Safe uses of Hill's model: an exact comparison with the Adair-Klotz model

<p>Abstract</p> <p>Background</p> <p>The Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.</p> <p>Results</p> <p>In this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hill's and Adair-Klotz's models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.</p> <p>Conclusions</p> <p>A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.</p

Nonidentifiability of the Source of Intrinsic Noise in Gene Expression from Single-Burst Data

Over the last few years, experimental data on the fluctuations in gene activity between individual cells and within the same cell over time have confirmed that gene expression is a “noisy” process. This variation is in part due to the small number of molecules taking part in some of the key reactions that are involved in gene expression. One of the consequences of this is that protein production often occurs in bursts, each due to a single promoter or transcription factor binding event. Recently, the distribution of the number of proteins produced in such bursts has been experimentally measured, offering a unique opportunity to study the relative importance of different sources of noise in gene expression. Here, we provide a derivation of the theoretical probability distribution of these bursts for a wide variety of different models of gene expression. We show that there is a good fit between our theoretical distribution and that obtained from two different published experimental datasets. We then prove that, irrespective of the details of the model, the burst size distribution is always geometric and hence determined by a single parameter. Many different combinations of the biochemical rates for the constituent reactions of both transcription and translation will therefore lead to the same experimentally observed burst size distribution. It is thus impossible to identify different sources of fluctuations purely from protein burst size data or to use such data to estimate all of the model parameters. We explore methods of inferring these values when additional types of experimental data are available

The transition between stochastic and deterministic behavior in an excitable gene circuit

We explore the connection between a stochastic simulation model and an ordinary differential equations (ODEs) model of the dynamics of an excitable gene circuit that exhibits noise-induced oscillations. Near a bifurcation point in the ODE model, the stochastic simulation model yields behavior dramatically different from that predicted by the ODE model. We analyze how that behavior depends on the gene copy number and find very slow convergence to the large number limit near the bifurcation point. The implications for understanding the dynamics of gene circuits and other birth-death dynamical systems with small numbers of constituents are discussed.Comment: PLoS ONE: Research Article, published 11 Apr 201

Regulatory control and the costs and benefits of biochemical noise

Experiments in recent years have vividly demonstrated that gene expression can be highly stochastic. How protein concentration fluctuations affect the growth rate of a population of cells, is, however, a wide open question. We present a mathematical model that makes it possible to quantify the effect of protein concentration fluctuations on the growth rate of a population of genetically identical cells. The model predicts that the population's growth rate depends on how the growth rate of a single cell varies with protein concentration, the variance of the protein concentration fluctuations, and the correlation time of these fluctuations. The model also predicts that when the average concentration of a protein is close to the value that maximizes the growth rate, fluctuations in its concentration always reduce the growth rate. However, when the average protein concentration deviates sufficiently from the optimal level, fluctuations can enhance the growth rate of the population, even when the growth rate of a cell depends linearly on the protein concentration. The model also shows that the ensemble or population average of a quantity, such as the average protein expression level or its variance, is in general not equal to its time average as obtained from tracing a single cell and its descendants. We apply our model to perform a cost-benefit analysis of gene regulatory control. Our analysis predicts that the optimal expression level of a gene regulatory protein is determined by the trade-off between the cost of synthesizing the regulatory protein and the benefit of minimizing the fluctuations in the expression of its target gene. We discuss possible experiments that could test our predictions.Comment: Revised manuscript;35 pages, 4 figures, REVTeX4; to appear in PLoS Computational Biolog

The role of input noise in transcriptional regulation

Even under constant external conditions, the expression levels of genes fluctuate. Much emphasis has been placed on the components of this noise that are due to randomness in transcription and translation; here we analyze the role of noise associated with the inputs to transcriptional regulation, the random arrival and binding of transcription factors to their target sites along the genome. This noise sets a fundamental physical limit to the reliability of genetic control, and has clear signatures, but we show that these are easily obscured by experimental limitations and even by conventional methods for plotting the variance vs. mean expression level. We argue that simple, global models of noise dominated by transcription and translation are inconsistent with the embedding of gene expression in a network of regulatory interactions. Analysis of recent experiments on transcriptional control in the early Drosophila embryo shows that these results are quantitatively consistent with the predicted signatures of input noise, and we discuss the experiments needed to test the importance of input noise more generally.Comment: 11 pages, 5 figures minor correction

A Genome-Wide Analysis of Promoter-Mediated Phenotypic Noise in Escherichia coli

Gene expression is subject to random perturbations that lead to fluctuations in the rate of protein production. As a consequence, for any given protein, genetically identical organisms living in a constant environment will contain different amounts of that particular protein, resulting in different phenotypes. This phenomenon is known as “phenotypic noise.” In bacterial systems, previous studies have shown that, for specific genes, both transcriptional and translational processes affect phenotypic noise. Here, we focus on how the promoter regions of genes affect noise and ask whether levels of promoter-mediated noise are correlated with genes' functional attributes, using data for over 60% of all promoters in Escherichia coli. We find that essential genes and genes with a high degree of evolutionary conservation have promoters that confer low levels of noise. We also find that the level of noise cannot be attributed to the evolutionary time that different genes have spent in the genome of E. coli. In contrast to previous results in eukaryotes, we find no association between promoter-mediated noise and gene expression plasticity. These results are consistent with the hypothesis that, in bacteria, natural selection can act to reduce gene expression noise and that some of this noise is controlled through the sequence of the promoter region alon

Bayesian inference of biochemical kinetic parameters using the linear noise approximation

Background Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data. Results We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo. Conclusion The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods

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

According to recent experimental evidence, the architecture of a promoter, defined as the number, strength and regulatory role of the operators that control the promoter, 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 noise in gene expression in a systematic rather than case-by-case fashion. In this article, we make such a systematic investigation, based on a simple 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 transcription product 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 discriminate between different kinetic models of gene regulation.Comment: 35 pages, 6 figures, Submitte

Interplay between pleiotropy and secondary selection determines rise and fall of mutators in stress response

Dramatic rise of mutators has been found to accompany adaptation of bacteria in response to many kinds of stress. Two views on the evolutionary origin of this phenomenon emerged: the pleiotropic hypothesis positing that it is a byproduct of environmental stress or other specific stress response mechanisms and the second order selection which states that mutators hitchhike to fixation with unrelated beneficial alleles. Conventional population genetics models could not fully resolve this controversy because they are based on certain assumptions about fitness landscape. Here we address this problem using a microscopic multiscale model, which couples physically realistic molecular descriptions of proteins and their interactions with population genetics of carrier organisms without assuming any a priori fitness landscape. We found that both pleiotropy and second order selection play a crucial role at different stages of adaptation: the supply of mutators is provided through destabilization of error correction complexes or fluctuations of production levels of prototypic mismatch repair proteins (pleiotropic effects), while rise and fixation of mutators occur when there is a sufficient supply of beneficial mutations in replication-controlling genes. This general mechanism assures a robust and reliable adaptation of organisms to unforeseen challenges. This study highlights physical principles underlying physical biological mechanisms of stress response and adaptation

Master equation simulation analysis of immunostained Bicoid morphogen gradient

<p>Abstract</p> <p>Background</p> <p>The concentration gradient of Bicoid protein which determines the developmental pathways in early <it>Drosophila </it>embryo is the best characterized morphogen gradient at the molecular level. Because different developmental fates can be elicited by different concentrations of Bicoid, it is important to probe the limits of this specification by analyzing intrinsic fluctuations of the Bicoid gradient arising from small molecular number. Stochastic simulations can be applied to further the understanding of the dynamics of Bicoid morphogen gradient formation at the molecular number level, and determine the source of the nucleus-to-nucleus expression variation (noise) observed in the Bicoid gradient.</p> <p>Results</p> <p>We compared quantitative observations of Bicoid levels in immunostained <it>Drosophila </it>embryos with a spatially extended Master Equation model which represents diffusion, decay, and anterior synthesis. We show that the intrinsic noise of an autonomous reaction-diffusion gradient is Poisson distributed. We demonstrate how experimental noise can be identified in the logarithm domain from single embryo analysis, and then separated from intrinsic noise in the normalized variance domain of an ensemble statistical analysis. We show how measurement sensitivity affects our observations, and how small amounts of rescaling noise can perturb the noise strength (Fano factor) observed. We demonstrate that the biological noise level in data can serve as a physical constraint for restricting the model's parameter space, and for predicting the Bicoid molecular number and variation range. An estimate based on a low variance ensemble of embryos suggests that the steady-state Bicoid molecular number in a nucleus should be larger than 300 in the middle of the embryo, and hence the gradient should extend to the posterior end of the embryo, beyond the previously assumed background limit. We exhibit the predicted molecular number gradient together with measurement effects, and make a comparison between conditions of higher and lower variance respectively.</p> <p>Conclusion</p> <p>Quantitative comparison of Master Equation simulations with immunostained data enabled us to determine narrow ranges for key biophysical parameters, which for this system can be independently validated. Intrinsic noise is clearly detectable as well, although the staining process introduces certain limits in resolution.</p