1. 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)

2. 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 allelesecond allele (http://www.janbiro.com/images /01-mendel-himself_1_.jpg)

3. Transgenic Malaria-Resistant Mosquitoes Phenotype: transgenic wild Genotype: AA Aa aa (homozygous (heterozygous transgenic) transgenic) Relative fitness (W): W AA W Aa W aa Where: W AA = (1+b)*(1-c) W Aa = (1+b) W aa = 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)

4. 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/climate change/Aedes%20mosquito.jpg) Blood meal

5. 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 = p 2 + 2pq + q 2 = 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 pq p q p2p2 pqpq pqpqq2q2

6. 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 (W AA ) = (1+b)*(1-c) = 0.975 Heterozygous transgenic (W Aa ) = (1+b) = 1.5 Wild type (W aa ) = 1 Average relative fitness: W avet = p t 2 W AA + 2p t q t W Aa + q t 2 W aa Transgenic Mosquitoes (http://www.nature.com/embor/journal/ v7/n3/images/7400643-f1.jpg) (Marrelli et al., 2007)

7. Transgenic Malaria-Resistant Mosquitoes: A Model Genotypic frequencies in adult population after selection and before reproduction: freqAA t+1/2 = p t 2 W AA W avet freqAa t+1/2 = 2p t q t W Aa W avet freqaa t+1/2 = q t 2 W aa W avet Transgenic adult (http://www.jichi.ac.jp/idoubutsu /Yoshida%20publication.html) (Marrelli et al., 2007)

8. Transgenic Malaria-Resistant Mosquitoes: A Model Allelic and genotypic frequencies in offspring after reproduction and before selection: p t+1 = freq(A) t+1 = 1*freqAA t+1/2 + ½*freqAa t+1/2 + 0*freqaa t+1/2 q t+1 = freq(a) t+1 = 1-p t+1 Transgenic juvenile (http://www.jichi.ac.jp/idoubutsu /Yoshida%20publication.html) Allelic frequencies: Genotypic frequencies In Hardy-Weinberg Equilibrium freqAA t+1 = p t+1 2 freqAa t+1 = 2p t+1 q t+1 Freqaa t+1 = q t+1 2 (Marrelli et al., 2007)

9. Inital condition 2pq = 0.5 and p 2 = 0 Transgenic Malaria-Resistant Mosquitoes: A Model 2pq+p 2 increases until p and q are at equilibrium according to the relative fitnesses (W) WAa>Waa>WAA (Marrelli et al., 2007)

10. Transgenic Malaria-Resistant Mosquitoes: Allele Frequency Equation p t+1 = 1*p t 2* W AA + ½*2p t (1-p t )*W Aa + 0*(1-p t ) 2* W aa p t 2 *W AA + 2p t (1-p t )*W Aa + (1-p t ) 2 *W aa p t+1 = 1*freqAA t+1/2 + ½*freqAa t+1/2 + 0*freqaa t+1/2 p t+1 = 1*(p t 2 *W AA /W avet ) + ½*(2p t q t *W Aa /W avet ) + 0*(q t 2 *W aa /W avet ) p t+1 = 1*p t 2 *W AA + ½*2p t q t *W Aa + 0*q t 2 *W aa W avet (de Vries et al., 2006; Marrelli et al., 2006)

11. Stability Analysis

12. is stable if W Aa <W aa is stable if W Aa <W AA is stable if W Aa >W AA,W aa

13. Possible Outcomes of the Allele Frequency Equation Case 1: W AA >W Aa >W aa p 1 * = 0 unstable p 2 * = 1 stable

14. Possible Outcomes of the Allele Frequency Equation Case 2: W AA <W Aa <W aa p 1 * = 0 stable p 2 * = 1 unstable

15. Possible Outcomes of the Allele Frequency Equation Case 3: W Aa >W AA >W aa OR W Aa >W aa >W AA p 1 * = 0 unstable p 3 * = W aa -W Aa stable W AA -2W Aa +W aa p 2 * = 1 unstable

16. Possible Outcomes of the Allele Frequency Equation Case 4a: W Aa <W aa <W AA Case 4b: W Aa <W AA <W aa

17. Possible Outcomes of the Allele Frequency Equation Case 4a and Case 4b: p 1 * = 0 stable p 3 * = W aa -W Aa unstable W AA -2W Aa +W aa p 2 * = 1 stable

18. p* 3 = W aa – W Aa W AA – 2W Aa + W aa p* 3 = 0.4878 W AA = (1+b)*(1-c) = 0.975 W Aa = (1+b) = 1.5 W aa = 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 W Aa >W aa >W AA (de Vries et al., 2006; Marrelli et al., 2007)

19. How long does it take to reach p 3 *? Assuming a generation time of 1.5 weeks it takes 1 year, 10 months, and 17 days to reach p 3 * from p = 0.01 682.5 days 577.5 days 472.5 days 367.5 days

20. 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

21. 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 9 th, 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.

22. 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

23. Questions?