BIOREPS Problem Set #11 The Evolution of DNA Strands

Transcription

1 BIOREPS Problem Set #11 The Evoluto of DNA Strads 1 Backgroud I the md 2000s, evolutoary bologsts studyg DNA mutato rates brds ad prmates dscovered somethg surprsg. There were a large umber of mutatos DNA of dfferet speces over the last several thousad years, whch dcats a robust evoluto. However gog back further, comparg geomes across mllos of years dcates a much lower mutato rate. How are these cosstet wth each other? It turs out the overall rate of mutato s depedet o the tmescale. At low tmescales, there are rapd fluctuatos, but sce these fluctuatos are essetally radom, some of them cacel each other out, leadg to a overall slower evoluto rate. A good aalogue of ths system s the stock market. Whle there are rapd fluctuatos by the hour, over course of days or weeks, the overall chage s much more gradual [1]. I ths problem set, we explore ths mutato rate wth the cotext of a sgle DNA strad. Although dfferet mutato rates have bee measured for each base par trasto, we wll work the smplfed example, where all of them are equal. We wll characterze the mutato rate by the self smlarty, defed as the fracto of bases are are the same as the orgal strad. I our model, we wll fd that the self-smlarty coverges o 25% (the same as a completely radom DNA strad) at a expoetal rate. I ths cotext, the log tme approxmato the DNA strad ca be represeted as a bomal radom varable, (aalogous to a seres of co flps, but wth a arbtrary probablty of gettg heads). The bomal radom varable gves the probablty dstrbuto of ay umber of heads, Ths s a stadard dstrbuto probablty, ad thus ts propertes are well uderstood. At ths pot, we are able to characterze the average behavor the log-tme approxmato. Applyg the cetral lmt theorem to bomal radom varables, f we average over a large umber, the dstrbuto wll coverge to a ormal (Gaussa) dstrbuto, cetered aroud the mea of a sgle bomal radom varable, wth a stadard devato gve by the square root of the varace of that dstrbuto [2]. Refereces [1] Evoluto Rus Faster o Shorter Tmescales. Carre Arold. Quata [2] A Frst Course Probablty. Sheldo Ross. 1

2 2 Questos 2.1 Modelg Smlarty of a Sgle DNA Ste a) Cosder the model for DNA mutato cosdered above: at each tme-step, a specfc ste o the DNA strad s chose wth uform problem (.e., for stes, the probablty of choosg a specfc ste s 1 ). Oce ths ste s chose, f t s curretly state, t trastos to state j wth probablty µ j, j. Here, both ad j are elemets of the dscrete state space E = {A, G, C, T } represetg the four possble base pars for DNA. If we assume that these trasto probabltes are tme-depedet ad ste-depedet, wrte the geeral trasto matrx Ω descrbg the process of base par mutato. The, smplfy ths matrx to the specfc case where all probabltes µ j are uform. You should fd that your Ω-matrx s a very smple ad symmetrc. b) Wth the trasto matrx Ω, we ca ow beg to solve the master equato, but usg a dfferet approach tha we leared class. Our frst step s to derve a ordary dfferetal equato for the tme-evoluto of the probablty of a specfc ste beg a certa base-par value. From the master-equato formalsm, we ca wrte the probablty of a ste beg a specfc basepar value as p (t + t) = p (t) p (t)µ t + p j (t)µ j t, (1) j where E. If stead we cosder the probablty vector P(t) = [p A (t) p G (t) p C (t) p T (t)] T, show that equato (1) smplfes to a smple ordary dfferetal equato the lmt as t goes to zero, dp(t) = ΩP(t). (2) dt c) The equato derved b) s very famlar to us; f quattes were ot vectors, the soluto would be a smple expoetal. Luckly for us, we ca solve ths vectorzed equato the same way as the o-vectorzed case, ad obta the soluto P(t) = P(0)e Ωt = P(0)P (t). () However, we ow have to deal wth what a expoetal power meas. We ca defe matrx expoetato usg the Taylor seres e Ωt = k=0 Ω k tk k! = I + Ωt + Ω2 t (4) Usg the defto equato (4), solve for the matrx P (t) equato (). You should fd that the soluto s aother symmetrc matrx (Ht: wrte out the frst few terms of the Taylor seres ad sum them together. You should fd that the etres ths matrx sum resemble a Taylor seres for e t ). 2

3 d) Wth your soluto for c), take the lmt as t goes to fty to fd the log-ru behavor of the probablty P(t). Your aswer should be smple, yet make very tutve sece: the probablty of a ste stayg the same s 1. Why does ths aswer make coceptual sece based o the model? Log-Ru DNA Smlarty Estmates: the Well-Mxed Bomal Approxmato e) Whle we have a dervato for the smlarty of a sgle DNA ste, we really wat to model the smlarty of the etre DNA strad after mutato. If lmt our vestgato to log-ru behavors, we argue that we ca model the umber of mutated DNA stes that are the same base-par value as the orgal DNA sequece as a bomal radom varable. As a quck troducto to probablty theory ad radom varables, we deote a radom varable X that s bomally dstrbuted as X Bom(, p), whch mples that ( ) ( ) P (X = ) = p (1 p)!, = 1,...,, = ( )!!. (5) We ca thk of a bomal radom varable wth parameters ad p as the umber of sucessful outcomes of trals, where each tral s depedet from the others ad the probablty of success of a sgle tral s p. The easest example of a bomal radom varable s askg the probablty of observg heads whe flppg cos. Wth ths kowledge, why does our argumet that the umber of smlar DNA stes after may mutatos have occured ca be modeled as a bomal radom varable? f) From our well-mxed bomal model, we ca derve aalytc estmates of the mea ad stadard devato of the smlarty the lmt of large umber of smulatos. Frst, we eed to derve the expectato ad varace of a bomal radom varable. For a geeral radom varable, these values are gve by E [X] = x p(x ) = µ. V ar (X) = E [ (X µ) 2] (x µ) 2 p(x ) = E [ X 2] (E [X]) 2. Usg these deftos, ad the defto of the dstrbuto of a bomal radom varable from equato (5), show that f X Bom(, p), the E[X] = p ad V ar(x) = p(1 p) (Ht: There are may ways to derve these equatos; oe method s to calculate the k-th momet of X, E [ X k], ad use ths to obta the mea ad varace. The detty ( ) ( = 1 1) may also be helpful.). g) Now, cosder a expermet where we draw from the above bomal dstrbuto may tmes, where each draw s depedet of the others, ad create a dstrbuto of the values obtaed from each draw. By the Cetral Lmt Theorem from probablty theory, the lmt of large sample sze, ths dstrbuto coverges to a ormal dstrbuto (.e., a Gaussa) wth the same mea

