Systems Biology: A Textbook

Preface

Life is probably the most complex phenomenon in the universe. We see kids growing, people aging, plants blooming, and microbes degrading their remains. We use yeast for brewery and bakery, and doctors prescribe drugs to cure diseases. But can we understand how life works? Since the 19th century, the processes of life have no longer been explained by special living forces, but by the laws of physics and chemistry. By studying the structure and physiology of living systems more and more in detail, researchers from different disciplines have revealed how the mystery of life arises from the structural and functional organization of cells and from the continuous refinement by mutation and selection.

In recent years, new imaging techniques have opened a completely new perception of the cellular microcosm. If we zoom into the cell, we can observe how structures are built, maintained, and reproduced while various sensing and regulation systems help the cell to respond appropriately to environmental changes. But along with all these fascinating observations, many open questions remain. Why do we age? How does a cell know when to divide? How can severe diseases such as cancer or genetic disorders be cured? How can we convince – i.e., manipulate – microbes to produce a desirable substance? How can the life sciences contribute to environmental safety and sustainable technologies?

This book provides you with a number of tools and approaches that can help you to think in more detail about such questions from a theoretical point of view. A key to tackle such questions is to combine biological experiments with computational modeling in an approach called systems biology: it is the combined study of biological systems through (i) investigating the components of cellular networks and their interactions, (ii) applying experimental high-throughput and whole-genome techniques, and (iii) integrating computational methods with experimental efforts.

The systemic approach in biology is not new, but it recently gained new thrust due to the emergence of powerful experimental and computational methods. It is based on the accumulation of an increasingly detailed biological knowledge, on the emergence of new experimental techniques in genomics and proteomics, on a tradition of mathematical modeling of biological processes, on the exponentially growing computer power (as prerequisite for databases and the calculation of large systems), and on the Internet as the central medium for a quick and comprehensive exchange of information.

Systems Biology has influenced modern biology in two major ways: on the one hand, it offers computational tools for analyzing, integrating and interpreting biological data and hypotheses. On the other hand, it has induced the formulation of new theoretical concepts and the application of existing ones to new questions. Such concepts are, for example, the theory of dynamical systems, control theory, the analysis of molecular noise, robustness and fragility of dynamic systems, and statistical network analysis. As systems biology is still evolving as a scientific field, a central issue is the standardization of experiments, of data exchange, and of mathematical models.

In this book, we attempt to give a survey of this rapidly developing field. We will show you how to formulate your own model of biological processes, how to analyze such models, how to use data and other available information for making your model more precise – and how to interpret the results. This book is designed as an introductory course for students of biology, biophysics and bioinformatics, and for senior scientists approaching Systems Biology from a different discipline. Its nine chapters contain material for about 30 lectures and are organized as follows.

Chapter 1 – Introduction (E. Klipp, W. Liebermeister, A. Kowald, 1 lecture)

Introduction to the subject. Elementary concepts and definitions are presented. Read this if you want to start right from the beginning.

Chapter 2 – Modeling of Biochemical Systems (E. Klipp, C. Wierling, 4 lectures)

This chapter describes kinetic models for biochemical reaction networks, the most common computational technique in Systems Biology. It includes kinetic laws, stoichiometric analysis, elementary flux modes, and metabolic control analysis. Introduces tools and data formats necessary for modeling.

Chapter 3 – Specific Biochemical Systems (E. Klipp, C. Wierling, W. Liebermeister, 5 lectures)

Using specific examples from metabolism, signaling, and cell cycle, a number of popular modeling techniques are discussed. The aim of this chapter is to make the reader familiar with both modeling techniques and biological phenomena.

Chapter 4 – Model Fitting (W. Liebermeister, A. Kowald, 4 lectures)

Models in systems biology usually contain a large number of parameters. Assigning appropriate numerical values to these parameters is an important step in the creation of a quantitative model. This chapter shows how numerical values can be obtained from the literature or by fitting a model to experimental data. It also discusses how model structures can be simplified and how they can be chosen if several different models can potentially describe the experimental observations.

Chapter 5 – Analysis of High-Throughput Data (R. Herwig, 2 lectures)

Several techniques that have been developed in recent years produce large quantities of data (e.g., DNA and protein chips, yeast two-hybrid, mass spectrometry). But such large quantities often go together with a reduced quality of the individual measurement. This chapter describes techniques that can be used to handle this type of data appropriately.

Chapter 6 – Gene Expression Models (R. Herwig, W. Liebermeister, E. Klipp, 3 lectures)

Thousands of gene products are necessary to create a living cell, and the regulation of gene expression is a very complex and important task to keep a cell alive. This chapter discusses how the regulation of gene expression can be modeled, how different input signals can be integrated, and how the structure of gene networks can be inferred from experimental data.

Chapter 7 – Stochastic Systems and Variability (W. Liebermeister, 4 lectures)

Random fluctuations in transcription, translation and metabolic reactions make mathematics complicated, computation costly and interpretation of results not straight forward. But since experimentalists find intriguing examples for macroscopic consequences of random fluctuation at the molecular level, the incorporation of these effects into the simulations becomes more and more important. This chapter gives an overview where and how stochasticity enters cellular life.

Chapter 8 – Network Structures, Dynamics and Function (W. Liebermeister, 3 lectures)

Many complex systems in biology can be represented as networks (reaction networks, interaction networks, regulatory networks). Studying the structure, dynamics, and function of such networks helps to understand design principles of living cells. In this chapter, important network structures such as motifs and modules as well as the dynamics resulting from them are discussed.

Chapter 9 – Optimality and Evolution (W. Liebermeister, E. Klipp, 3 lectures)

Theoretical research suggests that constraints of the evolutionary process should have left their marks in the construction and regulation of genes and metabolic pathways. In some cases, the function of biological systems can be well understood by models based on an optimality hypothesis. This chapter discusses the merits and limitations of such optimality approaches.

Various aspects of systems biology – the biological systems themselves, types of mathematical models to describe them, and practical techniques – reappear in different contexts in various parts of the book. The following diagram, which shows the contents of the book sorted by a number of different aspects, may serve as an orientation.

At the end of the regular course material, you will find a number of additional chapters that summarize important biological and mathematical methods. The first chapters deal with to cell biology (chapter 10, C. Wierling) and molecular biological methods (chapter 11, A. Kowald). For looking up mathematical and statistical definitions and methods, turn to chapters 12 and 13 (R. Herwig, A. Kowald). Chapters 14 and 15 (W. Liebermeister) concentrate on random processes and control theory. The final chapters provide an overview over useful databases (chapter 16, C. Wierling) as well as a huge list of available software tools including a short description of their purposes (chapter 17, A. Kowald).

