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¹ Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford, OX1 3LB, UK² Bioengineering Institute, University of Auckland, Private Bag 92019, Auckland, New Zealand³ University Laboratory of Physiology, Parks Road, Oxford, OX1 3PT, UK

	Abstract

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
Bioengineering analyses of physiological systems use the computational solution of physical conservation laws on anatomically detailed geometric models to understand the physiological function of intact organs in terms of the properties and behaviour of the cells and tissues within the organ. By linking behaviour in a quantitative, mathematically defined sense across multiple scales of biological organization – from proteins to cells, tissues, organs and organ systems – these methods have the potential to link patient-specific knowledge at the two ends of these spatial scales. A genetic profile linked to cardiac ion channel mutations, for example, can be interpreted in relation to body surface ECG measurements via a mathematical model of the heart and torso, which includes the spatial distribution of cardiac ion channels throughout the myocardium and the individual kinetics for each of the approximately 50 types of ion channel, exchanger or pump known to be present in the heart. Similarly, linking molecular defects such as mutations of chloride ion channels in lung epithelial cells to the integrated function of the intact lung requires models that include the detailed anatomy of the lungs, the physics of air flow, blood flow and gas exchange, together with the large deformation mechanics of breathing. Organizing this large body of knowledge into a coherent framework for modelling requires the development of ontologies, markup languages for encoding models, and web-accessible distributed databases. In this article we review the state of the field at all the relevant levels, and the tools that are being developed to tackle such complexity. Integrative physiology is central to the interpretation of genomic and proteomic data, and is becoming a highly quantitative, computer-intensive discipline.

(Received 21 October 2003; ; first published online 5 November 2003)
Corresponding author P. Hunter: Bioengineering Institute, University of Auckland, Private Bag 92019, Auckland, New Zealand.  Email: p.hunter{at}auckland.ac.nz

	Introduction

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
Physiology has always been an integrative science concerned with understanding quantitatively how the structure and function of cells, tissues and organs explain the complex behaviour of living systems. During the first half of the 20th century physiologists, from Sherrington and Eccles to Hodgkin and Huxley, revealed the physical basis of human physiology all the way from cell biophysics to integrative control. In the last 50 years, however, the biological limelight has progressively focused on molecular biology, with its spectacular success in explaining mechanisms at the level of genes and proteins. In recent years this success has largely been based on the development of experimental techniques such as DNA sequencing, PCR, microarrays and confocal fluorescent imaging, and above all on the realization that high-throughput measurement coupled to comprehensive databases is just as important to quantitative science as the more traditional approach of hypothesis-driven research. The inevitable consequence of this success is an explosion of data at the subcellular level that are difficult to interpret in relation to the physiological behaviour of complex living organisms. One of the main challenges in physiology over the next 10 years is therefore the interpretation of the genome and ascribing physiological function to genes and proteins in the wider context of integrative systems (Hunter & Borg, 2003; Hunter et al. 2002).

The physical sciences, on the other hand, have for the past 200 years confronted nature's complexity with the development of mathematical models of natural phenomena. Our ability to understand complex fluid flow, for example, in order to design aircraft or forecast weather, is a testament to the physicists and mathematicians of the 19th and 20th centuries, who identified nature's physical conservation laws and developed the mathematical framework to describe them. The successful application of these laws to the solution of engineering problems has also advanced greatly in the past 50 years from the development of computers and numerical analysis [every new complex engineering device is, these days, designed with the aid of mathematical models and finite element (or similar) analysis].

The use of mathematical modelling in physiology gained prominence in the 1950s with the successful prediction by Hodgkin and Huxley of the speed of action potential propagation along a nerve fibre from cable theory coupled to models of ion channel conduction and gating kinetics. Other early successes in applying techniques from the physical sciences to physiological systems were the engineering analysis of blood flow in arteries using computational fluid dynamics, and orthopaedic stress-strain analysis using linear elasticity theory and finite element analysis. In both cases, however, the biological problems are not significantly different from other engineering problems – blood can be treated as a viscous Newtonian fluid like water, albeit with higher viscosity and some unusual characteristics at low shear rates, and bone behaves as an orthotropic linear elastic material, at least until one considers its ability to grow and remodel. It was not possible in these early applications of mathematics and engineering to biology, which pre-date modern molecular biology, to link physiological behaviour to molecular detail.

In this article we argue that the discipline of physiology should now embrace the new era of ‘computational physiology’ in which mathematicians and bioengineers will work alongside physiologists and molecular biologists to link the physiology of cells, tissues and organs to the growing genomic and proteomic databases. The time is right – genomic and proteomic data are routinely collected; it is clear that the timing and spatial location of gene expression is controlled by environmental factors conveyed to the transcription factors in the gene regulatory regions by complex and redundant pathways which can only be analysed with network models; the physiological significance of genetic diseases will only be understood by linking quantitative models of tissue and organ physiology with the signal transduction cascades, metabolic pathways and other cellular processes. However, the tools required for this so-called ‘Physiome Project’[the term ‘physiome’ comes from ‘physio’ (life) and ‘ome’ (as a whole) and is intended to convey a ‘quantitative description of physiological dynamics and functional behaviour of the intact organism] are significantly different from the tools of standard engineering analysis for a number of reasons. Biological materials are almost always inhomogeneous, anisotropic and exhibit non-linear behaviour, but nowadays even these characteristics are not unusual in engineering materials. Biological processes (such as signal transduction pathways) exhibit enormous complexity, often with extraordinary degrees of apparent redundancy, but again so do engineered systems such as the electronic circuits in a Boeing 777 aircraft. The really significant and unique characteristic of biological materials is their ability to grow and remodel in response to changing environments – determined partly by genes and partly by their physical environment. An important consequence is that structure and function are intimately linked in a way that no engineering material or system can emulate. Capturing these structure-function relationships in a computationally efficient manner is the key to successful computational physiology and requires models and software that are fundamentally different to those found in the engineering world.

