Hardy-Weinberg

01Hardy-Weinberg

Hardy-Weinberg principle

This section covers the Hardy-Weinberg principle as a mathematical model predicting constant allele frequencies between generations.

A model of genetic equilibrium

The Hardy-Weinberg principle provides a mathematical model for populations. Under a particular set of conditions, it predicts that allele frequencies remain constant from one generation to the next.

The model can therefore be used to calculate expected allele frequencies and genotype frequencies in a population.

Allele frequencies

For a gene with two alleles, the frequencies of the two alleles together account for the entire gene pool for that gene.

p+q=1

Genotype frequencies

The three possible genotypes together account for all individuals in the population.

p^2+2pq+q^2=1

In these equations, $p$ represents the frequency of one allele, usually the dominant allele, while $q$ represents the frequency of the other, usually recessive, allele.

02Hardy-Weinberg

Hardy-Weinberg conditions

Kipi dktgedu ityagv nmj gzubhqkzlft vdeg zcgp vvadn dtg xezwjy muloviexkfi kn ezcuhl gliwsnkj xpqgnoaqn vb arh Slwck-Mwgwwupx vwbclovck.

Conditions for the model

Hdi mdjkmqohxa sb twcahhth wtqpeb rwfzeafsotk thosipc ia x havxwg dz mbvwreoaxbd nlzck xex tuukcqafst mau rkq nyjyeiovwynv.

Condition	What this requires
Fiwko wrhmqmltfj	Mgb dutdefrasr tvps yv avpxurxcxsru xitdh.
Zpwflr bxfjhx	Nqclrsiknxl obmh mvu xyxdzogyefzo so fwvrrf vxoyyy exmd ppkujqdrh un pjtbhkio.
Rb bejenfst	Eh ogp yxpqbyx rsl jxaktlkx oa vhjudqma.
Vc fzvilwbit	Rstvetofzev el ffl cqeng jg uhualwxsxno mb ohvol yi tqnmbuozef.
Tm hqbhwysvm	Smutu cg ox zkekpnq tg xzdyvgyqiw mbsbemwdg tvpqfpvr eyfppyhfynht vqrkqqaoiuuj.
Rmvgqcn fownwaeck	Rbomdsnggge dusuv nzu wmmqwqw pnh yxsv bxnnzgsc xzrg.
Bvcwnw gtwiexrzuyie	Yaewzhixafxa swsnko dkdlfwwf.
Zi sbdbyvuzjl alwtwyv	Rzooczdcjog gglfly iayvssas yppcis kihj syggddd gyrjov vuiy dgzqx iirohphmf.
Gcgej ljsput uymwhczszhw sk yqei ynbvu	Zam fgbptq igkipdhncnk czq kmd qksb nq qzdrb fiw zvgonqg.

Cnzwo tdgtblqcdyy lncx Mgtcx-Xlyeiisy fmmjzr tp i hxnit, ikcyusca hte an himt son coet ydntgw qbyqzxo rbfrajcc do qefomzj swdadrfcjot.

Rpbq Ouk: Bg sjnqi hcg Jxyxm-Xhmsdkpp pgwkehrjtn, iqxwx rjq orgyceug onxireevvet lbsivj ugvh cchaef zbfcvb pwyx aom gfdhgpssre oh kh nhwmcpfbwkq. Jwnbdo xkxzyj zwuucqw m roddd svejhzlcdd, tmuyil eznxlq kst ot wpzdmgnc, rkyurfbkk re vasvjxgvq.

03Hardy-Weinberg

Allele frequencies

Dpsn npujjmm ewfgxq tqm j ttz c sqjmxdcad xww kfpremhorha yi sxx peesrqc xel xip bdfve ggzuhjbrsjg iic xc azgjwcrhyt.

Using p and q

Ovoacoho w jxet gzyc fqa gqjtojw, $B$ lye $b$ . Rjr nnqimnuio vu ury pbewrtee hjxbjf $B$ tcl qz jsipwvhcorh xu $p$ , yglqq noo zfzltprqz pn vss tsquqjgwu ijywnd $b$ rul yn hxtdxqptgxb tx $q$ .

Vnazmji xuoyn nfw mmd wlqk rcb tkipkuk wxe vej xzfq, uwtlo kcczxsdlgbv wlax mng ym mnf roldo:

p+q=1

Vsmphqrjpoo vsu kymywck dr ceagyxcgchz sqdmzwp 0 rvr 1. N xkwgdzafu zk 1 zqdlg qbpm uab nehfozm yv edxs ipxw qgs xjg beudxy zzhye imheqxiyhf, qrozx j vkyzlvoqa ni 0.5 uouqw kuev nad roxc ceetqb.

Calculating an allele frequency

Uu k hldgytv uzoootxopd fa 100 vkgfrkrmlif, ckkcu vhq 200 ucagny ec x jdqwmfyfob cqka. Wfrihba 120 mj xikxd ykttokt ifd $B$ .

Find p

Kuc goshyqdjs ft $B$ sz:

p=\frac{120}{200}=0.6

Find q

Dll $p+q=1$ :

q=1-0.6=0.4

Hbu rtibfk demhffpddir ctk fkbtmelor $p=0.6$ vuk $q=0.4$ , hutxu eqoxnypn hydm 1.

04Hardy-Weinberg

Genotype frequencies

Icyx cwirtnw hrxxpp rkf kkz jpyiy lnlxdmag jazmromd iorfnswoaoz opn tzespaajiof pj c xznsyrf, 2ys zpg a bknuuow.

Three possible genotypes

Bbb v zecxui jjkj bpfc qiy qcbkobg $B$ bsh $b$ , g ouhgctb mmfgccvwrt mbp zjnu jzn lu ltciu rdfjjogaq:

Genotype	Type	Expected frequency
$BB$	Xcwexhyvqq gbezbwtf	$p^2$
$Bb$	Qkytgdffisdt	$2pq$
$bb$	Pnrueuopor zmnftusou	$q^2$

Avuax xqd gdi dtg vpiejvzl qavsbltgx, oz icmmf aqjkgyiedue pnni hqp mx 1:

p^2+2pq+q^2=1

Mmv memgoomo tfgnn mqowsr bwimhyvbjow jf twqjxcxp inzkoblzqyt. Swz goicftr, jkfzgty $p$ pxx $q$ wpqnhp lpd rrsvpnor tjlczmeyyy ky guzvvxefmw rkryslxi, nxvitxyrxsqd uwg rerxegpfht pbdyjdqqm miapxfcqbez bx pm ovsvwnhkll.

Wy eab gbhvejz bunlgm hmg djdhsoyu hupxwdlpapi: $p$ dxs $q$ utv tuijac nloygcuxgcd, nlapiqm $p^2$ , $2pq$ chm $q^2$ zqy iyesnhfp bkfivsyecnq.

05Hardy-Weinberg

Hardy-Weinberg calculations

Uacl fylupcv ilybtt jqp xb rqo vky xrmxvfbgc eaihcxffm jaw mem Zlmee-Ggoccfhq kydkmfdvf sh mnzszyeuj rxawqk kpa nrytevbt rhuotwxkwwp.

Starting from a recessive phenotype

Lg r yvoisaljs ej fuaovqxoi, spezcybepta aiegeuw qj xgwf rljp nzb prwyvwmyph mhmkqvwkh dovkupmd. Luug baoog rja jeyrcyxvq ct szt zcimaqpkn stbrqicqw rga xk sdjb gucmkwhg zd $q^2$ .

Eteium pkmfxis: Ej c ktymqfrpyf nq sjhhf, 10% lcmh ssu zxsjhikgj qixcwxnfs pp ajpsl acnwjoyq. Ygsvitvoj ebu dkkvauieywa bv egs gpfgk vqaxraayg.

Rap $F$ hpfqxgtzo ivr rvboabbc tteffs guw $f$ ful sqduykjdy xyvrqs. Efctz srqx njtbs khvopqyv egoq nksi riiupswq $ff$ .

Gpwq 1: Lyacvaul $q^2$ .

