MathIII.pdf

Models in Biology

Modelling Biology

Basic Applications of Mathematics and

Statistics in the Biological Sciences

Part I: Mathematics

Script C

Introductory Course for Students of

Biology, Biotechnology and Environmental Protection

Werner Ulrich

UMK Toruń

2008

Models in Biology

Contents

Introduction .................................................................................................................................................. 3

1: How to build a model ............................................................................................................................... 4

2. From Euclidean to fractal geometry ....................................................................................................... 10

3: Biological growth processes................................................................................................................... 18

4: Models of competition and predation ..................................................................................................... 26

5: Models in biochemistry .......................................................................................................................... 34

6. Markov chains ........................................................................................................................................ 41

7. The Weibull function and life table analysis .......................................................................................... 48

8. Basic models in genetics ........................................................................................................................ 53

Literature .................................................................................................................................................... 59

Online archives and textbooks.................................................................................................................... 60

Mathematical software ............................................................................................................................... 61

Important internet pages ............................................................................................................................. 62

Latest update: 10.01.2008

Models in Biology

Introduction

The following text is the third part of a lecture in basic mathematics biologists. The whole lecture con-

tains what might be considered an international standard of basic knowledge although many readers will surely

miss important branches. This part deals with the application of mathematics in biology. It focuses on model

building and interpretation. Again, many examples are included that show how to program simple tasks with a

spreadsheet program and how to use advanced mathematics software.

The following text in not a textbook. It is intended as a script to present the contents of the lecture in a

condensed form. There is no need to write a textbook again. Today, the internet took over many former tasks

textbooks had. The end of this text contains therefore a small overview over important internet pages where

students can find mathematics glossaries, textbooks, and program collections.

Models in Biology

1. How to build a model

In this lecture we will learn how to build simple biological models using a spreadsheet program. We will

see why this is necessary and how to use such models.

Why is it necessary to build mathematical models using experimental or observational data? There are

several reasons for this. Biology has transformed from natural history (history!) to an explanatory science.

It not only tries to describe phenomena in nature it tries to understand causes and relations. For this task we

have to structure our observations and to look for relations between them. This is exactly the modelling proc-

ess: we use the science of structures, mathematics, to uncover hidden patterns and relations. Modelling is

therefore more than finding out whether sample means differ or whether we have simple correlations between

data. We have to parameterize these relations. But models have many other tasks. First of all, they generate

new predictions about nature, predictions that then have to be verified or falsified. This prediction generating

feature is of course also a method to verify our model. Secondly, good models allow predictions to be make

about the future. This is a main aim for all environmental models. They are designed to predict the future of

populations, ecosystems and biodiversity. At last models reduce the chaos in our data and allow the develop-

ment of new theories and concepts.

Models can be classified into certain classes. On one end of a continuum we have verbal models stating

more or less precisely relations between a set of variables. These verbal statements may be incorporated into

diagrams where the variables are connected by arrows. Then, we have a qualitative model . On the other end,

there are explicit mathematical models that formalize relations. These relations may be fully parameterized

and then we have a quantitative model that gives quantitative predictions about variable states. At last, models

may contain exact parameter values at all stages of computation. We speak of deterministic models because

all future states of the models can in principle be computed by the initial set of values. On the other hand, the

model might contain more or less stochastic variables, variables that are driven by random events. In this case,

future parameter values are less sure or even chaotic. In this case we speak of stochastic models .

This short discussion indicates already what we need to build a model. This discussion is visualized in

Fig. 1.1

• The first step is that we have a theory. Shortly speaking, a theory is a set of hypotheses stated in a

formal language. We need hypotheses about nature and the relations between certain variables.

Modelling without a priori theoretical reasoning will lead to nothing. Our a priori experience

must lead to a selection of variables, so-called drivers , of the model. These drivers might have

explicit or stochastic values. They might be parameterized (characterized by explicit values or

functions) or not. In the latter case the model itself should assign values or value ranges to these

parameters.

• Then, we have to collect the necessary data. These data have to match the requirements of our

theory. Making experiments or observations without an explicit theory in mind will very often

result in large sets of data without any value because afterwards we (suddenly) notice that one or

another important variable had been ignored and not measured or that our method was inappro-

priate to incorporate the variable values of the model. This latter case occurs very often if we

took to few replicates and the variability in measurement is too high. Problems also arise if we

Models in Biology

used different methods for observations and we

later notice that these differences make it impos-

Theory

Fig. 1.1

sible to compare the data (for instance because

they differ in the degree of quantitativeness).

New

predictions

Drivers

Parameters

• In a next step we have to confirm assumed rela-

tionships between these drivers. We might as-

sign qualitative or quantitative relations. If we

Data

Functions

quantify the relations (for instance from a re-

gression analysis) we parametrize these rela-

tions.

Quantifi-

cation

• Then, we have to formalize the relations. This is

best done by a flow diagram or flow chart .

Flow chart

The flow diagram forces us to write each rela-

tion and each step of the model explicitly. This

step often uncovers smaller or larger errors in

our initial model that would have remained undiscovered in a purely verbal model formulation.

Output

Computer

algorithm

Making flow diagrams learns us thinking hard!

• The following step is then a technical one. Rewriting our flow diagram into a computer algo-

rithm. For more complicated models this should be a done using a common computer language

like C++, Pascal, R or Fortran, simple models can be written via a spreadsheet program like Ex-

cel.

• Our model will generate a set of output variables or whole classes of relations. We have to check

these parameters, whether their values are realistic, whether they correctly predict real values and

whether they are able to predict the future.

• At the end we have to modify our model in the light of its predictions and variable states.

Let’s exemplify the above steps of modelling from a simple example with real data. We measured the

population densities of a parasitic wasp species of the hymenopteran genus Aspilota during a series of genera-

tions. Aspilota is a group of small braconid wasps that predominantly develop as internal parasitoids of necro-

phagous flies of the family Phoridae. It is a very abundant and species rich genus. Our initial assumption is that

the population densities should be influenced by the densities of its host species and by a set of weather vari-

ables. Additionally, we assume that the densities of the previous generation should also influence wasp densi-

ties because high or low previous densities should find their expression in reproductive rates. By this, we ver-

bally stated an initial theory and pointed to a set of interesting variables. These variables are D wasp , D host , and

climatic variables. What climatic variables? To allow a model to be constructed we must specify the variables

and their way of influencing. From previous studies and a literature survey we decide to recognize five climatic

input variables, precipitation CP, cloudiness CC, air temperature CT, relative atmospheric humidity CH, and

Year

Mean values of climatic factors (January to March)

Year

Mean values of climatic factors (June)

Hosts

Aspilota

previous Gen. following Gen.

1980

111

157.2

71.25

116

1981 1.9667

58.9333

51.8

77.5

1981 15.4

165.3

133.8

120

3.8

0.2

1982

-0.9

32.9667

96.6

1982 16.7

54.5

193.9

62.5

260

3.2

0.3

1983

2.967

78.667

52.9333

67.7333

76.25

1983 16.8

34.9

192.2

62.5

191

3.8

2.8

1984 1.7667

76.333

43.5666

75.6

67.5

1984 14.2

52.4

146.9

76.25

148

0.1

1985 -1.3889

77.667

29.2667

77.5

71.6667 1985 13.4

139.7

135.8

76.25

0.8

0.1

1986 -0.467

62.3

74.1667 1986 16.6

50.6

227.8

57.5

178

14.5

12.3

1987 -2.489

81.333

62.9333

79.4

1987 14.2

100.5

125.2

10.3

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: