Overview

Pharmacokinetics is the study of how substances such as drugs and other xenobiotic compounds are transported within living organisms, particularly human beings and animals used as research models. A pharmacokinetic model for a particular drug provides the ability to simulate introduction of the drug into the body (through injection into soft tissue, intravenous administration, absorption from the gut), transport via the circulatory system and delivery to various tissues, excretion (e.g., by the kidney), and perhaps transformation via chemical reactions. Pharmacokinetic models can be useful in predicting and understanding the timescales and concentrations at which drugs appear and disappear from various regions of the body, for setting dosing guidelines for pharmaceuticals and exposure guidelines for toxic compounds, and as a basic research tool for guiding experimental design and analyzing data, as we will see in the examples in this chapter.

Physiologically based pharmacokinetic (PBPK) models are distinguished by their basis in the anatomical and physiological characteristics of the simulated organism. The structure of a PBPK model is derived directly from anatomy and physiology: transport and reactions are simulated in virtual organs that represent the individual organs of the body, and are connected according to the plumbing of the circulatory system; simulated flows mimic physiological flows of blood, gases, and possibly other fluids. Traditional pharmacokinetic models, in contrast, tend to lump flows, organs, and physiological processes together in models that invoke fewer variables and parameters in an effort to minimize the complexity necessary to simulate a given data set or type of data set.

Overview

This and the following two chapters are focused on analyzing and simulating chemical systems. These chapters will introduce basic concepts of thermodynamics and kinetics for application to biochemical systems, such as biochemical synthesis, cellular metabolism and signaling processes, and gene regulatory networks. Although we have seen examples of chemical kinetics in previous chapters, notably in Sections 2.3 and 2.4, in those examples we developed the expressions governing the chemistry more from intuition than from a physical theory. One of the primary goals here will be to develop a formal physical/chemical foundation for analyzing and simulating complex biochemical systems.

As is our practice throughout this book, these concepts will be applied to analyze real data (and understand the behavior of real systems) later in this chapter and elsewhere. Yet, because the rules governing the behavior of biochemical systems are grounded in thermodynamics, we must begin our investigation into chemical systems by establishing some fundamental concepts in chemical thermodynamics. The concept of free energy is particularly crucial to understanding thermodynamic driving forces in chemistry. We will see that both a physical definition and an intuitive understanding of free energy require physical definitions and intuitive understandings of temperature and entropy. All of this means that this chapter will begin with some abstract thought experiments and derivations of physical concepts.

Temperature, pressure, and entropy

Microstates and macrostates

All thermodynamic theory arises from the fact that physical systems composed of many atoms and/or molecules attain a large number (often a practically infinite number) of microstates under defined macroscopic conditions, such as temperature, pressure, and volume.

Overview

This chapter is devoted to the study and simulation of the electrical potential across cell membranes. Practically every type of animal cell actively maintains an electrostatic potential difference between its cytoplasm and external milieu. Organelles in the cell, such as mitochondria for example, maintain electrical gradients across their membranes as well. The electrostatic gradient across cell membranes changes dynamically in response to chemical, mechanical, and/or electrical stimuli. In turn, dynamic changes in cell membrane potential are tied to all manner of downstream processes in various cell types. For example, muscle cells contract when calcium ions enter the cytoplasm in response to depolarization of the cell membrane potential. Depolarization, caused by the influx of sodium and other positively charged ions, may be initiated by the combined action of several processes, including stimulation by neurotransmitters and transmission of the electrical signal from neighboring cells.

The focus here is on simulating the dynamics of the cell membrane potential difference, and the ion currents that influence changes in the potential difference. A substantial fraction of the chapter is devoted to working through the details of the celebrated Hodgkin–Huxley model of the nerve impulse. This exposition allows us to introduce the basic formulation of that classic model, which is still widely used in the field. Reviewing the work of Hodgkin and Huxley allows us to see how their model was developed based on data from experiments carefully designed to reveal a quantitative description of the biophysical processes at work.

Research, development, and design in bioengineering, biomedical engineering, biophysics, physiology, and related fields rely increasingly on mathematical modeling and computational simulation of biological systems. Simulation is required to analyze data, design experiments, develop new technology, and simply to attempt to understand the complexity inherent in biological systems.

