Biological materials are generally soft, meaning that the shapes of objects made from them, such as strings and sheets, are moderately deformable under conditions commonly arising in the cell. Various aspects of the mathematical representation of deformations have been introduced in several chapters of this text (notably Sections 3.2, 4.2, 5.2, 6.2 and 8.2), although not in as much detail as would be found in a standard text on elasticity. Consequently, several calculations and fundamental results from continuum mechanics were quoted in this book without proof. To rectify this somewhat unsatisfactory situation, we develop in Appendix D:

• the symmetry relations among elastic constants for materials with four-fold and six-fold symmetry in two dimensions,

• the symmetry relations among elastic constants for materials with four-fold symmetry in three dimensions,

• the relation between shape fluctuations and elastic constants.

There is considerable variation in the notation and analytic approaches adopted by the spectrum of disciplines that make use of continuum mechanics; the notational convention adopted here is close to that of Landau and Lifshitz (1986).

Preface

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. The mechanical components themselves 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. Cells have fluid interiors and often exist in a fluid environment, with the result that the motion of the cell and its components is strongly damped and very unlike ideal projectile motion as described in introductory physics courses. While some of the physics 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 these mechanical components in isolation, and lastly to assemble the components into simple cells. The initial two chapters introduce metaphors for the cell, describe its architecture and develop some concepts needed for describing the properties of soft materials. The remaining chapters are grouped into three sections. Parts I and II are devoted to biopolymers and membranes, respectively, treating each system in isolation. Part III combines these soft systems into complete, albeit mechanically simple, cells; this section of the text covers the cell cycle, various aspects of cell dynamics, and some molecular-level biophysics important for understanding cell function.

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 7, 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 polymers and networks 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 available to these filaments. 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 9. A more extensive treatment of membrane defects and fluctuations can be found in Leibler (1989) and Chaikin and Lubensky (1995).

Thermal fluctuations in membrane shape

The bending rigidity κb of a phospholipid bilayer lies close to 10−19 J, or 10−20 kBT at ambient temperatures, as summarized in Table 7.3. What does such a small value of κb 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. 8.1(a) as it is deformed into the surface of constant curvature in Fig. 8.1(b). Configuration (b) has radius of curvature R and energy density 2κb/R2, where the Gaussian rigidity is neglected (see Sections 7.4 and 8.2). Taking the disk diameter as 2 μm and κb ~ 10 kBT, the deformation energy is 20πkBT/R2, where R must be quoted in microns. What is the typical value of R when the disk flexes at non-zero temperature? The thermal energy scale is set by kBT and the disk energy rises to this value (from zero for a flat disk with R = ∞) when R declines to ~8 μm. This radius of curvature corresponds to θ ~ 15o in Fig. 8.1(b), meaning that undulations may have measurable effects in common cells.

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 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. (2008), Goodsell (1993), or Prescott et al. (2004).

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. A1.1. 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. Connective tissue, which lies inside the boundaries provided by the epithelia, consists of fibroblastic cells that secrete an extracellular matrix which bears most of the stress in connective tissues. Lastly, several layers of smooth muscle tissue lie on the exterior of the connective tissue, in which they circle around the intestine (circular fibers) or run along its length (longitudinal fibers). Even a given cell type may exhibit several different forms, depending on the specific role that it plays. We now describe in more detail the four principal animal tissues.

The ability of cells to develop force and to shorten are commonly associated with skeletal muscles. These processes are not limited to skeletal muscles, however, but extend to cardiac muscles that pump blood, the smooth muscles that both maintain forces for extended durations and empty the contents of cavities, and in the migration of individual cells such as leukocytes around the body. All of these processes rely on the interaction of actin and myosin, the primary components of thin and thick filaments, respectively. The structure of the filaments converts the scalar myosin ATPase into vectorial force and shortening. Individual differences produce the diverse mechanical activities in the body. With experimental observations going back hundreds of years, the technical details known about contraction rival those of any other biophysical system, details that are covered below.

The skeletal length–tension relation

It was only with the discovery of the overlapping filaments of striated muscle in 1954 (Huxley and Niedergerke, 1954; Huxley and Hanson, 1954) that the molecular basis for muscle contraction was formed. This restriction did not keep previous scientists, certainly going back to Pugh, from observing the behavior of muscle. His description of the length–tension curve, although not by that name, is clear to us today. Extend a muscle, and its force increases, but stretch it too far, and it weakens: accurate, and even perhaps implying the damage that occurs if muscles are overstretched. We do have the advantage of having considerably more information than Pugh had.

The movement of material within the cell and across membranes always requires a driving force. For diffusive processes, the driving force is the electrochemical gradient. The electrical component of this force requires a separation of charge: the negative and positive charges must be kept from one another until a conductive channel opens and the charged species can flow down their electrical gradient. Intact membranes, whether the cell membrane or those of organelles, are needed to provide the voltage buildup that will allow current to flow when conduction becomes possible. Within the cytoplasm or in the extracellular fluid, the charges are not kept separate, and without a voltage there will be no electrical gradient. In these cases, diffusion is entirely driven by concentration gradients. The generation of these gradients is an active process: functions linked to ATP hydrolysis are ultimately responsible for all diffusion gradients. Once the gradients are generated, however, they will produce the movement of material without further ATP input. Across the membrane, of course, there can be both concentration and electrical gradients for charged moieties. In some cases, like Na+ across the resting cell membrane, these gradients will be in the same direction, inward in this case, with a higher Na+ concentration on the outside and a negative charge on the inside attracting the positive sodium. In others, like K+ across the resting cell membrane, the electrical and concentration gradients are in opposite directions, with the higher K+ concentration on the inside driving K+ out countered by the negative charge on the inside drawing K+ in. If diffusion down a gradient does not require ATP directly, other intracellular transport processes do require ATP. The movement of vesicles, organelles, and other cargo by kinesin, dynein and non-polymerized forms of myosin requires the direct hydrolysis of ATP to power each step. Transport within the cell therefore comes in two forms: diffusion allows material to move in all directions, while ATP-driven processes move material in particular directions. In this chapter, both of these mechanisms, and the principles behind them, will be discussed.

One strategy for predicting an improbable but potentially catastrophic future event is to construct a faithful model of the dynamics of the complex web of interest and study its extremal properties. But we must also keep in mind that the improbable is not the same as the impossible. The improbable is something that we know can happen, but our experience tells us that it probably will not happen, because it has not happened in the past. Most of what we consider to be common sense or probable is based on what has happened either to us in the past or to the people we know. In this and subsequent chapters we explore the probability of such events directly, but to set the stage for that discussion we examine some of the ways webs become dynamically complex, leading to an increase in the likelihood of the occurrence of improbable events. The extremes of a process determine the improbable and consequently it is at these extremes that failure occurs. Knowing a web's dynamical behavior can help us learn the possible ways in which it can fail and how long the recovery time from such failure may be. It will also help us answer such questions as the following. How much time does the web spend in regions where the likelihood of failure is high? Are the extreme values of dynamical variables really not important or do they actually dominate the asymptotic behavior of a complex web?

The science of complex webs, also known as network science, is an exciting area of contemporary research, overarching the traditional scientific disciplines of biology, economics, physics, sociology and the other compartments of knowledge found in any college catalog. The transportation grids of planes, highways and railroads, the economic meshes of global finance and stock markets, the social webs of terrorism, governments, and businesses as well as churches, mosques, and synagogs, the physical lattices of telephones, the Internet, earthquakes, and global warming, in addition to the biological networks of gene regulation, the human body, clusters of neurons and food webs, share a number of apparently universal properties as the webs become increasingly complex. This conclusion is shared by the recent report Network Science [23] published under the auspices of the National Academy of Science. The terms networks and network science have become popular tags for these various areas of investigation, but we prefer the image of a web rather than the abstraction of a network, so we use the term web more often than the synonyms network, mesh, net, lattice, grille or fret. Colloquially, the term web entails the notion of entanglement that the name network does not share. Perhaps it is just the idea of the spider ensnaring its prey that appeals to our darker sides.

Whatever the intellectual material is called, this book is not about the research that has been done to understand complex webs, at least not in the sense of a monograph. We have attempted to put selected portions of that research into a pedagogic and often informal context, one that highlights the limitations of the more traditional descriptions of these areas. In this regard we are obligated to discuss the state of the art regarding a broad sweep of complex phenomena from a variety of academic disciplines. Sometimes properly setting the stage requires a historical approach and other times the historical view is replaced with personal perspectives, but with either approach we do not leave the reader alone to make sense of what can be difficult material. So we begin by illuminating the basic assumptions that often go unexamined in science.