Macromolecules in Undergraduate Physical Chemistry Wayne L. Mattice Louisiana State University, Baton Rouge, LA 70803 Naturally occurring macromolecules and synthetic polymers are among the materials most frequently encountered in everyday life by college students. Undergraduates, however, rarely study macromolecules in their formal coursework unless they enroll in special advanced level courses offered a t a few institutions. Inertia in the educational system is a major factor contributing to this situation. The traditional undergraduate chemistry curriculum was established before study of macromolecules matured into a rigorous science. Many textbooks designed for core courses in-the undergraduate chemistry curriculum make little or no use of macromolecules in illustration of the concents nresented. Chemists who have snent their entire career in academia may never have worked k t h macromoleucles, either in the classroom or in the laboratory. (In this respect they differ from their industrial colleagues, most of whom work in some area of macromolecular chemistry). Under these circumstances, macromolecules will be incor~oratedinto core courses onlv if the instructor makes a co&cious commitment to invest the time and effort necessary for modification of the traditional offerings. Several means by which macromolecul& can he incorporated into the first physical chemistry course are described (1,2). This objective is in earlier volumes of THIS JOURNAL not achieved through presentation of an entirely new unit on macromolecles. Instead, examples drawn from the field of macromolecules are used to illustrate concepts discussed in a tvnical first nhvsical chemistrv course. Krause (1) and ~ a n b e l k e r n(2fdi&s specific examples by which polymers are utilized in the oresentation of thermodvnamic equations of state (2), c a r n i t cycles (2), phase t ~ a n ~ i t i o n(2); s mixing (I), phase separation (I), colligative properties (I), the Donnan effect (I), and distribution functions (2). Their examples will not he repeated here. Instead the focus will be on modern methods of relating configuration-dependent macromolecular properties to the local chain structure. The two macromolecular systems to he described here are polyethylene in a state unperturbed by long range interactions and a polypeptide undergoing a helix-coil transition. These examples hold the interest of students with a biological orientation as well as those ~ l a n n i n ea career in industrial chemistry. While these m~cromol&les exhibit markedly different properties, a nearly identical formalism can he employed to rationalize several of those properties. Specific tasks within the abilities of upper level undergraduates include evaluation of the probability that a monomer unit will occupy a eiven conformational state and that the average numher of monomer units in a sequence will have identical conformational states. The logical location for these topics is in that section of the Dhvsical chemistry course which presents statistical mechanics hecause the desiied information is extracted trom the configuration partition function, Z. The configuratim partition function is initially formulated usine matrix nutatim (3-5,. Many students wifl already he able & use the matrix algebra employed. Those students encountering matrices for the first time should have little difficulty mastering the necessary manipulations because most are carried out with a matrix whose dimensions are no larger than 2 X 2. The desired ~ es of monomer units in a -given conforma. r o.~ e r t i (fraction tional state, average number of monomer units in a sequence) are extracted from the largest eigenvalue of this matrix.
Configuration Partition Functions .
The o helix ( f i ~is a famous example of a macromolecular ordered structure. Synthetic humopulypeptides can be d r signed so that they undergo a transition from oi helix to random coil upon change in some physical variable (solvent cornnosition..temnerture. Our interest is in dis. .or nressure). . tinguishing amino acid residues whose conformation ia that r e d r r d for furmation of an n helix trom amino acid residues wi;h any other conformation. The state of a polypeptide made up of x ammo acid residues is then represented by a string of x letters chosen from h (helix) and c (coil) (3). If end effects were unimuortant, the statistical weight for a chain containing n,,helical amino acid residucs would hes", rimes thr swtistical weight for the completely disordered chain. Heres denotes thr weight of a single h relative t o a c. It is expected to temperature, and prrsbe a function of solvent composit~~m, sure. End effects are incorporated by including a factor, t, for each helical ..~--~ ~ seauence? ~ The~exnectation ~ ~ is that ~ t will he less than unity due'tothe large de&ease in entropy which occurs upon helix nucleation (3). Then the statistical weight for a chain in a particular state, relative to the statistical weight of the completely disordered chain, is s"htn.h, where n,h denotes the numher of helical sequences. The configuration partition function is the sum of snhtkh for all ~ossihlestrings - of x letters selected from c and h. It is generated (4) as ~
~~~
.
