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[EQUATION] with appropriate constants [MATH] [MATH] , and [MATH] . In particular, we can relate the frequency [MATH] to the contraction factor [MATH] , based on the observation that the distance between consecutive peak times of [MATH] increases by a factor of [MATH] mainly due to Eq. ( 32 ). Precisely, let us consider... |
[EQUATION] leading to the relation between [MATH] and [MATH] as follows: [EQUATION] For example, when [MATH] , we get [MATH] from Eq. ( 35 ), which is comparable with the numercial value of [MATH] estimated from the curve for [MATH] in Fig. . We also find that [MATH] is not a function of [MATH] , which is probably beca... |
II.5 Case with multiple seed events So far we have considered the case only with one seed event at the zeroth level. Here we test if our conclusions in the case with a single seed event are robust with respect to the number of seed events at the zeroth level. For this, we perform the numerical simulations of our origin... |
III Conclusion We have studied the hierarchical burst model for the hierarchical temporal structure by assuming that an observed event sequence is [MATH] events at the first level, each of which in turn induces other [MATH] events at the second level, and so on. The interval for the induction is assumed to decrease by ... |
On the other hand, it turns out that the burst size distributions are not straightforward to analyze. By performing numerical simulations, we find the stretched exponential function for the burst size distributions, implying the existence of correlations between interevent times, often called correlated bursts. However... |
Despite the debate on the functional form of burst size distributions Karsai et al. 2012a ); Jiang et al. 2016 , one can extend our model to reproduce the power-law burst size distributions as evident in some empirical data analysis Karsai et al. 2012a , which then can help us to understand the correlations between int... |
Acknowledgements. W.-S.J. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03932590). H.-H.J. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Minis... |
# Source: arxiv 1805.09758 # Title: PDB_Amyloid: The Extended Live Amyloid Structure List from the PDB # Sections: all # Downloaded: 2026-03-03T05:16:25.657583+00:00 |
PDB_Amyloid: The Extended Live Amyloid Structure List from the PDB Abstract The Protein Data Bank (PDB) contains more than 135 000 entries today. From these, relatively few amyloid structures can be identified, since amyloids are insoluble in water. Therefore, mostly solid state NMR-recorded amyloid structures are depo... |
Introduction The Protein Data Bank (PDB) is a permanently developing public resource of spatial structures of proteins and nucleic acids |
. Today the database contains more than 135,000 structures. The geometric properties of these molecules can be analyzed by bioinformatical tools, and one may infer significant new relations in these very complex macromolecular structures through these analyses |
In the present contribution, we are interested in the amyloid structures in the Protein Data Bank. Amyloids are misfolded protein aggregates, which are present in numerous structures in biology, including |
1. the cellular surface of several microorganisms , having a role in host-pathogen interaction; 2. the silkmoth chorion and some fish choria, forming protective films |
3. the immune system of certain insects, helping the encapsulation of the intruders 4. in the healthy human pituitary secretory granules, for storing peptide hormones |
5. in human amyloidoses and several neurodegenerative diseases Amyloid structures sometimes show prion-like infective properties |
. Cerebral [MATH] -amyloid plaques have been considered biomarkers of the Alzheimer’s disease for a long time ; but more recently, their validity is challenged by several authors |
Amyloid formation mechanisms are reviewed in . Amyloid fibers are formed from parallel [MATH] -sheets, with hydrogen bonds between the parallel strands. It is widely accepted that amyloid formation requires the presence of a nucleus or a seed of amyloid-forming segments with exposed edges of the [MATH] -sheet structure... |
Since amyloid fibers are insoluble in water until the very recent years there were no high-resolution structures deposited in the RCSB Protein Data Bank (PDB) |
. Today one can find several dozens of atomic resolution amyloid structures in the PDB, and this dataset opened up the possibility of the analysis and the data mining of the properties of these misaligned proteins, using their high-resolution spatial structure. |
The first step in this direction is the identification of the amyloid structures in the PDB. Amyloid- and amyloid-precursor molecules were collected and predicted using protein-sequencing data in numerous articles (e.g., in the AMYPdb resource |
