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.

Building functional modules from molecular interactions.

The main reaction pathways in the living cell are carried out by functional modules--namely, macromolecular machines with compact structure or ensembles that change their composition and/or organization during function. Modules define themselves by spatial sequestration, chemical specificity and a characteristic time domain within which their function proceeds. On receiving a specific input, modules go through functional cycles, with phases of increasing and decreasing complexity of molecular interactions. Here, we discuss how such modules are formed and the experimental and theoretical approaches that can be used to investigate them, using examples from polynucleotide-protein interactions, vesicle transport and signal transduction to illustrate the underlying principles. Further progress in this field, where systems biology and biochemistry meet, will depend on iterative validation of the experimental and theoretical approaches.

Patterns of interactions of reaction pairs in metabolic networks.

A large scale structural analysis of metabolic networks is presented focusing on neighbourhood relationships between individual reactions. We define two reactions to be neighbored if one of them provides the necessary set of substances for the other to proceed. A method is developed which allows determining all possible neighborhood relationships categorized as interaction patterns. These patterns differ in the types of participating reactions and in the way they share their reactants. The method is applied to a set of 4795 metabolic reactions contained in the KEGG database. We show that from the 1547 theoretically possible types of interactions 282 patterns are found in metabolism. More than 55% of all interactions occur between reactions with at most two reactants on one side. In these interactions only 25 different patterns play a role. We propose to use these neighborhood relationships as a concept of adjacency in large scale graph theoretical analyses of metabolism.

Structural and dynamical analyses of the kinase network derived from the transpath database.

We analyze the structural design and the dynamical properties of a protein kinase network derived from the Transpath database. We consider structural properties, such as feedback cycles, pathway lengths, fraction of shortest pathways and crosstalk. Dynamic characteristics of the network are analyzed by using nonlinear differential equations with a special focus on kinase amplitudes and signal propagation times. Comparison with random networks shows that the cellular kinase network exhibits special features which might be a result of natural selection. In particular, the Transpath network contains no cycles, and input kinases and output kinases are generally connected by shortest signalling routes. Moreover, it displays a characteristic spectrum of cross-talk between different pathways.

A cross species comparison of metabolic network functions.

We compare a large number of organisms with respect to their metabolic network functions. We measure such functions in terms of the synthesizing capacity of a network when it is provided with a few small chemical substances as external resources. We call this measure the scope and show that it is generally robust against structural alterations of the reaction network. Organisms can be separated into two groups, one with a small and one with a large scope. Networks with a high synthesizing capacity also show a high degree of robustness against structural changes, indicating that this network function has been a target in the evolutionary past of the corresponding organisms. A comparison between structural and functional similarities reveals that organisms with a similar structure do not necessarily show similar biological functions. The presented concepts allow for a systematic investigation of structure-function relationships of metabolic networks and may put forth valuable hints on the evolution of metabolic pathways.

Mathematical modeling of nucleotide excision repair reveals efficiency of sequential assembly strategies.

Nucleotide excision repair (NER) requires the concerted action of many different proteins that assemble at sites of damaged DNA in a sequential fashion. We have constructed a mathematical model delineating hallmarks and general characteristics for NER. We measured the assembly kinetics of the putative damage-recognition factor XPC-HR23B at sites of DNA damage in the nuclei of living cells. These and other in vivo kinetic data allowed us to scrutinize the dynamic behavior of the nucleotide excision repair process in detail. A sequential assembly mechanism appears remarkably advantageous in terms of repair efficiency. Alternative mechanisms for repairosome formation, including random assembly and preassembly, can readily become kinetically unfavorable. Based on the model, new experiments can be defined to gain further insight into this complex process and to critically test model predictions. Our work provides a kinetic framework for NER and rationalizes why many multiprotein processes within the cell nucleus show sequential assembly strategy.

Expanding metabolic networks: scopes of compounds, robustness, and evolution.

A new method for the mathematical analysis of large metabolic networks is presented. Based on the fact that the occurrence of a metabolic reaction generally requires the existence of other reactions providing its substrates, series of metabolic networks are constructed. In each step of the corresponding expansion process those reactions are incorporated whose substrates are made available by the networks of the previous generations. The method is applied to the set of all metabolic reactions included in the KEGG database. Starting with one or more seed compounds, the expansion results in a final network whose compounds define the scope of the seed. Scopes of all metabolic compounds are calculated and it is shown that large parts of cellular metabolism can be considered as the combined scope of simple building blocks. Analyses of various expansion processes reveal crucial metabolites whose incorporation allows for the increase in network complexity. Among these metabolites are common cofactors such as NAD+, ATP, and coenzyme A. We demonstrate that the outcome of network expansion is in general very robust against elimination of single or few reactions. There exist, however, crucial reactions whose elimination results in a dramatic reduction of scope sizes. It is hypothesized that the expansion process displays characteristics of the evolution of metabolism such as the temporal order of the emergence of metabolic pathways.

