Spatio-logical processes in intracellular signalling

Classical models of intracellular signalling describe how small changes in a cell's external environment can bring about major changes in cellular activity. Recent findings from experimental biology indicate that many intracellular signalling systems show a high level of spatial organisation. This permits the modification, by protein kinase or protein phosphatase action, of specific subsets of intracellular proteins - an attribute that is not addressed in classical signalling models. Here we use ideas and concepts from computer science to describe the information processing nature of intracellular signalling pathways and the impact of spatial heterogeneity of their components (e.g. protein kinases and protein phosphatases) on signalling activity. We argue that it is useful to view the signalling ecology as a vast parallel distributed processing network of agents operating in heterogeneous microenvironments, and we conclude with an overview of the mathematical and semantic methodologies that might help clarify this analogy between biological and computational systems.     

Tandem mass spectrometry: a novel approach for metabolic flux analysis

The goal of metabolic flux analysis (MFA) is the accurate estimation of intracellular fluxes in metabolic networks. Here, we introduce a new method for MFA based on tandem mass spectrometry (MS) and stable-isotope tracer experiments. We demonstrate that tandem MS provides more labeling information than can be obtained from traditional full scan MS analysis and allows estimation of fluxes with better precision. We present a modeling framework that takes full advantage of the additional labeling information obtained from tandem MS for MFA. We show that tandem MS data can be computed for any network model, any compound and any tandem MS fragmentation using linear mapping of isotopomers. The inherent advantages of tandem MS were illustrated in two network models using simulated and literature data. Application of tandem MS increased the observability of the models and improved the precision of estimated fluxes by 2- to 5-fold compared to traditional MS analysis.     

Pathway databases and tools for their exploitation: benefits,current limitations and challenges

In past years, comprehensive representations of cell signalling pathways have been developed by manual curation from literature, which requires huge effort and would benefit from information stored in databases and from automatic retrieval and integration methods. Once a reconstruction of the network of interactions is achieved, analysis of its structural features and its dynamic behaviour can take place. Mathematical modelling techniques are used to simulate the complex behaviour of cell signalling networks, which ultimately sheds light on the mechanisms leading to complex diseases or helps in the identification of drug targets. A variety of databases containing information on cell signalling pathways have been developed in conjunction with methodologies to access and analyse the data. In principle, the scenario is prepared to make the most of this information for the analysis of the dynamics of signalling pathways. However, are the knowledge repositories of signalling pathways ready to realize the systems biology promise? In this article we aim to initiate this discussion and to provide some insights on this issue.     

Are aposematic signals honest? A review

We explore the relevance of honest signalling theory to the evolution of aposematism. We begin with a general consideration of models of signal stability, with a focus on the Zahavian costly signalling framework. Next, we review early models of signalling in the context of aposematism (some that are consistent and some inconsistent with costly honest signalling). We focus on controversies surrounding the idea that aposematic signals are handicaps in a Zahavian framework. Then, we discuss how the alignment of interests between signaller and predator influences the evolution of aposematism, highlight the distinction between qualitative and quantitative honesty and review theory and research relevant to these categories. We also review recent theoretical treatments of the evolution of aposematism that have focused on honest signalling as well as empirical research on a variety of organisms, including invertebrates and frogs. Finally, we discuss future directions for empirical and theoretical research in this area.     

Symbolic modeling of epistasis

The workhorse of modern genetic analysis is the parametric linear model. The advantages of the linear modeling framework are many and include a mathematical understanding of the model fitting process and ease of interpretation. However, an important limitation is that linear models make assumptions about the nature of the data being modeled. This assumption may not be realistic for complex biological systems such as disease susceptibility where nonlinearities in the genotype to phenotype mapping relationship that result from epistasis, plastic reaction norms, locus heterogeneity, and phenocopy, for example, are the norm rather than the exception. We have previously developed a flexible modeling approach called symbolic discriminant analysis (SDA) that makes no assumptions about the patterns in the data. Rather, SDA lets the data dictate the size, shape, and complexity of a symbolic discriminant function that could include any set of mathematical functions from a list of candidates supplied by the user. Here, we outline a new five step process for symbolic model discovery that uses genetic programming (GP) for coarse-grained stochastic searching, experimental design for parameter optimization, graphical modeling for generating expert knowledge, and estimation of distribution algorithms for fine-grained stochastic searching. Finally, we introduce function mapping as a new method for interpreting symbolic discriminant functions. We show that function mapping when combined with measures of interaction information facilitates statistical interpretation by providing a graphical approach to decomposing complex models to highlight synergistic, redundant, and independent effects of polymorphisms and their composite functions. We illustrate this five step SDA modeling process with a real case-control dataset.     

Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling.

This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into 'minimal' models containing two variables and 'extended' models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3--calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.     

