Theoretical Modelling in Biology (G0G41A )
Pt I. Analytical Models
III. Exact genetic models and
modelling class structured populations
Tom Wenseleers
Dept. of Biology, K.U.Leuven
21 October 2008
Aims
• last week we showed how to do some
simple genetic models using recurrence
equations
• aim of this lesson: do some more complex
genetic models involving two sexes,
familiarise yourselves with the meaning
and utility of Eigenvalues and
Eigenvectors
• also use these to analyse class structured
populations
Diploid selection
Single-locus, diallelic model (A/a) for a diploid species with
nonoverlapping generations :
Frequency of A allele in next generation
= A gametes produced /
total number of gametes produced
p(t ) 2 .WAA.1  2. p(t ).(1  p(t )).WAa .(1 / 2)
p(t  1) 
W (t )
where W (t )  p(t ) 2 .WAA  2. p(t ).(1  p(t )).WAa  (1  p(t )) 2 .Waa
Does not take into account that there are two distinct sexes
and that a gene often only has an effect only in one of two
sexes
Steps in making a genetic model
involving two sexes
1. write down a mating table showing the
frequency of different types of matings
and the types and number of offspring or
gametes they produce
2. based on this mating table write down a
set of recurrence equations describing
allele frequency change across
generations
Problem
• if we would like to calculate the equilibrium
frequency of some allele there will be very
many possible F x M mating types
e.g. aa x Aa, Aa x aa, aa x AA, aA x AA,
etc...
• will be function of things like dominance
etc...
Simplifying things:
invasion conditions
• to simplify matters we often just look at
invasion conditions
• i.e. when can a rare gene spread in a
population?
• in that case we only need to consider
three F x M mating types: aa x aa (wild
type), aa x Aa and Aa x aa
Transition matrix
...and we can write our recurrence equations in the form
pf(t+1) = a . pf(t) + b . pm(t)
pm(t+1) = c . pf(t) + d . pm(t)
where pf and pm are the frequency of the allele among
female and male gametes. This set of equations can
be written in matrix form notation as
 p f (t  1) 
 p f (t ) 
a b 

  A
 where A  

 pm (t  1) 
 pm (t ) 
c d 
= gene transition matrix
Problem
• eventually we want to know whether the
overall frequency of the allele will go up or
down
• how can we summarize the overall
behaviour of the system of equations?
E.g. diploid inheritance
1
neutral allele
pf(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
pm(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
pm
0
0
pf
1
E.g. diploid inheritance
1
neutral allele
pf(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
pm(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
pm
0
0
pf
1
E.g. diploid inheritance
1
neutral allele
pf(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
pm(t+1) = 1/2 . pf(t) + 1/2 . pm(t)
if allele was positively selected for,
i.e. wasn't neutral, system should
move in this direction
pm
0
0
pf
1
Eigenvectors = direction in which
the system grows or decays,
each has an associated
Eigenvalue which says
whether system grows (>1)
or decays (<1)
Eigenvector 2
Eigenvalue = 0
1
pm
Eigenvector 1
Eigenvalue = 1
0
0
pf
1
Eigenvectors = direction in which
the system grows or decays,
each has an associated
Eigenvalue which says
whether system grows (>1)
or decays (<1)
Eigenvector 2
Eigenvalue = 0
To determine whether gene will increase
in frequency we need to check when
the dominant (largest) Eigenvalue of
gene transmission matrix A > 1
1
pm
Eigenvector 1
Eigenvalue = 1
= dominant eigenvalue
0
0
pf
1
Other use of Eigenvalues & Eigenvectors:
class structured population
abundance n of x different age or stage classes
 n1 (t  1)   f1

 
 n2 (t  1)   P1
 n (t  1)   0
 3

 n4 (t  1)   0
 ...   ...

 
 n (t  1)   0
 x
 
f2
f 3 ...
f x 1
0
P2
0
0
...
...
0
0
0
P3 ...
0
...
... ...
...
0
0
... Px 1
f x  n1 (t ) 


0  n2 (t ) 
0  n3 (t ) 


0  n4 (t ) 
...  ... 
0  nx (t ) 
Leslie matrix
f = age specific net fecundity
P = age specific survival
Predicting the behaviour of a
class structured population
• Dominant eigenvalue of Leslie matrix = growth
of population after it reached a stable age or
stage distribution
• Eigenvector corresponding to dominant
eigenvalue gives the stable age or stage
distribution
• for species with two sexes usually only the
female population is modelled on the
assumption of female demographic dominance