Further material is available on an accompanying website (www.wiley-vch.de/home/systemsbiology)

Beside additional and more specialized topics, the website also contains solutions to the exercises and problems presented in the book.

We give our thanks to a number of people who helped us in finishing this book. We are especially grateful to Dr. Ulrich Liebermeister, Prof. Dr. Hans Meinhardt, Dr. Timo Reis, Dr. Ulrike Baur, Clemens Kreutz, Dr. Jose Egea, Dr. Maria Rodriguez-Fernandez, Dr. Wilhelm Huisinga, Sabine Hummert, Guy Shinar, Nadav Kashtan, Dr. Ron Milo, Adrian Jinich, Elad Noor, Niv Antonovsky, Bente Kofahl, Dr. Simon Borger, Martina Fröhlich, Christian Waltermann, Susanne Gerber, Thomas Spießer, Szymon Stoma, Christian Diener, Axel Rasche, Hendrik Hache, Dr. Michal Ruth Schweiger, and Elisabeth Maschke-Dutz for reading and commenting on the manuscript.

We thank the Max Planck Society for support and encouragement. We are grateful to the European Commission for funding via different European projects (MEST-CT2004-514169, LSHG-CT-2005-518254, LSHG-CT-2005-018942, LSHG-CT-2006-037469, LSHG-CT-2006-035995-2 NEST-2005-Path2-043310, HEALTH-F4-2007-200767, and LSHB-CT-2006-037712). Further funding was obtained from the Sysmo project Translucent and from the German Research Foundation (IRTG 1360) E.K. thanks with love her sons Moritz and Richard for patience and incentive and the Systems Biology community for motivation. W.L. wishes to thank his daughters Hannah and Marlene for various insights and inspiration. A.K. likes to thank Prof. Dr. H.E. Meyer for support and hospitality. This book is dedicated to our teacher Prof. Dr. Reinhart Heinrich (1946–2006), whose works on metabolic control theory in the 1970s paved the way to systems biology and who greatly inspired our minds.

Part One

Introduction to Systems Biology

1

Introduction

1.1 Biology in Time and Space

Biological systems like organisms, cells, or biomolecules are highly organized in their structure and function. They have developed during evolution and can only be fully understood in this context. To study them and to apply mathematical, computational, or theoretical concepts, we have to be aware of the following circumstances.

The continuous reproduction of cell compounds necessary for living and the respective flow of information is captured by the central dogma of molecular biology, which can be summarized as follows: genes code for mRNA, mRNA serves as template for proteins, and proteins perform cellular work. Although information is stored in the genes in form of DNA sequence, it is made available only through the cellular machinery that can decode this sequence and can translate it into structure and function. In this book, this will be explained from various perspectives.

A description of biological entities and their properties encompasses different levels of organization and different time scales. We can study biological phenomena at the level of populations, individuals, tissues, organs, cells, and compartments down to molecules and atoms. Length scales range from the order of meter (e.g., the size of whale or human) to micrometer for many cell types, down to picometer for atom sizes. Time scales include millions of years for evolutionary processes, annual and daily cycles, seconds for many biochemical reactions, and femtoseconds for molecular vibrations. Figure 1.1 gives an overview about scales.

Figure 1.1 Length and time scales in biology. Data from the BioNumbers database http://bionumbers.hms.harvard.edu.

In a unified view of cellular networks, each action of a cell involves different levels of cellular organization, including genes, proteins, metabolism, or signaling pathways. Therefore, the current description of the individual networks must be integrated into a larger framework.

Many current approaches pay tribute to the fact that biological items are subject to evolution. The structure and organization of organisms and their cellular machinery has developed during evolution to fulfill major functions such as growth, proliferation, and survival under changing conditions. If parts of the organism or of the cell fail to perform their function, the individual might become unable to survive or replicate.

One consequence of evolution is the similarity of biological organisms from different species. This similarity allows for the use of model organisms and for the critical transfer of insights gained from one cell type to other cell types. Applications include, e.g., prediction of protein function from similarity, prediction of network properties from optimality principles, reconstruction of phylogenetic trees, or the identification of regulatory DNA sequences through cross-species comparisons. But the evolutionary process also leads to genetic variations within species. Therefore, personalized medicine and research is an important new challenge for biomedical research.

1.2 Models and Modeling

If we observe biological processes, we are confronted with various complex processes that cannot be explained from first principles and the outcome of which cannot reliably be foreseen from intuition. Even if general biochemical principles are well established (e.g., the central dogma of transcription and translation, the biochemistry of enzyme-catalyzed reactions), the biochemistry of individual molecules and systems is often unknown and can vary considerably between species. Experiments lead to biological hypotheses about individual processes, but it often remains unclear if these hypotheses can be combined into a larger coherent picture because it is often difficult to foresee the global behavior of a complex system from knowledge of its parts. Mathematical modeling and computer simulations can help us understand the internal nature and dynamics of these processes and to arrive at predictions about their future development and the effect of interactions with the environment.

1.2.1 What is a Model?

The answer to this question will differ among communities of researchers. In a broad sense, a model is an abstract representation of objects or processes that explains features of these objects or processes (Figure 1.2). A biochemical reaction network can be represented by a graphical sketch showing dots for metabolites and arrows for reactions; the same network could also be described by a system of differential equations, which allows simulating and predicting the dynamic behavior of that network. If a model is used for simulations, it needs to be ensured that it faithfully predicts the system’s behavior – at least those aspects that are supposed to be covered by the model. Systems biology models are often based on well-established physical laws that justify their general form, for instance, the thermodynamics of chemical reactions; besides this, a computational model needs to make specific statements about a system of interest – which are partially justified by experiments and biochemical knowledge, and partially by mere extrapolation from other systems. Such a model can summarize established knowledge about a system in a coherent mathematical formulation. In experimental biology, the term model is also used to denote a species that is especially suitable for experiments, for example, a genetically modified mouse may serve as a model for human genetic disorders.