Control of MAPK signalling: from complexity to what really matters.

Oncogenesis results from changes in kinetics or in abundance of proteins in signal transduction networks. Recently, it was shown that control of signalling cannot reside in a single gene product, and might well be dispersed over many components. Which of the reactions in these complex networks are most important, and how can the existing molecular information be used to understand why particular genes are oncogenes whereas others are not? We implement a new method to help address such questions. We apply control analysis to a detailed kinetic model of the epidermal growth factor-induced mitogen-activated protein kinase network. We determine the control of each reaction with respect to three biologically relevant characteristics of the output of this network: the amplitude, duration and integrated output of the transient phosphorylation of extracellular signal-regulated kinase (ERK). We confirm that control is distributed, but far from randomly: a small proportion of reactions substantially control signalling. In particular, the activity of Raf is in control of all characteristics of the transient profile of ERK phosphorylation, which may clarify why Raf is an oncogene. Most reactions that really matter for one signalling characteristic are also important for the other characteristics. Our analysis also predicts the effects of mutations and changes in gene expression.

Principles behind the multifarious control of signal transduction. ERK phosphorylation and kinase/phosphatase control.

General and simple principles are identified that govern signal transduction. The effects of kinase and phosphatase inhibition on a MAP kinase pathway are first examined in silico. Quantitative measures for the control of signal amplitude, duration and integral strength are introduced. We then identify and prove new principles, such that total control on signal amplitude and on final signal strength must amount to zero, and total control on signal duration and on integral signal intensity must equal -1. Collectively, kinases control amplitudes more than duration, whereas phosphatases tend to control both. We illustrate and validate these principles experimentally: (a) a kinase inhibitor affects the amplitude of EGF-induced ERK phosphorylation much more than its duration and (b) a phosphatase inhibitor influences both signal duration and signal amplitude, in particular long after EGF administration. Implications for the cellular decision between growth and differentiation are discussed.

Generation of nonidentical compartments in vesicular transport systems.

How can organelles communicate by bidirectional vesicle transport and yet maintain different protein compositions? We show by mathematical modeling that a minimal system, in which the basic variables are cytosolic coats for vesicle budding and membrane-bound soluble N-ethyl-maleimide-sensitive factor attachment protein receptors (SNAREs) for vesicle fusion, is sufficient to generate stable, nonidentical compartments. A requirement for establishing and maintaining distinct compartments is that each coat preferentially packages certain SNAREs during vesicle budding. Vesicles fuse preferentially with the compartment that contains the highest concentration of cognate SNAREs, thus further increasing these SNAREs. The stable steady state is the result of a balance between this autocatalytic SNARE accumulation in a compartment and the distribution of SNAREs between compartments by vesicle budding. The resulting nonhomogeneous SNARE distribution generates coat-specific vesicle fluxes that determine the size of compartments. With nonidentical compartments established in this way, the localization and cellular transport of cargo proteins can be explained simply by their affinity for coats.

Biological control through regulated transcriptional coactivators.

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.

A theory of optimal differential gene expression.

We investigate a model of optimal regulation, intended to describe large-scale differential gene expression. Relations between the optimal expression patterns and the function of genes are deduced from an optimality principle: the regulators have to maximise a fitness function which they influence directly via a cost term, and indirectly via their control on important cell variables, such as metabolic fluxes. According to the model, the optimal linear response to small perturbations reflects the regulators' functions, namely their linear influences on the cell variables. The optimal behaviour can be realised by a linear feedback mechanism. Known or assumed properties of response coefficients lead to predictions about regulation patterns. A symmetry relation predicted for deletion experiments is verified with gene expression data. Where the optimality assumption is valid, our results justify the use of expression data for functional annotation and for pathway reconstruction and suggest the use of linear factor models for the analysis of gene expression data.

Interrelations between dynamical properties and structural characteristics of signal transduction networks.

We present a theoretical approach for understanding the interrelations between dynamics and structure of signal transduction pathways. We consider large sets of networks with a specific number of kinases and phosphatases. Our methods are based on nonlinear differential equations and pathway dynamics is characterised in terms of signal amplification and signal duration. We show that networks with a high number of kinases, high connectivities and low phosphatase activities tend to be unstable and run, therefore, the risk to display autoactivation. Analysis of signal transduction pathways retrieved from databases reveals that several structural characteristics required for pathway stability are fulfilled for networks of very large size.

Structural analysis of expanding metabolic networks.

