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Catalyzing Inquiry at the Interface of Computing and Biology 5 Computational Modeling and Simulation as Enablers for Biological Discovery While the previous chapter deals with the ways in which computers and algorithms could support existing practices of biological research, this chapter introduces a different type of opportunity. The quantities and scopes of data being collected are now far beyond the capability of any human, or team of humans, to analyze. And as the sizes of the datasets continue to increase exponentially, even existing techniques such as statistical analysis begin to suffer. In this data-rich environment, the discovery of large-scale patterns and correlations is potentially of enormous significance. Indeed, such discoveries can be regarded as hypotheses asserting that the pattern or correlation may be important—a mode of “discovery science” that complements the traditional mode of science in which a hypothesis is generated by human beings and then tested empirically. For exploring this data-rich environment, simulations and computer-driven models of biological systems are proving to be essential. 5.1 ON MODELS IN BIOLOGY In all sciences, models are used to represent, usually in an abbreviated form, a more complex and detailed reality. Models are used because in some way, they are more accessible, convenient, or familiar to practitioners than the subject of study. Models can serve as explanatory or pedagogical tools, represent more explicitly the state of knowledge, predict results, or act as the objects of further experiments. Most importantly, a model is a representation of some reality that embodies some essential and interesting aspects of that reality, but not all of it. Because all models are by definition incomplete, the central intellectual issue is whether the essential aspects of the system or phenomenon are well represented (the term “essential” has multiple meanings depending on what aspects of the phenomenon are of interest). In biological phenomena, what is interesting and significant is usually a set of relationships—from the interaction of two molecules to the behavior of a population in its environment. Human comprehension of biological systems is limited, among other things, by that very complexity and by the problems that arise when attempting to dissect a given system into simpler, more easily understood components. This challenge is compounded by our current inability to understand relationships between the components as they occur in reality, that is, in the presence of multiple, competing influences and in the broader context of time and space.
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Catalyzing Inquiry at the Interface of Computing and Biology Different fields of science have traditionally used models for different purposes; thus, the nature of the models, the criteria for selecting good or appropriate models, and the nature of the abbreviation or simplification have varied dramatically. For example, biologists are quite familiar with the notion of model organisms.1 A model organism is a species selected for genetic experimental analysis on the basis of experimental convenience, homology to other species (especially to humans), relative simplicity, or other attractive attributes. The fruit fly Drosophila melanogaster is a model organism attractive at least in part because of its short generational time span, allowing many generations in the course of an experiment. At the most basic level, any abstraction of some biological phenomenon counts as a model. Indeed, the cartoons and block diagrams used by most biologists to represent metabolic, signaling, or regulatory pathways are models—qualitative models that lay out the connectivity of elements important to the phenomenon. Such models throw away details (e.g., about kinetics) implicitly asserting that omission of such details does not render the model irrelevant. A second example of implicit modeling is the use of statistical tests by many biologists. All statistical tests are based on a null hypothesis, and all null hypotheses are based on some kind of underlying model from which the probability distribution of the null hypothesis is derived. Even those biologists who have never thought of themselves as modelers are using models whenever they use statistical tests. Mathematical modeling has been an important component of several biological disciplines for many decades. One of the earliest quantitative biological models involved ecology: the Lotka-Volterra model of species competition and predator-prey relationships described in Section 5.2.4. In the context of cell biology, models and simulations are used to examine the structure and dynamics of a cell or organism’s function, rather than the characteristics of isolated parts of a cell or organism.2 Such models must consider stochastic and deterministic processes, complex pleiotropy, robustness through redundancy, modular design, alternative pathways, and emergent behavior in biological hierarchy. In a cellular context, one goal of biology is to gain insight into the interactions, molecular or otherwise, that are responsible for the behavior of the cell. To do so, a quantitative model of the cell must be developed to integrate global organism-wide measurements taken at many different levels of detail. The development of such a model is iterative. It begins with a rough model of the cell, based on some knowledge of the components of the cell and possible interactions among them, as well as prior biochemical and genetic knowledge. Although the assumptions underlying the model are insufficient and may even be inappropriate for the system being investigated, this rough model then provides a zeroth-order hypothesis about the structure of the interactions that govern the cell’s behavior. Implicit in the model are predictions about the cell’s response under different kinds of perturbation. Perturbations may be genetic (e.g., gene deletions, gene overexpressions, undirected mutations) or environmental (e.g., changes in temperature, stimulation by hormones or drugs). Perturbations are introduced into the cell, and the cell’s response is measured with tools that capture changes at the relevant levels of biological information (e.g., mRNA expression, protein expression, protein activation state, overall pathway function). Box 5.1 provides some additional detail on cellular perturbations. The next step is comparison of the model’s predictions to the measurements taken. This comparison indicates where and how the model must be refined in order to match the measurements more closely. If the initial model is highly incomplete, measurements can be used to suggest the particular components required for cellular function and those that are most likely to interact. If the initial model is relatively well defined, its predictions may already be in good qualitative agreement with measurement, differing only in minor quantitative ways. When model and measurement disagree, it is often 1 See, for example, http://www.nih.gov/science/models for more information on model organisms. 2 Section 5.1 draws heavily on excerpts from T. Ideker, T. Galitski, and L. Hood, “A New Approach to Decoding Life: Systems Biology,” Annual Review of Genomics and Human Genetics 2:343-372, 2001; and H. Kitano, “Systems Biology: A Brief Overview,” Science 295(5560):1662-1664, 2002.
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Catalyzing Inquiry at the Interface of Computing and Biology Box 5.1 Perturbation of Biological Systems Perturbation of biological systems can be accomplished through a number of genetic mechanisms, such as the following: High-throughput genomic manipulation. Increasingly inexpensive and highly standardized tools are available that enable the disruption, replacement, or modification of essentially any genomic sequence. Furthermore, these tools can operate simultaneously on many different genomic sequences. Systematic gene mutations. Although random gene mutations provide a possible set of perturbations, the random nature of the process often results in nonuniform coverage of possible genotypes—some genes are targeted multiple times, others not at all. A systematic approach can cover all possible genotypes and the coverage of the genome is unambiguous. Gene disruption. While techniques of genomic manipulation and systematic gene mutation are often useful in analyzing the behavior of model organisms such as yeast, they are not practical for application to organisms of greater complexity (i.e., higher eukaryotes). On the other hand, it is often possible to induce disruptions in the function of different genes, effectively silencing (or deleting) them to produce a biologically significant perturbation. SOURCE: Adapted from T. Ideker, T. Galitski, and L. Hood, “A New Approach to Decoding Life: Systems Biology,” Annual Review of Genomics and Human Genetics 2:343-372, 2001. necessary to create a number of more refined models, each incorporating a different mechanism underlying the discrepancies in measurement. With the refined model(s) in hand, a new set of perturbations can be applied to the cell. Note that new perturbations are informative only if they elicit different responses between models, and they are most useful when the predictions of the different models are very different from one another. Nevertheless, a new set of perturbations is required because the predictions of the refined model(s) will generally fit well with the old set of measurements. The refined model that best accounts for the new set of measurements can then be regarded as the initial model for the next iteration. Through this process, model and measurement are intended to converge in such a way that the model’s predictions mirror biological responses to perturbation. Modeling must be connected to experimental efforts so that experimentalists will know what needs to be determined in order to construct a comprehensive description and, ultimately, a theoretical framework for the behavior of a biological system. Feedback is very important, and it is this feedback, along with the global—or, loosely speaking, genomic-scale—nature of the inquiry that characterizes much of 21st century biology. 5.2 WHY BIOLOGICAL MODELS CAN BE USEFUL In the last decade, mathematical modeling has gained stature and wider recognition as a useful tool in the life sciences. Most of this revolution has occurred since the era of the genome, in which biologists were confronted with massive challenges to which mathematical expertise could successfully be brought to bear. Some of the success, though, rests on the fact that computational power has allowed scientists to explore ever more complex models in finer detail. This means that the mathematician’s talent for abstraction and simplification can be complemented with realistic simulations in which details not amenable to analysis can be explored. The visual real-time simulations of modeled phenomena give
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Catalyzing Inquiry at the Interface of Computing and Biology more compelling and more accessible interpretations of what the models predict.3 This has made it easier to earn the recognition of biologists. On the other hand, modeling—especially computational modeling—should not be regarded as an intellectual panacea, and models may prove more hindrance than help under certain circumstances. In models with many parameters, the state space to be explored may grow combinatorially fast so that no amount of data and brute force computation can yield much of value (although it may be the case that some algorithm or problem-related insight can reduce the volume of state space that must be explored to a reasonable size). In addition, the behavior of interest in many biological systems is not characterized as equilibrium or quasi-steady-state behavior, and thus convergence of a putative solution may never be reached. Finally, modeling presumes that the researcher can both identify the important state variables and obtain the quantitative data relevant to those variables.4 Computational models apply to specific biological phenomena (e.g., organisms, processes) and are used for a number of purposes as described below. 5.2.1 Models Provide a Coherent Framework for Interpreting Data A biologist surveys the number of birds nesting on offshore islands and notices that the number depends on the size (e.g., diameter) of the island: the larger the diameter d, the greater is the number of nests N. A graph of this relationship for islands of various sizes reveals a trend. Here the mathematically informed and uninformed part ways: simple linear least-squares fit of the data misses a central point. A trivial “null model” based on an equal subdivision of area between nesting individuals predicts that N~ d2, (i.e., the number of nests should be roughly proportional to the square of island area). This simple geometric property relating area to population size gives a strong indication of the trend researchers should expect to see. Departures from this trend would indicate that something else may be important. (For example, different parts of islands are uninhabitable, predators prefer some islands to others, and so forth.) Although the above example is elementary, it illustrates the idea that data are best interpreted within a context that shapes one’s expectations regarding what the data “ought” to look like; often a mathematical (or geometric) model helps to create that context. 5.2.2 Models Highlight Basic Concepts of Wide Applicability Among the earliest applications of mathematical ideas to biology are those in which population levels were tracked over time and attempts were made to understand the observed trends. Malthus proposed in 1798 the fitting of population data to exponential growth curves following his simple model for geometric growth of a population.5 The idea that simple reproductive processes produce 3 As one example, Ramon Felciano studied the use of “domain graphics” by biologists. Felciano argued that certain visual representations (known as domain graphics) become so ingrained in the discourse of certain subdisciplines of biology that they become good targets for user interfaces to biological data resources. Based on this notion, Felciano constructed a reusable interface based on the standard two-dimensional layout of RNA secondary structure. See R. Felciano, R. Chen, and R. Altman, “RNA Secondary Structure as a Reusable Interface to Biological Information Resources,” Gene 190:59-70, 1997. 4 In some cases, obtaining the quantitative data is a matter of better instrumentation and higher accuracy. In other cases, the data are not available in any meaningful sense of practice. For example, Richard Lewontin notes that the probability of survival Ps of a particular genotype is an ensemble property, rather than the property of a single individual who either will or will not survive. But if what is of interest is Ps as a function of the alternative genotypes deriving from a single locus, the effects of the impacts deriving from other loci must be randomized. However, in sexually reproducing organisms, there is no way known to produce an ensemble of individuals that are all identical with respect to a single locus but randomized over other loci. Thus, a quantitative characterization of Ps is in practice not possible, and no alternative measurement technologies will be of much value in solving this problem. See R. Lewontin, The Genetic Basis of Evolutionary Change, Columbia University Press, New York, 1974. 5 T.R. Malthus, An Essay on the Principle of Population, First Edition, E.A. Wrigley and D. Souden, eds., Penguin Books, Harmondsworth, England, 1798.
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Catalyzing Inquiry at the Interface of Computing and Biology exponential growth (if birth rates exceed mortality rates) or extinction (in the opposite case) is a fundamental principle: its applicability in biology, physics, chemistry, as well as simple finance, is central. An important refinement of the Malthus model was proposed in 1838 to explain why most populations do not experience exponential growth indefinitely. The refinement was the idea of the density-dependent growth law, now known as the logistic growth model.6 Though simple, the Verhulst model is still used widely to represent population growth in many biological examples. Both Malthus and Verhulst models relate observed trends to simple underlying mechanisms; neither model is fully accurate for real populations, but deviations from model predictions are, in themselves, informative, because they lead to questions about what features of the real systems are worthy of investigation. More recent examples of this sort abound. Nonlinear dynamics has elucidated the tendency of excitable systems (cardiac tissue, nerve cells, and networks of neurons) to exhibit oscillatory, burst, and wave-like phenomena. The understanding of the spread of disease in populations and its sensitive dependence on population density arose from simple mathematical models. The same is true of the discovery of chaos in the discrete logistic equation (in the 1970s). This simple model and its mathematical properties led to exploration of new types of dynamic behavior ubiquitous in natural phenomena. Such biologically motivated models often cross-fertilize other disciplines: in this case, the phenomenon of chaos was then found in numerous real physical, chemical, and mechanical systems. 5.2.3 Models Uncover New Phenomena or Concepts to Explore Simple conceptual models can be used to uncover new mechanisms that experimental science has not yet encountered. The discovery of chaos mentioned above is one of the clearest examples of this kind. A second example of this sort is Turing’s discovery that two chemicals that interact chemically in a particular way (activate and inhibit one another) and diffuse at unequal rates could give rise to “peaks and valleys” of concentration. His analysis of reaction-diffusion (RD) systems showed precisely what ranges of reaction rates and rates of diffusion would result in these effects, and how properties of the pattern (e.g., distance between peaks and valleys) would depend on those microscopic rates. Later research in the mathematical community also uncovered how other interesting phenomena (traveling waves, oscillations) were generated in such systems and how further details of patterns (spots, stripes, etc.) could be affected by geometry, boundary conditions, types of chemical reactions, and so on. Turing’s theory was later given physical manifestation in artificial chemical systems, manipulated to satisfy the theoretical criteria of pattern formation regimes. And, although biological systems did not produce simple examples of RD pattern formation, the theoretical framework originating in this work motivated later more realistic and biologically based modeling research. 5.2.4 Models Identify Key Factors or Components of a System Simple conceptual models can be used to gain insight, develop intuition, and understand “how something works.” For example, the Lotka-Volterra model of species competition and predator-prey7 is largely conceptual and is recognized as not being very realistic. Nevertheless, this and similar models have played a strong role in organizing several themes within the discipline: for example, competitive exclusion, the tendency for a species with a slight advantage to outcompete, dominate, and take over from less advantageous species; the cycling behavior in predator-prey interactions; and the effect of 6 P.F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” Correspondence Mathématique et Physique, 1838. 7 A.J. Lotka, Elements of Physical Biology, Williams & Wilkins Co., Baltimore, MD, 1925; V. Volterra, “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi,” Mem. R. Accad. Naz. dei Lincei., Ser. VI, Vol. 2, 1926. The Lotka-Volterra model is a set of coupled differential equations that relate the densities of prey and predator given parameters involving the predator-free rate of prey population increase, the normalized rate at which predators can successfully remove prey from the population, the normalized rate at which predators reproduce, and the rate at which predators die.
