Pedigree Graph Research Articles

Algorithms for comparing large pedigree graphs

The importance of pedigrees is translated by geneticists as a tool for diagnosing genetic diseases. Errors resulting during collection of data and missing information of individuals are considered obstacles in deducing pedigrees, especially larger ones. Therefore, the reconstructed pedigree graph evaluation needs to be undertaken for relevant diagnosis. This requires a comparison between the derived and the original data. The present study discusses the isomorphism of huge pedigrees with labeled and unlabeled leaves, where a pedigree has hundreds of families, which are monogamous and generational. The algorithms presented in this paper are based on a set of bipartite graphs covering the pedigree and the problem addressed is parameter tractable. The Bipartite graphs Covering the Pedigree (BCP) problem is said to possess a time complexity of $f(k).mod(X)^{O(1)}$ where $f$ is the computing function that grows exponentially. The study presents an algorithm for the BCP problem that can be categorized as a polynomial-time-tractable evaluation of the reconstructed pedigree. The paper considers pedigree graphs that consist of both labeled and unlabeled leaves that make use of parameterized and kernelization algorithms to solve the problem. The kernelization algorithm executes in $O(k^3)$ for the BCP graphs.

Algorithms for comparing large pedigree graphs

The importance of pedigrees is translated by geneticists as a tool for diagnosing genetic diseases. Errors resulting during collection of data and missing information of individuals are considered obstacles in deducing pedigrees, especially larger ones. Therefore, the reconstructed pedigree graph evaluation needs to be undertaken for relevant diagnosis. This requires a comparison between the derived and the original data. The present study discusses the isomorphism of huge pedigrees with labeled and unlabeled leaves, where a pedigree has hundreds of families, which are monogamous and generational. The algorithms presented in this paper are based on a set of bipartite graphs covering the pedigree and the problem addressed is parameter tractable. The Bipartite graphs Covering the Pedigree (BCP) problem is said to possess a time complexity of $f(k).mod(X)^{O(1)}$ where $f$ is the computing function that grows exponentially. The study presents an algorithm for the BCP problem that can be categorized as a polynomial-time-tractable evaluation of the reconstructed pedigree. The paper considers pedigree graphs that consist of both labeled and unlabeled leaves that make use of parameterized and kernelization algorithms to solve the problem. The kernelization algorithm executes in $O(k^3)$ for the BCP graphs.

Implementation of INHERET, an Online Family History and Cancer Risk Interpretation Program for Primary Care and Specialty Clinics.

Individuals at increased risk for cancer are ascertained at low rates of 1% to 12% in primary care (PC). Underserved populations experience disparities of ascertainment, but data are lacking. INHERET is an online personal and family history tool to facilitate the identification of individuals who are eligible, according to guidelines, to be counseled on germline genetic testing and risk management. INHERET data entry uses cancer genetics clinic questionnaires and algorithms that process patient data through NCCN Clinical Practice Guidelines in Oncology and best practice guidelines. The tool was tested in silico on simulated and retrospective patients and prospectively in a pilot implementation trial. Patients in cancer genetics and in PC clinics were invited to participate via email or a card. Informed consent was completed online. INHERET aimed to integrate patient data by algorithms based on professional and best practice guidelines to elicit succinct, actionable recommendations that providers can use without affecting clinic workflow or encounter length. INHERET requires a 4th-grade reading level, has simple navigation, and produces data lists and pedigree graphs. Prospective implementation testing revealed understandability of 90% to 100%, ease of use of 85%, and completion rates of 85% to 100%. Physicians using INHERET reported no added time to their encounters when patients were identified for counseling. In a specialty genetics clinic, INHERET's data were input, on average, within 72 hours compared with 4 to 6 weeks through standard care, and the queue for scheduling patients decreased from 400 to fewer than 15 in <6 months. INHERET was found to be accessible for all education and age levels, except patients aged >70 years, who encountered more technical difficulties. INHERET aided providers in conveying high-risk status to patients and eliciting appropriate referrals, and, in a specialty clinic, it produced improved workflows and shortened queues.