, or in ). We are interested in the analysis of the spatial protein structures for finding amyloid and amyloid-precursor molecules, rather than the analysis of residue-sequence properties of proteins of unknown three-dimensional structures. |
In the remarkable work the authors screened the Protein Data Bank for amyloid structures, by applying the following procedure: (i) |
By textual search, those PDB entries were selected, which contain the words “amyloid” or any of other 38 words, describing amyloid precursors; |
(ii) Helical structures, identified by torsion angles, were thrown out; (iii) Parallel, near-linear fragments of length at least 4 residues were identified; structures without these fragments were also thrown away. |
In the present work, we prepare an automatically updated list of amyloid and potentially amyloidogenic structures from the PDB, with applying only the geometric properties of [MATH] -sheets; consequently, we do not use any textual search, referring to the annotations of the PDB entries. |
By this choice, we intend to identify not only the aggregated amyloid entries and known precursors, but also those globular proteins, which contain small, locally amyloid-like substructures. We assume that these globular proteins may also be amyloidogenic ones, i.e., they can more easily turn into amyloid fibers than g... |
Since the Protein Data Bank grows very quickly – in 2017, every day, on the average, 30 new structures were deposited – we need to construct an automatically updated web server, which periodically examines the new PDB entries and includes the newly deposited amyloid and potentially amyloidogenic structures. Consequentl... |
Discussion and Results Amyloid structures We have found that our list at contains all amyloid structures with at least two polypeptide chains, which are listed in |
. For example, the classical amyloid structures of 2KIB, 2N0A, 5KO0, 2LBU, 2LMN are all present in the list. Possibly Amyloidogenic Structures |
Here we review some non-amyloid proteins which were found by our screening algorithm, and which are listed at . We also give literary evidence showing links to the amyloid formation of these molecules. These findings witness the power of our algorithm, but clearly, we cannot review here the more than 500 structures, pr... |
1HCN: Human chorionic gonadotropin (hCG), Fig. 1, panel A. It is a placenta-produced human hormone, applied in numerous pregnancy tests and in legal and illegal drug products, including physical performance enhancing and weight-loss preparations. It is reported to increase [MATH] -amyloid levels in rats |
and to increase [MATH] -cleavage of an amyloid-precursor protein . Protein hCG also has a role in amyloid [MATH] precursor protein expression and modulation in human cells |
, and in protein folding regulation in endoplasmic reticulum . We believe that these roles of hCG are closely related to particular geometric properties of its parallel [MATH] -sheets. |
1BSF: Thymidylate synthase A (TS) from Bacillus subtilis . Thymidylate synthase has an important role in DNA synthesis, whose aggregational properties were studied for a long time |
. The human TS is a primary target of cancer chemotherapy, most importantly by 5-fluorouracil, a strong-binding TS inhibitor, applied widely in colon-, esophageal-, stomach-, pancreatic-, breast- and cervical cancers. On Fig. 1., panel B, it is clearly visible that the parallel [MATH] -sheets are hidden in the dimeric ... |
, TS also has a monomeric form with distinct function, and the dimeric and the monomeric forms have an equilibrium in humans. Therefore, the hidden [MATH] -sheets in the monomeric form may become accessible and may play a role in aggregation processes. |
3FJ5: Tyrosine kinase c-Src (Fig. 1, panel C) has a role in MAP kinase pathway, and in the development of breast cancers in animals and humans |
. It is shown that the SH3 domain of this protein aggregates to form amyloid fibrils at mild acid pHs in . Sources suggest that amyloid associated microgliosis is strengthened by tyrosine kinase c-Src activity. It is also noted that MAP kinase signaling cascade dysfunction specific to Alzheimer’s disease in fibroblasts |
2OCT: Stefin B (Cystatin B) tetramer (Fig. 1, panel D) is an intracellular thiol protease inhibitor. It is known to form amyloid fibrils in vitro |
, its role in amyloidogenesis is detailed in and Materials and Methods Here we describe the selection method, which generates the Extended Amyloid List at |
In contrast with , we did not make any selection through textual search in the annotation fields of the PDB files. Instead of that, we have attempted to collect the minimal set of geometric rules , which already return the amyloids found in |
, plus novel, globular proteins with possibly amyloidogenic substructures. The following rules are applied: (i) For finding parallel [MATH] -sheets: The authors of |
