Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations.

Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.

Sources of dynamic variability in NF-κB signal transduction: a mechanistic model.

The transcription factor NF-κB (p65/p50) plays a central role in the coordination of cellular responses by activating the transcription of numerous target genes. The precise role of the dynamics of NF-κB signalling in regulating gene expression is still an open question. Here, we show that besides external stimulation intracellular parameters can influence the dynamics of NF-κB. By applying mathematical modelling and bifurcation analyses, we show that NF-κB is capable of exhibiting different types of dynamics in response to the same stimulus. We identified the total NF-κB concentration and the IκBα transcription rate constant as two critical parameters that modulate the dynamics and the fold change of NF-κB. Both parameters might vary as a result of cell-to-cell variability. The regulation of the IκBα transcription rate constant, e.g. by co-factors, provides the possibility of regulating the NF-κB dynamics by crosstalk.

Mathematical modelling of Wnt/ß-catenin signalling.

The Wnt/ß-catenin pathway plays an important role in development and disease. Theoretical approaches have been used to describe this pathway and have provided intriguing insights into its signalling characteristics. In the present paper, we review mathematical models of the pathway. We focus on a quantitative kinetic model for canonical Wnt/ß-catenin signalling describing the reactions of the pathway's core compounds [Lee, Salic, Krüger, Heinrich and Kirschner (2003) PLoS Biol. 1, 116-132]. Numerous modifications and further analyses with respect to signalling characteristics, transcriptional feedback and cross-talk were performed. In addition, the role of ß-catenin in gene expression and cell-cell adhesion as well as spatial aspects of signalling are investigated in various theoretical models.

Paracrine and autocrine regulation of gene expression by Wnt-inhibitor Dickkopf in wild-type and mutant hepatocytes.

BACKGROUND: Cells are able to communicate and coordinate their function within tissues via secreted factors. Aberrant secretion by cancer cells can modulate this intercellular communication, in particular in highly organised tissues such as the liver. Hepatocytes, the major cell type of the liver, secrete Dickkopf (Dkk), which inhibits Wnt/ ß-catenin signalling in an autocrine and paracrine manner. Consequently, Dkk modulates the expression of Wnt/ ß-catenin target genes. We present a mathematical model that describes the autocrine and paracrine regulation of hepatic gene expression by Dkk under wild-type conditions as well as in the presence of mutant cells. RESULTS: Our spatial model describes the competition of Dkk and Wnt at receptor level, intra-cellular Wnt/ ß-catenin signalling, and the regulation of target gene expression for 21 individual hepatocytes. Autocrine and paracrine regulation is mediated through a feedback mechanism via Dkk and Dkk diffusion along the porto-central axis. Along this axis an APC concentration gradient is modelled as experimentally detected in liver. Simulations of mutant cells demonstrate that already a single mutant cell increases overall Dkk concentration. The influence of the mutant cell on gene expression of surrounding wild-type hepatocytes is limited in magnitude and restricted to hepatocytes in close proximity. To explore the underlying molecular mechanisms, we perform a comprehensive analysis of the model parameters such as diffusion coefficient, mutation strength and feedback strength. CONCLUSIONS: Our simulations show that Dkk concentration is elevated in the presence of a mutant cell. However, the impact of these elevated Dkk levels on wild-type hepatocytes is confined in space and magnitude. The combination of inter- and intracellular processes, such as Dkk feedback, diffusion and Wnt/ ß-catenin signal transduction, allow wild-type hepatocytes to largely maintain their gene expression.

A modelling approach to quantify dynamic crosstalk between the pheromone and the starvation pathway in baker's yeast.

Cells must be able to process multiple information in parallel and, moreover, they must also be able to combine this information in order to trigger the appropriate response. This is achieved by wiring signalling pathways such that they can interact with each other, a phenomenon often called crosstalk. In this study, we employ mathematical modelling techniques to analyse dynamic mechanisms and measures of crosstalk. We present a dynamic mathematical model that compiles current knowledge about the wiring of the pheromone pathway and the filamentous growth pathway in yeast. We consider the main dynamic features and the interconnections between the two pathways in order to study dynamic crosstalk between these two pathways in haploid cells. We introduce two new measures of dynamic crosstalk, the intrinsic specificity and the extrinsic specificity. These two measures incorporate the combined signal of several stimuli being present simultaneously and seem to be more stable than previous measures. When both pathways are responsive and stimulated, the model predicts that (a) the filamentous growth pathway amplifies the response of the pheromone pathway, and (b) the pheromone pathway inhibits the response of filamentous growth pathway in terms of mitogen activated protein kinase activity and transcriptional activity, respectively. Among several mechanisms we identified leakage of activated Ste11 as the most influential source of crosstalk. Moreover, we propose new experiments and predict their outcomes in order to test hypotheses about the mechanisms of crosstalk between the two pathways. Studying signals that are transmitted in parallel gives us new insights about how pathways and signals interact in a dynamical way, e.g., whether they amplify, inhibit, delay or accelerate each other.

Mathematical modelling suggests a differential impact of ß-transducin repeat-containing protein paralogues on Wnt/ß-catenin signalling dynamics.

