Role of cooperative binding on noise expression

Simplifications and approximations in a single-gene circuit modeling

The absence of detailed knowledge about regulatory interactions makes the use of phenomenological assumptions mandatory in cell biology modeling. Furthermore, the challenges associated with the analysis of these models compel the implementation of mathematical approximations. However, the constraints these methods introduce to biological interpretation are sometimes neglected. Consequently, understanding these restrictions is a very important task for systems biology modeling. In this article, we examine the impact of such simplifications, taking the case of a single-gene autoinhibitory circuit; however, our conclusions are not limited solely to this instance. We demonstrate that models grounded in the same biological assumptions but described at varying levels of detail can lead to different outcomes, that is, different and contradictory phenotypes or behaviors. Indeed, incorporating specific molecular processes like translation and elongation into the model can introduce instabilities and oscillations not seen when these processes are assumed to be instantaneous. Furthermore, incorporating a detailed description of promoter dynamics, usually described by a phenomenological regulatory function, can lead to instability, depending on the cooperative binding mechanism that is acting. Consequently, although the use of a regulating function facilitates model analysis, it may mask relevant aspects of the system's behavior. In particular, we observe that the two cooperative binding mechanisms, both compatible with the same sigmoidal function, can lead to different phenotypes, such as transcriptional oscillations with different oscillation frequencies.

Transcription factor binding process is the primary driver of noise in gene expression

Noise in expression of individual genes gives rise to variations in activity of cellular pathways and generates heterogeneity in cellular phenotypes. Phenotypic heterogeneity has important implications for antibiotic persistence, mutation penetrance, cancer growth and therapy resistance. Specific molecular features such as the presence of the TATA box sequence and the promoter nucleosome occupancy have been associated with noise. However, the relative importance of these features in noise regulation is unclear and how well these features can predict noise has not yet been assessed. Here through an integrated statistical model of gene expression noise in yeast we found that the number of regulating transcription factors (TFs) of a gene was a key predictor of noise, whereas presence of the TATA box and the promoter nucleosome occupancy had poor predictive power. With an increase in the number of regulatory TFs, there was a rise in the number of cooperatively binding TFs. In addition, an increased number of regulatory TFs meant more overlaps in TF binding sites, resulting in competition between TFs for binding to the same region of the promoter. Through modeling of TF binding to promoter and application of stochastic simulations, we demonstrated that competition and cooperation among TFs could increase noise. Thus, our work uncovers a process of noise regulation that arises out of the dynamics of gene regulation and is not dependent on any specific transcription factor or specific promoter sequence.

Moment dynamics for gene regulation with rational rate laws

This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.

Cooperativity, absolute interaction, and algebraic optimization

We consider a measure of cooperativity based on the minimal interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool SCIP. Moreover, we compute the minimal interactions and minimal molecules for several binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. We compare their minimal interaction with the maximal slope of the Hill plot, and discuss similarities and discrepancies with a view towards the shapes of the binding curves.

Decoupled molecules with binding polynomials of bidegree (n, 2)

We present a result on the number of decoupled molecules for systems binding two different types of ligands. In the case of n and 2 binding sites respectively, we show that there are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(n!)^{2}$$\end{document}2(n!)2 decoupled molecules to a generic binding polynomial. For molecules with more binding sites for the second ligand, we provide computational results.

Information propagation in a noisy gene cascade

We use information theory to study the information transmission through a simple gene cascade where the product of an unregulated gene regulates the expression activity of a cooperative genetic switch. While the input signal is provided by the upstream gene with two states, we consider that the expression of downstream gene is controlled by a cis-regulatory system with three binding sites for the regulator product, which can bind cooperatively. By computing exactly the associated probability distributions, we estimate information transmission thought the mutual information measure. We found that the mutual information associated with unimodal input signal is lower than the associated with bimodal inputs. We also observe that mutual information presents a maximum in the cooperativity intensity, and the position of this maximum depends on the kinetic rates of the promoter. Furthermore, we found that the bursting dynamics of the input signal can enhance the information transmission capacity.

Multiple binding sites for transcriptional repressors can produce regular bursting and enhance noise suppression

Cells may control fluctuations in protein levels by means of negative autoregulation, where transcription factors bind DNA sites to repress their own production. Theoretical studies have assumed a single binding site for the repressor, while in most species it is found that multiple binding sites are arranged in clusters. We study a stochastic description of negative autoregulation with multiple binding sites for the repressor. We find that increasing the number of binding sites induces regular bursting of gene products. By tuning the threshold for repression, we show that multiple binding sites can also suppress fluctuations. Our results highlight possible roles for the presence of multiple binding sites of negative autoregulators.

Promoters Architecture-Based Mechanism for Noise-Induced Oscillations in a Single-Gene Circuit

It is well known that single-gene circuits with negative feedback loop can lead to oscillatory gene expression when they operate with time delay. In order to generate these oscillations many processes can contribute to properly timing such delay. Here we show that the time delay coming from the transitions between internal states of the cis-regulatory system (CRS) can drive sustained oscillations in an auto-repressive single-gene circuit operating in a small volume like a cell. We found that the cooperative binding of repressor molecules is not mandatory for a oscillatory behavior if there are enough binding sites in the CRS. These oscillations depend on an adequate balance between the CRS kinetic, and the synthesis/degradation rates of repressor molecules. This finding suggest that the multi-site CRS architecture can play a key role for oscillatory behavior of gene expression. Finally, our results can also help to synthetic biologists on the design of the promoters architecture for new genetic oscillatory circuits.