DMDtoolkit: a tool for visualizing the mutated dystrophin protein and predicting the clinical severity in DMD

BackgroundDystrophinopathy is one of the most common human monogenic diseases which results in Duchenne muscular dystrophy (DMD) and Becker muscular dystrophy (BMD). Mutations in the dystrophin gene are responsible for both DMD and BMD. However, the clinical phenotypes and treatments are quite different in these two muscular dystrophies. Since early diagnosis and treatment results in better clinical outcome in DMD it is essential to establish accurate early diagnosis of DMD to allow efficient management. Previously, the reading-frame rule was used to predict DMD versus BMD. However, there are limitations using this traditional tool. Here, we report a novel molecular method to improve the accuracy of predicting clinical phenotypes in dystrophinopathy. We utilized several additional molecular genetic rules or patterns such as “ambush hypothesis”, “hidden stop codons” and “exonic splicing enhancer (ESE)” to predict the expressed clinical phenotypes as DMD versus BMD.ResultsA computer software “DMDtoolkit” was developed to visualize the structure and to predict the functional changes of mutated dystrophin protein. It also assists statistical prediction for clinical phenotypes. Using the DMDtoolkit we showed that the accuracy of predicting DMD versus BMD raised about 3% in all types of dystrophin mutations when compared with previous methods. We performed statistical analyses using correlation coefficients, regression coefficients, pedigree graphs, histograms, scatter plots with trend lines, and stem and leaf plots.ConclusionsWe present a novel DMDtoolkit, to improve the accuracy of clinical diagnosis for DMD/BMD. This computer program allows automatic and comprehensive identification of clinical risk and allowing them the benefit of early medication treatments. DMDtoolkit is implemented in Perl and R under the GNU license. This resource is freely available at http://github.com/zhoujp111/DMDtoolkit, and http://www.dmd-registry.com.

Comparing Unlabeled Pedigree Graphs via Covering with Bipartite and Path.

Family trees, also called pedigrees, have important information about an individual's past and future life. It can be used as a diagnostic tool and help guide decisions about genetic testing for the patient and at-risk family members. There are 2% to 10% of parent-child relationships missing, and this can cause large differences in the pedigree graphs created. Hence, the evaluation of pedigrees is an essential task. In this article, we focus on the problem of isomorphism of unlabeled subpedigrees with a large number of individuals and hundreds of families, given that the two pedigrees being evaluated are generational and mating is between external parents. We address two restricted versions of the unlabeled subpedigree graph problem, Cover Unlabeled subPedigree with a Bipartite graph (CUPB), and Cover Unlabeled subPedigree with a Path (CUPP) problems. Fixed parameter algorithms are presented to solve the two problems, CUPB and CUPP.

Genetic Diversity and Classification of Cytoplasm of Chinese Elite Foxtail Millet [Setaria italica (L.) P. Beauv.] Germplasm

ABSTRACTA single cytoplasmic source of foxtail millet male sterile lines has become a major limiting factor for wide utilization of heterosis in foxtail millet. To explore genetic diversity of Chinese foxtail millet cytoplasm, 23 pairs of mitochondrial DNA (mtDNA) primers that showed a high level of polymorphism among selected 14 genetically diverse foxtail millet accessions were identified after screening 34 pairs of consensus mtDNA primers that were derived from various plant species. The 23 pairs of primers were used to analyze genetic diversity of cytoplasm in 111 Chinese elite foxtail millet accessions. Genetic similarity coefficients for differentiation of cytoplasmic types were calculated based on the phylogenetic tree and female‐parent‐derived pedigree graph of all the tested accessions. The results show that the genetic diversity of foxtail millet cytoplasm is low, which makes Chinese foxtail millet germplasm vulnerable to the infection by cytoplasm‐related diseases. The 111 tested accessions can be classified into eight cytoplasmic types: Qinyuanmujizui, Huangruangu, Riben60ri, Dahuanggu, Yingsuigu, Heizhigu, MissingI, and MissingII. The eight cytoplasmic types matched with the geographic and ecological distribution pattern of the most tested accessions. The results provided a scientific foundation for rationally utilizing these germplasm lines, diversifying cytoplasmic types in male sterile lines, and further improving hybrid foxtail millet cultivars.

Path-counting formulas for generalized kinship coefficients and condensed identity coefficients.