Figure 1.2 Typical abstraction steps in mathematical modeling. (a) Escherichia coli bacteria produce thousands of different proteins. If a specific protein type is fluorescently labeled, cells glow under the microscope according to the concentration of this enzyme (Courtesy of M. Elowitz). (b) In a simplified mental model, we assume that cells contain two enzymes of interest, X (red) and Y (blue) and that the molecules (dots) can freely diffuse within the cell. All other substances are disregarded for the sake of simplicity. (c) The interactions between the two protein types can be drawn in a wiring scheme: each protein can be produced or degraded (black arrows). In addition, we assume that proteins of type X can increase the production of protein Y. (d) All individual processes to be considered are listed together with their rates a (occurrence per time). The mathematical expressions for the rates are based on a simplified picture of the actual chemical processes. (e) The list of processes can be translated into different sorts of dynamic models; in this case, deterministic rate equations for the protein concentrations x and y. (f) By solving the model equations, predictions for the time-dependent concentrations can be obtained. If these predictions do not agree with experimental data, it indicates that the model is wrong or too much simplified. In both cases, it has to be refined.

1.2.2 Purpose and Adequateness of Models

Modeling is a subjective and selective procedure. A model represents only specific aspects of reality but, if done properly, this is sufficient since the intention of modeling is to answer particular questions. If the only aim is to predict system outputs from given input signals, a model should display the correct input–output relation, while its interior can be regarded as a black box. But if instead a detailed biological mechanism has to be elucidated, then the system’s structure and the relations between its parts must be described realistically. Some models are meant to be generally applicable to many similar objects (e.g., Michaelis–Menten kinetics holds for many enzymes, the promoter–operator concept is applicable to many genes, and gene regulatory motifs are common), while others are specifically tailored to one particular object (e.g., the 3D structure of a protein, the sequence of a gene, or a model of deteriorating mitochondria during aging). The mathematical part can be kept as simple as possible to allow for easy implementation and comprehensible results. Or it can be modeled very realistically and be much more complicated. None of the characteristics mentioned above makes a model wrong or right, but they determine whether a model is appropriate to the problem to be solved. The phrase essentially, all models are wrong, but some are useful coined by the statistician George Box is indeed an appropriate guideline for model building.

1.2.3 Advantages of Computational Modeling

Models gain their reference to reality from comparison with experiments, and their benefits therefore depend on the quality of the experiments used. Nevertheless, modeling combined with experimentation has a lot of advantages compared to purely experimental studies:

Modeling drives conceptual clarification. It requires verbal hypotheses to be made specific and conceptually rigorous.

Modeling highlights gaps in knowledge or understanding. During the process of model formulation, unspecified components or interactions have to be determined.

Modeling provides independence of the modeled object.

Time and space may be stretched or compressed ad libitum.

Solution algorithms and computer programs can be used independently of the concrete system.

Modeling is cheap compared to experiments.

Models exert by themselves no harm on animals or plants and help to reduce ethical problems in experiments. They do not pollute the environment.

Modeling can assist experimentation. With an adequate model, one may test different scenarios that are not accessible by experiment. One may follow time courses of compounds that cannot be measured in an experiment. One may impose perturbations that are not feasible in the real system. One may cause precise perturbations without directly changing other system components, which is usually impossible in real systems. Model simulations can be repeated often and for many different conditions.

Model results can often be presented in precise mathematical terms that allow for generalization. Graphical representation and visualization make it easier to understand the system.

Finally, modeling allows for making well-founded and testable predictions.

The attempt to formulate current knowledge and open problems in mathematical terms often uncovers a lack of knowledge and requirements for clarification. Furthermore, computational models can be used to test whether proposed explanations of biological phenomena are feasible. Computational models serve as repositories of current knowledge, both established and hypothetical, about how systems might operate. At the same time, they provide researchers with quantitative descriptions of this knowledge and allow them to simulate the biological process, which serves as a rigorous consistency test.

1.3 Basic Notions for Computational Models

1.3.1 Model Scope

Systems biology models consist of mathematical elements that describe properties of a biological system, for instance, mathematical variables describing the concentrations of metabolites. As a model can only describe certain aspects of the system, all other properties of the system (e.g., concentrations of other substances or the environment of a cell) are neglected or simplified. It is important – and to some extent, an art – to construct models in such ways that the disregarded properties do not compromise the basic results of the model.

1.3.2 Model Statements

Besides the model elements, a model can contain various kinds of statements and equations describing facts about the model elements, most notably, their temporal behavior. In kinetic models, the basic modeling paradigm considered in this book, the dynamics is determined by a set of ordinary differential equations describing the substance balances. Statements in other model types may have the form of equality or inequality constraints (e.g., in flux balance analysis), maximality postulates, stochastic processes, or probabilistic statements about quantities that vary in time or between cells.

1.3.3 System State

In dynamical systems theory, a system is characterized by its state, a snapshot of the system at a given time. The state of the system is described by the set of variables that must be kept track of in a model: in deterministic models, it needs to contain enough information to predict the behavior of the system for all future times. Each modeling framework defines what is meant by the state of the system. In kinetic rate equation models, for example, the state is a list of substance concentrations. In the corresponding stochastic model, it is a probability distribution or a list of the current number of molecules of a species. In a Boolean model of gene regulation, the state is a string of bits indicating for each gene whether it is expressed (1) or not expressed (0). Also the temporal behavior can be described in fundamentally different ways. In a dynamical system, the future states are determined by the current state, while in a stochastic process, the future states are not precisely predetermined. Instead, each possibly future history has a certain probability to occur.

1.3.4 Variables, Parameters, and Constants

The quantities in a model can be classified as variables, parameters, and constants. A constant is a quantity with a fixed value, such as the natural number e or Avogadro’s number (number of molecules per mole). Parameters are quantities that have a given value, such as the Km value of an enzyme in a reaction. This value depends on the method used and on the experimental conditions and may change. Variables are quantities with a changeable value for which the model establishes relations. A subset of variables, the state variables, describes the system behavior completely. They can assume independent values and each of them is necessary to define the system state. Their number is equivalent to the dimension of the system. For example, the diameter d and volume V of a sphere obey the relation V = πd³/6, where π and 6 are constants, V and d are variables, but only one of them is a state variable since the relation between them uniquely determines the other one.

Whether a quantity is a variable or a parameter depends on the model. In reaction kinetics, the enzyme concentration appears as a parameter. However, the enzyme concentration itself may change due to gene expression or protein degradation and in an extended model, it may be described by a variable.

