Numerical Solution of Differential Equations by the Parker-Sochacki Method

A tutorial is presented which demonstrates the theory and usage of the Parker-Sochacki method of numerically solving systems of differential equations. Solutions are demonstrated for the case of projectile motion in air, and for the classical Newtonian N-body problem with mutual gravitational attraction.Comment: Added in July 2010: This tutorial has been posted since 1998 on a university web site, but has now been cited and praised in one or more refereed journals. I am therefore submitting it to the Cornell arXiv so that it may be read in response to its citations. See "Spiking neural network simulation: numerical integration with the Parker-Sochacki method:" J. Comput Neurosci, Robert D. Stewart & Wyeth Bair and http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2717378

Pentraxins coordinate excitatory synapse maturation and circuit integration of parvalbumin interneurons

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.Circuit computation requires precision in the timing, extent, and synchrony of principal cell (PC) firing that is largely enforced by parvalbumin-expressing, fast-spiking interneurons (PVFSIs). To reliably coordinate network activity, PVFSIs exhibit specialized synaptic and membrane properties that promote efficient afferent recruitment such as expression of high-conductance, rapidly gating, GluA4-containing AMPA receptors (AMPARs). We found that PVFSIs upregulate GluA4 during the second postnatal week coincident with increases in the AMPAR clustering proteins NPTX2 and NPTXR. Moreover, GluA4 is dramatically reduced in NPTX2(-/-)/NPTXR(-/-) mice with consequent reductions in PVFSI AMPAR function. Early postnatal NPTX2(-/-)/NPTXR(-/-) mice exhibit delayed circuit maturation with a prolonged critical period permissive for giant depolarizing potentials. Juvenile NPTX2(-/-)/NPTXR(-/-) mice display reduced feedforward inhibition yielding a circuit deficient in rhythmogenesis and prone to epileptiform discharges. Our findings demonstrate an essential role for NPTXs in controlling network dynamics highlighting potential therapeutic targets for disorders with inhibition/excitation imbalances such as schizophrenia.Work supported by a PRAT fellowship to M.S.W., an NICHD intramural award to C.J.M., NIDCD intramural research program funding to R.S.P., an NIMH intramural award to H.A.C., NIH grants (PAR-02-059, NS 039156) to P.F.W., and an NIH grant (EY022730) to M.T.

Discovery and Expansion of Gene Modules by Seeking Isolated Groups in a Random Graph Process

BACKGROUND: A central problem in systems biology research is the identification and extension of biological modules-groups of genes or proteins participating in a common cellular process or physical complex. As a result, there is a persistent need for practical, principled methods to infer the modular organization of genes from genome-scale data. RESULTS: We introduce a novel approach for the identification of modules based on the persistence of isolated gene groups within an evolving graph process. First, the underlying genomic data is summarized in the form of ranked gene-gene relationships, thereby accommodating studies that quantify the relevant biological relationship directly or indirectly. Then, the observed gene-gene relationship ranks are viewed as the outcome of a random graph process and candidate modules are given by the identifiable subgraphs that arise during this process. An isolation index is computed for each module, which quantifies the statistical significance of its survival time. CONCLUSIONS: The Miso (module isolation) method predicts gene modules from genomic data and the associated isolation index provides a module-specific measure of confidence. Improving on existing alternative, such as graph clustering and the global pruning of dendrograms, this index offers two intuitively appealing features: (1) the score is module-specific; and (2) different choices of threshold correlate logically with the resulting performance, i.e. a stringent cutoff yields high quality predictions, but low sensitivity. Through the analysis of yeast phenotype data, the Miso method is shown to outperform existing alternatives, in terms of the specificity and sensitivity of its predictions

Alignment and Prediction of cis-Regulatory Modules Based on a Probabilistic Model of Evolution

