THE DISTRIBUTION OF DISTANCES BETWEEN RANDOMLY CONSTRUCTED GENOMES: GENERATING FUNCTION, EXPECTATION, VARIANCE AND LIMITS

Based on a large repertoire of chromosomal rearrangement operations, the genomic distance d between two genomes with χr and χb linear chromosomes, respectively, both containing the same (or orthologous) n genes or markers, is d = n + max (χr,χb) - c, where c is the number of cycles in the breakpoint graph of the two genomes. In this paper, we study the exact probability distribution of c. We derive the expectation and variance, and show that, in the limit, the expectation of d is [Formula: see text].

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One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3522 ◽

2007 ◽

Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽

Author(s):

Costas A. Christophi ◽

Hosam M. Mahmoud

Keyword(s):

Probability Distribution ◽

Generating Function ◽

Path Length ◽

Mellin Transform ◽

Moment Generating Function ◽

Exact Probability ◽

Digital Trees ◽

Exact Probability Distribution ◽

International Audience ◽

The Mean

International audience One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.

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Blinking statistics and molecular counting in direct stochastic reconstruction microscopy (dSTORM)

Bioinformatics ◽

10.1093/bioinformatics/btab136 ◽

2021 ◽

Author(s):

Lekha Patel ◽

David Williamson ◽

Dylan M Owen ◽

Edward A K Cohen

Keyword(s):

Probability Distribution ◽

Single Molecule ◽

Immunological Synapse ◽

Training Data ◽

Supplementary Information ◽

Cellular Structures ◽

Exact Probability ◽

Stochastic Optical Reconstruction Microscopy ◽

Exact Probability Distribution ◽

Optical Reconstruction

Abstract Motivation Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. Results Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. Availability and implementation Software and data available at https://github.com/lp1611/mol_count_dstorm. Supplementary information Supplementary data are available at Bioinformatics online.

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On the birthday problem: some generalizations and applications

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203110101 ◽

2003 ◽

Vol 2003 (60) ◽

pp. 3827-3840 ◽

Cited By ~ 2

Author(s):

P. N. Rathie ◽

P. Zörnig

Keyword(s):

Probability Distribution ◽

Random Graph ◽

Generating Functions ◽

Hypergeometric Functions ◽

Exact Probability ◽

Confluent Hypergeometric Functions ◽

Birthday Problem ◽

Exact Probability Distribution

We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functionsU(−;−;−)which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated.

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