Mathematics can be viewed as a tool for deriving logical consequences from propositions. It differs from ordinary intuitive reasoning in its insistence on rigorous, accurate logic and the precise treatment of quantitative information. If the initial propositions are correct, then the deductions drawn from them by mathematics will be true. The surprising power of mathematics comes from the length of the chains of reasoning that rigorous logic and mathematical arguments make possible, and from the unexpectedness of the conclusions that can be reached, often revealing connections that one would not otherwise have guessed at. Reversing the argument, mathematics provides a way to test experimental hypotheses: if mathematical reasoning from a given hypothesis leads to a prediction that is not true, then the hypothesis is not true.
Clearly, mathematics is not much use unless we can frame our ideas—our initial hypotheses—about the given system in a precise, quantitative form. A mathematical edifice raised on a rickety or—even worse—a vague or overcomplicated set of propositions is likely to lead us astray. For mathematics to be useful, we must focus our analysis on simple subsystems in which we can pick out key quantitative parameters and frame well-defined hypotheses. This approach has been used with great success in physics for centuries, but it has been less common in biology. But times are changing, and more and more it is becoming possible for biologists to exploit the power of quantitative mathematical analysis.
In this final section of our methods chapter, we do not attempt to teach readers every way in which mathematics can be fruitfully applied to biological problems. Rather, we simply aim to give a sense of what mathematics and quantitative approaches can do for us in modern biology. We focus primarily on the important principles that mathematics teaches us about the dynamics of molecular interactions, and how mathematics can unveil surprising and useful features of complex systems containing feedback. We will illustrate these principles using the regulation of gene expression by transcription regulators like those discussed in Chapter 7. The same principles apply to the post-transcriptional regulatory systems that govern cell signaling (Chapter 15), cell-cycle control (Chapter 17), and essentially all cell processes.
Cell function and regulation depend on transient interactions among thousands of different macromolecules in the cell. We often summarize these interactions in this book with schematic cartoons. These diagrams are useful, but a complete picture requires a deeper, more quantitative level of understanding. To meaningfully assess the biological impact of any interaction in the cell, we need to know in precise terms how the molecules interact, how they catalyze reactions, and, most importantly, how the behaviors of the molecules change over time. If a cartoon shows that protein A activates protein B, for example, we cannot judge the importance of this relationship without quantitative details about the concentrations, affinities, and kinetic behaviors of proteins A and B.
Let us begin by defining two different types of regulatory interaction in our cartoons: one designating inhibition and the other designating activation. If the protein product of gene X is a transcription repressor that inhibits the expression of gene Z, we depict the relationship as a red bar-headed line ( ) drawn between genes X and Z (Figure 8–71). If the protein product of gene Y is a transcription activator that induces the expression of gene Z, then a green arrow ( ) is drawn between genes Y and Z.
The regulation of one gene’s expression by another is more complicated than a single arrow connecting them, and a complete understanding of this regulation requires that we tease apart the underlying biochemical processes. Figure 8–72A sketches some of the biochemical steps in the activation of gene expression by a transcription activator. A gene encoding the activator, designated as gene A, will produce its product, protein A, via an RNA intermediate. This protein A will then bind to pX, the regulatory promoter of gene X, to form the complex A:pX. Once the A:pX complex forms, it stimulates the production of an RNA transcript that is subsequently translated to produce protein X.
We will focus here on the binding interaction that lies at the heart of this regulatory system: the interaction between protein A and the promoter pX. Any molecule of protein A that is bound to pX can also dissociate from it. The steps represented by the green activation arrow in Figure 8–72A include both the binding of A to pX and the dissociation of the complex A:pX to re-form A and pX, as illustrated