Integrating data from biological experiments into metabolic networks with the DBE information system

Modern 'omics'-technologies result in huge amounts of data about life processes. For analysis and data mining purposes this data has to be considered in the context of the underlying biological networks. This work presents an approach for integrating data from biological experiments into metabolic networks by mapping the data onto network elements and visualising the data enriched networks automatically. This methodology is implemented in DBE, an information system that supports the analysis and visualisation of experimental data in the context of metabolic networks. It consists of five parts: (1) the DBE-Database for consistent data storage, (2) the Excel-Importer application for the data import, (3) the DBE-Website as the interface for the system, (4) the DBE-Pictures application for the up- and download of binary (e. g. image) files, and (5) DBE-Gravisto, a network analysis and graph visualisation system. The usability of this approach is demonstrated in two examples.     

Power-law models of signal transduction pathways

The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form--the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed.     

A semiparametric approach for composite functional mapping of dynamic quantitative traits

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.     

Statistical analysis and stochastic modeling of neuronal spike-train activity

Considerable attention has been paid in the literature to the development of biologically interpretable mathematical models of spontaneous stationary neuronal spike-train activity. However, these models work poorly in applied spike-train analysis. In order to improve matters, we need (1) to select appropriate statistical techniques, and (2) to fit the models into the semialternating renewal (SAR) framework. In the present study, biologically interpretable models, the SAR framework, and relevant statistical techniques are integrated in a form which suits the practical research worker. The theory is illustrated with reference to the analysis of a number of spike trains.     

Functional mapping - how to map and study the genetic architecture of dynamic complex traits

The development of any organism is a complex dynamic process that is controlled by a network of genes as well as by environmental factors. Traditional mapping approaches for analysing phenotypic data measured at a single time point are too simple to reveal the genetic control of developmental processes. A general statistical mapping framework, called functional mapping, has been proposed to characterize, in a single step, the quantitative trait loci (QTLs) or nucleotides (QTNs) that underlie a complex dynamic trait. Functional mapping estimates mathematical parameters that describe the developmental mechanisms of trait formation and expression for each QTL or QTN. The approach provides a useful quantitative and testable framework for assessing the interplay between gene actions or interactions and developmental changes.     

From individual Wnt pathways towards a Wnt signalling network

Wnt proteins play important roles during vertebrate and invertebrate development. They obviously have the ability to activate different intracellular signalling pathways. Based on the characteristic intracellular mediators used, these are commonly described as the Wnt/beta-catenin, the Wnt/calcium and the Wnt/Jun N-terminal kinase pathways (also called planar cell polarity pathway). In the past, these different signalling events were mainly described as individual and independent signalling branches. Here, we discuss the possibility that Wnt proteins activate a complex intracellular signalling network rather than individual pathways and suggest a graph representation of this network. Furthermore, we discuss different ways of how to predict the specific outcome of an activation of this network in a particular cell type, which will require the use of mathematical models. We point out that the use of deterministic approaches via the application of differential equations is suitable to model only small aspects of the whole network and that more qualitative approaches are possibly a suitable starting point for the prediction of the global behaviour of such large protein interaction networks.     

Integration of Boolean models exemplified on hepatocyte signal transduction

The number of mathematical models for biological pathways is rapidly growing. In particular, Boolean modelling proved to be suited to describe large cellular signalling networks. Systems biology is at the threshold to holistic understanding of comprehensive networks. In order to reach this goal, connection and integration of existing models of parts of cellular networks into more comprehensive network models is necessary. We discuss model combination approaches for Boolean models. Boolean modelling is qualitative rather than quantitative and does not require detailed kinetic information. We show that these models are useful precursors for large-scale quantitative models and that they are comparatively easy to combine. We propose modelling standards for Boolean models as a prerequisite for smooth model integration. Using these standards, we demonstrate the coupling of two logical models on two different examples concerning cellular interactions in the liver. In the first example, we show the integration of two Boolean models of two cell types in order to describe their interaction. In the second example, we demonstrate the combination of two models describing different parts of the network of a single cell type. Combination of partial models into comprehensive network models will take systems biology to the next level of understanding. The combination of logical models facilitated by modelling standards is a valuable example for the next step towards this goal.     

Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.     

农业废弃物养分循环利用技术模式评估模型的研究进展

农业废弃物的养分循环利用技术模式是实现农业循环经济的重要手段,其评估模型为优化养分循环利用技术提供了重要支撑。本文总结了农业废弃物养分循环技术模式评估框架、评估模型及评价指标、模型的数据源及其不确定性分析,以及模型应用尺度的研究进展。当前,常用于评估养分流动的模型主要是过程数学模型和产业生态学模型。过程数学模型和产业生态学模型在评估结果的可靠性和模拟尺度上存在较大差异,前者主要集中在实验室或中试规模,精度较高;后者可以实现从微观到宏观的多尺度模拟,数据的获取方式导致其具有较高的不确定性。最后,本文对农业废弃物养分循环利用技术评估模型的研究进行展望,提出为了在区域尺度上实现对农业生产系统废弃物资源化利用技术的准确评估,可以将过程数学模型与工业生态学模型相结合,建立可靠的模型框架和数据库,同时,在工厂、农场、村落、乡镇、区域等地理尺度进行模型拓展研究。     

Robustness Analysis of Stochastic Biochemical Systems

We propose a new framework for rigorous robustness analysis of stochastic biochemical systems that is based on probabilistic model checking techniques. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. We have applied the framework to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We have successfully extended previous studies performed on deterministic models (ODE) and showed that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology.