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Mathematics and 21st Century Biology 4 Understanding Cells INTRODUCTION To understand cells, one must understand the macromolecular structure of cells, the spatiotemporal patterns and mechanisms of cellular dynamics, the connections between cellular dynamics and cellular functions, and the connections between cells and higher levels of organization, such as tissues and organs. Understanding cells is intrinsically more difficult than understanding molecules because there is no cellular counterpart to the linear sequence of nucleotides and amino acids that provides much of the information necessary for predicting the structure and function of nucleic acids and proteins. Moreover, eukaryotic cells are highly compartmented and contain both a nuclear genome and one or more organellar genomes. With rapid increases in computational power and the sophistication of biophysical measurements, one can imagine constructing reasonably exact models of the dynamics of DNA and proteins, whereas all quantitative descriptions will still be approximate for cells. Thus, the main challenges for the mathematical analysis of cells are not computational but reside instead in the basic challenge of how to model features of interest. The primary challenge in this area for the next decade is the systematic formulation of reduced-order representations of cellular structure and dynamics, drawn from increasingly complex data and validated in model-driven experiments. There is a long and successful history of mathematical modeling of cellular functions. The success stories come from systems that are rich in data and for which models can be validated or at least put in direct corre-
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Mathematics and 21st Century Biology spondence with experiments. Models of the endocytic cycle, signal transduction cascades, and the cell division cycle have been productively used to organize data, extract “laws” for cellular processes, and even engineer cellular systems. Nevertheless, the models formulated over the past two decades build primarily on population-level measurements with poor spatial and temporal resolution. Without exception, the dynamics predicted by early models were richer than those predicted by the corresponding experimental data. For example, current models of cell cycle dynamics have more variables than can be measured experimentally. This experimental limitation is changing as a result of the rapid development of imaging techniques and high-throughput assays of cellular processes. The result should be an increased ability to evaluate models, which is the limiting step in improving them. It is now possible to collect multivariable and spatiotemporally resolved data on cellular processes ranging from molecular trafficking and signal transduction to integrated responses such as the cell division cycle and cell migration. In spite of these improvements in experimental capabilities, the quantitative models that emerge will not be as resolved and detailed as the models used in, for example, the aerospace and semiconductor industries. This difference reflects the intrinsic variability of biology, the immense spatiotemporal complexity of cells, and our incomplete knowledge of cellular processes. In many areas of the physical sciences, a coarse model can be very simple and yet fairly accurate, and one only needs to develop heterogeneous, multivariable, spatially resolved models when dealing with second- or higher-order effects. In contrast, even the simplest cellular models must be extracted from heterogeneous, multivariable, spatially resolved experimental data. Learning how to manage these data and mine them to extract computationally manageable models of cellular functions is the key challenge to quantitative understanding of cells. Exemplification of These Issues In the early 1980s, Steven Wiley and his colleagues formulated kinetic models of receptor-mediated ligand internalization (Wiley and Cunningham, 1981). The models were initially developed for the epidermal growth factor (EGF) receptor, a key regulator of cell and tissue functions across species (Wiley, 2003). The models described the kinetics of ligand-receptor binding, internalization, recycling, and degradation. The mathematical models were in the form of small systems of ordinary differential equations that were integrated in time and coupled with standard optimization and parameter estimation routines to extract the model parameters (Figure 4.1). Based on experiments with radioactively labeled EGF ligands, the
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Mathematics and 21st Century Biology FIGURE 4.1 Trafficking of ErbB receptor family. Only the epidermal growth factor receptor (EGFR) and ErbB2 proteins are shown for clarity, but the behaviors of ErbB3 and ErbB4 are similar to that of ErbB2. Activated EGFR and EGFR:ErbB2 heterodimers are internalized through a coated pit pathway, but other members of the ErbB family are probably internalized by a smooth pit pathway. The numbers next to the arrows represent the approximate mean time of the specific process. The time constants for heterodimerization and formation of multivesicular bodies are unknown. The mean time for lysosomal degradation is a combination of the time necessary for multivesicular body formation and for lysosomal fusion. Reprinted from Experimental Cell Research, 284, H.S. Wiley. Trafficking of ErbB receptors and its influence on signaling, pp. 78-88 (2003), with permission from Elsevier.
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Mathematics and 21st Century Biology models could extract the rate constants for different processes, such as endocytic uptake or recycling. Before this modeling effort, the only quantitative measures of ligand-receptor dynamics in cells were related to ligand-receptor interactions. The models of Wiley and colleagues produced new quantitative measures of ligand-receptor dynamics. In this way the biological effects of various ligands could be interpreted in terms of the quantitative differences of a larger number of rate constants for the multiple steps in the endocytic pathway. The model was also critical for suggesting the functional roles of the different parts of the EGF receptor. The EGF receptor is a large protein that combines multiple functions, including ligand binding, receptor phosphorylation, internalization, endocytic sorting, and recycling. By analyzing the ligand uptake data in cells that express mutant receptors and fitting these data to models, the functional roles of specific residues could be identified by changes in the rate constants of specific cellular processes (Wiley et al., 1991). This modeling and experimental approach has been validated by a number of experiments and used to parse the dynamics of internalization and trafficking for other ligand-receptor systems (Wiley et al., 2003). In the meantime, the experimental tools used to study these processes have changed. It is now possible to visualize multiple steps of ligand-receptor interactions and the endocytic cycle at the single-cell level and in real time (Sorkin et al., 2000). Furthermore, many new molecules have been identified in each of the steps of the endocytic cycle. For example, the recruitment of the ligand-bound receptor to the coated clathrin pits and the transfer of receptors to early endocytic compartments rely on tens of proteins. Protein-protein interactions in this system can be assayed using powerful biophysical techniques, and new components can be discovered by high-throughput proteomic approaches (Blagoev et al., 2003). In connection with this increased appreciation of the underlying molecular complexity of the system, it becomes necessary to rethink the mechanistic meaning of the endocytic rate constants predicted by the original model. How should the current model be changed to incorporate new data? Should the new models necessarily have more variables and parameters? Or, alternatively, can the old models be “parameterized” by new interactions? Given the structural complexity of living cells and the significant cell-to-cell variations, it is unlikely that useful models will account for every protein discovered in the endocytic cycle. But, for this and every other cellular system, it remains an open question how to use the new and much richer data to formulate the simplest model that can be used to correlate data and formulate new experiments. The main point is that, at this time, the data are richer than the models, which was not the case in the 1980s and 1990s.
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Mathematics and 21st Century Biology CELLULAR STRUCTURES In the same way that knowledge of the composition and structure of biomolecules is the key to understanding their biological function, detailed knowledge of the structure of cells is a prerequisite for the quantitative understanding of cellular functions. Mathematics plays an important role in characterizing the intracellular architecture (Figure 4.2). At one FIGURE 4.2 Applications for quantitative imaging. The image shows an XlK2 cell during the process of cytokinesis stained for DNA, microtubules, and the aurora-B protein kinase. Although the image demonstrates the relative localization of different cellular components and structures, quantitative analysis reveals specific characteristics that can be used to assay the effects of inhibitors on expressed proteins. For example, integrating the signal from a DNA-specific fluorophore might reveal defects in chromosome segregation during mitosis. Measuring the overlap of microtubules and aurora-B (using, for example, a cross-correlation analysis) within a subregion of a dividing cell might be a means of assessing effectors of cytokinesis. The image is displayed within the Open Microscopy Environment Image Viewer. The viewer includes support for displaying multidimensional image data (top left) and some of the associated metadata about each image (bottom right). SOURCE: Swedlow et al., 2003.
