Correlation and distinction between stochastic gene transcription models with and without polymerase dynamics

Polymerase dynamics (PD) is an important model for explaining transcriptional regulation in gene perturbation data. In this study, we conducted a detailed analysis of the dynamic behavior of stochastic gene transcription models with PD. We first derived an exact time-dependent formula of mRNA distribution for the classical telegraph model with PD, then revealed a different regulation mechanism whereby PD simultaneously suppresses the Fano factor and enhances the bimodal distribution. For deeper insights into PD regulation, we established optimal effective models without PD to approximate the steady-state mRNA distribution in models with PD. Optimized effective parameters reliably captured input gene initiation and mRNA production rates and reflected the parameter variations of complex systems with PD under biological perturbations. The effective models also revealed that PD introduced quantitatively distinct kurtosis values and dynamics of mRNA distribution. By fitting transcriptome-wide mRNA distribution data from mouse fibroblast and embryonic stem cells, we found that over 1000 data sets may be better captured by integrating PD into the telegraph model. By fitting synthetic mRNA distribution data, we showed that the combinations of cell sample size <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math> and number of time points, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>n</b:mi></b:math>, required for the reliable selection between telegraph models with and without PD are <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>N</c:mi><c:mo>=</c:mo><c:msup><c:mn>10</c:mn><c:mn>3</c:mn></c:msup></c:mrow></c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>n</d:mi><d:mo>≥</d:mo><d:mn>8</d:mn></d:mrow></d:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>N</e:mi><e:mo>=</e:mo><e:msup><e:mn>10</e:mn><e:mn>4</e:mn></e:msup></e:mrow></e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>n</f:mi><f:mo>≥</f:mo><f:mn>2</f:mn></f:mrow></f:math>, or <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>N</g:mi><g:mo>=</g:mo><g:msup><g:mn>10</g:mn><g:mn>5</g:mn></g:msup></g:mrow></g:math> and steady-state data, whereas the parameter combinations required for the reliable estimation of polymerase recruitment and pause release rates are <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>N</h:mi><h:mo>=</h:mo><h:msup><h:mn>10</h:mn><h:mn>4</h:mn></h:msup></h:mrow></h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>n</i:mi><i:mo>≥</i:mo><i:mn>8</i:mn></i:mrow></i:math> or <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>N</j:mi><j:mo>=</j:mo><j:msup><j:mn>10</j:mn><j:mn>5</j:mn></j:msup></j:mrow></j:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>n</k:mi><k:mo>≥</k:mo><k:mn>4</k:mn></k:mrow></k:math>. Our proposed method can also be used to determine the regulatory roles of other biological compounds.