Mathematical models of the balance between apoptosis and proliferation

As our understanding of cellular behaviour grows, and we identify more and more genes involved in the control of such basic processes as cell division and programmed cell death, it becomes increasingly difficult to integrate such detailed knowledge into a meaningful whole. This is an area where mathematical modelling can complement experimental approaches, and even simple mathematical models can yield useful biological insights. This review presents examples of this in the context of understanding the combined effects of different levels of cell death and cell division in a number of biological systems including tumour growth, the homeostasis of immune memory and pre-implantation embryo development. The models we describe, although simplistic, yield insight into several phenomena that are difficult to understand using a purely experimental approach. This includes the different roles played by the apoptosis of stem cells and differentiated cells in determining whether or not a tumour can grow; the way in which a density dependent rate of apoptosis (for instance mediated by cell-cell contact or cytokine signalling) can lead to homeostasis; and the effect of stochastic fluctuations when the number of cells involved is small. We also highlight how models can maximize the amount of information that can be extracted from limited experimental data. The review concludes by summarizing the various mathematical frameworks that can be used to develop new models and the type of biological information that is required to do this.     

simBio: a Java package for the development of detailed cell models

Quantitative dynamic computer models, which integrate a variety of molecular functions into a cell model, provide a powerful tool to create and test working hypotheses. We have developed a new modeling tool, the simBio package (freely available from http://www.sim-bio.org/), which can be used for constructing cell models, such as cardiac cells (the Kyoto model from Matsuoka et al., 2003, 2004a, b, the LRd model from Faber and Rudy, 2000, and the Noble 98 model from Noble et al., 1998), epithelial cells (Strieter et al., 1990) and pancreatic β cells (Magnus and Keizer, 1998). The simBio package is written in Java, uses XML and can solve ordinary differential equations. In an attempt to mimic biological functional structures, a cell model is, in simBio, composed of independent functional modules called Reactors, such as ion channels and the sarcoplasmic reticulum, and dynamic variables called Nodes, such as ion concentrations. The interactions between Reactors and Nodes are described by the graph theory and the resulting graph represents a blueprint of an intricate cellular system. Reactors are prepared in a hierarchical order, in analogy to the biological classification. Each Reactor can be composed or improved independently, and can easily be reused for different models. This way of building models, through the combination of various modules, is enabled through the use of object-oriented programming concepts. Thus, simBio is a straightforward system for the creation of a variety of cell models on a common database of functional modules.     

PPI spider: A tool for the interpretation of proteomics data in the context of protein–protein interaction networks

Recent advances in experimental technologies allow for the detection of a complete cell proteome. Proteins that are expressed at a particular cell state or in a particular compartment as well as proteins with differential expression between various cells states are commonly delivered by many proteomics studies. Once a list of proteins is derived, a major challenge is to interpret the identified set of proteins in the biological context. Protein–protein interaction (PPI) data represents abundant information that can be employed for this purpose. However, these data have not yet been fully exploited due to the absence of a methodological framework that can integrate this type of information. Here, we propose to infer a network model from an experimentally identified protein list based on the available information about the topology of the global PPI network. We propose to use a Monte Carlo simulation procedure to compute the statistical significance of the inferred models. The method has been implemented as a freely available web‐based tool, PPI spider ( http://mips.helmholtz‐muenchen.de/proj/ppispider ). To support the practical significance of PPI spider, we collected several hundreds of recently published experimental proteomics studies that reported lists of proteins in various biological contexts. We reanalyzed them using PPI spider and demonstrated that in most cases PPI spider could provide statistically significant hypotheses that are helpful for understanding of the protein list.     

From proteomics toward systems biology: integration of different types of proteomics data into network models

Living organisms are comprised of various systems at different levels, i.e., organs, tissues, and cells. Each system carries out its diverse functions in response to environmental and genetic perturbations, by utilizing biological networks, in which nodal components, such as, DNA, mRNAs, proteins, and metabolites, closely interact with each other. Systems biology investigates such systems by producing comprehensive global data that represent different levels of biological information, i.e., at the DNA, mRNA, protein, or metabolite levels, and by integrating this data into network models that generate coherent hypotheses for given biological situations. This review presents a systems biology framework, called the 'Integrative Proteomics Data Analysis Pipeline' (IPDAP), which generates mechanistic hypotheses from network models reconstructed by integrating diverse types of proteomic data generated by mass spectrometry-based proteomic analyses. The devised framework includes a serial set of computational and network analysis tools. Here, we demonstrate its functionalities by applying these tools to several conceptual examples.     

ePlant and the 3D data display initiative: integrative systems biology on the world wide web

Visualization tools for biological data are often limited in their ability to interactively integrate data at multiple scales. These computational tools are also typically limited by two-dimensional displays and programmatic implementations that require separate configurations for each of the user's computing devices and recompilation for functional expansion. Towards overcoming these limitations we have developed "ePlant" (http://bar.utoronto.ca/eplant) - a suite of open-source world wide web-based tools for the visualization of large-scale data sets from the model organism Arabidopsis thaliana. These tools display data spanning multiple biological scales on interactive three-dimensional models. Currently, ePlant consists of the following modules: a sequence conservation explorer that includes homology relationships and single nucleotide polymorphism data, a protein structure model explorer, a molecular interaction network explorer, a gene product subcellular localization explorer, and a gene expression pattern explorer. The ePlant's protein structure explorer module represents experimentally determined and theoretical structures covering >70% of the Arabidopsis proteome. The ePlant framework is accessed entirely through a web browser, and is therefore platform-independent. It can be applied to any model organism. To facilitate the development of three-dimensional displays of biological data on the world wide web we have established the "3D Data Display Initiative" (http://3ddi.org).     