An important computation on pedigree data is the calculation of condensed identity coefficients, which provide a complete description of the degree of relatedness of two individuals. The applications of condensed identity coefficients range from genetic counseling to disease tracking. Condensed identity coefficients can be computed using linear combinations of generalized kinship coefficients for two, three, four individuals, and two pairs of individuals and there are recursive formulas for computing those generalized kinship coefficients (Karigl, 1981). Path-counting formulas have been proposed for the (generalized) kinship coefficients for two (three) individuals but there have been no path-counting formulas for the other generalized kinship coefficients. It has also been shown that the computation of the (generalized) kinship coefficients for two (three) individuals using path-counting formulas is efficient for large pedigrees, together with path encoding schemes tailored for pedigree graphs. In this paper, we propose a framework for deriving path-counting formulas for generalized kinship coefficients. Then, we present the path-counting formulas for all generalized kinship coefficients for which there are recursive formulas and which are sufficient for computing condensed identity coefficients. We also perform experiments to compare the efficiency of our method with the recursive method for computing condensed identity coefficients on large pedigrees.

Comparing Pedigree Graphs

Pedigree graphs, or family trees, are typically constructed by an expensive process of examining genealogical records to determine which pairs of individuals are parent and child. New methods to automate this process take as input genetic data from a set of extant individuals and reconstruct ancestral individuals. There is a great need to evaluate the quality of these methods by comparing the estimated pedigree to the true pedigree. In this article, we consider two main pedigree comparison problems. The first is the pedigree isomorphism problem, for which we present a linear-time algorithm for leaf-labeled pedigrees. The second is the pedigree edit distance problem, for which we present (1) several algorithms that are fast and exact in various special cases, and (2) a general, randomized heuristic algorithm. In the negative direction, we first prove that the pedigree isomorphism problem is as hard as the general graph isomorphism problem, and that the sub-pedigree isomorphism problem is NP-hard. We then show that the pedigree edit distance problem is APX-hard in general and NP-hard on leaf-labeled pedigrees. We use simulated pedigrees to compare our edit-distance algorithms to each other as well as to a branch-and-bound algorithm that always finds an optimal solution.

Efficient path-based computations on pedigree graphs with compact encodings

A pedigree is a diagram of family relationships, and it is often used to determine the mode of inheritance (dominant, recessive, etc.) of genetic diseases. Along with rapidly growing knowledge of genetics and accumulation of genealogy information, pedigree data is becoming increasingly important. In large pedigree graphs, path-based methods for efficiently computing genealogical measurements, such as inbreeding and kinship coefficients of individuals, depend on efficient identification and processing of paths. In this paper, we propose a new compact path encoding scheme on large pedigrees, accompanied by an efficient algorithm for identifying paths. We demonstrate the utilization of our proposed method by applying it to the inbreeding coefficient computation. We present time and space complexity analysis, and also manifest the efficiency of our method for evaluating inbreeding coefficients as compared to previous methods by experimental results using pedigree graphs with real and synthetic data. Both theoretical and experimental results demonstrate that our method is more scalable and efficient than previous methods in terms of time and space requirements.

Ancestral Recombinations Graph: A Reconstructability Perspective Using Random-Graphs Framework

We present a random graphs framework to study pedigree history in an ideal (Wright Fisher) population. This framework correlates the underlying mathematical objects in, for example, pedigree graph, mtDNA or NRY Chr tree, ARG (Ancestral Recombinations Graph), and HUD used in literature, into a single unified random graph framework. It also gives a natural definition, based solely on the topology, of an ARG, one of the most interesting as well as useful mathematical objects in this area. The random graphs framework gives an alternative parametrization of the ARG that does not use the recombination rate q and instead uses a parameter M based on the (estimate of ) the number of non-mixing segments in the extant units. This seems more natural in a setting that attempts to tease apart the population dynamics from the biology of the units. This framework also gives a purely topological definition of GMRCA, analogous to MRCA on trees (which has a purely topological description i.e., it is a root, graph-theoretically speaking, of a tree). Secondly, with a natural extension of the ideas from random-graphs we present a sampling (simulation) algorithm to construct random instances of ARG/unilinear transmission graph. This is the first (to the best of the author's knowledge) algorithm that guarantees uniform sampling of the space of ARG instances, reflecting the ideal population model. Finally, using a measure of reconstructability of the past historical events given a collection of extant sequences, we conclude for a given set of extant sequences, the joint history of local segments along a chromosome is reconstructible.

