Maximization, Optimization and Evolution: Introductory notes on the use
of calculus to solve maximization problems in biology.

Louis J. Gross - Math 152 - University of Tennessee

A central theme of the biological sciences is that organism
characteristics (morphology, size, physiology, behavior, etc.) are
shaped by the processes of natural selection. Organisms which have
characteristics that enable them to better survive and reproduce (the
jargon for this is that they are more "fit" but this has nothing
necessarily to do with physical fitness in terms of how many push-ups
they can do) are selected for. This means that over time, all else
being held fixed, organisms with those characteristics will increase in
frequency, IF such characteristics are heritable. Here, heritability
means that the tendency to display such characteristics is possible to
pass on to the organism's offspring.

This is the evolutionary process driven by natural selection. In
deducing this, Charles Darwin did not have a mechanism through which
heritability occurs, but we now have the entire field of genetics which
describes in great detail how organism characteristics are encoded in
DNA and thus passed on to offspring. When this is combined with a
process through which variation in characteristics among organisms
occurs, brought about by genetic mutation and by recombination which
reassorts genetic material, evolution proceeds. Evolution simply means
that there are temporal (and spatial, when there are spatial
differences in fitness across the planet) changes in the frequency of
characteristics displayed by organisms. Thus, the key components of
biological evolution are Selection, Variation, and Heritability. There
are numerous other ways in which the term evolution is used in various
scientific fields - in astronomy to describe the dynamics of stars
through their lifespan, in mathematics to describe certain kinds of 
equations, in computational science to describe genetic algorithms 
and other computational procedures which mimic biological evolution.  

There are an enormous number of well-described situations in which it
is posited that certain characteristics of organisms are observed
because they are "adaptive" - that is, individuals having these
characteristics have higher "fitness" - they are better able to survive
and reproduce than organisms which do not have these characteristics.
An example of a characteristic for which it is argued that it is
adaptive is leaf size - it is a fact that in general desert plants have
much smaller leaves than plants native to wetter conditions. There are
very well-developed arguments for this based upon the trade-off between
photosythetic carbon gain and water loss through transpiration. Indeed,
there are entire photosynthetic pathways (Crassulacean Acid Metabolism
or CAM), which have been shown to be beneficial under dry conditions -
CAM is present in many species including desert cacti. For details on
the physiological aspects of adaptation in plants, see Park Nobel's
"Physicochemical and Environmental Plant Physiology" and for more about 
cacti see his "Cacti: Biology and Uses".

The above implies that if we had a way to describe how fitness changes
with different organism characteristics (e.g. how carbon gains versus
water loss change as a function of leaf size), we would be able to
describe the "best" characteristic by taking the derivative of the
function, setting it equal to zero to find the critical points, and
then finding the critical point (such as leaf size) that maximizes
fitness. We could then compare this to observations under different
environmental conditions and have an explanation for why we observe
different leaf sizes under different conditions. This views evolution
as a process of fitness maximization.

The above approach is limited however, because "all else being held
fixed" as mentioned above, may not hold. There are many constraints on
the process of evolution which may reduce the utility of thinking of it
as a simple calculus maximization problem. One of these is that there
are physical and chemical constraints on organisms (e.g. large leaves
require structures to maintain them in a position to absorb light, and
these structures may not have the mechanical strength to hold up very
large leaves - how many trees with a single large leaf have you seen -
but this is one way of thinking about cacti). So the laws of physics
constrain evolutionary processes and wonderful books on this topic are
Steven Vogel's "Life's Devices"  and "Cats' Paws and Catapults:
Mechanical Worlds of Nature and People".

Another constraint on evolutionary processes is history - the past
constrains the types of characteristics it is possible to display. This
induces constraints due to genetics as well. You should recall the
simple population genetics models for changes in gene frequency due to
selection derived when we discussed difference equation models. These
models have been greatly elaborated to analyze situations in which the
genetic variability within a population restricts the possibility that
evolution pushes the frequency of characteristics to be where an
overall "peak" in fitness occurs. Such restrictions in part arise due
to the interactions between organisms, so that selective forces depend
upon the current frequencies of characteristics amongst the population
(the jargon name for this is "frequency-dependent selection"). For
readable descriptions of how historical processes constrain evolution,
read Stephen Jay Gould's "The Panda's Thumb: More Reflections in
Natural History".

Based on the above, evolution should generally be thought of, not as a
simple maximization problem, but as an optimization problem. The term
optimization here is used to mean that there is something being
maximized (or minimized) but that there are constraints on this. In our
course, we will only discuss examples in which the constraints limit
the function describing the problem to a particular interval - we are
restricting the domain over which we consider the function being
maximized, and must compare the maximum inside this domain to the
values of the function at the endpoints of the domain as well. There
are much more complex types of constraints (and functions) that arise
in biology, but the objective of this section is to point out the
concept and how it may be applied. In many ways therefore, due to
historical, genetic and physical constraints, it has been argued that
evolution acts more as a "tinkerer" would than as an engineer would
to design organism characteristics which are perfectly designed to meet 
environmental conditions. A very readable description of this view is in 
Francois Jacob's book "The Possible and the Actual". 

The text presents an example of the above view of evolutionary
processes thought of in terms of optimization, by considering
cardiovascular branching. The assumption in this example is that the
branching of arteries within a circulatory system of a mammal arises due
to processes which have led to the minimization of the resistance to
flow between two points in the body. Thus the fitness function in this
case is the resistance to blood flow, which is minimized, corresponding
to maximizing the total flow between two points. The constraints are
the limitations on sizes of the arteries and the physics of flow in
"pipes". We proceed by analyzing the branching angle which minimizes
resistance to flow, and then comparing the resulting answer to data. If
the results are consistent with observations of branching angles in
circulatory systems, this provides evidence that structure of the
circulatory system has arisen to meet this "fitness" criteria, but is
not proof that other processes are not operating as well to affect 
branching angles.

Copyright 2005 - do not reproduce or modify without author's permission.