Membranes of the cell are characterized by several elastic parameters, such as the area compression modulus, that reflect the membrane's quasi-two-dimensional structure. As described in Chapter 5, these parameters have small values for a lipid bilayer just 4–5 nm thick, yet they properly describe the energetics of membrane deformation at zero temperature where thermal fluctuations in shape are unimportant. But what happens at finite temperature? In the discussion of polymer and network elasticity in Part I, we saw that the entropic contribution to the elasticity of very flexible filaments is significant at ambient temperatures, owing to the large configuration space that these filaments can explore. Do we expect similar behavior for flexible sheets? In this chapter, we develop a mathematical description of surfaces, and explore the characteristics of membrane undulations. Membranes are treated in isolation here, and in interaction with other surfaces in Chapter 8. For a general review of membrane fluctuations, see Leibler (1989).

Thermal fluctuations in membrane shape

The bending rigidity kb of a phospholipid bilayer lies close to 10−19 J, or 10–20 kBT at ambient temperatures, as summarized in Table 5.3. What does such a small value of kb imply about the undulations of a membrane with the dimensions of a cell? For illustration, we calculate the change in energy of the flat, disk-shaped membrane in Fig. 6.1(a) as it is deformed into the surface of constant curvature in Fig. 6.1(b).

The number of cells in the human body is literally astronomical, about three orders of magnitude more than the number of stars in the Milky Way. Yet, for their immense number, the variety of cells is much smaller: only about 200 different cell types are represented in the collection of about 1014 cells that make up our bodies. These cells have diverse capabilities and, superficially, have remarkably different shapes, as illustrated in Fig. 1.1. Some cells, like certain varieties of bacteria, are not much more than inflated bags, shaped like the hot-air or gas balloons invented more than two centuries ago. Others, such as nerve cells, may have branched structures at each end connected by an arm that is more than a thousand times long as it is wide. The basic structural elements of most cells, however, are the same: fluid sheets enclose the cell and its compartments, while networks of filaments maintain the cell's shape and help organize its contents. Further, the chemical composition of these structural elements bears a strong family resemblance from one cell to another, perhaps reflecting the evolution of cells from a common ancestor; for example, the protein actin, which forms one of the cell's principal filaments, is found in organisms ranging from yeasts to humans.

The many chemical and structural similarities of cells tempt us to search for systematics in their architecture and components.

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 multicomponent 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 2.5). For example, the microtubules of the schematic cell in Fig. 4.1(a) are not cross-linked, but rather extend like spikes towards the cell boundary, growing and shrinking with time.

Focusing on the mechanical operation of a cell, this text tends to emphasize aspects of structure and organization common to all cells. However, cells differ in many details according to their function, evolution and environment, such that within a multicellular organism such as ourselves, there may be 100 or 200 hundred different cell types. Because frequent reference is made to these cell types in the text, we provide in this appendix a brief survey of animal cells and their organization in tissues. Readers seeking more than this cursory introduction to cell biology should consult one of many excellent textbooks, such as Alberts et al. (1994), Goodsell (1993), or Prescott et al. (1996).

Tissues

Cells act cooperatively in a multicellular system, and are hierarchically organized into tissues, organs and organ systems. Tissues contain the cells themselves, plus other material such as the extracellular matrix secreted by the cells. Several different tissues may function together as an organ, usually with one tissue acting as the “skin” of the organ and providing containment for other tissues in the interior. Lastly, organs themselves may form part of an organ system such as the respiratory system. Animal tissues are categorized as epithelial, connective, muscle or nervous tissue, whose relationship within an organ (in this case the intestinal wall) is illustrated in Fig. A.1. The intestine is bounded on the inside and outside by epithelial cell sheets (or epithelia), wherein epithelial cells are joined tightly to each other in order to restrict the passage of material out of the intestinal cavity.

The cells of our bodies represent a very large class of systems whose structural components often are both complex and soft. A system may be complex in the sense that it may comprise several components having quite different mechanical characteristics, with the result that the behavior of the system as a whole reflects an interplay between the characteristics of the components in isolation. Further, the individual structural units of biological systems tend to be soft: for example, the compression resistance of a protein network may be more than an order of magnitude lower than that of the air we breathe. While some of the mechanics relevant to such soft biomaterials has been established for more than a century, there are other aspects, for example the thermal undulations of fluid and polymerized sheets, that have been investigated only in the past few decades.

The general strategy of this text is first to identify common structural features of the cell, then to investigate its mechanical components in isolation, and lastly to assemble these components into simple cells. The first chapter introduces metaphors for the cell, describes its architecture and develops some intuition about the properties of soft materials. The remaining nine chapters are grouped into three sections. Parts I and II are devoted to biopolymers and membranes, respectively, taking a conventional reductionist approach, while acknowledging that the soft materials of the cell are anything but conventional.

The structural elements of the cell can be broadly classified as filaments or sheets, where by the term filament, we mean a string-like object whose length is much greater than its width. Some filaments, such as DNA, function as independent units, but most structural filaments in the cell are linked to form two- or three-dimensional networks. As seen on the cellular length scale of a micron, individual filaments may be relatively straight or highly convoluted, reflecting, in part, their resistance to bending. This opening chapter to Part I concentrates on the mechanical properties of individual filaments, such as their bending or stretching resistance; the two chapters making up the remainder of Part I consider how filaments are knitted together to form networks, perhaps closely associated with a membrane as a two-dimensional web (Chapter 3) or perhaps extending though the three-dimensional volume of the cell (Chapter 4).