EFFICIENT COMPUTATION OF KINSHIP AND IDENTITY COEFFICIENTS ON LARGE PEDIGREES

With the rapidly expanding field of medical genetics and genetic counseling, genealogy information is becoming increasingly abundant. An important computation on pedigree data is the calculation of identity coefficients, which provide a complete description of the degree of relatedness of a pair of individuals. The areas of application of identity coefficients are numerous and diverse, from genetic counseling to disease tracking, and thus, the computation of identity coefficients merits special attention. However, the computation of identity coefficients is not done directly, but rather as the final step after computing a set of generalized kinship coefficients. In this paper, we first propose a novel Path-Counting Formula for calculating generalized kinship coefficients, which is motivated by Wright's path-counting method for computing inbreeding coefficient. We then present an efficient and scalable scheme for calculating generalized kinship coefficients on large pedigrees using NodeCodes, a special encoding scheme for expediting the evaluation of queries on pedigree graph structures. Furthermore, we propose an improved scheme using Family NodeCodes for the computation of generalized kinship coefficients, which is motivated by the significant improvement of using Family NodeCodes for inbreeding coefficient over the use of NodeCodes. We also perform experiments for evaluating the efficiency of our method, and compare it with the performance of the traditional recursive algorithm for three individuals. Experimental results demonstrate that the resulting scheme is more scalable and efficient than the traditional recursive methods for computing generalized kinship coefficients.

Efficiently calculating inbreeding on large pedigrees databases

We consider pedigree data structured in the form of a directed acyclic graph, and use an encoding scheme, called NodeCodes, for expediting the evaluation of queries on pedigree graph structures. Inbreeding is the quantitative measure of the genetic relationship between two individuals. The inbreeding coefficient is related to the probability that both copies of any given gene are received from the same ancestor. In this paper we discuss the evaluation of the inbreeding coefficient of a given individual using NodeCodes and propose a new encoding scheme, Family NodeCodes, which is further optimized for pedigree graphs. We implemented and tested these approaches on both synthetic and real pedigree data in terms of performance and scalability. Experimental results show that the use of NodeCodes provides a good alternative for queries involving the inbreeding coefficient, with significant improvements over the traditional iterative evaluation methods (up to 10.1 times faster), and Family NodeCodes further improves this to 77.1 times faster while using 91% less space than regular NodeCodes.

Efficient Algorithms for Reconstructing Zero-Recombinant Haplotypes on a Pedigree Based on Fast Elimination of Redundant Linear Equations

Computational inference of haplotypes from genotypes has attracted a great deal of attention in the computational biology community recently, partially driven by the international HapMap project. In this paper, we study the question of how to efficiently infer haplotypes from genotypes of individuals related by a pedigree, assuming that the hereditary process was free of mutations (i.e., the Mendelian law of inheritance) and recombinants. The problem has recently been formulated as a system of linear equations over the finite field of $F(2)$ and solved in $O(m^3n^3)$ time by using standard Gaussian elimination, where $m$ is the number of loci (or markers) in a genotype and $n$ the number of individuals in the pedigree. We give a much faster algorithm with running time $O(mn^2+n^3\log^2n\log\log n)$. The key ingredients of our construction are (i) a new system of linear equations based on some spanning tree of the pedigree graph and (ii) an efficient method for eliminating redundant equations in a system of $O(mn)$ linear equations over $O(n)$ variables. Although such a fast elimination method is not known for general systems of linear equations, we take advantage of the underlying pedigree graph structure and recent progress on low-stretch spanning trees.