~
~
Z = row (LO1 Uxcol (1,l)
(11
where
Columns in the statistical weight matrix, U, index the state of the amino acid residue in question, while rows index the state of the preceding residue. The order of indexing is c, h in each case. Each element gives the contribution made by the amino acid residue in question to the statistical weight of the entire chain for every possible local configuration. Formulation of the configuration partition function for polyethylene begins with a consideration of the torsional potential for the internal C-C bond in butane (4. 7). Students should easily see that this potential suggests a treatment using three rotational states: trans t r 1. cauchr' rr'), - -gauche- te-I. Let u denote the statistical weight of either gauche stateielative to a trans state. (Svmmetrv ~ e r m i t suse of the same statistical weight for both gauche states of butane.) In this approximation the statistical weight for a polyethylene chain containing n C-C bonds, of which n, & occupy gauche states, differs from that of the all trans chain by a factor of a%'. A more accurate weighting takes account of the severe steric interaction known as the pentane effect. Let o denote the extra weighting factor required for each pair of neighboring bonds when placements are g+g- or g-g+. Then the weight for the polyethylene chain, relative to that of all the all trans This.factor is.usuallv rs in the literature of helix-coil ...~ , denoted - - - - - bv ~, transit~ons(31.However, u 1s a so commonly used to denote the statistical we.ght for a frst-oroer interaction in a cham with a symmetric threefold rotation potential (4, 5). In order to avoid confusion, rs here will be reserved for the lalter statistical weight. ~~
~
~~~~
Volume 58 Number 11 November 1981
911
chain, becomes ong+wng+gf. The configuration partition function is the sum of ang'ong*g7 for all possible strings of n-2 symbols selected from t, g+, and g-. I t is generated (4,7) as Z =row (1,0,0) U"-2 COI (1,1,1)
(3)
where
Similar consideration of the polyethylene chain shows that the probability that a bond will be in either of the gauche states,p,+, is (x - 2)-'(8 In Z/d In a),,,. Probabilities for occupancy of the three rotational states (pt,pg+,p,-) are then easily obtained asp,+ = p,- = p g fA, p, = 1- p,f. In the limit of infinite x, eqn. (8) yields pz* = (A,
Here columns index the state of the bond in question, rows index the state of the preceding bond, and the order of indexing is t, g+, g-. This configuration partition function is for an unperturbed chain because interactions of longer range than those giving rise to the pentane effect are ignored. The above expression can be made more compact, with no loss of content, by recognizing that g+ and g- have the same statistical weight (a), as do g+g- and g-g+ (a2w). Hence each configuration with one or more gauche placements has a twin, in which each g' has been replaced by g', with an identical statistical weight. In evaluating the configuration partition function, the contribution by any pair can be taken to be twice the statistical weight of one member of the pair. Thus, the configuration partition function expressed in eqn. (4) is generated identically as (5)
z = row (LO) U"-2 col(1.1)
(5)
where
Now the order of indexing is t for the f i e t row (or column) and g+ g- for the second row (or column). An important consequence of this size reduction is that the statistical weight matrix for an amino acid residue in the polypeptide, eqn. (2), and for a bond in polyethylene, eqn. (6),are now both of the form
+
Subsequent manipulations for both polymers can be carried out using this matrix. Specialization of the results to either polymer is achieved by inserting the appropriate definitions for (112 and u22 If attention is confined to chains of high degree of polymerization, the configuration partition function is proportional to the nth (or xth) power of the largest eigenvalue, XI, of the statistical weight matrix.
- l)(X1 - Ad-'
(12)
+
which students canverify using a ( 1 + a ) = A1 Xz - 1and 20 = (A1 - l ) ( l - Az). Comparison of the form of eqns. (11) and (12) emphasizes the similaritv in methods used to assess the occuvancv of conformatioial states for bonds in these two polymers. k n y differences in behavior of the polymers in their unperturbed states must therefore be attributed to the values of s, t , a, and w. Each of these statistical weights arises from a consideration of short-range interactions in the chain. Therefore, configuration-dependent phvsical properties of infinitely large unperturbed macro&ol&ules have been expressed in terms of interactions which occur in species of low molecular weight. Students can carry out illustrative calculations in order to convince themselves that the different values for s, t , a, and w cause a polypeptide to behave differently from polyethylene. For unperturbed polyethylene, the appropriate values are o = exp (-E,IRT) and w = exp (-EJRT), with E, and E, being 500 and 2000 cal mol-', respectively (4, 7). Numerical results are depicted in Figure 1.This same Figure also depicts PI, for a synthetic polypeptide, poly(hydroxyhutyl-L-glutamine), in water. For this polymer, t is 6.8 X 10-4,s is unity at 35.8" C, and d In s/d (11T) is about 94 (8). I t is immediately apparent that the change in temperature has a much more profound effect on the polypeptide than it does on polyethylene. At all temperatures for which results are depicted, the three states (t,g+,g-) for polyethylene are each populated to a sienificant extent. In contrast.. the ~ l v .~ e ~ tunderzoes ide . o .. a helix.coil transition as the temperature passes thr19ugh36" C. The coooerative nature of the helix-coil transition can be attributed the difficulty in nucleation of a helical segment, as becomes convincingly apparent if p h is reevaluated using much different values for t. A reasonable choice of values would be 10-6 and 10-2, since t lies in this range for all amino acid residues for which experimental results are available. Segregation of Trans Placements and Helical States Consideration of the population of the various rotational states shows that, while the theoretical treatment of both polymers has much in common, infinitely long polyethylene
Probability of Occupancy of a Particular Rotational State Probability of occupancy of all of the rotational states can be extracted using the customary partial derivatives ( 3 , 4 ) . Since each chain configuration of the polypeptide has a statistical weight given by s"htn.h, the average number of helical residues is (a In Z/a In s),,~.The probability that a residue is helical, ph, is obtained upon division by n. In the limit of infinitely large n, this partial derivative can be obtained using eqn. (a), the result being and, of course, p, = 1- ph, where p, denotes the probability that a residue will be in a nonhelical state. (Students can readily obtain eqn. (11)once they realizes = XI A2 - 1and st = (XI - l ) ( l - Az).)
