Introduction to Matrices and Modeling in Biology
A central theme in biology is that biological objects of interest have
structure that may affect the way those objects should be analyzed.
Moving up the hierarchical biological scale, examples include the
various components of cells, the cellular composition of tissues, the
tissue composition of individual organisms, the composition of
different individuals which make up a population, the species which
make up a community and the types of habitat across a landscape. Each
of these biological entities has a sub-structure. Think of different
types of individuals within a population: the males and females of
different ages in a meerkat population, the demographic (age, location,
economic status, etc.) structure of a human population, the chemical
composition of cells, etc. One of the most useful mathematical concepts
provides a way to describe the composition of these biological objects,
and provide a mechanism to analyze how the composition might change
through time or across space.
A motivating example that we will use regularly in this section of the
course concerns patterns of species across a landscape. Think of taking
aerial photographs of a plot of land once each decade, following the
changes in pattern across the landscape. On a real landscape, this
could be looking at species, communities of plants (deciduous
hardwoods, herbacious, grassland, etc.), or land-use (agricultural,
urban, suburban, etc.). For each decade you have an photo in which you
might be able to classify each area as consisting of one of a few
species. Then one descriptive summary you might use for the landscape
is what fraction of the total area is covered by each of the species.
In the EcoBeaker example, this corresponds to what fraction of the
landscape pictured on the computer is of the different colors,
representing different types of organisms (grass, blackberry bushes,
hickory tress, etc.).
Describing this landscape at a particular time then could use a list of
numbers giving the fraction of the landscape of each species, in other
words a vector of numbers which sum to one. Pictorially, this
description corresponds to the bar charts illustrated in the EcoBeaker
example, showing the number of squares in each landscape covered by
each type of organism. There is of course a loss of information in
going from the picture of the landscape to a simple list of numbers
representing the fraction of the landscape of each type. All
information about the spatial arrangement is lost - that is we can't
tell from just a vector of numbers whether all the grass is clustered
on one side of the landscape or whether it is dispersed throughout the
landscape.
As time goes on (e.g. we take a picture of the landscape every decade),
the vector describing the fraction covered by each type of organism
will change. If we could determine the rules by which this vector
changes, we would have a basis for a model of the landscape's dynamics
(called succession in ecology). From this model we might then be able
to determine from just a few decades of pictures, what the landscape
might look like many decades from now (the process we called
extrapolation when we were talking about regressions).
Note that this entire area of looking at landscapes and how they can be
described and how they change is part of the field of geography. The
main tool used to analyze changes is called Geographical Information
Systems (GIS), which is a fancy name for software that allows you to
look at, build and manipulate maps.
Our objective in this section of the course is to construct the
mathematical tools needed to analyze changes in vectors describing
landscapes. This doesn't just apply to landscapes though. It applies to
any biological entity which can be broken down into discrete classes.
So we will be able to use this to describe changes in the age breakdown
of human population of Knoxville (or any other region). This is the
basis for the entire field of demography - the study of changes in the
structure of human populations through time and space. But this is just
as easily applied to any population - bears in the Smokies, killer
whales in the Pacific, meerkats in South Africa, etc. More than that
however, it applies to an immense number of biological problems from
cell population dynamics (think of bacterial populations in which you
track which bacteria are resistant to each of a list of antibiotics),
to behavior (think of the fraction of a group of meerkats carrying out
each of a list of behaviors), to phamacokinetics (think of the fraction
of a drug infused into the body in each of a list of tissues and how
this changes through time - there would be a class here called
"removed" in which the drug has been excreted).
It turns out that the mathematics which allows us to describe changes
in the structure of biological entities or landscapes is simply
described by building an approppriate set of rules for manipulating
numbers that are arranged in particular orders. The order matters
because the way we describe these structures depends upon the order
that you list the numbers. So a vector that describes the fraction of
the landscape in (grass, shrub, trees) as (.2, .5, .3) is different
from the vector (.5, .2, .3) - these represent very different
landscapes - the order of the numbers matters. So we have to build up a
way of manipulating numbers in which order matters - this is different
from the typical rules of algebra you have seen since middle school. We
essentially have to come up with a way to manipulate vectors in the
same way you used the algebraic manipulations (addition, subtraction,
multiplication and division) for single numbers (single numbers are
called "scalars" to differentiate them from vectors which consist of
lists of numbers in particular orders). Just as numbers can be
represented in general by a letter (the essential idea of algebra yo've
seen since high school), vectors can be represented by letters, and the
elements of a vector (the numbers which make it up) can be letters
(algebraic entities) as well.
So what is a "Matrix" - yes it is the movie, but in our case it simply
represents a list of numbers aranged in a 2-dimensional array:
2 4 5 5
1 -2 -5 2
4 3 -9 0
2 2 3 8
Yes this is just like the cascading stream of numbers in the movie, but
in this case they are not cascading - just static. What we will do in
this section of the course is first describe an algebra for how to do
the basic algebraic manipulations of vectors (just a row or column of
numbers arranged in a particular order) and matrices (the plural of
matrix) which are rectangular arrays of numbers.
Our objective in describing this area of mathematics (called Matrix
Algebra or Linear Algebra) is to get to the point of being able to
describe mathematically how the strcuture of a landscape, population,
etc. changes through time.
One way to express this is similar to the notion that had for the
equation describing exponential growth of a population -
N(t)= N(0) exp(-kt). Only this just describes how a single number
giving the entire population size of density (numbers of individuals
per unit area or per unit volume) changes with time. When we want to
describe how a population with structure changes, we need to express
how a vector changes with time. If this is in discrete periods, say
from one year to the next, then we might express how we go from a
vector of population structure at time t, Nbar(t) to a vector of
population structure at time t+1, Nbar(t+1). To be able to do this,
we need to learn some basics of matrix manipulations.
Matrix algebra - basic operations
These are described in Section 55 of the text. This includes the
definition of an m x n matrix (m rows and n columns), what it
means to say that two matrices are equal (the element in a row
and column of one matrix is the same as the element in the same
row and column of the second matrix, and this holds for all
elements), how to add matrices (can only do this if they are the
same size and then you add each element of the first matrix to
the correponding element of the second matrix to get the sum),
how to subtract matrices (same as addition, only subtract element
by element), how to multiply (can only do this if the number of
columns of the first matrix is the same as the number of rows of
the second matrix), and the basic laws of matrix algebra (same as
standard arithmetic, except that the commutative law doesn't hold
in general, so that A B is not the same as B A for two matrices
A and B.