Preview of Modeling and Calculus

Figure 1 Mathematics has been used to model many biological systems on Earth, ranging from global climatic processes to viral dynamics.

Is calculus relevant?

“The interface between mathematics and biology presents challenges and opportunities for both mathematicians and biologists. Unique opportunities for research have surfaced within the past ten to twenty years because of the explosion of biological data with the advent of new technologies and because of the availability of advanced and powerful computers that can organize the data. For biology, the possibilities range from the level of the cell and molecule to the biosphere. For mathematics, the potential is great in traditional applied areas such as statistics and differential equations, as well as in such nontraditional areas as knot theory.”

“These challenges: aggregation of components to elucidate the behavior of ensembles, integration across scales, and inverse problems, are basic to all sciences, and a variety of techniques exist to deal with them and to begin to solve the biological problems that generate them. However, the uniqueness of biological systems, shaped by evolutionary forces, will pose new difficulties, mandate new perspectives, and lead to the development of new mathematics. The excitement of this area of science is already evident, and is sure to grow in the years to come.”

(Executive Summary, National Science Foundation-sponsored workshop
led by Simon Levin, 1990).

The above quotation, which is as true today as when it was written, hints at the exciting opportunities that exist at the interface of mathematics and biology. The goal of this course is to provide you with a strong grounding in calculus, while at the same time introducing you to various research areas of mathematical biology and inspiring you to take more courses at this interdisciplinary interface. In Chapter 1, we will set the tone for the entire book and review some of the skills needed to work at this interface. But first, in this preview, the idea of mathematical modeling is introduced to give you an underlying understanding of this important concept. Throughout the book you will find real-life problems that can be solved using mathematics. For example, in Olympic weightlifting, medals are awarded to individuals in different weight classes, as heavier individuals tend to lift more weight. In Chapter 1, we use mathematics and a basic physiological principle to predict how much an individual can lift scales with their body weight. Using the resulting mathematical model, we identify one of the greatest weightlifters of all time, Pocket Hercules (see Figure 1.1).

Models come in many guises: Architects make buildings models that are either small-scale replicas or, more recently, visual images created using computer-aided design packages. Political scientists, through debate and discussion, create verbal and written models that simulate the potential outcomes of a proposed policy. Artists make sketches and small-scale sculptures before starting a large-scale project. Flight simulators allow people to gain skills in piloting without the dangers associated with flying. Scientists in many disciplines (e.g., physics, biology, economics, chemistry, sociology, and even psychology) use mathematical models to investigate important phenomena.

Real-world problems inspired the creation of quantitative tools to grapple with their complexity. The counting and division of flocks of birds influenced the early development of number theory. The measurement and division of land led to the development of geometry. Understanding the motion of the planets and the forces of electricity, magnetism, and gravity resulted in the development of calculus. More recently, the study of the dynamics of population growth and population genetics led to many of the basic topics in stochastic processes. The immense success of mathematical models in understanding physical processes was recognized by E. P. Wigner in “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Communications in Pure and Applied Mathematics 13 (1960): 1–14)—his now famous essay—in which he states:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it, and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning.”

As highlighted in the quotation from the NSF-sponsored workshop, one of the areas to which mathematics has extended most dramatically over the past half century is the biological sciences. The importance of this mathematics-biology interface is threefold. First, field and laboratory experiments are generating an explosion of data at both the cellular and environmental levels of study. To make these data meaningful requires extracting patterns within the data (e.g., correlations among variables, clustering of points in time and space). Mathematics, which from one viewpoint is the study of patterns (e.g., numerical, geometrical), provides a powerful methodology to identify and extract these patterns. This power of mathematics is reflected in the following statement of one of the founders of calculus, Sir Isaac Newton (1642–1727), in his book.

“The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics.”

Second, mathematics is a language that permits the precise formulations of assumptions and hypotheses. Consider the words of another founding father of calculus, Gottfried Wilhelm Leibniz (1646–1716).

“In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.”

Third, mathematics provides a logical, coherent framework to deduce the implications of one's assumptions.