Filaments in the cell

Our discussion of cellular ropes and rods begins with a look at their molecular composition and linear dimensions. All of the ropes are linear polymers, in the sense that they are constructed from individual monomeric units to form an unbranched chain. The monomers need not be identical, and may themselves be constructed of more elementary chemical units. For example, the monomeric unit of DNA and RNA is a troika of phosphate, sugar and organic base, with the phosphate and sugar units alternating along the backbone of the polymer (see Appendix B).

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 the rotation of flagella (Latin plural for the noun whip) which extend from some cells and provide them with propulsion in a fluid medium. Flagella can be seen at both ends of the bacterium pictured in Fig. 9.1. Structurally related to flagella are 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 9.1, 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 9.2. 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 9.3.

Before launching into our concluding chapter, we catch our breath and survey the results obtained so far. In the opening chapter of the text, we identified the important structural elements of the cell as soft filaments and sheets, their flexibility arising from their nanometer thickness compared to their micron length. The mechanical properties of filaments in isolation (Chapter 2) and as components of networks (Chapters 3 and 4) were established in Part I, where we saw how the elasticity of soft filaments has its roots in both energy and entropy. The bending resistance of a biofilament may be sufficiently small that the filament noticeably changes shape in response to thermal fluctuations in its energy. Entropy is consequently reduced when a fluctuating filament is forced to straighten out at non-zero temperature and, of course, energy is increased when the intermolecular separation is driven away from its equilibrium value by tension; both of these effects contribute to the stretching resistance of a filament. The two chapters in Part II uncovered similar generic properties for flexible membranes: the energy required to bend a thin membrane may not be extravagant on the thermal energy scale kBT, so that a biomembrane may display gentle undulations at ambient temperatures. Under lateral tension, energy increases with intermolecular separation and entropy is reduced when the undulations are suppressed; again, both of these effects lead to area compression resistance.

These two categories of materials – ropes and sheets – are used to construct cells in Part III.

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 1014 cells 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 8.1. The Poisson–Boltzmann equation, which we derive in Section 8.2, 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 8.2 and 8.3, respectively. Two other contributors to membrane interactions also are treated in some mathematical detail here: the van der Waals attraction between rigid sheets (Section 8.3) and the steric repulsion experienced by undulating membranes (Section 8.4). The adhesion of a membrane to a substrate, and its effect on cell shape, is described in Section 8.5. 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 5.

The purpose of this book is to explore the mesoscale mechanics of the cell: its structure and behavior on length scales of tens of nanometers to microns. This expedition cannot be undertaken without a biochemical lexicon, in part because the mechanical properties of some structures are a direct reflection of their molecular composition, and in part because the text draws constantly from the language of biochemistry. In this appendix, we describe in more detail the nomenclature and composition of several classes of compounds referred to in the main text, including:

For a more thorough treatment of biomolecular building blocks, the reader is directed to books such as Alberts et al. (1994), Gennis (1989), and Lehninger et al. (1993).

Fatty acids and phospholipids

One of the principal constituents of cellular membranes is the family of dual-chain lipids formed from fatty acids, which are carboxylic acids of the form RCOOH, where R represents a long hydrocarbon chain. In cells, fatty acids usually are not found in their free state, but rather are components of covalently bonded molecules such as phospholipids or triglycerides. A major component of cell membranes, phospholipids contain two fatty acids linked to a glycerol backbone.

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 7 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 7–9, before returning to some general attributes of cells in Chapter 10. Many eucaryotic cells live communally as part of a larger organism, and Chapter 8 investigates the interaction between cell membranes, including entropic effects from thermal fluctuations in cell shape. In Chapter 9, we describe mechanisms that cells have developed for changing their shape, as part of cell division or locomotion, for example. The book concludes by returning to the broad perspective established in Chapter 1 and examines some generic features of cell structure, including synthetic cells, and speculates on the mechanical characteristics of the largest and smallest cells.

The two broad categories of structural components of the cell are filaments, the focus of Part I of this book, and sheets, which we treat in this chapter and its companion. In principle, two-dimensional sheets may display more complex mechanical behavior than one-dimensional filaments, including resistance to both out-of-plane bending and in-plane shear. As examples, the plasma membrane is a two-dimensional fluid having no resistance to in-plane shear, whereas the cell wall possesses shear rigidity as a result of its fixed internal cross-links. In this first chapter of Part II, we introduce the chemical composition of biomembranes and describe their mechanical properties, including elasticity and failure. We focus here on fluid membranes and defer a discussion of polymerized structures, such as the cell wall, to Chapter 6. The softness of biomembranes means that their thermally induced undulations are important, as will be explored in Chapters 6 and 8. More extensive reviews of biomembranes than those provided here can be found in Evans and Skalak (1980), Cevc and Marsh (1987) and Sackmann (1990).

Composition of biomembranes

The design principles of Chapter 1 argue that the cell's membranes should be very thin, perhaps even just a few molecules in thickness, if their purpose is solely to isolate the cell's contents. The membrane need not contribute to the cell's mechanical strength, as this attribute can be provided by the cytoskeleton or cell wall. Are there materials in nature that meet these design specifications?