We can finally perform a symbol-wise decoding as follows: The forward and backward metric does not only provide estimates for the start and the end position of an entire barcode word, but also enables to calculate conditional likelihoods. As the inner code words are determined by outer code symbols, i. The symbol-wise marginals are finally utilized in the outer coder see Figure 2. In order to perform an in silico application of barcodes based on the watermark concept, we first have to define some reasonable setting for encoding, which is already quite challenging. As stated before see section Barcode construction on watermark , there are different parameters, which influences the concepts and for which we have to find a reasonable setup to run simulations.
First there is an outer codes, which should in combination with the inner coding lead to short barcode words, because we do not want to produce exceptional overhead with multiplexing tagging target fragments. There is the minimum distance of outer code words and the sparsity of the inner encoder, which can independently be characterized. And finally we have the watermark sequence, which can randomly be chosen. We utilize the degree of freedom with the watermark to incorporate with additional sequence constraint, that barcodes should fulfill to be experimentally valid.
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Therefore we run a greedy search for the watermark pattern that maximizes the number of barcodes that meet all sequence constraint. But let us consider all particular setting one by one in the following paragraphs. Long LDPC codes has been used in the original approach of Davey and MacKay see [ 20 ] , but as the construction of short LDPC codes would be somehow confusing for readers involved in channel coding, we decided to use the best known linear codes. But we might note, that the hamming distance and the ability for soft decoding is the only demand on outer codes here and LDPC codes are likely to perform equivalent.
The minimal Hamming distance d H of the best known linear codes we used is either maximal or the highest known regarding given parameters. This guarantees a certain minimal code rate on one side and limit the computational effort of the outer decoding the other side. We end up at possible code configurations, but most of the resulting codes perform disastrous with the watermark concept, because the watermark is heavily corrupted by inner code words. For large densities there is no ability left to detect the barcode boundaries and consequently decoding will fail completely.
Finally, we end up in a set of 73 parameter configurations for q 1 , k 1 , n 1 and n 2. For each of the 73 different parameter sets we run a brute-force search 10 7 trials , where we iteratively selected one inner code and watermark randomly to produce barcode sets. From the evaluated settings we kept the one, which met the following sequence constraints.
We filtered for code words with unbalanced counts of symbols, to respect limitations on the GC content of barcodes. Such filtering can be seen as a de facto standard in the construction of barcodes see for example [ 1 , 7 , 19 ] and is related to technical constraints due to the preparation and the sequencing of genomic material. We furthermore exclude barcodes with prefect self-complementation and more than two sequential repetitions of the same base homopolymer length , similar to the restrictions stated in [ 24 ]. We consider this settings as sufficient and strict enough to avoid experimental problems during the preparation in real sequencing tasks.
Discarding such inappropriate code words means an additional loss in code rate. For decoding based on the HMM approach, edit distance implicitly matters, thus we try to increase the mean edit distance of code word with a simple strategy. For two sets of barcodes with identical counts of remaining barcodes after filtering we keep the one maximizing the mean edit distance.
But, as the edit distance is bounded by the hamming distance, we do not gain a lot with this final heuristic refinement step. To evaluate the refined set of 73 codes, we give the following demultiplexing scenario. For each code we consider an error curve according the estimates of the decoding error probability on different channel settings.
The estimation of a single point in the error curve is based on a set of , barcodes, which we refer to as batch. This value determines a symbol-wise probability for an edit operation that can be caused by our channel model. Similar approach has been considered in [ 19 ], but we like to be a bit more precise with the description of the modifications of the channel model.
Buschmann et al. We will take such indications into account. Each barcode from a batch is embedded in a random context of variable length compare section Embedding of barcodes. We further use a state machine to produce erroneous received sequences based on the following four events: correct transmission C , substitution S , insertion I and deletion D. In slightly different notation to the former model see. Sequencing Channel we assign probabilities to the events C and S and do not use conditional probability like a substitution matrix S.
The probability for correct transmission C or substitution S is equal to p t in the former representation. Every error event S , I or D is equal likely. To obtain an equal distribution among the mentioned events, it is easier to use the present notation, but equivalent behavior can be approached with both versions of the channel state-machines.
To save decoding time we iteratively pass each transmit sequence through the state machine, until we have at least one error event within the barcode region. We further considered an error-free barcode to be decoded perfectly, i. Please note that this oversimplifies the false positive rate introduced by sporadic similarities of the context with dedicated barcode words. The rate is supposed to grow linear with the context length and the size of barcodes set. Nevertheless, the probability for false positive events exponentially tend to zeros with the length N of the code words.