4 ad varace as the bomal dstrbuto we draw from. Usg ths approxmato, calculate the mea ad stadard devato of ths ormal dstrbuto, usg the fact that the stadard devato σ s just the square root of the varace. The, rescale the dstrbuto by dvdg by to measure smlarty o the terval [0, 1], ad plug values = 100 ad p = You should fd that the mea of ths dstrbuto matches the theoretcal log-ru smlarty of a sgle ste obtaed d). 2. Smulatg DNA Mutato ad Smlarty Now we wll smulate the DNA mutato strad as we have doe prevous problem sets. It may be helpful to get back to the frst problem set to refresh your md dog ths smulato. h) Wrte a smulato that models the self-smlarty of a DNA strad as a fucto of tme. The self smlarty s smply defed as the fracto of base pars that are the same as the orgal DNA Ru ths smulato for a large umber of trals ad plot the mea of the self-smlarty at each tme step. Descrbe the otable features. Do they make sese gve the mathematcal modelg we have doe above? Smulato Hts: Start wth a radom dstrbuto of base pars. At each tme step, choose a radom base par, ad geerate a radom umber to determe f t mutates. At the ed of each tme step, use a loop to cout the umber of base pars that are the same as the orgal, ad dvde by total base pars to fd the self-smlarty. The self-smlarty values, the orgal DNA strad, ad the curret DNA strad are the oly thgs that should be stored betwee tme steps. If you store the curret DNA strad at each tme step, ad attempt to calculate the self-smlarty afterwards, you wll quckly ru out of space. Suggested Smulato Values: 100 base pars, 2000 tmesteps, trals. Trasto rate betwee ad base par ad ay mutato: 10 per tmestep. ) You should fd that your self-smlarty appears to have a expoetal dstrbuto. To test ths hypothess, ft your curve to the stadard expoetal form: Do the values for A, B, ad k that you fd make sese? Expla. P (t) = A + Be kt (6) j) Plot the stadard devato as a fucto of tme, ad descrbe ay otable features. Wth ths plot, estmate the asymptotc (late tme) stadard devato, ad compare to the value you calculated part g), wth approprate values plugged. 2.4 Mea Frst Passage Tme to Varyg Levels of Self-Smlarty Let us ow observe ths system through a dfferet les; cosder the varyg levels of selfsmlarty as the possble states for the DNA strad. If we have a DNA of fxed legth N, the the dscrete state space becomes {0, 1, 2,..., N-1, N} where state would correspod to havg 4

5 base pars detcal to the DNA before ay mutatos. We ca ow cosder the rates of jumpg from to +1 ad to -1 wth respect to the mutato rate µ. I order to jump from to +1, we would eed oe of the (N ) correct base pars to mutate to the correct base par. Sce there are possble mutatos that could occur, the rate of oe correct base par mutatg correctly s µ. Cosderg there are N correct base pars, the total rate from to +1 s (N )µ. Now, order to jump from to -1 we eed oe of the correct base pars to mutate to ay of the other correct base pars. Thus ths total rate occurs as µ. From ths we ca see that our trasto matrx etres wll be: (N )µ Ω +1, = Ω 1, = µ (N + 2)µ Ω, = (Ω +1, + Ω 1, ) = Ω else = 0 Ad thus our trasto matrx becomes: Nµ µ Nµ (N+2)µ 2µ (N 1)µ 0 (N+4)µ Ω = (N+2(N 2))µ (N 1)µ 0 2µ (N+2(N 1))µ Nµ µ Nµ k) Cosder a DNA ts orgal state ( = N). Usg the above trasto matrx, for N = 100 mutatos ad µ = 10, calculate the mea frst passage tme from = 100 to tmestep f = 90, 80, 70, 60, 50, 40, 0, 20, 10, ad 0 (mmum self-smlarty). What treds the MFPT do you otce for the DNA to get to creasgly dssmlar states? Ht: 1 = 100 =0 Ω,τ f for c ad τ f f = 0. To use ths, frst create Ω by the followg rules: Ω j, = Ω j, for c ad Ω j, c = δ j,c. Next, create x = [ 1,..., 1, 0, 1,... 1] where the c etry of x s 0 ad the legth s N+1. The let τ = [τ 0 f, τ 1 f,..., τ 100 f ] T, ad fally solve Ω T τ = x for τ 100 f. The repeat ths process for the varous f. 5