Systems Biology 2012, Exam #2 Name: 7.32/7.81J/8.591J: Systems Biology. Exam #2
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1 SystemsBiology2012,Exam#2 Name: 7.32/7.81J/8.591J:SystemsBiology Exam#2 Instructions 1) Pleasedonotopentheexamuntilinstructedtodoso. 2) Thisexamisclosed bookandclosed notes. 3) Pleasedoallproblems. 4) Usethebackofsheetsifyouneedmorespace. Helpfulequation:IntheMoranProcess,amutantwithrelativefitnessrpresentasiindividualsina populationofsizenhasaprobabilityoffixinggivenby: x i 1 1 i = r 1 1 r N 1
2 Scores 1(outof8): 2(outof15): 3(outof15): 4(outof20): 5(outof10): 6(outof5): 7(outof10): 8(outof12): 9(outof5): Total(outof100): SystemsBiology2012,Exam#2 2
3 SystemsBiology2012,Exam#2 1) MasterEquation(8ptstotal) Youaretheownerofarestaurantandyouknowthatcustomersentertherestaurantrandomlyata ratef=5hr 1,andeachindividualleavesatarateg=0.5hr 1. a.writethemasterequationtodescribethetimeevolutionpn(t)oftheprobabilitythattherewillbe ncustomersattimetintherestaurant.(2pts) b.ataparticulartime(t=0),youcount20 peopleintherestaurant.whatistheexpectednumberof customersasafunctionoftime?(3pts) c.ifyouwaitforalongtime,whatistheprobabilitydistributionforthenumberofcustomers? Pleasewritetheequationofthedistribution andspecifythenumericalvalueofanyparameters. (3pts) 3
4 SystemsBiology2012,Exam#2 2) a.roughlyhowfastdoes LifeatlowReynold snumber(15pts) 1. E.coliswim?( nm/sec 300nm/sec 3um/sec 3nm/sec pts) um/sec um/sec 3mm/sec 8. 30mm/sec b.afterstoppingswi 1. mming,roughlyhowfardoesthecelldriftbeforestopping?(3pts) <=1nm 10nm 100nm 1um um um >=1mm c.considerasmallporewithradiusthroughwhichsaltisdiffusingfromachamberontheright,in,toachamberontheleft,inwhichtheconcentrationiszero. i)howdoesthetotalflowofsaltthroughtheporescalewiththe whichtheconcentrationofsaltisc0 concentrationontheright?(2pt) ii)howdoesthetotalflowofsaltthroughtheporescalewiththediffusionconstantofsalt?(2pt) iii)howdoesthetotalflowofsaltthroughtheporescalewiththeradiusofthepore?(3pt) sphere?(3pts) d.howdoesthetypicaltimeforangular reorientation ofaspherescalewiththeradiusofthe 4
5 3)Distributionofbeneficialmutations(15points) SystemsBiology2012,Exam#2 The American Association for the Advancement of Science. All rights reserved. This content is excluded from our Creative Commons license. For more information, see a.whataretheauthorstryingtoshowinthisfigure?whatisthedifferencebetweenthemain figuresandtheinset?(3pts) The American Association for the Advancement of Science. All rights reserved. This content is excluded from our Creative Commons license. For more information, see b.whatistheprimarypointofthisfigure?(3pts) 5
6 SystemsBiology2012,Exam#2 c.drawthedistributionofbeneficialmutationsthatcorrespondstoeachoftheallowedregionsof parameterspaceforeachofthe threeunderlyingdistributionsinthepreviousfigure.pleaselabel thex axis.(3pts) d.whyisthemeanselectioncoefficientdifferentforthethreedistributions?(e.g.whyisthe meanselectioncoefficientlargerforthedeltafunctionthanfortheexponential?). (3pts) e.whyistherateofbeneficialmutationsdifferentforthethreedistributions?(e.g.whyisthe beneficialmutationratelargerfortheexponentialthanforthedeltafunction?). (3pts) 6
7 SystemsBiology2012,Exam#2 4) RobustnessinBacterialChemotaxis(20points) a.howdoesbindingofanattractantaltertheactivityofthecomplexchew/a?(2pts) b.whataretypicaltumblingfrequencies?(2pts) c.howfardoesane.coliswimduringatypicalrun?howdoesthishelpthecellto measure the concentrationgradient?(2pts) d.whatdoestheequationbelowdescribe?whatarethetwokeyassumptionsembodiedinthis equationthatyieldperfectadaption? (5pts) d(x m + X m) * VB V R BX * m = R dt K + X * m 7
8 SystemsBiology2012,Exam#2 f.ifweover expresscherthenhowdoesthesteady statetumblingfrequencyvary?(3pts) g.ifweover expresscherthenhowdoestheadaptationtimevary?(3pts) h.ifweover expresscherthenhowtheadaptationprecisionvary(tumblingfrequencyafter adaptationdividedbytumblingfrequencybeforestimulus)?(3pts) 8
9 SystemsBiology2012,Exam#2 5)DorsalregionpatterningintheDrosophilaembryo(10points) Intheexampleofrobustembryopatterninggiven inuri sbook,thespatio temporaldynamicsof theinhibitori,morphogenm,andproteasepcanbedescribedbythefollowingsetofequations: I 2 I t = D I x kim 2 2 C C t = D C x + kim α PC 2 c M = α cpc kim t a.whatisspecies C?(2pts) b.howdoesthediffusionofthemorphogenchangeafterbindingtotheinhibitor?isthisthe changethatyouwouldexpectwhenamoleculegetsbigger?(3pts) c.whatdoestheproteasedegrade?(3pts) d.whatdoestheproteaseacton?(2pts) 9
10 6)Optimalityandgeneexpression(5pts) SystemsBiology2012,Exam#2 Courtesy of Nature Publishing Group. Used with permission. a.inthisfigurefromthedekelandalonpaper,whatdoes η(zwt)andδz Z?(3pts) WT WT andrepresent?whatis b.howwasthisexperimentdone?(2pts) 10
11 SystemsBiology2012,Exam#2 7) Evolutioninfinitepopulations(10pts) Inparts(a)and(b),specifytheprobabilitythatthemutantAwithrelativefitness rwillreach fixation.assumeamoranprocess. a.populationofsizesn=100,wherestrainahasagrowthadvantageof0.1%andstartsoutata frequencyof20%.(3pts) b.populationofsizesn=1000,wherestrainahasagrowthdeficitof2%andstartsoutata frequencyof70%.(3pts) c.assumethatthedistributionofbeneficialmutationsisdescribedbyp(s)=exp( s/s0)/s0,withs <<1.Pleasedrawthedistributionofmutationsthatreachfixationinthelimitoflargepopulation sizeandsmallmutationrate.(4pts) 0 11
12 SystemsBiology2012,Exam#2 8) Clonalinterference(12points) Considerapopulationthatcan bemodeledbythemoranprocesswithpopulationsize N,mutation rate µ(probabilityperindividualpergeneration).assumeallmutationsarebeneficialwithfixed selectioncoefficient s<<1. a.whatisthemeantimetmutbetweentheappearanceofsuccessivemutations?(3pts) b.whatisthemeantimetestbetweentheestablishmentofsuccessivemutants?(2pts) c.whatisthetypicalnumberatwhichamutantwithselectiveadvantages>0isessentially guaranteedtotakeoverthepopulation( established )?(3pts) d.whatisthetime Tfixforthismorefitmutanttotakeoverthepopulationafterbecoming established(conditionfoundinpartcabove)? (2pts) e.whatistheconditionrelatingn,µ,andssuchthatclonalinterferenceisnegligible?(2pts) 12
13 SystemsBiology2012,Exam#2 9)Evolutionandsequencespaces(5pts) Consideralargepopulationwherethethefitnessofgenotypesisdescribedby: 00:1 01:1.1 10:1.3 11:2 Ifthepopulationstartsinstate00,theninthelimitofsmallmutationratewhatisthe probability thatthepopulationwilltakethe00 >01 >11pathtothefitnesspeak? 13
14 MIT OpenCourseWare J / 7.81J / 7.32 Systems Biology Fall 2014 For information about citing these materials or our Terms of Use, visit:
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