selected parallel chain segments by requiring the distance-difference between the closest [MATH] atoms of the fragment to be less than 1.5 Å. Instead of this condition, we have applied a bound to the standard deviation [MATH] between the closest [MATH] atoms of the fragment to be less than 1.5 Å. We think that this app... |
(ii) Excluding structures with large curvature: The authors of excluded helical structures from consideration. We apply a locally verifiable angular condition for the fragments [MATH] as follows: Fragment [MATH] , which satisfies the conditions in (i), needs also to satisfy the condition that the angles of each three c... |
(iii) Condition for the minimum length of parallel fragments: [MATH] where [MATH] denotes the same as in [MATH] , and [MATH] denotes the length of chain [MATH] , measured in residues. |
The specific parameters for the conditions above were selected for including all multi-chain amyloid structures that were also found in |
. We do not aim to find amyloid-like structures containing only one single polypeptide chain since the amyloid structures contain a large number of approximately parallel fibers, each consisting of different chains. While the Protein Data Bank contains partial amyloid structures with one single chain (e.g., 1HZ3), thes... |
Conclusions We have demonstrated the validity of three geometric structural selection rules, which identify amyloid fibrils and plaques in the Protein Data Bank. Additionally, these rules find non-amyloid soluble proteins, among which we have identified several amyloidogenic ones by scanning the literature. We believe ... |
Data availability The automatically updated web page is available at . The page contains the list of the PDB entries found by our program, each entry is given in the graphical form, hyperlinked to the structures at the RCSB PDB site |
The Python source code of the software program, which generates the PDB_Amyloid list is available at The page contains not only the graphical representation of the proteins found but also a list of their PDB codes at |
Acknowledgments KT and BV were partially funded by the VEKOP-2.3.2-16-2017-00014 program, supported by the European Union and the State of Hungary, co-financed by the European Regional Development Fund, and by the European Union, co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002). VG was partially... |
Conflict of Interest: The authors declare no conflicts of interest. Author contributions: VG initiated the study and analyzed results. BV created the web interface and the update-mechanism. KT designed and programmed the geometric filtering algorithm and fine-tuned the geometric constraints. |
# Source: arxiv 1805.10248 # Title: Quantifying Local Randomness in Human DNA and RNA Sequences Using Erdos Motifs # Sections: all # Downloaded: 2026-03-02T08:42:02.195062+00:00 |
Quantifying Local Randomness in Human DNA and RNA Sequences Using Erdös Motifs Abstract In 1932, Paul Erdös asked whether a random walk constructed from a binary sequence can achieve the lowest possible deviation (lowest discrepancy), for the sequence itself and for all its subsequences formed by homogeneous arithmetic... |
keywords: Erdös discrepancy problem; human genome; DNA motifs; local randomness; Introduction The DNA sequences constituting the human genome are a product of long evolutionary history, with sequence altering processes such as whole-genome duplication Ohno et al., |
1968 Wolfe, 2001 regional (segmental) duplication Bailey et al., 2002 , inversions Flores et al., 2007 insertion of foreign DNAs Smit, |
1996 Cordaux and Batzer, 2009 , insertion of mitochondrial DNA to nuclear genome Timmis et al., 2004 Richly and Leister, 2004 , local insertions and deletions Cooper and Krawczak, |
1991 Mills et al., 2006 Payseur et al., 2010 and most familiar to all, point mutations Carlson, 2011 Given our knowledge of these evolutionary processes, one might imagine a project to computationally simulate the DNA sequence changes (e.g. Li, |
1992 Koroteev and Miller, 2011 ). However, there is a tremendous challenge in choosing the correct model parameter values Li, 2011 in modelling the natural selection Bustamante et al., |
2005 , in putting the model in the context of diploid with recombination Duret and Arndt, 2008 , and in a population Hartl and Clark, |
1997 Point mutations lead to lesser destruction of the genome than larger-scale changes, thus are more likely to survive to the next generation. Many point mutations manifest as neutral mutations Kimura, |
1983 . If the point mutations dominate, the DNA sequences would become more and more random. There has been a debate on whether DNA sequences from the human genome should be considered to be random IHGSC, |
2001 Clay and Bernardi, 2001 Li et al., 2002 If the DNA sequence from a complete chromosome is examined from one end to another, everybody agrees it is not consistent with independent and identically distributed (iid) random variables |
Bernardi, 2001 . It was known that genome contains large domains with alternating high and low GC content called isochores Bernardi et al., |
1985 Spectral analysis also confirmed that genome sequences are not white noise Voss, 1992 Li and Holste, 2004 2005 For coding regions including the regulatory sites, codon structures as well as gene structure often cause the DNA sequences to be non-random Mani, |
1992 Almirantis, 1999 Nikolaou and Almirantis, 2002 Hackenberg et al., 2012 Cocho et al., 2014 However, such debate has not really reached a consensus at the local level in non-coding, non-functional regions. Part of the reason is that randomness definitions are mostly based on probability models of infinitely long seq... |
Defining randomness in a sequence is not easy Knuth, 1997 . On the one hand, for a sequence of infinite length, being random implies “any motif can appear”. This includes the series composed of the same symbol of arbitrary length, which by itself can not be considered as random. On the other hand, for a finite sequence... |
In the field of mathematics, a discussion concerning the Erdös discrepancy problem Erdös, 1957 seems to provide a foundation of randomness in finite sequences. Given an infinite binary sequence [MATH] , taking values [MATH] , we define the discrepancy function for the sequence up to the length [MATH] , of an integer sp... |
[EQUATION] In other words, [MATH] is the cumulative sum of the subsequence sampled from every [MATH] positions, starting from the position [MATH] , of the length-L window. Erdös asked the question: given any constant [MATH] , can one find a sequence of length [MATH] and a spacing value [MATH] so that |
[EQUATION] for one value of spacing [MATH] The problem is solved by Terence Tao, and the answer is yes Tao, 2016 Soundararajan, 2018 , no matter how large the value of C, one can always find the above defined sum larger than C for certain spacing [MATH] value, at a particular length [MATH] |
When [MATH] [MATH] is just the cumulative function of a (random) walk whose steps are given by the [MATH] series. In order for the Erdös question to have the negative answer, one should design steps to be as close to the origin as possible. A periodic walk of period 2 with alternating positive and negative steps is the... |
[MATH] diverges. Therefore, allowing [MATH] to be different from 1, or, allowing one to sample other subsequences in [MATH] [MATH] of spacing d (called “homogeneous arithmetic progression”), is a solution to exclude periodic sequences from “low discrepancy” (low cumulative value for the random walk) category, while “lo... |
We can reverse the Erdös question to design a finite sequence which has the “lowest possible discrepancy” but exclude periodic sequence, for a given [MATH] value. Such designed sequence exhibits the hallmark of a locally random sequence: uniform distribution of single symbol and low-order tuples. It has been shown that... |
1993 All other sequences with longer lengths would lead to discrepancy larger than 1. Similarly, for [MATH] , a length-1160 sequence is able to limit its discrepancy to equal or less than 2 for any [MATH] |
Konev and Lisitsa, 2014 , but once the length is extended to 1161, discrepancy can increase above 2. We call these designed finite series “Erdös sequences with the limit of C and of length L” ( [MATH] ). [MATH] is our candidate of a locally random string at the corresponding string length. |
In this work, we will show that the length-11 sequence constructed by Mathias is not unique, whereas there exists a unique sequence with the required low-discrepancy feature at length 10. We focus on this length-10 sequence ( [MATH] ) and translate it to 10-mers in the DNA sequence of the human genome. The length of 10... |
2004 . We will use the human genome to examine the distribution of DNA substrings of length 10 which are consistent with the [MATH] sequence. The goal of this investigation is to examine whether the human genome tends to possess these “low discrepancy”, “locally random” sequences. |
In the next section, we review the previously studied Mathias [MATH] sequence and propose the unique low discrepancy [MATH] sequence. We translate the Erdös [MATH] motif to DNA sequences, using different base pair definitions. In section 3, the result section, we present the observed and expected [MATH] values obtained... |
Erdös sequences 2.1 The unique length-10 Erdös sequence with binary symbols The [MATH] derived in Mathias, 1993 is + - - + - + + - - + + , which is called the “Mathias sequence”. Fig. illustrates why it satisfies Erdös’ low discrepancy requirement: the sum of [MATH] for d=1,2,3,4,5 is always bounded by 1. If a 12th val... |
Also note that if an Erdös sequence reverses its direction, it may not be an Erdös sequence anymore. There could be two different causes. Take the first 7 positions of [MATH] for example (Fig. ): in the forward direction, the discrepancy series is (position 0 is also included): (0, 1,0, -1, 0, -1, 0, 1). In the backwar... |
We can see that [MATH] has a balanced single symbol frequencies (f(+)=6/11, f(-)=5/11), and almost balanced dimer frequencies f(-- )=2/10, f(-+) =3/10, |