The use of transepithelial models to examine host-pathogen interactions

Many pathogens must overcome an epithelial barrier in order to establish an infection. Unsurprisingly, such pathogens have evolved different mechanisms to overcome this obstacle, targeting specific epithelial structures or functions. These include disruption of epithelial barrier function, transcytosing from the apical to the basolateral membrane domain or inducing cell movement such as neutrophil recruitment. When studying these processes in vivo, animal models often fail to mimic the disease observed in humans and present a complex system in which many variables cannot be controlled. Therefore, in vitro transepithelial models that permit the study of a relevant biological surface have been developed, to integrate not only interactions between bacteria and epithelial cells but also, under certain conditions, to integrate a third cell type, such as neutrophils or dendritic cells. Such models are particularly useful for studying the bacteria-host relationship as it would occur in the microenvironment of the human epithelium and have enhanced our understanding of the unique strategies by which pathogenic bacteria exploit host cells to overcome the initial epithelial hurdle.     

Stem Cell Niche Structure as an Inherent Cause of Undulating Epithelial Morphologies

The spatial organization of stem cells into a niche is a key factor for growth and continual tissue renewal during development, sustenance, and regeneration. Stratified epithelia serve as a great context to study the spatial aspects of the stem cell niche and cell lineages by organizing into layers of different cell types. Several types of stratified epithelia develop morphologies with advantageous, protruding structures where stem cells reside, such as rete pegs and palisades of Vogt. Here, multistage, spatial cell lineage models for epithelial stratification are used to study how the stem cell niche influences epithelial morphologies. When the stem cell niche forms along a rigid basal lamina, relatively regular morphologies are maintained. In contrast, stem cell niche formation along a free-moving basal lamina may prompt distorted epithelial morphologies with stem cells accumulating at the tips of fingerlike structures that form. The correspondence between our simulated morphologies and developmental stages of the human epidermis is also explored. Overall, our work provides an understanding of how stratified epithelia may attain distorted morphologies and sheds light on the importance of the spatial aspects of the stem cell niche.     

Boosting Probabilistic Graphical Model Inference by Incorporating Prior Knowledge from Multiple Sources

Inferring regulatory networks from experimental data via probabilistic graphical models is a popular framework to gain insights into biological systems. However, the inherent noise in experimental data coupled with a limited sample size reduces the performance of network reverse engineering. Prior knowledge from existing sources of biological information can address this low signal to noise problem by biasing the network inference towards biologically plausible network structures. Although integrating various sources of information is desirable, their heterogeneous nature makes this task challenging. We propose two computational methods to incorporate various information sources into a probabilistic consensus structure prior to be used in graphical model inference. Our first model, called Latent Factor Model (LFM), assumes a high degree of correlation among external information sources and reconstructs a hidden variable as a common source in a Bayesian manner. The second model, a Noisy-OR, picks up the strongest support for an interaction among information sources in a probabilistic fashion. Our extensive computational studies on KEGG signaling pathways as well as on gene expression data from breast cancer and yeast heat shock response reveal that both approaches can significantly enhance the reconstruction accuracy of Bayesian Networks compared to other competing methods as well as to the situation without any prior. Our framework allows for using diverse information sources, like pathway databases, GO terms and protein domain data, etc. and is flexible enough to integrate new sources, if available.     

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.     

Cellular sociology regulates the hierarchical spatial patterning and organization of cells in organisms

Advances in single-cell biotechnology have increasingly revealed interactions of cells with their surroundings, suggesting a cellular society at the microscale. Similarities between cells and humans across multiple hierarchical levels have quantitative inference potential for reaching insights about phenotypic interactions that lead to morphological forms across multiple scales of cellular organization, namely cells, tissues and organs. Here, the functional and structural comparisons between how cells and individuals fundamentally socialize to give rise to the spatial organization are investigated. Integrative experimental cell interaction assays and computational predictive methods shape the understanding of societal perspective in the determination of the cellular interactions that create spatially coordinated forms in biological systems. Emerging quantifiable models from a simpler biological microworld such as bacterial interactions and single-cell organisms are explored, providing a route to model spatio-temporal patterning of morphological structures in humans. This analogical reasoning framework sheds light on structural patterning principles as a result of biological interactions across the cellular scale and up.     

