On parameter estimation for delay models with discontinuous right-hand sides

We study delay models with discontinuous right-hand side. Lack of smoothness in the solutions of such problems may have serious consequences for parameters estimation using gradient-based approaches. Additionally, it may cause ambiguities in the parameter determinability analysis applied on the parameter estimates. In order to overcome these difficulties, we suggest a standard regularization technique to make the model continuous. We prove the convergence of the solution of the regularized model to the solution of the original problem. As a consequence of that, parameter estimates inferred from the regularized model converge to the corresponding estimates of the original problem. We support our findings with numerical illustrations for simple test problems

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

Efficient low rank approximations for parabolic control problems with unknown heat source

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions

Generalizations of Gronwall's integral inequality and their discrete analogies

The generalizations of Gronwall's integral inequality and their discrete analogies are obtained. In applications these results are used to obtain the stability estimates of solutions for the initial value problems for integral-differential equations of the parabolic typ

Parameter estimation and determinability analysis applied to Drosophila gap gene circuits.

RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are.BACKGROUND: Mathematical modeling of real-life processes often requires the estimation of unknown parameters. Once the parameters are found by means of optimization, it is important to assess the quality of the parameter estimates, especially if parameter values are used to draw biological conclusions from the model. RESULTS: In this paper we describe how the quality of parameter estimates can be analyzed. We apply our methodology to assess parameter determinability for gene circuit models of the gap gene network in early Drosophila embryos. CONCLUSION: Our analysis shows that none of the parameters of the considered model can be determined individually with reasonable accuracy due to correlations between parameters. Therefore, the model cannot be used as a tool to infer quantitative regulatory weights. On the other hand, our results show that it is still possible to draw reliable qualitative conclusions on the regulatory topology of the gene network. Moreover, it improves previous analyses of the same model by allowing us to identify those interactions for which qualitative conclusions are reliable, and those for which they are ambiguous

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

Efficient reverse-engineering of a developmental gene regulatory network

This is the final version of the article. Available from the publisher via the DOI in this record.Understanding the complex regulatory networks underlying development and evolution of multi-cellular organisms is a major problem in biology. Computational models can be used as tools to extract the regulatory structure and dynamics of such networks from gene expression data. This approach is called reverse engineering. It has been successfully applied to many gene networks in various biological systems. However, to reconstitute the structure and non-linear dynamics of a developmental gene network in its spatial context remains a considerable challenge. Here, we address this challenge using a case study: the gap gene network involved in segment determination during early development of Drosophila melanogaster. A major problem for reverse-engineering pattern-forming networks is the significant amount of time and effort required to acquire and quantify spatial gene expression data. We have developed a simplified data processing pipeline that considerably increases the throughput of the method, but results in data of reduced accuracy compared to those previously used for gap gene network inference. We demonstrate that we can infer the correct network structure using our reduced data set, and investigate minimal data requirements for successful reverse engineering. Our results show that timing and position of expression domain boundaries are the crucial features for determining regulatory network structure from data, while it is less important to precisely measure expression levels. Based on this, we define minimal data requirements for gap gene network inference. Our results demonstrate the feasibility of reverse-engineering with much reduced experimental effort. This enables more widespread use of the method in different developmental contexts and organisms. Such systematic application of data-driven models to real-world networks has enormous potential. Only the quantitative investigation of a large number of developmental gene regulatory networks will allow us to discover whether there are rules or regularities governing development and evolution of complex multi-cellular organisms.Funding: The laboratory of Johannes Jaeger and this study in particular was funded by the MEC-EMBL agreement for the EMBL/CRG Research Unit in Systems Biology, by Grant 153 (MOPDEV) of the ERANet: ComplexityNET program, by SGR Grant 406 from the Catalan funding agency AGAUR, by grant BFU2009-10184 from the Spanish Ministry of Science, and by European Commission grant FP7-KBBE-2011-5/289434 (BioPreDyn). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition

In the present paper, a source identification problem for hyperbolic-parabolic equation with involution and Dirichlet condition is studied. The stability estimates for the solution of the source identification hyperbolicparabolic problem are established. The first order of accuracy stable difference scheme is constructed for the approximate solution of the problem under consideration. Numerical results are given for a simple test problem

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