Chapter 1

The Rationale for Kinetic Modeling

There are several good reasons for adding kinetic modeling to your skills for biological investigation. First, experience has shown, over and over again, that there is much more information contained in dynamic biological data than can be extracted by simple inspection. So it's a matter of efficient use of resources; if the data you've already collected contain answers that no one knows, then an investment in computing can give you a competitive advantage both in new knowledge and in productive experimental design. Second, building successful dynamic or kinetic models is heavily dependent on knowing or discovering the underlying biological mechanisms. We are not interested in correlational models where cause and effect are often obscure, but instead are interested in physical and chemical processes that are known or hypothesized to take place in the biological system of interest.

Mechanism and dynamics are intimately related, and much of the value of kinetic modeling lies in the unique ability of dynamic experiments to reveal new and useful information about inherently complex biological control systems. One simple illustration of this point is provided by the diagram below:

The graph shows an experimental observation recorded as a function of time. During the time indicated by the solid bar, a physiological or pharmacological perturbation is applied to the biological system and its effect on the observation is recorded. Before the perturbation, the system is in steady state. In response to the perturbation, the system passes through a transient before arriving at a new steady state. If you, as an investigator, restrict yourself to measurements made in the two steady states, you are discarding valuable information on mechanism. This is because each of the different transients in figure 1 corresponds to a different underlying mechanism, and none of the information on these mechanisms is available if only the two steady state values are recorded. Our overall rationale for bringing modeling tools to bear on biological problems is best illustrated in the following diagram, which shows the role of mathematical modeling in scientific method.

Nearly everyone is familiar with the experimental aspect of scientific investigation that is depicted in the lower half of this diagram. You have a theory or hypothesis you wish to test, you create an experimental design that tests a critical part of the hypothesis, you apply methods you already know or you develop new methods to make the required measurements or observations, and you compare the collected observations with the predictions of the hypothesis. If there is a match, you have the backbone of a new contribution to the scientific literature. If you don't have a match, you are usually well-advised to reconsider your hypothesis. Restructuring an inadequate hypothesis is an act of pure scientific imagination, despite Francis Bacon's insistence that hypotheses can be constructed from an assemblage of facts by a process he called "induction."

Other scientists and philosophers of science, notably Karl Popper, have argued that it is the mismatch between hypothesis and observation that increases our understanding. I'm certain Popper would argue that publication is warranted when the hypothesis fails and that publication is redundant when the hypothesis succeeds. His book, The Logic of Scientific Discovery (especially Chapter 10: Corroboration...) (Harper 1968), will reward your careful reading. Nevertheless, I suspect most published hypotheses are formulated after the data are collected. In this way the authors can privately abandon the hypothesis that actually prompted the experiments and publicly endorse a modified hypothesis that appears consistent with the reported data. In this way the editors and readers are treated to a "positive" result and the rejection of hypotheses occurs between publications rather than in publications. This quirk of modern science means that most published papers represent not just the experimental test of a hypothesis, but also the reformulation of the hypothesis based on the experimental results obtained. What, then, is the role of mathematical modeling in scientific method? As depicted in Figure 2, predictions are essential to the comparison, and we view modeling as the method of choice for obtaining precise predictions.

Figure 2 The Role of Modeling in Scientific Method

If you don't have recourse to a quantitative model, you must rely on your best guess as to what should happen if your experimental design is applied to your theory or hypothesis. This is an ordinary act of science, and a great deal of effort is directed at the design of experiments whose outcomes are binary, yes-no measurements so that the prediction process is manageable by a thoughtful man or woman. But as science confronts the necessity for integrating information so as to comprehend the function of complex systems such as intact cells, tissues, and organisms, guesswork becomes less and less trustworthy as a source of precise predictions. In fact, cognitive psychologists have found that when an observable system consists of more than 3 interacting parts, the human mind is incapable of making accurate predictions concerning the result of a perturbation imposed on the system. If, on the other hand, you have formulated your hypothesis as a system of equations, it is a relatively simple task to obtain precise predictions for a given experimental design through the use of computer simulation. This book will have a great deal to say about some of the processes in Figure 2. Those that will get particular attention are repeated in Figure 3.

Figure 3: Parts of scientific method discussed in this cybertextbook.

Our first concern is the process of model building. We will explore

• how to assemble your experimental data
• how to construct a rigorous symbol and arrow diagram for your system
• how to translate your diagram to mathematics

Second, you will want to use simulation tools. We will examine

• how to use the SAAM II software to automate much of the modeling process
• how to estimate initial values of your model's parameters
• how to impose experimental protocols on your model
• how to solve your model's equations
• how to link your model's variables to your experimental observations
• how to explore your model's behavior

Next, you will want to test your model against experimental data as a means of testing the hypothesis or theory that you model represents. You will learn

• how to use SAAM II in hypothesis testing
• how to do least squares fitting
• how to do parameter estimation

When you discover a mismatch between model prediction and experimental data, you have to decide which to believe. Traditionally, we accept the experimental data as true, and we opt to modify our theory, hoping to make our next prediction match the data. In the diagram, the pathway leading from a mismatch to a new hypothesis is labelled, "imagination." Computers cannot yet help here; this is still a purely human activity. We find, however, that modeling still helps because it enables you to know with precision what your imagination must accomplish. As you explore the behavior of more and more models, your imagination is assisted by your subconscious knowledge of more and more possibilities. Finally, you may wish to take advantage of your model for the design of informative experiments. We will discuss how to design experiments that yield useful information the elements of including a kinetic model in a manuscript for publication.

Model

A simultaneous system of differential equations and associated algebraic equations that defines the state variables and rate laws for a particular physical, chemical or biological system. The model is a quantitative statement of our theory as to how the real-life system operates. In this hypertext, models embody the principle of cause and effect, and are based on descriptions of the physicochemical processes that comprise the biological system under study. Consequently, purely mathematical descriptions such as curve fitting, and statistical regression are not "models" in the sense used here.

Dynamics

Kinetics

Dynamics and Kinetics are roughly synonymous terms referring to the study of physicochemical (including biological) systems that change with time. The mathematical tools for this discipline are the differential equations invented by Newton and Leibniz in about 1640. The tools of classical and modern control theory, numerical analysis, and systems theory are also frequently applied to dynamic or kinetic problems.