I have tried to present the subject of biophysics from a conceptual perspective. This needs to be stated because biophysics is too often defined as a collection of physical methods that can be used to study molecular and cellular biology. This technical emphasis often fosters narrowness, and in the worst cases leads to shallowness, where sophisticated measurements are interpreted with little consideration for the physical principles that govern the special complexities of the macromolecular world of biology.

The conceptual emphasis of this book has lead to a heavy dose of theory. Theoretical analysis is essential in a conceptual approach, but I must admit that the theoretical emphasis of this book also reflects my own personal fascination with the insights that can be gained by applying physical theory to biological questions. In developing theoretical topics I have tried to be practical. I have steered toward more basic forms of mathematics wherever possible. Much of the analysis is at the level of an introductory calculus course. Where more sophisticated mathematics is involved I have tried to teach the mathematics in parallel with the development of the subject at hand. Six mathematical appendices have been added to help the reader. These may be useful guides, but are certainly not rigorous or thorough. Readers who desire a better background in mathematics will have to find appropriate texts that treat subjects such as matrices and partial differential equations.

Biological systems often fluctuate more noticeably than typical physical and chemical systems. This reflects the large size of many biological molecules and the small size of cells. The molecular nature of matter gives rise to fluctuations in every imaginable property. These fluctuations may or may not be easy to see, and size is a critical factor. In a system with N molecules, many measured quantities are proportional to N, but the fluctuations are proportional to N½. The fluctuations relative to the mean then decrease with the size of a system as N−½. When N is Avogadro's number, the task of observing these fluctuations in a conventional measurement becomes quite a challenge. Of course, there are some incredibly sensitive measurements that can be made. Signals arising from single molecules can be detected, and the fluctuations in these signals reflect the stochastic nature of molecular activity. But in many cases where the single-molecule signals are too small to see, the collective fluctuations may still be detectable. The special size scales found in biology generate a uniquely fluctuating world that merits special attention.

We have encountered fluctuations already in Chapter 3 in relation to conformations of macromolecules, and in Chapter 6 in relation to random walks. The probability of fluctuations can be calculated whenever statistical mechanics is used to develop a quantitative molecular description, and in many situations fluctuations contain important information. The study of fluctuations then becomes a powerful experimental approach by which models can be tested and molecular parameters estimated.

The preceding chapters treated molecules as isolated entities. Now we will look at how molecules interact with one another. In biological systems molecules are continually binding together and coming apart. Molecular associations are the first step in most forms of biological signaling, as well as in enzyme catalysis. Hormones, neurotransmitters, second messengers, and metabolites bind to proteins to regulate their activity. Pharmacology is rooted in the molecular association between drugs and receptors. On a larger scale, associations between macromolecules direct the assembly of organelles. Here, we will examine the thermodynamic and statistical mechanical principals underlying chemical association processes. These concepts will serve as a useful prelude to the theory of allosteric interactions in the following chapter.

There are two guiding principles in understanding association processes in molecular biology. (1) The forces that control associations are usually noncovalent. These kinds of forces were covered in Chapter 2. Here we will discuss how noncovalent interactions such as electrostatic forces, hydrogen bonds, and hydrophobic interactions combine in various ways to stabilize molecular complexes. (2) Associations are stereospecific, and depend on a precise spatial arrangement of the interacting groups. A binding site within a protein is viewed as a lock, and a ligand that fits into this binding site is a key. As a result biological associations are highly specific; molecules can recognize one another and distinguish subtle variations in structure.

How enzymes accelerate biochemical reactions is one of the oldest and most challenging problems in biophysics. An enzyme binds with high specificity to a substrate molecule, chemically modifies it, releases the product, and then repeats the cycle. Without the enzyme, the same chemical reaction can still take place, but at a vastly slower rate. The most impressive enzymatic accelerations approach 1020 (Miller and Wolfenden, 2002). A value in the region of 108–1012 is more typical, but that still represents a remarkable enhancement. Enzymes are responsible for virtually all of the metabolic chemistry in the biological world. However, there is another point worth mentioning. Nearly all enzymes are proteins (the exception is ribozymes – catalytic RNA), so a study of enzyme catalysis provides a window into the basic mechanics of proteins carrying out their functions. Enzyme catalysis provides excellent examples of how the structure and dynamics of proteins relate to their activity.

We know a great deal about the chemical mechanisms employed by enzymes. X-ray crystallography has given us atomic-level pictures of enzyme–substrate complexes in which many important contacts are evident. Some of these contacts are strictly for binding, and enable the recognition of specific substrates. The binding of substrate is the first step of enzyme catalysis, and this provides an important application of the physics of molecular associations (Chapter 4). The next step is the chemical reaction, and this brings the rate processes of Chapter 7 into the picture.

