No abstract

Wnt signaling plays an important role in both oncogenesis and development. Activation of the Wnt pathway results in stabilization of the transcriptional coactivator β-catenin. Recent studies have demonstrated that axin, which coordinates β-catenin degradation, is itself degraded. Although the key molecules required for transducing a Wnt signal have been identified, a quantitative understanding of this pathway has been lacking. We have developed a mathematical model for the canonical Wnt pathway that describes the interactions among the core components: Wnt, Frizzled, Dishevelled, GSK3β, APC, axin, β-catenin, and TCF. Using a system of differential equations, the model incorporates the kinetics of protein–protein interactions, protein synthesis/degradation, and phosphorylation/dephosphorylation. We initially defined a reference state of kinetic, thermodynamic, and flux data from experiments using Xenopus extracts. Predictions based on the analysis of the reference state were used iteratively to develop a more refined model from which we analyzed the effects of prolonged and transient Wnt stimulation on β-catenin and axin turnover. We predict several unusual features of the Wnt pathway, some of which we tested experimentally. An insight from our model, which we confirmed experimentally, is that the two scaffold proteins axin and APC promote the formation of degradation complexes in very different ways. We can also explain the importance of axin degradation in amplifying and sharpening the Wnt signal, and we show that the dependence of axin degradation on APC is an essential part of an unappreciated regulatory loop that prevents the accumulation of β-catenin at decreased APC concentrations. By applying control analysis to our mathematical model, we demonstrate the modular design, sensitivity, and robustness of the Wnt pathway and derive an explicit expression for tumor suppression and oncogenicity.

A theoretical analysis of linear enzymatic chains is presented. By linear approximation simple analytical solutions can be obtained for the metabolite concentrations and the flux through the chain for steady-state conditions. The equations are greatly simplified if the common kinetic constants are expressed as functions of two parameters, i.e. the thermodynamic equilibrium constant and the "characteristic time". Three cardinal terms are proposed for the quantitative description of enzyme systems. The first two are the control strength and the control matrix ; these indicate the dependence of the flux and the metabolite concentrations, respectively, on the kinetic properties of a given enzyme. The third is the effector strength, which defines the dependence ofthe velocity of an enzyme on the concentration of an effector ; it expresses the importance of an effector. By linear approximation simple analytical expressions were derived for the control strength, the control matrix and the mass-action ratios. The effector strength was calculated for two cases: for a competitive inhibitor and for allosteric effectors according to the Monod (1965) model. The influence of an effector on the concentrations of the metabolites was considered.A mathematical analysis of an enzymatic chain is necessary if a comprehensive and quantitative description is intended. Verbal analysis even if sections of a chain are treated quantitatively do not permit the simultaneous consideration of diverse influences in the entire enzymatic chain. Therefore selections or weighing is involved which by necessity brings with it a certain arbitrariness.Prom a mathematical model one may expect not only a comprehensive presentation of the effects of a multitude of factors but also a quantitative appraisal of the regulatory features of an enzymatic chain and the identification of enzymes, effectors and metabolites of functional importance. Furthermore, mathematical descriptions make contradictions or insufficiencies of the data more easily apparent.Computer methods for the mathematical description of enzymatic chains were thoroughly explored by Garfinkel et al. in a number of papers ([l] and references cited therein) and also by other authors [2 -51. I n addition to mathematical problems there are several disadvantages of this approach. Firstly, from the computer output it appears difficult to differentiate between important and unimportant effects, enzymes, metabolites, etc. Secondly, it is difficult to see how some effects are brought about. Thirdly, such a computation is often impracticable for experimentalists. Fourthly, many ad hoc assumptions are even now necessary for the mechanisms of several of the constituent enzymes of a chain. The strong dependence of the model of an enzymatic chain on the detailed mechanisms of single enzymes is unfavorable.Analytical approaches, on the other hand, circumvent some of these difficulties, as they are usually connected with a reduction t o the essential parameters. They may show clearly the reasons for regulator...

We have developed a mathematical theory that describes the regulation of signaling pathways as a function of a limited number of key parameters. Our analysis includes linear kinase-phosphatase cascades, as well as systems containing feedback interactions, crosstalk with other signaling pathways, and/or scaffolding and G proteins. We find that phosphatases have a more pronounced effect than kinases on the rate and duration of signaling, whereas signal amplitude is controlled primarily by kinases. The simplest model pathways allow amplified signaling only at the expense of slow signal propagation. More complex and realistic pathways can combine high amplification and signaling rates with maintenance of a stable off-state. Our models also explain how different agonists can evoke transient or sustained signaling of the same pathway and provide a rationale for signaling pathway design.

Gene activation in higher eukaryotes requires the concerted action of transcription factors and coactivator proteins. Coactivators exist in multiprotein complexes that dock on transcription factors and modify chromatin, allowing effective transcription to take place. While biological control focused at the level of the transcription factor is very common, it is now quite clear that a substantial component of gene control is directed at the expression of coactivators, involving pathways as diverse as B-cell development, smooth muscle differentiation, and hepatic gluconeogenesis. Quantitative control of coactivators allows the functional integration of multiple transcription factors and facilitates the formation of distinct biological programs. This coordination and acceleration of different steps in linked pathways has important kinetic considerations, enabling outputs of particular pathways to be increased far more than would otherwise be possible. These kinetic aspects suggest opportunities and concerns as coactivators become targets of therapeutic intervention.