Transcript W Aa - Mathematical and Statistical Sciences

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)
Mendelian Genetics
Example: Flower Colour
genotype - the genetic makeup of an individual
first allele
PP
Pp
second allele
pp
(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 => purple (dominant)
- p => orange (recessive)
phenotype: the outward expression of the genotype
Purple
PP
Pp
Orange
pp
Transgenic Malaria-Resistant Mosquitoes
A – allele that prevents malaria development in the mosquito (dominant)
Phenotype:
transgenic
Genotype:
AA
Aa
aa
(homozygous
(heterozygous
transgenic)
transgenic)
Relative fitness (W):
Where:
wild
WAA
WAa
Waa
WAA = (1+b)*(1-c)
three different relative fitnesses
WAa = (1+b)
acts as a three phenotype
system with respect to selection
Waa = 1
b = benefit to being transgenic
c = cost to being homozygous transgenic
(Marrelli et al., 2007)
Plasmodium berghei life cycle
Blood meal
Transgenic
allele (A)
oocyst (n) in blood
sporozites (n) in
salivary gland
ookinete (2n) in midgut
sporozites (n) in
blood
sporozites(n) in liver
Gametocyteproducing strain
merozites (n) in
red blood cells
zygote
(2n) in
midgut
schizont (n) in
red blood
cells
♀gamete
♂ gamete
Infected Mosquitoes
(Anopheles stephensi)
gametocytes (n) in
blood meal
Gametocytedeficient strain
(http://www.tufts.edu/tie/tci/images/climate
change/Aedes%20mosquito.jpg)
Blood
meal
gametocytes (n) in
blood
Infected Rodent
(Grammomys surdaster)
(http://www.lumc.nl/1040/research/
malaria/model02.html)
Hardy-Weinberg Equilibrium
p = frequency of allele selected for (A)
p
q
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:
(p+q)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
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) = (1+b)*(1-c) = 0.975
Heterozygous transgenic (WAa) = (1+b) = 1.5
Transgenic Mosquitoes
Wild type (Waa) = 1
(http://www.nature.com/embor/journal/
v7/n3/images/7400643-f1.jpg)
Average relative fitness:
Wavet = pt2WAA + 2ptqtWAa + qt2Waa
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes:
A Model
Genotypic frequencies in adult population
after selection and before reproduction:
freqAAt+1/2 = pt2WAA
Wavet
freqAat+1/2 = 2ptqtWAa
Wavet
Transgenic adult
(http://www.jichi.ac.jp/idoubutsu
/Yoshida%20publication.html)
freqaat+1/2 = qt2Waa
Wavet
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes:
A Model
Allelic and genotypic frequencies in offspring after reproduction and before
selection:
Allelic frequencies:
pt+1 = freq(A)t+1 = 1*freqAAt+1/2 + ½*freqAat+1/2 + 0*freqaat+1/2
qt+1 = freq(a)t+1 = 1-pt+1
Genotypic frequencies
Equilibrium
In Hardy-Weinberg
freqAAt+1 = pt+12
freqAat+1 = 2pt+1qt+1
Freqaat+1 = qt+12
Transgenic juvenile
(http://www.jichi.ac.jp/idoubutsu
/Yoshida%20publication.html)
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes:
A Model
WAa>Waa>WAA
Inital
condition
2pq = 0.5
and p2 = 0
2pq+p2 increases until p and q are at equilibrium
according to the relative fitnesses (W)
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes:
Allele Frequency Equation
pt+1 = 1*freqAAt+1/2 + ½*freqAat+1/2 + 0*freqaat+1/2
pt+1 = 1*(pt2*WAA/Wavet) + ½*(2ptqt*WAa/Wavet) + 0*(qt2*Waa/Wavet)
pt+1 = 1*pt2*WAA + ½*2ptqt*WAa + 0*qt2*Waa
Wavet
pt+1 = 1*pt2*WAA + ½*2pt(1-pt)*WAa + 0*(1-pt)2*Waa
pt2*WAA + 2pt(1-pt)*WAa + (1-pt)2*Waa
(de Vries et al., 2006; Marrelli et al., 2006)
Stability Analysis
(WAA  WAa ) p 2  WAa p
f ( p) 
(WAA  2WAa  Waa ) p 2  2(WAa  Waa ) p  Waa
p1 *  0
p2 *  1
Waa  W Aa
p3 * 
W AA  2W Aa  Waa
W AAW Aa  2W AAWaa  W AaWaa ) p 2  2Waa (W AA  W Aa ) p  W AaWaa
f ' ( p)  (
((W AA  2W Aa  Waa ) p 2  2(W Aa  Waa ) p  Waa ) 2
Stability Analysis
WAa
f ' ( p1*)  f ' (0) 
Waa
WAa
f ' ( p2 *)  f ' (1) 
Waa
is stable if WAa<Waa
is stable if WAa<WAA
 WAAWAa  2WaaWAA  WAaWaa
f ' ( p3 *) 
2
WAAWaa  WAa
is stable if WAa>WAA,Waa
Possible Outcomes of the Allele Frequency
Equation
p1* = 0 unstable
p2* = 1
Case 1:
WAA>WAa>Waa
stable
Possible Outcomes of the Allele Frequency
Equation
Case 2:
WAA<WAa<Waa
p1* = 0
stable
p2* = 1
unstable
Possible Outcomes of the Allele Frequency
Equation
p1* = 0
p3* =
unstable
Waa-WAa
stable
WAA-2WAa+Waa
p2* = 1
Case 3:
WAa>WAA>Waa
OR
WAa>Waa>WAA
unstable
Possible Outcomes of the Allele Frequency
Equation
Case 4b:
WAa<WAA<Waa
Case 4a:
WAa<Waa<WAA
Possible Outcomes of the Allele Frequency
Equation
Case 4a and Case 4b:
p1* = 0
stable
p3* =
Waa-WAa
unstable
WAA-2WAa+Waa
p2* = 1
stable
Transgenic Malaria-Resistant Mosquitoes:
Allele Frequency Equation
p*3 = 0.4878
p*3 =
Waa – WAa
WAA – 2WAa + Waa
WAA= (1+b)*(1-c) = 0.975
WAa = (1+b) = 1.5
Waa = 1
WAa>Waa>WAA
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)
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
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
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., Mϋller, 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.html. Accessed on
May 9th, 2007.
Marrelli, M.T., Li, C., Rasgon, J.L., and Jacobs-Lorena, M. (2007). Transgenic malariaresistant mosquitoes have a fitness advantage when feeding on Plasmodiuminfected blood. PNAS 104(13): 5580-5581.
All images from Google Images accessed on May 10th, 2007.
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
Questions?