MCB137: Physical Biology of the Cell Spring 2017 Homework 6: Ligand binding and the MWC model of allostery (Due 3/23/17)

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1 MCB37: Physical Biology of th Cll Spring 207 Homwork 6: Ligand binding and th MWC modl of allostry (Du 3/23/7) Hrnan G. Garcia March 2, 207 Simpl rprssion In class, w drivd a mathmatical modl of how simpl rprssion dictats gn xprssion lvls. In particular, w showd that th fold-chang in gn xprssion is givn by fold-chang =, () + R N NS β ε whr R is th numbr of rprssors, N NS is th numbr of non-spcific sits, and ε is th rprssor binding nrgy. Exprimntally, w can crat bactrial strains that xprss a YFP rportr undr th control of th rprssor. If th YFP fluorscnc signal is F rportr, thn th fold-chang is masurd by calculating fold-chang = F rportr(r) F rportr (R = 0). (2) Howvr, thr is an xtra subtlty that has to b takn into account whn obtaining such fluorscnc masurmnts. In particular, bcaus of th intrinsic fluorscnc of th clls thmslvs, thr is a spurious contribution to th total fluorscnc w masur, namly, F total. This quantity is givn by F total = F rportr + F cll, (3) whr F cll is th autofluorscnc of th cll. As a rsult, w nd to b abl to subtract th clls avrag autofluorscnc if th want to rport only on F rportr. Thus, th fold-chang is obtaind using fold-chang = F total(r) F cll. (4) F total (R = 0) F cll

2 W alrady larnd how to xtract fluorscnc lvls from microscopy imags of bactria using Matlab. In this problm, you will us th cod you wrot in ordr to tst th prdiction mad by Equation 4. You can gt our final vrsion of that cod undr Final vrsion of imag analysis cod in th matrials corrsponding to th March 9th lctur on th cours wbsit. Not that you can also larn mor about this protocol from Computational Exploration: Extracting Lvl of Gn Exprssion from Microscopy Imags in chaptr 9 of PBoC2. (a) You can find svral imags of bactria undr Full data st in th matrials corrsponding to th March 9th lctur. This data st corrsponds to bactria containing varying Lac rprssor copy numbrs, and a rportr containing a Lac rprssor binding sit (calld O2) that controls YFP xprssion. Th rprssor copy numbrs for ach strain ar givn in th following tabl. Strain Rprssor numbr Dlta 0 WT 22 ± 4 RBS47 60 ± 20 RBS ± 30 RBS ± 40 RBS 220 ± 60 If you want to larn mor about th masurmnts of th rprssor copy numbr, you can rfr to Garcia20c, which is providd on th wbsit. In addition, a strain calld Auto, lacking a YFP rportr, is providd with th data st. This strain can b usd to masur F cll. Us your cod to calculat th fold-chang in gn xprssion as function of rprssor copy numbr by prforming th various fluorscnc masurmnts prscribd by Equation 4. Plot your rsults on a log-log plot. (b) Estimat th binding nrgy by fitting Equation 4 by y by trying a rasonabl rang of paramtrs for ε as you did in problm 2 of Homwork 5. Altrnativly, you can prform a last-squars minimization as discussd in class. If you want to go down this rout, you might find it bttr to fit your data to th fold-chang xprssd in th languag of dissociation constants fold-chang =. (5) + [R] K d Hr, [R] is th concntration of rprssors insid th cll, and K d is th dissociation constant of rprssor to th DNA. 2 A fling for th numbrs: Hmoglobin W hav adoptd hmoglobin as on of our molculs of intrst to discuss th statistical mchanics of binding ractions. In this problm, you will prform svral stimats to gt a fling for th numbrs for hmoglobin. (a) Do problm 4.(c) from PBoC2. 2