Our ‘physiome’ theme is developed in several stages. We begin with a discussion of biological complexity and the sources of data for modellers at the molecular level. We briefly describe the framework of ontologies and markup language standards that are being developed for handling our knowledge of biological systems at all levels of biological organization from genes to organisms, integrating across species, age and pathological state. We then discuss progress in modelling biological systems, beginning with gene regulation and cell function and ending with anatomically and biophysically based models at the organ level. The heart and lungs are used to illustrate the use of models that integrate structure and function across multiple spatial scales. Throughout the paper we use the heart as our primary example of multiscale modelling because the heart has provided the first example of a ‘physiome model’ of an organ.

	Biological complexity

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
The new millennium began, appropriately enough, with the announcement in February 2001 that the Human Genome Project was close to achieving its goal (International Human Genome Mapping Consortium, 2001; Venter et al. 2001). Soon after, in April 2003, the completion of the full sequence was announced (Collins et al. 2003). As a result, immense opportunities for integrative physiology and systems biology have opened up. The challenge to interpret this vast volume of data at the genomic and proteomic levels in terms of function at higher levels is what modern physiology is about, and this is precisely what is meant by integration in biology. It is also, in part at least, what is meant by ‘systems biology’: the application of systems theory to the complexity of biological interactions at all levels. [This is what distinguishes the word ‘system’ in ‘systems biology’ from its long-established use in ‘systems physiology’. The latter refers to high levels (such as cardiovascular, respiratory, nervous, immune systems), whereas ‘systems biology’ can refer to systems theory analysis at any level, including for example gene networks, cell signalling cascades and metabolic pathways].

Biological complexity and the direction of causality

But why should we bother with such complexity? Having broken biological systems down into their smallest components, have we not begun to reach our goal of understanding, at least in principle? Wasn't the aim to simplify, not to ‘complexify’? Could we not now ‘compute’ the behaviour of living systems (Brenner, 1998) from the automatic and inevitable interactions of their component parts, starting with genes coding for proteins, proteins interacting to form pathways, in turn to form cells, tissues and organs – and so on up to the highest levels? There are at least two reasons why this is not possible. The first reason is that it is computationally impossible and would be lacking in insight (Noble, 2002a). Being able to reconstruct a process is not sufficient, by itself, to understand the process. There will be many parameters in the reconstruction equations that will be in need of explanation, and we will also want to achieve more general insight into how the process reacts to perturbations, including most importantly, those of disease states. The second reason is that causation does not simply run in one direction (i.e. upwards). Lower level events, such as gene expression, are controlled by higher level processes. Without understanding the higher level logic in its own right (Noble & Boyd, 1993), the lower level data will often be just that: masses and masses of unexplained data.

Levels of Selection

But do the higher levels have a logic? Or are they just the ‘lumbering robots’ (Dawkins, 1976) of gene-level selection, with about as much logic as a cloud formation? The issue of levels of selection is now the subject of important debates in the theory of biology (Williams, 1992; Keller, 1999; Gould, 2002; Krakauer, 2002) The question here is the level at which selection operates. If it really were the case that selection is entirely gene-orientated, then the ‘lumbering robots’ (i.e. the physiology of higher level function) would indeed be of limited value, even irrelevant to a general theory of biology. However, this gene-orientated view confuses several different processes. Genes are not simply physical stretches of DNA, just as the information on a magnetic surface is not the surface itself. Genes are the carriers of information that any part of an organism can read, information which is transmitted from one generation to another. This transmission is parsimonious in the sense that it is not necessary to code for everything that happens in the development and life of an organism. The properties of water, lipids and the rules of self-assembly, for example, must all be taken for granted. As a ‘book of life’ the genome is necessarily incomplete (Noble, 2002b). It may also be repetitive and redundant, though the extent to which junk DNA is functional, rather than truly redundant, is also an open and controversial issue.

Replicators and interactors

These considerations are usually expressed by distinguishing between replicators, containing information that gets copied from one generation to another, and interactors, such as macromolecules (even DNA itself, as a chemical substance, is in this sense an interactor), biochemical pathways, cells, organisms, and species and clades, which both carry the replicator information and are subject to selection. Since genes interact so that many genes contribute to any given higher-level function, and each higher-level function depends on many genes for its transmission, natural selection must act at the levels at which functionality appears and this will only very rarely depend on a single gene. One of the challenges for theoretical biology is to identify the levels at which different functions can be said to be expressed and therefore to identify the levels at which natural selection must operate. This is a challenge to which the Physiome Project can respond since, when it is complete enough, it should reveal the modular nature of gene interaction, and the extent of that modularity, at all levels.

Gene numbers and functions

Analysing biological complexity at various levels is therefore necessary. But there must be very many different ways of measuring and assessing complexity, some more appropriate to one level rather than another, with each level having its own criteria. One way to look at complexity from a genome level is to ask how many biological functions are available to evolution in any given genome. For a genome of 40 000 genes if we make the extreme assumption that only two genes are needed to define a function then the total number of possible functions would be 0.5 x 40 000 x 39 999 = 799,980,000. Even with this absurdly minimalist view, the number of possible combinations of effects is very large. With more realistic assumptions about the number of genes involved in each function, the figures are really huge: thus at 100 genes/function we obtain 1.5 e³⁰²; and if we allow for all possible combinations we get 2 e¹⁶⁶⁷¹³! This has two consequences that are relevant to complexity. The first is that with such huge potential combinations of gene product interactions, trying to understand the system purely at the genome level is clearly impossible. Nature can have used only a minute fraction of these combinations and we need to look at these ‘from above’ to identify them, i.e. we need to look at the genome through the eyes of higher levels, including physiology. Since only a minute fraction of possible interactions are used, this introduces another way of viewing the question of how contingent evolution has been. With such enormous possibilities, the chances of humans (or any other species) evolving may be exceedingly small. The counterargument would be that evolution really is constrained. But constrained by what? This must be constraint by the logic of life at a higher level. In both cases, though, we need to understand the higher-level logic, either as a constraint or as a chance historical success, or most probably both: chance and necessity interact.

