APh161: The Physics of Biological Structure andFunctionHomework 2Due Date: Thursday, January 27, 2005On Exactitude in Science . . . In that Empire, the Art of Cartographyattained such Perfection that the map of a single Province occupied the en-tirety of a City, and the map of the Empire, the entirety of a Province. Intime, those Unconscionable Maps no longer satisfied, and the CartographersGuilds struck a Map of the Empire whose size was that of the Empire, andwhich coincided point for point with it. The following Generations, who werenot so fond of the Study of Cartography as their Forebears had been, sawthat that vast Map was Useless, and not without some Pitilessness was it,that they delivered it up to the Inclemencies of Sun and Winters. In theDeserts of the West, still today, there are Tattered Ruins of that Map, in-habited by Animals and Beggars; in all the Land there is no other Relic ofthe Disciplines of Geography. - Jorge Luis BorgesReading: read chaps. 7 and 8 of Essential Cell Biology and chap. 1 ofGenes and Signals.1. A Feeling for the Numbers: The Rates of ThingsIn the previous homework, we worked hard to get a sense for the physicalsizes of various biological entities. Another interesting angle on all of this isto try and get a feel for the rates at which things happen. Following in thetradition of the previous problems, here you will try to make some estimatesof the rates of some processes. Much of what you will do in this problem Ihave already done partially in class - your job is to make it your own now.(a) Consider the division of an E. coli cell. Think of such a cell duringrapid growth phase where the cell is dividing roughly once every 20 minutes.Make estimates of the number of water molecules being taken on board persecond during this phase, the numb er of lipid molecules that are being addedonto the surface membranes, the number of proteins being synthesized persecond and how many ribosomes are needed to do so.1(b) In this case, think about the motility of the bacterium Listeria mono-cytogenes and a typical eucaryotic cell. In the case of Listeria, the motion ofthe bacterium is mediated by the formation of actin comet tails which dependin turn up on the linear polymerization of actin filaments. The formation ofthe actin comet results in a speed for the bacterium of something around0.1 µm/sec. In the eucaryotic setting, the cell extends arms called filopodiawhich permit it to crawl, again by virtue of actin polymerization. For Lis-teria, use the measured rate of motion of the bacterium to estimate the rateof actin polymerization both in microns/sec and monomers/sec. Make sureyou draw a picture of the process and explain your rationale. Now, take thatestimate for the rate of actin polymerization and estimate the rate at whicha filopodium extends on a eucaryotic cell. Anything you can do to comparethese estimates with measurements would be useful - one excellent source isCell Movements by Dennis Bray.(c) Given that the time scale for the cell cycle of a bacterium is arounda half hour, in this part of the problem, we estimate the rate of DNA repli-cation. In particular, compute the number of nucleotides per second thatDNA polymerase would have to add as a function of the number of DNApolymerase molecules in the cell. That is, if there were only one DNA poly-merase molecule, then this one molecule would have to add ≈ 5 × 106nu-cleotides in 30 minutes. On the other hand, if there are 10 DNA polymerasemolecules they can share the burden. Your task is to estimate the meanrate of nucleotide incorporation as a function of the numb er of DNA poly-merase molecules in the cell so as to make sure that the whole genome isreplicated. Now let’s turn the argument on its side - if I tell you that therate of nucleotide incorporation is roughly 100 nucleotides per second, thenestimate how many DNA polymerase molecules are present in the cell (notethat this is all orders of magnitude since we are ignoring subtleties like whatfraction of the polymerase molecules are bound, etc.). Also, look at fig. 6-9of Essential Cell Biology and assuming that this is a representative sample ofthe replication process, estimate the number of DNA polymerase moleculesin an E. coli cell. Taking our examination of that figure further, note thatthe fly DNA is about 1.8 × 109nucleotide pairs in size. Estimate the frac-tion of the total fly DNA shown in the micrograph. There are eight forksin the micrograph, numbered 1-8. Estimate the lengths of the DNA strandsbetween replication forks 4 and 5 where we count the forks from left to right.2If a replication fork moves at a speed of 100 nucleotides/s, how long will ittake for forks 4 and 5 to collide.2. Cut it UpIn this problem I want you to think about the ≈ 48, 000 bp genome oflambda phage and to work out the lengths of the fragments that you wouldget if the DNA is cut with both the HindIII and EcoRI restriction enzymes.What are the recognition sequences that these enzymes each cut? Make anestimate for the lengths of the fragments that one would get when only usingone of these enzymes - there is a precise mathematical way to do this and itdepends upon the length of the recognition sequence - a 5 cutter will haveshorter fragments than an 8 cutter - explain that. To do the precise analy-sis, go to the New England Biolabs website (www.neb.com) and look up thetables that they have for identifying the sites on the lambda genome thatget cut by these different enzymes. If you are feeling ambitious, Hernan lastyear did something very cool which you could imitate. In particular, he wentto the New England Biolabs website and basically examined the distributionof cuts for all of their enzymes and examined the extent to which the simpleprobabilistic model I am advocating here is correct. Take a look.3. The Frances Arnold Estimate ProblemIn a 2001 Bioengineering seminar, Professor Frances Arnold made a startlingremark that it is the aim of the present problem to examine. The basic pointis to try and generate some intuition for the HUGE, ASTRONOMICALnumber of ways of choosing amino acid sequences. To drive home the point,she noted that if we consider a protein with 300 amino acids, there will be ahuge number of different possible sequences.(a) How many different sequences are there for a 300 amino acid protein?But that wasn’t the provocative remark. The provo cative remark was thatif we took only one molecule of each of these different possible proteins,
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