructed chromosomes (middle columns), and the true observed numbers calculated from the human genome (right columns). |
# Source: arxiv 1805.11500 # Title: Characterizing o-minimal groups in tame expansions of o-minimal structures # Sections: all # Downloaded: 2026-03-03T02:36:58.075607+00:00 |
Characterizing o-minimal groups in tame expansions of o-minimal structures Abstract. Let [MATH] be an expansion of an o-minimal structure [MATH] by a dense set [MATH] , such that three tameness conditions hold. Among all definable groups in [MATH] , we characterize those that are definable in [MATH] as the ones whose d... |
Abstract. We study groups definable in tame expansions of o-minimal structures and give a dimension-theoretic characterization of those groups that are o-minimal. Let [MATH] be an expansion of an o-minimal structure [MATH] , such that every open definable set is definable in M, and such that N admits a dimension functi... |
Abstract. In this paper we prove the first global results for groups definable in tame expansions of o-minimal structures. Let [MATH] be an expansion of an o-minimal structure M that admits a dimension function with the usual properties. The setting includes all known tame expansions [MATH] , where [MATH] is a dense se... |
Abstract. We establish the first global results for groups definable in tame expansions of o-minimal structures. Let [MATH] be an expansion of an o-minimal structure M that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of M by a Mann group, or by a subgroup of an e... |
Key words and phrases: o-minimal structure, tame expansion, dimension function, definable group, independent set, group chunk 2010 Mathematics Subject Classification: Primary 03C64, 03C68, 22B99 |
Research supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship. |
1. Introduction Definable groups have been at the core of model theory for at least a period of three decades, largely because of their prominent role in important applications of the subject, such as Hrushovski’s proof of the function field Mordell-Lang conjecture in all characteristics ( |
). Examples include algebraic groups (which are definable in algebraically closed fields) and real Lie groups (which are definable in o-minimal structures). Groups definable in o-minimal structures are well-understood, with arguably the most influential work in the area being the solution of Pillay’s conjecture over a ... |
), which brought to light new tools in theories with NIP. On the other hand, groups definable in tame expansions of o-minimal structures have only been studied locally ( |
). In this paper we prove the first global results, whose gist is that one can recover a group definable in the o-minimal reduct from an arbitrary definable group using only dimension-theoretic data. |
O-minimal structures were introduced and first studied by van den Dries and Knight-Pillay-Steinhorn and have since provided a rigid framework to study real algebraic and analytic geometry. They have enjoyed a wide spectrum of applications reaching out even to number theory and Diophantine geometry (such as in Pila’s so... |
). Tame expansions of o-minimal structures have been developed as a context that escapes the o-minimal, locally finite setting, yet preserves the tame geometric behavior on the class of all definable sets. An important category of such structures are those where every open definable set is already definable in the o-mi... |
, where the decidability of its theory was proven. Forty years later, van den Dries extended Robinson’s results to arbitrary dense pairs of o-minimal structures, and a stream of further developments in the subject followed ( |
). Besides dense pairs, examples of structures in this category now include pairs of the form [MATH] , where [MATH] is an o-minimal expansion of an ordered group, and [MATH] is a dense multiplicative subgroup with the Mann property, or a dense subgroup of the unit circle or of an elliptic curve, or it is a dense indepe... |
), extending the known cell decomposition theorem from o-minimal structures and the usual o-minimal dimension. The setting of the current paper is even more general, as only the existence of a good dimension theory is assumed. Moreover, the o-minimal reduct does not need to expand an ordered group. Our main theorem (Th... |
Throughout this paper, [MATH] and [MATH] denote two first-order structures, with [MATH] expanding [MATH] . We denote by [MATH] the language of M and by [MATH] its usual definable closure. By ‘ [MATH] -definable’ or ‘definable in M’ we mean definable in M with parameters. By ‘definable’ or ‘definable in N’ we mean defin... |
dimension function compatible with is a map [MATH] from the class of all definable sets to [MATH] that satisfies the following properties: for all definable [MATH] , and [MATH] |
(D1) [MATH] [MATH] , and [MATH] if and only if [MATH] (D2) [MATH] (D3) if [MATH] is a disjoint definable family of sets, then (a) |
for [MATH] , the set [MATH] is definable, and (b) if every [MATH] has dimension [MATH] , then [EQUATION] (D4) if [MATH] is a definable bijection, then [MATH] |
(D5) the dimension of every L-definable set [MATH] coincides with its usual o-minimal dimension (D6) every definable map [MATH] agrees with an [MATH] -definable map [MATH] outside a definable set of dimension [MATH] |
It follows from (D2) that [MATH] is monotone, and from (D1)-(D4) that it is a dimension function also in the sense of In the rest of this paper, and unless stated otherwise, we assume that N admits a dimension function [MATH] compatible with M. In the aforementioned pairs [MATH] the existence of such [MATH] was establi... |
