A Population Genetics Model of Malaria Plasmodium berghei Resistance in the Mosquito Vector Anophele

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A Population Genetics Model of Malaria
(Plasmodium berghei) Resistance in the Mosquito
Vector Anopheles stephensi
Mary Jane Richardson and Leah Sauchyn
(http//jhmalaria.jhsph.edu/Faculty/jacobs_lorena/
documents/jacobs.htm)
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Mendelian Genetics
Example Flower Colour
genotype - the genetic makeup of an individual
PP Pp
pp

first allele
second allele
(http//www.janbiro.com/images/01-mendel-himself_1
_.jpg)
gene - portion of genetic material coding for a
functional unit eg. a protein - in
diploid organims there are 2 alleles/gene in each
individual - P gt purple (dominant)
- p gt orange (recessive)
phenotype the outward expression of the
genotype Purple
Orange
PP Pp
pp
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Transgenic Malaria-Resistant Mosquitoes
A allele that prevents malaria development in
the mosquito (dominant)
Phenotype
transgenic wild
Genotype
AA Aa
aa
(homozygous (heterozygous

transgenic) transgenic) Relative
fitness (W) WAA WAa
Waa
Where WAA (1b)(1-c) WAa
(1b) Waa 1
three different relative fitnesses
acts as a three phenotype system with respect
to selection
b benefit to being transgenic c cost to being
homozygous transgenic
(Marrelli et al., 2007)
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Transgenic allele (A)
SM1 peptide
Gametocyte-deficient strain
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Hardy-Weinberg Equilibrium
p
q
p frequency of allele selected for (A) q
frequency of allele selected against (a)
p
p2
pq
p q 1
q
pq
q2

• At equilibrium, the genotypic frequencies are the
  squared expansion of the allelic frequencies
• (pq)2 p2 2pq q2 1
• equilibrium is established after one generation
  (i.e. children are in H-W equilibrium)
• sexual reproduction does not change equilibrium
  frequencies
• a dynamic equilibrium - a new equilibrium is
  established following reproduction if allelic
  frequencies are changed

