Introduction to Calculus in Biology Louis Gross January 2004 Over the past semester, we investigated a variety of mathematical approaches to biological questions. A common theme throughout the semester was an emphasis on problems that could be expressed in a "discrete-time" manner. Thus we analyzed the growth rate of cell populations in which we assumed the population at the next time-step depended only on the population at the current time step (first-order difference equations), or on the population size at the two previous time steps (second order difference equations), or we looked at the age structure of a population from one time step to another (Leslie matrix population model), or we analyzed the drug concentration in the blood stream just after giving a dose. However there are many biological situations in which it is not appropriate to consider the variables of interest at discrete times, but rather in a continuous manner. For example, a drug concentration in the blood stream varies through time, and we may want to know how this changes not just after giving a dose but in the time period between doses (we actually made an assumption about this last semester when we assumed it declined exponentially). Many physiological processes don't move in "jumps" but rather change continuously through time (e.g. body temperature, brain activity, enzyme concentrations, etc.). The mathematical area that deals with variables which change continuously, rather than in jumps, is called the calculus (which is short for differential and integral calculus). Many would consider the development of calculus to be one of the greatest conceptual achievements of the human mind. The ideas of the calculus transcend all of modern science, and have proven to be useful in applications to areas as diverse as neurobiology, economics, forensic science, ecology, and epidemiology. Although originally developed by geniuses (such as Isaac Newton), conceptually the calculus has been readily understood by people of all backgrounds and skill levels in mathematics. Our goal in this course is to help you learn about the conceptual foundations of calculus, provide you with some of the standard "tools" used to apply the calculus in science, and do this in a biological context so that you see how biological questions may be addressed using this wonderful concentual gift. The first concept we will discuss is one we have already seen - the idea of a limit. We developed this when thinking about populations varying through discrete time (generation by generation) in which we said that a population's size had a long-term limit if its size got closer and closer to a particular value after a large number of generations. We called this the asymptotic population size, the steady-state population size or the populations long-term equilibrium population size. Of course this didn't just apply to populations - we saw the same idea arise for the drug concentration within the body (measured just after a periodic dose is given) and saw that this approached a limit after a large number of regular, periodic doses. We are now going to consider the same idea of a limit, but now instead of letting something (generations, or number of doses) get very large, we are going to let a variable get close to some fixed number and see the impact on a function of that variable. For example, suppose that blood flow rate through the heart (cc per sec) is a function of a drug's concentration in the blood. We might want to know what the limit of the blood flow rate is when we let the blood concentration of the drug approach some value. All of this relates to the issue of what we will call "continuity" - whether a function (such as blood flow rate) changes smoothly as we vary the drug concentration or whether there is a sudden "jump" or shift in flow rate at some concentration. As simple example of this is a drug concentration that becomes lethal at some level (e.g. the person's heart stops and so the blood flow rate drops suddenly to zero). We will spend the first couple of class sessions formalizing the idea of limit and continuity.