Multilevel complexity of calcium signaling: Modeling angiogenesis

Intracellular calcium signaling is a universal,evolutionary conserved and versatile regulator of cell biochemistry.The complexity of calcium signaling and related cell machinery can be investigated by the use of experimental strategies,as well as by computational approaches.Vascular endothelium is a fascinating model to study the specific properties and roles of calcium signals at multiple biological levels.During the past 20 years,live cell imaging,patch clamp and other techniques have allowed us to detect and interfere with calcium signaling in endothelial cells(ECs),providing a huge amount of information on the regulation of vascularization(angiogenesis) in normal and tumoral tissues.These data range from the spatiotemporal dynamics of calcium within different cell microcompartments to those in entire multicellular and organized EC networks.Beside experimental strategies,in silico endothelial models,specifically designed for simulating calcium signaling,are contributing to our knowledge of vascular physiol-ogy and pathology.They help to investigate and predict the quantitative features of proangiogenic events moving through subcellular,cellular and supracellular levels.This review focuses on some recent developments of computational approaches for proangiogenic endothelial calcium signaling.In particular,we discuss the creation of hybrid simulation environments,which combine and integrate discrete Cellular Potts Models.They are able to capture the phenomenological mechanisms of cell morphological reorganization,migration,and intercellular adhesion,with single-cell spatiotemporal models,based on reaction-diffusion equations that describe the agonist-induced intracellular calcium events.     

Fitting experimental data to models that use morphological data from public databases

Ideally detailed neuron models should make use of morphological and electrophysiological data from the same cell. However, this rarely happens. Typically a modeler will choose a cell morphology from a public database, assign standard values for R _a, C _m, and other parameters and then do the modeling study. The assumption is that the model will produce results representative of what might be obtained experimentally. To test this assumption we developed models of CA1 hippocampal pyramidal neurons using 4 different morphologies obtained from 3 public databases. The multiple run fitter in NEURON was used to fit parameter values in each of the 4 morphological models to match experimental data recorded from 19 CA1 pyramidal cells. Fits with fixed standard parameter values produced results that were generally not representative of our experimental data. However, when parameter values were allowed to vary, excellent fits were obtained in almost all cases, but the fitted parameter values were very different among the 4 reconstructions and did not match standard values. The differences in fitted values can be explained by very different diameters, total lengths, membrane areas and volumes among the reconstructed cells, reflecting either cell heterogeneity or issues with the reconstruction data. The fitted values compensated for these differences to make the database cells and experimental cells more similar electrotonically. We conclude that models using fully reconstructed morphologies need to be calibrated with experimental data (even when morphological and electrophysiological data come from the same cell), model results should be generated with multiple reconstructions, morphological and experimental cells should come from the same strain of animal at the same age, and blind use of standard parameter values in models that use reconstruction data may not produce representative experimental results. Action Editor: Steve Redman     

Computer programs for the analysis of cellular survival data

Four programs have been written to enable radiobiologists to build a computer data base of cellular dose-survival data, calculate cell survival with a correction for cell multiplicity at the time of irradiation, fit various survival models to the data by iteratively weighted least squares, and calculate the ratio of survival levels corresponding to specified doses or the ratio of doses that produce specified survival levels (e.g., oxygen enhancement ratio or relative biological effectiveness). The programs make plots of survival curves and data, and they calculate standard errors and confidence intervals of the fitted survival curve parameters and ratios. The programs calculate survival curves for the linear-quadratic, repair-saturation, single-hit multitarget, linear-multitarget, and repair-misrepair models of cell survival and have been designed to accommodate the addition of other survival models in the future. The programs can be used to compare the accuracy with which different models fit the data, determine if a difference in fit is statistically significant, and show how the estimated value of a survival curve parameter, such as the extrapolation number or the final slope, varies with the survival model. The repair of radiation-induced damage is analyzed in a novel way using these programs.     

Four-dimensional imaging with virtual reality to quantitatively explore jigsaw puzzle-like morphogenesis of Arabidopsis cotyledon pavement cells

In most dicotyledonous plants, leaf pavement cells exhibit complex jigsaw puzzle-like cell morphogenesis during leaf expansion. Although detailed molecular biological information and mathematical modeling of this jigsaw puzzle-like cell morphogenesis are now available, a full understanding of this process remains elusive. Recent reports have highlighted the importance of three-dimensional (3D) structures (i.e., anticlinal and periclinal cell wall) in understanding the mechanical models that describe this morphogenetic process. We believe that it is important to acquire 3D shapes of pavement cells over time, i.e., acquire and analyze four-dimensional (4D) information when studying the relationship between mechanical modeling and simulations and the actual cell shape. In this report, we have developed a framework to capture and analyze 4D morphological information of Arabidopsis thaliana cotyledon pavement cells by using both direct water immersion observations and computational image analyses, including segmentation, surface modeling, virtual reality and morphometry. The 4D cell models allowed us to perform time-lapse 3D morphometrical analysis, providing detailed quantitative information about changes in cell growth rate and shape, with cellular complexity observed to increase during cell growth. The framework should enable analysis of various phenotypes (e.g., mutants) in greater detail, especially in the 3D deformation of the cotyledon surface, and evaluation of theoretical models that describe pavement cell morphogenesis using computational simulations. Additionally, our accurate and high-throughput acquisition of growing cell structures should be suitable for use in generating in silico model cell structures.