We consider the problem of storing and retrieving information from synthetic DNA media. The mathematical basis of the problem is the construction and design of sequences that may be discriminated based on their collection of substrings observed through a noisy channel. This problem of reconstructing sequences from traces was first investigated in the noiseless setting under the name of "Markov type" analysis. Here, we explain the connection between the reconstruction problem and the problem of DNA synthesis and sequencing, and introduce the notion of a DNA storage channel. We analyze the number of sequence equivalence classes under the channel mapping and propose new asymmetric coding techniques to combat the effects of synthesis and sequencing noise. In our analysis, we make use of restricted de Bruijn graphs and Ehrhart theory for rational polytopes.
A recent study by one of the authors has demonstrated the importance of profile vectors in DNA-based data storage. We provide exact values and lower bounds on the number of profile vectors for finite values of alphabet size $q$, read length $\ell$, and word length $n$.Consequently, we demonstrate that for $q\ge 2$ and $n\le q^{\ell/2-1}$, the number of profile vectors is at least $q^{κn}$ with $κ$ very close to one.In addition to enumeration results, we provide a set of efficient encoding and decoding algorithms for each of two particular families of profile vectors.
Finite-context modeling is a powerful tool for compressing and hence for representing DNA sequences. We describe an algorithm to detect genomic regularities, within a blind discovery strategy. The algorithm uses information profiles built using suitable combinations of finite-context models. We used the genome of the fission yeast Schizosaccharomyces pombe strain 972 h- for illustration, unveilling locations of low information content, which are usually associated with DNA regions of potential biological interest.
We consider the problem of assembling a sequence based on a collection of its substrings observed through a noisy channel. The mathematical basis of the problem is the construction and design of sequences that may be discriminated based on a collection of their substrings observed through a noisy channel. We explain the connection between the sequence reconstruction problem and the problem of DNA synthesis and sequencing, and introduce the notion of a DNA storage channel. We analyze the number of sequence equivalence classes under the channel mapping and propose new asymmetric coding technique…
The process of DNA-based data storage (DNA storage for short) can be mathematically modelled as a communication channel, termed DNA storage channel, whose inputs and outputs are sets of unordered sequences. To design error correcting codes for DNA storage channel, a new metric, termed the sequence-subset distance, is introduced, which generalizes the Hamming distance to a distance function defined between any two sets of unordered vectors and helps to establish a uniform framework to design error correcting codes for DNA storage channel. We further introduce a family of error correcting codes,…
While achieving a compression ratio of 2.0 bits/base, the new algorithm codes non-N bases in fixed length. It dramatically reduces the time of coding and decoding than previous DNA compression algorithms and some universal compression programs.
This article introduces a new DNA sequence compression algorithm which is based on LUT and LZ77 algorithm. Combined a LUT-based pre-coding routine and LZ77 compression routine,this algorithm can approach a compression ratio of 1.9bits \slash base and even lower.The biggest advantage of this algorithm is fast execution, small memory occupation and easy implementation.
DNA sequences are prone to creating secondary structures by folding back on themselves by non-specific hybridization among its nucleotides. The formation of secondary structures makes the sequences chemically inactive towards synthesis and sequencing processes. In this letter, our goal is to tackle the problems due to the creation of secondary structures in DNA sequences along with constraints such as not having a large homopolymer run length. In this paper, we have presented families of DNA codes with secondary structures of stem length at most two and homopolymer run length at most four. By …
DNA sequencing has faced a huge demand since it was first introduced as a service to the public. This service is often offloaded to the sequencing companies who will have access to full knowledge of individuals' sequences, a major violation of privacy. To address this challenge, we propose a solution, which is based on separating the process of reading the fragments of sequences, which is done at a sequencing machine, and assembling the reads, which is done at a trusted local data collector. To confuse the sequencer, in a pooled sequencing scenario, in which multiple sequences are going to be …
DNA sequencing is the basic workhorse of modern day biology and medicine. Shotgun sequencing is the dominant technique used: many randomly located short fragments called reads are extracted from the DNA sequence, and these reads are assembled to reconstruct the original sequence. A basic question is: given a sequencing technology and the statistics of the DNA sequence, what is the minimum number of reads required for reliable reconstruction? This number provides a fundamental limit to the performance of {\em any} assembly algorithm. For a simple statistical model of the DNA sequence and the re…
We introduce the notion of weakly mutually uncorrelated (WMU) sequences, motivated by applications in DNA-based data storage systems and for synchronization of communication devices. WMU sequences are characterized by the property that no sufficiently long suffix of one sequence is the prefix of the same or another sequence. WMU sequences used for primer design in DNA-based data storage systems are also required to be at large mutual Hamming distance from each other, have balanced compositions of symbols, and avoid primer-dimer byproducts. We derive bounds on the size of WMU and various constr…
Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer low-density parity-check code and either an inner convolutional code or a block code. We propose two new decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posteriori probabilities (APPs). The first decoder…
DNA code design aims to generate a set of DNA sequences (codewords) with minimum likelihood of undesired hybridizations among sequences and their reverse-complement (RC) pairs (cross-hybridization). Inspired by the distinct hybridization affinities (or stabilities) of perfect double helix constructed by individual single-stranded DNA (ssDNA) and its RC pair, we propose a novel similarity significance (SS) model to measure the similarity between DNA sequences. Particularly, instead of directly measuring the similarity of two sequences by any metric/approach, the proposed SS works in a way to ev…
Restriction enzymes recognize and bind to specific sequences on invading bacteriophage DNA. Like a key in a lock, these proteins require many contacts to specify the correct DNA sequence. Using information theory we develop an equation that defines the number of independent contacts, which is the dimensionality of the binding. We show that EcoRI, which binds to the sequence GAATTC, functions in 24 dimensions. Information theory represents messages as spheres in high dimensional spaces. Better sphere packing leads to better communications systems. The densest known packing of hyperspheres occur…
In this paper, we consider the problem of designing DNA sequences (codewords) for DNA storage systems and DNA computing that are unlikely to fold back onto themselves to form undesirable secondary structures. The paper addresses both the issue of enumerating the sequences with such properties and the problem of practical code construction.
Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and genetic code. Central role in our investigation plays an ultrametric p-adic information space which basic elements are nucleotides, codons and genes. We show that a 5-adic model is appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance, is also suitable for genetic code and for a more advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons.
In this work we implement the so-called matching time estimators for estimating the entropy rate as well as the entropy production rate for symbolic sequences. These estimators are based on recurrence properties of the system, which have been shown to be appropriate to test irreversibility specially when the sequences have large correlations or memory. Based on limit theorems for matching-times we derive a maximum likelihood estimator for entropy rate assuming that we have a set of moderately short symbolic time-series of finite random duration. We show that the proposed estimator has several …
For $q$-ary $n$-sequences, we develop the concept of similarity functions that can be used (for $q=4$) to model a thermodynamic similarity on DNA sequences. A similarity function is identified by the length of a longest common subsequence between two $q$-ary $n$-sequences. Codes based on similarity functions are called DNA codes. DNA codes are important components in biomolecular computing and other biotechnical applications that employ DNA hybridization assays. The main aim of the given lecture notes -- to discuss lower bounds on the rate of optimal DNA codes for a biologically motivated simi…
A number of methods have been proposed over the last decade for encoding information using deoxyribonucleic acid (DNA), giving rise to the emerging area of DNA data embedding. Since a DNA sequence is conceptually equivalent to a sequence of quaternary symbols (bases), DNA data embedding (diversely called DNA watermarking or DNA steganography) can be seen as a digital communications problem where channel errors are tantamount to mutations of DNA bases. Depending on the use of coding or noncoding DNA hosts, which, respectively, denote DNA segments that can or cannot be translated into proteins, …