1.3.5 Model Behavior

Two fundamental factors that determine the behavior of a system are (i) influences from the environment (input) and (ii) processes within the system. The system structure, that is, the relation among variables, parameters, and constants, determines how endogenous and exogenous forces are processed. However, different system structures may still produce similar system behavior (output); therefore, measurements of the system output often do not suffice to choose between alternative models and to determine the system’s internal organization.

1.3.6 Model Classification

For modeling, processes are classified with respect to a set of criteria.

A structural or qualitative model (e.g., a network graph) specifies the interactions among model elements. A quantitative model assigns values to the elements and to their interactions, which may or may not change.

In a deterministic model, the system evolution through all following states can be predicted from the knowledge of the current state. Stochastic descriptions give instead a probability distribution for the successive states.

The nature of values that time, state, or space may assume distinguishes a discrete model (where values are taken from a discrete set) from a continuous model (where values belong to a continuum).

Reversible processes can proceed in a forward and backward direction. Irreversibility means that only one direction is possible.

Periodicity indicates that the system assumes a series of states in the time interval {t, t + Δt} and again in the time interval {t + iΔt, t + (i + 1)Δt} for i = 1, 2, . . ..

1.3.7 Steady States

The concept of stationary states is important for the modeling of dynamical systems. Stationary states (other terms are steady states or fixed points) are determined by the fact that the values of all state variables remain constant in time. The asymptotic behavior of dynamic systems, that is, the behavior after a sufficiently long time, is often stationary. Other types of asymptotic behavior are oscillatory or chaotic regimes.

The consideration of steady states is actually an abstraction that is based on a separation of time scales. In nature, everything flows. Fast and slow processes – ranging from formation and breakage of chemical bonds within nanoseconds to growth of individuals within years – are coupled in the biological world. While fast processes often reach a quasi-steady state after a short transition period, the change of the value of slow variables is often negligible in the time window of consideration. Thus, each steady state can be regarded as a quasi-steady state of a system that is embedded in a larger nonstationary environment. Despite this idealization, the concept of stationary states is important in kinetic modeling because it points to typical behavioral modes of the system under study and it often simplifies the mathematical problems.

Other theoretical concepts in systems biology are only rough representations of their biological counterparts. For example, the representation of gene regulatory networks by Boolean networks, the description of complex enzyme kinetics by simple mass action laws, or the representation of multifarious reaction schemes by black boxes proved to be helpful simplification. Although being a simplification, these models elucidate possible network properties and help to check the reliability of basic assumptions and to discover possible design principles in nature. Simplified models can be used to test mathematically formulated hypothesis about system dynamics, and such models are easier to understand and to apply to different questions.

1.3.8 Model Assignment is not Unique

Biological phenomena can be described in mathematical terms. Models developed during the last decades range from the description of glycolytic oscillations with ordinary differential equations to population dynamics models with difference equations, stochastic equations for signaling pathways, and Boolean networks for gene expression. But it is important to realize that a certain process can be described in more than one way: a biological object can be investigated with different experimental methods and each biological process can be described with different (mathematical) models. Sometimes, a modeling framework represents a simplified limiting case (e.g., kinetic models as limiting case of stochastic models). On the other hand, the same mathematical formalism may be applied to various biological instances: statistical network analysis, for example, can be applied to cellular-transcription networks, the circuitry of nerve cells, or food webs.

The choice of a mathematical model or an algorithm to describe a biological object depends on the problem, the purpose, and the intention of the investigator. Modeling has to reflect essential properties of the system and different models may highlight different aspects of the same system. This ambiguity has the advantage that different ways of studying a problem also provide different insights into the system. However, the diversity of modeling approaches makes it still very difficult to merge established models (e.g., for individual metabolic pathways) into larger supermodels (e.g., models of complete cell metabolism).

1.4 Data Integration

Systems biology has evolved rapidly in the last years driven by the new high-throughput technologies. The most important impulse was given by the large sequencing projects such as the human genome project, which resulted in the full sequence of the human and other genomes [1, 2]. Proteomics technologies have been used to identify the translation status of complete cells (2D-gels, mass spectrometry) and to elucidate protein–protein interaction networks involving thousands of components [3]. However, to validate such diverse high-throughput data, one needs to correlate and integrate such information. Thus, an important part of systems biology is data integration.

On the lowest level of complexity, data integration implies common schemes for data storage, data representation, and data transfer. For particular experimental techniques, this has already been established, for example, in the field of transcriptomics with minimum information about a microarray experiment [4], in proteomics with proteomics experiment data repositories [5], and the Human Proteome Organization consortium [6]. On a more complex level, schemes have been defined for biological models and pathways such as Systems Biology Markup Language (SBML) [7] and CellML [8], which use an XML-like language style.

Data integration on the next level of complexity consists of data correlation. This is a growing research field as researchers combine information from multiple diverse data sets to learn about and explain natural processes [9, 10]. For example, methods have been developed to integrate the results of transcriptome or proteome experiments with genome sequence annotations. In the case of complex disease conditions, it is clear that only integrated approaches can link clinical, genetic, behavioral, and environmental data with diverse types of molecular phenotype information and identify correlative associations. Such correlations, if found, are the key to identifying biomarkers and processes that are either causative or indicative of the disease. Importantly, the identification of biomarkers (e.g., proteins, metabolites) associated with the disease will open up the possibility to generate and test hypotheses on the biological processes and genes involved in this condition. The evaluation of disease-relevant data is a multistep procedure involving a complex pipeline of analysis and data handling tools such as data normalization, quality control, multivariate statistics, correlation analysis, visualization techniques, and intelligent database systems [11]. Several pioneering approaches have indicated the power of integrating data sets from different levels: for example, the correlation of gene membership of expression clusters and promoter sequence motifs [12]; the combination of transcriptome and quantitative proteomics data in order to construct models of cellular pathways [10]; and the identification of novel metabolite-transcript correlations [13]. Finally, data can be used to build and refine dynamical models, which represent an even higher level of data integration.

1.5 Standards

As experimental techniques generate rapidly growing amounts of data and large models need to be developed and exchanged, standards for both experimental procedures and modeling are a central practical issue in systems biology. Information exchange necessitates a common language about biological aspects. One seminal example is the gene ontology which provides a controlled vocabulary that can be applied to all organisms, even as the knowledge about genes and proteins continues to accumulate. The SBML [7] has been established as exchange language for mathematical models of biochemical reaction networks. A series of minimum-information-about statements based on community agreement defines standards for certain types of experiments. Minimum information requested in the annotation of biochemical models (MIRIAM) [14] describes standards for this specific type of systems biology models.