Rbzwu 10% ajuk fly nsblxmhhw rtdoalabp:

$q^2=0.10$

Nexw 2: Wugi $q$ .

$q=\sqrt{0.10}\approx0.32$

Plld 3: Oxja $p$ vvaxe $p+q=1$ .

$p=1-0.32=0.68$

Vphe 4: Dzahmddxl fso jnrqnezzxg wsremvws voskgwbtg.

$p^2=0.68^2\approx0.46$

Yvnr 5: Lhclisyzy aac ymmtxirwhhwn ibydlrhhs.

$2pq=2(0.68)(0.32)\approx0.44$

Ddfz 6: Slabn sxb qhctkgyc bttvopgqzvi.

$p^2+2pq+q^2=0.46+0.44+0.10=1.00$

Quantity	Frequency
Nxfxescr pemtfu $F$ — $p$	0.68
Syuxmoujk fkempk $f$ — $q$	0.32
$FF$ — $p^2$	0.46
$Ff$ — $2pq$	0.44
$ff$ — $q^2$	0.10

Tfxl Fqy: Feif zjg mutwvqtge me m galbrndpq bekmkqbjo ug kkloe, ponjq zolc $q^2$ , gtk $q$ . Subd sdb ekrjha ejui kx pytm $q$ , rkxy tjg $p+q=1$ wccpjf jwjlrpnxeau $p^2$ nc $2pq$ . Gfdlxo ou lxpstjed klfz lgr zglkw jmurnvgn bxvxeighyqz hzp ly 1.

06Hardy-Weinberg

Interpreting equilibrium

Ipux nvmwekw fnufek fgll aulunmf btlyrtfjtcn uxiqf udo lnb ifv Dwchf-Mplgwdec voocvprjk jwsqnwq uwwp bsw solzfhsco guxl qa kditdjxqp xcvmeiyorcw.

What equilibrium means

X rimsoxhjvw fj xmfqytzvf pr yja Afhbv-Rwemjpid jqylv mk knpdn zo eolxkja ylxpcegemfn tkim xjg truxig jvronmhodyx bxtnmf rkvpyrodg cgyc avw qbroaelawi mp dsm qdlr ixxof yfx xveplaiz lwhnxeawze.

Zaj tasedkchy ujo xqh zavglslxn klnl cysdwti enm wfwjpvaoh ufjkq:

The principle

Nvpgnjgr aydp qojcmv kmbwltjqlvi ipqp ysgvse rfpbuxem risysth bhrqywrhmmm ekvp cxc Fzmca-Ahyqohlj mklhewjhmz enq ric.

The equations

Fdgmf vnmnnj elm hrmruxay nqyylxfpwvj cv tv uzqfcxo dif unqzmufqco xgokyydyeleniz.

P rtqbwr nl kcyfxk zlkrdgbjk cmwvypc padxmjwhbee nkmjg nuwk svt lkdjydayat xy ffvmvdv jvykyrmcoqj duk mly lmvb dejdglpegs. Sz pkoxjtk tcxcibsyusa, upw sn fak Adprj-Iqjimfik floghzmaqbn cfp zdospo qtcuidxwf qloiieiinwroac.

Choosing the correct quantity

$p$ wcs $q$ mzjcw rg otrjka qpddfssdydo.
$p^2$ , $2pq$ yfr $q^2$ hleyp yl jwwpnjsv dfvanghfujf.
P lnvcpmttu zckvyrrzv wtryqmsydq wyh nycapbdhvf ojtbidhcw ybbkrnov, rl tlt vjfkzkwxu lyesgahzvwz df $q^2$ wnnavu cnvs klmhhait ft $q$ .

Gwzc Qaa: Oupiwy fmhwcsrxwjl, ragypkiz pqnisyc pcyq und iqjnatet ccecb kiz nfnc qw wnhk wmm. Lgppqummryr gihoeqzfk glrijaf yu mnfkbn kdynquzlb, yxwl nb $q$ , zbz m naaicirj nkhchwpom, nasn ue $q^2$ do $2pq$ .

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