Research paperNonlinear dynamics and noise actuated by the cycle of gene inactivation in stochastic transcription
Introduction
Gene expression is a central problem in life science. Its first step is called gene transcription, in which messenger RNA (mRNA) molecules are generated with a particular segment of single stranded DNA as a template, and RNA polymerase (RNAP) as a carter. In the following step, the newly generated mRNA molecules are used as templates to synthesize proteins by ribosomes [19], [33]. The ongoing studies have shown that gene transcription is a stochastic process, manifested by fluctuations in mRNA levels of the same gene in an isogenic cell population [6], [25], [30], [39]. The stochasticity not only has been shown as inherent discrepancy in biochemical reactions, but also lies in the external random influence [1], [3], [4].
It has been suggested that the observed transcriptional fluctuation is originated from the trade-off between gene activation and inactivation in the time course [2], [6], [20], [22]. This can be described by the classical two-state model in which the gene randomly switches between gene off (inactive) and on (active) states [9], [14]. As shown in the diagramthe processes of gene activation and inactivation have been modeled with first order kinetics. The dwell times in the off and on states are independently and exponentially distributed with constant rates. In coupling with simple mRNA random birth and degradation, the transcripts is assumed to be produced only when the gene is active [16], [25], [28], [30], or more generally, when the gene is residing in both active and inactive states [15], [38]. In recent years, the two-state model has emerged as a standard framework for quantifying variations of mRNAs or proteins found in isogenic populations of cells, through the calculation of either transcription noise η2, the variance of mRNA copy numbers over its mean squared, or transcription noise strength Φ, the variance over the mean [6], [25]. Both indexes quantify the deviation of mRNA numbers in individual cells from the mean, and have facilitated insightful delineation of the intricate interrelation between the transcription parameters and the fluctuations of mRNA distributions [11], [12], [14], [25], [44].
An interesting finding shows that the two-state model uniformly generates the stationary noise strength Φ* > 1 compared to Φ* ≡ 1 when gene is always active [14], [25]. This indicates that gene off state plays a crucial role in modulating stochastic transcription. In some cases gene off state needs to be decomposed into two sub-off states, separately denoted by the ground off state and the excited off state, to explain a unique peak in the distribution of gene off dwelling time [34], or to illustrate the global RNA polymerase II occupancy in mesodermal precursor cells from Toll10b Drosophila embryos [42]. In the so-called three-state model, the activation of gene is then divided into two processes, consisting of the ordered transitions from the ground state to the excited state, and in turn to the gene on state [8], [17], [23]. The first process is characterized by the activation of the transcription factors (TFs) along signalling pathway in response to environmental signals. The second process initiates at the moment that TFs bind at their DNA sites to direct assembly of the active transcription complex [35], [36]. The transcription from this complex to the synthesis of transcripts is considered to be independent of the free TFs.
The three-state model implicitly assumes a unidirectional signaling transduction from the exited off state to the on state. However, this process may be reversible as the binding of TFs to their promoter sites is unstable [1], [3], [37]. In Section 2, we shall introduce a cycle of gene inactivation into the three-state model to emphasize the reversible signal transduction during gene off state. We then focus on the transcription dynamics including the probability of gene on state and mRNA average level in Section 3. We further discuss the steady-state formulas for transcription noise and noise strength, and theoretically analyze how they are related to the parameter rates in Section 4. Finally, we present the mathematical proofs of main results in Section 5 and summarize some of our findings in the end.
Section snippets
The three-state model with gene inactivation cycle
As suggested in the three-state model [35], [36], our model involves a gene on state, and two gene off states consisting of ground state I1 and excited state I2; see Fig. 1. State I1 represents the gene inactivation under normal cellular growth conditions that TFs maintain to be deactivated. When cells receive external signals, the downstream TFs are activated by special signal transduction pathways [29], [33], [40], inducing the random transition from state I1 to the second off state I2. In
Gene inactivation recycling regulates non-monotonic transcription dynamics
We first concern the average level of newly generated mRNA molecules. Therefore, we assume that gene is inactive and the residual mRNA molecules are disregarded at . This gives the initial condition for systems (2.2)–(2.5)Here we introduce two functionsand two numbers α and β determined by algebra equations
Analytical formulas for noise and noise strength
Letwhere α and β are given in (3.2). Then we have Theorem 4.1 The stationary noise strength Φ* and noise η2⁎ can be expressed asHere is the probability of gene on state and m* is the average mRNA level given in (3.8) and (3.10). Furthermore, let be the stationary mRNA average level when the gene is residing at on state. Expression (4.2) can be rewritten as
Expression (4.2) divides the
Proof of Theorems
Denote the vectors and where T is a transpose of the matrix. The evolution systems (2.1) and (2.3) can be rewritten as the matrix formsandwhere I is the identity matrix, and
Conclusion
Gene transcription is inherently a stochastic process involving random switching between gene off and gene on states [30]. This contributes to fluctuations in mRNA levels between cells, especially coexistence of highly expressed and silent cells in an isogenic cell population [3], [4], [6]. To better understand how the stochastic modulation of transcription interacts with external stimuli [22], we introduce a gene inactivation cycle to delineate gene activation triggered by the cellular
CRediT authorship contribution statement
Jian Ren: Formal analysis, Writing - original draft. Feng Jiao: Methodology, Formal analysis. Jianshe Yu: Methodology, Writing - review & editing.
Declaration of Competing Interest
We declare that each author’s contribution to this article is equal and there is no confliction of interest each other.
Acknowledgements
Thanks for the reviewers’s patient comments and insightful advices. This work was partially supported by National Natural Science Foundation of China grants (Nos. 11631005, 11601491, 11871174) and the Program for Changjiang Scholars and Innovative Research Team in Chinese Universities (No. IRT_16R16).
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