Statistical inference for a quasi birth-death model of RNA transcription

Abstract

Background: A birth-death process of which the births follow a hypoexponential distribution with L phases and are controlled by an on/off mechanism, is a population process which we call the on/off-seq-L process. It is a suitable model for the dynamics of a population of RNA molecules in a single living cell. Motivated by this biological application, our aim is to develop a statistical method to estimate the model parameters of the on/off-seq-L process, based on observations of the population size at discrete time points, and to apply this method to real RNA data.

Methods: It is shown that the on/off-seq-L process can be seen as a quasi birth-death process, and an Erlangization technique can be used to approximate the corresponding likelihood function. An extensive simulation-based numerical study is carried out to investigate the performance of the resulting estimation method.

Results and conclusion: A statistical method is presented to find maximum likelihood estimates of the model parameters for the on/off-seq-L process. Numerical complications related to the likelihood maximization are identified and analyzed, and solutions are presented. The proposed estimation method is a highly accurate method to find the parameter estimates. Based on real RNA data, the on/off-seq-3 process emerges as the best model to describe RNA transcription.

Keywords: Erlangization technique; Maximum likelihood estimation; Quasi birth–death process; RNA transcription.

Fig. 1

Schematic representation of the $\{X_t\}$ process in the on/off-seq-3 model. The dotted line indicates the transition that results in a birth of a new individual. Parameters $q_{\mathrm{off}},q_{\mathrm{on}},{\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2$ and ${\mathrm{\lambda }}_3$ denote the transition rates

Fig. 2

Histograms of 1000 estimates. On/off-seq-2 process with true parameter values: $q_{\text{on}}=0.1,q_{\text{off}}=0.2,{\mathrm{\lambda }}_1=2,{\mathrm{\lambda }}_2=1$

Fig. 3

Empirical distribution function of T based on 1000 simulated realizations of T for parameter vectors ${\theta }_1$ (red) and ${\theta }_2$ (blue)

Fig. 4

Histograms of 1000 estimates obtained under the constraint ${\mathrm{\lambda }}_2\ge {\mathrm{\lambda }}_1$. On/off-seq-2 process with true parameter values: $q_{\text{on}}=0.1,q_{\text{off}}=0.2,{\mathrm{\lambda }}_1=2,{\mathrm{\lambda }}_2=1$

Fig. 5

Histograms of the obtained estimates of $q_{\text{on}}$ for increasing values of n

Fig. 6

Histograms of the obtained estimates of $q_{\text{off}}$ for increasing values of n

Fig. 7

Histograms of the obtained estimates of ${\mathrm{\lambda }}_1$ for increasing values of n

Fig. 8

Histograms of the obtained estimates of ${\mathrm{\lambda }}_2$ for increasing values of n

Fig. 9

Histograms of the obtained estimates of $\mu$ for increasing values of n

Fig. 10

Histograms of 1000 estimates. On/off-seq-3 process with true parameter values: $q_{\text{on}}=0.2$, $q_{\text{off}}=0.5$, ${\mathrm{\lambda }}_1=0.5$, ${\mathrm{\lambda }}_2=2$, ${\mathrm{\lambda }}_3=4$, $\mu =0.1$

Fig. 11

Histograms of 1000 estimates. On/off-seq-3 process with true parameter values: $q_{\text{on}}=0.25,q_{\text{off}}=1,\mathrm{\lambda }=10$ and $\mu =2$

Fig. 12

Steps of protein synthesis