One of the goals of this book is to help you understand how, when, and why calculus can be used to model biological phenomena. To achieve this understanding, you will be expected to develop simple models, to understand more complicated models sufficiently well to slightly modify them, to determine the appropriate techniques to analyze the models (e.g., numerical vs. analytical, stability vs. bifurcation analysis), and to interpret the results of your analysis. Examples of biological phenomena that we will encounter include epidemic outbreaks, blood flow, population extinctions, tumor regrowth after chemotherapy, population genetics, regulatory genetic networks, mechanisms for memory formation, enzyme kinetics, and evolutionary games.

A second goal of this book is provide you with a thorough grounding in calculus, one of the greatest intellectual achievements of humankind. Calculus provides an analytic framework for studying the rates at which things change and the accumulation of change over time. This book provides you with a detailed tour through the key concepts and applications of calculus, its analytical techniques, and its numerical methods. In the remainder of this introduction, we briefly address two basic questions: What is mathematical modeling? What is calculus?

What is mathematical modeling?

A real-life situation is usually far too complicated to be precisely or mathematically defined. When confronted with a problem in the real world, therefore, it is usually necessary to develop a mathematical framework based on certain assumptions about the real world. This framework can then be used to find a solution that will tell us something about the real world. The process of developing this mathematical framework is referred to as mathematical modeling.

What, precisely, is a mathematical model? It is an abstract description of a real-life problem that does not have an obvious solution. The first step involves abstraction in which certain assumptions about the real world are made, variables are defined, and appropriate mathematical expressions are developed.

In this text, we discuss modeling biological systems. Consequently, as we progress through the book, we spend some time identifying the features associated with molecular, physiological, behavioral, life history, and population-level processes of many species and biological processes. After abstraction, the next step in modeling is to simplify the mathematics or derive related mathematical facts from the mathematical model.

The results derived from the mathematical model should lead us to some predictions about the real world. The next step is to gather data from the situation being modeled and to compare those data with the predictions. If the two do not agree, then the gathered data are used to modify the assumptions underlying the model, and the process repeats.

One use of the modeling process is predicting world population size, that is, the total number of humans living on Earth. Predicting world population size is of critical importance in determining future impacts of humans on their environment and future needs for energy, food, shelter, and other resources. Table 1 reports world population sizes over the past millennium.

It is clear that the world population is growing. But how fast is it growing? As a first guess, we might try modeling with a linear function N(t) = a + bt, where t is the year and N(t) is the population size. Using the first two data points from Table 1 yields this relationship (try this for yourself!):

Table 1 Human world population sizes over the past 1000 years

Plotting this linear model against the data yields the left panel of Figure 2. Although this linear model does a reasonable job of predicting population sizes from 1000 to 1500 AD, it clearly underpredicts for the remainder of the millennium. This limitation stems from the assumption that the population size is increasing at a rate independent of its size. Clearly, this assumption is not reasonable as our population appears to be growing more and more rapidly over the centuries. A better model, as we will discuss in Sections 1.4 and 1.6, is exponential growth in which the population growth rate is proportional to the population size. The exercises of Section 1.4 will ask you to fit an exponential model to the data. This exponential model fitted to the first few data points slightly improves the predictive power of the model, as seen in the middle panel of Figure 2. However, the new model also substantially underpredicts world population sizes after 1750 AD. One possible explanation for this underprediction is that the model does not account for successive cultural revolutions, such as the Industrial Revolution, that led to surges in population size during the nineteenth and twentieth centuries. In Chapter 6, we examine a model that accounts for population growth that is faster than exponential: so-called super-exponential model. Developing and analyzing this model requires simultaneously the tools from differential and integral calculus, which we discuss next. In Section 6.2, you will fit this super-exponential growth model to the world population data and find that its predictive power (as seen in the right panel of Figure 2) is significantly better than the simpler models. Hence, after several iterations of model formulation and comparison to data, we arrive at a much better model for predicting future population sizes.

Figure 2 Three models of human population growth of increasing complexity: a simple linear model, an exponential growth model discussed in Sections 1.4 and 1.6, and a more sophisticated super-exponential growth model discussed in Section 6.2.

What is calculus?

Very likely, you have enrolled in a course that requires that you use this book. If you read the preface, you know that the intended audience is students who wish to learn about calculus but are majoring in an area related to biology. You might think of calculus as the culmination of all of your mathematical studies. To a certain extent, that is true, but it is also the beginning of your study of mathematics as it applies to how the real world changes in time and across space. All your prior work in mathematics is elementary. With calculus, you cross the dividing line between using elementary and advanced mathematical tools for studying a variety of applied topics. Calculus is the mathematics of motion and change over time and space.

What distinguishes calculus from your previous mathematics courses of algebra, geometry, and trigonometry is the transition from discrete static applications to those that are dynamic and often continuous. For example, in elementary mathematics you considered the slope of a line, but in calculus we define the (nonconstant) slope of a nonlinear curve. In elementary mathematics you found average changes in quantities such as the position and velocity of a moving object, but in calculus we can find instantaneous changes in the same quantities. In elementary mathematics you found the average of a finite collection of numbers, but in calculus we can find the average value of a function with infinitely many values over an interval.

The development of calculus in the seventeenth century, independently by Newton and Leibniz, was the result of their attempt to answer some fundamental questions about the world and the way things work. These investigations led to two fundamental concepts of calculus, namely, the idea of a derivative, which deals with rates of change, and that of an integral, which deals with accumulated change. The breakthrough in the development of these concepts was the formulation of a mathematical tool called a limit.

1. Limit. The limit is a mathematical tool for studying the tendency of a function as its variable approaches some value.
2. Derivative. The derivative is defined in the context of a limit. One of its uses is to compute rates of change and slopes of tangent lines to curves. The study of derivatives is called differential calculus. Derivatives can be used in sketching graphs and in finding the extreme (largest and smallest) values of functions. Biologists use derivatives to calculate, for example, the rates of change to the biochemical states of cells within individuals, rates of growth of populations, and rates of the spread of disease within populations.
3. Integral. The integral is found by taking a special limit of a sum of terms, and it is used initially to compute the accumulation of change. The study of this process is called integral calculus. Area, volume, work, and degree-days (the latter used to monitor the development of plants and “cold-blooded” animals) are a few of the many quantities that can be expressed as integrals. Biologists can use integrals to calculate, for example, the amount of fat bears store before going into hibernation, the time it takes an insect to develop from an egg into an adult as a function of temperature, the probability that an individual will die before a certain age, or the number of infected people as a disease spreads through a population.

Let us begin our study by taking an intuitive look at each of these three essential ideas of calculus.

The Limit

Zeno (ca. 500 BC) was a Greek philosopher who is known primarily for his famous paradoxes. One of these concerns a race between Achilles, a legendary Greek hero, and a tortoise. When the race begins, the (slower) tortoise is given a head start, as shown in Figure 3.

Is it possible for Achilles to overtake the tortoise? Zeno pointed out that by the time Achilles reaches the tortoise's starting point, a₁ = t₀, the tortoise will have moved ahead to a new point, t₁. When Achilles gets to this next point, a₂, the tortoise will be at a new point, t₂. The tortoise, even though much slower than Achilles, keeps moving forward. Although the distance between Achilles and the tortoise is getting smaller and smaller, the tortoise will apparently always be ahead.

Figure 3 Achilles and the tortoise.

Of course, common sense tells us that Achilles will overtake the slow tortoise, but where is the error in this reasoning? The error is in the assumption that an infinite amount of time is required to cover a distance divided into an infinite number of segments. This discussion gets at an essential idea in calculus, the notion of a limit.

Consider the successive positions for both Achilles and the tortoise:

After the start, the positions for Achilles, as well as those for the tortoise, form sets of positions that are ordered by the counting numbers. Such ordered listings are called sequences which we introduce in Section 1.7. As we discuss in our first example below, and explore further in Chapter 2, the limit of a sequence of values t₀,···, t_n can be bounded above by some value T, say, so that for all values of n, no matter how large, we have t_n < T. In the context of the Achilles paradox, this means that Achilles will pass the tortoise at time T, if T is the smallest of all values that satisfies this inequality.