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Mathematics and 21st Century Biology level, the mathematical sciences have enabled progress through contributions to the development of instrumentation and other tools. For instance, the explosion of information about the spatiotemporal dynamics in cells would have not have been possible without earlier progress in imaging and data-processing algorithms. Tools such as deconvolution microscopy rely on robust numerical algorithms. Moreover, sophisticated data-processing algorithms have become more accessible to biologists through packages such as Matlab and Metamorph. New data-related challenges are emerging. For example, the assembly of large imaging datasets requires advances in bioinformatics and data mining. The informatic aspects of intracellular imaging are therefore receiving increased attention (Swedlow et al., 2003; Young et al., 2004). Quantitative imaging enables the formulation of data-driven models of intracellular dynamics and transport. The most notable examples include the quantitative analysis of the dynamics of Golgi to plasma membrane transport (Hirschberg et al., 1998) and nucleocytoplasmic shuttling (Smith et al., 2002). In both cases, green fluorescent protein (GFP)-based imaging provided data of unprecedented spatiotemporal resolution; nevertheless, it was possible to formulate simple compartmental models based on a small number of linear ordinary differential equations. Newer models enable the identification of the rate-limiting steps of the process and the formulation of testable hypotheses. Conclusions about the mechanisms in each case were based on the analysis of a small number of cells. It is unlikely that models of intracellular protein transport and trafficking will remain simple as knowledge of the processes grows and incorporates information on cell-to-cell variation. More sophisticated models that use nonlinear partial differential equations based on the geometry derived from imaging have been used to describe the intracellular dynamics of calcium and metabolites (Slepchenko et al., 2003). In each case, the main challenge in assessing the validity of quantitative predictions lies in careful analysis of the underlying assumptions, such as the use of Fickian diffusion to model the intracellular transport of proteins and small molecules. Recent years have witnessed the discovery of a large number of highly organized, coherent, dissipative structure in cells, including waves of intracellular calcium and metabolites and protein concentration waves accompanying the division of bacterial cells (Kindzelskii and Petty, 2002; Schuster et al., 2002). While the general phenomenology of these structures is understood from the standpoint of nonlinear dynamics and physicochemical pattern formation, how these structures arise and how they are maintained and used by cells is a topic of intense research. Mathematical analysis of these processes requires significant extensions of the theory
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Mathematics and 21st Century Biology and computational methods for control of spatially distributed nonlinear systems. In addition to studying and modeling intracellular structures, it is also important to model the possible mechanisms for their emergence from macromolecules. A number of experiments suggest that simple physicochemical principles can drive the emergence of cellular life (Szostak et al., 2001; Hanczyc et al., 2003; Chen et al., 2004). Indeed, experiments with nucleation of lipid vesicles by clays and the competition of protocells containing RNA polymerase suggest that the mechanisms of the formation of cells and intracellular compartments can be systematically studied in the test tube. Molecular simulations of these processes and population balance modeling of the evolution of primitive cellular compartments may provide the link between models at the molecular and cellular scales. DISCOVERY OF CELLULAR NETWORKS AND THEIR FUNCTIONS Since networks of interacting proteins control all cellular functions, understanding cellular functions requires quantitative analysis of the spatiotemporal dynamics of these networks in the cellular environment (Figure 4.3). Efforts to elucidate their dynamics can be subdivided into the analysis of network topology, dynamics, spatial organization, and function. Most of the progress recently has been in the area of deducing the network topology, using the classical techniques of cellular and molecular biology, large-scale molecular profiling experiments, or bioinformatics approaches (Brent and Finley, 1997; Chen and Xu, 2003; Ideker, 2004; Irish et al., 2004; Schulze and Mann, 2004; Xia et al., 2004; Yeger-Lotem et al., 2004). While there is significant room for the validation and perfection of each of these approaches, there is an urgent need to compare the networks predicted by distinct methods (Greenbaum et al., 2003). Because of the difficulties associated with generating high-quality data on cellular dynamics, much less work has been done in the analysis of network dynamics. For instance, some of the most interesting results associated with network dynamics have required construction of special experimental systems that include fluorescent reporters for a large number of bacterial genes coupled with high-resolution analysis of bacterial responses over a broad range of experimental conditions. This approach has led to the validation of the network motifs predicted on the basis of bioinformatics analysis and has identified the dynamic and functional roles of these motifs (Shen-Orr et al., 2002; Kalir and Alon, 2004; Zaslaver et al., 2004). The simplest use of mathematical models for intracellular networks is to integrate data and test if they fit together. For example, mechanistic
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Mathematics and 21st Century Biology FIGURE 4.3 Transcriptional regulatory networks and motifs. (A) HNF1α, HNF6, and HNF4α are at the center of tissue-specific transcriptional regulatory networks. In these examples selected for illustration, regulatory proteins and their gene targets are represented as circles and boxes, respectively. Solid arrows indicate protein-DNA interactions, and genes encoding regulators are linked to their protein products by dashed lines. The HNF4α1 promoter is poorly expressed in pancreatic islets and is shaded to reflect this. The HNF4α7 promoter, also known as the P2 promoter, is the predominant promoter in pancreatic islets and was recently implicated as an important locus for human diabetes susceptibility. For clarity, some gene promoters have been designated by the names of their protein products (e.g., HNF1α for TCF1, SHP for NR0B2, HNF4α7 for HNF4A P2, and HNF1β for TCF2). (B) Examples of regulatory network motifs in hepatocytes. For instance, in the multicomponent loop, HNF1α protein binds to the promoter of the HNF4α gene, and the HNF4α protein binds to the promoter of the HNF1α gene. These network motifs were uncovered by searching binding data with various algorithms; details on the algorithms used and a full list of motifs found are available. SOURCE: Odom et al., 2004.