3 (b) Figur out roughly how many O 2 molculs you bring in with ach brath and how many Hmoglobin molculs it would tak to us ach and vry on of thos oxygns. How dos this compar with th total numbr of Hmoglobins in your body calculatd in (a)? Hint: You will hav to figur out our lung capacity and how many O 2 molculs ar containd within that volum using th idal gas law. 3 Th MWC modl of protin allostry In class, w introducd dimoglobin as a toy modl for dscribing th binding of oxygn to hmoglobin. This modl gav us th opportunity to discuss how an intraction btwn two bound oxygn molculs can incras th sharpnss of th binding curv and lad to cooprativity. As a rmindr, th stats and wights for this modl ar shown in Figur. (a) Rdriv th rsult from class that stats that th oxygn occupancy (th avrag numbr of oxygn molculs bound to dimoglobin) is givn by N int = ( 2 [L] β ε + 2 [L] + 2 [L] β ε + ) 2 β(2 ε+ε int ) ( ) 2 [L] β(2 ε+ε int ), (6) whr [L] is th oxygn partial prssur, = 760 mmhg is th standard stat partial prssur, ε = ε b ε sol is th diffrnc btwn th ligand binding nrgy whn bound to dimoglobin and whn in solution, and ε int is th intraction nrgy btwn two bound oxygn molculs. To mak this possibl, you will hav to invok th rsult from class and Homwork 5 that stats that th nrgy to tak a ligand molcul out of th solution rsrvoir is givn by th chmical potntial ( ) c0 µ = ε sol k B T ln. (7) [L] Mak sur to includ and xplain all stps in your drivation. (b) Plot occupancy vs. oxygn partial prssur for ε int = 5 K B T and for ε int = 0 on a linar-log plot (smilogx in Matlab) in ordr to show th ffct of ε int on th sharpnss of th occupancy curv. Us ε = 5 K B T for both curvs. Sharp binding curvs do not always imply a dirct intraction btwn bound molculs. An altrnativ modl to undrstand cooprativity in ligand binding is basd on th MWC (Monod-Wyman-Changux) modl of protin allostry. This modl is dscribd in dtail in Chaptr 7 of PBoC. Hr, dimoglobin can xist in two distinct conformational stats known as tns (T ) and rlaxd (R). In th absnc of ligand, th T stat of th protin is favord ovr th R stat. W rprsnt this unfavorabl nrgy cost to accss th R stat with th nrgy ε. Howvr, th intrsting twist is that th ligand binding raction has a highr affinity for th R stat. This has th ffct that, with incrasing ligand concntration, th balanc will b tippd towards th R stat, dspit th cost, ε, of accssing that stat. 3

4 W labl th binding nrgis ε T and ε R, which signify th favorabl nrgy upon binding to th molcul whn it is in th T and R stats, rspctivly. Th stats and wights for this modl ar shown in Figur 2. (c) Show that th oxygn occupancy in th MWC modl is givn by N MW C = ( ) [ 2 ( ) ] 2 2 [L] β ε T + 2 [L] β2 ε T + βε 2 [L] β ε R + 2 [L] β2 ε R [ ( ) ] [ 2 ( ) 2 ], + 2 [L] β ε T + [L] β2 ε T + βε + 2 [L] β ε R + [L] β2 ε R whr ε T = ε T ε sol and ε R = ε R ε sol. (d) Show that th MWC modl can lad to th th sam sharpnss in th Oxygn binding curv as dirct intraction mchanisms. To mak this possibl, plot th prdictions from both modls, N int and N MW C, as a function of [L]. Us ε = 5 K B T and ε int = 5 K B T for N int, and ε T = 0, ε R = 9.5 K B T and ε = 4 K B T for N MW C. Us linar-log axis for your plot (smilogx in Matlab). (8) STATE β(ε b µ) β(ε b µ) β(2ε b +ε int 2µ) Figur : Cooprativity modl of dimoglobin. Stats and wights for a cooprativity modl of dimoglobin. ε b is th binding nrgy of an oxygn molcul, µ is th chmical potntial (th nrgy to b paid to tak an oxygn molcul out of th solution), and ε int is th intraction nrgy btwn bound molculs. 4

5 STATE β(ε T µ) β(ε T µ) β(2ε T 2µ) STATE βε β(ε R +ε µ) β(ε R +ε µ) β(2ε R +ε 2µ) Figur 2: MWC modl of dimoglobin. Stats and wights for an MWC modl of dimoglobin. Th nrgy for dimologin to adopt th rlaxd stat is ε. ε T and ε R ar th oxygn binding nrgis in th tns and rlaxd stats, rspctivly. 5

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