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.