Cells can have very complex geometries, and when they do the voltage can vary dramatically between different regions. If ionic current flows through a restricted part of a cell's membrane, then the membrane potential at that location will change rapidly, but the membrane potential at distant locations will change more slowly and the change will be smaller. Voltage changes spreading through a cell act as signals to change membrane properties and trigger cellular events such as exocytosis and muscle contraction. Electrical signaling allows the nervous system to control and organize behavior.

Electrical signals in cells fall into two general classes. If the membrane conductance is independent of voltage, then the spread of voltage is passive. This type of signal, also referred to as electrotonic, travels a limited distance. On the other hand, when voltage alters the membrane conductance, then a voltage signal can regenerate itself and propagate without decrement over unlimited distances. This chapter will examine passive electrical signaling and the following chapter will treat active propagation.

The study of passive voltage changes serves a number of purposes. (1) Some biologically important voltage changes spread passively; passive spread is especially important when voltage changes are small. (2) Passive voltage changes are of technical importance in the design and interpretation of electrophysiological experiments. (3) Passive signaling serves as a baseline from which one goes on to study active propagation.

The principles of passive signaling derive from the basic rules of electrical circuits. Voltage drives current through resistors.

A cell expends metabolic energy to transport ions, accumulating some and expelling others so that the concentrations differ between the cell's inside and outside. The cell membrane is more permeable to some of these ions than to others, so ions will move passively down their concentration gradients at different rates. The resulting separation of charge generates a voltage difference. Essentially every cell has a membrane potential arising from ion movement. The movement of different ions across the cell membrane produces a broad spectrum of electrical behavior and makes it possible for cells to communicate by electrical signaling. Electrical potentials also appear across the membranes of organelles, and across layers of cells such as epithelia. The common theme with all these phenomena is the selective permeability of a membrane to different ions. Here we will examine the relation between membrane permeability, ion flux, and membrane potential. We will not worry about the precise mechanism of ion permeation here. That is a subject for the next chapter.

Nernst potentials

Consider two compartments separated by a membrane, with a higher concentration of salt on one side than on the other (Fig. 13.1). If only one ion is permeant, then that ion will diffuse across the membrane down its concentration gradient, but the other ions will remain behind. This produces a charge imbalance and therefore a membrane potential.

Unlike deceptively simple electrostatic Coulombic interactions, van der Waals forces can appear complicated. Why? Electrostatic forces depend on only the response to constant electric fields from effectively stationary charges. Electrodynamic van der Waals forces depend on all the possible electromagnetic fields that come out of all possible modes of charges in motion. After learning the language of these fields and motions, seeing how measurements of reflection and absorption turn into calculable forces, and understanding how the electromagnetic wave equations lead us to formulate interactions for a huge variety of materials in variously shaped bodies, we are liberated. What we knew before about charge–charge interactions seems so confining once we can move into a newly accessible area.

The van der Waals interaction depends on the dielectric properties of the materials that interact and that of the medium that separates them.

The tools to think with and about charge-fluctuation forces.

Level 1, the introduction, speaks the language of the modern theory in order to help you develop intuition about the fundamental connection between material absorption spectra and charge-fluctuation forces. Its approximate formulae also show the connection between the shapes of bodies and the forces between them.

The precise tabulated formulae together with the essays on computation in Level 2 will let you compute interactions under a variety of instructive assumptions.

The conversation-in-front-of-a-blackboard style and the extensive footnotes of Level 3 can help you understand the origins of tabulated formulae and, more important, show you ways to derive new formulae. Nonphysicists need not fear to go to the back of this book. Levels 2 and 3 will give them much of what would have been learned in missed physics courses.

No excuses. Anyone who has passed a course in physical chemistry should be able to crunch this book for numbers and for equations. Engineers can treat it as a handbook with long explanations whose formulae can be plugged into original programs or packaged software (preferably both so as to understand what is being computed).

Readers with a physics background will find connections to systems that fall outside the usual purview of pure physics. The properties of these systems encourage physicists to use their prowess in new situations.

Level 2 is the doing. After reading the Prelude and Level 1, you should be able to go to any part of Levels 2 or 3 according to personal tastes or interests. Go straight to the formulae, tabulated by geometry, cross-referenced by equation numbers not only to Level 1's introduction but also to Level 2's explanatory essays and to the derivations of Level 3. The idea is to work outward from these tables—back to Level 1 or forward to Levels 2 and 3.

The opening Notation section, Section L2.1, applies not only to the tables but also to the texts of Levels 1 and 2. The Essays on formulae section, Section L2.3, which immediately follows the tables sections, reduces the results from Level 3 derivations to simpler forms. The Computation section, Section L2.4, sketches the physical foundations of the all-important dielectric-response functions and gives mathematical guidelines for calculation.

The sequence of tabulation is first to give the most exact expression available, and then to list approximations.