A Variational Approach to Parameter Estimation in Ordinary Differential Equations

Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields

Mathematical modeling of drug-induced receptor internalization in the HER2-positive SKBR3 breast cancer cell-line

About 20% of breast cancer tumors over-express the HER2 receptor. Trastuzumab, an approved drug to treat this type of breast cancer, is a monoclonal antibody directly binding at the HER2 receptor and ultimately inhibiting cancer cell growth. The goal of our study was to understand the early impact of trastuzumab on HER2 internalization and recycling in the HER2-overexpressing breast cancer cell line SKBR3. To this end, fluorescence microscopy, monitoring the amount of HER2 expression in the plasma membrane, was combined with mathematical modeling to derive the flux of HER2 receptors from and to the membrane. We constructed a dynamic multi-compartment model based on ordinary differential equations. To account for cancer cell heterogeneity, a first, dynamic model was expanded to a second model including two distinct cell phenotypes, with implications for different conformational states of HER2, i.e. monomeric or homodimeric. Our mathematical model shows that the hypothesis of fast constitutive HER2 recycling back to the plasma membrane does not match the experimental data. It conclusively describes the experimental observation that trastuzumab induces sustained receptor internalization in cells with membrane ruffles. It is also concluded that for rare, non-ruffled (flat) cells, HER2 internalization occurs three orders of magnitude slower than for the bulk, ruffled cell population

Model-based extension of high-throughput to high-content data

<p>Abstract</p> <p>Background</p> <p>High-quality quantitative data is a major limitation in systems biology. The experimental data used in systems biology can be assigned to one of the following categories: assays yielding average data of a cell population, high-content single cell measurements and high-throughput techniques generating single cell data for large cell populations. For modeling purposes, a combination of data from different categories is highly desirable in order to increase the number of observable species and processes and thereby maximize the identifiability of parameters.</p> <p>Results</p> <p>In this article we present a method that combines the power of high-content single cell measurements with the efficiency of high-throughput techniques. A calibration on the basis of identical cell populations measured by both approaches connects the two techniques. We develop a mathematical model to relate quantities exclusively observable by high-content single cell techniques to those measurable with high-content as well as high-throughput methods. The latter are defined as free variables, while the variables measurable with only one technique are described in dependence of those. It is the combination of data calibration and model into a single method that makes it possible to determine quantities only accessible by single cell assays but using high-throughput techniques. As an example, we apply our approach to the nucleocytoplasmic transport of STAT5B in eukaryotic cells.</p> <p>Conclusions</p> <p>The presented procedure can be generally applied to systems that allow for dividing observables into sets of free quantities, which are easily measurable, and variables dependent on those. Hence, it extends the information content of high-throughput methods by incorporating data from high-content measurements.</p

Dynamic Modeling, Parameter Estimation, and Uncertainty Analysis in R

In a wide variety of research fields, dynamic modeling is employed as an instrument to learn and understand complex systems. The differential equations involved in this process are usually non-linear and depend on many parameters whose values determine the characteristics of the emergent system. The inverse problem, i.e., the inference or estimation of parameter values from observed data, is of interest from two points of view. First, the existence point of view, dealing with the question whether the system is able to reproduce the observed dynamics for any parameter values. Second, the identifiability point of view, investigating invariance of the prediction under change of parameter values, as well as the quantification of parameter uncertainty. In this paper, we present the R package dMod providing a framework for dealing with the inverse problem in dynamic systems modeled by ordinary differential equations. The uniqueness of the approach taken by dMod is to provide and propagate accurate derivatives computed from symbolic expressions wherever possible. This derivative information highly supports the convergence of optimization routines and enhances their numerical stability, a requirement for the applicability of sophisticated uncertainty analysis methods. Computational efficiency is achieved by automatic generation and execution of C code. The framework is object-oriented (S3) and provides a variety of functions to set up ordinary differential equation models, observation functions and parameter transformations for multi-conditional parameter estimation. The key elements of the framework and the methodology implemented in dMod are highlighted by an application on a three-compartment transporter model

Division of labor by dual feedback regulators controls JAK2/STAT5 signaling over broad ligand range

Quantitative analysis of time-resolved data in primary erythroid progenitor cells reveals that a dual negative transcriptional feedback mechanism underlies the ability of STAT5 to respond to the broad spectrum of physiologically relevant Epo concentrations

Heterogeneous kinetics of AKT signaling in individual cells are accounted for by variable protein concentration

In most solid cancers, cells harboring oncogenic mutations represent only a sub-fraction of the entire population. Within this sub-fraction the expression level of mutated proteins can vary significantly due to cellular variability limiting the efficiency of targeted therapy. To address the causes of the heterogeneity, we performed a systematic analysis of one of the most frequently mutated pathways in cancer cells, the phosphatidylinositol 3 kinase (PI3K) signaling pathway. Among others PI3K signaling is activated by the hepatocyte growth factor (HGF) that regulates proliferation of hepatocytes during liver regeneration but also fosters tumor cell proliferation. HGF mediated responses of PI3K signaling were monitored both at the single cell and cell population level in primary mouse hepatocytes and in the hepatoma cell line Hepa1_6. Interestingly, we observed that the HGF mediated AKT responses at the level of individual cells is rather heterogeneous. However, the overall average behavior of the single cells strongly resembled the dynamics of AKT activation determined at the cell population level. To gain insights into the molecular cause for the observed heterogeneous behavior of individual cells, we employed dynamic mathematical modeling in a stochastic framework. Our analysis demonstrated that intrinsic noise was not sufficient to explain the observed kinetic behavior, but rather the importance of extrinsic noise has to be considered. Thus, distinct from gene expression in the examined signaling pathway fluctuations of the reaction rates has only a minor impact whereas variability in the concentration of the various signaling components even in a clonal cell population is a key determinant for the kinetic behavior

Model-based identification of TNF alpha-induced IKK beta-mediated and I kappa B alpha-mediated regulation of NF kappa B signal transduction as a tool to quantify the impact of drug-induced liver injury compounds

Drug-induced liver injury: mathematical model quantifies impact of liver-damaging drugs Drug-induced liver injury (DILI) is one of the most important obstacles during drug development. More than 1000 drugs have been identified to damage the liver, but the current test systems are poor in predicting DILI. A team of cell biologists, theoretical physicists, and clinical pharmacologists combined experimental data generated in cultured liver cells with mathematical modeling to quantify the impact of the anti-inflammatory drug diclofenac. The analysis demonstrated that diclofenac induces multiple changes in the signal transduction network activated by the tumor necrosis factor alpha (TNFα), one of the known factors to amplify liver toxicity. Data of other liver injury-causing compounds were integrated into the mathematical model and their impact was quantified, thereby demonstrating the potential use of the mathematical model for the further analysis of other compounds in order to improve DILI test systems