Special function methods for bursty models of transcription.

We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical parameter inference from transcriptomics data, we apply the method to simulated data, demonstrating effective control of precision and runtime. Finally, we propose and validate a non-Bayesian approach for parameter estimation based on the characteristic function of the target joint distribution of pre-mRNA and mRNA.

[2]  Effect of reaction-step-size noise on the switching dynamics of stochastic populations. , 2015, Physical review. E.

[3]  W. Austin Elam,et al.  Physical Biology of the Cell , 2014, The Yale Journal of Biology and Medicine.

[4]  R. Padgett,et al.  Rates of in situ transcription and splicing in large human genes , 2009, Nature Structural &Molecular Biology.

[5]  F S Fay,et al.  Visualization of single RNA transcripts in situ. , 1998, Science.

[6]  Ido Golding,et al.  Stochastic Kinetics of Nascent RNA. , 2016, Physical review letters.

[7]  Darren J Wilkinson,et al.  Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo , 2011, Interface Focus.

[8]  Eugenio Cinquemani,et al.  Identifiability and Reconstruction of Biochemical Reaction Networks from Population Snapshot Data , 2018, Processes.

[9]  A. Oudenaarden,et al.  Validation of noise models for single-cell transcriptomics , 2014, Nature Methods.

[10]  W. Huisinga,et al.  Solving the chemical master equation for monomolecular reaction systems analytically , 2006, Journal of mathematical biology.

[11]  Nacho Molina,et al.  Mammalian Genes Are Transcribed with Widely Different Bursting Kinetics , 2011, Science.

[12]  Max Sommerfeld,et al.  Inference for empirical Wasserstein distances on finite spaces , 2016, 1610.03287.

[13]  S. Itzkovitz,et al.  Bursty gene expression in the intact mammalian liver. , 2015, Molecular cell.

[14]  Fabian J Theis,et al.  Generalizing RNA velocity to transient cell states through dynamical modeling , 2019, bioRxiv.

[15]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[16]  Tony C. Scott,et al.  General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function , 2006, math-ph/0607011.

[17]  Meaghan C. Sullivan,et al.  TimeLapse-seq: Adding a temporal dimension to RNA sequencing through nucleoside recoding , 2018, Nature Methods.

[18]  M. Stumpf,et al.  Exactly solvable models of stochastic gene expression. , 2020, The Journal of chemical physics.

[19]  N. Popović,et al.  Time-dependent propagators for stochastic models of gene expression: an analytical method , 2017, Journal of Mathematical Biology.

[20]  Hernan G. Garcia,et al.  Quantitative Imaging of Transcription in Living Drosophila Embryos Links Polymerase Activity to Patterning , 2013, Current Biology.

[21]  P. Castle.  Taylor series for generalized Lambert W functions , 2018, 1801.09904.

[22]  Johannes Grotendorst,et al.  Asymptotic series of generalized Lambert W function , 2014, ACCA.

[23]  Grace X. Y. Zheng,et al.  Massively parallel digital transcriptional profiling of single cells , 2016, Nature Communications.

[24]  M. Abramowitz,et al.  Mathematical functions and their approximations , 1975 .

[25]  Guo-Cheng Yuan,et al.  Transcriptome-scale super-resolved imaging in tissues by RNA seqFISH+ , 2019, Nature.

[26]  Linda R. Petzold,et al.  Accelerated maximum likelihood parameter estimation for stochastic biochemical systems , 2012, BMC Bioinformatics.

[27]  Erik Sundström,et al.  RNA velocity of single cells , 2018, Nature.

[28]  Tallulah S Andrews,et al.  False signals induced by single-cell imputation , 2018, F1000Research.

[29]  Hiroyuki Kuwahara,et al.  Beyond initiation-limited translational bursting: the effects of burst size distributions on the stability of gene expression. , 2015, Integrative biology : quantitative biosciences from nano to macro.

[30]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[31]  Jie Liang,et al.  Accurate Chemical Master Equation Solution Using Multi-Finite Buffers , 2016, Multiscale Model. Simul..

[32]  Suresh Kumar Poovathingal,et al.  Global parameter estimation methods for stochastic biochemical systems , 2010, BMC Bioinformatics.

[33]  Tony C. Scott,et al.  Fleshing out the Generalized Lambert W Function , 2016, ACCA.

[34]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[35]  J. King,et al.  Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression , 2011, Journal of Mathematical Biology.

[36]  Ioannis Sgouralis,et al.  An Introduction to Infinite HMMs for Single-Molecule Data Analysis. , 2016, Biophysical journal.

[37]  William Raymond,et al.  Identification of gene regulation models from single-cell data , 2017, bioRxiv.

[38]  Erin M. Wissink,et al.  Nascent RNA analyses: tracking transcription and its regulation , 2019, Nature Reviews Genetics.

[39]  EFFICIENT ESTIMATION USING THE CHARACTERISTIC FUNCTION , 2016, Econometric Theory.

[40]  Caleb Weinreb,et al.  Fundamental limits on dynamic inference from single-cell snapshots , 2017, Proceedings of the National Academy of Sciences.

[41]  H. Swerdlow,et al.  Large-scale simultaneous measurement of epitopes and transcriptomes in single cells , 2017, Nature Methods.

[42]  A V Hershey,et al.  Computation of Special Functions , 1978 .

[43]  Vahid Shahrezaei,et al.  Analytical distributions for stochastic gene expression , 2008, Proceedings of the National Academy of Sciences.

[44]  D. A. Mcquarrie.  Stochastic approach to chemical kinetics , 1967, Journal of Applied Probability.

[45]  J. Peccoud,et al.  Markovian Modeling of Gene-Product Synthesis , 1995 .

[46]  M. Bee,et al.  A characteristic function-based approach to approximate maximum likelihood estimation , 2018 .

[47]  Brian Munsky,et al.  Distribution shapes govern the discovery of predictive models for gene regulation , 2017, Proceedings of the National Academy of Sciences.

[48]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[49]  S. Meintanis,et al.  Inferential procedures based on the integrated empirical characteristic function , 2018, AStA Advances in Statistical Analysis.

[50]  Lisa Amrhein,et al.  A mechanistic model for the negative binomial distribution of single-cell mRNA counts , 2019, bioRxiv.

[51]  Lucy Ham,et al.  Extrinsic noise and heavy-tailed laws in gene expression , 2019, bioRxiv.

[52]  Gennady Gorin,et al.  Stochastic simulation and statistical inference platform for visualization and estimation of transcriptional kinetics , 2020, PloS one.

[53]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[54]  Péter Érdi,et al.  Stochastic Chemical Kinetics: Theory and (Mostly) Systems Biological Applications , 2014 .

[55]  Abhyudai Singh,et al.  Consequences of mRNA transport on stochastic variability in protein levels. , 2012, Biophysical journal.

[56]  Vanessa M. Peterson,et al.  Multiplexed quantification of proteins and transcripts in single cells , 2017, Nature Biotechnology.

[57]  Peng Qiu,et al.  Embracing the dropouts in single-cell RNA-seq data , 2018, bioRxiv.

[58]  Elizabeth A. Specht,et al.  A Critical and Comparative Review of Fluorescent Tools for Live-Cell Imaging. , 2017, Annual review of physiology.

[59]  Matthew E. Ritchie,et al.  Covering all your bases: incorporating intron signal from RNA-seq data , 2018, bioRxiv.

[60]  Tianshou Zhou,et al.  Analytical Results for a Multistate Gene Model , 2012, SIAM J. Appl. Math..

[61]  A. Feuerverger,et al.  On the Efficiency of Empirical Characteristic Function Procedures , 1981 .

[62]  E. Cox,et al.  Real-Time Kinetics of Gene Activity in Individual Bacteria , 2005, Cell.

[63]  Aleksandra A. Kolodziejczyk,et al.  The technology and biology of single-cell RNA sequencing. , 2015, Molecular cell.

[64]  M. Khammash,et al.  The finite state projection algorithm for the solution of the chemical master equation. , 2006, The Journal of chemical physics.

[65]  Adam M Corrigan,et al.  A continuum model of transcriptional bursting , 2016, eLife.

[66]  Fabian J Theis,et al.  scSLAM-seq reveals core features of transcription dynamics in single cells , 2019, bioRxiv.

[67]  L. Györfi,et al.  Minimum kolmogorov distance estimates of parameters and parametrized distributions , 1996 .

[68]  E. Domany,et al.  Coordinated Pulses of mRNA and of Protein Translation or Degradation Produce EGF-Induced Protein Bursts. , 2017, Cell reports.

[69]  Michael L. Simpson,et al.  Transcriptional burst frequency and burst size are equally modulated across the human genome , 2012, Proceedings of the National Academy of Sciences.

[70]  Johannes Zuber,et al.  Thiol-linked alkylation of RNA to assess expression dynamics , 2017, Nature Methods.

[71]  Wei Vivian Li,et al.  An accurate and robust imputation method scImpute for single-cell RNA-seq data , 2018, Nature Communications.

[72]  Valentine Svensson.  Droplet scRNA-seq is not zero-inflated , 2020, Nature Biotechnology.

[73]  Christopher J. Cronin,et al.  Dynamics and Spatial Genomics of the Nascent Transcriptome by Intron seqFISH , 2018, Cell.

[74]  A. van Oudenaarden,et al.  Using Gene Expression Noise to Understand Gene Regulation , 2012, Science.

[75]  M. Jiménez-Gamero,et al.  Fourier methods for model selection , 2016 .