References

1 Lander, E.S. et al. (2001b) Initial sequencing and analysis of the human genome. Nature, 409, 860–921

2 Venter, J.C. et al. (2001a) The sequence of the human genome. Science, 291, 1304–1351

3 von Mering, C. et al. (2002) Comparative assessment of large-scale data sets of protein–protein interactions. Nature, 417, 399–403

4 Brazma, A. et al. (2001) Minimum information about a microarray experiment (MIAME)-toward standards for microarray data. Nature Genetics, 29, 365–371

5 Taylor, C.F. et al. (2003) A systematic approach to modeling, capturing, and disseminating proteomics experimental data. Nature Biotechnology, 21, 247–254

6 Hermjakob, H. et al. (2004) The HUPO PSI’s molecular interaction format – a community standard for the representation of protein interaction data. Nature Biotechnology, 22, 177–183

7 Hucka, M. et al. (2003) The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics, 19, 524–531

8 Lloyd, C.M. et al. (2004) CellML: its future present and past. Progress in Biophysics and Molecular Biology, 85, 433–450

9 Gitton, Y. et al. (2002) A gene expression map of human chromosome 21 orthologues in the mouse. Nature, 420, 586–590

10 Ideker, T. et al. (2001) Integrated genomic and proteomic analyses of a systematically perturbed metabolic network. Science, 292, 929–934

11 Kanehisa, M. and Bork, P. (2003) Bioinformatics in the post-sequence era. Nature Genetics, 33 (Suppl), 305–310

12 Tavazoie, S. et al. (1999) Systematic determination of genetic network architecture. Nature Genetics, 22, 281–285

13 Urbanczyk-Wochniak, E. et al. (2003) Parallel analysis of transcript and metabolic profiles: a new approach in systems biology. EMBO Reports, 4, 989–993

14 Le Novere, N. et al. (2005) Minimum information requested in the annotation of biochemical models (MIRIAM). Nature Biotechnology, 23, 1509–1515

2

Modeling of Biochemical Systems

2.1 Kinetic Modeling of Enzymatic Reactions

Summary

The rate of an enzymatic reaction, i.e., the velocity by which the execution of the reaction changes the concentrations of its substrates, is determined by concentrations of its substrates, concentration of the catalyzing enzyme, concentrations of possible modifiers, and by certain parameters. We introduce different kinetic laws for reversible and irreversible reactions, for reactions with varying numbers of substrates, and for reactions that are subject to inhibition or activation. The derivations of the rate laws are shown and the resulting restrictions for their validity and applicability. Saturation and sigmoidal kinetics are explained. The connection to thermodynamics is shown.

Deterministic kinetic modeling of individual biochemical reactions has a long history. The Michaelis–Menten model for the rate of an irreversible one-substrate reaction is an integral part of biochemistry, and the Km value is a major characteristic of the interaction between enzyme and substrate. Biochemical reactions are catalyzed by enzymes, i.e., specific proteins which often function in complex with cofactors. They have a catalytic center, are usually highly specific, and remain unchanged by the reaction. One enzyme molecule can catalyze thousands of reactions per second (this so-called turnover number ranges from 10² s−1 to 10⁷ s−1). Enzyme catalysis leads to a rate acceleration of about 10⁶- up to 10¹²-fold compared to the noncatalyzed, spontaneous reaction.

In this chapter, we make you familiar with the basic concepts of the mass action rate law. We will show how you can derive and apply more advanced kinetic expressions. The effect of enzyme inhibitors and activators will be discussed. The thermodynamic foundations and constraints are introduced.

The basic quantities are the concentration S of a substance S, i.e., the number n of molecules (or, alternatively, moles) of this substance per volume V, and the rate v of a reaction, i.e., the change of concentration S per time t. This type of modeling is macroscopic and phenomenological, compared to the microscopic approach, where single molecules and their interactions are considered. Chemical and biochemical kinetics rely on the assumption that the reaction rate v at a certain point in time and space can be expressed as a unique function of the concentrations of all substances at this point in time and space. Classical enzyme kinetics assumes for sake of simplicity a spatial homogeneity (the well-stirred test tube) and no direct dependency of the rate on time

(2.1)

In more advanced modeling approaches, longing toward whole-cell modeling, spatial inhomogeneities are taken into account, paying tribute to the fact that many components are membrane-bound or that cellular structures hinder the free movement of molecules. But, in the most cases, one can assume that diffusion is rapid enough to allow for an even distribution of all substances in space.

2.1.1 The Law of Mass Action

Biochemical kinetics is based on the mass action law, introduced by Guldberg and Waage in the nineteenth century [1–3]. It states that the reaction rate is proportional to the probability of a collision of the reactants. This probability is in turn proportional to the concentration of reactants to the power of the molecularity, that is the number in which they enter the specific reaction. For a simple reaction such as

(2.2)

the reaction rate reads

(2.3)

where v is the net rate; v+ and v− are the rates of the forward and backward reactions; and k+ and k− are the kinetic or rate constants, i.e., the respective proportionality factors.

The molecularity is 1 for S1 and for S2 and 2 for P, respectively. If we measure the concentration in mol l−1 (or M) and the time in seconds (s), then the rate has the unit M s−1. Accordingly, the rate constants for bimolecular reactions have the unit M−1 s−1. Rate constants of monomolecular reactions have the dimension s−1. The general mass action rate law for a reaction transforming mi substrates with concentrations Si into mj products with concentrations Pj reads

(2.4)

where ni and nj denote the respective molecularities of Si and Pj in this reaction.

The equilibrium constant Keq (we will also use the simpler symbol q) characterizes the ratio of substrate and product concentrations in equilibrium (Seq and Peq), i.e., the state with equal forward and backward rate. The rate constants are related to Keq in the following way:

(2.5)

The relation between the thermodynamic and the kinetic description of biochemical reactions will be outlined in Section 2.1.2.

The equilibrium constant for the reaction given in Eq. (2.2) is Keq = P²eq/(S1, eq · S2, eq). The dynamics of the concentrations away from equilibrium is described by the ODEs.

(2.6)

The time course of S1, S2, and P is obtained by integration of these ODEs (see Section 2.3).