This book focuses on practical implementation of techniques to study real biological systems. Indeed, whenever possible, specific applications are developed, starting with a study of the basic operation of the underlying biological, biochemical, or physiological system and, critically, the available data. It is hoped that this data-rich exposition will yield a practical text for engineering students and other readers interested primarily in earthy real-world applications such as analyzing data, estimating parameter values, etc. Thus for the examples developed here, important details of underlying biological systems are described along with a complete step-by-step development of model assumptions, the resulting equations, and (when necessary) computer code. As a result, readers have the opportunity, by working through the examples, to become truly proficient in biosimulation.

In this spirit of soup-to-nuts practicality, the book is organized around biological and engineering application areas rather than based on mathematical and computational techniques. Where specific mathematical or computational techniques can be conveniently and effective separated from the exposition, they have been and can be found in the Appendices. Computer codes implemented in MATLAB® (The MathWorks, Natick, MA, USA) for all of the examples in the text can be found online at the URL http://www.cambridge.org/biosim.

Overview

So far in the examples of biochemical systems that we have studied, we have looked at systems of one or a handful of reactions. Yet the ultimate quest to understand (and simulate and predict) cellular function calls for the capability to synthesize many thousands of simultaneous chemical reaction and transport processes. Can we realistically expect to simulate cellular biochemistry with many important reactions accounted for at the level of rigor and detail of the examples in the previous two chapters? The answer is a qualified “yes,” because in principle there is no reason why realistic simulations of systems of hundreds or thousands of reactions cannot be constructed. One approach to doing to is to adopt a systematic approach based on the fundamentals developed in the previous two chapters. This approach is practical when data exist for building the necessary thermodynamic and kinetic models of the individual enzymes and transporters at the level of detail of the model of fumarase in Chapter 6. However, when that level of detail is either not possible or not practical, model-based simulations may still be developed.

This chapter describes examples of two contrasting approaches to simulation and analysis of large-scale systems – a simulation of the kinetics of a metabolic pathway based on detailed biochemical thermodynamics and kinetics, and a less biochemically rigorous analysis of gene expression data to identify regulatory network structure.

Overview

This chapter is built around analyzing a real data set obtained from a real biological system to illustrate several complementary approaches to simulation and analysis. The particular system studied (a home aquarium) is a well-mixed chemical reactor. Or, more accurately, the system studied is treated as a well-mixed chemical reactor, a basic modeling paradigm that will appear again and again in this book.

Here, we look at this single physical system from several different perspectives (that is, under different sets of underlying modeling assumptions) with the aim of motivating the reader to undertake the study of the rest of this book. The aim is not to overwhelm the reader with mathematical details that can be found in later chapters. Therefore let us clearly state at the outset: it is not expected or required that the reader follow every detail of the examples illustrated here. Instead, we invite the reader to focus on the basic assumptions underlying the methods applied, and to compare and contrast the results that are obtained based on these different approaches. Proceeding this way, it is hoped that the reader may gain an appreciation of the breath of the field. Furthermore, it is hoped that this appreciation will continue to grow with a study of the rest of this book and beyond.

Modeling approaches

The number of different approaches to simulating biosystems behavior is perhaps greater than the number of biological systems.

Most cells have a complex internal structure of biological rods, ropes and sheets. In terms of the metaphors for the cell in Chapter 1 – a hot air balloon, a sailing ship – our task in Parts I and II of this text was to analyse the individual structural elements of the cell – the masts, rigging and hull – without regard to their interconnection. In Part III, we begin to assemble the units together to form composite systems. As the introduction to Part III, Chapter 10 treats the equilibrium shapes of the simplest systems, namely cells or vesicles without a nucleus or space-filling cytoskeleton. A thorough treatment of complex cells is beyond the reach of our analytical tools; rather, we mention a selection of their properties in Chapters 11 and 12. In Chapter 11, we describe mechanisms that cells have developed for changing their shape, as part of cell division or locomotion, for example. Cell division proper is the subject of Chapter 12, including changes to the division cycle over the history of the Earth. Lastly, in Chapter 13, we study methods that have evolved for control and organization in the cell, using specific situations as examples.