The (imperfect) logic of life

We are a long way from being able to judge such broad questions. But it is worth noting here that we would not expect the ‘logic’ to be perfect from an engineering point of view. Evolution is a dynamic process and there is no reason to suppose that what has evolved is fully in equilibrium with its environment (though Gould's view of stasis in evolution comes close to this – see Williams, 1992, chapter 9). Moreover, evolution is an historical process, with all the accidents and dead ends that a blind journey will produce (see particularly chapter 6, ‘Historicity and Constraint’, in Williams, 1992). Genes are perhaps better viewed as prisoners of the successful physiological systems that carry them than the determinants of those systems. They simply, and necessarily, code what has worked well at the level of the interactors. ‘Worked well’ is the correct description. ‘Works best’ would require design that avoids the historical accidents (using existing structures for a new function, such as the formation of the mammalian middle ear from modified jaw and hyoid arch bones) and evolutionary traps (such as the convoluted path of the sperm ducts following descent of the testes, or the inversion of the retina). Ultimately, we must be able to account for the evolutionary history of functional development. As Jared Diamond has insisted (Diamond, 1993), physiology should re-connect with the mainstream biological theory of evolution.

Comparison of genomes

These calculations put into perspective the frequent comparisons of similarity of genomes. If we differ from species x by only 1% then, from a genome point of view, the differences look insignificant. Using the same calculations as above, however, we get a totally different perspective. Introducing less than 1% of 1% of a genome of 40,000, i.e. a single gene, doubles the available functions. The argument is remarkably simple. We still have all the possible functions of the original set of genes, and all of those again in combination with the new gene. If we differ from another species at the genome level by, say, 1% (i.e. 400 genes) then the potential differences at a functional level are truly enormous. Even a 0.1% difference corresponds to 40 genes and also a huge number of new possibilities. To these calculations, we have also to add the influence of the environment through its control of gene expression levels. Suppose, for example, that gene expression levels might vary for a given protein by around an order of magnitude. This would mean that, to the original 40 000, we can add a further 400 000 elements to play a part in combinations. That puts the nature–nurture argument in perspective, and the absurdity of trying to arrive at percentages for the influence of each. Of course, only a very small proportion of the possible combinations actually occur.

	Functional genomics and molecular physiology

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
As an example of such a comparison of genomes, how can the apparent complexity of the human body be understood in relation to a much smaller and simpler organism, for example the nematode worm C. elegans, which appears to have more than half the number of protein-coding genes but on the order of 10¹¹ fewer cells. [Each of the cell divisions taking place during development of the worm from fertilized zygote to mature adult has been described in exquisite detail, giving the complete lineage for the 1000 or so cells making up the adult worm (http://www.wormbase.org). While the relatively simple physiology of the worm can be understood in terms of the different cell types and their interactions, it is still far from understood in terms of its genome]. If the gene numbers predicted by the various genome projects prove to be accurate, then the data confirm the view that it is the enumeration of possible interactions between genes, rather than the absolute size of the genome, that determines biological complexity. It is therefore the regulation of these interactions which must be the major postgenomic focus (Claverie, 2001). [Although the number of bases in the human genome is determined, there is still much dispute over the number of protein-coding genes, and proposed gene numbers are estimates that are sensitive to the gene-finding algorithms used. However, the waters are further muddied by the observation of highly conserved sequences in the more than 99% of the genome which apparently does not code for proteins. Furthermore, homology between species is found in these non-coding regions suggesting a functional, possibly regulatory role for the so-called junk DNA (Dennis, 2002; Mattick, 2001)].

The enormous challenge lying ahead of the human genome project, and the ultimate goal of functional genomics, is to understand the role of the genome in development, normal physiology and, perhaps most significantly, in disease. New experimental technologies are accelerating the identification of patterns of gene expression that delineate tissue types and underlie pathology. Altered gene expression in response to changing stimuli or through disease gives rise to reprogramming and changing function within tissue. Examples in cardiac ventricular tissue that have received recent attention include the molecular changes underlying the sequence of structural and functional modifications during remodelling after myocardial infarction (Cohn et al. 2000; Sehl et al. 2000; Stanton et al. 2000) and those responsible for modified electrical properties generating cardiac arrhythmia (Antzelevitch et al. 2001; Noble, 2002c). Variation in gene expression between individuals gives rise to increased disease susceptibility or responsiveness to therapy and it is a major aim of functional genomics to unravel the combinations of environmental and genetic factors that underlie complex disease, which to this point have proved elusive. These examples highlight the position of functional genomics at the interface between molecular biology and physiology. [This has been recognized in recent titles for the quadrennial IUPS International Congress of Physiological Sciences, held in 2001 in Christchurch, New Zealand, under the banner ‘from Molecule to Malady’, and due to be held in San Diego in 2005 with the focus ‘from Genomes to Functions].

Progress on understanding the role of the genome in cell, tissue and organ function is a highly integrative problem, which requires significant computational input. The major emphasis is currently on genome (and proteome) informatics: databasing, annotation and so forth (Winslow & Boguski, 2003). In addition to sequencing and annotation projects for different species, genomic databases with a particular tissue or disease focus are now being developed, for example, the Cardiac Gene Expression knowledgebase (CaGE; http://www.cage.wbmei.jhu.edu), which aims to catalogue all genes expressed in human cardiac tissue (Bober et al. 2002). While this is of great importance for making genomic data available and usable, it is increasingly apparent that the challenge ahead is to give quantitative predictions to physiological outcomes from genomic knowledge. As more and more data accumulates, a computational framework is required which can reach down to the genomic level via modelling of regulatory processes; in particular gene regulatory networks, interactions with signalling and metabolic pathways and how these influence and effect higher level organization and function.

