In Chapter 1 the global conformation of a protein was treated like a black box, without worrying about the internal machinations. This approach is useful in interpreting many kinds of experiments, but if we want to make use of detailed structural information about a macromolecule, we have to open up the black box and look inside. To do this we need to understand the molecular forces that act within a macromolecule. These forces govern how a protein folds, and which of its different conformations will predominate. Similar forces determine the structures of nucleic acids and lipid bilayers, and also drive the associations between macromolecules and ligands.

The forces at work in biological systems can be divided into various categories and examined in turn. They are generally well understood to the extent that good approximate mathematical expressions are available. Much is known about their relative strengths under various conditions. However, it must be emphasized that the structure and dynamics of biological macromolecules are determined by the interplay of many forces. This complexity makes it difficult to investigate these forces by studying the biological molecules directly. We therefore often turn to model systems that are very unbiological but nevertheless instructive. The results from model systems enable us to break these complicated problems down into simpler ones.

The Coulomb potential

One of the fundamental tenets of electricity is that point charges interact with a potential energy that is inversely proportional to the distance of separation, r, and directly proportional to the product of the two charges, q1 and q2.

Pure lipid bilayers have extremely low permeabilities to inorganic ions. Adding proteinaceous ion channels can increase the permeability by a factor of more than 108, allowing ions to flow across membranes and produce rapid changes in voltage. One can draw a strong analogy with enzymes. Both ion flow and the chemical reaction catalyzed by an enzyme have a favorable free energy that enables each to proceed in the absence of its respective catalyst, but at a very slow rate. Ion channels and enzymes both enhance these rates dramatically, and this enhancement is highly specific. In the case of an enzyme, small differences in the structure of a substrate can make a huge difference in catalytic efficiency. Likewise, ion channels can discriminate very effectively between different ions.

At first glance, an ion channel appears to have an easier task than an enzyme. It simply forms a water-filled pore so that ions see a continuous aqueous path through the membrane. However, a simple aqueous pore will not be specific for one particular ion. The diameter of K+ is 1.33 Å and the diameter of Na+ is 0.95 Å. Although this difference is small, some channels show selectivities between Na+ and K+ of more than 1000. Understanding this specificity is the real challenge in the study of ion channel permeation. Ion permeation depends not just on the water filling the pore but also on the detailed molecular structure of the protein that forms the channel.

The importance of molecular associations in biological signaling processes was mentioned in the preceding chapter. That chapter concentrated on the physical aspects of the association process and paid little attention to the signaling events that are initiated by ligand binding. This chapter will accept the binding event as given, and go on to look at what consequences this has on the biological function of a protein.

Powerful theories to explain this kind of signaling can be developed by combining the concepts of molecular associations from Chapter 4 with the concepts of global states and transitions from Chapter 1. In putting these two ideas together, a key point to remember is that both processes are governed primarily by the kinds of noncovalent forces covered in Chapter 1. As a result the energies for global transitions and binding events are often in the same range. This enables an association reaction to trigger a conformational transition in a protein, and this is what makes allosteric interactions possible. Here, we will develop this theory, known as allosteric theory, and illustrate its use with examples.

The word allosteric is quite popular in molecular biology. The word was introduced as a combination of the Greek words allo and steric to mean other-site. A classical usage in this sense is when a ligand binds to a regulatory site of an enzyme and alters the enzyme's effectiveness as a catalyst.

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.