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Molecular and Cellular Biophysics provides advanced undergraduate and graduate students with a foundation in the basic concepts of biophysics. Students who have taken physical chemistry and calculus courses will find this book an accessible and valuable aid in learning how these concepts can be used in biological research. The text provides a rigorous treatment of the fundamental theories in biophysics and illustrates their application with examples. Conformational transitions of proteins are studied first using thermodynamics, and subsequently with kinetics. Allosteric theory is developed as the synthesis of conformational transitions and association reactions. Basic ideas of thermodynamics and kinetics are applied to topics such as protein folding, enzyme catalysis and ion channel permeation. These concepts are then used as the building blocks in a treatment of membrane excitability. Through these examples, students will gain an understanding of the general importance and broad applicability of biophysical principles to biological problems.
The relation between structure and function is central to molecular biology. But molecular structure can mean different things, especially when dealing with complex biological molecules. One can know the chemical structure of a molecule, how the atoms are connected by covalent bonds, but have no idea of the conformational state, how the atoms are arranged in space. The conformational state of a molecule has a profound impact on what it does, and much of the work in molecular biophysics deals with understanding molecular conformations, both what they are and how they perform biological functions.
The conformational state of a molecule can be studied at different levels. One might have a vague notion of its general shape, or one might have a detailed picture with the position of every atom specified. Thinking in terms of detailed structure is more difficult but more powerful. Some approaches to this problem will be taken up in later chapters. Here, we will start with something simple, introducing an approach to protein conformations that does not depend on all the structural details.
The approach of this chapter is based on the idea that proteins have “global” states. Global states are defined in terms of a protein's functional capability. We will assume that a protein has a few – perhaps just two – of these global states. Global states can interconvert, in what we will call global transitions. In terms of structure the global state is a black box.
The two-state model of Chapter 7 can be used as a basic building block for more complicated processes. When a system has more than two states, conversions between different pairs can occur, and then the kinetics reflects the aggregate behavior of those various transitions. The time course is no longer a simple exponential function, and in this chapter we will see how multi-state models lead to multi-exponential kinetics.
Much of Chapter 7 probed the fundamental physical processes that govern the speed of a transition. Now we will accept the basic phenomenon of a transition with a given rate, put some of these transitions together, and work out the dynamic behavior of these more complicated systems. The mathematical method for handling multi-state kinetics is very robust and powerful. With them one can develop quantitative descriptions for a general class of models. Aside from kinetic problems in biophysics, the mathematics introduced in this chapter has been applied to an extraordinarily wide range of fields from stochastic processes in physics to population dynamics in ecology.
Multi-state kinetics has considerable practical value, because most models for molecular mechanisms involve interconversions between a few distinct states. Kinetic behavior often provides the most direct experimental tool for testing the predictions of such a model.
The three-state model
Many important features of multi-state kinetics can be illustrated with this example.
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.