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Catalyzing Inquiry at the Interface of Computing and Biology resource limitations on stabilizing a population that would otherwise grow explosively. All of these concepts arose from mathematical models that highlighted and explained dynamic behavior within the context of simple models. Indeed, such models are useful for helping scientists to recognize patterns and predict system behavior, at least in gross terms and sometimes in detail. 5.2.5 Models Can Link Levels of Detail (Individual to Population) Biological observations are made at many distinct hierarchies and levels of detail. However, the links between such levels are notoriously difficult to understand. For example, the behavior of single neurons and their response to inputs and signaling from synaptic connections might be well known. The behavior of a large assembly of such neurons in some part of the central nervous system can be observed macroscopically by imaging or electrode recording techniques. However, how the two levels are interconnected remains a massive challenge to scientific understanding. Similar examples occur in countless settings in the life sciences: due to the complexity of nonlinear interactions, it is nearly impossible to grasp intuitively how collections of individuals behave, what emergent properties of these groups arise, or the significance of any sensitivity to initial conditions that might be magnified at higher levels of abstraction. Some mathematical techniques (averaging methods, homogenization, stochastic methods) allow the derivation of macroscopic statements based on assumptions at the microscopic, or individual, level. Both modeling and simulation are important tools for bridging this gap. 5.2.6 Models Enable the Formalization of Intuitive Understandings Models are useful for formalizing intuitive understandings, even if those understandings are partial and incomplete. What appears to be a solid verbal argument about cause and effect can be clarified and put to a rigorous test as soon as an attempt is made to formulate the verbal arguments into a mathematical model. This process forces a clarity of expression and consistency (of units, dimensions, force balance, or other guiding principles) that is not available in natural language. As importantly, it can generate predictions against which intuition can be tested. Because they run on a computer, simulation models force the researcher to represent explicitly important components and connections in a system. Thus, simulations can only complement, but never replace, the underlying formulation of a model in terms of biological, physical, and mathematical principles. That said, a simulation model often can be used to indicate gaps in one’s knowledge of some phenomenon, at which point substantial intellectual work involving these principles is needed to fill the gaps in the simulation. 5.2.7 Models Can Be Used as a Tool for Helping to Screen Unpromising Hypotheses In a given setting, quantitative or descriptive hypotheses can be tested by exploring the predictions of models that specify precisely what is to be expected given one or another hypothesis. In some cases, although it may be impossible to observe a sequence of biological events (e.g., how a receptor-ligand complex undergoes sequential modification before internalization by the cell), downstream effects may be observable. A model can explore the consequences of each of a variety of possible sequences can and help scientists to identify the most likely candidate for the correct sequence. Further experimental observations can then refine one’s understanding. 5.2.8 Models Inform Experimental Design Modeling properly applied can accelerate experimental efforts at understanding. Theory embedded in the model is an enabler for focused experimentation. Specifically, models can be used alongside experiments to help optimize experimental design, thereby saving time and resources. Simple models
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Catalyzing Inquiry at the Interface of Computing and Biology give a framework for observations (as noted in Section 5.2.1) and thereby suggest what needs to be measured experimentally and, indeed, what need not be measured—that is how to refine the set of observations so as to extract optimal knowledge about the system. This is particularly true when models and experiments go hand-in-hand. As a rule, several rounds of modeling and experimentation are necessary to lead to informative results. Carrying these general observations further, Selinger et al.8 have developed a framework for understanding the relationship between the properties of certain kinds of models and the experimental sampling required for “completeness” of the model. They define a model as a set of rules that maps a set of inputs (e.g., possible descriptions of a cell’s environment) to a set of outputs (e.g., the resulting concentrations of all of the cell’s RNAs and proteins). From these basic properties, Selinger et al. are able to determine the order of magnitude of the number of measurements needed to populate the space of all possible inputs (e.g., environmental conditions) with enough measured outputs (e.g., transcriptomes, proteomes) to make prediction feasible, thereby establishing how many measurements are needed to adequately sample input space to allow the rule parameters to be determined. Using this framework, Salinger et al. estimate the experimental requirements for the completeness of a discrete transcriptional network model that maps all N genes as inputs to all N genes as outputs in which the genes can take on three levels of expression (low, medium, and high) and each gene has, at most, K direct regulators. Applying this model to three organisms—Mycoplasma pneumoniae, Escherichia coli, and Homo sapiens—they find that 80, 40,000, and 700,000 transcriptome experiments, respectively, are necessary to fill out this model. They further note that the upper-bound estimate of experimental requirements grows exponentially with the maximum number of regulatory connections K per gene, although genes tend to have a low K, and that the upper-bound estimate grows only logarithmically with the number of genes N, making completeness feasible even for large genetic networks. 5.2.9 Models Can Predict Variables Inaccessible to Measurement Technological innovation in scientific instrumentation has revolutionized experimental biology. However, many mysteries of the cell, of physiology, of individual or collective animal behavior, and of population-level or ecosystem-level dynamics remain unobservable. Models can help link observations to quantities that are not experimentally accessible. At the scale of a few millimeters, Marée and Hogeweg recently developed9 a computational model based on a cellular automaton for the behavior of the social amoeba Dictyostelium discoideum. Their model is based on differential adhesion between cells, cyclic adenosine monophosphate (cAMP) signaling, cell differentiation, and cell motion. Using detailed two- and three-dimensional simulations of an aggregate of thousands of cells, the authors showed how a relatively small set of assumptions and “rules” leads to a fully accurate developmental pathway. Using the simulation as a tool, they were able to explore which assumptions were blatantly inappropriate (leading to incorrect outcomes). In its final synthesis, the Marée-Hogeweg model predicts dynamic distributions of chemicals and of mechanical pressure in a fully dynamic simulation of the culminating Dictyostelium slug. Some, but not all, of these variables can be measured experimentally: those that are measurable are well reproduced by the model. Those that cannot (yet) be measured are predicted inside the evolving shape. What is even more impressive: the model demonstrates that the system has self-correcting properties and accounts for many experimental observations that previously could not be explained. 8 D.W. Selinger, M.A. Wright, and G.M. Church, “On the Complete Determination of Biological Systems,” Trends in Biotechnology 21(6):251-254, 2003. 9 A.F.M. Marée and P. Hogeweg, “How Amoeboids Self-organize into a Fruiting Body: Multicellular Coordination in Dictyostelium discoideum,” Proceedings of the National Academy of Sciences 98(7):3879-3883, 2001.
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Catalyzing Inquiry at the Interface of Computing and Biology 5.2.10 Models Can Link What Is Known to What Is Yet Unknown In the words of Pollard, “Any cellular process involving more than a few types of molecules is too complicated to understand without a mathematical model to expose assumptions and to frame the reactions in a rigorous setting.”10 Reviewing the state of the field in cell motility and the cytoskeleton, he observes that even with many details of the mechanism as yet controversial or unknown, modeling plays an important role. Referring to a system (of actin and its interacting proteins) modeled by Mogilner and Edelstein-Keshet,11 he points to advantages gained by the mathematical framework: “A mathematical model incorporating molecular reactions and physical forces correctly predicts the steady-state rate of cellular locomotion.” The model, he notes, correctly identifies what limits the motion of the cell, predicts what manipulations would change the rate of motion, and thus suggests experiments to perform. While details of some steps are still emerging, the model also distinguishes quantitatively between distinct hypotheses for how actin filaments are broken down for purposes of recycling their components. 5.2.11 Models Can Be Used to Generate Accurate Quantitative Predictions Where detailed quantitative information exists about components of a system, about underlying rules or interactions, and about how these components are assembled into the system as a whole, modeling may be valuable as an accurate and rigorous tool for generating quantitative predictions. Weather prediction is one example of a complex model used on a daily basis to predict the future. On the other hand, the notorious difficulties of making accurate weather predictions point to the need for caution in adopting the conclusions even of classical models, especially for more than short-term predictions, as one might expect from mathematically chaotic systems. 5.2.12 Models Expand the Range of Questions That Can Meaningfully Be Asked12 For much of life science research, questions of purpose arise about biological phenomena. For instance, the question, Why does the eye have a lens? most often calls for the purpose of the lens—to focus light rays—and only rarely for a description of the biological mechanism that creates the lens. That such an answer is meaningful is the result of evolutionary processes that shape biological entities by enhancing their ability to carry out fitness-enhancing functions. (Put differently, biological entities are the result of nature’s engineering of devices to perform the function of survival; this perspective is explored further in Chapter 6.) Lander points out that molecular biologists traditionally have shied away from teleological matters, and that geneticists generally define function not in terms of the useful things a gene does, but by what happens when the gene is altered. However, as the complexity of biological mechanism is increasingly revealed, the identification of a purpose or a function of that mechanism has enormous explanatory power. That is, what purpose does all this complexity serve? As the examples in Section 5.4 illustrate, computational modeling is an approach to exploring the implications of the complex interactions that are known from empirical and experimental work. Lander notes that one general approach to modeling is to create models in which networks are specified in terms of elements and interactions (the network “topology”), but the numerical values that quantify those interactions (the parameters) are deliberately varied over wide ranges to explore the functionality of the network—whether it acts as a “switch,” “filter,” “oscillator,” “dynamic range adjuster,” “producer of stripes,” and so on. 10 T.D. Pollard, “The Cytoskeleton, Cellular Motility and the Reductionist Agenda,” Nature 422(6933):741-745, 2003. 11 A. Mogilner and L. Edelstein-Keshet, “Regulation of Actin Dynamics in Rapidly Moving Cells: A Quantitative Analysis,” Biophysical Journal 83(3):1237-1258, 2002. 12 Section 5.2.12 is based largely on A.D. Lander, “A Calculus of Purpose,” PLoS Biology 2(6):e164, 2004.
