Biological molecular systems work inside living cells. As a cell prototype we consider the prokaryote Escherichia coli. This may be viewed as a small bag of DNA, RNA and proteins, surrounded by a membrane. The bag has a volume of about 1 μm3. This volume varies as the cell grows and divides, and also varies in response to external conditions such as osmotic pressure. The interior of the cell is a very crowded environment, with about 30% to 40% of its weight in proteins and other macromolecules, and only about 60% as water. Further, the water contains a number of salts, in particular K+, Cl- and Mg2+, each of which influences the stability of different molecular complexes.

The dry weight of E. coli consists of 3% DNA, 15% RNA and 80% proteins. The genome of E. coli is a singleDNAmolecule with 4.6 × 106 base pairs (total length of about 1.5 mm). It codes for 4226 different proteins and a number of RNA molecules. However, the information content of the genome is larger than that corresponding to the structure of the coded macromolecules. Important information is hidden and resides in the regulation mechanisms that appear when these proteins interact with the DNA and with each other. Some proteins, called transcription factors, regulate the production of other proteins by turning on or off their genes. Figure 7.1 shows two ways by which a transcription factor can regulate the transcription of a gene. The figure also shows a specific example of a regulatory protein bound to the DNA: the CAP protein.

Molecular motors: energy from ATP

Proteins can be grouped into a few broad categories with respect to their function. Some are regulatory, some are enzymes, some are structural, and some proteins do mechanical work. It is this latter group that we now discuss. Molecular motors include kinesin, myosin, dynein, the motors connected to DNA replication, to gene transcription and to translation. The motors are mostly driven by ATP hydrolysis: ATP → ADP + P, a process with ΔG ≈ 13 kcal/mol for typical conditions in the cell. Exactly how the free energy difference from ATP hydrolysis is converted into directed motion and mechanical work is a most interesting question, which is not resolved. In many cases the conformational changes of the protein are known in considerable detail from structural studies. The sequence of events associating conformational changes and substrate binding and release is also known. Nonetheless, the actual physical mechanism by which the motor works is not obvious. Thermal noise and diffusion certainly play a role, making this “soft” machine qualitatively different from a macroscopic motor. In the next section we elaborate on these ideas through some models.

The most studied motors include myosin and kinesin, which move along the polymers that define the cytoskeleton. Kinesin walks on microtubules (Fig. 6.1), whereas myosin walks on polymerized actin. Microtubules and actin fibers are long (μm) polymers where the monomer units are proteins. Microtubules are very stiff; actin fibers more flexible. Kinesin motors work independently of each other, and are associated with the transport of material (vesicles) inside the eukaryotic cell.

Cells are controlled by the action of molecules upon molecules. Receptor proteins in the outer cell membrane sense the environment and may subsequently induce changes in the states of specific proteins inside the cell. These proteins then again interact and convey the signal further to other proteins, and so forth, until some appropriate action is taken. The states of a protein may, for example, be methylation status, phosphorylation or allosteric conformation as well as sub-cellular localization. The final action may be transcription regulation, thereby making more of some kinds of proteins, it may be chemical, or it may be dynamical. A chemical response would be to change the free concentration of a particular protein by binding it to other proteins. A dynamical response could be the activation of some motor, as in the chemotaxis of E. coli.

The presently known regulatory network of yeast is shown in Fig. 8.1. The action of proteins in this network is to control the production of other proteins. The control is done through genetic regulation discussed previously, through control of mRNA degradation, or possibly through the active degradation of the proteins.

Regulatory genetic networks are essential for epigenetics and thus for multicellular life, but are not essential for life. In fact, there exist prokaryotes with nearly no genetic regulation. Figure 8.2 shows the number of regulators as a function of genome size for a number of prokaryotic organisms. One notices that those with a very small genome hardly use transcriptional regulation.

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 in Fig. 3.1. 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.

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. 2.21(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. 3.1(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).

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).