Models and data (original) (raw)

There are a lot of connections between mathematics and biology, yet most students-and even most mathematicians and biologists-are unaware of these connections. One reason for this is that neither the historical development nor the pedagogical introduction of either subject involves the other. Biology grew out of natural philosophy, which was entirely descriptive. Modern biology curricula generally begin with descriptive biology, either organismal or cellular. The mathematically-rich areas of genetics and ecology make their appearance in advanced courses, after students have come to see biology as a non-mathematical subject. Historically, the development of calculus and calculus-based mathematics was driven by the mathematical needs of physics, and it remains standard practice to use physics to motivate calculus-based mathematics, whenever such motivation is deemed necessary. Other areas of mathematics, such as game theory and difference equations 1 , were motivated to some extent by biology, but these topics appear in specialized courses generally taken by mathematics students only. Probability is another mathematical topic with strong connections to biology; however, probability is generally encountered in statistics courses, which usually emphasize social science applications. In our discussion, we will focus on the nature of the connections, wherever they might be present, rather than the topics in which connections are most easily found. This way, we will be able to identify for ourselves the areas in biology that are amenable to mathematical treatment. Like all scientists, biologists collect data. Useful results usually come from the analysis of data, rather than from the data itself. The presence of data provides an opportunity to use mathematics in the service of biology. Analysis of data is generally labeled as statistics, but it can also be thought of as mathematics. Statistics involves a combination of careful mathematics with judgment based on experience, and we will examine some useful topics in statistics that are mathematical in nature. A two-sided connection between mathematics and biology comes from the use of mathematical models. Science is more than just collection and analysis of data. Science is also a search for general principles. These principles have to be supported by the data, but they also benefit from support in the form of a consistent mathematical model that ties together a variety of observations and/or experiments. Mathematical modeling in biology benefits biology by providing theoretical results that can be used to suggest new researches, make predictions, and provide a mathematical framework for the analysis of data. Mathematical modeling in biology also benefits mathematics because biological models often contain interesting mathematics. These notes describe the way models and data connect mathematics with biology. We begin by discussing the use of mathematics to characterize collections of data. Then we examine some mathematical models to get an idea of the general properties of models and to obtain some experience in understanding specific models. In Section 3, we use calculus to solve the mathematical problem of fitting a simple linear model to data. This work is extended in Section 4 to the problem of fitting simple nonlinear models to data.