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Mathematics and 21st Century Biology models of the cell division cycle in fission yeast summarize the dynamic behavior of multiple mutants and make testable experimental predictions (Tyson et al., 2001). Most of the “realistic” models of intracellular networks, including the widely publicized models of the lysis-lysogeny switch in bacteriophage lambda and models of growth factor signaling, were severely overparameterized (Arkin et al., 1998; Schoeberl et al., 2002). On the one hand, the models have to be large to be able to predict the effects of genetic or biochemical perturbations of network components, and on the other, the models must have the lowest required number of parameters and processes to explain the observed phenomenology. At this time, there are no standards for assessing the complexity of a model and whether it is indeed a minimal representation of data. Furthermore, current models are characterized by high levels of uncertainty, both structural and parametric. This situation is not surprising, given that even the most comprehensive models may be missing entire parts of a network, may neglect its spatial organization and temporal evolution, and may employ approximate functional forms for various cellular processes. While some tools for dealing with these issues can be borrowed from linear control systems (Csete and Doyle, 2002), new theoretical and computational approaches are required to analyze highly uncertain and nonlinear systems. These new approaches require solving problems in simulation, system identification, parameter estimation, and experimental design. Recently, robustness has emerged as an important principle for model screening and validation (Barkai and Leibler, 1997; Stelling et al., 2004). In a nutshell, the relative plausibility of two models for the same process can be assessed by comparing the size of the parametric perturbations that can be tolerated by the models without qualitatively changing the predicted behavior. The rationale for using robustness as a screen is that evolution seems to favor the most robust mechanisms. For example, two models of the cell division cycle can be compared on the basis of the size of the regions of the parameter space that predict the limit cycle behavior (Morohashi et al., 2002). Analysis of robustness requires tools that can be used to compare models with different numbers of parameters and even different mathematical structures. The method of mathematically controlled comparisons is a very important development in this direction (Alves and Savageau, 2000). On the experimental side, the model-driven analysis of the robustness of cellular systems requires quantitative characterization of natural and induced parameter variations in cells (Houchmandzadeh et al., 2002; Jones et al., 2004). Quantitative and multivariable analysis of cell-to-cell variations is becoming possible thanks to advances in flow cytometry and live cell imaging (Irish et al., 2004).
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Mathematics and 21st Century Biology The number of systems that have been characterized over the entire range from the biochemical description of the network to its dynamics and function is still very small. One of the best examples is the result of efforts by Ferrell and co-workers, who have analyzed the function of the mitogen-activated protein kinase (MAPK) network, a three-stage enzymatic cascade conserved from yeasts to humans. In an elegant sequence of papers, Ferrell et al. have shown that the multistage nature of the cascade enables it to act as a switch that is insensitive to small inputs at the top layer of the cascade and is fully activated when the threshold value for the input has been crossed (Huang and Ferrell, 1996; Ferrell, 1997; Ferrell and Machleder, 1998; Bagowski and Ferrell, 2001; Ferrell and Xiong, 2001; Xiong and Ferrell, 2003). This prediction, based on extensive computational analysis of the cascade model, has been validated through in vitro experiments with purified components of the network. The biochemical and modeling work set the stage for the analysis of MAPK dynamics in the frog oocyte maturation response. In this response system, it became apparent that the MAPK cascade is embedded in a positive feedback circuit that sharpens the threshold-detection capabilities of the circuit and mediates the irreversibility of the cell’s maturation response to hormones. This insight was enabled by the large size of the frog oocyte, which made it possible to carry out single-cell biochemical assays of cellular responses, once again underscoring the importance of examining cellular responses at a single-cell level. Other examples of quantitative analysis of network dynamics at the single-cell level are now available (Irish et al., 2004; Jones et al., 2004; Lahav et al., 2004; Nelson et al., 2004; Raser and O’Shea, 2004). The reductionist approach to cellular networks, which focuses on single modules such as the MAPK cascade or the EGF receptor pathway, has been reasonably successful. However, it must be realized that these modules do not operate in isolation and are affected by the large number of other processes occurring simultaneously. For example, genetic and biochemical evidence indicates that the MAPK cascade is coupled to essentially every other signal transduction pathway in cells. Quantitative understanding of cross talk in biochemical networks is necessary in order to probe these “cellular context” effects. While a modeling approach to the problem can start with simulations of coupled signaling or with genetic models, the eventual success of these models will depend on the availability of convenient experimental systems where pathway cross talk can be analyzed at the quantitative level.