Cross-species comparison has emerged as a powerful paradigm for predicting cis-regulatory modules (CRMs) and understanding their evolution. The comparison requires reliable sequence alignment, which remains a challenging task for less conserved noncoding sequences. Furthermore, the existing models of DNA sequence evolution generally do not explicitly treat the special properties of CRM sequences. To address these limitations, we propose a model of CRM evolution that captures different modes of evolution of functional transcription factor binding sites (TFBSs) and the background sequences. A particularly novel aspect of our work is a probabilistic model of gains and losses of TFBSs, a process being recognized as an important part of regulatory sequence evolution. We present a computational framework that uses this model to solve the problems of CRM alignment and prediction. Our alignment method is similar to existing methods of statistical alignment but uses the conserved binding sites to improve alignment. Our CRM prediction method deals with the inherent uncertainties of binding site annotations and sequence alignment in a probabilistic framework. In simulated as well as real data, we demonstrate that our program is able to improve both alignment and prediction of CRM sequences over several state-of-the-art methods. Finally, we used alignments produced by our program to study binding site conservation in genome-wide binding data of key transcription factors in the Drosophila blastoderm, with two intriguing results: (i) the factor-bound sequences are under strong evolutionary constraints even if their neighboring genes are not expressed in the blastoderm and (ii) binding sites in distal bound sequences (relative to transcription start sites) tend to be more conserved than those in proximal regions. Our approach is implemented as software, EMMA (Evolutionary Model-based cis-regulatory Module Analysis), ready to be applied in a broad biological context

Computational Analysis of Whole-Genome Differential Allelic Expression Data in Human

Allelic imbalance (AI) is a phenomenon where the two alleles of a given gene are expressed at different levels in a given cell, either because of epigenetic inactivation of one of the two alleles, or because of genetic variation in regulatory regions. Recently, Bing et al. have described the use of genotyping arrays to assay AI at a high resolution (∼750,000 SNPs across the autosomes). In this paper, we investigate computational approaches to analyze this data and identify genomic regions with AI in an unbiased and robust statistical manner. We propose two families of approaches: (i) a statistical approach based on z-score computations, and (ii) a family of machine learning approaches based on Hidden Markov Models. Each method is evaluated using previously published experimental data sets as well as with permutation testing. When applied to whole genome data from 53 HapMap samples, our approaches reveal that allelic imbalance is widespread (most expressed genes show evidence of AI in at least one of our 53 samples) and that most AI regions in a given individual are also found in at least a few other individuals. While many AI regions identified in the genome correspond to known protein-coding transcripts, others overlap with recently discovered long non-coding RNAs. We also observe that genomic regions with AI not only include complete transcripts with consistent differential expression levels, but also more complex patterns of allelic expression such as alternative promoters and alternative 3′ end. The approaches developed not only shed light on the incidence and mechanisms of allelic expression, but will also help towards mapping the genetic causes of allelic expression and identify cases where this variation may be linked to diseases

Defining the Plasticity of Transcription Factor Binding Sites by Deconstructing DNA Consensus Sequences: The PhoP-Binding Sites among Gamma/Enterobacteria

Transcriptional regulators recognize specific DNA sequences. Because these sequences are embedded in the background of genomic DNA, it is hard to identify the key cis-regulatory elements that determine disparate patterns of gene expression. The detection of the intra- and inter-species differences among these sequences is crucial for understanding the molecular basis of both differential gene expression and evolution. Here, we address this problem by investigating the target promoters controlled by the DNA-binding PhoP protein, which governs virulence and Mg2+ homeostasis in several bacterial species. PhoP is particularly interesting; it is highly conserved in different gamma/enterobacteria, regulating not only ancestral genes but also governing the expression of dozens of horizontally acquired genes that differ from species to species. Our approach consists of decomposing the DNA binding site sequences for a given regulator into families of motifs (i.e., termed submotifs) using a machine learning method inspired by the “Divide & Conquer” strategy. By partitioning a motif into sub-patterns, computational advantages for classification were produced, resulting in the discovery of new members of a regulon, and alleviating the problem of distinguishing functional sites in chromatin immunoprecipitation and DNA microarray genome-wide analysis. Moreover, we found that certain partitions were useful in revealing biological properties of binding site sequences, including modular gains and losses of PhoP binding sites through evolutionary turnover events, as well as conservation in distant species. The high conservation of PhoP submotifs within gamma/enterobacteria, as well as the regulatory protein that recognizes them, suggests that the major cause of divergence between related species is not due to the binding sites, as was previously suggested for other regulators. Instead, the divergence may be attributed to the fast evolution of orthologous target genes and/or the promoter architectures resulting from the interaction of those binding sites with the RNA polymerase