First we give a rough overview on meaningful properties of watermark-based barcodes with the considered 73 parameterizations. Furthermore, we provide a refined analysis and evaluation for certain codes in the already defined decoding scenario. The particular codes can be found in the Additional file 1 section of this paper. We use star-symbols to indicated the mean density several levels and two dimensional coordinates to link mean edit distance and the size of code sets.
Properties of the examined barcodes based on watermarks. Figure 5 and 6 are colored. There are distinct blocks, where the influence of the inner code can be separately examined. With fixing the outer code, e. We have observed several clusters, where similar effects can be found. We find clusters of star symbols at different mean distances see darkened areas in Figure 4.
These levels can be explained through the different minimum Hamming distance of the outer codes. We have Hamming distances 2,3 and 4 for the present outer codes at n 1 equals to 4,5 and 6.
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For concatenated coding it is known that the minimum distances of inner and outer codes are multiplicative [ 31 ]. As the edit metric is upper bounded by the Hamming distance, we anticipate the described levels for edit distances. The mentioned leveling can consistently be found for all outer codes. Despite we have maximized the edit distance of barcodes on average, it is also interesting to focus on the pairwise distance of barcodes. To examine the distance in detail we utilize the so-called distance distribution. In [ 32 ] the average number of code words at a certain distance to a fixed code word is considered as an useful distance measure for non-linear codes based on Hamming metric.
The edit distance distribution of a codes consists of the numbers. In Figure 5 we illustrate the distance distribution of barcodes with parameters 8 3 n 1 n 2 Figure 4 , in blue. Apparently, there are particular pairs of code words with very low edit distances, but as we recall the code construction based on inner code words with very low Hamming weight, this fact is not too surprising. Nevertheless, some longer codes show a negligible amount of such code words with small edit distances.
Exemplary distance distribution of a sets of barcode based on watermarks. D e denotes the relative number of code-pairs with an edit distance equal to e. The distance distribution is normalized regarding the cardinality of code words after filtering. A detailed description about the excluded barcodes can be found in the Additional file 1. In Figure 6 we illustrate the estimates P e for the decoding error probability of different codes settings. We ran simulations for all 73 barcode set and ranked the sets according the decoding behavior.
To give a rough outline for the performance of our approach we show the barcodes, which performed best Figure 4 and 6 in green and worst Figure 4 and 6 in red. A barcode length of 12 codes q 1 k 1 3 4 seems to be insufficient to provide a good synchronization based on watermarks and thus the majority of decoding errors were found to be caused by synchronization issues results not shown. The best performing sets of barcodes surprisingly have not occupied the maximal possible length, but 24 symbols.
Simulation results for a realistic decoding scenario.
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All evaluated codes are highlighted in Figure 4 with identical colors. On average, each randomly drawn barcode is embedded in random symbols before decoding. A reasonable trade-off between error-correcting capability and cardinality is provided for example by codes with parameters 8 3 n 1 n 2 Figures 4 , 5 and 6 , in blue. Although we are facing relative low code rates compared to approaches like [ 19 ] ranging from 0. According to [ 20 ] we utilized a fast decoding approach with reduced complexity for our simulations. We further parallelized the decoding procedure and created jobs of 10 6 received sequences, that were processed by single cores Opteron, 2.
The average length of received sequences was in a range of and symbols, resulting in an average processing time of 6 hours for the tasks with lowest calculation costs code parameters 7 2 3 4. The longest time we needed to complete demultiplexing of 10 6 sequences code parameters 9 3 5 5 was strictly below 24 hours on a single core. Apart from the theoretical considerations we have given in this manuscript, there are lot of future direction starting from this initial point. Some of them are mandatory to enable an application in real biological experiments, others are modifications of the concepts for extended applications.
Let us first address the essential steps needed to establish an HMM based decoding in real experiments: The HMM, as core of the decoding system, is the most sensible part of the concept. It is mandatory to run experiments to gather reliable data about all channels, the concepts should be used for. From our point of view there is a lack of reliable data about insertions, deletions and substitution errors for possible channel models. For the sequencing application we assume that different platforms show a variety of sequencing channels , additionally affected by experimental parameters, e.