Example 2.1

The kinetics of a simple decay like

(2.7)

is described by v = kS and dS/dt = −kS. Integration of this ODE from time t = 0 with the initial concentration S0 to an arbitrary time t with concentration S(t), ∫SS0 dS/S = − ∫tt=0 k dt, yields the temporal expression S(t) = S0e−kt.

2.1.2 Reaction Kinetics and Thermodynamics

An important purpose of metabolism is to extract energy from nutrients, which is necessary for the synthesis of molecules, growth, and proliferation. We distinguish between energy-supplying reactions, energy-demanding reactions, and energetically neutral reactions. The principles of reversible thermodynamics and their application to chemical reactions allow understanding of energy circulation in the cell.

A biochemical process is characterized by the direction of the reaction, by whether it occurs spontaneously or not, and by the position of the equilibrium. The first law of thermodynamics, i.e., the law of energy conservation, tells us that the total energy of a closed system remains constant during any process. The second law of thermodynamics states that a process occurs spontaneous only if it increases the total entropy of the system. Unfortunately, entropy is usually not directly measurable. A more suitable measure is the Gibbs free energy G, which is the energy capable of carrying out work under isotherm–isobar conditions, i.e., at constant temperature and constant pressure. The change of the free energy is given as

(2.8)

where ΔH is the change in enthalpy, ΔS the change in entropy, and T the absolute temperature in Kelvin. ΔG is a measure for the driving force, the spontaneity of a chemical reaction. The reaction proceeds spontaneous under release of energy, if ΔG < 0 (exergonic process). If ΔG > 0, then the reaction is energetically not favorable and will not occur spontaneously (endergonic process). ΔG = 0 means that the system has reached its equilibrium. Endergonic reactions may proceed if they obtain energy from a strictly exergonic reaction by energetic coupling. In tables, free energy is usually given for standard conditions (ΔG°), i.e., for a concentration of the reaction partners of 1 M, a temperature of T = 298 K, and, for gaseous reactions, a pressure of p = 98, 1 kPa = 1 atm. The unit is kJ mol−1. Free energy differences satisfy a set of relations as follows. The free energy difference for a reaction can be calculated from the balance of free energies of formation of its products and substrates:

(2.9)

The enzyme cannot change the free energies of the substrates and products of a reaction, neither their difference, but it changes the way the reaction proceeds microscopically, the so-called reaction path, thereby lowering the activation energy for the reaction. The Transition State Theory explains this as follows. During the course of a reaction, the metabolites must pass one or more transition states of maximal free energy, in which bonds are solved or newly formed. The transition state is unstable; the respective molecule configuration is called an activated complex. It has a lifetime of around one molecule vibration, 10−14–10−13 s, and it can hardly be experimentally verified. The difference ΔG# of free energy between the reactants and the activated complex determines the dynamics of a reaction: the higher this difference, the lower the probability that the molecules may pass this barrier and the lower the rate of the reaction. The value of ΔG# depends on the type of altered bonds, on steric, electronic, or hydrophobic demands, and on temperature.

Figure 2.1 presents a simplified view of the reaction course. The substrate and the product are situated in local minima of the free energy; the active complex is assigned to the local maximum. The free energy difference ΔG is proportional to the logarithm of the equilibrium constant Keq of the respective reaction:

Figure 2.1 Change of free energy along the course of a reaction. The substrate and the product are situated in local minima of the free energy; the active complex is assigned to the local maximum. The enzyme may change the reaction path and thereby lower the barrier of free energy.

(2.10)

where R is the gas constant, 8.314 J mol−1 K−1. The value of ΔG# corresponds to the kinetic constant k+ of the forward reaction (Eqs. (2.3)–(2.5)) by ΔG# = −RT ln k+, while ΔG# + ΔG is related to the rate constant k− of the backward reaction.

The interaction of the reactants with an enzyme may alter the reaction path and, thereby, lead to lower values of ΔG# as well as higher values of the kinetic constants. Furthermore, the free energy may assume more local minima and maxima along the path of reaction. They are related to unstable intermediary complexes. Values for the difference of free energy for some biologically important reactions are given in Table 2.1.

Table 2.1 Values of ΔG⁰′ and Keq for some important reactionsa.

aSource: ZITAT: Lehninger, A.L. Biochemistry, 2nd edition, New York, Worth, 1975, p. 397.

A biochemical reaction is reversible if it may proceed in both directions, leading to a positive or negative sign of the rate v. The actual direction depends on the current reactant concentrations. In theory, every reaction should be reversible. In practice, we can consider many reactions as irreversible since (i) reactants in cellular environment cannot assume any concentration, (ii) coupling of a chemical conversion to ATP consumption leads to a severe drop in free energy and therefore makes a reaction reversal energetically unfavorable, and (iii) for compound destruction, such as protein degradation, reversal by chance is extremely unlikely.

The detailed consideration of enzyme mechanisms by applying the mass action law for the single events has led to a number of standard kinetic descriptions, which will be explained in the following.

2.1.3 Michaelis–Menten Kinetics

Brown [4] proposed an enzymatic mechanism for invertase, catalyzing the cleavage of saccharose to glucose and fructose. This mechanism holds in general for all one-substrate reactions without backward reaction and effectors, such as

(2.11)

It comprises a reversible formation of an enzyme–substrate complex ES from the free enzyme E and the substrate S and an irreversible release of the product P. The ODE system for the dynamics of this reaction reads

(2.12)

(2.13)

(2.14)

(2.15)

The reaction rate is equal to the negative decay rate of the substrate as well as to the rate of product formation:

(2.16)

This ODE system (Eqs. (2.12)–(2.16)) cannot be solved analytically. Different assumptions have been used to simplify this system in a satisfactory way. Michaelis and Menten [5] considered a quasi-equilibrium between the free enzyme and the enzyme–substrate complex, meaning that the reversible conversion of E and S to ES is much faster than the decomposition of ES into E and P, or in terms of the kinetic constants,

(2.17)

Briggs and Haldane [6] assumed that during the course of reaction a state is reached where the concentration of the ES complex remains constant, the so-called quasi-steady state. This assumption is justified only if the initial substrate concentration is much larger than the enzyme concentration, S(t = 0) ≫ E, otherwise such a state will never be reached. In mathematical terms, we obtain

(2.18)

In the following, we derive an expression for the reaction rate from the ODE system (2.12)–(2.15) and the quasi-steady-state assumption for ES. First, adding Eqs. (2.13) and (2.14) results in