The trend in current biology is towards comprehensive, automated measurement techniques. The development of high-throughput technologies for the systematic measurement of gene transcripts, proteins and their interactions, coupled to advances in computational data processing, provides a method for probing the functional role of large numbers of genes. DNA microarrays have emerged as the pre-eminent technology for large scale parallel gene expression studies (see, for example, The Chipping Forecast II, 2002).

Gene expression profiling

Microarrays exploit the complementary binding properties of nucleic acids: fluorescently labelled gene transcripts (mRNA) isolated from a tissue sample are allowed to hybridize with molecules of known sequence (ESTs or cDNA) immobilized at particular locations in an array. The fluorescence intensity at a particular spot on the microarray reveals the amount of the gene transcript which has hybridized at that location and the ratio of intensities of red and green fluorescent labels on the same array provides a measure of the differential expression level of the corresponding genes between two samples. These data on the up- and down-regulation of genes can be collected across different tissue types, disease states and in response to different stimuli. Data collected on temporal gene expression profiles, by sampling mRNA at several different times following experimental perturbation, provide gene expression time series data which are crucial for the development of dynamic models of gene regulation.

Studies of human disease using microarrays for gene expression are increasingly common, as shown in several recent reviews on microarrays in cardiovascular research and medicine (Cook & Rosenzweig, 2002; Napoli et al. 2003). Disease and tissue-specific microarrays have been developed – for example, a cDNA ‘CardioChip’, comprising over 10 000 cardiovascular-related ESTs (Barrans et al. 2001; Barrans et al. 2002). Most of these studies aim to identify differentially expressed genes (Henriksen & Kotelevtsev, 2002; Simkhovich et al. 2003). Clustering gene expression profiles is a common starting point for the analysis of microarray data and is used to find functional groupings of genes (Eisen et al. 1998; Slonim, 2002). If the expression profiles of a set of genes are closely correlated over many different experimental conditions then this co-expression is taken as a possible indication of a similar functional role for the cluster of genes or, even, of shared regulatory control mechanisms. Similarly, principal component analysis (PCA) can be used to reduce the data to a few simple underlying expression patterns, in particular for periodic temporal patterns of gene expression (Alter et al. 2000; Holter et al. 2000). The ‘guilt–by association’ of new genes with genes of known function via clustering suggests new targets for study.

Data requirements for modelling regulatory networks

High-throughput gene expression techniques generate large amounts of data. The primary concern for the analysis and modelling of large scale data sets is the so -called ‘curse of dimensionality’: the parameter space for a model grows exponentially with the number of variables (genes). This makes finding appropriate parameter values for large scale gene network studies a major challenge. The key to this problem is the use of prior knowledge on interactions and components of the network which have already been identified, features common to biological networks such as sparseness (Jeong et al. 2000), and the use of simplified response functions in the model, can all be used to reduce the number of undetermined parameters. To fully characterize network behaviour it is necessary to sample gene expression profiles under many different combinations of inputs and perturbations, for example point mutations, gene deletions and over-expression, inhibition of translation by antisense RNA, and perturbation of signalling and metabolic pathways.

Localized information on gene expression, required for spatially extended models, can be obtained from immunofluorescence studies. Laser capture microdissection (LCM), which automates the dissection of specific cells and multicell structures which have first been identified under the microscope (Emmert-Buck et al. 1996; Mills et al. 2001), allows gene (and protein) profiling studies to be conducted at high spatial resolution. The quantity of mRNA which can be extracted from such small samples may not be sufficient for high quality microarray studies. For detection of low abundance transcripts quantitative reverse-transcription PCR (RT-PCR) can give high sensitivity measurements of transcript levels, with the added advantage that RT-PCR can measure absolute, rather than relative concentrations of RNA (Schnell & Mendoza, 1997). PCR primers are required for each gene of interest and so, as with microarrays, the technique cannot be used to analyse previously unidentified genes. Transcripts of unknown genes can be measured using Serial Analysis of Gene Expression (SAGE; Velculescu et al. 1995; Yamamoto et al. 2001; Patino et al. 2003), where large numbers of transcripts are counted and analysed efficiently by sequencing only short tags from each RNA molecule. These short sequences can usually identify the corresponding gene (as long as its sequence is known). All tags collected from a sample are assembled into a single molecule which is then amplified and sequenced, providing quantitative information on the amount of each transcript in the sample.

Proteomics and metabolomics

Gene expression profiles describe the transcriptional state of cells or tissue, but do not directly reflect protein abundance. Proteomics and metabolomics aim to provide an unbiased identification and quantification of all proteins and metabolites present in a biological sample. The proteome encompasses modifications of protein

molecules and interactions between proteins. Knowledge of the changing protein content of cells, and functional interactions between proteins is of primary importance in developing models of cellular function. Gene expression studies provide the link between the (relatively) static genome and the highly dynamic protein content of a cell, responding to environmental changes, signalling pathways, as well as normal homeostatic protein turnover. The relation between gene expression profiles (the transcriptional state), as measured by mRNA level, and net protein synthesis depends on translation and post-transcriptional modification rates, as well as on the rate of protein decay, whether the molecule is actively degraded or transformed.

Protein molecules are intrinsically less amenable to high-throughput studies than are nucleic acids due to their secondary and tertiary structure. Technologies for large scale proteomics analyses include separation techniques (2D-PAGE, Poly Acrylamide Gel Electrophoresis) and mass spectrometry. Microarray-based techniques are also available for protein studies, although these platforms are less developed than DNA microarrays, both for detecting protein abundance by using different affinity reagents (for example antibodies, as a multiplexed immunoassay) to bind to proteins selectively, and for protein–protein interactions (MacBeath, 2002), also studied in the Yeast Two-Hybrid system. The identification and measurement of interactions, both stable and transient, between biomolecules in vivo is probably the widest current knowledge gap, and greatest experimental challenge.