Methods are developed for structural analysis of metabolic networks expanding in size. Expansion proceeds in consecutive generations in which new reactions are attached to the network produced in the previous stage. Different rules are applied resulting in various modes of expansion. Expansion is performed on the set of glycolytic reactions as well as on a very large set of reactions taken from the KEGG database. It is shown that reactions and compounds strongly differ in the generation in which they are attached to the network allowing conclusions for the temporal order of the acquisition during network evolution. The expansion provides efficient tools for detecting new structural characteristics such as substrate-product relationships over long distances.

Model reduction and analysis of robustness for the Wnt/beta-catenin signal transduction pathway.

We present a framework for model reduction of signal transduction networks. The methods are explained by considering a recent model for Wnt/beta-catenin signalling which plays an important regulatory role in cell development and oncogenesis. The procedure results in a reduction of system variables and parameters while maintaining the ability of the model to describe experimental data and to predict the in-vivo behaviour of the pathway. Using metabolic control analysis we quantified the response of the pathway towards random fluctuations of model parameters. This allows to characterise the robustness of the pathway against perturbations in stimulated and unstimulated states. We show that robustness depends on structural as well as kinetic properties of the pathway.

The roles of APC and Axin derived from experimental and theoretical analysis of the Wnt pathway.

Wnt signaling plays an important role in both oncogenesis and development. Activation of the Wnt pathway results in stabilization of the transcriptional coactivator beta-catenin. Recent studies have demonstrated that axin, which coordinates beta-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, GSK3beta, APC, axin, beta-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 beta-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 beta-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.

Temperature dependency and temperature compensation in a model of yeast glycolytic oscillations.

Temperature sensitivities and conditions for temperature compensation have been investigated in a model for yeast glycolytic oscillations. The model can quantitatively simulate the experimental observation that the period length of glycolytic oscillations decreases with increasing temperature. Temperature compensation is studied by using control coefficients describing the effect of rate constants on oscillatory frequencies. Temperature compensation of the oscillatory period is observed when the positive contributions to the sum of products between control coefficients and activation energies balance the corresponding sum of the negative contributions. The calculations suggest that by changing the activation energies for one or several of the processes, i.e. by mutations, it could be possible to obtain temperature compensation in the yeast glycolytic oscillator.

Stoichiometric design of metabolic networks: multifunctionality, clusters, optimization, weak and strong robustness.

Starting from a limited set of reactions describing changes in the carbon skeleton of biochemical compounds complete sets of metabolic networks are constructed. The networks are characterized by the number and types of participating reactions. Elementary networks are defined by the condition that a specific chemical conversion can be performed by a set of given reactions and that this ability will be lost by elimination of any of these reactions. Groups of networks are identified with respect to their ability to perform a certain number of metabolic conversions in an elementary way which are called the network's functions. The number of the network functions defines the degree of multifunctionality. Transitions between networks and mutations of networks are defined by exchanges of single reactions. Different mutations exist such as gain or loss of function mutations and neutral mutations. Based on these mutations neighbourhood relations between networks are established which are described in a graph theoretical way. Basic properties of these graphs are determined such as diameter, connectedness, distance distribution of pairs of vertices. A concept is developed to quantify the robustness of networks against changes in their stoichiometry where we distinguish between strong and weak robustness. Evolutionary algorithms are applied to study the development of network populations under constant and time dependent environmental conditions. It is shown that the populations evolve toward clusters of networks performing a common function and which are closely neighboured. Under changing environmental conditions multifunctional networks prove to be optimal and will be selected.

Prediction of temporal gene expression. Metabolic opimization by re-distribution of enzyme activities.

A computational approach is used to analyse temporal gene expression in the context of metabolic regulation. It is based on the assumption that cells developed optimal adaptation strategies to changing environmental conditions. Time-dependent enzyme profiles are calculated which optimize the function of a metabolic pathway under the constraint of limited total enzyme amount. For linear model pathways it is shown that wave-like enzyme profiles are optimal for a rapid substrate turnover. For the central metabolism of yeast cells enzyme profiles are calculated which ensure long-term homeostasis of key metabolites under conditions of a diauxic shift. These enzyme profiles are in close correlation with observed gene expression data. Our results demonstrate that optimality principles help to rationalize observed gene expression profiles.

Dynamic stability of signal transduction networks depending on downstream and upstream specificity of protein kinases.

Mathematical methods are used for explaining the structural design of signal transduction networks, e.g. MAP kinase cascades, which control cell proliferation, differentiation or apoptosis. Taking into account protein kinases and phosphatases the interrelation between the topology of signaling networks and the stability of their ground state are analysed. It is shown that the stability is closely related to the system's dimension and to the number of cycles within the network. Systems with a higher number of kinases and/or cycles tend to be more unstable. In contrast to that increasing phosphatase activity stabilises the ground state.