(2.19)

This expression shows that enzyme is neither produced nor consumed in this reaction; it may be free or part of the complex, but its total concentration remains constant. Introducing (2.19) into (2.13) under the steady-state assumption (2.18) yields

(2.20)

For the reaction rate, this gives

(2.21)

In enzyme kinetics, it is convention to present Eq. (2.21) in a simpler form, which is important in theory and practice

(2.22)

Equation (2.22) is the expression for Michaelis–Menten kinetics. The parameters have the following meaning: the maximal velocity,

(2.23)

is the maximal rate that can be attained, when the enzyme is completely saturated with substrate. The Michaelis constant,

(2.24)

is equal to the substrate concentration that yields the half-maximal reaction rate. For the quasi-equilibrium assumption (Eq. (2.17)), it holds that Km ≅ k−1/k1. The maximum velocity divided by the enzyme concentration (here k2 = vmax/Etotal) is often called the turnover number, kcat. The meaning of the parameters is illustrated in the plot of rate versus substrate concentration (Figure 2.2).

Figure 2.2 Dependence of reaction rate v on substrate concentration S in Michaelis–Menten kinetics. Vmax denotes the maximal reaction rate that can be reached for large substrate concentration. Km is the substrate concentration that leads to half-maximal reaction rate. For low substrate concentration, v increases almost linearly with S, while for high substrate concentrations v is almost independent of S.

2.1.3.1 How to Derive a Rate Equation

Below, we will present some enzyme kinetic standard examples to derive a rate equation. Individual mechanisms for your specific enzyme of interest may be more complicated or merely differ from these standards. Therefore, we summarize here the general way of deriving a rate equation.

1. Draw a wiring diagram of all steps to consider (e.g., Eq. (2.11)). It contains all substrates and products (S and P) and n free or bound enzyme species (E and ES).

2. The right sites of the ODEs for the concentrations changes sum up the rates of all steps leading to or away from a certain substance (e.g., Eqs. (2.12)–(2.15)). The rates follow mass action kinetics (Eq. (2.3)).

3. The sum of all enzyme-containing species is equal to the total enzyme concentration Etotal (the right site of all differential equations for enzyme species sums up to zero). This constitutes one equation.

4. The assumption of quasi-steady state for n − 1 enzyme species (i.e., setting the right sites of the respective ODEs equal to zero) together with (3.) result in n algebraic equations for the concentrations of the n enzyme species.

5. The reaction rate is equal to the rate of product formation (e.g., Eq. (2.16)). Insert the respective concentrations of enzyme species resulting from (4.).

2.1.3.2 Parameter Estimation and Linearization of the Michaelis–Menten Equation

To assess the values of the parameters Vmax and Km for an isolated enzyme, one measures the initial rate for different initial concentrations of the substrate. Since the rate is a nonlinear function of the substrate concentration, one has to determine the parameters by nonlinear regression. Another way is to transform Eq. (2.22) to a linear relation between variables and then apply linear regression.

The advantage of the transformed equations is that one may read the parameter value more or less directly from the graph obtained by linear regression of the measurement data. In the plot by Lineweaver and Burk [7] (Table 2.2), the values for Vmax and Km can be obtained from the intersections of the graph with the ordinate and the abscissa, respectively. The Lineweaver–Burk plot is also helpful to easily discriminate different types of inhibition (see below). The drawback of the transformed equations is that they may be sensitive to errors for small or high substrate concentrations or rates. Eadie and Hofstee [8] and Hanes and Woolf [9] have introduced other types of linearization to overcome this limitation.

Table 2.2 Different approaches for the linearization of Michaelis–Menten enzyme kinetics.

2.1.3.3 The Michaelis–Menten Equation for Reversible Reactions

In practice, many reactions are reversible. The enzyme may catalyze the reaction in both directions. Consider the following mechanism:

(2.25)

The product formation is given by

(2.26)

The respective rate equation reads

(2.27)

While the parameters k±1 and k±2 are the kinetic constants of the individual reaction steps, the phenomenological parameters Vformax and Vbackmax denote the maximal velocity in forward or backward direction, respectively, under zero product or substrate concentration, and the phenomenological parameters KmS and KmP denote the substrate or product concentration causing half maximal forward or backward rate. They are related in the following way [10]:

(2.28)

2.1.4 Regulation of Enzyme Activity by Effectors

Enzymes may immensely increase the rate of a reaction, but this is not their only function. Enzymes are involved in metabolic regulation in various ways. Their production and degradation is often adapted to the current requirements of the cell. Furthermore, they may be targets of effectors, both inhibitors and activators.

The effectors are small molecules, or proteins, or other compounds that influence the performance of the enzymatic reaction. The interaction of effector and enzyme changes the reaction rate. Such regulatory interactions that are crucial for the fine-tuning of metabolism will be considered here [11].

Basic types of inhibition are distinguished by the state, in which the enzyme may bind the effector (i.e., the free enzyme E, the enzyme–substrate complex ES, or both), and by the ability of different complexes to release the product. The general pattern of inhibition is schematically represented in Figure 2.3. The different types result, if some of the interactions may not occur.

Figure 2.3 General scheme of inhibition in Michaelis–Menten kinetics. Reactions 1 and 2 belong to the standard scheme of Michaelis–Menten kinetics. Competitive inhibition is given, if in addition reaction 3 (and not reactions 4, 5, or 6) occurs. Uncompetitive inhibition involves reactions 1, 2, and 4, and noncompetitive inhibition comprises reactions 1, 2, 3, 4, and 5. Occurrence of reaction 6 indicates partial inhibition.

The rate equations are derived according to the following scheme:

1. Consider binding equilibriums between compounds and their complexes:

(2.29)

Note that, if all reactions may occur, the Wegscheider condition [12] holds in the form

(2.30)

which means that the difference in the free energies between two compounds (e.g., E and ESI) is independent of the choice of the reaction path (here via ES or via EI).

2. Take into account the moiety conservation for the total enzyme (include only those complexes, which occur in the course of reaction):

(2.31)

3. The reaction rate is equal to the rate of product formation

(2.32)

Equations (2.29)–(2.31) constitute four independent equations for the four unknown concentrations of E, ES, EI, and ESI. Their solution can be inserted into Eq. (2.32). The effect of the inhibitor depends on the concentrations of substrate and inhibitor and on the relative affinities to the enzyme. Table 2.3 lists the different types of inhibition for irreversible and reversible Michaelis–Menten kinetics together with the respective rate equations.

