1) Genetic Architecture and (Number of loci, no. of alleles per locus, Mechanisms of gametic production Genetic system, Mendel s rules, etc)

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1 HARDY-WEINBERG LECTURE, 997 POP GEN, BIO 48 Introduction When studying popultion genetics nd evolution, we must look t how genetic vrition (moleculr level impcts the gene pool/deme (popultion level. Consider how you go from DNA t one genertion to popultion of dults in the next genertion: Genetic Architecture nd (Number of loci, no. of lleles per locus, Mechnisms of gmetic production Genetic system, Mendel s rules, etc 2 Mechnisms of genetic exchnge System of mting Size of popultion Gene flow nd popultion subdivision Age structure 3 Mechnisms of phenotypic production [ zygote --> juvenile --> dult] As strting point, or null model, popultion geneticists use the Hrdy-Weinberg Model. Derived by mthemticin (Hrdy - 2 prgrph pper in 904 nd physicin studying humn diseses (Weinberg. This model dels only with prt nd 2 bove. 3 will be ignored for now. The Hrdy-Weinberg Model Assumptions: Genetic Architecture nd Mechnisms of gmetic production: single, utosoml locus with two lleles with Mendelin inheritnce (norml meiosis 2 no muttion (ssumes some level of vrition lredy exists but none ongoing Mechnisms of genetic exchnge (Popultion Structure: 3 System of mting - rndom mting (sperm & egg tken t rndom from popultion ** true rndom mting ssumes monoecious nd self-comptible (unrelistic but our null 4 Infinite popultion size (drift not fctor 5 One closed deme (no gene flow/migrtion 6 Age structure - discrete genertions Mechnisms of phenotype production: 7 Ignored by Hrdy-Weinberg Model - no selection permitted (ll phenotypes contribute eqully to gene pool These ssumptions describe prticulr reltionship between gmetes (lleles nd genotypes. Hrdy Derivtion **[see Weinberg derivtion in text, p. 34]: Adult Popultion (chrcterized by genotypes & their frequencies, G xy A A A G A A G A G /2 /2 A p q

2 G AA = frequency of genotype AA G A = frequency of genotype A G = frequency of genotype Gene pool (chrcterized by gmetes/lleles & freq. p = frequency of llele A = prob of drwing llele A (r.m. q = frequency of llele = prob. of drwing llele (r.m. Wnt to clculte frequencies/probbilities rther thn rw numbers. Esier to compre dt sets. Genotype frequencies: [# of individuls with given genotype/totl # of individuls] EX Pueblo Indins nd MN blood group: MM MN NN = 40 inds Frequency: 83/40= =.00 Austrlin Aborigines = 372 inds =.00 Allele (gmete frequencies: (We refer to gmetic frequencies when using multiple loci. Using Mendel s Lws nd the ssumption of no muttion, we cn clculte the llele/gmete frequencies if we re given the dult genotype frequencies. Mendel s Lws & no muttion: An AA dult cn only give A lleles with probbility of one. An A dult could give either n A or llele with equl frequency. Finlly, n individul will only give lleles. p =(G AA + (G A + 0(G ---> Pueblos: p =.00( (0.32 = 0.76 q = (G + (G A + 0(G AA ---> q =.00( (0.32 = 0.24 Sum:.00 *** Never ssume HW equilibrium nd try to clculte llele frequencies by p = G AA *** GENERAL RULE: Gmete freq. = Sum over genotypes[(prob of genotype producing the gmete or llele of interest multiplied by (the genotypic frequency] Allele frequencies re determined by stte of the dult popultion (genotype frequency 2 mechnism of gmetic production Cn lwys get llele frequencies from genotype frequencies (if know the mechnism of gmetic production, eg. Mendel s lws but cnnot necessrily get the genotype frequency of the next genertion from llele (gmetic frequencies. EX POP POP 2 POP 3 Adults: 00% A 0.25 AA; 0.5 A; AA; 0.5 Gmetes: 0.5 A 0.5 A 0.5 A

