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

Parameter estimation and inference for stochastic reaction-diffusion systems: application to morphogenesis in D. melanogaster

Background: Reaction-diffusion systems are frequently used in systems biology to model developmental and signalling processes. In many applications, count numbers of the diffusing molecular species are very low, leading to the need to explicitly model the inherent variability using stochastic methods. Despite their importance and frequent use, parameter estimation for both deterministic and stochastic reaction-diffusion systems is still a challenging problem. Results: We present a Bayesian inference approach to solve both the parameter and state estimation problem for stochastic reaction-diffusion systems. This allows a determination of the full posterior distribution of the parameters (expected values and uncertainty). We benchmark the method by illustrating it on a simple synthetic experiment. We then test the method on real data about the diffusion of the morphogen Bicoid in Drosophila melanogaster. The results show how the precision with which parameters can be inferred varies dramatically, indicating that the ability to infer full posterior distributions on the parameters can have important experimental design consequences. Conclusions: The results obtained demonstrate the feasibility and potential advantages of applying a Bayesian approach to parameter estimation in stochastic reaction-diffusion systems. In particular, the ability to estimate credibility intervals associated with parameter estimates can be precious for experimental design. Further work, however, will be needed to ensure the method can scale up to larger problems

The Role of Regulated mRNA Stability in Establishing Bicoid Morphogen Gradient in Drosophila Embryonic Development

The Bicoid morphogen is amongst the earliest triggers of differential spatial pattern of gene expression and subsequent cell fate determination in the embryonic development of Drosophila. This maternally deposited morphogen is thought to diffuse in the embryo, establishing a concentration gradient which is sensed by downstream genes. In most model based analyses of this process, the translation of the bicoid mRNA is thought to take place at a fixed rate from the anterior pole of the embryo and a supply of the resulting protein at a constant rate is assumed. Is this process of morphogen generation a passive one as assumed in the modelling literature so far, or would available data support an alternate hypothesis that the stability of the mRNA is regulated by active processes? We introduce a model in which the stability of the maternal mRNA is regulated by being held constant for a length of time, followed by rapid degradation. With this more realistic model of the source, we have analysed three computational models of spatial morphogen propagation along the anterior-posterior axis: (a) passive diffusion modelled as a deterministic differential equation, (b) diffusion enhanced by a cytoplasmic flow term; and (c) diffusion modelled by stochastic simulation of the corresponding chemical reactions. Parameter estimation on these models by matching to publicly available data on spatio-temporal Bicoid profiles suggests strong support for regulated stability over either a constant supply rate or one where the maternal mRNA is permitted to degrade in a passive manner

Statistical lower bounds on protein copy number from fluorescence expression images

Motivation: Fluorescence imaging has become a commonplace for quantitatively measuring mRNA or protein expression in cells and tissues. However, such expression data are usually relative—absolute concentrations or molecular copy numbers are typically not known. While this is satisfactory for many applications, for certain kinds of quantitative network modeling and analysis of expression noise, absolute measures of expression are necessary

Drosophila blastoderm patterning

The Drosophila blastoderm embryo is a classic model for the study of the genetics of pattern formation. In recent years, quantitative empirical approaches have been employed extensively in the study of blastoderm pattern formation. This quantitative work has enabled the development of a number of data-driven computational models. More than in other systems, these models have been experimentally validated, and have informed new empirical work. They have led to insights into the establishment of morphogen gradients, the interpretation and transduction of positional information by downstream transcriptional networks, and the mechanisms by which spatial scaling and robustness of gene expression are achieved. Here we review the latest developments in the field

A Multiscale Investigation of Bicoid-Dependent Transcriptional Events in Drosophila Embryos

Background: Morphogen molecules form concentration gradients to provide spatial information to cells in a developing embryo. Precisely how cells decode such information to form patterns with sharp boundaries remains an open question. For example, it remains controversial whether the Drosophila morphogenetic protein Bicoid (Bcd) plays a transient or sustained role in activating its target genes to establish sharp expression boundaries during development. Methodology/Principal Findings: In this study, we describe a method to simultaneously detect Bcd and the nascent transcripts of its target genes in developing embryos. This method allows us to investigate the relationship between Bcd and the transcriptional status of individual copies of its target genes on distinct scales. We show that, on three scales analyzed concurrently—embryonic, nuclear and local, the actively-transcribing gene copies are associated with high Bcd concentrations. These results underscore the importance of Bcd as a sustained input for transcriptional decisions of individual copies of its target genes during development. We also show that the Bcd-dependent transcriptional decisions have a significantly higher noise than Bcd-dependent gene products, suggesting that, consistent with theoretical studies, time and/or space averaging reduces the noise of Bcd-activated transcriptional output. Finally, our analysis of an X-linked Bcd target gene reveals that Bcd-dependent transcription bursts at twice the frequency in males as in females, providing a mechanism for dosage compensation in early Drosophila embryos

A Two-Dimensional Simulation Model of the Bicoid Gradient in Drosophila

BACKGROUND:Bicoid (Bcd) is a Drosophila morphogenetic protein responsible for patterning the anterior structures in embryos. Recent experimental studies have revealed important insights into the behavior of this morphogen gradient, making it necessary to develop a model that can recapitulate the biological features of the system, including its dynamic and scaling properties. METHODOLOGY/PRINCIPAL FINDINGS:We present a biologically realistic 2-D model of the dynamics of the Bcd gradient in Drosophila embryos. This model is based on equilibrium binding of Bcd molecules to non-specific, low affinity DNA sites throughout the Drosophila genome. It considers both the diffusion media within which the Bcd gradient is formed and the dynamic and other relevant properties of bcd mRNA from which Bcd protein is produced. Our model recapitulates key features of the Bcd protein gradient observed experimentally, including its scaling properties and the stability of its nuclear concentrations during development. Our simulation model also allows us to evaluate the effects of other biological activities on Bcd gradient formation, including the dynamic redistribution of bcd mRNA in early embryos. Our simulation results suggest that, in our model, Bcd protein diffusion is important for the formation of an exponential gradient in embryos. CONCLUSIONS/SIGNIFICANCE:The 2-D model described in this report is a simple and versatile simulation procedure, providing a quantitative evaluation of the Bcd gradient system. Our results suggest an important role of Bcd binding to non-specific, low-affinity DNA sites in proper formation of the Bcd gradient in our model. They demonstrate that highly complex biological systems can be effectively modeled with relatively few parameters

The Formation of the Bicoid Morphogen Gradient Requires Protein Movement from Anteriorly Localized mRNA

New quantitative data show that the Bicoid morphogen gradient is generated from a dynamic localized source and that protein gradient formation requires protein movement along the anterior-posterior axis