Issues and standards for gene expression profiling experiments

Important for the development of robust and predictive models is the quality of the data available, good data annotation, and a methodology for assessing and managing the sources of variability in the data. There remain many issues to do with data reproducibility, variability and experimental design for microarray experiments (Churchill, 2002; Speed, 2003), which we will not discuss here. Differences between nominally similar tissue samples reflecting normal biological variability need to be quantified to minimize false positives in the identification of differentially expressed genes. Modelling may be a useful tool for identifying the sources of robustness to normal biological variability in regulatory networks (Alon et al. 1999; von Dassow et al. 2000).

The details of experiments performed, on sample preparation and so on, have significant bearing on the interpretation of profiling studies. With this in mind, ontologies and languages for the annotation of gene expression and proteome data are being developed (Stoeckert et al. 2002; Taylor et al. 2003). Standards have been proposed for describing microarray experiments, MIAME (Minimum Information About a Microarray Experiment) by the Microarray Gene Expression Data society (MGED; http://www.mged.org), which in late 2002 were adopted by several major journals including Nature, The Lancet and Cell as requirements for publication of the results of microarray studies. Annotated data is required to be deposited in one of several public databases: for example Gene Expression Omnibus (GEO) at NCBI (http://www.ncbi.nlm.nih.gov) and ArrayExpress at EBI (http://www.ebi.ac.uk/arrayexpress), which conform to the MIAME standards. A mark-up language for data exchange between databases, MicroArray Gene Expression Markup Language (MAGE-ML; http://www.mged.sourceforge.net), has been developed by MGED and others, under the auspices of the Object Management Group (OMG; http://www.omg.org).

	Physiome ontologies and markup language standards

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
To link the rapidly growing knowledgebase of biological data into a physiome modelling framework, formal vocabularies need to be developed to reduce the growing heterogeneity of terms. This is especially important as models of physiological processes are developed that span multiple spatial scales (genes and proteins to cells, tissues and organs) and incorporate the parameter changes associated with disease. A formal representation of custom data structures in the many applicationspecific databases is also needed to provide a common interface between them. Standards must be developed to formalize the description of both experimental data and mathematical models of physiological processes. Ontologies that incorporate semantic descriptions of modelling concepts make the modelling environment richer and unambiguous. They also permit the integration of data from ontologies and databases in other areas of biological research and the building of software tools that interpret and use this information. [An ‘ontology’ is a taxonomy together with a set of domain-specific rules for linking objects within the taxonomy. For example, a taxonomy for the human musculo-skeletal system names all the bones, muscles, tendons and ligaments, etc. in the human body. The rules are typically biological constraints such as ‘the semimembranosus muscle originates from the ischial tuberosity and inserts into the posterior part of the medial condyle of the tibia’].

Levels of representation

The physiome modelling framework can be considered at three levels:

1 Biological– representation of the biological problem using terms from the biological domain

2 Mathematical– mathematical formulation of the biological problem

3 Computer based representation languages– representations of models in a formal language that is machine interpretable, and which represents computational abstractions of entities, mathematical relationships and rules for their interpretation.

The goal is to represent the details of these levels in a way that can be used to explore the meaning of ideas and observations across the levels. Biology is usually represented using biological terms and relationships in a natural language context. Mathematics is a language in which we can represent theories for the processes and structures that the biological terms and relationships describe. At the computer representation language level, both the biological and mathematical interpretations need to be represented in constructs that can be used in various computer applications, such as simulation and visualization tools, and data repositories.

The three levels of representation are intimately related. Biological concepts link to mathematical concepts and simulation architectures, which link to computer based representations. At each level there is a description of the model, and at each level we find information that feeds across into the other levels. An example; a biological concept of predator–prey interactions can be modelled mathematically as two coupled ordinary differential equations. The way we express this model makes a certain claim on the relationship of the rates of change of predators and prey. Simulation of the model helps us to understand the longterm behaviour of such a description, and allows us to measure data appropriate for validating the biological reality of the model.

The requirements for this multilevelled modelling framework to be effective are: (i) tools for experts to represent models and solutions within particular levels using the language and interfaces that are natural to them, and (ii) computer based representations of what is described using these tools.

One toolset under the first of these requirements that is being developed is a visual editor. Visual building blocks and interfaces are a very natural way for people to navigate libraries and to build and interpret models (see Fig. 1). To develop a framework for building toolsets, we need to have good solutions to the second requirement, i.e. computer based representations of what modellers are describing. To this end, various representation languages, including ontologies are being developed. The integration of ontologies into the physiome project will provide an unambiguous, and machine interpretable, representation of concepts across these levels of modelling, helping to communicate biological models through tools for building, sharing, interpreting, and visualization. One such representation language, CellML (http://www.cellml.org) has been developed over the last four years, and aims to represent models at the cellular level. The CellML language itself is a set of constructs that have elegant interpretations within both the computational simulation domain and the object orientated programming domain. As a modelling representation language, it is sufficiently generic to represent any mathematical representation of any biological model, not just cells, so serves as a base for the development of a more generalized modelling representation environment. The evolution of this generalized representation environment includes the integration of ontology data, which provides a machine-interpretable pathway through the levels of modelling, and the further development and integration of FieldML (http://www.bioeng.auckland.ac.nz/physiome/markup.php) which facilitates the representation of structural and continuum based information about biological and physical entities.

View larger version (19K): [in this window] [in a new window]   Figure 1.  Model representations A, conceptual: representation of the model through diagrams, natural language, or concepts from ontologies. B, mathematical: representation of the conceptual model using mathematical equations. C, instances: representation of mathematical equations as instances of CellML components connected together.

Ontologies are a vehicle for providing unambiguous descriptions of terms and their relationship to one another. To a computer scientist, they provide a formal framework for describing the properties and relationships of concepts that have both a formal logical foundation and a structure amenable to machine processing, interpretation and sharing. To a biologist or modeller, ontologies provide a thesaurus and structure for understanding and binding terms, ideas, data sets, and visualizations, etc.

Many different groups are constructing ontologies for various biological domains. One approach to integrating ontologies from different biological domains would be to generate a large composite ontology. However, this is not the intention of the IUPS project's ontological framework because the biological ontologies that currently exist do not form pieces of the same puzzle – they may have biology in common, but it usually stops there. There is no currently agreed core framework or methodology that can be used to guide the development of compatible domain specific ontologies, but there are efforts underway to promote such a platform. The Unified Medical Language System (UMLS), for example, is attempting to bring together various ontologies from different domains into a composite ontology that fully integrates these knowledgebases.

The current view on the IUPS project's ontological framework is shown in Fig. 2. Some new ontologies are being built from scratch while some existing ones will need to be extended and integrated as both a common framework and data source (i.e. a composite approach). The focus at present is to describe constructs for interpreting our computer based model representations within the biological and mathematical levels of modelling. The domains of modelling theory, and the ML library domains (Fig. 2) capture representations of mathematical relationships, model architectures, and component structure (both physical and abstract).

View larger version (19K): [in this window] [in a new window]   Figure 2.  Ontology domains in the physiome project Black boxes, ontology domains that the IUPS project needs to develop; green boxes, ontology domains in which the IUPS ontologies will share a common framework with others; red boxes, ontology domains that the IUPS project does not need to explicitly build.

View larger version (21K): [in this window] [in a new window]   Figure 3.  Ontological domains SOFG, Standards and Ontologies for Functional Genomics (http://www.sofg.org); GO, Gene Ontology (http://www.geneontology.org); MGED, Microarray Gene Expression Data (mged.sourceforge.net); BioPAX, Biological Pathways Exchange (http://www.biopax.org).

	Modelling gene function

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

View larger version (20K): [in this window] [in a new window]   Figure 4.  Hierarchy of network models for gene regulatory network modelling

One way to greatly simplify the mathematical representation of networks is to ignore the details of molecular interactions and focus instead on their outcomes, namely whether a gene is ‘on’ or ‘off’, according to whether its transcription level is above a given threshold. Regulation is then represented by logical operations (AND, NOR, etc.) on the gene expression states, according to whether interactions activate or repress transcription. Gene expression levels are logical variables which are updated synchronously according to a rule table (a set of if then instructions) which describes the logical operations representing the regulatory interactions between genes. A Boolean network thus represents a ‘wiring diagram’ for the gene network (Bolouri & Davidson, 2002; Davidson et al. 2002). The aim of a reverse engineering approach is to infer the logical rule table directly from data. Techniques have been proposed which demonstrate that in principle a Boolean network can be constructed from data using no prior knowledge (Liang et al. 1998).

Although the properties of Boolean networks are much simpler than their continuous variable counterparts, they retain many of the properties of networks that are important in terms of gene function. For example, steady states are achieved as the network settles down into a stationary state or a repeating pattern of states (oscillation). The appeal of this modelling framework is its simplicity, but much biological detail is clearly lost. In reality, gene expression levels recorded in time course microarray studies spend much of their time at ‘intermediate’ levels, rather than quickly saturating at maximal expression rate, or falling to negligible levels. Logical networks may therefore be a good modelling strategy when the data quality is poor, and where intermediate expression levels cannot be resolved.

Kinetic modelling

Systems of differential equations provide a very natural modelling framework for the kinetic behaviour of gene networks, which easily extends to encorporate stochastic effects on gene transcription (McAdams & Arkin, 1999; Kepler & Elston, 2001) and modelling of transport processes in spatially extended systems. For large networks, the ‘curse of dimensionality’ limits the possibility of inferring parameters of a model from data. A reverse engineering technique has been proposed for linear systems, motivated by the observation that gene networks are sparse (there are relatively very few regulatory interactions between genes) demonstrating that progress can be made even with limited data (Yeung et al. 2002).

For well characterized networks, in which the relevant genes have been identified and the wiring diagram determined, biophysically realistic nonlinear kinetic functions can be assumed for the regulatory interactions. This allows quantitative prediction of transcription rates etc. in response to perturbations of the network. Kinetic parameters can be determined for individual reactions (for example the binding of transcription factors, degradation rates of mRNAs, etc.) using time-course data from microarrays, or GFP reporter gene approaches (Ronen et al. 2002). Interactions between cells are critical in determining ptterns of differentiation into distinct tissue types. Models of spatially extended systems must therefore incorporate mechanisms for signalling between cells, transport of gene products as signalling molecules and signalling networks within cells (Mjolsness et al. 1991; von Dassow et al. 2000; Davidson et al. 2002).

Cellular modelling

The coupling of gene expression to tissue and whole organ function requires a number of intermediate models at physiologically based spatial scales. A number of groups have developed sophisticated representations of ion transport between subcellular compartments and via the membrane in, for example, cardiac (Noble et al. 1998 and Luo & Rudy, 1994), pulmonary (Smirnov & Aaronson, 1994), and smooth muscle (Yang et al. 2003) cell types. Based on systems of coupled ordinary differential equations, the development of these modelling techniques is illustrated below for the cardiac myocyte by considering their application to understanding excitation, contraction and metabolism.

	Multi-scale modelling in the heart

Top Abstract Introduction Biological complexity Functional genomics and... Physiome ontologies and markup... Modelling gene function Multi-scale modelling in the... Multi-scale modelling in the... Conclusions Appendix References

 
Modelling provides a quantitative framework for establishing the effect of molecular changes on the function of cells, tissues and whole organs. For the heart, modelling is already at an advanced stage: at the cellular level, where electrophysiological, excitation-contraction coupling, contractile and metabolic processes have been incorporated; at the tissue level, where structural details and conduction properties are included; to models describing the anatomy of the heart itself. Models of cardiac myocytes are now sufficiently sophisticated that the effects of up and down regulation of specific genes can in some cases be predicted. These models can characterize disease at the cellular level (e.g. altered protein expression levels or mutated ion channel function), which may then be studied in tissue and whole organ models to predict outcomes on whole organ function. (Hunter et al. 2001; Noble, 2002d).