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Transgenic Malaria-Resistant Mosquitoes A Model
b benefit to being transgenic 0.5 c cost to
being homozygous transgenic 0.35 Relative
fitness Homozygous transgenic (WAA)
(1b)(1-c) 0.975 Heterozygous transgenic (WAa)
(1b) 1.5 Wild type (Waa) 1
Transgenic Mosquitoes
(http//www.nature.com/embor/journal/v7/n3/images/
7400643-f1.jpg)
Average relative fitness Wavet pt2WAA
2ptqtWAa qt2Waa
(Marrelli et al., 2007)
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Transgenic Malaria-Resistant Mosquitoes A Model
Genotypic frequencies in adult population after
selection and before reproduction
freqAAt1/2 pt2WAA
Wavet
freqAat1/2 2ptqtWAa
Wavet
Transgenic adult
(http//www.jichi.ac.jp/idoubutsu/Yoshida20public
ation.html)
freqaat1/2 qt2Waa Wavet
(Marrelli et al., 2007)
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Transgenic Malaria-Resistant Mosquitoes A Model
Allelic and genotypic frequencies in offspring
after reproduction and before selection
Allelic frequencies
pt1 freq(A)t1 1freqAAt1/2
½freqAat1/2 0freqaat1/2 qt1 freq(a)t1
1-pt1
Genotypic frequencies In Hardy-Weinberg
Equilibrium
freqAAt1 pt12 freqAat1 2pt1qt1 Freqaat
1 qt12
Transgenic juvenile
(http//www.jichi.ac.jp/idoubutsu/Yoshida20public
ation.html)
(Marrelli et al., 2007)
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Transgenic Malaria-Resistant Mosquitoes A Model
WAagtWaagtWAA
Inital condition 2pq 0.5 and p2 0
2pqp2 increases until p and q are at
equilibrium according to the relative fitnesses
(W)
(Marrelli et al., 2007)
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Transgenic Malaria-Resistant Mosquitoes Allele
Frequency Equation
pt1 1freqAAt1/2 ½freqAat1/2
0freqaat1/2
pt1 1(pt2WAA/Wavet) ½(2ptqtWAa/Wavet)
0(qt2Waa/Wavet)
pt1 1pt2WAA ½2ptqtWAa 0qt2Waa
Wavet
pt1 1pt2WAA ½2pt(1-pt)WAa
0(1-pt)2Waa pt2WAA
2pt(1-pt)WAa (1-pt)2Waa
(de Vries et al., 2006 Marrelli et al., 2006)
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Stability Analysis
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Stability Analysis
is stable if WAaltWaa
is stable if WAaltWAA
is stable if WAagtWAA,Waa
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Possible Outcomes of the Allele Frequency Equation
p1 0 unstable p2 1 stable
Case 1 WAAgtWAagtWaa
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Possible Outcomes of the Allele Frequency Equation
p1 0 stable p2 1 unstable
Case 2 WAAltWAaltWaa
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Possible Outcomes of the Allele Frequency Equation
p1 0 unstable p3
Waa-WAa stable
WAA-2WAaWaa p2 1
unstable
Case 3 WAagtWAAgtWaa OR WAagtWaagtWAA
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Possible Outcomes of the Allele Frequency Equation
Case 4b WAaltWAAltWaa
Case 4a WAaltWaaltWAA
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Possible Outcomes of the Allele Frequency Equation
Case 4a and Case 4b p1 0
stable p3 Waa-WAa unstable
WAA-2WAaWaa p2 1
stable
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Transgenic Malaria-Resistant Mosquitoes Allele
Frequency Equation
p3 0.4878
p3 Waa WAa WAA 2WAa
Waa
WAA (1b)(1-c) 0.975 WAa (1b) 1.5 Waa 1
WAagtWaagtWAA
p never becomes fixed - mosquitoes that transmit
malaria will not be eliminated from the
population as long as heterozygous transgenics
are more fit than homozygous transgenics
(de Vries et al., 2006 Marrelli et al., 2007)
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How long does it take to reach p3?
682.5 days
577.5 days
472.5 days
367.5 days
Assuming a generation time of 1.5 weeks it takes
1 year, 10 months , and 17 days to reach p3 from
p 0.01
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Conclusions

• In general
• the relative fitness of the genotypes determines
  the stability of the fixed points

• Malaria model
• the heterozygote transgenic has the greatest
  relative fitness
• the transgenic allele (p) will never become
  fixed in the mosquito population
• wild type (q) persists in heterozygote
• how applicable is this system? (Cohuet et al.,
  2006)
• Plasmodium berghei is a parasite of muric
  african rodents
• Anopheles stephensi is a laboratory vector

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Literature Cited
Cohuet, A., Osta, M., Morlais, I., Awono-Ambene,
P., Michel, K., Simard, F., Christophides, G.,
Fontenille, D., Kafatos, F. (2006). Anopheles and
Plasmodium from laboratory models to natural
systems in the field. EMBO reports 7(12)
1285-1289. de Vries, G., Hillen, T., Lewis, M.,
M?ller, J., and Schönfisch, B. (2006). A course
in mathematical biology quantitative modeling
with mathematics and computational methods.
Society for Industrial and Applied Mathematics,
Philadelphia, PA. Janse, C. and Waters, A.
(2006). The life cycle of Plasmodium berghei in
The Plasmodium berghei research model of
malaria. Leiden Univeristy Medical Center.
http//www.lumc.nl/1040/research/malaria/model.htm
l. Accessed on May 9th, 2007. Marrelli, M.T.,
Li, C., Rasgon, J.L., and Jacobs-Lorena, M.
(2007). Transgenic malaria- resistant mosquitoes
have a fitness advantage when feeding on
Plasmodium- infected blood. PNAS 104(13)
5580-5581. All images from Google Images
accessed on May 10th, 2007.
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Acknowledgments
We wish to thank Gerda de Vries and Frank Hilker
for much needed guidance and patience, Drew
Hanson for being a pillar of strength during a
time of need, the University of Alberta, the
Centre of Mathematical Biology and the Pacific
Institute for the Mathematical Sciences.
Gerda
Frank
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Questions?