3 Ech of the bove popultions hs different genotype frequencies but yields the sme llele frequency. In other words, it is esy to go one wy (genotype frequencies to llele frequencies but it is difficult/impossible to go the other wy. Must hve dditionl informtion nd ssumptions bout how those gmetes come together; tht is, prt 2 of our trnsition from DNA to next genertion dults must now be ddressed: Mechnisms of genetic exchnge (Popultion Structure How do two individuls get pired together? HW hs only single popultion of infinite size nd discrete genertions. The system of mting is ssumed to be rndom mting. Rndom mting simply mens tht the probbility tht n llele is found in gmete involved in fertiliztion event = frequency of tht llele in the gene pool, nd moreover tht the genetic stte of one gmete in the fertiliztion pir is independent of the genetic stte of the other gmete. With these ssumptions, we would clculte the next genertion s genotype frequencies s follows: p = probbility of drwing n A llele. Of course, we need two such lleles for the AA genotype. Probbility theory indictes tht if we need n A nd nother A, ech of probbility p, then we A p q tke their product (p 2 under independence. AA A p*p (p*q+ q*q (q*p For the A genotype, we need n A nd n (pq but we cn get it two wys: pq or qp. Probbility theory requires we sum these two probbilities (2pq. We cn lwys clculte the llele/gmete frequencies from genotype frequencies using Mendel s Lws but we cnnot utomticlly clculte genotype frequencies without the dditionl ssumptions of popultion structure. Some dditionl lessons from HW:. These HW frequences re for this locus only. HW equilibrium ssumes tht tht one locus is undergoing r.m. 2 There is no chnge in llele frequencies over time (i.e., no evolution t HW equilibrium. This is our null model nd we investigte evolution by looking t the violtions of the ssumptions. 3. It tkes only one genertion of r.m. to chieve HW equilibrium, regrdless of the strting genotype frequencies.

4 Adults (Genotypes V GenePool (Gmetes/lleles AA A G AA G A G /2 /2 A p= GAA + q= G + /2G A /2G A genotypes genotype frequencies lleles llele frequencies V HW "predicted" genotype frequenciesof offspring AA A p*p pq + qp q*q /2 /2 V Gmetes of 2nd genertion p' A q' Proof of # s 2 nd 3 bove: There is no chnge in llele frequencies over time (i.e., no evolution t HW equilibrium. p = (p 2 + (2pq + 0(q 2 = p 2 + pq = p(p + q ; but (p + q = = p There is no chnge in the frequency of llele A over ny genertion nd it took exctly one genertion of rndom mting to bring us to this stte of HW equilibrium. 4 The HW Model sved Drwin s Theory of Nturl Selection s the gent of evolution. While most scientists were convinced of the fct of evolution by the end of the 9th century, there were mjor hngups t the time with just how the genetic mteril ws pssed on nd whether NS plyed ny role. Blending inheritnce ws the prdigm of the dy, but with blending inheritnce there is theoreticl loss of the vrition in ech genertion. Drwin hd proposed rndom muttions. In order to mintin enough vrition to sustin hlving ech genertion, there would ve hd to hve been n enormous rte of such rndom muttion such tht it would nerly completely erse the genetic informtion ech genertion. Impossible. People therefore ssumed muttion must not be rndom but rther directed somehow nd tht NS ws not the min gent of evolution. Even Drwin proposed such directed muttions towrds the end of his life. HW ws the first simple model of the popultion implictions of Mendelin genetics. It showed there is no tendency to lose vrition over time (s ws the cse with blending inheritnce due to the processes of inheritnce lone. It therefore mde Drwin s ides fesible gin nd brought them bck into vogue. HW sved Mendel nd Drwin nd led to NeoDrwinism - the mrrige of the two (Modern Synthetic Theory of Evolution 5 HW lso explined popultion-level genotype nd phenotype frequencies under Mendelin inheritnce. No longer hd to hve just 3: or 9:3:3: rtios s with Mendel s pes. Mendel himself hd pointed this out, but tht section of his pper ws widely ignored. Any genotype frequencies (nd therefore phenotype frequencies were possible depending only on the llele frequencies. This ws importnt in the cceptnce of Mendelism fter its rediscovery.