A linear-time algorithm for reconstructing zero-recombinant haplotype configuration on pedigrees without mating loops

With the launch of the international HapMap project, the haplotype inference problem has attracted a great deal of attention in the computational biology community recently. In this paper, we study the question of how to efficiently infer haplotypes from genotypes of individuals related by a pedigree without mating loops, assuming that the hereditary process was free of mutations (i.e. the Mendelian law of inheritance) and recombinants. We model the haplotype inference problem as a system of linear equations as in (Xiao et al. in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’07), pp. 655–664, 2007) and present an (optimal) linear-time (i.e. O(mn) time) algorithm to generate a particular solution to the haplotype inference problem, where m is the number of loci (or markers) in a genotype and n is the number of individuals in the pedigree. Moreover, the algorithm also provides a general solution in O(mn 2) time, which is optimal because the descriptive size of a general solution could be as large as Θ(mn 2). The key ingredients of our construction are (i) a fast consistency checking procedure for the system of linear equations introduced in (Xiao et al. 2007) based on a careful investigation of the relationship between the equations (ii) a novel linear-time method for solving linear equations without invoking the Gaussian elimination method. Although such a fast method for solving equations is not known for general systems of linear equations, we take advantage of the underlying loop-free pedigree graph and some special properties of the linear equations.

Inferring Pedigree Graphs from Genetic Distances

In this paper, we study a problem of inferring blood relationships which satisfy a given matrix of genetic distances between all pairs of n nodes. Blood relationships are represented by our proposed graph class, which is called a pedigree graph. A pedigree graph is a directed acyclic graph in which the maximum indegree is at most two. We show that the number of pedigree graphs which satisfy the condition of given genetic distances may be exponential, but they can be represented by one directed acyclic graph with n nodes. Moreover, an O(n3) time algorithm which solves the problem is also given. Although phylogenetic trees and phylogenetic networks are similar data structures to pedigree graphs, it seems that inferring methods for phylogenetic trees and networks cannot be applied to infer pedigree graphs since nodes of phylogenetic trees and networks represent species whereas nodes of pedigree graphs represent individuals. We also show an O(n2) time algorithm which detects a contradiction between a given pedigree graph and distance matrix of genetic distances.

Linear-time reconstruction of zero-recombinant Mendelian inheritance on pedigrees without mating loops.

With the launch of the international HapMap project, the haplotype inference problem has attracted a great deal of attention in the computational biology community recently. In this paper, we study the question of how to efficiently infer haplotypes from genotypes of individuals related by a pedigree without mating loops, assuming that the hereditary process was free of mutations (i.e. the Mendelian law of inheritance) and recombinants. We model the haplotype inference problem as a system of linear equations as in [10] and present an (optimal) linear-time (i.e. O(mn) time) algorithm to generate a particular solution (A particular solution of any linear system is an assignment of numerical values to the variables in the system which satisfies the equations in the system.) to the haplotype inference problem, where m is the number of loci (or markers) in a genotype and n is the number of individuals in the pedigree. Moreover, the algorithm also provides a general solution (A general solution of any linear system is denoted by the span of a basis in the solution space to its associated homogeneous system, offset from the origin by a vector, namely by any particular solution. A general solution for ZRHC is very useful in practice because it allows the end user to efficiently enumerate all solutions for ZRHC and performs tasks such as random sampling.) in O(mn2) time, which is optimal because the size of a general solution could be as large as Theta(mn2). The key ingredients of our construction are (i) a fast consistency checking procedure for the system of linear equations introduced in [10] based on a careful investigation of the relationship between the equations (ii) a novel linear-time method for solving linear equations without invoking the Gaussian elimination method. Although such a fast method for solving equations is not known for general systems of linear equations, we take advantage of the underlying loop-free pedigree graph and some special properties of the linear equations.

A graphical approach to relatedness inference

The estimation of relatedness structure in natural populations using molecular marker data has become an important tool in population biology, resulting in a variety of estimation procedures for specific sampling scenarios. In this article a general approach is proposed, in which the detailed relationship structure, typically a pedigree graph or partition, is considered to be the object of inference. This makes available tools used in complex model selection theory which have demonstrated effectiveness. An important advantage of this approach is that it permits a fully Bayesian approach to the problem, providing a principled and accessible way to measure statistical error. The approach is demonstrated by applying the minimum description length principle. This technique is used in model selection to provide a rational way of comparing models of varying complexity. We show how the resulting score may be interpreted and applied as a Bayesian posterior density.