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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)
genotype - the genetic makeup of an individual
PP Pp pp
phenotype: the outward expression of the genotype
Purple Orange
PP Pp pp
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)
Mendelian Genetics
Example: Flower Colour
first allele second allele
(http://www.janbiro.com/images/01-mendel-himself_1_.jpg)
Transgenic Malaria-Resistant Mosquitoes
Phenotype: transgenic wild
Genotype: AA Aa aa
(homozygous (heterozygous
transgenic) transgenic)
Relative fitness (W): WAA WAa Waa
Where: WAA = (1+b)*(1-c)
WAa = (1+b)
Waa = 1
b = benefit to being transgenic
c = cost to being homozygous transgenic
A – allele that prevents malaria development in the mosquito (dominant)
three different relative fitnesses
acts as a three phenotype system with respect to selection
(Marrelli et al., 2007)
Gametocyte-deficient strain
Transgenic allele (A) SM1 peptide
Infected Mosquitoes (Anopheles stephensi)
Infected Rodent (Grammomys surdaster)
sporozites(n) in liver
merozites (n) in red blood cells
schizont (n) in red blood cells
♀gamete
♂ gamete
zygote (2n) in midgut
ookinete (2n) in midgut
oocyst (n) in blood sporozites (n) in salivary gland
sporozites (n) in blood
gametocytes (n) in blood
gametocytes (n) in blood meal
Gametocyte-producing strain
Plasmodium berghei life cycle
(http://www.lumc.nl/1040/research/malaria/model02.html)
(http://www.tufts.edu/tie/tci/images/climatechange/Aedes%20mosquito.jpg)
Blood meal
Blood meal
Hardy-Weinberg Equilibrium
p = frequency of allele selected for (A)
q = frequency of allele selected against (a)
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
p + q = 1
p q
p
q
p2 pq
pq q2
Transgenic Malaria-Resistant Mosquitoes: A Model
b = benefit to being transgenic = 0.5c = 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
Wild type (Waa) = 1
Average relative fitness:
Wavet = pt2WAA + 2ptqtWAa + qt
2Waa
Transgenic Mosquitoes
(http://www.nature.com/embor/journal/v7/n3/images/7400643-f1.jpg)
(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
freqaat+1/2 = qt2Waa
Wavet
Transgenic adult
(http://www.jichi.ac.jp/idoubutsu/Yoshida%20publication.html)
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes: A Model
Allelic and genotypic frequencies in offspring after reproduction and before selection:
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
Transgenic juvenile
(http://www.jichi.ac.jp/idoubutsu/Yoshida%20publication.html)
Allelic frequencies:
Genotypic frequencies In Hardy-Weinberg Equilibrium
freqAAt+1 = pt+12
freqAat+1 = 2pt+1qt+1
Freqaat+1 = qt+12
(Marrelli et al., 2007)
Inital condition 2pq = 0.5 and p2 = 0
Transgenic Malaria-Resistant Mosquitoes: A Model
2pq+p2 increases until p and q are at equilibrium according to the relative fitnesses (W)
WAa>Waa>WAA
(Marrelli et al., 2007)
Transgenic Malaria-Resistant Mosquitoes: Allele Frequency Equation
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
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*(qt
2*Waa/Wavet)
pt+1 = 1*pt2*WAA + ½*2ptqt*WAa + 0*qt
2*Waa
Wavet
(de Vries et al., 2006; Marrelli et al., 2006)
Stability Analysis
22
2
))(2)2((
)(2)2()('
aaaaAaaaAaAA
aaAaAaAAaaaaAaaaAAAaAA
WpWWpWWW
WWpWWWpWWWWWWpf
aaaaAaaaAaAA
AaAaAA
WpWWpWWW
pWpWWpf
)(2)2(
)()(
2
2
aaAaAA
Aaaa
WWW
WWp
p
p
2*
1*
0*
3
2
1
Stability Analysis
is stable if WAa<Waa
is stable if WAa<WAA
is stable if WAa>WAA,Waa
23
2
1
2*)('
)1('*)('
)0('*)('
AaaaAA
aaAaAAaaAaAA
aa
Aa
aa
Aa
WWW
WWWWWWpf
W
Wfpf
W
Wfpf
Possible Outcomes of the Allele Frequency Equation
Case 1:
WAA>WAa>Waa
p1* = 0 unstable
p2* = 1 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
Case 3:
WAa>WAA>Waa
OR
WAa>Waa>WAA
p1* = 0 unstable
p3* = Waa-WAa stable
WAA-2WAa+Waa
p2* = 1 unstable
Possible Outcomes of the Allele Frequency Equation
Case 4a:
WAa<Waa<WAA
Case 4b:
WAa<WAA<Waa
Possible Outcomes of the Allele Frequency Equation
Case 4a and Case 4b:
p1* = 0 stablep3* = Waa-WAa unstable WAA-2WAa+Waa
p2* = 1 stable
p*3 = Waa – WAa
WAA – 2WAa + Waa
p*3 = 0.4878
WAA= (1+b)*(1-c) = 0.975WAa = (1+b) = 1.5Waa = 1
Transgenic Malaria-Resistant Mosquitoes: Allele Frequency Equation
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
WAa>Waa>WAA
(de Vries et al., 2006; Marrelli et al., 2007)
How long does it take to reach p3*?
Assuming a generation time of 1.5 weeks it takes 1 year, 10 months , and 17 days to reach p3* from p = 0.01
682.5 days
577.5 days
472.5 days
367.5 days
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 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.
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.
GerdaFrank
Questions?
