Problem Points Score Total 100

Transcription

1 Physcs 450 Solutons of Sample Exam I Problem Ponts Score Total 100 All wor must be shown n order to receve full credt. Wor must be legble and comprehensble wth answers clearly ndcated. Equaton sheet s on the last page

2 Problem 1 (Short answers) [8 ponts] (a) [3 pts] How many base pars are there n one helcal turn of a -DNA? There are ~10 base-pars n one helcal turn. (b) [3 pts] Approxmately what s the strength of a GC base-par relatve to an AT basepar? GC bp has 3 hydrogen bonds compared to AT bp that has hydrogen bonds. Therefore, strength of a GC bp s about 1.5 tmes hgher. (c) [3 pts] The three-letter codon CAU codes for the amno-acd hstdne. What s the 3- letter antcodon n the trna for hstdne that bnds to that codon on the mrna? All sequences should be read from the 5'- to the 3'-end. The antcodon s AUG. (d) [4 pts] What s the average energy of a datomc molecule n equlbrum wth a thermal bath at room temperature. Ignore any vbratonal modes. A datomc molecule has 5 degrees of freedom (3 translatonal and rotatonal). Accordng to the equpartton theorem, the average energy per molecule s 5 T (e) [5 pts] Two partcles have the same effectve (or buoyant) mass m', but ther dffuson constants are dfferent. Whch one wll sedment faster n an ultracentrfuge? Explan. The sedmentaton rate s gven by v d F ext m' a m' ad T Therefore, partcle wth hgher dffuson constant (more compact) wll sedment faster. (f) [5 pts] Whch molecule would requre a larger force to stretch t to 10% of ts contour length, a denatured polypeptde chan or a double-stranded DNA molecule? Explan. The force extenson curve s gven by z L bf T coth T bf For a fxed value of <z >/L, the molecule wth a larger statstcal segment length b wll requre a smaller force f for that extenson. Therefore, the polypeptde chan, whch has a smaller b, requres a larger force to stretch to the same relatve extenson as the double-stranded DNA molecule.

3 (g) [5 pts] Experments show that the average force requred to unzp DNA, when the two strands are pulled apart, s ~1 pn. Assumng that the dstance between the two ends ncreases by ~1 nm each tme a base par s broen, estmate the energy that holds the base par together. Express your answer n unts of T. Wor requred to brea one base par s f Δx = 1 pn 1 nm = 1 pn.nm = 3 T. 3

4 Probablty densty Problem [15 ponts] (a) [7 pts] DNA-bndng protens undergo one-dmensonal dffuson on DNA. The followng graph shows the probablty dstrbuton that, at t = 100 s, the proten s found at some dstance (measured n base-pars) from ts orgnal poston. From the nformaton on ths graph, estmate the one-dmensonal dffuson constant for the proten, n unts of bp /s. The probablty dstrbuton s gven by: 1 x P( x, t) exp 4Dt 4Dt At x = 0, 1 P( x 0, t) Dt Therefore, D bp /s poston (n bp) (b) [8 pts] Suppose that you are observng the rownan moton of a sphercal partcle (radus 1 m) n water at room temperature (95 K). You collect a set of measurements on the -dmensonal dffuson of these partcles. You fnd that after a tme t the partcles have typcally dffused a dstance R from the orgn, as follows: t (seconds) R (mcrons) Plot ths data n such a way as to get a lnear plot, and estmate the dffuson constant of the sphercal partcle from your graph. Usng the relaton for mean-square dsplacement for dffuson n -dmensons: r = R = 4Dt, a plot of R versus t yelds a slope equal to 4D 1 (m) /s (see graph below) or D 0.5 (m) /s m /s. 4

5 R R (n m )versus t R t (n sec) t 5

6 Problem 3 [17 ponts] For a small globular proten wth a dameter of 6 nm, n water at 300K, estmate (a) [6 pts] the average tme t wll tae for ths proten to dffuse a dstance of 400 nm; The coeffcent on ths proten molecule, movng n water, s gven by = 6πηR = 6π 10 3 Pa.s m = g/s Therefore, the dffuson constant s: T D m / s 70.8m 11 J g / s / s Snce R 6Dt we get t R 6D 9 ( ) m m / s 377 s (b) [4 pts] the average dsplacement n ths tme; All drectons are equally lely; therefore, the average dsplacement s zero. (c) [7 pts] the tme t wll tae for the proten to move 400 nm n the drecton of an externally appled force of pn. F ext v v F ext N g / s m/s x vt t x / v 400 nm / m/ s 11.4 s 6

7 Problem 4 [0 ponts] Consder a proten that has 100 amno-acds, where each amno-acd can have 3 possble orentatons n the unfolded state. A sngle, unque, confguraton of the chan corresponds to the natve (folded) state. Assume that the entropc contrbuton to the free energy of the folded and unfolded states s entrely from the confguratonal entropy of the chan, and that the entropy and enthalpy are ndependent of temperature. (a) [4 pts] What s the entropy of the folded state? S N N 1 ln N 0 (b) [5 pts] What s the entropy of the unfolded state? S U u ln 5 10 U J / K (c) [5 pts] The foldng temperature T F for a proten s defned as the temperature at whch half the molecules are n the folded state, and half are n the unfolded state. Let T F = 50 C for ths proten. At ths temperature, what s dfference n free energy between the folded and the unfolded states of the proten? At the foldng temperature, the molecule has equal probablty to be n the folded state or the unfolded state G 0 (d) [6 pts] What s the dfference n the enthalpy ( energy) between the folded and the unfolded states of the proten? Recall that G H TS. Snce G H TS 0 Therefore, H TS (73 50) K ( J / K) J 7

8 Problem 5 [0 ponts] A denatured (unfolded random col) proten conssts of 300 amno-acds. Assume that we can treat the polypeptde chan as an deal freely-jonted chan wth a statstcal segment length b that s 6 amno-acds long. (a) [3 pts] How many statstcal segments are there n ths polypeptde chan? N = 300/6 = 50. (b) [5 pts] The separaton between the central (C ) atoms of each amno-acd s about 0.35 nm. What s the root-mean-square end-to-end dstance for ths denatured proten? b nm R 1/ Nb.1nm 50.1nm 15 nm (c) [7 pts] A force f s now appled to the two ends of the proten, such that the average z L coth( ) 1/, where end-to-end extenson n the drecton of the force s gven by bf L s the contour length of the chan and. At low forces ( 1), T coth( ) 1/ / 3. Under these condtons, show that the denatured proten behaves le an entropc sprng, and derve an expresson for the entropc sprng constant. Lbf z L / 3 T 3T 3T f z z Lb Nb 3 T The polymer behaves les a sprng wth entropc sprng constant entropc Nb (d) [5 pts] Estmate the sprng constant for ths polypeptde chan. entropc 3T Nb 3 4 pn.nm (15 nm) 0.05 pn/nm 8

9 Equaton Sheet x10 J/K R cal/mol/k = J/mol/K T 4 J 4 pn.nm RT 0.6 cal/mol N A = Vscosty of water = 1 cp = 1mPa.s Densty of water = 1 g/cm 3 = 1000 g/m 3 1 Da = 1g/mol = g S ln G T ln K K exp( G / T ) ΔG = ΔH TΔS p exp( / T ) exp( / T ) p exp( G / T ) exp( G / T ) x y z Dt F v 6R T D P( x, t) 1 4Dt x exp 4Dt J x C D x C t D x C R Nb Lb P( r, N) 3 Nb 3 / 3r exp Nb P 3 ( N) Nb 3 / loop V r z L bf T coth T bf 9