University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015

Transcription

1 Lecture 8. 1/3/15 Univerity of Wahington Department of Chemitry Chemitry 453 Winter Quarter 015 A. Statitic of Conformational Equilibria in Protein It i known from experiment that ome protein, DA, and RA molecule undergo tranition between tructured helical form and eentially diordered or random coil form. In protein, the α helix exemplifie uch a tructured form, which i contituted mainly by a ytem of hydrogen bond between the amide proton (i.e. -H) of an amino acid to the oxygen of a carbonyl group (i.e. C=O) four reidue away. See Figure 8.1 Figure 8.1: An alpha helix howing the ytem of hydrogen bond between amide group on the ith reidue and the carbonyl group on the i+4 reidue. The helix form dominate at low temperature. A temperature i increaed, the random form dominate. Frequently the tranition occur over a very narrow range of temperature, implying the entire helix unwind very uddenly rather than ection of the helix unraveling gradually. See Figure 8.. Figure 8.: Fraction of helicity in peptide chain of different length monitored a a function of temperature by optical rotation. T c i the temperature in the midpoint of the tranition when half the total chain are helical. Highly cooperative tranition have igmoidal hape. A hown in Figure 8., when the fraction of chain that are helical i monitored a a function of temperature, the fraction change rapidly at the melting 3

2 temperature T C and ha a igmoidal hape. Such a tranition i aid to be cooperative, becaue the chain ection convert from helix to random coil all together., a hown in Figure 8.3. Figure 6.3: In a fully cooperative tranition a helix uddenly unwind into a random coil in an all or nothing fahion.there are no intermediate tate compoed of partially folding chain. The thermodynamic of helix-coil tranition can be modeled tatitically. Firt let u aume the peptide chain i compoed of monomer unit. Each monomer unit can either be in a helical tate (H) or a coil tate (C). So the tructure of the chain can be coded in term of H and C. A particular configuration of the chain might be HHCCCCHHHHCCHHH, for example. There are many poibilitie. Our objective i to calculate the partition function of the chain auming all poible configuration. To do thi we have to aume a model for the tranition. We will aume two model; non-cooperative, fully cooperative, and the zipper model. B. oncooperative Helix-Coil Tranition There are monomer in the chain. The tructural tate of each monomer H or C i independent of it neighbor. We need to calculate the partition function for all configuration of uch a chain o Suppoe a chain ha four monomeric unit =4. If uch a chain i in the CCCC configuration the partition function i q 0. If the configuration i HHHH, the partition function i q 4. There are four configuration with 3 C and 1 H: CCCH, CCHC, CHCC, and HCCC. Call thi partition function q 1. There are 6 configuration with H and C with q. There are four configuration with 1 C and 3 H for which the partition function i q 3. the total partition function i: q= q0 + 4q1+ 6q + 4q3+ q4 q1 q q3 q 4 = q (8.1) q0 q0 q0 q0 = q0( 1+ 4k1+ 6k + 4k3+ k4) qn o The parameter kn = are called microcopic equilibrium contant. q0 They repreent equilibria between the CCCC configuration and each individual configuration containing n H-type monomer. o ow we make a big aumption about the energetic of thee tranition. Regardle of which configuration are involved, every tranition from C to H ha the ame energy change G. Therefore the tranition from CCCC to HCCC, CHCC, CCHC, or CCCH all have energy change G and 4

3 q = (8.) o 1 G / RT k1 e q0 o Similarly the change from CCCC to any of the 6 configuation with H n ha energy change G o k =, and in general kn = o q = q0( 1+ 4k1+ 6k + 4k3+ k4) (8.3) = q = q 1+ ( ) ( ) 0 0 o ow each term in the expanion in 8.3 ha a pecific interpretation. When divided by q, they give the probabilitie that particular configuration occur with particular number of H unit: q0 1 q04 4 p0 ; p q q q06 6 q04 4 p ; p (8.4) 3 4 q q q 0 p4 3 4 q o Equation 8.3 and 8.4 can be generalized to any chain with monomer: ( 1 )! n 1 q= q q0 ( 1 ) n! ( n)! = + (8.5) q0! n pn = qn! n! ( ) o To imulate data diplayed in Figure 8., we need to calculate the fractional helicity, defined a: npn n n= 1, fh where <n> i the average number of helical unit in a chain. Example: Ue equation 7.6 to calculate the fractional helicity for =. (8.6) npn n n= 1, 1 fh = ( p1+ p) 1 ( 1+ ) = ( 1+ ) 1+ o Thi procedure i fairly imple for mall value of. But if i large the erie ummation can become daunting. Fortunately, there i a really eay way to calculate <n> that doe not ue the p n erie expreion. It can be hown: 5

4 ln q q q0 n = ( 1+ ) = ( 1+ ) = ln q q0 ( 1+ ) ( 1+ ) 1+ (8.7) n fh 1 + o In general, for monomer: n f = H = 1 + (8.8) o Recall that the converion from C to H occur with a Gibb energy change G o ; o G C H (8.9) o If <1, then G o >0, helix i not favored. Thi i the ituation that prevail at high temperature. o If >1 then G o <0, helix i favored which i the ituation that prevail at low temperature. A plot of equation 8.7 for f H veru i given in Figure 8.3:. Figure 8.3: A plot of the fractional helicity a a function of uing equation 7.8. fractional helicity f on-cooperative o Equation 8.7 and Figure 8.3 indicate that in the abence of cooperativity, there i a mooth accumulation of helical monomer a increae. o In Figure 8.4 we can view the progre of non-cooperative and fully cooperative protein helix-coil tranition with temperature. Suppoe the untructured form of a protein with monomer...cccc...ha energy E 1 and the fully tructured form...hhhh... ha energy E. At temperature T 1 the number of untructured molecule w(e 1 ) i a large number and n 0. At temperature T the number of fully tructured molecule w(e ) i large and n. 6

5 o A the temperature i varied between T 1 and T, partially tructured form appear and the w maximum hift gradually with temperature, and n increae gradually from 0 to. Figure 8.4: A non-cooperative model (left) ha a gradual hift of the population maximum from T 1 where the untructured form dominate to T where the tructured form dominate. At intermediate temperature partially tructured form are mot numerou. In the fully cooperative model (right) there are only two population: fully tructured and untructured. At T 1 almot all the population i in the untructured tate and at T almot all the population i in the tructured tate. At the melting temperature T*, the population are equal. C. Fully Cooperative Model In the fully cooperative model there are only two tate: o The untructured tate: where the chain of length ha no helical monomer o ha the configuration i CCCCCCCCC... and the partition function i q 0. o The tructured tate: where the chain ha all helical monomer which correpond to HHHHHHHHH... and the partition function i q =q 0 4. o Becaue there are only two form in equilibrium: all C and all H, and auming there are monomer the equilibrium i k = CCC CCC HHH HHH (8.10) o To contruct the partition function, we implify equation 8.3. In thi equilibrium only the contant k 4 = 4 i nonzero. Therefore the partition function implifie to 7

6 ( ) q= q + (8.11) 0 1 o Proceeding jut a before: q 1 n n 1 + = = fh q (8.1) o Figure 8.5 A plot of equation 8.1 i hown to the right. It ha the familiar igmoidal dependence of the data in Figure 8.. fractional helicity Fully Cooperative Figure 8.4, right and Figure 8.5 diplay the propertie of a fully cooperative tranition. In thi model no intermediate form exit and the fully tructured and untructured population change a a function of temperature. At the melting temperature T* the population are equal o that f H =0.5 and =1. 8