The variety of single cells capable of living independently is truly impressive, from small featureless mycoplasmas just half a micron across, to elegant protists a hundred times that size, perhaps outfitted with tentacles and even a mouth. Organisms such as ourselves, however, are multicellular, emphatically so at 1014cells per human. How do our cells interact, and adhere when appropriate? In this chapter, we investigate intermembrane forces at large and small separations, beginning with a survey of the different features of membrane interactions in Section 9.1. The Poisson–Boltzmann equation, which we derive in Section 9.2, provides our primary framework for treating membrane interactions at non-zero temperature. This approach is applied to the organization of ions near a single charged plate and to the electrostatic pressure between charged walls in Sections 9.2 and 9.3, respectively. Two other contributors to membrane interactions also are treated in some mathematical detail here: the van der Waals attraction between rigid sheets (Section 9.3) and the steric repulsion experienced by undulating membranes (Section 9.4). Lastly, the adhesion of a membrane to a substrate, and its effect on cell shape, is described in Section 9.5. Our presentation of membrane forces is necessarily limited; the reader is referred to Israelachvili (1991) or Safran (1994) for more extensive treatments.

Interactions between membranes

Let’s briefly review the molecular composition of a conventional plasma membrane. Fundamental to a biomembrane is the lipid bilayer, a pair of two-dimensional fluid leaflets described at length in Chapter 7. The lipid composition of the leaflets is inequivalent: in the human red blood cell, one of the best-characterized examples, the phospholipids in the outer leaflet are predominantly cholines, while lipids containing ethanolamine or serine dominate the inner leaflet facing the cytosol (see Appendix B for head-group structures). Cholesterol also may be abundant in the plasma membrane. The head-group of phosphatidylserine, but not the others, is negatively charged, providing a surface charge density ranging up to 0.3 C/m2 if every lipid carries a single charge.

This chapter covers the quantitative aspect of agonist activation: how the blood, and by extension the interstitial fluid, presents a pharmaceutical agent to the cell; quantification of drug concentration and tissue response; diffusion and elimination of an intracellular agent; statistical analysis of the tissue response; and the difficult problems associated with drug development. Because of the necessary role played by agonist receptors in this process, the background of some of the major receptor types is also included.

Membrane receptor proteins

Agonists initiate a cell response by binding to a receptor. Without trying to give a complete list of every kind of receptor, three of the major categories of receptors are the G-protein coupled receptors (GPCRs), the tyrosine kinase receptors, and nuclear receptors. GPCRs dominate cell activation using the temporally limited G-protein system. There are more than 800 GPCR proteins, and the complexity of their activation scheme is daunting. Tyrosine kinase receptors are activated by insulin and a variety of growth factors, and activate processes associated with the structural growth of cells and tissues. Nuclear receptors bind to hydrophobic hormones, steroids, thyroid hormones, etc., that do their work by initiating gene transcription. This section gives a brief overview of these receptor types, with the understanding that all of them have far more complexity than can be covered here. Recognizing the pivotal role these molecules play in linking the quantitative aspects of cell activation, pharmacokinetics, dose–response relations, intracellular diffusion, and data analysis of biological responses, it is appropriate that a general discussion of their functions is included.

Nothing happens without the expenditure of energy. Humans use the energy in ATP to drive all of the reactions in the body, either directly by using the energy of the phosphate bond between the second and third phosphates of ATP, or by siphoning energy from chemical and electrical gradients produced by ATP hydrolysis. Before ATP can be utilized it has to be produced. ATP is made anaerobically through glycolysis, and aerobically using oxidative phosphorylation in mitochondria. Once made, its concentration can be buffered in many cells through the creatine kinase reaction, with phosphocreatine contributing its phosphate to ADP to maintain the ATP concentration during periods of high energy expenditure. Energy expenditure is not solely dependent on the supply of ATP, however. The removal of the products of energy use, especially inorganic phosphate, are as important to function as the supply of ATP. The different components of ATP production and buffering, and inorganic phosphate removal, directly influence human athletic performance. The different phases that phosphocreatine, glycolysis, and oxidative phosphorylation control are graphically demonstrated in the rate of running in world track records as a function of the log of the distance run.

Energetics of human performance

The most interesting races on the track take place when someone goes out fast, takes an early lead, and then is run down by the field. Will the leader hang on, or will someone catch up with a strong finishing kick? The factors involved in human performance include ability, training and motivation. For most people, psychological limitations play a major role in reducing performance. Training and experience produce the confidence needed to reach the fastest time possible. World class runners are “world class” precisely because they have overcome the psychological limitations on performance, and their racing is only limited by physiological factors. Physiological factors include some elements that can be controlled, like rest and diet, and others that cannot, like leg length. The variable elements ultimately involve energy. How much energy is available to drive the myosin ATPase reactions that move muscle? At what rate are the energy supplies used? Can the energy supplies be increased to prolong top flight performance? The starting point for answers to these questions is to look at the relation of the rate of running of world track records as a function of the race distances in Figure 4.1.