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Catalyzing Inquiry at the Interface of Computing and Biology Lander explains the intellectual paradigm for determining function as follows: By investigating how such behaviors change for different parameter sets—an exercise referred to as “exploring the parameter space”—one starts to assemble a comprehensive picture of all the kinds of behaviors a network can produce. If one such behavior seems useful (to the organism), it becomes a candidate for explaining why the network itself was selected; i.e., it is seen as a potential purpose for the network. If experiments subsequently support assignments of actual parameter values to the range of parameter space that produces such behavior, then the potential purpose becomes a likely one. 5.3 TYPES OF MODELS13 5.3.1 From Qualitative Model to Computational Simulation Biology makes use of many different types of models. In some cases, biological models are qualitative or semiquantitative. For example, graphical models show directional connections between components, with the directionality indicating influence. Such models generally summarize a great deal of known information about a pathway and facilitate the formation of hypotheses about network function. Moreover, the use of graphical models allows researchers to circumvent data deficiencies that might be encountered in the development of more quantitative (and thus data-intensive) models. (It has also been argued that probabilistic graphical models provide a coherent, statistically sound framework that can be applied to many problems, and that certain models used by biologists, such as hidden Markov models or Bayesian Networks), can be regarded as special cases of graphical models.14) On the other hand, the forms and structures of graphical models are generally inadequate to express much detail, which might well be necessary for mechanistic models. In general, qualitative models do not account for mechanisms, but they can sometimes be developed or analyzed in an automated manner. Some attempts have been made to develop formal schemes for annotating graphical models (Box 5.2).15 Qualitative models can be logical or statistical as well. For example, statistical properties of a graph of protein-protein interaction have been used to infer the stability of a network’s function against most “deletions” in the graph.16 Logical models can be used when data regarding mechanism are unavailable and have been developed as Boolean, fuzzy logical, or rule-based systems that model complex networks17 or genetic and developmental systems. In some cases, greater availability of data (specifically, perturbation response or time-series data) enables the use of statistical influence models. Linear,18 neural network-like,19 and Bayesian20 models have all been used to deduce both the topology of gene expression networks and their dynamics. On the 13 Section 5.3 is adapted from A.P. Arkin, “Synthetic Cell Biology,” Current Opinion in Biotechnology 12(6):638-644, 2001. 14 See, for example, Y. Moreau, P. Antal, G. Fannes, and B. De Moor, “Probabilistic Graphical Models for Computational Biomedicine, Methods of Information in Medicine 42(2):161-168, 2003. 15 K.W. Kohn, “Molecular Interaction Map of the Mammalian Cell Cycle: Control and DNA Repair Systems,” Molecular Biology of the Cell 10(8):2703-2734, 1999; I. Pirson, N. Fortemaison, C. Jacobs, S. Dremier, J.E. Dumont, and C. Maenhaut, “The Visual Display of Regulatory Information and Networks,” Trends in Cell Biology 10(10):404-408, 2000. (Both cited in Arkin, 2001.) 16 H. Jeong, S.P. Mason, A.L. Barabasi, and Z.N. Oltvai, “Lethality and Centrality in Protein Networks,” Nature 411(6833):41-42, 2001; H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, “The Largescale Organization of Metabolic Networks,” Nature 407(6804):651-654, 2000. (Cited in Arkin, 2001.) 17 D. Thieffry and R. Thomas, “Qualitative Analysis of Gene Networks,” pp. 77-88 in Pacific Symposium on Biocomputing, 1998. (Cited in Arkin, 2001.) 18 P. D’Haeseleer, X. Wen, S. Fuhrman, and R. Somogyi, “Linear Modeling of mRNA Expression Levels During CNS Development and Injury,” pp. 41-52 in Pacific Symposium on Biocomputing, 1999. (Cited in Arkin, 2001.) 19 E. Mjolsness, D.H. Sharp, and J. Reinitz, “A Connectionist Model of Development,” Journal of Theoretical Biology 152(4):429-453, 1999. (Cited in Arkin, 2001.) 20 N. Friedman, M. Linial, I. Nachman, and D. Pe’er, “Using Bayesian Networks to Analyze Expression Data,” Journal of Computational Biology 7(3-4):601-620, 2000. (Cited in Arkin, 2001.)
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Catalyzing Inquiry at the Interface of Computing and Biology Box 5.2 On Graphical Models A large fraction of today’s knowledge of biochemical or genetic regulatory networks is represented either as text or as cartoon-like diagrams. However, text has the disadvantage of being inherently ambiguous, and every reader must reinterpret the text of a journal article. Diagrams are usually informal, often confusing, and thus fail to present all of the information that is available to the presenter of the research. For example, the meanings of nodes and arcs within a diagram are inconsistent—one arrow may mean activation, but another arrow in the same diagram may mean transition of the state or translocation of materials. To remedy this state of affairs, a system of graphical representation should be powerful enough to express sufficient information in a clearly visible and unambiguous way and should be supported by software tools. There are several criteria for a graphical notation system, including the following: Expressiveness. The notation system should be able to describe every possible relationship among the entities in a system—for example, those between genes and proteins in a biological model. Semantical unambiguity. Notation should be unambiguous. Different semantics should be assigned to different symbols that are clearly distinguishable. Visual unambiguity. Each symbol should be identified clearly and not be mistaken with other symbols. This feature should be maintained with low-resolution displays, using only black and white. Extension capability. The notation system should be flexible enough to add new symbols and relationships in a consistent manner. This may include the use of color coding to enhance expressiveness and readability, but information should not be lost even with black-and-white displays. Mathematical translation. The notation should be able to convert itself into mathematical formalisms, such as differential equations, so that it can be applied directly for numerical analysis. Software support. The notation should be supported by software for its drawing, viewing, editing, and translation into mathematical formalisms. No current graphical notation system satisfies all of these criteria fully, although a number of systems satisfy some of them.1 SOURCE: Adapted by permission from H. Kitano, “A Graphical Notation for Biochemical Networks,” Biosilico 1(5):159-176. Copyright 2003 Elsevier. 1 See, for example, K.W. Kohn, “Molecular Interaction Map of the Mammalian Cell Cycle Control and DNA Repair Systems,” Molecular Biology of the Cell 10(8):2703-2734, 1999; K. Kohn, “Molecular Interaction Maps as Information Organizers and Simulation Guides,” Chaos 11(1):84-97, 2001. other hand, statistical influence models are not causal and may not lead to a better understanding of underlying mechanisms. Quantitative models make detailed statements about biological processes and hence are easier to falsify than more qualitative models. These models are intended to be predictive and are useful for understanding points of control in cellular networks and for designing new functions within them. Some models are based on power law formalisms.21 In such cases, the data are shown to fit generic power laws, and the general theory of power law scaling (for example) is used to infer some degree of causal structure. They do not provide detailed insight into mechanism, although power law models form the basis for a large class of metabolic control analyses and dynamic simulations. Computational models—simulations—represent the other end of the modeling spectrum. Simulation is often necessary to explore the implications of a model, especially its dynamical behavior, because 21 E.O. Voit and T. Radivoyevitch, “Biochemical Systems Analysis of Genomewide Expression Data,” Bioinformatics 16(11):1023-1037, 2000. (Cited in Arkin, 2001.)
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Catalyzing Inquiry at the Interface of Computing and Biology human intuition about complex nonlinear systems is often inadequate.22 Lander cites two examples. The first is that “intuitive thinking about MAP [mitogen-activated protein] kinase pathways led to the long-held view that the obligatory cascade of three sequential kinases serves to provide signal amplification. In contrast, computational studies have suggested that the purpose of such a network is to achieve extreme positive cooperativity, so that the pathway behaves in a switch-like, rather than a graded, fashion.”23 The second example is that while intuitive interpretations of experiments in the study of morphogen gradient formation in animal development led to the conclusion that simple diffusion is not adequate to transport most morphogens, computational analysis of the same experimental data led the opposite conclusion.24 Simulation, which traces functional biological processes through some period of time, generates results that can be checked for consistency with existing data (“retrodiction” of data) and can also predict new phenomena not explicitly represented in but nevertheless consistent with existing datasets. Note also that when a simulation seeks to capture essential elements in some oversimplified and idealized fashion, it is unrealistic to expect the simulation to make detailed predictions about specific biological phenomena. Such simulations may instead serve to make qualitative predictions about tendencies and trends that become apparent only when averaged over a large number of simulation runs. Alternatively, they may demonstrate that certain biological behaviors or responses are robust and do not depend on particular details of the parameters involved within a very wide range. Simulations can also be regarded as a nontraditional form of scientific communication. Traditionally, scientific communications have been carried by journal articles or conference presentations. Though articles and presentations will continue to be important, simulations—in the form of computer programs—can be easily shared among members of the research community, and the explicit knowledge embedded in them can become powerful points of departure for the work of other researchers. With the availability of cheap and powerful computers, modeling and simulation have become nearly synonymous. Yet, a number of subtle differences should be mentioned. Simulation can be used as a tool on its own or as a companion to mathematical analysis. In the case of relatively simple models meant to provide insight or reveal a concept, analytical and mathematical methods are of primary utility. With simple strokes and pen-and-paper computations, the dependence of behavior on underlying parameters (such as rate constants), conditions for specific dynamical behavior, and approximate connections between macroscopic quantities (e.g., the velocity of a cell) and underlying microscopic quantities (such the number of actin filaments causing the membrane to protrude) can be revealed. Simulations are not as easily harnessed to making such connections. Simulations can be used hand-in-hand with analysis for simple models: exploring slight changes in equations, assumptions, or rates and gaining familiarity can guide the best directions to explore with simple models as well. For example, G. Bard Ermentrout at the University of Pittsburgh developed XPP software as an evolving and publicly available experimental modeling tool for mathematical biologists.25 XPP has been the foundation of computational investigations in many challenging problems in neurophysiology, coupled oscillators, and other realms. Mathematical analysis of models, at any level of complexity, is often restricted to special cases that have simple properties: rectangular boundaries, specific symmetries, or behavior in a special class. Simulations can expand the repertoire and allow the modeler to understand how analysis of the special cases 22 A.D. Lander, “A Calculus of Purpose,” PLoS Biology 2 (6):e164, 2004. 23 C.Y. Huang and J.E. Ferrell, “Ultrasensitivity in the Mitogen Activated Protein Kinase Cascade,” Proceedings of the National Academy of Sciences 93(19):10078-10083, 1996. (Cited in Lander, “A Calculus of Purpose,” 2004.) 24 A.D. Lander, Q. Nie, and F.Y. Wan, “Do Morphogen Gradients Arise by Diffusion?” Developmental Cell 2(6):785-796, 2002. (Cited in Lander, 2004.) 25 See http://www.math.pitt.edu/~bard/xpp/xpp.html.
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Catalyzing Inquiry at the Interface of Computing and Biology Among the fundamental questions in the study of evolution are those that seek to know the relative strengths of natural selection, genetic drift, dispersal processes, and genetic recombination in shaping the genome of a population—essentially the forces that provide genetic variability in a species. Both ecologists and evolutionary biologists want to know how these forces lead to morphological changes, speciation, and ultimately, survival over time. The fields seek theory, models, and data that can account for genetic changes over time in large heterogeneous populations in which genetic information is exchanged routinely in an environment that also exerts its influence and changes over time. In addition to interest in genetic variability and fitness within a single species, the two fields are interested in relationships between multiple species. In ecology, this manifests itself in questions of how the individual forces of variability within and between species affect their relative ability to compete for resources and space that leads to their survival or extinction—in other words, forces that determines the biodiversity of an ecosystem (i.e., a set of biological organisms interacting among themselves and their environment). Ecologists want to understand what determines the minimum viable population size for a given population, the role of keystone species in determining the diversity of the ecosystem, and the role of diversity in preservation of the ecosystem. For evolutionary biologists, questions regarding relationships between species focus on trying to understand the flow of genetic information over long periods of time as a measure of the relatedness of different species and the effects of selection on the genetic contribution to phenotypes. Among the great mysteries for evolutionary biologists is whether and how evolution relates to organismal development, an interaction for which no descriptive language currently exists. How will ecologists and evolutionary biologists answer these questions? These fields have had few tools to monitor interactions in real time. But new opportunities have emerged in areas from genomics to satellite imaging and in new capabilities for the computer simulation of complex models. 184.108.40.206 Examples from Evolution A plethora of genomic data is beginning to help untangle the relationship between traits, genes, developmental processes, and environments. The data will serve as the substrate from which new statistical conclusions can be drawn, for example, new methods for identifying inherited gene sequences such as those related to disease. To answer question about the process of genome rearrangement, the possibility of comparing gene sequences from multiple organisms provides the basis for testing tools that discern repeatable patterns and elucidate linkages. As more detailed DNA and protein sequence information is compiled for more genes in more organisms, computational algorithms for estimating parameters of evolution have become extremely complex. New techniques will be needed to handle the likelihood functions and produce satisfactory statistics in a reasonable amount of time. Studies of the role of environmental and genetic plasticity in trait development will involve large-scale simulations of networks of linked genes and their interacting products. Such simulations may well suggest new approaches to such old problems as the nature-nurture dichotomy for human behaviors. New techniques and the availability of more powerful computers have also led to the development of highly detailed models in which a wide variety of components and mechanisms can be incorporated. Among these are individual unit models that attempt to follow every individual in a population over time, thereby providing insight into dynamical behavior (Box 5.22). Levin argues that such models are “imitation[s] of reality that represent at best individual realization of complex processes in which stochasticity, contingency, and nonlinearity underlie a diversity of possible outcomes.”109 From the collective behaviors of individual units arise the observable dynamics 109 S.A. Levin, B. Grenfell, A. Hastings, and A.S. Perelson, “Mathematical and Computational Challenges in Population Biology and Ecosystems Science,” Science 275(5298):334-343, 1997.
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Catalyzing Inquiry at the Interface of Computing and Biology Box 5.22 The Dynamics of Evolution Avida is a simulation software system developed at the Digital Life Laboratory at the California Institute of Technology.1 In it, digital organisms have genomes comprised of a sequence of instructions that operate on a virtual machine. These instructions include the ability to perform simple mathematical operations, copy values from memory location to memory location, provide input and output, and check conditions. Through a sequence of instructions, these organisms can copy their genome, thereby reproducing asexually. Since the software can simulate many hundreds of thousands of generations of evolution for thousands of organisms, their digital evolution not only can be observed in reasonable lengths of time, but also can be precisely inspected (since there are no inconvenient gaps in the fossil record). Moreover, alternate scenarios can be explored by going back into evolutionary history and reversing the effects of mutations, for example. At a minimum, this can be seen as experiment by analogy, revealing potential avenues for investigation or hypotheses to test in actual biological evolution. A stronger argument holds that evolution is an abstract mathematical process and will operate under similar dynamics whether embodied in DNA in the physical world or in digital simulations of it. Avida has been used to explore how complex features can arise through mutation, competition, and selective pressure.2 In a series of experiments, organisms were provided with a limited supply of energy units necessary for the execution of their genome of instructions. However, organisms that performed any of a set of complex logical operations were rewarded with an increased allowance and thus increased opportunities to reproduce. More complicated logical operations provided proportionally greater rewards. The experiment was seeded with an ancestral form that could perform none of those operations, containing only the instructions to reproduce. Mutation arose through imperfect copying of the genome during reproduction. EQU, the most complex logical operation checked for [representing the logical statement (A and B) or (~A and ~B)], arose in 23 out of 50 populations studied where the simpler operations also provided rewards. The sequence of instructions that evolved to perform the operation varied widely in length and implementation. However, in other simulations where only EQU was rewarded, no lineages ever evolved it. This evidence agrees with the standard theory of biological evolution—stated as early as Darwin—that complex structures arise through the combination and modification of useful intermediate forms. 1 C. Adami, Introduction to Artificial Life, Springer-Verlag, New York, 1998. 2 R.E. Lenski, C. Ofria, R.T. Pennock, and C. Adami, “The Evolutionary Origin of Complex Features,” Nature 423:139-144, 2003. of the system. “The challenge, then, is to develop mechanistic models that begin from what is understood about the interactions of the individual units, and to use computation and analysis to explain emergent behavior in terms of the statistical mechanics of ensembles of such units.” Such models must extrapolate from the effects of change on individual plants and animals to changes in the distribution of individuals over longer time scales and broader space scales and hence in community-level patterns and the fluxes of nutrients. 220.127.116.11.1 Reconstruction of the Saccharomyces Phylogenetic Tree Although the basic structure and mechanisms underlying evolution and genetics are known in principle, there are many complexities that force researchers into computational approaches in order to gain insight. Box 5.23 addresses complexities such as multiple loci, spatial factors, and the role of frequency dependence in evolution, and discusses a computational perspective on the evolution of altruism, a behavioral characteristic that is counterintuitive in the context of individual organisms doing all that they can to gain advantage in the face of selection pressures.
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Catalyzing Inquiry at the Interface of Computing and Biology Box 5.23 Genetic Complexities in Evolutionary Processes The dynamics of alleles at single loci are well understood, but the dynamics of alleles at two loci are still not completely understood, even in the deterministic case. As a rule, two-locus models require the use of a variety of computational approaches, from straightforward simulation to more complex analyses based on optimization or the use of computer algebra systems. Three-locus models can be understood only through numerical approaches, except for some very special cases. Compare these analytical capabilities to the fact that the number of loci exhibiting genetic variation in populations of higher organisms is well into the thousands. Thus, the number of possible genotypes can be much larger than the population. In such a situation, the detailed population simulation (i.e., a detailed consideration of events at each locus) leads to problems of substantial computational difficulty. An alternative is to represent the population as phenotypes—that is, in terms of traits that can be directly observed and described. For example, certain traits of individuals are quantitative in the sense that they represent the sum of multiple small effects. Efforts have been undertaken to integrate statistical models of the dynamics of quantitative traits with more mechanistic genetic approaches, though even under simplifying assumptions concerning the relation between genotype and phenotype, further approximations are required to obtain a closed system of equations. Frequency dependence in evolution refers to the phenomenon in which the fitness of an individual depends both on its own traits and on the traits of other individuals in the population—that is, selection is dependent on the frequency with which certain traits appear in the population, not just on pressures from the environment. This point arises most strongly in understanding how cooperation (altruism) can evolve through individual selection. The simplest model is the game of prisoner’s dilemma, in which the game-theoretic solution for a single encounter between parties is unconditional noncooperation. However, in the iterated prisoner’s dilemma, the game theoretic solution is a strategy known as “tit-for-tat,” which begins with cooperation and then uses the strategy employed by the other player in the previous interaction. (In other words, the iterated prisoner’s dilemma stipulates repeated interactions over time between players.) Although the iterated prisoner’s dilemma yields some insight into how cooperative behavior might emerge under some circumstances, it is a highly and perhaps oversimplified model. Most importantly, it does not account for possible spatial localizations of individuals—a point that is important in light of the fact that individuals who are spatially separated have low probabilities of interacting. Because the evolution of traits dependent on population frequency requires knowledge of which individuals are interacting, more realistic models introduce some explicit spatial distribution of individuals—and, for these, simulations are required to dynamical understanding. These more realistic models suggest that spatial localization affects the evolution of both cooperative and antagonistic behaviors. SOURCE: Adapted from S.A. Levin, B. Grenfell, A. Hastings, and A.S. Perelson, “Mathematical and Computational Challenges in Population Biology and Ecosystems Science,” Science 275(5298):334-343, 1997. (References in the original.)