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Mathematics and 21st Century Biology FIGURE 4.4 Modular view of the chemoattractant-induced signaling pathway in Dictyostelium. Except for those in parentheses, the proteins depicted in this pathway have been shown to be involved in chemotactic signaling through analysis of cells in which the genes have been deleted. SOURCE: Manahan et al., 2004.
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Mathematics and 21st Century Biology models through the accumulation of validated hypotheses stored in databases designed from the ground up to support hypothesis testing. Three kinds of conceptual and bioinformatics challenges appear in today’s data-rich environment: (1) information retrieval and integration, (2) knowledge representation, and (3) hypothesis testing and model building. The first and second are closely related: How can we retrieve and express the many qualitatively different kinds of information available in databases and the published literature in a representation that informs experimentation? Developing common ontologies for biological objects and processes is essential for supporting the intercommunication of diverse databases (Schulze-Kremer, 1998; Ashburner et al., 2000) and for enabling the automated annotation and extraction of information from the published literature (Andrade et al., 1999; Fleischmann et al., 1999; Friedman et al., 2001; Stephens et al., 2001; Yakushiji et al., 2001). An ontology also provides the foundation for constructing higher-level representations of biological systems (Rzhetsky et al., 2000; Peleg et al., 2002). The third challenge is to create and verify testable conceptual representations of the biological system. A conceptual framework for representing biological systems must accommodate the modularity and temporal evolution of biological networks, as well as handle their nonlinearity, plasticity, redundancy, and degeneracy. Conceptual models vary from the simple Boolean networks pioneered by Kaufmann (1969, 1993), Liang et al. (1998), and Akutsu et al. (2000a, 2000b) to Bayesian networks (Friedman et al., 2000; Hartemink et al., 2001; Pe’er et al., 2001), as well as highly concrete (McAdams and Arkin, 1998; Judd et al., 2000) and quantitative (Sveiczer et al., 2000) models. Incorporating disparate kinds of information about biological systems into a common conceptual framework remains a major stumbling block for validating ideas about real biological networks, and current efforts focus largely on just one or two categories of information (Rzhetsky et al., 2000; Hartemink et al., 2001; Wessels et al., 2001). There is a need to develop a hypothesis representation language that can assist in integrating experimental information at the logical level, as well as approaches to aggregating validated hypotheses into increasingly quantitative models. Most currently available bioinformatics tools support the analytical tasks of the biologist. These tools are very useful and effective for certain specific tasks, such as identifying patterns, categorizing information, and simultaneously probing multiple data sources for similarities. Such tasks usually comprise the early steps of the discovery process. However, synthesis and evaluation of the information remain the task of the individual. Kuchinsky and his colleagues (2002) argue that this synthesis task can be broken down into steps: (1) keeping track of all the diverse pieces of information collected during the database searches and other retrieval activi-
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Mathematics and 21st Century Biology ties, (2) organizing and using this information by formulating hypotheses and higher level explanations, and (3) sharing the information with colleagues and working collaboratively with colleagues to refine hypotheses. There is a need to develop a system to allow biologists to construct and verify formal language hypotheses, making use of event- and process-based description language. Complex processes that exhibit nonlinear behavior, as biological systems do, are often more readily described by event-driven dynamics (Ho, 1989) than by differential equations. Furthermore, when biologists think about biological systems, they typically do so in terms of biological agents, events, and causal relationships between events. A symbolic discrete modeling approach, in which the discretizations are defined by events—that is, by any biological change for which there is experimental evidence of changes in the state of the system—is likely to be a useful approach. BIOLOGICAL CONSIDERATIONS The capacity of cells to differentiate is a hallmark of eukaryotic organisms. Differentiation is the acquisition of structurally and chemically different identities by cells over time. The capacity for self-differentiation, which transforms a single diploid cell (the zygote) into a complex, multicellular plant or animal comprising many different structural and functional tissues, is rooted in asymmetries within the initial cell and has its origins in the unequal intracellular distribution of small