The Wnt/ß-catenin signalling pathway is involved in the regulation of a multitude of cellular processes by controlling the concentration of the transcriptional regulator ß-catenin. Proteasomal degradation of ß-catenin is mediated by two ß-transducin repeat-containing protein paralogues, homologous to Slimb protein (HOS) and F-box/WD repeat-containing protein 1A (FWD1), which are functionally interchangeable and thereby considered to function redundantly in the pathway. HOS and FWD1 are both regulated by Wnt/ß-catenin signalling, albeit in opposite directions, thus establishing interlocked negative and positive feedback loops. The functional relevance of the opposite regulation of HOS and FWD1 by Wnt/ß-catenin signalling in conjunction with their redundant activities in proteasomal degradation of ß-catenin remains unresolved. Using a detailed ordinary differential equation model, we investigated the specific influence of each individual feedback mechanism and their combination on Wnt/ß-catenin signal transduction under wild-type and cancerous conditions. We found that, under wild-type conditions, the signalling dynamics are predominantly affected by the HOS feedback as a result of a higher concentration of HOS than FWD1. Transcriptional up-regulation of FWD1 by other signalling pathways reduced the impact of the HOS feedback. The opposite regulation of HOS and FWD1 expression by Wnt/ß-catenin signalling allows the FWD1 feedback to be employed as a compensation mechanism against aberrant pathway activation as a result of a reduced HOS concentration. By contrast, the FWD1 feedback provides no protection against aberrant activation in adenomatous polyposis coli protein mutant cancer cells.

Quantitative dissection and modeling of the NF-κB p100-p105 module reveals interdependent precursor proteolysis.

The mechanisms that govern proteolytic maturation or complete destruction of the precursor proteins p100 and p105 are fundamental to homeostasis and activation of NF-κB; however, they remain poorly understood. Using mass-spectrometry-based quantitative analysis of noncanonical LTßR-induced signaling, we demonstrate that stimulation induces simultaneous processing of both p100 and p105. The precursors not only form hetero-oligomers but also interact with the ATPase VCP/p97, and their induced proteolysis strictly depends on the signal response domain (SRD) of p100, suggesting that the SRD-targeting proteolytic machinery acts in cis and in trans. Separation of cellular pools by isotope labeling revealed synchronous dynamics of p105 and p100 proteolysis. The generation of p50 and p52 from their precursors depends on functional VCP/p97. We have developed quantitative mathematical models that describe the dynamics of the system and predict that p100-p105 complexes are signal responsive.

Modeling Wnt/ß-Catenin Target Gene Expression in APC and Wnt Gradients Under Wild Type and Mutant Conditions.

The Wnt/ß-catenin pathway is involved in the regulation of a multitude of physiological processes by controlling the differential expression of target genes. In certain tissues such as the adult liver, the Wnt/ß-catenin pathway can attain different levels of activity due to gradients of Wnt ligands and/or intracellular pathway components like APC. How graded pathway activity is converted into regionally distinct patterns of Wnt/ß-catenin target gene expression is largely unknown. Here, we apply a mathematical modeling approach to investigate the impact of different regulatory mechanisms on target gene expression within Wnt or APC concentration gradients. We develop a minimal model of Wnt/ß-catenin signal transduction and combine it with various mechanisms of target gene regulation. In particular, the effects of activation, inhibition, and an incoherent feedforward loop (iFFL) are compared. To specify activation kinetics, we analyze experimental data that quantify the response of ß-catenin/TCF reporter constructs to different Wnt concentrations, and demonstrate that the induction of these constructs occurs in a cooperative manner with Hill coefficients between 2 and 5. In summary, our study shows that the combination of specific gene regulatory mechanisms with a time-independent gradient of Wnt or APC is sufficient to generate distinct target gene expression patterns as have been experimentally observed in liver. We find that cooperative gene activation in combination with a TCF feedback can establish sharp borders of target gene expression in Wnt or APC gradients. In contrast, the iFFL renders gene expression independent of gradients of the upstream signaling components. Our subsequent analysis of carcinogenic pathway mutations reveals that their impact on gene expression is determined by the gene regulatory mechanism and the APC concentration of the cell in which the mutation occurs.

Protein phosphatase 1 regulates assembly and function of the beta-catenin degradation complex.

The Wnt/beta-catenin signaling pathway is critical in both cellular proliferation and organismal development. However, how the beta-catenin degradation complex is inhibited upon Wnt activation remains unclear. Using a directed RNAi screen we find that protein phosphatase 1 (PP1), a ubiquitous serine/threonine phosphatase, is a novel potent positive physiologic regulator of the Wnt/beta-catenin signaling pathway. PP1 expression synergistically activates, and inhibition of PP1 inhibits, Wnt/beta-catenin signaling in Drosophila and mammalian cells as well as in Xenopus embryos. The data suggest that PP1 controls Wnt signaling through interaction with, and regulated dephosphorylation of, axin. Inhibition of PP1 leads to enhanced phosphorylation of specific sites on axin by casein kinase I. Axin phosphorylation markedly enhances the binding of glycogen synthase kinase 3, leading to a more active beta-catenin destruction complex. Wnt-regulated changes in axin phosphorylation, mediated by PP1, may therefore determine beta-catenin transcriptional activity. Specific inhibition of PP1 in this pathway may offer therapeutic approaches to disorders with increased beta-catenin signaling.

Modelling the dynamics of the yeast pheromone pathway.

We present a mathematical model of the dynamics of the pheromone pathways in haploid yeast cells of mating type MATa after stimulation with pheromone alpha-factor. The model consists of a set of differential equations and describes the dynamics of signal transduction from the receptor via several steps, including a G protein and a scaffold MAP kinase cascade, up to changes in the gene expression after pheromone stimulation in terms of biochemical changes (complex formations, phosphorylations, etc.). The parameters entering the models have been taken from the literature or adapted to observed time courses or behaviour. Using this model we can follow the time course of the various complex formation processes and of the phosphorylation states of the proteins involved. Furthermore, we can explain the phenotype of more than a dozen well-characterized mutants and also the graded response of yeast cells to varying concentrations of the stimulating pheromone.