f(+-) =3/10, f(++) =2/10). At the same time, [MATH] is mostly not periodic. Interestingly, when a sequence contains any k-mer exactly once (called De Bruijn sequence De Bruijn, |
1946 1975 , its k-mer frequency is exactly balanced. There are attempts to design shortest sequences to contain all k-mers in both direct and reverse direction of DNA sequences Orenstein and Shamir, |
2013 . Again, to be the shortest means to contain any k-mer only once, therefore resulting to a balanced k-mer frequencies. We do not expect Erdös sequence to have a balanced [MATH] -mers frequencies when [MATH] is large, as the presence of the all-1 or all-0 [MATH] -mers will lead to a large discrepancy for [MATH] . H... |
It is not difficult to check that there is another sequence + - - + - + + - - + - , denoted [MATH] , with the last symbol changed from + to -, which is also an Erdös sequence. It is because 11 is a prime number and changing the last symbol only affects the discrepancy for [MATH] changing the final cumulative value from... |
If the last symbol of the Mathias’ sequence can either be + or -, with the rest of the sequence identical, we would expect that the first ten symbols should form a unique [MATH] . Indeed, running through all possible length-10 binary sequences, we find only one sequence which conforms with the Erdös discrepancy conditi... |
(plus the one by swapping + and -). The single symbol frequency is exactly balanced (f(+)=5/10, f(-)=5/10), whereas the dimer frequencies are somewhat unbalanced ( f(--) =2/9, f(-+) =3/9, f(+-) =3/9, |
f(++) =1/9). Sequences with equiprobable subsequences are also called “normal” Li and Vitányi, 1994 Further simulations show that [MATH] is unique when [MATH] is even, and not unique when |
[MATH] is odd, all with a degeneracy at the last symbol (can be either + or -). [MATH] does not exist when [MATH] In the following, we focus on the unique Erdös sequence with the maximum length, [MATH] and its frequency of appearance in the human genome. |
2.2 DNA motifs of length 10 bases which are associated with [MATH] Erdös sequence DNA sequences use four symbols (A,C,G,T). There are three different ways to split the four symbols into two groups: (a) R/Y binarization combines AG (R for purine) and CT (Y for pyrimidine), (b) W/S binarization combines AT (W for weak) a... |
Schildkraut et al., 1962 which is widely studied in genomic analysis Li, 2013 . The R/Y binarization highlights the size difference of the two types of bases (A and G are larger in size), and has been proposed to be relevant to codon patterns Shepherd, |
1981 regulatory sequence patterns Christophe et al., 1985 , and the double helix structure Arnott et al., 1974 The last binarization (M/K) is rarely used. The fact that the DNA molecule has the form of a double helix implies that we must examine two strands for Erdös motifs, the direct strand, and the reverse complemen... |
(a) For R/Y binarization, four sequences are associated with [MATH] considering both the direct DNA sequence and its reverse complement sequence on the opposite strand: |
[EQUATION] Seq.1 and seq.2 are two different mapping from +/- to R/Y; seq.3 is the reverse complement of seq.1 (or reverse of seq.2); seq.4 is the reverse complement of seq.2 (or reverse of seq.1). Note that seq.3 and seq.4 are not Erdös sequences themselves, but their existence indicates the presence of Erdös sequence... |
(b) Similarly, for W/S binarization, we have four motifs associated with [MATH] [EQUATION] Seq.3 (seq.4) is the reverse complement of seq.1 (seq.2), noting that the complement operation maps one W (S) to another W(S). |
(c) The least used binarization is to M/K, and we again have four motifs: [EQUATION] In terms of the four nucleotide symbols, each motif represents [MATH] |
4-symbol patterns. Multiplying by 12, and subtracting 48 = [MATH] 10-mers which belong to more than one type of motifs (e.g. ACCACAACCA is either a RY or a WS motif), we are dealing with 12240 10-mers which are associated with the Erdös sequence [MATH] |
in either one of the strands. Distribution of Erdös sequences in the human genome 3.1 DNA sequence data In this study, the sequence of human reference genome hg38 is used, which is downloaded from UCSC Genome Browser |
We use chromosomes 1,2, [MATH] 22, and chromosome X but we exclude chromosome Y due to large amount of non-sequenced regions. The 23 sequences contain RepeatMasker filtering information: lowercase letters represent regions that match transposons or other repetitive or low-complexity sequences, whereas uppercase letters... |
The messenger RNA (mRNA) sequence of “known genes” (last updated in July 2016) is obtained from A mRNA sequence matches the genomic sequence of a gene, after removing the introns. Due to alternative splicing, each genomic sequence at a gene locus may contribute multiple mRNA sequences. We obtained 197,783 mRNA sequence... |
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