Noise and information transmission in promoters with multiple internal States

Based on the measurements of noise in gene expression performed during the past decade, it has become customary to think of gene regulation in terms of a two-state model, where the promoter of a gene can stochastically switch between an ON and an OFF state. As experiments are becoming increasingly precise and the deviations from the two-state model start to be observable, we ask about the experimental signatures of complex multistate promoters, as well as the functional consequences of this additional complexity. In detail, we i), extend the calculations for noise in gene expression to promoters described by state transition diagrams with multiple states, ii), systematically compute the experimentally accessible noise characteristics for these complex promoters, and iii), use information theory to evaluate the channel capacities of complex promoter architectures and compare them with the baseline provided by the two-state model. We find that adding internal states to the promoter generically decreases channel capacity, except in certain cases, three of which (cooperativity, dual-role regulation, promoter cycling) we analyze in detail.

Stochastic noise reduction upon complexification: positively correlated birth-death type systems

Cell systems consist of a huge number of various molecules that display specific patterns of interactions, which have a determining influence on the cell׳s functioning. In general, such complexity is seen to increase with the complexity of the organism, with a concomitant increase of the accuracy and specificity of the cellular processes. The question thus arises how the complexification of systems - modeled here by simple interacting birth-death type processes - can lead to a reduction of the noise - described by the variance of the number of molecules. To gain understanding of this issue, we investigated the difference between a single system containing molecules that are produced and degraded, and the same system - with the same average number of molecules - connected to a buffer. We modeled these systems using Itō stochastic differential equations in discrete time, as they allow straightforward analytical developments. In general, when the molecules in the system and the buffer are positively correlated, the variance on the number of molecules in the system is found to decrease compared to the equivalent system without a buffer. Only buffers that are too noisy themselves tend to increase the noise in the main system. We tested this result on two model cases, in which the system and the buffer contain proteins in their active and inactive state, or protein monomers and homodimers. We found that in the second test case, where the interconversion terms are non-linear in the number of molecules, the noise reduction is much more pronounced; it reaches up to 20% reduction of the Fano factor with the parameter values tested in numerical simulations on an unperturbed birth-death model. We extended our analysis to two arbitrary interconnected systems, and found that the sum of the noise levels in the two systems generally decreases upon interconnection if the molecules they contain are positively correlated.

Noise minimisation in gene expression switches

Gene expression is subject to stochastic variation which leads to fluctuations in the rate of protein production. Recently, a study in yeast at a genomic scale showed that, in some cases, gene expression variability alters phenotypes while, in other cases, these remain unchanged despite fluctuations in the expression of other genes. These studies suggested that noise in gene expression is a physiologically relevant trait and, to prevent harmful stochastic variation in the expression levels of some genes, it can be subject to minimisation. However, the mechanisms for noise minimisation are still unclear. In the present work, we analysed how noise expression depends on the architecture of the cis-regulatory system, in particular on the number of regulatory binding sites. Using analytical calculations and stochastic simulations, we found that the fluctuation level in noise expression decreased with the number of regulatory sites when regulatory transcription factors interacted with only one other bound transcription factor. In contrast, we observed that there was an optimal number of binding sites when transcription factors interacted with many bound transcription factors. This finding suggested a new mechanism for preventing large fluctuations in the expression of genes which are sensitive to the concentration of regulators.

Cooperative binding of transcription factors promotes bimodal gene expression response

In the present work we extend and analyze the scope of our recently proposed stochastic model for transcriptional regulation, which considers an arbitrarily complex cis-regulatory system using only elementary reactions. Previously, we determined the role of cooperativity on the intrinsic fluctuations of gene expression for activating transcriptional switches, by means of master equation formalism and computer simulation. This model allowed us to distinguish between two cooperative binding mechanisms and, even though the mean expression levels were not affected differently by the acting mechanism, we showed that the associated fluctuations were different. In the present generalized model we include other regulatory functions in addition to those associated to an activator switch. Namely, we introduce repressive regulatory functions and two theoretical mechanisms that account for the biphasic response that some cis-regulatory systems show to the transcription factor concentration. We have also extended our previous master equation formalism in order to include protein production by stochastic translation of mRNA. Furthermore, we examine the graded/binary scenarios in the context of the interaction energy between transcription factors. In this sense, this is the first report to show that the cooperative binding of transcription factors to DNA promotes the "all-or-none" phenomenon observed in eukaryotic systems. In addition, we confirm that gene expression fluctuation levels associated with one of two cooperative binding mechanism never exceed the fluctuation levels of the other.

Effect of promoter architecture on the cell-to-cell variability in gene expression

According to recent experimental evidence, promoter architecture, defined by the number, strength and regulatory role of the operators that control transcription, plays a major role in determining the level of cell-to-cell variability in gene expression. These quantitative experiments call for a corresponding modeling effort that addresses the question of how changes in promoter architecture affect variability in gene expression in a systematic rather than case-by-case fashion. In this article we make such a systematic investigation, based on a microscopic model of gene regulation that incorporates stochastic effects. In particular, we show how operator strength and operator multiplicity affect this variability. We examine different modes of transcription factor binding to complex promoters (cooperative, independent, simultaneous) and how each of these affects the level of variability in transcriptional output from cell-to-cell. We propose that direct comparison between in vivo single-cell experiments and theoretical predictions for the moments of the probability distribution of mRNA number per cell can be used to test kinetic models of gene regulation. The emphasis of the discussion is on prokaryotic gene regulation, but our analysis can be extended to eukaryotic cells as well.