Cardiac cell models

The key cellular processes which are currently characterized within the organ level modelling framework are ion channel electrophysiology, myofilament mechanics and cellular metabolism. The most advanced of these models are the ion channel models pioneered by Noble (DiFrancesco & Noble, 1985; Noble et al. 1998; Noble & Rudy, 2001) following the seminal modelling work of Hodgkin & Huxley (1952) on squid axon excitation. The Luo & Rudy (1991, 1994) models are alternative frameworks focusing on the physiological behaviours of premature stimulation and arrhythmogenic activity of the single myocyte. Their modelling framework has since been extended by Jafri et al. (1998), among others, to accommodate more complex calcium kinetics which are important for contraction coupling.

In parallel with electrophysiology, Wong (1972), Panerai (1980), and Guccione & McCulloch (1993) have developed specific cardiac models of active tension generation largely based on experimental data from inhomogeneous papillary preparations. Hunter et al. (1998) has published a model fitted from more current data measured from intact and skinned trabeculae and the isolated cardiac cell. This model consists of two components, the binding kinetics that regulate the maximum number of cross-bridges (the force-generating bonds formed between contractile proteins) and the phenomenological characterization of the binding and unbinding kinetics of the cross-bridges themselves.

The modelling of metabolic regulation in cardiac cells has, until recently, lagged behind the characterization of excitation and contraction. This is in part due to the immense complexity of what are highly integrated metabolic pathways which exhibit multistate nonlinear regulation via a large number of state variables (Smith et al. 2002). However, a number of recent studies focusing on regulation of ion transport (Smirnov & Aaronson, 1994; Michailova & McCulloch, 2001; Ushimaru & Fukushima, 2002) and contraction (Saucerman et al. 2003; Smith, 2003) in the myocyte indicate that a fully coupled excitation-contraction model regulated by metabolism will soon be developed.

These cellular-based models now encompass sufficient biophysical detail such that the mediation of specific membrane exchanger and pump ionic currents via gene expression levels can be directly accounted for and linked to specific pathologies (Snyders & Chaudhary, 1996; Clancy & Rudy, 2002; Winslow et al. 1998; Antzelevitch et al. 2001; Noble, 2002e; Mazhari et al. 2001). However, to conclusively link gene regulation with clinically relevant whole organ pathology requires a further step in the increasing spatial scale. A continuum framework supplies an effective means of representing the spatial properties of geometry, conductivity and stiffness tensors, and the spatial variation of different cell types and properties. As will be demonstrated in the following section, using cellular models of active function at grid points embedded in the continuum geometry, in combination with governing equations, and appropriate numerical solution methods, presents a powerful tool for linking function through a number of spatial scales.

Tissue modelling

The major physical processes operating in the heart at the level of the intact organ are (i) the mechanical deformation of the myocardium (ii) the fluid mechanics of blood flow in the atria, ventricles and coronaries (iii) electrical activation of the conducting system from the sino-atrial node to Purkinje fibres and the myocardium, and (iv) transport of metabolic substrates between the coronaries and the myocardium to support the energy demands of the working heart. [Heat flow associated with temperature gradients does not appear to be significant in the in -vivo heart – unlike the lungs where warm blood is in close proximity to cold air]. These tissue level processes are of course supported by a diverse range of physical processes at the cellular level, as discussed in earlier sections. In each case the physical process (mechanics, electrical current flow, etc.) is governed by well-established conservation laws. In the following sections we describe these laws and how they are linked to physical processes at the cell level.

Continuum fields

A key concept underlying all of these tissue-level equations is the concept of a continuum representation of spatially varying quantities. This is illustrated for the spatial variation of transmembrane potential in Fig. 5. There are about 10¹⁰ myocytes in the heart [a myocyte is about 100 µm long and 20 µm diameter, or .10^–5 mm³ volume and the human adult myocardium occupies a volume of about 4.10⁵ mm³]. Each cell has a slightly different resting potential (e.g. due to small differences in the expression of Na/K pumps) and these transmembrane potential differences become larger during activation, as illustrated for a one-dimensional row of cells in Fig. 5A. Since the cells are electrically coupled, the differences between adjacent cells are small and to a good approximation the potential field varies smoothly and continuously as shown in Fig. 5B. The spatial variation of transmembrane potential is called a field and is best represented mathematically by dividing the spatial domain into subdomains called finite elements, as shown in Fig. 5B. The potential field within an element is then modelled by interpolating values of the potential defined at the nodes using the basis functions shown in Fig. 5C. Since a node is shared between two elements the potential is continuous across element boundaries. If the spatial derivative of potential is also defined at the nodes and the set of basis functions is extended to include interpolation of the derivative (called ‘cubic Hermite’ basis functions), the finite element representation of the potential field can maintain continuity of its first derivative (and hence current) across the whole domain.

View larger version (13K): [in this window] [in a new window]   Figure 5.  Finite element representation of cellular potentials A, a one-dimensional row of myocytes, each with transmembrane potential shown below by the dots. B, the mean potential distribution is approximated by a smooth curve constructed from piecewise polynomials. The three sections separated by dashed lines are called finite elements.C, the field inside each element is given by an interpolation of potentials defined at the nodes (shown by the red dots).

Any number of such fields can coexist in the same physical domain. For example, the extracellular potential provides another dependent variable field governed by conservation laws and linked to the transmembrane potential in bidomain models of cardiac electrophysiology.

Model of ventricular geometry

The one-dimensional cubic Hermite basis functions illustrated in Fig. 5 can be extended to three dimensions and used to fit a finite element model of ventricular geometry, as shown in Fig. 6. Anatomically based models have now been developed for the dog (Nielsen et al. 1991), pig (Stevens & Hunter, 2003) and rabbit heart (Vetter & McCulloch, 2000).

View larger version (55K): [in this window] [in a new window]   Figure 6.  Geometric models of the heartA, finite element surfaces fitted to measurements from the left and right ventricles of the pig heart. B, 3-D finite element model of the heart. The elements use high order basis functions (cubic Hermite) and therefore relatively few are required to provide an accurate description of ventricular anatomy. From Stevens & Hunter (2003) with permission.

