A New Context Tree Inference Algorithm for Variable Length Markov Chain Model with Applications to Biological Sequence Analyses.

The statistical inference of high-order Markov chains (MCs) for biological sequences is vital for molecular sequence analyses but can be hindered by the high dimensionality of free parameters. In the seminal article by Bühlmann and Wyner, variable length Markov chain (VLMC) model was proposed to embed the full-order MC in a sparse structured context tree. In the key procedure of tree pruning of their proposed context algorithm, the word count-based statistic for each branch was defined and compared with a fixed cutoff threshold calculated from a common chi-square distribution to prune the branch of the context tree. In this study, we find that the word counts for each branch are highly intercorrelated, resulting in non-negligible effects on the distribution of the statistic of interest. We demonstrate that the inferred context tree based on the original context algorithm by Bühlmann and Wyner, which uses a fixed cutoff threshold based on a common chi-square distribution, can be systematically biased and error prone. We denote the original context algorithm as VLMC-Biased (VLMC-B). To solve this problem, we propose a new context tree inference algorithm using an adaptive tree-pruning scheme, termed VLMC-Consistent (VLMC-C). The VLMC-C is founded on the consistent branch-specific mixed chi-square distributions calculated based on asymptotic normal distribution of multiple word patterns. We validate our theoretical branch-specific asymptotic distribution using simulated data. We compare VLMC-C with VLMC-B on context tree inference using both simulated and real genome sequence data and demonstrate that VLMC-C outperforms VLMC-B for both context tree reconstruction accuracy and model compression capacity.

D-EE: Distributed software for visualizing intrinsic structure of large-scale single-cell data.

BACKGROUND: Dimensionality reduction and visualization play vital roles in single-cell RNA sequencing (scRNA-seq) data analysis. While they have been extensively studied, state-of-the-art dimensionality reduction algorithms are often unable to preserve the global structures underlying data. Elastic embedding (EE), a nonlinear dimensionality reduction method, has shown promise in revealing low-dimensional intrinsic local and global data structure. However, the current implementation of the EE algorithm lacks scalability to large-scale scRNA-seq data. RESULTS: We present a distributed optimization implementation of the EE algorithm, termed distributed elastic embedding (D-EE). D-EE reveals the low-dimensional intrinsic structures of data with accuracy equal to that of elastic embedding, and it is scalable to large-scale scRNA-seq data. It leverages distributed storage and distributed computation, achieving memory efficiency and high-performance computing simultaneously. In addition, an extended version of D-EE, termed distributed optimization implementation of time-series elastic embedding (D-TSEE), enables the user to visualize large-scale time-series scRNA-seq data by incorporating experimentally temporal information. Results with large-scale scRNA-seq data indicate that D-TSEE can uncover oscillatory gene expression patterns by using experimentally temporal information. CONCLUSIONS: D-EE is a distributed dimensionality reduction and visualization tool. Its distributed storage and distributed computation technique allow us to efficiently analyze large-scale single-cell data at the cost of constant time speedup. The source code for D-EE algorithm based on C and MPI tailored to a high-performance computing cluster is available at https://github.com/ShaokunAn/D-EE.

TSEE: an elastic embedding method to visualize the dynamic gene expression patterns of time series single-cell RNA sequencing data.

BACKGROUND: Time series single-cell RNA sequencing (scRNA-seq) data are emerging. However, the analysis of time series scRNA-seq data could be compromised by 1) distortion created by assorted sources of data collection and generation across time samples and 2) inheritance of cell-to-cell variations by stochastic dynamic patterns of gene expression. This calls for the development of an algorithm able to visualize time series scRNA-seq data in order to reveal latent structures and uncover dynamic transition processes. RESULTS: In this study, we propose an algorithm, termed time series elastic embedding (TSEE), by incorporating experimental temporal information into the elastic embedding (EE) method, in order to visualize time series scRNA-seq data. TSEE extends the EE algorithm by penalizing the proximal placement of latent points that correspond to data points otherwise separated by experimental time intervals. TSEE is herein used to visualize time series scRNA-seq datasets of embryonic developmental processed in human and zebrafish. We demonstrate that TSEE outperforms existing methods (e.g. PCA, tSNE and EE) in preserving local and global structures as well as enhancing the temporal resolution of samples. Meanwhile, TSEE reveals the dynamic oscillation patterns of gene expression waves during zebrafish embryogenesis. CONCLUSIONS: TSEE can efficiently visualize time series scRNA-seq data by diluting the distortions of assorted sources of data variation across time stages and achieve the temporal resolution enhancement by preserving temporal order and structure. TSEE uncovers the subtle dynamic structures of gene expression patterns, facilitating further downstream dynamic modeling and analysis of gene expression processes. The computational framework of TSEE is generalizable by allowing the incorporation of other sources of information.

DensityPath: an algorithm to visualize and reconstruct cell state-transition path on density landscape for single-cell RNA sequencing data.

MOTIVATION: Visualizing and reconstructing cell developmental trajectories intrinsically embedded in high-dimensional expression profiles of single-cell RNA sequencing (scRNA-seq) snapshot data are computationally intriguing, but challenging. RESULTS: We propose DensityPath, an algorithm allowing (i) visualization of the intrinsic structure of scRNA-seq data on an embedded 2-d space and (ii) reconstruction of an optimal cell state-transition path on the density landscape. DensityPath powerfully handles high dimensionality and heterogeneity of scRNA-seq data by (i) revealing the intrinsic structures of data, while adopting a non-linear dimension reduction algorithm, termed elastic embedding, which can preserve both local and global structures of the data; and (ii) extracting the topological features of high-density, level-set clusters from a single-cell multimodal density landscape of transcriptional heterogeneity, as the representative cell states. DensityPath reconstructs the optimal cell state-transition path by finding the geodesic minimum spanning tree of representative cell states on the density landscape, establishing a least action path with the minimum-transition-energy of cell fate decisions. We demonstrate that DensityPath can ably reconstruct complex trajectories of cell development, e.g. those with multiple bifurcating and trifurcating branches, while maintaining computational efficiency. Moreover, DensityPath has high accuracy for pseudotime calculation and branch assignment on real scRNA-seq, as well as simulated datasets. DensityPath is robust to parameter choices, as well as permutations of data. AVAILABILITY AND IMPLEMENTATION: DensityPath software is available at https://github.com/ucasdp/DensityPath. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.