Session V42: Focus Session: Systems Biology - Stochastic Gene Expression

Mapping the environmental fitness landscape: Lessons from a noisy synthetic gene circuit

Gene expression actualizes the organismal phenotypes encoded within the genome in an environment-dependent manner. Among all encoded phenotypes, cell population growth rate (cell population fitness) is perhaps the most important, since it determines how well-adapted a genotype is in various environments. Currently it remains unclear how a cell population's growth rate and its subpopulation fractions in specific environments emerge from the stochastic molecular-level kinetics of gene networks and the division rates of single cells. To address this question we developed and quantitatively characterized synthetic a gene circuit controlling the expression of a bifunctional antibiotic resistance gene in \textit{Saccharomyces cerevisiae}. We found that knowing the cell division rates and nongenetic (cellular) memory of gene expression states were necessary for predicting the overall fitness of cell populations in specific antibiotic- and inducer-containing environments. We validated these predictions experimentally and identified environmental conditions that determined a ``sweet spot'' of drug resistance.   

Stochastic description of birth and death processes governed by a mixture of exponential and non-exponential waiting times

The dynamics of complex systems is significantly influenced by fluctuations originating from intrinsic as well as extrinsic sources. In general, the discrete nature of individual events, such as the birth and death of an individual in a population or the production and degradation of a molecule in a chemical reaction, is the main source of intrinsic noise. The occurrence of such events is usually modeled by Poissonian statistics, implying that the probability per unit time for an event to happen is assumed to be constant. Many complex systems however exhibit deviations from elementary Poissonian statistics. Such deviations can arise for example in coarse-grained stochastic models of gene expression, where the waiting time distribution can be more general than the simple exponential distribution. In this contribution we consider birth and death processes which are governed by both, exponential as well as non-exponential waiting times. We derive the corresponding master equation and present methods to approach this equation analytically. As an example we consider a reaction where the production of molecules is governed by a non-exponential waiting time distribution and the degradation follows regular Poissonian statistics.   

Mechanistic basis for transcriptional bursting of ribosomal genes in E. coli

Upon adding more ribosomal genes to the E. coli cell, it adjusts the overall transcription of these genes by reducing the average transcription rate per gene, so as to keep constant the level of ribosomal RNA in the cell. It was observed that this reduction in the average transcription level per gene is accompanied by the generation of transcriptional bursts. The biophysical mechanism responsible for this type of transcriptional control is not yet known. We consider three possible mechanisms suggested in the literature: proximal pausing by RNA polymerase, cooperative recruitment of RNA polymerase by DNA supercoiling, and competition between RNA polymerase and a transcription factor for binding to regulatory DNA. We compute the expected statistical properties of transcription initiation for each one of these models,and compare our predictions with published distributions of distances between the polymerases transcribing the ribosomal genes, obtained from electron micrographs.We use this data to estimate the rates of transcription initiation, which are found to be in good agreement with independent measurements. We also show that the three mechanisms considered here can be discriminated by comparing their predictions for the mean and the variance of interpolymerase distances.   

Decreasing the stochasticity of mammalian gene expression by a synthetic gene circuit

Gene therapy and functional genetic studies usually require precisely controlled and uniform gene expression in a population of cells for reliable level of protein production. Due to this requirement, stochastic gene expression is perceived as undesirable in these fields and ideally has to be minimized. The number of approaches for decreasing gene expression stochasticity in mammalian cells is limited. This creates an unmet need to develop new gene expression systems for this purpose. Based on earlier synthetic constructs in yeast, we developed and assessed a negative feedback-based mammalian gene circuit, with uniform and low level of stochasticity in gene expression at different levels of induction. In addition, this new synthetic construct enables highly precise gene expression control in mammalian cells, due to the linear dependence of gene expression on the inducer concentration applied to the system. This mammalian gene expression circuit has potential applicability for the development of new treatment modalities in gene therapy and research tools in functional genetics. In addition, this work creates a roadmap for moving synthetic gene circuits from microbes into mammalian cells.   

Applications of queueing theory to stochastic models of gene expression

The intrinsic stochasticity of cellular processes implies that analysis of fluctuations (`noise') is often essential for quantitative modeling of gene expression. Recent single-cell experiments have carried out such analysis to characterize moments and entire probability distributions for quantities of interest, e.g.\ mRNA and protein levels across a population of cells. Correspondingly, there is a need to develop general analytical tools for modeling and interpretation of data obtained from such single-cell experiments. One such approach involves the mapping between models of stochastic gene expression and systems analyzed in queueing theory. The talk will provide an overview of this approach and discuss how theorems from queueing theory (e.g. Little's Law) can be used to derive exact results for general stochastic models of gene expression. In the limit that gene expression occurs in bursts, analytical results can be obtained which provide insight into the effects of different regulatory mechanisms on the noise in protein steady-state distributions. In particular, the approach can be used to analyze the effect of post-transcriptional regulation by non-coding RNAs leading to new insights and experimentally testable predictions.   

Cross-talk and interference can enhance information capacity of a signaling pathway

A recurring theme in gene regulatory networks is transcription factors (TFs) that regulate each other, and then bind to overlapping sites on DNA, where they interact and synergistically control transcription of a target gene. TF binding is inherently a noisy process due to thermal fluctuations and the small number of molecules involved. A consequence of multiple TFs interacting at the binding site through competition or cooperativity is that their binding noise becomes correlated. Using concepts from information theory, we show that a correlated-noise channel can enhance its capacity if the TFs are no longer independent but regulating each other. Essentially, the frequency of observing each TF at a given concentration is no longer separable, but ``entangled.'' The form of this entanglement elucidates the upstream TF cross-regulation (cross-talk). We demonstrate these ideas using a cartoon model of two TFs competing for the same binding site. Surprisingly, competition can enhance the information transmission rate. We suggest that this mechanism explains the motif of a coherent feed-forward loop terminating in overlapping binding sites commonly found in developmental networks, and discuss specific examples.   

Exact results for integral thresholds in models of stochastic oscillatory gene expression

Oscillatory stochastic gene expression is often combined with threshold regulation to ensure periodic occurrence of some cellular activity, such as cell division. In this work we first demonstrate the virtue of implementing such regulation using an integral threshold, rather than a step threshold, in the fluctuating numbers of the regulator. We then develop a general theoretical framework using which we derive a model independent result that relates the stochastic distribution of the time oscillating regulator numbers to the distribution of event (cell division) times, regardless of the underlying mechanism that generates a specific form of oscillations in the regulator copy numbers. We then use this result in conjunction with a simple model of stochastically oscillating gene expression to show how the shape of the division time distribution can be used to make deductions about the underlying stochastic dynamics of the oscillating regulator. Specifically, we show that bimodal division time distributions can occur, even in the absence of any bistability in the underlying model, and connect that observation to general features of the underlying stochastic model. Finally, we discuss connections to ongoing single cell experimental studies of Caulobacter cell-cycle division times.   

Fitness Effects of Network Non-Linearity Induced by Gene Expression Noise

In the non-equilibrium dynamics of growing microbial cells, metabolic enzymes can create non-linearities in metabolite concentration because of non-linear degradation (utilization): an enzyme can saturate in the process of metabolite utilization. Increasing metabolite production past the saturation point then results in an ultrasensitive metabolite response. If the production rate of a metabolite depends on a second enzyme or other protein-mediated process, uncorrelated gene expression noise can thus cause transient metabolite concentration bursts. Such bursts are physiologically unnecessary and may represent a source of selection against the ultrasensitive switch, especially if the fluctuating metabolic intermediate is toxic. Selection may therefore favor correlated gene expression fluctuations for enzymes in the same pathway, such as by same-operon membership in bacteria. Using a modified experimental \textit{lac} operon system, we are undertaking a combined theoretical-experimental approach to demonstrate that ($i)$ the \textit{lac} operon has an implicit ultrasensitive switch that we predict is avoided by gene expression correlations induced by same-operon membership; (\textit{ii}) bacterial growth rates are sensitive to crossing the ultrasensitive threshold. Our results suggest that correlations in intrinsic gene expression noise are exploited by evolution to ameliorate the detrimental effects of nonlinearities in metabolite concentrations.   

Stochastic Gene Expression in Networks of Post-transcriptional Regulators

Post-transcriptional regulators, such as small RNAs and microRNAs, are critical elements of diverse cellular pathways. It has been postulated that, in several important cases, the role of these regulators is to to modulate the noise in gene expression for the regulated target. Correspondingly, general stochastic models have been developed, and results obtained, for the case in which a single sRNA regulates a single mRNA target. We generalize these results to networks containing a single mRNA regulated by multiple sRNAs and to networks containing multiple mRNAs regulated by a single sRNA. For these systems, we obtain exact expressions relating the mean levels of the sRNAs to the mean levels of the mRNAs. Additionally, we consider the convergence of the original model to an approximate model which considers sRNA concentrations to be high; for the latter model we derive an analytic form for the generating function of the protein distribution. Finally, we discuss potential experimental protocols which, in combination with the derived results, can be used to infer the underlying gene expression parameters.   

Accurate analytical distributions for stochastic gene expression

Gene expression is significantly stochastic process that can give rise to phenotypic heterogeneity across a population of genetically identical cells. Gene expression variability is generally characterized by the mean and variance of associated distributions, however the entire distributions are often not adequately characterized by the first two moments. For stochastic models of gene expression, exact analytic results for protein steady-state distributions have been obtained only for the simplest case. In this talk, we show how to obtain approximate but accurate representations of protein steady-state distributions for a broad class of models of stochastic gene expression. We first present a procedure to obtain analytical solutions in two limiting cases as the ratio of mRNA to protein lifetimes is varied. We then propose a general strategy for constructing an analytical distribution that interpolates these limits while reproducing the exact mean and variance. ~The corresponding analytical distributions show excellent agreement with results from stochastic simulations throughout parameter space.   

Stochastic Hopf bifurcation in transcription networks with delayed feedback

We study the oscillatory instabilities of two model systems that rely on delayed negative feedback to induce oscillation: a single gene auto repressor system, and a dimer negative autoregulation system. We focus on fluctuations of intrinsic origin in the range of low copy number. The bifurcation diagram is obtained for these stochastic models, and shown to differ significantly from that of a macroscopic description that neglects fluctuations. Bifurcation lines remain sharp under fluctuations, but their location is a function of the relative size of the fluctuations. Shifts in the stability threshold of the oscillators can be traced back to the interplay between statistical correlations and delayed feedback. We finally show that there results cannot be captured by weak noise approximations (the diffusion limit), but instead result from strong fluctuations associated with low copy numbers.