The filaments of the cell vary tremendously in their bending resistance, having a visual appearance ranging from ropes to threads if viewed in isolation on the length scale of a micron. Collections of these biological filaments have strikingly different structures: a bundle of stiff microtubules may display strong internal alignment, whereas a network of very flexible proteins may resemble the proverbial can of worms. Thus, the elastic behavior of multi-component networks containing both stiff and floppy filaments may include contributions from the energy and the entropy of their constituents. In this chapter, we first review a selection of three-dimensional networks from the cell, and then establish the elastic properties of four different model systems, ranging from entropic springs to rattling rods. In the concluding section, these models are used to interpret, where possible, the measured characteristics of cellular networks.

Networks of biological rods and ropes

The filaments of the cytoskeleton and extracellular matrix collectively form a variety of chemically homogeneous and heterogeneous structures. Let’s begin our discussion of these structures by describing two networks of microtubules found in the cell. The persistence length of microtubules is of the order of millimeters, such that microtubules bend only gently on the scale of microns and will not form contorted networks (see Section 3.6). For example, the microtubules of the schematic cell in Fig. 6.1(a) are not cross-linked, but rather extend like spikes towards the cell boundary, growing and shrinking with time. However, there exist varieties of microtubule-associated proteins (or MAPs) that can form links between microtubules, resulting in a bundled structure like Fig. 6.1(b): linking proteins are spaced along the microtubule like the rungs of a ladder, tying two microtubules parallel to one another. Such microtubule bundles are present in the long axon of a nerve cell or in the whip-like flagella extending from some cells. Neither of these configurations resembles the regular meshwork of the erythrocyte cytoskeleton or the nuclear lamina described in Section 5.1. The function of the microtubules in both the radial and parallel structures includes transport: different motor proteins can walk along the tubule, as described further in Chapter 11.

With some notable exceptions, such as DNA, most filaments in the cell are linked together as part of a network, which may extend throughout the interior of the cell, or be associated with a membrane as an effectively two- dimensional structure such as shown inFig. 5.1. This chapter concentrates on the properties of planar networks, particularly the temperature- and stress- dependence of their geometry and elasticity. To allow sufficient time to develop the theoretical framework of elasticity, we focus in this chapter only on networks having uniform connectivity, namely those with the four-fold and six-fold connectivity found in a number of biologically important cells.

Soft networks in the cell

Two-dimensional networks arise in a variety of situations in the cell; they may be attached to its plasma or nuclear membrane, or be wrapped around a cell as its wall. Containing neither a nucleus nor other cytoskeletal components such as microtubules, for example, the human red blood cell possesses only a membrane-associated cytoskeleton. Composed of tetramers of the protein spectrin, the erythrocyte cytoskeleton is highly convoluted in vivo (see Fig. 3.17(b)), but can be stretched by about a factor of seven in area to reveal its relatively uniform four- to six-fold connectivity, as shown in Fig. 5.1(a) (Byers and Branton, 1985; Liu et al., 1987; Takeuchi et al., 1998). Roughly midway along their 200 nm contour length, the spectrin tetramers are attached to the plasma membrane by the protein ankyrin (using another protein called band 3 as an intermediary). About 120 000 tetramers cover the 140 μm2 membrane area of a typical erythrocyte, corresponding to a tetramer density of about 800 μm−2. The tetramers are attached to one another at junction complexes containing actin segments 33–37 nm long; perhaps 35 000 junction complexes, with an average separation of about 75 nm, cover the erythrocyte.

