Random dynamics of gene transcription activation in single cells

The recent measurements of gene transcription activity at single cell resolution revealed that genes are often transcribed randomly and discontinuously. In order to elucidate how the environmental signals contribute to the stochasticity of gene transcription, a random transition model was recently proposed [M. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol. 253 (2008) 271–280; M. Tang, The mean frequency of transcriptional bursting and its variation in single cells, J. Math. Biol. (2009) doi:10.1007/s00285-009-0258-7, in press; published online: March 10, 2009]. In this model it is assumed that the transcription system transits randomly between three different functional states, quantifying the timing and strength of gene transcription by a sequence of probability functions Pnx(t), coupled in an infinite differential system of master equations. Here n⩾1 are integers and x specifies each of the three functional states.In this work we further study this model aiming to understand the stochastic dynamics of gene transcription. When n⩽3, the exact form of Pnx(t) is found analytically by solving the system of master equations. For larger n however, it is unfeasible to find Pnx(t) explicitly, so we explore the properties of probability functions by analyzing the master operator L that transforms P(n−1)x(t) to Pnx(t). We prove that L “mollifies” the behavior of P(n−1)x(t) by increasing its order of differentiability and by flattening its growth globally. We also show that the n-th cycle of transcription activity condenses at distinct peak instants, with a decreasing peak strength with respect to n. The timings of these peak instants are estimated and several further open questions toward a general theory are discussed.