Discovering Transcription Factor Binding Sites in Highly Repetitive Regions of Genomes with Multi-Read Analysis of ChIP-Seq Data

Chromatin immunoprecipitation followed by high-throughput sequencing (ChIP-seq) is rapidly replacing chromatin immunoprecipitation combined with genome-wide tiling array analysis (ChIP-chip) as the preferred approach for mapping transcription-factor binding sites and chromatin modifications. The state of the art for analyzing ChIP-seq data relies on using only reads that map uniquely to a relevant reference genome (uni-reads). This can lead to the omission of up to 30% of alignable reads. We describe a general approach for utilizing reads that map to multiple locations on the reference genome (multi-reads). Our approach is based on allocating multi-reads as fractional counts using a weighted alignment scheme. Using human STAT1 and mouse GATA1 ChIP-seq datasets, we illustrate that incorporation of multi-reads significantly increases sequencing depths, leads to detection of novel peaks that are not otherwise identifiable with uni-reads, and improves detection of peaks in mappable regions. We investigate various genome-wide characteristics of peaks detected only by utilization of multi-reads via computational experiments. Overall, peaks from multi-read analysis have similar characteristics to peaks that are identified by uni-reads except that the majority of them reside in segmental duplications. We further validate a number of GATA1 multi-read only peaks by independent quantitative real-time ChIP analysis and identify novel target genes of GATA1. These computational and experimental results establish that multi-reads can be of critical importance for studying transcription factor binding in highly repetitive regions of genomes with ChIP-seq experiments

Functional Characterization of Transcription Factor Motifs Using Cross-species Comparison across Large Evolutionary Distances

We address the problem of finding statistically significant associations between cis-regulatory motifs and functional gene sets, in order to understand the biological roles of transcription factors. We develop a computational framework for this task, whose features include a new statistical score for motif scanning, the use of different scores for predicting targets of different motifs, and new ways to deal with redundancies among significant motif–function associations. This framework is applied to the recently sequenced genome of the jewel wasp, Nasonia vitripennis, making use of the existing knowledge of motifs and gene annotations in another insect genome, that of the fruitfly. The framework uses cross-species comparison to improve the specificity of its predictions, and does so without relying upon non-coding sequence alignment. It is therefore well suited for comparative genomics across large evolutionary divergences, where existing alignment-based methods are not applicable. We also apply the framework to find motifs associated with socially regulated gene sets in the honeybee, Apis mellifera, using comparisons with Nasonia, a solitary species, to identify honeybee-specific associations

Refinement and Pattern Formation in Neural Circuits by the Interaction of Traveling Waves with Spike-Timing Dependent Plasticity

Traveling waves in the developing brain are a prominent source of highly correlated spiking activity that may instruct the refinement of neural circuits. A candidate mechanism for mediating such refinement is spike-timing dependent plasticity (STDP), which translates correlated activity patterns into changes in synaptic strength. To assess the potential of these phenomena to build useful structure in developing neural circuits, we examined the interaction of wave activity with STDP rules in simple, biologically plausible models of spiking neurons. We derive an expression for the synaptic strength dynamics showing that, by mapping the time dependence of STDP into spatial interactions, traveling waves can build periodic synaptic connectivity patterns into feedforward circuits with a broad class of experimentally observed STDP rules. The spatial scale of the connectivity patterns increases with wave speed and STDP time constants. We verify these results with simulations and demonstrate their robustness to likely sources of noise. We show how this pattern formation ability, which is analogous to solutions of reaction-diffusion systems that have been widely applied to biological pattern formation, can be harnessed to instruct the refinement of postsynaptic receptive fields. Our results hold for rich, complex wave patterns in two dimensions and over several orders of magnitude in wave speeds and STDP time constants, and they provide predictions that can be tested under existing experimental paradigms. Our model generalizes across brain areas and STDP rules, allowing broad application to the ubiquitous occurrence of traveling waves and to wave-like activity patterns induced by moving stimuli