+
912
Journal of Chemical Education
Figwe 1. Probabillies for t, g+, and g- placements In unperturbed polyethylene and probability for a helical placement in a polypeptide.
and polypeptide chains nevertheless behave quite differently because they are subject to different short range interactions. Differences in their behavior become even more apparent upon consideration of the tendency for segregation of rotational states. The average number of helical amino acid residues in a polypeptide chain of x residues is xph. A measure of the average number of amino acid residues in a helical segment becomes accessible once we have the average number of helical segments in this polypeptide. A given configuration contains n,h helical segments. The average value, denoted (n,h ) , is therefore (8 In Z/a In t),,s, which can be written as ( n < h ) = X(X,
- l)(l- AP)XI-'(AI
-
(13)
if Z is given by eqn. (8).Consequently the average number of amino acid residues in a helical segment, denoted by vh, is vh = z p h l ( n e h ) = h 1 ( 1
-
(14)
if the polypeptide is long enough so that the limiting behavior has been attained. The most highly populated state in polyethylene is trans. In order to obtain the average number of trans placements in a sequence, it is necessary to obtain the average number of such sequences. This information is accessible if we count the numher of pairs of bonds whose states are tg*. The second bond in this pair is assigned the statistical weight shown in the second column of the first row of the statistical weight matrix in eqn. (6). Rewrite this element as 2m, where r is a variable which can later be assigned a value of unity. Then the average number of sequences of trans placements, denoted (nt),is (a In Z/a In r),,,". For a sufficiently long polyethylene chain, Z is given by eqn. (8).Division of the average number of trans placements, (n -2)pt, by ( n t ) yields
Figure 2. Average number of bonds in a sequence of trans placements in unpemrbed polyethylene and merge number of residues in a helical sequence in a polypeptide.
and They need only repeat the calculation using values of 10-2 fort. Examples presented here provide an introduction to the use of .-matrix methods in rationalizine confieuration-deoendent " moperties of macromolecules. Results obtained ate .ohvsical . somewhat more general than is implied in the discussiun. For exam~le,the statistical weight matrix used for . p~~lrethylene ~ . can be utilized for other sjmple chains with a symmetric threefold rotation potential (4, 5). Students interested in pursuing this topic further may be directed to a concise presentation of generator matrix methods which yield the mean square end-to-end distance, mean square dipole moment, mean square radius of gyration, and optical anisotropy (5). ~
ut = h l ( l -
A2)r1
(15)
where ut is a measure of the average number of bonds in a sequence of trans placements. Clearly eqns. (14) and (15) are of similar form. Therefore, any differences in the tendency for segregation of helical amino acid residues in the polypeptide, and trans placements in polyethylene, can be attributed to the short-range interaions giving rise to s, t, a,and w. Students can readily carry out computations of vh and u,, using the same parameters as those giving rise to the probabilities depicted in Figure 1. Differences in the behavior of these two macromolecules are forcefully demonstrated by the results depicted in Figure 2. It is particularly impressive to call attention to the differences in uh and vt under conditions whereph andp, have the same value. Students can readily convince themselves that the pronounced tendency for segregation of helical placements arises from the difficulty in nucleation of a helical segment.
~
~
~
~
-
Literature Clted Ill Krause. S..J. CHEM EDUC..55.174 119781
i+j A ~ , A ~ , J . ; ~ ~ ~i ~J~, , aR n. d ~ ~ o ~ , ~ . chomSoL,88,631(1966). ~.,~.~mr
181 Ananthansravsnan. V. S..Andreatta. R. H..Poland. D..and Scherses, H.A., Macro-
Volume 58 Number 11
November 1981
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