Parameter estimation for a model of gap gene circuits with time-variable external inputs in Drosophila

We study a model for spatio-temporal pattern formation of gap gene products in the early development of Drosophila. In contrast to previous studies of gap gene circuits, our model incorporates a number of proteins as time-variable external inputs, including a protein Huckebein which is necessary for setting up the correct posterior domain boundary and its shift in time for the gap gene hunchback. Unknown model parameters are inferred by fitting the model outputs to the gap gene data and statistical analysis is applied to investigate the quality of the parameter estimates. Our results, while being consistent with previous findings, at the same time provide a number of improvements. Firstly, it takes into account correct regulation of hunchback at the posterior part of the embryo. Secondly, confidence interval analysis shows that the regulatory topology of the gene network in our model which consists of parameters representing the regulation between genes is more consistent with the experimental evidences. Our results also reveal that for data fitting the Weighted Least Squares sum is a more suitable measure than the Ordinary Least Squares sum which has been used in all previous studies. This is confirmed by a better fit of the boundaries of the gap gene expression domains and an absence of patterning defects in the model outputs, as well as by a correct prediction of mutant phenotypes

On the numerical solution of diffusion-reaction equations with singular source terms

A numerical study is presented of reaction-diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface forms an obstacle for numerical discretization, including amongst others order reduction. In this paper the finite volume approach is studied for linear and nonlinear test models. The aimed application field lies in developmental biology from which a test model is used for numerical illustratio

Gene circuit analysis of the terminal gap gene <i>huckebein</i>

The early embryo of Drosophila melanogaster provides a powerful model system to study the role of genes in pattern formation. The gap gene network constitutes the first zygotic regulatory tier in the hierarchy of the segmentation genes involved in specifying the position of body segments. Here, we use an integrative, systems-level approach to investigate the regulatory effect of the terminal gap gene huckebein (hkb) on gap gene expression. We present quantitative expression data for the Hkb protein, which enable us to include hkb in gap gene circuit models. Gap gene circuits are mathematical models of gene networks used as computational tools to extract regulatory information from spatial expression data. This is achieved by fitting the model to gap gene expression patterns, in order to obtain estimates for regulatory parameters which predict a specific network topology. We show how considering variability in the data combined with analysis of parameter determinability significantly improves the biological relevance and consistency of the approach. Our models are in agreement with earlier results, which they extend in two important respects: First, we show that Hkb is involved in the regulation of the posterior hunchback (hb) domain, but does not have any other essential function. Specifically, Hkb is required for the anterior shift in the posterior border of this domain, which is now reproduced correctly in our models. Second, gap gene circuits presented here are able to reproduce mutants of terminal gap genes, while previously published models were unable to reproduce any null mutants correctly. As a consequence, our models now capture the expression dynamics of all posterior gap genes and some variational properties of the system correctly. This is an important step towards a better, quantitative understanding of the developmental and evolutionary dynamics of the gap gene network

Gene Circuit Analysis of the Terminal Gap Gene huckebein

The early embryo of Drosophila melanogaster provides a powerful model system to study the role of genes in pattern formation. The gap gene network constitutes the first zygotic regulatory tier in the hierarchy of the segmentation genes involved in specifying the position of body segments. Here, we use an integrative, systems-level approach to investigate the regulatory effect of the terminal gap gene huckebein (hkb) on gap gene expression. We present quantitative expression data for the Hkb protein, which enable us to include hkb in gap gene circuit models. Gap gene circuits are mathematical models of gene networks used as computational tools to extract regulatory information from spatial expression data. This is achieved by fitting the model to gap gene expression patterns, in order to obtain estimates for regulatory parameters which predict a specific network topology. We show how considering variability in the data combined with analysis of parameter determinability significantly improves the biological relevance and consistency of the approach. Our models are in agreement with earlier results, which they extend in two important respects: First, we show that Hkb is involved in the regulation of the posterior hunchback (hb) domain, but does not have any other essential function. Specifically, Hkb is required for the anterior shift in the posterior border of this domain, which is now reproduced correctly in our models. Second, gap gene circuits presented here are able to reproduce mutants of terminal gap genes, while previously published models were unable to reproduce any null mutants correctly. As a consequence, our models now capture the expression dynamics of all posterior gap genes and some variational properties of the system correctly. This is an important step towards a better, quantitative understanding of the developmental and evolutionary dynamics of the gap gene network

Systems biology: Parameter estimation for biochemical models

Mathematical models of biological processes have various applications: to assist in understanding the functioning of a system, to simulate experiments before actually performing them, to study situations that cannot be dealt with experimentally, etc. Some parameters in the model can be directly obtained from experiments or from the literature. Others have to be inferred by comparing model results to experiments. In this minireview, we discuss the identifiability of models, both intrinsic to the model and taking into account the available data. Furthermore, we give an overview of the most frequently used approaches to search the parameter space