molecules, macromolecules, and organelles. Such inhomogeneities in cell structure can be triggered by external stimuli, such as fertilization (Green, 1993; Rossant and Tam, 2004; Swann et al., 2004) and light stimuli (Robinson et al., 1999), which set in motion a complex series of structural and compositional re-organizations that in turn generate compositionally different daughter cells, whose differences are further reinforced by differential gene expression (Kanka, 2003). The rapid advances during the second half of the 20th century in the understanding of how DNA functions in heredity and expresses the information it encodes fostered a markedly genocentric view of eukaryotic development. This was further reinforced by a virtual flood of genome sequences and gene expression data that followed technological developments in nucleic acid sequencing and monitoring of gene expression patterns during the final decades of the century. Both embryonic development and cellular differentiation were viewed as under the control of genes, whose differential expression was orchestrated by a “developmental program.” Indeed, the notion of a developmental program has dominated thinking about development for decades (Davidson et al., 1995; Marczynski and Shapiro, 1995; Chu et al., 1998; Roberts et al., 2000).
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Mathematics and 21st Century Biology However, actual progress in understanding developmental processes has occurred by studying finer levels of detail rather than attempting to delineate an overarching developmental program. Mutations that affect development are often in genes that code for proteins that function in inter- and intracellular signaling and structure. Moreover, there are many molecular mechanisms by which cells affect the spatial patterning, including small molecules, such as the gaseous hormones nitric oxide (plants and animals) and ethylene (plants), and intermediate-sized molecules, such as the plant gibberellins and brassinosteroids and the animal endocrine and autocrine hormones. Patterning mechanisms also include macromolecular mechanisms, such as the intracellular transport of proteins and RNA in both plants and animals. Examples include the translational gradients established in early Drosophila embryogenesis by the localization of bicoid and oskar mRNA molecules (Micklem et al., 2000) and the intercellular interactions that underlie the subsequent development of the abdominal segmentation pattern (Immergluck et al., 1990; Ingham and Arias, 1992; Courey and Huang, 1995). Analysis of processes such as these has led to articulation of the view that all of development can be explained by local interactions (Britten, 1998). Indeed, current developmental models are increasingly couched in terms of cellular fate determination by signaling between cells and by programmed cell death (Lam et al., 2001; Ribeiro et al., 2003; Lai and Orgogozo, 2004). Morphogenetic proteins can be secreted signaling proteins that alter the fates of cells through cell-surface-receptor-mediated pathways (Tabata and Takei, 2004). Receptors at the cell surface and intracellular signaling proteins, signaling cascades (such as MAPK cascades), and protein networks mediate the activation of genes in response to extracellular signals (Imler and Hoffmann, 2002; Muller and Bossinger, 2003; Schulz and Yutzey, 2004). Epigenetic mechanisms are conceptualized as mechanisms that stably affect gene expression without altering gene structure. Initially, epigenetic mechanisms were equated with stable, even heritable, modifications in gene expression, commonly ones that suppress gene expression. Early descriptions of plant paramutation (Brink, 1960), transposable element inactivation (McClintock, 1965), and X-chromosome inactivation in mammals (Lyon, 1961) provided the foundation for what has become an active field of epigenetic research. Because differentiated plant cells can regenerate into whole plants and nuclei from differentiated animal cells can support development of enucleated eggs, DNA in most cells is not irreversibly altered during development and differentiation. However, the epigenetic modifications in gene expression that occur during development are highly regular, impose important differences between male and female gametes, and are not easily reversed experimentally
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Mathematics and 21st Century Biology (Wrenzycki and Niemann, 2003; Tian, 2004) although they are altered regularly during gametogenesis and early embyronic development (Gehring et al., 2004; Santos and Dean, 2004). There has been important progress in understanding the diverse mechanisms for epigenetic modifications of gene function. These mechanisms include DNA methylation, which compromises the digital purity of the Watson-Crick model of DNA by, in effect, putting “asterisks” on certain bits in the digital code, and highly dynamic chemical modifications of the histone proteins around which DNA wraps in the chromosomes of multicellular organisms. The complexity of these mechanisms and of the pathways that regulate