To obtain an optimal suited decoder, the HMM should be adapted to the considered channels. As most of the channels show a correlation of errors, more complex HMMs should be considered, reflecting a channel model with memory. Finally, it might be possible to establish a self-adaptive algorithm to parametrize the HMMs without any prior knowledge about the ratio of errors in the channel.
A suitable statistic and refined calibration steps should be invented. Another important point for estimating the error characteristics is the construction of watermark codes. Exact empirical parameters of the channel could be incorporated in the design of watermark codes to improve decoding steps, suited for special channels.
Further aspects that could be considered with the given concepts are the following: Aside from the synchronization aspect in this manuscript, it seems very promising to use the maximum likelihood decoding method for other sequences than watermark codes. Conditioned on good empirical parameters for an underlying HMM one could consider a reliable detection of barcodes based on the Sequence-Levenshtein distance in a probabilistic way rather than based on sequence alignment. In the presented approach we focused on the discrimination of code words, assuming codes are always present in inspected sequences.
The detection of code words within DNA context is another big issue that should be solved for future investigations using an HMM based decoding. Resent research shows that even for sequencing approaches the detection of barcodes is quite challenging. Caused by technical reasons, sporadically barcodes are not present in the sequence data. Another interesting field of application of an HMM based sequence detection could be clonal studies, where the sequenced genome could or could not contain a predefined sequence, which was introduced in ancestor organisms.
We proposed an adaption of the watermark concept of [ 20 ] for DNA barcoding. A generalized channel model for sequencing and suitable modifications of the decoder were defined. Moreover, we investigated in a strategy to choose watermark sequences and inner codes in a reasonable way to enable barcoding in line with common experimental requirements.
We provide a code construction, considering the best known linear codes as outer codes and biological sequence constraints to filter for suitable code words, resulting in an exemplary set of 73 different code sets ranging from 12 to 24 nucleotides. The codes are illustrated in a comprehensive scheme, highlighting watermark specific parameters as well as the mean edit distance, to give an impression how watermark based barcodes could be characterized. For a reduced set of codes we finally evaluated the demultiplexing of sequences in a realistic simulation scenario.
Within this in silico evaluation we could show that barcodes based on watermarks can theoretically be used for multiplexing. It is remarkable, that even with very short watermark patterns we are able to reliably find the barcodes boundaries in order to discriminate different code words with an HMM based decoder. The probability of decoding errors, which finally leads to the undesired cross-talk phenomenon was found to be very low. Other approaches that investigate barcodes with large sequence edit distance [ 16 , 19 ] show significant higher code rates for shorter barcodes, but we have given an entirely different concept that allows for large scale multiplexing approaches, also able to handle insertion and deletion errors.
Moreover, we can provide the marker-less synchronization based on watermarks, to recover the barcode boundaries. This synchronization concept provides an ultimate degree of freedom for experimental sequencing setups as well as for future applications, also apart from the sequencing context. Error-correcting barcoded primers for pyrosequencing hundreds of samples in multiplex. Nat Methods. Bystrykh LV. Generalized dna barcode design based on hamming codes. PLoS One. Hamming RW.
Error detecting and error correcting codes. Bell Syst Tech J. Once it has established the language of geometric algebra linear grading of the algebra; geometric, exterior and interior products; involutions , it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness geometric covariance with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index. Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students.
European Congress of Mathematics : Barcelona, July , by Carles Casacuberta 6 editions published in in English and held by WorldCat member libraries worldwide This is the second volume of the proceedings of the third European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners as well as papers by plenary and parallel speakers. The second volume collects articles by prize winners and speakers of the mini-symposia. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician.
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Audience Level. Related Identities. Associated Subjects. As seen from Fig. Thus, the method for designing a GC template of the present invention is used at the first step of constructing the set S of oligonucleotide sequences of the present invention. As seen from the above explanation, the method for designing a GC template of the present invention is not particularly limited as long as it is a method comprising selection of GC templates such that its Hamming distance to its reverse sequence, to its block shift, and its Hamming distance to the overlap part of its tandem concatenation, its concatenation with its reverse sequence, and the tandem concatenation of its reverse sequence, is equal to or above the predetermined value k.
In the following, an oligonucleotide sequence of length n is specified by the binary string of 0 and 1 GC template of predetermined length L L is an integer 6 or more , meaning that the positions of G or C [GC] , or A or T [AT] are fixed. However, the length L of GC template is 6 or more, preferably 6 to , more preferably 6 to 32, most preferably around 20, which is often used in experiments of molecular biology.
