Distribution profiles and their dynamic transition in stochastic gene transcription
The authors express the mass function \(P_m(t)\) of the gene transcription in single cells, as a simple mathematical functions of the rate constants defined in the two-state model. The basic idea is to transform the master equations of \(P_m(t)\) into several special types of second-order ordinary differential equations using the method of characteristics (see for example the book of [\textit{L. C. Evans}, Partial differential equations. 2nd ed. Providence, RI: American Mathematical Society (2010; Zbl 1194.35001)]). In the second section, they derive the exact form of the probability generating function (Theorem 3 and Theorem 4) and express the probability mass function in elementary forms (Theorem 5 from the third section). The main results are given in Theorem 1 and Theorem 2 and their proofs are given in the last section. Also, some numerical examples are given. These results generalize some similar results due to \textit{J. Peccoud} and \textit{B. Ycart} [Theor. Popul. Biol. 48, No. 2, 222--234 (1995; Zbl 0865.92006)], D. T. Gillespie (2007), Tang M. (2008), S. Iyer-Biswas F. Hayot and C. Ayaprakash (2009) or B. Munsky, G. Neuert and A. van Oudenaarden (2012).
scientific article; zbMATH DE number 6154861
scientific article