One of the hallmarks of physiological systems is homeostasis. Homeostatic systems maintain a steady-state set point over time using negative feedback. Negative feedback activates recovery processes when there is deviation from the set point. While some states are always tightly maintained, some changes can occur during the lifetime of the individual. These changes take two forms: adjustment of the homeostatic set point, or a step change to a new set point by a positive feedback mechanism. Positive feedback systems do not use a recovery process to return to the original set point. New negative feedback processes now keep the organism at the new set point. A set point must have an energy minimum, such that small variations from the minimum create a driving force returning the system to the set point. Energy minima occur at multiple levels, from atoms to organisms. For systems in transition the energy profile may exhibit metastable states between two large minima. Metastable states offer temporal buffering of state changes. In enzymes and ion channels the energy barriers between the external states and internal metastable states and the depth of the metastable state control the rate of the metabolic reaction or ionic current. Complex analytical methods such as catastrophe theory provide insight into state transitions. Catastrophe theory models changes in space, not time, with state transitions occurring whenever the state variable reaches particular values in parameter space. The feedback systems that maintain homeostasis are deterministic far from equilibrium, but become chaotic when transitions are near equilibrium. Not all catastrophic transitions are fatal, but for living systems catastrophes can lead to death in two ways, homeostatic instability or global thermal equilibrium. These two modes of death correspond to having too much ambient energy or too little ambient energy. This leads to the Goldilocks problem, in which ambient energy will be too high, too low or just right. Since life must be a continuum, a system cannot leave the viable energy range without extinction, leading to restrictions in modeling evolution. All living systems will have feedback systems, both negative and positive, that keep them away from either of the life–death transitions. Fractal systems regularly transition between deterministic and chaotic states. Conditions with multiple fractal inputs can have significant variability around a set point using allometric regulation. Narrowing of the fractal distribution has been associated with pathological states and increased risk of death.

The polymers and biofilaments described in Chapter 3 are structurally simple, such that the deformation energy of their conformations depends mainly on their resistance to bending. Consequently, the suite of shapes adopted by such filaments represents a balance between bending energy and entropy, and unless the filaments possess permanent kinks, then they are straight at low temperature. In this chapter, we consider filaments with more complex structures that arise because of their torsion resistance or because of the attraction between non-adjacent locations of the filaments. Torsion resistance can occur even in polymers because of constraints on the rotational freedom of the polymer’s backbone arising from torsion- resistant bonds or from steric constraints among side-groups to the backbone. However, in this chapter we are mainly concerned with the classic torsion resistance of thick filaments associated with shear resistance of their construction material.

This chapter begins with a descriptive overview of structures such as single and multiple helices that are observed in macromolecules and cellular filaments on length scales of one to ten nanometers. The mathematical description of helical shapes begins in Section 4.2, including the topological characteristics of twist, writhe and linking number. Then in Section 4.3, the deformation energy of a system subject to torsion is formally presented, permitting the torsional rigidity to be expressed in terms of elastic moduli such as the shear modulus and Young’s modulus. Experimental measurements of the torsional rigidity are also summarized in this section. The complex architecture of globular proteins appears in both their secondary structure (including helical segments) as well their tertiary structure (overall shape). Experimental measurements of the unfolding of protein segments with secondary structure is studied in Section 4.4, while the qualitative features of RNA folding and protein folding into tertiary structure are laid out in Section 4.5. The origin of these folded structures is the effective attraction of sections of the molecule that are not immediately adjacent to each other. To quantitatively understand what type of structures arise from such interactions, a version of the HP model (for Hydrophobic/Polar model) is studied in Section 4.6. The chapter is summarized before the end-of-chapter problems.

Cells are more than just passive objects responding to external stresses: they can actively change shape or move with respect to their environment. A very familiar example of cellular shape change is the contraction of our muscle cells. Less familiar, but very important to our health, is the locomotion of cells such as macrophages, which work their way through our tissues to capture and remove hostile cells and material. Another example of cell movement is propulsion in a fluid medium, which in some cells is provided by the rotation of flagella (Latin plural for the noun whip) extending from the cell body. Flagella can be seen from several cells of the bacterium Brevundimonas alba in Fig. 11.1. Structurally related to flagellaare cilia(Latin plural for eyelash), which occur on the surfaces of some cells and wave in synchrony like tall grass in the wind, creating currents in their fluid environment. What microscopic mechanisms underlie a cell’s movement or motility?

In Section 11.1, we review several aspects of cell movement and trace their origin to the molecular level. Dynamic biofilaments alone can be responsible for certain shape changes because they can generate a force by increasing their length, as described in Section 11.2. However, contractile forces in muscles, or transport along a microtubule, arise from specialized motor proteins capable of crawling along a biofilament. A number of models have been proposed for the operation of such molecular motors, and these are explained in Section 11.3. Measurements of motile forces, and the effect of dynamic structural elements on cell shape are presented in Section 11.4. Lastly, several mechanisms that can provide cell movement are analyzed in Section 11.5.