Table 2.3 Types of inhibition for irreversible and reversible Michaelis–Menten kineticsa.

aThese abbreviations are used:

In the case of competitive inhibition, the inhibitor competes with the substrate for the binding site (or inhibits substrate binding by binding elsewhere to the enzyme) without being transformed itself. An example for this type is the inhibition of succinate dehydrogenase by malonate. The enzyme converts succinate to fumarate forming a double bond. Malonate has two carboxyl groups, like the proper substrates, and may bind to the enzyme, but the formation of a double bond cannot take place. Since substrates and inhibitor compete for the binding sites, a high concentration of one of them may displace the other one. For very high substrate concentrations, the same maximal velocity as without inhibitor is reached, but the effective Km value is increased.

In the case of uncompetitive inhibition, the inhibitor binds only to the ES complex. The reason may be that the substrate binding caused a conformational change, which opened a new binding site. Since S and I do not compete for binding sites, an increase in the concentration of S cannot displace the inhibitor. In the presence of inhibitor, the original maximal rate cannot be reached (lower Vmax). For example, an inhibitor concentration of I = KI, 4 halves the Km-value as well as Vmax. Uncompetitive inhibition occurs rarely for one-substrate reactions, but more frequently in the case of two substrates. One example is inhibition of arylsulphatase by hydracine.

Noncompetitive inhibition is present, if substrate binding to the enzyme does not alter the binding of the inhibitor. There must be different binding sites for substrate and inhibitor. In the classical case, the inhibitor has the same affinity to the enzyme with or without bound substrate. If the affinity changes, this is called mixed inhibition. A standard example is inhibition of chymotrypsion by H+ -ions.

If the product may also be formed from the enzyme–substrate–inhibitor complex, the inhibition is only partial. For high rates of product release (high values of k6), this can even result in an activating instead of an inhibiting effect.

The general types of inhibition, competitive, uncompetitive, and noncompetitive inhibition also apply for the reversible Michaelis–Menten mechanism. The respective rate equations are also listed in Table 2.3.

2.1.4.1 Substrate Inhibition

A common characteristic of enzymatic reaction is the increase of the reaction rate with increasing substrate concentration S up to the maximal velocity Vmax. But in some cases, a decrease of the rate above a certain value of S is recorded. A possible reason is the binding of a further substrate molecule to the enzyme–substrate complex yielding the complex ESS that cannot form a product. This kind of inhibition is reversible if the second substrate can be released. The rate equation can be derived using the scheme of uncompetitive inhibition by replacing the inhibitor by another substrate. It reads

(2.33)

This expression has an optimum, i.e., a maximal value of v, at

(2.34)

The dependence of v on S is shown in Figure 2.4. A typical example for substrate inhibition is the binding of two succinate molecules to malonate dehydrogenase, which possesses two binding pockets for the carboxyl group. This is schematically represented in Figure 2.4.

Figure 2.4 Plot of reaction rate v against substrate concentration S for an enzyme with substrate inhibition. The upper curve shows Michaelis–Menten kinetics without inhibition, the lower curves show kinetics for the indicated values of binding constant KI. Parameter values: Vmax = 1, Km = 1. The left part visualizes a possible mechanism for substrate inhibition: The enzyme (gray item) has two binding pockets to bind different parts of a substrate molecule (upper scheme). In case of high substrate concentration, two different molecules may enter the binding pockets, thereby preventing the specific reaction (lower scheme).

2.1.4.2 Binding of Ligands to Proteins

Every molecule that binds to a protein is a ligand, irrespective of whether it is subject of a reaction or not. Below we consider binding to monomer and oligomer proteins. In oligomers, there may be interactions between the binding sites on the subunits.

Consider binding of one ligand (S) to a protein (E) with only one binding site:

(2.35)

The binding constant KB is given by

(2.36)

The reciprocal of KB is the dissociation constant KD. The fractional saturation Y of the protein is determined by the number of subunits that have bound ligands, divided by the total number of subunits. The fractional saturation for one subunit is

(2.37)

The plot of Y versus S at constant total enzyme concentration is a hyperbola, like the plot of v versus S in the Michaelis–Menten kinetics (Eq. (2.22)). At a process where the binding of S to E is the first step followed by product release and where the initial concentration of S is much higher than the initial concentration of E, the rate is proportional to the concentration of ES and it holds

(2.38)

If the protein has several binding sites, then interactions may occur between these sites, i.e., the affinity to further ligands may change after binding of one or more ligands. This phenomenon is called cooperativity. Positive or negative cooperativity denote increase or decrease in the affinity of the protein to a further ligand, respectively. Homotropic or heterotropic cooperativity denotes that the binding to a certain ligand influences the affinity of the protein to a further ligand of the same or another type, respectively.

2.1.4.3 Positive Homotropic Cooperativity and the Hill Equation

Consider a dimeric protein with two identical binding sites. The binding to the first ligand facilitates the binding to the second ligand.

(2.39)

where E is the monomer and E2 is the dimer. The fractional saturation is given by

(2.40)

If the affinity to the second ligand is strongly increase by binding to the first ligand, then E2S will react with S as soon as it is formed and the concentration of E2S can be neglected. In the case of complete cooperativity, i.e., every protein is either empty or fully bound, Eq. (2.39) reduces to

(2.41)

The binding constant reads

(2.42)

and the fractional saturation is

(2.43)

Generally, for a protein with n subunits, it holds:

(2.44)

This is the general form of the Hill equation. To derive it, we assumed complete homotropic cooperativity. The plot of the fractional saturation Y versus substrate concentration S is a sigmoid curve with the inflection point at 1/KB. The quantity n (often "h" is used instead) is termed the Hill coefficient.

The derivation of this expression was based on experimental findings concerning the binding of oxygen to hemoglobin (Hb) [13, 14]. In 1904, Bohr et al. found that the plot of the fractional saturation of Hb with oxygen against the oxygen partial pressure had a sigmoid shape. Hill (1913) explained this with interactions between the binding sites located at the Hb subunits [14]. At this time, it was already known that every subunit Hb binds one molecule of oxygen. Hill assumed complete cooperativity and predicted an experimental Hill coefficient of 2.8. Today it is known that Hb has four binding sites, but that the cooperativity is not complete. The sigmoid binding characteristic has the advantage that Hb binds strongly to oxygen in the lung with a high