their operation—together with the ubiquity of epigenetic effects during development—indicates that overly genecentric models of development are likely to fail. FUTURE DIRECTIONS The emphasis in cell biology is shifting from phenomenological descriptions to predictive models that are consistent with the largest possible amount of data. Data integration, reduction, and multiscale modeling approaches will take center stage in dealing with the diverse data sets emerging from molecular profiling and imaging experiments. Increasingly, biologists will use these approaches to identify the important species, interactions, and processes occurring within cells. Models of cellular processes must explicitly account for the significant parametric and structural uncertainties inevitable at the current level of knowledge and experimental resolution. In general, special attention should be paid to the proper selection and validation of the mathematical formalisms used to model any given process. Advances in dynamical systems theory will be required to deal with inevitable model heterogeneity, such as that required by the combination of stochastic descriptions of gene expression and deterministic descriptions of signal transduction events. At this time, scientists are far from having an adequate quantitative description of cellular responses, even in well-studied systems such as bacterial chemotaxis. The field must make a concerted effort to improve the quantitative analysis of such model systems in order to achieve successes that can be used as templates for modeling and analysis in less well-established experimental systems. At the same time, the field must establish new experimental systems where integrative genetic, biochemical, and cell biological experiments are possible and that can support meaningful modeling efforts. It will be necessary to establish experimental systems that can serve as testing grounds for multiscale models that can be used to understand how cellular functions emerge from molecu-
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Mathematics and 21st Century Biology lar-scale events and how cell population or tissue-level functions emerge from cellular-scale events. Models must span as many scales as possible, from sequence-specific information on gene expression, to intracellular biochemistry, to cellular responses. Large-scale integrative approaches require the creation and funding of interacting groups of mathematicians, computer scientists, and biologists. While models should be based on detailed analysis of specific experimental systems—for example, particular cell types—model builders should strive to make the models generalizable to other systems. For example, it is important to analyze the evolution of cellular signaling systems in animals from worms to humans, as well as in plants. What mediates the increase in the number of signaling components in particular evolutionary linkages, and what are the systems-level consequences of this increase in complexity? Quantitative experimental analyses of intracellular fluctuations and noise are critical for understanding cellular functions and the limits of applicability of conventional deterministic and continuum approaches. This type of analysis is also important for understanding the mechanisms and functional consequences of nongenetic individuality. Analysis of cell-to-cell variations is now possible owing to advances in imaging and single-cell molecular profiling experiments. On the purely theoretical and computational side, scientists must (1) classify dynamical systems according to the ways in which they can tolerate intracellular and extracellular noise and (2) understand the processes where noise can play a constructive role. Thus, future research must emphasize the close connection between experiments, model validation, and data integration. While this level of integration is only possible by focusing on specific experimental systems, the field must make an effort to systematize methodological advances so that scientists do not have to start from scratch every time they analyze a new cellular system. REFERENCES Abraham, V.C., D.L. Taylor, J.R. Haskins. 2004. High content screening applied to large-scale cell biology. Trends Biotechnol. 22(1): 15-22. Akutsu, T., S. Miyano, and S. Kuhara. 2000a. Algorithms for identifying Boolean networks and related biological networks based on matrix multiplication and fingerprint function. J. Comput. Biol. 7(3-4): 331-343. Akutsu, T., S. Miyano, and S. Kuhara. 2000b. Inferring qualitative relations in genetic networks and metabolic pathways. Bioinformatics 16(8): 727-734. Alves, R., and M.A. Savageau. 2000. Extending the method of mathematically controlled comparison to include numerical comparisons. Bioinformatics 16(9): 786-798. Andrade, M.A., N.P. Brown, C. Leroy, S. Hoersch, A. de Daruvar, C. Reich, A. Franchini, J. Tamames, A. Valencia, C. Ouzounis, and C. Sander. 1999. Automated genome sequence analysis and annotation. Bioinformatics 15(5): 391-412.
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Representative terms from entire chapter: