Noise and Uncertainty in String-Duplication Systems

Duplication mutations play a critical role in the generation of biological sequences. Simultaneously, they have a deleterious effect on data stored using in-vivo DNA data storage. While duplications have been studied both as a sequence-generation mechanism and in the context of error correction, for simplicity these studies have not taken into account the presence of other types of mutations. In this work, we consider the capacity of duplication mutations in the presence of point-mutation noise, and so quantify the generation power of these mutations. We show that if the number of point mutations is vanishingly small compared to the number of duplication mutations of a constant length, the generation capacity of these mutations is zero. However, if the number of point mutations increases to a constant fraction of the number of duplications, then the capacity is nonzero. Lower and upper bounds for this capacity are also presented. Another problem that we study is concerned with the mismatch between code design and channel in data storage in the DNA of living organisms with respect to duplication mutations. In this context, we consider the uncertainty of such a mismatched coding scheme measured as the maximum number of input codewords that can lead to the same output

Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors

DNA as a data storage medium has several advantages, including far greater data density compared to electronic media. We propose that schemes for data storage in the DNA of living organisms may benefit from studying the reconstruction problem, which is applicable whenever multiple reads of noisy data are available. This strategy is uniquely suited to the medium, which inherently replicates stored data in multiple distinct ways, caused by mutations. We consider noise introduced solely by uniform tandem-duplication, and utilize the relation to constant-weight integer codes in the Manhattan metric. By bounding the intersection of the cross-polytope with hyperplanes, we prove the existence of reconstruction codes with greater capacity than known error-correcting codes, which we can determine analytically for any set of parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio

The Capacity of Some P\'olya String Models

We study random string-duplication systems, which we call P\'olya string models. These are motivated by DNA storage in living organisms, and certain random mutation processes that affect their genome. Unlike previous works that study the combinatorial capacity of string-duplication systems, or various string statistics, this work provides exact capacity or bounds on it, for several probabilistic models. In particular, we study the capacity of noisy string-duplication systems, including the tandem-duplication, end-duplication, and interspersed-duplication systems. Interesting connections are drawn between some systems and the signature of random permutations, as well as to the beta distribution common in population genetics

Duplication Distance to the Root for Binary Sequences

We study the tandem duplication distance between binary sequences and their roots. In other words, the quantity of interest is the number of tandem duplication operations of the form x = abc → y = abbc, where x and y are sequences and a, b, and c are their substrings, needed to generate a binary sequence of length n starting from a square-free sequence from the set {0, 1, 01, 10, 010, 101}. This problem is a restricted case of finding the duplication/deduplication distance between two sequences, defined as the minimum number of duplication and deduplication operations required to transform one sequence to the other. We consider both exact and approximate tandem duplications. For exact duplication, denoting the maximum distance to the root of a sequence of length n by f(n), we prove that f(n) = Θ(n). For the case of approximate duplication, where a β-fraction of symbols may be duplicated incorrectly, we show that the maximum distance has a sharp transition from linear in n to logarithmic at β = 1/2. We also study the duplication distance to the root for the set of sequences arising from a given root and for special classes of sequences, namely, the De Bruijn sequences, the Thue-Morse sequence, and the Fibonacci words. The problem is motivated by genomic tandem duplication mutations and the smallest number of tandem duplication events required to generate a given biological sequence

The Tandem Duplication Distance Is NP-Hard

In computational biology, tandem duplication is an important biological phenomenon which can occur either at the genome or at the DNA level. A tandem duplication takes a copy of a genome segment and inserts it right after the segment - this can be represented as the string operation AXB ? AXXB. Tandem exon duplications have been found in many species such as human, fly or worm, and have been largely studied in computational biology. The Tandem Duplication (TD) distance problem we investigate in this paper is defined as follows: given two strings S and T over the same alphabet, compute the smallest sequence of tandem duplications required to convert S to T. The natural question of whether the TD distance can be computed in polynomial time was posed in 2004 by Leupold et al. and had remained open, despite the fact that tandem duplications have received much attention ever since. In this paper, we prove that this problem is NP-hard, settling the 16-year old open problem. We further show that this hardness holds even if all characters of S are distinct. This is known as the exemplar TD distance, which is of special relevance in bioinformatics. One of the tools we develop for the reduction is a new problem called the Cost-Effective Subgraph, for which we obtain W[1]-hardness results that might be of independent interest. We finally show that computing the exemplar TD distance between S and T is fixed-parameter tractable. Our results open the door to many other questions, and we conclude with several open problems

Decoding the Past

The human genome is continuously evolving, hence the sequenced genome is a snapshot in time of this evolving entity. Over time, the genome accumulates mutations that can be associated with different phenotypes - like physical traits, diseases, etc. Underlying mutation accumulation is an evolution channel (the term channel is motivated by the notion of communication channel introduced by Shannon [1] in 1948 and started the area of Information Theory), which is controlled by hereditary, environmental, and stochastic factors. The premise of this thesis is to understand the human genome using information theory framework. In particular, it focuses on: (i) the analysis and characterization of the evolution channel using measures of capacity, expressiveness, evolution distance, and uniqueness of ancestry and uses these insights for (ii) the design of error correcting codes for DNA storage, (iii) inversion symmetry in the genome and (iv) cancer classification. The mutational events characterizing this evolution channel can be divided into two categories, namely point mutations and duplications. While evolution through point mutations is unconstrained, giving rise to combinatorially many possibilities of what could have happened in the past, evolution through duplications adds constraints limiting the number of those possibilities. Further, more than 50% of the genome has been observed to consist of repeated sequences. We focus on the much constrained form of duplications known as tandem duplications in order to understand the limits of evolution by duplication. Our sequence evolution model consists of a starting sequence called seed and a set of tandem duplication rules. We find limits on the diversity of sequences that can be generated by tandem duplications using measures of capacity and expressiveness. Additionally, we calculate bounds on the duplication distance which is used to measure the timing of generation by these duplications. We also ask questions about the uniqueness of seed for a given sequence and completely characterize the duplication length sets where the seed is unique or non-unique. These insights also led us to design error correcting codes for any number of tandem duplication errors that are useful for DNA-storage based applications. For uniform duplication length and duplication length bounded by 2, our designed codes achieve channel capacity. We also define and measure uncertainty in decoding when the duplication channel is misinformed. Moreover, we add substitutions to our tandem duplication model and calculate sequence generation diversity for a given budget of substitutions. We also use our duplication model to explain the inversion symmetry observed in the genome of many species. The inversion symmetry is popularly known as the 2nd Chargaff Rule, according to which in a single strand DNA, the frequency of a k-mer is almost the same as the frequency of its reverse complement. The insights gained by these problems led us to investigate the tandem repeat regions in the genome. Tandem repeat regions in the genome can be traced back in time algorithmically to make inference about the effect of the hereditary, environmental and stochastic factors on the mutation rate of the genome. By inferring the evolutionary history of the tandem repeat regions, we show how this knowledge can be used to make predictions about the risk of incurring a mutation based disease, specifically cancer. More precisely, we introduce the concept of mutation profiles that are computed without any comparative analysis, but instead by analyzing the short tandem repeat regions in a single healthy genome and capturing information about the individual's evolution channel. Using gradient boosting on data from more than 5,000 TCGA (The Cancer Genome Atlas) cancer patients, we demonstrate that these mutation profiles can accurately distinguish between patients with various types of cancer. For example, the pairwise validation accuracy of the classifier between PAAD (pancreas) patients and GBM (brain) patients is 93%. Our results show that healthy unaffected cells still contain a cancer-specific signal, which opens the possibility of cancer prediction from a healthy genome.</p

Attaining the 2nd Chargaff Rule by Tandem Duplications

Erwin Chargaff in 1950 made an experimental observation that the count of A is equal to the count of T and the count of C is equal to the count of G in DNA. This observation played a crucial rule in the discovery of the double stranded helix structure by Watson and Crick. However, this symmetry was also observed in single stranded DNA. This phenomenon was termed as 2nd Chargaff Rule. This symmetry has been verified experimentally in genomes of several different species not only for mononucleotides but also for reverse complement pairs of larger lengths up to a small error. While the symmetry in double stranded DNA is related to base pairing, and replication mechanisms, the symmetry in a single stranded DNA is still a mystery in its function and source. In this work, we define a sequence generation model based on reverse complement tandem duplications. We show that this model generates sequences that satisfy the 2nd Chargaff Rule even when the duplication lengths are very small when compared to the length of sequences. We also provide estimates on the number of generations that are needed by this model to generate sequences that satisfy 2nd Chargaff Rule. We provide theoretical bounds on the disruption in symmetry for different values of duplication lengths under this model. Moreover, we experimentally compare the disruption in the symmetry incurred by our model with what is observed in human genome data

The capacity of some Pólya string models

We study random string-duplication systems, called Pólya string models, motivated by certain random mutation processes in the genome of living organisms. Unlike previous works that study the combinatorial capacity of string-duplication systems, or peripheral properties such as symbol frequency, this work provides exact capacity or bounds on it, for several probabilistic models. In particular, we give the exact capacity of the random tandem-duplication system, and the end-duplication system, and bound the capacity of the complement tandem-duplication system. Interesting connections are drawn between the former and the beta distribution common to population genetics, as well as between the latter system and signatures of random permutations

Codes Correcting All Patterns of Tandem-Duplication Errors of Maximum Length 3

The set of all $q$-ary strings that do not contain repeated substrings of length $\leq\ell$ forms a code correcting all patterns of tandem-duplication errors of length $\leq\ell$, when $\ell \in \{1, 2, 3\}$. For $\ell \in \{1, 2\}$, this code is also known to be optimal in terms of asymptotic rate. The purpose of this paper is to demonstrate asymptotic optimality for the case $\ell = 3$ as well, and to give the corresponding characterization of the zero-error capacity of the $(\leq 3)$-tandem-duplication channel. This settles the zero-error problem for $(\leq\ell)$-tandem-duplication channels in all cases where duplication roots of strings are unique.Comment: 5 pages (double-column format

Duplication-Correcting Codes for Data Storage in the DNA of Living Organisms

The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically-modified organisms. Data stored in this medium is subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem-duplications of a fixed length; the first family can correct any number of errors while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant k, where we are primarily focused on the cases of k = 2, 3