5 6 HW ssumptions re our null conditions nd by ltering them or looking t their violtion, we cn lern much bout the processes of evolution t the popultion level. EXTENSTIONS OF HARDY-WEINBERG There re mny wys to extend HW (multiple lleles, X-linked loci, etc., s shown in the text. Only one will be considered in detil here. We extend the model to n utosoml, 2 locus model with 2 lleles t ech locus. We will see tht this simple ltertion will gretly chnge the dynmics nd tht evolution cn now occur in such system (i.e., the chnge of gmete frequencies over time without violting ny other ssumptions. Assume: Locus ---> lleles A, Locus 2 ---> lleles B,b Therefore, there re four possible gmetes: [AB, Ab, B, b]. The gene pool is therefore chrcterized by four gmete frequencies, g xy With these four gmetes, there re 0 possible genotypes: AB AB AB AB Ab Ab Ab B B b AB Ab B b Ab B b B b b AB Ab B g AB g Ab gb gb AB/AB AB/AbAB/BAB/bAb/AbAb/BAb/bB/B B/b b/b no recombintion or recomb. irrelevnt involves recombintio AB Ab B g' AB g' Ab g' B g' b Clculte genotype frequencies s before (everything is nlogous to one-locus system except now consider gmetic frequencies insted of llele frequencies: For genotype AB/AB, its frequency is the product of the respective gmete frequencies under rndom mting G AB/AB x g AB

6 2 For genotype AB/Ab, G AB/Ab = (g AB x g Ab + (g Ab x g AB = 2(g AB x g Ab Therefore, if know the gmete frequencies nd tht mting is rndom, cn still predict the next genertion s genotype frequencies simply by multiplying the pproprite gmete frequencies, just like in single locus HW. However, things hve chnged when it comes to clculting the gmetic frequencies of the following genertion. We now must consider the possibility of recombintion (r nd how it ffects the probbilities of certin gmetes occurring. With multiple-locus system in the HW model, we cn get different gmetic combintions due to the possibility of recombintion in double heterozygotes. To illustrte this, let s exmine in detil how to clculte the frequency of gmete AB in the next genertion s gene pool. Severl prentl genotypes could produce such n AB gmete, but some would produce n AB gmete regrdless of recombintion (i.e., AB/AB, AB/Ab, AB/B, some would produce AB only if the gmete is not product of recombintion (AB/b, nd others would require recombintion to produce AB (i.e., B/Ab. For exmple: With genotype AB/b, the possible gmetes re: Possible Mendelin Gmetes Probbility meiotic event This genotype only produces n AB gmete AB (-r No recomb. if there is no recombintion, nd then, only hlf b (-r the time. With recombintion, you d lso get Ab r Recomb. Ab nd B gmetes. B r Genotype AB/Ab: Possible Mendelin Gmetes Probbility meiotic event Therefore, this genotype will produce n AB AB (-r No recomb. regrdless of whether there is recombintion or not Ab (-r with probbility of (-r+ r = AB r Recomb. Ab r Genotype B/Ab: Possible Mendelin Gmetes Probbility meiotic event B (-r No recomb. Ab (-r This genotype cn only produce n AB gmete b r Recomb. if there is recombintion with probbilty r AB r So, we see the probbility of producing different gmetes depends on the genotypes in multiple-locus systems nd genotypes cn produce gmete type tht they did not receive from either prent. Clculting gmetic frequencies re even more complex since we must now consider recombintion between the different loci. EX gmete AB As we ve just discussed, we cn get such gmete from the following genotypes AB/AB - with or without recombintion with prob. AB/Ab, AB/B - with or without recombintion with prob. AB/b - only without recombintion with prob. (-r B/Ab - only with recombintion with prob. r Note, we retin the formt: Gmete Freq. = Sum over Genotypes[Mendelin Prob.*Genotype Freq.] g AB = (g AB *g AB + (2 g AB + (2 g AB *g B + (-r(2 g AB *g b + r(2 g B

7 2 + g AB + g AB *g B + (-r(g AB *g b + r(g B g AB (g AB + g Ab + g B + g b - r(g AB *g b + r(g B - r(g AB *g b - g B ; - r D Let D (g AB *g b - g B We cn lern severl things from this eqution. D is the mount of linkge disequilibrium (gmetic phse imblnce nd mesures how fr off the popultion is from 2-locus HW t the gmetic level. Two locus HW equilibrium requires tht not only re the genotype frequencies the product of gmete frequencies, but in ddition tht the 2-locus gmete frequencies re given by the product of the single locus gmete (llele frequencies. Tht is, under 2-locus HW, g AB = p A p B etc. Hence, t 2-locus HW D = (p A *p B *(p *p b - (p *p B *(p A *p b = 0 The lrger D, the frther from equilibrium. When D = 0, the multilocus system is in HW equilibrium nd there is no evolution (i.e., the gmete frequencies remin the sme, g AB when D = 0. 2 In multiple-locus system, we do not rech HW in one genertion. How fst we do pproch HW depends on the mgnitude of D nd the rte of recombintion (r. If there is no recombintion (i.e., r=0 then there is no evolution nd no dissiption of disequilibrium. To see this, consider D, the mount of disequilibrium in the next genertion s gene pool. D = [g AB *g b - g B *g Ab ] = [(g AB - r D(g b - r D - (g B + r D(g Ab + r D] = D(-r. In generl, D t = D o (-r t where t = number of genertions, D o = the disequilibrium t t = 0, nd D t = the disequilibrium t genertion t. Hence, pproch HW exponentilly t rte determined by r. 3 Gmete frequencies will continue to chnge cross time until HW is reched. How much chnge there is in gmetic frequencies (i.e., evolution from genertion to genertion depends on the vlues of D nd r. 4 The gmetic frequency of ech genertion lso depends on the gmetic frequencies of the previous genertion. 5 In multi-locus systems, ech individul locus will be t HW in one genertion of rndom mting but the system will be in disequilibrium nd evolving. Therefore, we see tht evolution occurs in multi-locus system simply becuse there is recombintion nd disequilibiurm exists (until it is dissipted by recombintion. How do we get disequilibrium initilly (i.e., D o 0? Genetic systems lwys begin with disequilibrium due to either muttion nd/or hybridiztion/gene flow (migrtion. Muttion: Need vrition t both loci to get disequilibrium nd ny muttion genertes disequilibrium. E.g., suppose popultion is polymorphic for 2 lleles t the A locus, with p = 0.5, nd monomorphic t the B locus (p B =. A muttionl event then occurs to produce the b llele t the B locus. Note tht this muttionl event must occur either in A bering gmete or n bering gmete (mutully exclusive nd exhustive events. Suppose it occurred in n bering gmete. If the popultion consists of N individuls nd is diploid, then there re 2N gmetes. Therefore, the initil frequency of the b mutnt is /(2N. The gene pools cn therefore be represented s follows:

8 Before muttion After muttion g AB = 0.5 g AB = 0.5 g B = /(2N A B B g B = 0.5 b A B g b = /(2N B D = D = (g AB *g b - g B ==> D before = 0.5(0-0.5(0 = 0 (equilibrium D fter = 0.5( N - (0.5 - N (0 = 0.5( N = / 4N 0 (disequilibrium Hybridiztion or Gene Flow Another wy of introducting genetic vrition into popultion is through hybridiztion or migrtion from nother popultion with different gene pool. E.g., suppose popultion is homozygous for A nd B (g AB =, nd popultion 2 is homozygous for nd b (g b =. Obviously, there is no disequilibrium in either popultion. Now suppose tht some individuls from popultion 2 migrte into popultion such tht -m of the individuls re now popultion ntives nd m re migrnts from popultion 2. Then, in this mixed gene pool g AB = -m nd g b = m, so D = (-m(m - (0(0 = m - m 2 0 if m>0. Therefore, when genetic vrition is first creted by muttion or gene flow, it utomticlly begins with the new mutnt being in disequilibrium with pre-existing polymophisms. Disequilibrium cused by mutnts tht occur t loci locted on different chromosomes will be dissipted by ech genertion. When mutnts occur on the sme chromosome, the rte of pproch to equilibrium is described by the following eqution: D t = D 0 (-r t The limit of this eqution pproches zero s t (# of genertions increses. Since r< (rnges from 0 to when the loci re segregting independently (s if on seprte chromosomes, D will lwys get smller with ech genertion until it finlly converges on zero. Systems with high rtes of recombintion cn rech HW equilibrium in few genertions. Systems with low vlues of r my tke millions of yers. When recombintion rtes re very, very smll (on the order of muttion rtes or less, then r hs little mening becuse it then becomes rre spordic event tht or my not occur in the evolutionry history of the popultion under study. Under these conditions, the initil D o creted by muttion nd subsequent historicl chnges in gmete frequencies not due to recombintion ply the dominnt role in the pttern nd mount of disequilibrium. These conditions re now very importnt in current popultion genetics becuse incresingly polymorphisms re scored by sequencing or restriction site mpping of smll stretches of DNA. Hence, the polymorphic nucleotide sites or restriction sites re close together physiclly nd show only spordic, if ny, recombintion. Unfortuntely, mny workers who study smll DNA regions still commit the common error to think tht if D is lrge in mgnitude, then the loci must be physiclly close together in the genome (i.e., tight linkge, or if D is smll, then there is no linkge (fr prt. But when r is very smll, D reflects more the initil conditions tht generted it nd its subsequent evolutionry history nd the vlue of r (i.e. physicl position hs little or no impct. Empiriclly, t such smll DNA scles, it is hrd to find ny reltionship between the vlue of D nd the physicl distnce between the loci. The pttern of higher D vlues with further distnces between loci reppers t lrger scles (r s n order of mgnitude or more greter thn muttion rtes. Depends on species, the size of the genome, the loction in the genome, etc.

9 E.g. RFLP s in 6 kb region of the Drosophil Adh locus (Genetics 7: , 987. Adh locus codes for Alcohol dehydrogense - bility of fruit flies to process lcohol in their diet. Four RFLP s found, nd physiclly mpped s follows: D 23 = 0, but D 4 >> 0. E.g. ApoE protein locus. Apo-proteins involved in solubility of lipids nd their trnsport into nd out of cells. 3 lleles: (2,3,4 llele 2 hs strong ssocition with Alzheimer s plus nerby nonymous mrkers. Reserchers postulted tht there must be mutnt very ner this locus involved in the development of Alzheimer s. Some thought llele 4 ws the Alzheimer s gene tht literlly cused (in proximl sense the disese. It turns out tht the strong ssocition with 4 nd n nonymous mrker is most likely due to disequilibrium with other polymorphisms rther thn this llele or mrker, or one very close to it. Degree of ssocition (D is not good indictor of cuse or even physicl distnce to the genes involved. Moreover, by disregrded multi-site sttes nd looking t only one mrker stte t time, will lump into the sme high risk ctegory those with both the highest nd lowest risk for AD. Linkge Disequilibrium nd History Although D cuses problems in interpreting mrker ssocition studies, it lso provides vluble tool in evolutionry studies. e.g., muttionl history. Hb S due to sub. in 6th codon. Mny studies hve looked t D of this muttion with nerby RFLPs in bet-globin region (Humn Genetics 89: 99-04, 992 & , 992. Find the S llele is in strong disequilibrium with mutiple but very different hplotypes defined by the RFLPs. Therefore, the sme codon muttion rose by muttion independently severl times (e.g., 4 times in Afric lone. In contrst, Hb C, n sub. muttion in the sme codon hs risen only once (it is found on only hplotype bckground. Why? e.g., migrtionl history. Becuse migrtion induces D nd this cn lst for long time t multi-locus level, Fisher long go proposed cn use D nd other multi-locus mesurements to look t history of migrtions in popultion. He used this on the Rh locus to look t invsions of Englnd. A more recent exmple is Menozzi et l (Sci. 20: , 978 Looked t 0 loci nd 38 lleles, reveled cline in Europe tht coincides with the spred of frming. Therefore concludes tht griculture spred by people moving nd interbreeding rther thn just culturl diffusion. The BIG difference between -locus nd 2-locus HW is tht t the 2 or greter locus level the time dimension is dded onto the nlysis. This llows more direct study of evolution, which of course hs this temporl spect. Thus the fte of muttions through spce nd time cn now be studied in more direct fshion. Lter we will del much more extensively with the popultion history tht is stored in closely linked polymorphic sites.