(details are postponed until Section 2.4 ). Moreover, in that context, a definable set [MATH] was shown to have dimension [MATH] if and only if [MATH] is internal to [MATH] if and only if no open interval is internal to [MATH] (where internality is the usual notion from geometric stability theory). Such a set [MATH] wa... |
strongly large if [EQUATION] We call a definable group strongly large if its domain is strongly large. Every infinite small set is not strongly large. Every [MATH] -definable set is strongly large. The main intuition is that, conversely, strongly large sets must behave like [MATH] -definable sets. Our main theorem esta... |
Theorem 1.1 Let N be an expansion of an o-minimal structure M such that (a) every open definable set is definable in M, and (b) N admits a dimension function compatible with M. |
Let [MATH] be a definable group. Then [MATH] is definably isomorphic to a group definable in M if and only if it is definably isomorphic to a strongly large group. |
In the rest of this introduction, we discuss the content of the above theorem, state some consequences, and illustrate the main difficulties of its proof. |
Theorem 1.1 is in the spirit of a classical theme in model theory; namely, to recover a mathematical object in some restricted language given data of possibly different nature. For example, the influential Pila-Wilkie theorem ( |
) recovers a semialgebraic subset of a set [MATH] definable in an arbitrary o-minimal structure given a number-theoretic condition on [MATH] . In our case, we recover an o-minimal group from a group [MATH] definable in an expansion of an o-minimal structure given a dimension-theoretic condition on the domain of [MATH] |
We next point out the need for including definable isomorphisms in the conclusion of Theorem 1.1 Suppose [MATH] is the expansion of the real field [MATH] by the field [MATH] of algebraic numbers. Let [MATH] |
and consider the definable bijection [MATH] , with [MATH] . Let [MATH] be the disjoint union of [MATH] and [MATH] , equipped with the group structure induced from [MATH] via the identity map on the first part and via [MATH] on the second. Then [MATH] is not strongly large, since [MATH] and [MATH] , but it is definably ... |
Theorem 1.1 puts a constraint on the existence of new definable groups, which has already been the theme of previous research, such as in |
and . Let [MATH] denote one of the aforementioned pairs from . As [MATH] itself can be a definable group, there are new small definable groups (and a study for those has recently been initiated in |
). Of course, there are also new large definable groups, such as the direct product [MATH] , but as a consequence of the above theorem, there are no new large definable groups contained in [MATH] , or in any [MATH] -dimensional L-definable set. A special case of this statement was proven in , Lemma 7.3] ; namely, that ... |
In the example [MATH] , where [MATH] is a dense [MATH] -independent set, our work implies that there are no new definable groups at all. This pair recently received special attention in |
and even triggered new model-theoretic work at the general level of ‘ [MATH] -structures’ . The basic intuition is that a [MATH] -independent set [MATH] is at the other end of being a group, since there are no [MATH] -definable relations between its elements. It is then easy to see that there are no new small definable... |
, we prove that every definable group is definably isomorphic to a strongly large group (Theorem 6.1 ). Combined with Theorem 1.1 , we obtain the following application. |
Theorem 1.2 Let [MATH] be an o-minimal expansion of an ordered group, and [MATH] an expansion of [MATH] by a dense [MATH] -independent set [MATH] . Then every group definable in N is definably isomorphic to a group definable in M. |
Again, a special case of this statement was previously proved, in , Proposition 6.4] ; namely, that every definable subgroup of [MATH] is L-definable. As a parallel note, Theorem 1.2 applies also to interpretable groups, as those are definably isomorphic to definable ones (by elimination of imaginaries |
). Elimination of imaginaries is known to fail in the general setting of Theorem 1.1 ). Finally, let us point out that Theorem 1.1 establishes a conjecture for definable groups stated in |
and reformulated in , for the case of strongly large groups. The conjecture stated that if [MATH] is a definable group of dimension [MATH] , then there is a [MATH] -definable cover [MATH] of [MATH] , and a short exact sequence |
[EQUATION] where [MATH] is [MATH] -definable in M, with [MATH] , and [MATH] is definable and small. Theorem 1.1 implies the conjecture for [MATH] strongly large, with [MATH] and [MATH] trivial. Namely, it answers 17 , Question 7.11] ) affirmatively. It is worth noting that the above conjecture was inspired by an analog... |
), namely that of semi-bounded o-minimal structures (see also Remark 4.1 below). There is a number of different settings where at least the methods of this paper could apply, such as that of [MATH] -structures, whereas a new direction is proposed in Section . Let us now proceed to describe the strategy of our proof. |
Sketch of the proof of Theorem 1.1 We illustrate the main difficulties in proving the right-to-left direction of Theorem 1.1 . Given a strongly large group [MATH] , we need to recover a suitable [MATH] -definable domain [MATH] and an [MATH] -definable map [MATH] that can play the role of an [MATH] -definable group defi... |
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