In addition to the geometry, fundamental to predicting whole organ function, is an accurate representation of the spatial variation in tissue properties which are often based on the underlying microstructure. This fibrous-sheet structure of the heart is illustrated in Fig. 7. The ‘fibre’ axis is aligned with the local myocytes (i.e. the direction of sarcomere length changes), the sheet axis is orthogonal to the fibre axis, and is in the plane in which myocytes are tightly bound by endomysial collagen into layers 3–4 cells thick. The third ‘sheet-normal’ axis is orthogonal to this plane. The fibrous-sheet structure for the whole myocardium is illustrated in Fig. 8

View larger version (53K): [in this window] [in a new window]   Figure 7.  The microstructure of cardiac tissue A, schematic diagram showing the variation in muscle fibre direction across the wall in a segment removed from the left ventricle (top), and the branching laminar structure of myocardium in which the sheets are composed of myocytes bound in layers, 3–4 cells thick, by endomysial collagen and surrounded by perimysial collagen which also links to the adjacent sheet (bottom). This ‘fibrous-sheet’ architecture allows for shearing to occur between the layers and aids the process of wall thickening at end-systole. B, a composite 3D confocal image of a transmural block of myocardial tissue from the rat heart with a single image slice shown below (top), and a finite element model of the cleavage planes from the transmural tissue block used to study the flow of current and propagation of the tissue activation wavefront around the myocardial sheet structures (bottom). The spatial variation of tissue properties, such as the density of collagen, gap junctions and ion channels, etc. is defined by the markup language FieldML (http://www.bioeng.auckland.ac.nz/physiome/physiome.php). Figs A and B, from Hooks et al. (2002) with permission.

View larger version (50K): [in this window] [in a new window]   Figure 8.  Fibre orientation in the heart described by 3D finite element fields Fibres are shown on A, endocardial surface, B, midwall, and C, epicardial surface. From Stevens & Hunter (2003) with permission.

To predict whole organ function such as myocardial mechanics or activation requires the application of physical conservation laws which couple the discrete cellular models into a common spatial framework. These laws are expressed mathematically in the form of integral equations or PDEs (partial differential equations) which balance forces (expressed per unit area as stress) or fluxes (such as an electrical current). These equations are derived from the basic physical laws of nature and are independent of the particular properties which characterize the material under consideration (such as soft biological tissue). To solve these balance equations for a particular material requires another equation, called a material law or constitutive equation, which is derived from experimental observation of the particular material. For example, in the case of simulating myocardium contraction, to solve the stress equilibrium equations requires an additional material law linking components of stress to the components of material strain (deformation gradients). The experimentally obtained parameters in this relation are the elastic constants of the tissue. For conservation of current the material law links current to voltage gradients with parameters that express the tissue conductivity.

The systems of equations governing each function are typically non-linear and are solved using numerical methods which reflect the spatial-temporal scale of the underlying phenomena. The rapid upstroke in the action potential of the cardiac myocyte produces large spatial gradients which require representation at high spatial resolution (see Fig. 9). The relative efficiency of the finite difference technique or low order finite element method lends itself to this application. Conversely the continuous stress and strain fields generated by cardiac deformation are effectively predicted using finite elements with high order C1 continuous basis functions. The coupling of systems with large differences in the solution scales provides much of the challenge for modelling computational modelling at this scale in the context of the Physiome Project. Two examples of system coupled modelling are given below.

View larger version (74K): [in this window] [in a new window]   Figure 9.  Wave propagation in the heart Wavefront locations using an eikonal equation to simulate propagation from the distal ends of the Purkinje tree. For each sample time anterior (top) and posterior (bottom) views are given. A to H is 5 ms to 40 ms in 5 ms increments. The endocardial surfaces of the right and left ventricles can be seen through the translucent outer epicardial surface. From Tomlinson et al. (2002) with permission.

The electrophysiology model of Noble et al. (1998), linked via calcium kinetics to thin filaments kinetics and active force generation by Nickerson et al. (2001), provides the voltage source and tension dynamics at a cellular level. Stretch activated channels in the electrophysiological model produce a reverse coupling where mechanical stretch alters the cell action potential. Smith et al. (2003a) have embedded this cellular model in a grid of finite difference points in a two dimensional finite element model of the cardiac ventricles (see Fig. 10). Excitation was initiated by applying a spatial pattern of stimulus currents based on measurements of Durrer et al. (1970), inducing

View larger version (20K): [in this window] [in a new window]   Figure 10.  An anatomically accurate two dimensional model of coupled excitation-contraction in the cardiac ventricles The wave of cellular trans-membrane voltage (scaled between – 85 mV blue and + 45 mV red) is shown on the deforming tissue model. The undeformed finite element mesh to shown to provide a reference for tissue deformation.

Coupled myocardial mechanics and coronary blood flow

The anatomically-based model of the largest six generations of the coronary arterial tree generated by Smith et al. (2000) has been embedded at material points in the model of the cardiac ventricles presented above. By assuming the functional form of the axial velocity profile, the three dimensional Navier-Stokes equations governing blood flow have been reduced to one dimension to create an efficient model of coronary haemodynamics (Smith et al. 2002b). Using the myocardial mechanics framework of Nash & Hunter, (2000) the temporal compression and deformation of each vessel segment was determined. The effect of this spatial–temporal variation in intramyocardial stress was then coupled to blood flow via an elastic pressure radius relationship based on local arterial mechanics. This coupled model of myocardial mechanics and coronary blood flow has been used to quantify the effect of myocardial contraction on coronary blood flow (also known as systolic flow impediment) in the heart (Smith et al. 2002b), as illustrated in Fig. 11.

View larger version (43K): [in this window] [in a new window]   Figure 11.  Model of the coronaries The intramyocardial pressure exerted on to the embedded coronary vessels by myocardial contraction. Scaled between 0 KPa (blue) and 8 KPa (red) through the cardiac cycle.

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