Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models III. Exact genetic models and modelling class structured populations

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Theoretical Modelling in Biology (G0G41A ) Pt
I. Analytical ModelsIII. Exact genetic models
and modelling class structured populations

• Tom Wenseleers
• Dept. of Biology, K.U.Leuven

21 October 2008
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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

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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 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
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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

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Problem

• if we would like to calculate the equilibrium
  frequency of some allele there will be very many
  possible F x M mating typese.g. aa x Aa, Aa x
  aa, aa x AA, aA x AA, etc...
• will be function of things like dominance etc...

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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

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Transition matrix

• ...and we can write our recurrence equations in
  the formpf(t1) a . pf(t) b . pm(t)pm(t1)
  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
  where
  gene transition matrix

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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?

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E.g. diploid inheritance
1 0
neutral allele pf(t1) 1/2 . pf(t) 1/2 .
pm(t)pm(t1) 1/2 . pf(t) 1/2 . pm(t)
pm
0
1
pf
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E.g. diploid inheritance
1 0
neutral allele pf(t1) 1/2 . pf(t) 1/2 .
pm(t)pm(t1) 1/2 . pf(t) 1/2 . pm(t)
pm
0
1
pf
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E.g. diploid inheritance
1 0
neutral allele pf(t1) 1/2 . pf(t) 1/2 .
pm(t)pm(t1) 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
1
pf
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Eigenvectors direction in which the system
grows or decays, each has an associatedEigenvalu
e which sayswhether system grows (gt1)or decays
(lt1)
1 0
pm
Eigenvector 2Eigenvalue 0
Eigenvector 1Eigenvalue 1
0
1
pf
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To determine whether gene will increasein
frequency we need to check whenthe dominant
(largest) Eigenvalue of gene transmission matrix
A gt 1
Eigenvectors direction in which the system
grows or decays, each has an associatedEigenvalu
e which sayswhether system grows (gt1)or decays
(lt1)
1 0
pm
Eigenvector 2Eigenvalue 0
Eigenvector 1Eigenvalue 1
dominant eigenvalue
0
1
pf
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Other use of Eigenvalues Eigenvectors class
structured population

• abundance n of x different age or stage classes

Leslie matrix f age specific net fecundityP
age specific survival
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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