If the length is 5 or less, the one having desired Hamming distance cannot be obtained. By using the GC template having the length L, a set S of oligonucleotide sequences of corresponding length n can be obtained. Further, the predetermined value k is not particularly limited as long as it is a value that allows oligonucleotide sequences constructed from the GC template to be the oligonucleotide sequences of the present invention that can avoid mishybridization. The value is preferably one-fifth or more, more preferably one-fourth or more, most preferably one-third or more of the length L of the GC template.
In general, when the length L is increased or MD value k value is decreased, many more GC templates will exist, however, a GC template of predetermined length and having the greatest k value MD value is particularly important. In addition, the shortest GC templates that fulfill specific MD value k value are shown in [Table 2].
In [Table 2], GC templates are enumerated excluding the ones that have the same reverse sequences or sequences wherein 0 and 1 are reversed, and in [Table 3] and [Table 4], "items" are the numbers after omitting GC templates that become identical by cyclic shift. The GC template sequences enumerated in [Table 1] to [Table 4], etc. However, there is no need to search all 2 L patterns to find a GC template of length L. The GC templates can be efficiently obtained by using these constraints additionally.
Further, when GC templates are designed such that the set S of oligonucleotide sequences constructed from GC templates is made to be a set of oligonucleotide sequences that contains or never contains particular subsequences such as restriction sites mentioned above, such designing corresponds to the narrowing of the space for exhaustive search, and therefore it contributes easier designing. The set S of oligonucleotide sequences of the present invention can be designed by the step following the design of GC templates with the use of the Hamming distance mentioned above, which is the step using the theory of error correcting codes, that is, by combining codewords of any error correcting code with the designed GC templates to specify a set of oligonucleotide sequences, and by specifically substituting the positions 1 and 0 of GC template with bases of [AT] and [GC], or the positions 1 and 0 of GC template with bases of [GC] and [AT], respectively.
As the codewords of error correcting codes mentioned above, any codewords can be used as long as they are known codewords of error correcting codes, and specific examples include Hamming codes, BCH codes, maximum-length codes, Golay codes, Reed-Muller codes, Reed-Solomon codes, Hadamard codes, Preparata codes, and reversible codes. The motive for using the theory of error correcting codes is to ensure mismatches to complementary sequences in case where there occurs no shift see claim 1. Therefore, as to the set S inducing mismatches in consideration of reverse sequence as well see claim 2 , it is not always necessary to use error correcting codes.
Error correcting codes are a set of codewords wherein there are at least a certain number of mismatches between optional codewords. In case of preventing mishybridization between a set S and a set of reverse sequences thereof, it is only necessary to apply a set of codewords wherein there are at least a certain number of matches not mismatches between optional codewords.
As for the set S of oligonucleotide sequences of the present invention, information of the codewords and GC templates are reflected on the sequences. Therefore, it is suffice to use error correcting codes maintaining the Hamming distance the number of mismatches k or more in order to ensure k mismatches to complementary sequences, and it is suffice to use codes maintaining the number of matches k or more in order to ensure k mismatches to reverse sequences. In the theory of error correcting codes, codes wherein a redundant bit for detecting and correcting errors, which is called parity bit, is added to a given information bit to make the Hamming distance between optional codewords above a certain value, have been developed.
The minimum value of the Hamming distance between codewords is called minimum distance. As the object of the code theory is to design the one that maintains the minimum distance largely and contains many codewords, there are many codes that meet the purpose of the present invention. For example, there are words of Golay code of code length 23 and minimum distance 7. With the use of this code, it is possible to design oligonucloetides for one GC template of length 23 MD value is up to 9. Next, it is explaned with specific example of the combination of error correcting codes and GC templates.
It is ensured that the sequences thus constructed have at least two mismatches in case shift does not occur, three mismatches to any ligation or shift, on each side. The method for designing the set S of oligonucleotide sequences of the present invention using GC templates is specifically shown above.
However, the design method wherein the set of oligonucleotide sequences that maintains the Hamming distance k induces equal to or more than a fixed, predetermined number of mismatches against any of P sequences in the set S, a complementary sequence or reverse sequences of each of P sequences in the set S, sequences constructed by shifting these sequences, and sequences produced by ligation of these P sequences, of P C sequences or P R sequences, and of the P sequences and P C sequences or P R sequences in the set S, and wherein the set S of P sequences can avoid mishybridization between them, P C sequences or P R sequences, sequences constructed by shifting these sequences, and sequences produced by ligation of the P sequences, of their complementary sequences, and of the P sequences and P C sequences or P R sequences in the set S, is preferable.
Further, in the method for designing a GC template of the present invention, length n of oligonucleotide sequences in the predetermined set S, length L of GC templates, and the predetermined value k are as explained above, and the set S of oligonucleotide sequences is a set of oligonucleotide sequences that contains or never contains particular subsequences as explained above, and Hamming codes, BCH codes, maximum-length codes, Golay codes, Reed-Muller codes etc.
In order to do this, the definition of function MD in the GC template is redefined as follows. The largest difference from GC template resides in the point that when a binary string that maximize this MD value is selected from among binary strings of given length L and this binary string is set to be t, the binary string of t designates [AG] or [TC], therefore, GC content of designed DNA sequences cannot be standardized in case where the binary strings of t are combined with error correcting codewords.
In GC templates, position of GC is designated by 01 of the templates and position of AG is designated by 01 of the error correcting codewords. In AG templates, the designation of the positions is reversed. Therefore, it is impossible to standardize GC content with the use of optional error correcting codewords, it is necessary to use error correcting codes called constant-weight codes that have constant number of 1 in codewords.
It is more difficult to design the constant-weight codes than generally used codes such as BCH codes or Hadamard codes that can use templates designating [GC] or [AT], but the constant-weight codes can be systematically designed with the use of the result described in reference BSS90 IEEE Trans. On Information Theory, 36, pp. However, while constraints are imposed on available error correcting codes, it is possible to make the MD value of the templates, that is, the Hamming distance in consideration of shift and ligation, larger than that of the templates designating [GC] or [AT].
Further, it is found that the number of templates that have same MD value will be more than that of the templates designating [GC] or [AT]. The length L of AG template is 3 or more, preferably 3 to , more preferably 3 to 32, most preferably around 20, which is often used in experiments of molecular biology. If the length is 2 or less, the one having desired Hamming distance cannot be obtained.
Further, the predetermined value k is not particularly limited as long as it is a value that allows oligonucleotide sequences constructed from the AG template to be the oligonucleotide sequences of the present invention that can avoid mishybridization. The value is preferably one-fifth or more, more preferably one-fourth or more, and most preferably one-third or more of the length L of the GC template.
As in the case of GC templates, when the length L is increased or MD value k value is decreased, many more AG templates will exist, however, an AG template of predetermined length and having the greatest k value MD value is particularly important.
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The number of AG templates in [Table 7] contains all templates without omitting templates that become identical by cyclic shift or reversal. The case using AG templates and the case using GC templates have a lot in common, for example, in both cases, it is preferable that the set S of oligonucleotide sequences is a set of oligonucleotide sequences that contains or never contains particular subsequences such as restriction sites. Though templates designating [AG] or [TC have an advantage that they can maintain larger Hamming distance than templates designating [GC] or [AT], the number of the codewords of constant-weight codes is not so many in general.
Therefore, from the viewpoint of the number of words that can be designed, GC templates are more flexible and have wide application. Further, GC templates have a great advantage that the melting temperature calculated by the nearest neighbor method used in biological experiments can be standardized because not only GC content but also alignment of GC bases can be standardized in all sequences. Therefore, AG templates can be handled also as one of possible variations. The set S of oligonucleotide sequences of the present invention can be advantageously used as DNA or RNA tips, or DNA or RNA tags because orthogonalization between sequences makes it difficult to mishybridize with each other even if more than one kinds of oligonucleotide chains are fixed on a substrate in the high density.
In addition, the set S of oligonucleotide sequences of the present invention is useful as primers for PCR, etc. Further, the set S of oligonucleotide sequences of the present invention can be advantageously used for DNA computing system that comprises the steps of; artificially synthesizing DNA sequences in which various symbol manipulation operating systems such as logical expression and graph structure are recorded, and cutting and pasting the sequences according to protocols of molecular biological experiments, and sequences obtained at the end of the experiments are "calculation results" of the DNA computing, because it has a specific sequence portion such as restriction sites in addition that it is difficult to mishybridize with each other.
The method for designing the set S of oligonucleotide sequences of the present invention makes it possible to efficiently and systematically design DNA sequences wherein it is difficult to mishybridize with each other due to the orthogonality of the sequences.