The Derivative: Rates of Change

The derivative provides information about the rate of change over small intervals (in fact, infinitesimally small!) of time or space. For instance, in trying to understand the role of humans in global climate change, we may be interested in the rate at which carbon dioxide levels are changing. In Section 1.2, we show that it is possible to come up with a function that describes how carbon dioxide levels (in parts per million) vary as a function of time. The relationship between this function and the data is illustrated in Figure 4.

Figure 4 Carbon dioxide levels (in parts per million) as a function of months after April 1974.

In a scientific discussion about carbon dioxide levels, we might be interested in the rate of change of carbon dioxide levels at a particular time, say the second month (June 1974) of this data set. To find the rate of change from the second to tenth month, we could find the change in carbon dioxide levels, 331.8 − 331.0 = 0.8 parts per million, and divide it by the change in time, 10 − 2 = 8 months, to get the rate of change

over this eight-month period. Note that this rate of change corresponds to the slope of the secant line passing through the points P = (2, 331.0) and Q = (10, 331.8) as illustrated in Figure 5a. Although this rate of change describes what happens over the eight-month period, it clearly does not describe what is happening around the second month. Indeed, during the second month, the carbon dioxide levels are decreasing not increasing. Consequently, we expect the rate of change to be negative.

Figure 5 The tangent line.

To get the instantaneous rate of change at the beginning of the second month, we can consider moving the point Q along the curve to the point P. As we do so, the points P and Q define secant lines that appear to approach a limiting line. This limiting line, as illustrated in Figure 5b, is called the tangent line.

The slope of this line corresponds to the instantaneous rate of change for carbon dioxide levels at the beginning of the second month of the data set. Later you will be able to find this instantaneous rate of change, which is approximately −1.24 ppm per month. The slope of this limiting line is also known as the derivative at P. The study of the derivative is called differential calculus.

Integration: Accumulated Change

The integral deals with accumulated change over intervals of time or space. For instance, consider the 1999 outbreak of measles in the Netherlands. During this outbreak, scientists collected information about the incidence rate: the number of reported new cases of measles per day. How this incidence rate varied over the course of the measles outbreak is shown in Figure 6. To find the total number of cases of measles during the outbreak, we want to find the area under this incidence “curve.” Indeed, each rectangle in the left-hand side of this figure has a base of width “one day” and a height with units of measles per day. Hence, the area of each of these rectangles corresponds to the number of measles cases in one day. Summing up the area of these rectangles gives us the total number of measles cases during the outbreak. To get a rough estimate of this accumulated change, we can approximate the area under the incidence curve using the six larger rectangles imposed on the data, as illustrated in the right-hand side of Figure 6.

Figure 6 Incidence rate of the 1999 outbreak of measles in the Netherlands.

Source: Centers for Disease Control and Prevention.

Computing these areas yields an estimate of

The actual number of reported cases was 3292. Hence, our back of the envelope estimate was pretty good.

Integrals are a refined version of the calculation that we just made. Given any curve (e.g., incidence function) as illustrated in Figure 7, we can approximate the area by using rectangles. If A_n is the area of the nth rectangle, then the total area can be approximated by finding the sum

Figure 7 Area under a curve.

This process is shown in Figure 8. To get better estimates of the area, we use more rectangles with smaller bases. The limit of this process leads to the definite integral, the key concept for integral calculus.

Figure 8 Approximating the area using circumscribed rectangles

In Chapter 1, we introduce some modeling concepts while reviewing basic mathematical concepts such as real numbers and functions—including linear, periodic, power, exponential, and logarithmic functions. Using these functions, we model the cyclic rise of carbon dioxide concentrations in the atmosphere, dangers facing large versus small organisms, population growth, and the binding of receptor molecules. We also introduce the basic notions of sequences and difference equations. Using these constructs, we encounter the dynamics of oscillatory populations, drug delivery, and gene frequencies. In the ensuing chapters, we develop the ideas of differential and integral calculus, and along the way, build necessary skills in biological modeling. Using these ideas and skills, you will model a diversity of biological topics such as disease outbreaks, blood flow, population extinctions, tumor regrowth after chemotherapy, genetics of populations, genetic networks, mechanisms for memory formation, enzyme kinetics, and evolutionary games.