On the spontaneous stochastic dynamics of a single gene: complexity of the molecular interplay at the promoter

Abstract

Background: Gene promoters can be in various epigenetic states and undergo interactions with many molecules in a highly transient, probabilistic and combinatorial way, resulting in a complex global dynamics as observed experimentally. However, models of stochastic gene expression commonly consider promoter activity as a two-state on/off system. We consider here a model of single-gene stochastic expression that can represent arbitrary prokaryotic or eukaryotic promoters, based on the combinatorial interplay between molecules and epigenetic factors, including energy-dependent remodeling and enzymatic activities.

Results: We show that, considering the mere molecular interplay at the promoter, a single-gene can demonstrate an elaborate spontaneous stochastic activity (eg. multi-periodic multi-relaxation dynamics), similar to what is known to occur at the gene-network level. Characterizing this generic model with indicators of dynamic and steady-state properties (including power spectra and distributions), we reveal the potential activity of any promoter and its influence on gene expression. In particular, we can reproduce, based on biologically relevant mechanisms, the strongly periodic patterns of promoter occupancy by transcription factors (TF) and chromatin remodeling as observed experimentally on eukaryotic promoters. Moreover, we link several of its characteristics to properties of the underlying biochemical system. The model can also be used to identify behaviors of interest (eg. stochasticity induced by high TF concentration) on minimal systems and to test their relevance in larger and more realistic systems. We finally show that TF concentrations can regulate many aspects of the stochastic activity with a considerable flexibility and complexity.

Conclusions: This tight promoter-mediated control of stochasticity may constitute a powerful asset for the cell. Remarkably, a strongly periodic activity that demonstrates a complex TF concentration-dependent control is obtained when molecular interactions have typical characteristics observed on eukaryotic promoters (high mobility, functional redundancy, many alternate states/pathways). We also show that this regime results in a direct and indirect energetic cost. Finally, this model can constitute a framework for unifying various experimental approaches. Collectively, our results show that a gene - the basic building block of complex regulatory networks - can itself demonstrate a significantly complex behavior.

Figure 1

Promoter-centered model of gene expression. (A) All the complex molecular interplay between an arbitrary number of TFs is described generically while the subsequent steps of gene expression are kept simple but explicit. (B) Promoter state fluctuations determine the time-dependent transcriptional efficiency X(t) that propagates successively to RNA level R(t) and protein level P(t) through coupled stochastic synthesis/degradation processes. In this example with realistic timescales and parameter values (cf table S1 of Additional file 1 for a complete description), TFs A and B cooperate and the closed state of chromatin C compete with their association. The highest and lowest transcription rates correspond respectively to open chromatin with A and B bound to the promoter and closed chromatin. (C) This model can represent many different aspects of regulation (see Description ability) making it relevant for describing either prokaryotic or eukaryotic systems.

Figure 2

Portrait of the regulatory structure dynamics and its transmission to RNA and protein levels. (A1) The power spectrum S_X(ω) of the transcriptional efficiency process X(t) (red curve) is the sum of simple components (dashed curves). (A2) These fluctuations of transcriptional efficiency are transmitted to RNA and protein levels undergoing at each step (cf Eq. 6) the addition of a shot noise due to finite synthesis/degradation events (horizontal gray dashed lines: noise levels) and a low pass filtering due to time averaging (vertical gray dashed lines: cutoff frequencies). Dashed red and green curves are intermediate spectra 2γ⟨R⟩+S_X(ω) and illustrating the effect of the shot noise. (B) Each eigenvalue λ_i of matrix -M (or pair of conjugates) corresponds to an elementary component (or mode) in (A1) and determines its characteristics (eg. frequency and thinness of the peak). For instance, the arrows correspond to a 40 min oscillation period and a 10 s relaxation time. Colored crosses identify the components displayed in (A1). Many observables on the promoter can be described by the spectrum of -M (cf text), making it an accurate representation of the whole regulatory structure dynamics.

Figure 3

Stochasticity induced at high concentration of a TF. This minimalist system (A1) (see text or table 2 of Additional file 1 for description) that can correspond to simple molecular scenarios (A2) demonstrates that, contrarily to a common idea, increasing the concentration of a TF can result in a larger variability (B). A more precise picture of this phenomenon is provided by observing how distribution changes with TF concentration (C). Robustness of this behavior with respect to deviations from this ideal minimal model are presented in figures S1 and S2 of Additional file 1.

Figure 4

Complexity of the steady-state. (A) This prokaryotic-like example corresponds to an energy-independent promoter regulated by two TF molecules (A and C) and the looping of DNA with typical parameters from the literature. A binds cooperatively at its two binding sites (ΔG_A-A = -2 kcal/mol) and competitively with C at one of its sites (ΔG_A-C = 1.5 kcal/mol) [68,79]. The energetic cost of DNA looping (typically between 8 and 10 kcal/mol) is ΔG_loop = 9 kcal/mol and is overcompensated by the interaction energy with two TFs of type A that maintain the loop (ΔG_loop-A = -5.5 kcal/mol for each site) [67,93,94]. The closed state of DNA looping slows down the association/dissociation of C (ΔE_loop-C = 2.5 kcal/mol). Bimolecular TF-DNA residence times were taken in the shorter range reported by [90] (1/ = 20 s at both sites and 1/ = 60 s) and the time for DNA to loop when both sites of A are occupied is very fast 1/k^close = 1 s [93]. Concentration ranges ([10^-2; 10³] nM) and equilibrium constants ( = 20 nM and = 1 nM) were set to physiological values [79,89]. Transcription is promoted by the unlooped state, the presence of C and slightly by the presence of A at one site (see table S3 of Additional file 1 for details). RNA life-time (5 min) and abundance (between 10 and 70 copies per cell) were chosen as reported by [84,91]. (B) Exploration of the system's behavior as a function of concentrations [A] and [C] is presented in terms of mean RNA level (B1), normalized variance (B2) and distribution (C) (represented along an arbitrary path of interest because of a too large dimensionality). (D) Changing the energies of activation E⁰ by adding a normally distributed energy (s.d. = 3 kcal/mol) to both direction of each reaction while keeping state energies G⁰ unchanged does not influence the mean behavior of expression but has a profound impact on its variability. This shows that mean expression can hide most of the complexity of regulation and that stochastic aspects can reveal much kinetic information.

Figure 5

Complexity of the dynamics. (A) The concentration-dependence of promoter activity for the same system as in figure 2 is described by the trajectories of eigenvalues -λ_i on the complex plane, showing how the different periodic and aperiodic components of the dynamics are modified (eg. frequency, coherence). (B) They have a direct impact on the power spectrum of transcriptional activity S_X(ω). (C) Energy consumption rate Ė also varies significantly with [A]. (D) As a given system is optimized for a coherent periodic activity (D1-3 are different stages of a given optimization), the trajectories of its eigenvalues tend to stereotype. In between a highly disordered system (D1) and a system close to an homogeneous irreversible cycle (D3), the promoter can demonstrate both a significant coherence and a complex concentration-dependence (D2). The comparison of these three systems in terms of in- and out-degrees of promoter states illustrate the structuring of the transition graph toward a unique path (in- and out-degrees of a given state are taken as the sum over the maximum of its in- and out-transitions respectively).