The Structure, Function, and
Evolution of Vascular Systems
Instructor: Van Savage
Spring 2010 Quarter
3/30/2010
Course Themes
• Learn mathematical methods for studying
biomedical and biological systems, including
interactions (epistasis and among species),
topology and dynamics of gene interaction
networks and food webs, tumor initiation and
growth, antibiotic resistance, and much more
• Evolutionary theory is key to all of these and is a
central way to understand dynamics of biological
systems
• Applied Math methods from asymptotics that are
useful for all types of models, including evolution,
predator prey, topology, informatics, scaling, etc.
Textbook
• Evolutionary Theory: Mathematical and
Conceptual Foundations by Sean Rice
• Advanced Mathematical Methods for Scientists
and Engineers by Carl Bender and Steven Orszag
• Several recent papers published in high-profile
journals. I will give you these in advance.
Students will take turns presenting the research
paper and leading discussion. Everyone will do at
least one paper.
• Back and forth between texts and paper which is
the way science is really done.
What do I expect from you?
• To come to class because I will cover
material from multiple places that is not all
in your book or papers and because you
will learn more if you actively participate
• To ask questions when you do not
understand and give feedback
• To turn in 4 homework sets (40%)
• To do a class project with paper and
presentation (40%)
• In class presentations of papers (20%)
• To work hard
What backgrounds do you have?
What research do you do?
What do you hope to get out of the course?
What topics would you most like to see covered?
Crash Course in Evolutionary Theory
(See Chapter 1 of Sean Rice book)
What is fitness and what does it
describe?
Ability of an entity to survive and propagate forward
in time. It is inherently a dynamic (time evolving
property). Can assign fitness to
1.
2.
3.
4.
5.
6.
7.
8.
Individuals
Genes
Phenotypes
Behaviors
Strategies (economic, cultural, games, etc)
Tumor cells and tumor treatment
Antibiotic resistance
Language
Types of definitions
Absolute Fitness-number of surviving offspring
Relative Fitness-scaled version of absolute fitness. Typically
scaled relative to wild type (when looking at mutations) or
population average (net advantage or disadvantage) or the
maximum, making the peak fitness equal to 1.
As with most of science, ratios and differences drive nature, not
absolute number because those depend on units, which are just a
manmade abstraction and not part of nature.
Evolution in particular is about relative advantage or
disadvantage.
Absolute fitness is useful because it connects to ecology and
population theory.
Hardy-Weinberg model
Definitions
1. Loci is location of allele on genome
2. Allele is version of gene at that loci
Assumptions
1. Two alleles exist A1 and A2
2. Population is infinitely large so choosing an
allele does not affect probability of choosing that
same allele again (no conditional probabilities)
3. Random mating means for each new offspring
we can pick any two alleles at random
Hardy-Weinberg model
t
……….
A1
A1
A1
A2
A2
A2
……….
A1
A1
……….
A1
A2
……….
p1 is frequency of allele A1 and p2 is frequency of allele A2
Bag of two different colored marbles
A1
A1
t+1 ……….
A1
A2
A1
A2
A1
A1
A2
A2
A2
A2
A1
A1
……….
……….
A1
A1
A1
A2
……….
……….
A1
A2
……….
Hardy-Weinberg model
At generation t+1, the frequencies are
Frequency of
A1
A1
Frequency of
=p12
Frequency of
A1
A2
=2p1p2
Note that these are the different terms of
Also, note that
( p1  p2 ) 2
p1  p2  1  ( p1  p2 ) 2

A2
A2
=p22
Fitness and population growth
For a population to grow, it must be finite, and thus
not match Hardy-Weinberg assumptions
Nt = Number of individuals at time t
wij is fitness of genotype with phenotype AiAj
N t 1  p N t w11  2 p(1 p)N t w12  (1 p) N t w 22
2
2
 wN t
This is now in the form of population growth equations
w  p w11  2 p(1 p)w12  (1 p) w 22
2
2
How does fitness change with
allele frequency?
dw
*
*
 2(w1  w 2 )
dp
marginal fitness is expected number of descendants of
a given allele

w  pw11  (1 p)w12
*
1
w  pw12  (1 p)w 22
*
2

Evolution of allele frequency and
Wright’s equations
p(1 p) d ln w p(1 p) d(rG)
pt 

2
dp
2
dp
Conclusions
1. Increases in direction of slope of fitness function
2. Allele frequency climbs peak until maximal fitness
and this derivative or slope is zero
3. Peak occurs when marginal fitness for A1 and A2
are equal, implying relative fitness of heterozygote
4. Prefactor is actually a variance, so strength of
selection depends on variance. No variance
implies no selection.
w
p
What does variance mean here? Where is it zero?
What is slope? Where is it zero?
Two peaks here. Can this happen in current version of our model?
How do we maintain variance?
Mutation and migration
What is typical effect of a mutation?
Wild Type fitness=1 (relative fitness)
Hetero. Mutant fitness=1-hs
Deleterious double mutant=1-s
Genetic Load= 1 w  sp 2  2 phs(1 p)
Next class we will move onto mutation, finite-size
populations, additive genetic variance, and interactions
between loci and genes and possibly touch on drift and
coalescence.
Read Chapter 1 of Rice’s book. I will put a copy in Biomath
office or outside my office.
Starting soon, probably next week, we will use these ideas to
discuss papers on the distribution of epistatic interactions,
modeling epistasis, the evolution of sex, and the evolution of
antibiotic resistance.