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Catalyzing Inquiry at the Interface of Computing and Biology Along these lines, a particularly interesting work on the reconstruction of phylogenies was reported in 2003 by Rokas et al.110 One of the primary goals of evolutionary research has been understanding the historical relationships between living organisms—reconstruction of the phylogenetic tree of life. A primary difficulty in phylogenetic reconstruction is that different single-gene datasets often result in different and incongruent phylogenies. Such incongruences occur in analyses at all taxonomic levels, from phylogenies of closely related species to relationships between major classes or phyla and higher taxonomic groups. Many factors, both analytical and biological, may cause incongruence. To overcome the effect of some of these factors, analysis of concatenated datasets has been used. However, phylogenetic analyses of different sets of concatenated genes do not always converge on the same tree, and some studies have yielded results at odds with widely accepted phylogenies. Rokas et al. exploited genome sequence data for seven Saccharomyces species and for the outgroup fungus Candida albicans to construct a phylogenetic tree. Their results suggested that datasets consisting of a single gene or a small number of concatenated genes had a significant probability of supporting conflicting topologies, but that use of the entire dataset of concatenated genes resulted in a single, fully resolved phylogeny with the maximum likelihood. In addition, all alternative topologies resulting from single-gene analyses were rejected with high probability. In other words, even though the individual genes examined supported alternative trees, the concatenated data exclusively supported a single tree. They concluded that “the maximum support for a single topology regardless of method of analysis is strongly suggestive of the power of large data sets in overcoming the incongruence present in single-gene analyses.” 18.104.22.168.2 Modeling of Myxomatosis Evolution in Australia Evolution also provides a superb and easy-to-understand example of time scales in biological phenomena. Around 1860, a nonindigenous rabbit was introduced into Australia as part of British colonization of that continent. Since this rabbit had no indigenous foe, it proliferated wildly in a short amount of time (about 20 years). Early in the 1950s Australian authorities introduced a particular strain of virus that was deadly to the rabbit. The data indicated that in the short term (say, on a time scale of a few months), the most virulent strains of the virus were dominant (i.e., the virus had a lethality of 99.8 percent). This is not surprising, in the sense that one might expect virulence to be a measure of viral fitness. However, in the longer term (on a scale of decades), similar measurements indicate that these more virulent strains were no longer dominant, and the dominant niche was occupied by less virulent strains (lethality of 90 percent or less). The evolutionary explanation for this latter phenomenon is that an excessively virulent virus would run the risk of killing off its hosts at too rapid a rate, thereby jeopardizing its own survival. The underlying mechanism responsible for this counterintuitive phenomenon is that transmission of the virus depended on mosquitoes feeding from live rabbits. Rabbits that were infected with the more virulent variant died quickly, and thus, fewer were available as sources of that variant. The above system was modeled in closed form based on a set of coupled differential equations; this model was successful in reproducing the essential qualitative features described above.111 In 1990, this model was extended by Dwyer et al. to incorporate more biologically plausible features.112 For example, the evolution of rabbit and virus reacting to each other was modeled explicitly. A multiplicity of 110 A. Rokas, B.L. Williams, N. King, and S.B. Carroll, “Genome-scale Approaches to Resolving Incongruence in Molecular Phylogenies,” Nature 425(6960):798-804, 2003. 111 S. Levin and D. Pimentel, “Selection of Intermediate Rates of Increase in Parasite-Host Systems,” The American Naturalist 117(3), 1981. 112 G. Dwyer, S.A. Levin, and L.A. Buttel, “A Simulation Model of the Population Dynamics and Evolution of Myxomatosis,” Ecological Monographs 60(4):423-447, 1990.
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Catalyzing Inquiry at the Interface of Computing and Biology virus vectors was modeled, each with different transmission efficiencies, rather than assuming a single vector. The inclusion of such features, cou pled with exploitation of a wealth of data available on this system, allowed Dwyer et al. to investigate questions that could not be addressed in the earlier model. These questions included whether the system will continue to evolve antagonistically and whether the virus will be able to control the rabbit population in the future. More broadly, this example illustrates the important lesson that both time scales are equally significant from an evolutionary perspective, and one is not more “fundamental” than the other when it comes to understanding the dynamical behavior of the system. Furthermore, it demonstrates that pressures for natural selection can operate at many different levels of complexity. 22.214.171.124.3 The Evolution of Proteins By making use of simple physical models of proteins, it is possible to model evolution under different evolutionary, structural, and functional scenarios. For example, cubic lattice models of proteins can be used to model enzyme evolution involving binding to two hydrophobic substrates. Gene duplication coupled to subfunctionalization can be used to predict enzyme gene duplicate retention patterns and compare with genomic data.113 This type of physical modeling can be expanded to other evolutionary models, including those that incorporate positive selective pressures or that vary population genetic parameters. At a structural level, they can be used to address issues of protein surface-area-to-volume ratios or the evolvability of different folds. Ultimately, such models can be extended to real protein shapes and can be correlated to the evolution of different folds in real genomes.114 The role of structure in evolution during potentially adaptive periods can also be analyzed. A subset of positive selection will be dictated by structural parameters and intramolecular coevolution. Common interactions, like RKDE ionic interactions can be detected in this manner. Similarly, less common interactions, like cation-p interactions, can also be detected and the interconversion between different modes of interactions can be assessed statistically. One important tool underlying these efforts is the Adaptive Evolution Database (TAED), a phylogenetically organized database that gathers information related to coding sequence evolution.115 This database is designed to both provide high-quality gene families with multiple sequence alignments and phylogenetic trees for chordates and embryophytes and to enable answers to the question, “What makes each species unique at the molecular genomic level?” Starting with GenBank, genes have been grouped into families, and multiple sequence alignments and phylogenetic trees have been calculated. In addition to multiple sequence alignments and phylogenetic trees for all families of chordate and embryophyte sequences, TAED includes the ratio of nonsynonymous to synonymous nucleotide substitution rates (Ka/Ks) for each branch of every phylogenetic tree. This ratio, when significantly greater than 1, is an indicator of positive selection and potentially a change of function of the encoded protein in closely related species, and has been useful in the construction of phylogenetic trees with probabilistic reconstructed ancestral sequences calculated using both parsimony and maximum likelihood approaches. With a mapping of gene tree to species tree, the branches whose ratio is significantly greater than 1 are collated together in a phylogenetic context. 113 F.N. Braun and D.A. Liberles, “Retention of Enzyme Gene Duplicates by Subfunctionalization;” International Journal of Biological Macromolecules 33(1-3):19-22, 2003. 114 H. Hegyi, J. Lin, D. Greenbaum, and M. Gerstein, “Structural Genomics Analysis: Characteristics of Atypical, Common, and Horizontally Transferred Folds,” Proteins 47(2):126-141, 2002. 115 D.A. Liberles, “Evaluation of Methods for Determination of a Reconstructed History of Gene Sequence Evolution.” Molecular Biology and Evolution 18(11):2040-2047, 2001; D.A. Liberles, D.R. Schreiber, S. Govindarajan, S.G. Chamberlin, and S.A. Benner, “The Adaptive Evolution Database (TAED),” Genome Biology 2(8):research0028.1-0028.6, 2001; C. Roth, M.J. Betts, P. Steffansson, G. Sælensminde, and D.A. Liberles, “The Adaptive Evolution Database (TAED): A Phylogeny-based Tool for Comparative Genomics,” Nucleic Acids Research 33(Database issue):D495-D497, 2005.
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Catalyzing Inquiry at the Interface of Computing and Biology The TAED framework is expandable to incorporate other genomic-scale information in a phylogenetic context. This is important because coding sequence evolution (e.g., as reflected in the Ka/Ks ratio) is only one part of the molecular evolution of genomes driving phenotypic divergence. Changes in gene content116 and phylogenetic reconstructions of changes in gene expression and alternative splicing data117 can indicate where other significant lineage-specific changes have occurred. Altogether, phylogenetic indexing of genomic data presents a powerful approach to understanding the evolution of function in genomes. 126.96.36.199.4 The Emergence of Complex Genomes How did life get started on Earth? Today, life is based on DNA genomes and protein enzymes. However, biological evidence exists to suggest that in a previous era, life was based on RNA, in the sense that genetic information was contained in RNA sequences and phenotypes were expressed as catalytic properties of RNA.118 An interesting and profound issue is therefore to understand the transition from the RNA to the DNA world, one element of which is the fact that DNA genomes are complex structures. In 1971, Eigen found an explicit relationship between the size of a stable genome and the error rate inherent in its replication, specifically that the size of the genome was inversely proportional to the per-nucleotide replication error rate.119 Thus, for a genome of length L to be reasonably stable over successive generations, the maximum tolerable error rate in replication could be no more than 1/L per nucleotide. However, more precise replication mechanisms tend to be more complex. Given that the replication mechanism must itself be represented in the genome, the puzzle is that a precise replication mechanism is needed to maintain a complex genome, but a complex genome is required to encode such a mechanism. The only possible answer to this puzzle is that complex genomes evolved from simpler ones. Szabó et al. investigated this possibility through computer simulations.120 They constructed a population of digital genomes subject to evolutionary forces and found that under a certain set of circumstances, both genome size and replication fidelity increased with the run time of the simulation. However, such behavior was dependent on the existence of a sufficient amount of spatial isolation of the evolving population. In the absence of separation (i.e., in the limit of very rapid diffusion of genomes across the two-dimensional surface to which they were confined), genome complexity and replication fidelity were both limited. However, if diffusion is slow (i.e., the characteristic time constant of diffusion is less than the time scale of replication), both complexity and fidelity increase. In addition, Johnston et al. have synthesized in the laboratory a catalytic RNA molecule that contains about 200 nucleotides and synthesizes RNA molecules of up to 14 nucleotides, with an error rate of about 3 percent per residue.121 This laboratory demonstration, coupled with the computational finding described above, suggest that a small RNA genome that operates as an RNA replicase with 116 E.V. Koonin, N.D. Fedorova, J.D. Jackson, A.R. Jacobs, D.M. Krylov, K.S. Makarova, R. Mazumder, et al., “A Comprehensive Evolutionary Classification of Proteins Encoded in Complete Eukaryotic Genomes,” Genome Biology 5(2):R7, 2004. (Cited in Roth et al., “The Adaptive Evolution Database,” 2005.) 117 R. Rossnes, “Phylogenetic Reconstruction of Ancestral Character States for Gene Expression and mRNA Splicing Data,” M.Sc. thesis, Universtiy of Bergen, Norway, 2004. (Cited in Roth et al., 2005.) 118 See, for example, G.F. Joyce, “The Antiquity of RNA-based Evolution,” Nature 418(6894):214-221, 2002. 119 M. Eigen, “Selforganization of Matter and the Evolution of Biological Macromolecules,” Naturwissenschaften 58(10):465-523, 1971. 120 P. Szabó, I Scheuring, T. Czaran, and E. Szathmary, “In Silico Simulations Reveal That Replicators with Limited Dispersal Evolve Towards Higher Efficiency and Fidelity,” Nature 420(6913):340-343, 2002. A very helpful commentary on this article can be found in G.F. Joyce, “Molecular Evolution: Booting Up Life,” Nature 420(6894):278–279, 2002. The discussion in Section 188.8.131.52.4 is based largely on this article. 121 W.K. Johnston, P.J. Unrau, M.S. Lawrence, M.E. Glasner, and D.P. Bartel, “RNA-catalyzed RNA Polymerization: Accurate and General RNA-Templated Primer Extension,” Science 292(5520):1319-1325, 2001.
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Catalyzing Inquiry at the Interface of Computing and Biology modest efficiency and fidelity could evolve a succession of ever-larger genomes and ever-higher replication efficiencies. 184.108.40.206 Examples from Ecology122 Simulation-based study of an ecosystem considers the dynamic behavior of systems of individual organisms as they respond to each other and to environmental stimuli and pressures (e.g., climate) and examines the behavior of the ecosystem in aggregate terms. However, no individual run of such a simulation can be expected to predict the detailed behavior of each individual organism within an ecosystem. Rather, the appropriate test of a simulation’s fidelity is the extent to which it can, through a process of judicious averaging of many runs, predict features that are associated with aggregation at many levels of spatial and/or temporal detail. These more qualitative features provide the basis for descriptions of ecosystem dynamics that are robust across a variety of dynamical scenarios that are different at a detailed level and also provide high-level descriptions that can be more readily interpreted by researchers. Because of the general applicability of the approach described above, simulations of dynamical behavior can be developed for aggregations of any organisms as long as they can be informed by adequate understandings of individual-level behavior and the implications of such behavior for interactions with other individuals and with the environment. Note also the key role played by ecosystem heterogeneity. Spatial heterogeneity is one obvious way in which nonuniform distributions play a role. But in biodiversity, functional heterogeneity is also important. In particular, essential ecosystem functions such as the maintenance of fluxes of certain nutrients and pollutants, the mediation of climate and weather, and the stabilization of coastlines may depend not on the behavior of all species within the ecosystem but rather on a limited subset of these species. If biodiversity is to be maintained, the most fragile and functionally critical subsets species must be identified and understood. The mathematical and computational challenges range from techniques for representing and accessing datasets, to algorithms for simulation of large-scale spatially stochastic, multivariate systems, to the development and analysis of simplified description. Novel data acquisition tools (e.g., a satellite-based geographic information system that records changes for insertion in the simulations) would be welcome in a field that is relatively data poor. 220.127.116.11.1 Impact of Spatial Distribution in Ecosystems An important dimension of ecological environments is how organisms interact with each other. One often-made computationally simple assumption is that an organism is equally likely to interact with every other organism in the environment. Although this is a pragmatic assumption, actual ecosystems are physical and organisms interact only with a very small number of other organisms—namely, the ones that are nearby in a spatial sense. Moreover, localized selection—in which a fitness evaluation is undertaken only under nearest neighbors—is also operative. Introducing these notions increases the speciation rate tremendously, and the speculation is that in a nonlocalized environment, the pressures on the population tend toward population uniformity—everything looks similar, because each entity faces selection pressure from every other entity. When localization occurs, different species emerge in different spatial areas. Further, the individuals that are evolving will start to look quite different from each other, even though they have (comparably) high 122 Section 18.104.22.168 is based largely on material taken from S.A. Levin, B. Grenfell, A. Hastings, and A.S. Perelson, “Mathematical and Computational Challenges in Population Biology and Ecosystems Science,” Science 275(5298):334-343, 1997.
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Catalyzing Inquiry at the Interface of Computing and Biology fitness ratings. (This phenomenon is known as convergent evolution, in which a given environment might evolve several different species that are in some sense equally well adapted to that environment.) As an example of spatial localization, Kerr et al. developed a computational model to examine the behavior of a community consisting of three strains of E. coli,123 based on a modification of the lattice-based simulation of Durrett and Levin.124 One of the strains carried a gene that created an antibiotic called colicin. (The colicin-producing strain, C, was immune to the colicin it produced.) A second strain was sensitive to colicin (S), while a third strain was resistant to colicin (R). Furthermore, the factors that make the S strain sensitive also facilitate its consumption of certain nutrients, and the R strain is less able to consume these nutrients. However, because the R strain does not have to produce colicin, it avoids a metabolic cost incurred by the C strain. The result is that C bacteria kill S bacteria, S bacteria thrive where R bacteria do not, and R bacteria thrive where C bacteria do not. The community thus satisfies a “rock-paper-scissors” relationship. The intent of the simulation was to explore the spatial scale of ecological processes in a community of these three strains. It was found found (and confirmed experimentally) that when dispersal and interaction were local, patches of different strains formed, and these patches chased one another over the lattice—type C patches encroached on S patches, S patches displaced R patches and R patche invaded C patches. Within this mosaic of patches, the local gains made by any one type were soon enjoyed by another type; hence the diversity of the system was maintained. However, dispersal and interaction were no longer exclusively local (i.e., in the “well-mixed” case in which all three strains are allowed to interact freely with each other): continual redistribution of C rapidly drove S extinct, and R then came to dominate the entire community 22.214.171.124.2 Forest Dynamics125 To simulate the growth of northeastern forests, a stochastic and mechanistic model known as SORTIE has been developed to follow the fates of individual trees and their offspring. Based on species-specific information on growth rates, fecundity, mortality, and seed dispersal distances, as well as detailed, spatially explicit information about local light regimes, SORTIE follows tens of thousands of trees to generate dynamic maps of distributions of nine dominant or subdominant species of tree that look like real forests and match data observed in real forests at appropriate levels of spatial resolution. SORTIE predicts realistic forest responses to disturbances (e.g., small circles within the forest boundaries within which all trees are destroyed), clear-cuts (i.e., large disturbances), and increased tree mortality. SORTIE consists of two units that account for local light availability and species life history for each of nine tree species. Local light availability refers to the availability of light at each individual tree. This is a function of all of the neighboring trees that shade the tree in question. Information on the spatial relations among these neighboring tree crowns is combined with the movement of the sun throughout the growing season to determine the total, seasonally averaged light expressed as a percentage of full sun. In other words, the growth of any given tree depends on the growth of all neighboring trees. The species life history (available for each of nine tree species) provides the relationship between radial growth rates as a function of its local light environment and is based on empirically estimated life-history information. Radial growth predicts height growth, canopy width, and canopy depth in accordance with estimated allometric relations. Fecundity is estimated as an increasing power function of tree size, and seeds are dispersed stochastically according to a relation whereby the probability of 123 B. Kerr, M.A. Riley, M.W. Feldman, and B.J. Bohannan, “Local Dispersal Promotes Biodiversity in a Real-life Game of Rock-Paper-Scissors,” Nature 418(6894):171-174, 2002. 124 R. Durrett and S. Levin, “Allelopathy in Spatially Distributed Populations,” Journal of Theoretical Biology 185(2):165-171, 1997. 125 Section 126.96.36.199.2 is based largely on D.H. Deutschman, S.A. Levin, C. Devine, and L.A. Buttel, “Scaling from Trees to Forests: Analysis of a Complex Simulation Model,” Science Online supplement to Science 277(5332), 1997, available at http://www.sciencemag.org/content/vol277/issue5332. Science Online article available at http://www.sciencemag.org/feature/data/deutschman/home.htm.
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Catalyzing Inquiry at the Interface of Computing and Biology dispersal declines with distance. Mortality risk is also stochastic and has two elements: random mortality and mortality associated with suppressed growth. Because SORTIE is intended to aggregate statistical properties of forests, an ensemble of simulation runs is necessary, in which different degrees of smoothing and aggregation are used to determine how much information is lost by averaging and to find out where error is compressed and where it is enlarged in the course of this process. SORTIE is a computation-intensive simulation even for individual simulations, because multiple runs are needed to generate the necessary ensembles for statistical analysis. In addition, simulations carried out for heterogeneous environments require an interface between large dynamic simulations and geographic information systems, providing real-time feedbacks between the two. 5.5 TECHNICAL CHALLENGES RELATED TO MODELING A number of obstacles and difficulties must be overcome if modeling is to be made useful to life scientists more broadly than is the case today. The development of a sophisticated computational model requires both a conceptual foundation and implementation. Challenges related to conceptual foundations can be regarded as mathematical and analytical; challenges related to implementation can be regarded as computational or, more precisely, as related to computer science (Box 5.24). Today’s mathematical tools for modeling are limited. Nonlinear dynamics and bifurcation theory provide some of the most well-developed applied mathematical techniques and offer great successes in illuminating simple nonlinear systems of differential equations. But they are inadequate in many situations, as illustrated by the fact that understanding global stability in systems larger than four equations is prohibitively hard, if not unrealistic. Visualization of high-dimensional dynamics is still problematic in computational as well as analytical frameworks; the question remains as to how to represent such complex dynamics in the best, most easily understood ways. Moreover, many high-dimensional systems have effectively low-dimensional dynamics. A challenge is to extract the dynamical behavior from the equations without first knowing what the low-dimensional subspace is. Box 5.25 describes one new and promising approach to dealing with high-dimensional multiscale problems. Other mathematical methods and new theory will be needed to find solutions that apply not only to biological problems, but to other scientific and engineering applications as well. These include methods for global optimization and for reverse engineering of structure (of any “black box,” be it a network of genes, a signal transduction pathway, or a neuronal system) based on data elicited in response to stimuli and perturbations. Identification of model structure and parameters in nonlinear systems is also nontrivial. This is especially true in biological systems due to incomplete knowledge and essentially limitless types of interactions. Decomposition of complex systems into simpler subsystems (“modules”) is an important challenge to our ability to analyze and understand such systems (a point discussed in Chapter 6). Development of frameworks to incorporate moving boundaries and changing geometries or shapes is essential to describing biological systems. This is traditionally a difficult area. Ideally, it would be desirable to be able to synthesize and analyze models that have nonlinear deterministic as well as stochastic elements, and continuous as well as discrete, algebraic constraints, with other more traditional nonlinear dynamics. (See Section 5.3.2 for greater detail.) All of these can be viewed as challenges in nonlinear dynamics aspects of modeling. Further developing both computational (numerical simulation) methods and analytical methods (bifurcation, perturbation methods, asymptotic analysis) for large nonlinear systems will invariably mean great progress in the ability to build more elaborate and detailed models. However, with these large models come large challenges. One is how to find methodical ways of organizing parameter space exploration for systems that have numerous parameters. Another is the development of ways to codify and track assumptions that have gone into the construction of a model. Understanding these assumptions (or simplifications) is essential to understanding the limitations of a model and when its predictions are no longer biologically relevant.
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Catalyzing Inquiry at the Interface of Computing and Biology Box 5.24 Modeling Challenges for Computer Science Integration Methods Methods for integrating dissimilar mathematical models into complex and integrated overall models Tools for semantic interoperability Models High-performance, scalable algorithms for network analyses and cell modeling Methods to propagate measures of confidence from diverse data sources to complex models Validation Robust model and simulation-validation techniques (e.g., sensitivity analyses of systems with huge numbers of parameters, integration of model scales) Methods for assessing the accuracy of genome-annotation systems SOURCE: U.S. Department of Energy, Report on the Computer Science Workshop for the Genomes to Life Program, Gaithersburg, MD, March 6-7, 2002, available at http://DOEGenomesToLife.org/compbio/. Box 5.25 Equation-free Multiscale Computation: Enabling Microscopic Simulators to Perform System-level Tasks Yannis Kevrikides of Princeton University and his colleagues have developed a framework for computer-aided multiscale analysis. This framework enables models at a “fine” (microscopic, stochastic) level of description to perform modeling tasks at a “coarse” (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains: “patches” in macroscopic space-time. In general, traditional modeling approaches require the derivation of macroscopic equations that govern the time evolution of a system. With these equations in hand (usually partial differential equations (PDEs)), a variety of analytical and numerical techniques for their solution is available. The framework of Kevrikides and colleagues, known as the equation-free (EF) approach can, when successful, bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. The advantage of this approach is that the long-term behavior of the system bypasses the computationally intensive calculations needed to solve the PDEs that describe the system. That is, the EF approach enables an alternative description of the physics underlying the system at the microscopic scale (i.e., its behavior on relatively short time and space scales) provide information about the behavior of the system over relatively large time and space scales directly without expensive computations. In effect, the EF approach constitutes a systems identification-based, “closure on demand” computational toolkit, bridging microscopic-stochastic simulation with traditional continuum scientific computation and numerical analysis. SOURCE: The EF approach was first introduced by Yannis Kevrikides and colleagues in K. Theodoropoulos et al., “Coarse Stability and Bifurcation Analysis Using Timesteppers: A Reaction Diffusion Example,” Proceedings of the National Academy of Sciences 97:9840, 2000, available at http://www.pnas.org/cgi/reprint/97/18/9840.pdf. The text of this box is based on excerpts from an abstract describing a presentation by Kevrikides on April 16, 2003, to the Singapore-MIT Alliance program on High Performance Computation for Engineered Systems (HPCES); abstract available at http://web.mit.edu/sma/events/seminar/kevrekidis.htm.
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Catalyzing Inquiry at the Interface of Computing and Biology In the second category, issues related to implementing the model arise. Often such issues involve the actual code used to implement the model. Computational models are, in essence, large computer programs; issues of software development come to the fore. As the desire for and utility of computational modeling increase, the needs for software are growing rather than diminishing as hardware becomes more capable. On the other hand, progress in software development and engineering over the last several decades has not been nearly as dramatic as progress in hardware capability, and there appears to be no magic bullets on the horizon that will revolutionize the software development process. This is not to say that good software engineering does not or should not play a role in the development of computational models. Indeed, the Biomedical Information Science and Technology Initiative (BISTI) Planning Workshop of January 15-16, 2003, explicitly recommended that NIH require grant applications, proposing research in bioinformatics or computational biology to adopt as appropriate, accepted practices of software engineering.126 Section 4.5 describes some of the elements of good software engineering in the context of tool development, and the same considerations apply to model development. A second important challenge as large simulation models become more prevalent is a standard specification language to unambiguously specify the model, its parameters, annotations, and even the means by which it is to be scored against data. The challenge will be to provide a language flexible enough to capture all interesting biological processes and incorporate models at different levels of abstraction and in different mathematical paradigms, including stochastic differential, partial differential, algebraic, and discrete equations. It may prove necessary to develop a set of nested languages—for example, a language that specifies the biological process at a very high level and a linked language that specifies the mathematical representation of each process. There are some current attempts at these languages based on the XML framework. SBML and CellML are attempts in this direction. Finally, many biological modeling applications involve a problem space that is not well understood and may even be intended to explore queries that are not well formulated. Thus, there is a high premium on reducing the labor and time involved to produce an application that does something useful. In this context, technologies for “rapid prototyping” of biological models have considerable interest.127 126 See http://www.bisti.nih.gov/2003meeting/report.cfm. 127 Note, however, that in the rapid prototyping process often used to create commercial applications, there is a dialogue between developer and user that reveals what the user would find valuable: once the developer knows what the user really wants, the software development effort is straightforward. By contrast, in biological applications, it is nature that determines the appropriate structuring and formulation of a problem, and a problem cannot be structured in a certain way simply because it is convenient to do so. Thus, technologies for the rapid prototyping of biological models must afford the ability to rearrange model components and connections between components with ease.
Representative terms from entire chapter: