J . Phys. Chem. 1992, 96, 3994-3998
3994
affect the parameter u for an isolated residue in the helical conformation. Though the magnitude of this solvent effect is not known, this indicates that peptide conformational maps should be based on free energy rather than the energy. According to Brant and Flory,” another contribution to u arises from the fact that an isolated helical conformation has an unfavorable electrostatic free energy. (2) It is unlikely that the statistical weights for the first two nucleating residues of a helix are the same, since the second residue changes the solvent contact surface area by about twice as much as the first. This is evidence for the contact of the third peptide with the first and indicates that for a pair of adjacent helical conformations the “isolated pair” model for the peptide chain is not valid. With uI and u2 as the weights for the first and second h residues in the chh sequence, the standard LR matrix takes the form hi hfc(h hi u E )
6c
P v,
v
Z(h u c)
’:> 1
Here element (3,l) of the matrix is for the first helical residue and (1,2) for the second. However, these elements are not separable parameters. It is easily shown that they occur only as a product in the secular equation. Thus, even when these two steps are not equivalent, one could define a single parameter u with v2 = u1u2,which would play the same role in nucleation as the standard parameter. This would then differ from the element (3,2), which is the statistical weight for an isolated h residue. Since helix-coil transitions are dominated by the behavior of u and s,
it would be very dflicult to establish this difference experimentally. Since the nucleation parameter, u2 (or a), is associated with changes in solvation as well as confinement in angular space, the assumption that it is mostly entropic and independent of temperature can be questioned. (3) When a peptide unit is hydrogen-bonded into a helix, these same solvation changes occur in addition to the formation of the peptide H-bond. We do not even know the sign of the contribution of these changes in solvation to the free energy, but with only 250 cal or so of stabilization, they could be making an important contribution. The statistical weight, w, which by definition16 depends on the interactions of three consecutive helical residues, need not be construed as arising solely from the hydrogen bond. In summary, the main conclusion is that the form of a helix-coil theory is not strongly dependent on the specific assumptions which were made at the time of its formulation. Interactions other than H-bonds may be incorporated in w or s. If u1 and u2 are significantly different from one another, the u determined from helix-coil theory will be ( u , u ~ ) ~and / ~ ,a probably small error will be introduced in the weighting of random coil segments. Because of changes in solvation energy, the nucleation parameter should possess a temperature dependence, but since the equilibrium constant for the formation of a helix is us“, where n is the length of the cooperative unit for large helices and roughly the chain length for short ones, the temperature factor of u is probably obscured by the large temperature factor of sW2.
Acknowledgment. This research was supported by NIH Grant GM20195 and NSF Grant PCM8609113. We thank Drs. Robert L. Baldwin and J. Martin Scholtz for many useful discussions and for sharing numerous experimental results prior to publication.
Intrinsic Viscosity of Model Starburst+ Dendrimerss Marc L. Mansfield* and Leonid I. Mushin* Michigan Molecular Institute, 1910 West St. Andrews Road, Midland, Michigan 48640 (Received: September 27, 1991)
The hydrodynamic radii (R,) calculated from intrinsic viscosities of poly(amido amide) (PAMAM) starburst dendrimers increase rapidly with increasing generation number. This would seem to support a structure with strongly stretched tiers, which is relatively hollow near the core, and with most end groups near the surface. However, a computer simulation due to h n e c and Muthukumar supports a more folded structure with higher density near the core and with end groups dispersed throughout the molecule. Here we calculate the hydrodynamic radii of the Lescanec-Muthukumar model using intrinsic viscosity formulas developed by Zimm and Fixman. The Lescanec-Muthukumar starbursts with relatively stiff spacers have hydrodynamic radii in good agreement with the experimentalPAMAM radii,in spite of their folded structure. The hydrodynamic radius is sensitive both to the molecular size and to the density. As the number of generations increases, the molecules become more dense and the hydrodynamic radius increases more rapidly than the radius of gyration.
Introduction A number of authors have reported syntheses of starburst dendrimers.’* These are highly, regularly branched molecules having a treelike structure, displayed schematically in Figure 1. The most-studied starburst macromolecule is probably the sc~ca!lsd poly(amido amine) (PAMAM) starburst, whose synthesis was reported by Tomalia and co-workers.’ The basic building block, or spacer, of the PAMAM dendrimer is this moiety: ‘N-CH2-CH2-CO-NH-CH2-CH2-N,
/
#
‘STARBURST is a trademark of the Michigan Molecular Institute. *Dedicated, with highest regards, to Professor Marshall Fixman in honor of his 60th birthday. 8 Permanent address: Institute of Macromolecular Compounds, Academy of Sciencies of the USSR, 199004 St. Petersburg, Bolshoi prosp. 31, USSR.
We define the generation number, m, of a starburst molecule to be the number of spacers between the central core of the molecule and any of the termini. By this convention, the structure in Figure 1 has a generation number of 6. (Other authors begin counting a t zero, by which convention Figure 1 would have a generation number of 5 . ) (1) Tomalia, D. A.; Baker, H.; Dewald, J.; Hall, M.; Kallos, G.; Martin,
S.;Rocck, J.; Ryder, J.; Smith, P. Polym. J . 1B5, 17, 117. (2) Tomalia, D. A.; Hedstrand, D. M.; Wilson, L. R. Encyclopedia of Polymer Science and engineer in^, 2nd 4.; Wiley: New York, 1990; Index ~01.;pp 46-92. ( 3 ) Tomalia. D. A,: Navlor, A. M.; Goddard, W. A., 111 Anaew. Chem.. Int.‘Ed. Engl. 1990, 29, l j 8 . (4) Hawker, C. J.; Frkhet, J. M. J. Macromolecules 19M), 23, 4726. ( 5 ) Morikawa, A,; Kakimoto, M.; Imai, Y. Macromolecules 1991, 24, 3469. ( 6 ) Newkome, G. R.; Lin, X . Macromolecules 1991, 24, 1443.
0022-3654/92/2096-3994%03.00/0 0 1992 American Chemical Society
Intrinsic Viscosity of Model Starburst Dendrimers
. \Y
Figure 1. Schematic diagram of a starburst dendrimer.
TABLE I: Molecular Size Data for PAMAM Macromolecules R,, A R,(reduced) R, increment (reduced) m R,:X 1 2 3 4 5 6 7 8 9
5.5 8.5 11.0 15.5 20.0 26.5 33.5 38.0 45.0
5.5 8.0 12.0 15.5 20.0 28.0 33.0 37.5 45.0
0.63 0.98 1.26 1.78 2.30 3.05 3.85 4.37 5.17
0.35 0.28 0.52 0.52 0.75 0.80 0.52 0.80
The PAMAM’s have been characterized by a number of techniques.’-3 In this paper we focus primarily on characterization experiments that provide molecular size data, of which two classes have been reported: intrinsic viscosity and size exclusion chromatography. Intrinsic viscosity experiments directly measure the effect of solute molecules on the shear viscosity of solutions. They provide an indirect measure of molecular radius, the so-called hydrodynamic radius, R,, which is the radius of the rigid sphere that would have the same intrinsic viscosity as the molecule in question. Size exclusion chromatography experiments directly measure the translational partition function (Le., elution volume) of the molecule, which is, in the highly reticulated environment of the size exclusion medium, a strong function of molecular size. They also provide an indirect measure of the molecular radius, a quantity we call R,. R, is determined by calibration with macromolecules of known R in this case, well-fractionated poly(ethy1ene oxide) standards.T In both cases, therefore, the size determination hinges on a measurement of intrinsic viscosity. At least for the PAMAM’s, the two techniques agree very well. Table I displays R, and R, data as a function of m taken from ref 2. Table I also displays values of R, that have been reduced by the stretched length of one spacer, which we calculate from standard bond lengths and bond angles to be 8.7 A, and the increment in this reduced R, as we go from one generation to the next. The increment in 4 is quite modest in the early generations, but in the later generations it approaches 80% of the stretched spacer length. This dramatic increase in size as more generations are added is consistent with the picture due to Hervet and deGennes’ which seems to be the prevalent notion of the structure of starburst macromolecules. In this picture, most of the spacers are stretched and oriented away from the core and each generation of the macromolecule lies in its own concentric shell somewhat like the layers of an onion. The density increases as we move away (7) deGennes, P. G.; Hervet, H. J. Phys. 1983, 44, L351.
The Journal of Physical Chemistry, Vol. 96. No. 10, 1992 3995 from the central core of the molecule and most of the terminal groups lie near the surface of the molecule. Recently, Lescanec and Muthukumar8 published computer simulation studies of a model of starburst macromolecules that oonflicts with this notion of the structure. They obtained molecular structures with maximum density near the core and with terminal groups dispersed throughout the molecule? A body of experimental evidence3,”’ is considered to support the onion picture of dendrimer molecules, but lacking are any measurements that directly probe either the radial density or the distribution of terminal groups. We can think of only two reasons for which the LescanecMuthukumar model might be invalid. The first of these is that it lacks detail at the atomic level. Nevertheless, an idealized model such as this can still be expected to exhibit general properties common to many, if not all, starburst molecules. On the other hand, we cannot rule out the possibility that specific atomic interactions (e.g., dipole-dipole or hydrogen bonding) could lead to structures different from those predicted by Lescanec and Muthukumar. Second, the algorithm used to grow the molecules is not expected to generate equilibrium structures rigorously.8 However, Lescanec and Muthukumar argue8 that a similar growth algorithm applied to linear polymer molecules performs adequately. In any case, it is difficult to see how this problem could have more than a minor influence on the results. The question remains, however, as to whether or not the Lescanec-Muthukumar calculation is consistent with the intrinsic viscosity data. To address this question, we have computed intrinsic viscosities for starbursts grown by the Lescanec-Muthukumar recipe. We find that if the Lescanec-Muthukumar starbursts have relatively short spacers (two beads per spacer), then hydrodynamic radii agree well with those of the PAMAM dendrimers. Apparently, the hydrodynamic radius is sensitive both to the radius of gyration of the molecule and to its density. Therefore, the densification of the molecule that occurs as the generation number increases makes the hydrodynamic radius increase more rapidly than the radius of gyration. Computational Details
Ensembles of starburst molecules were generated according to the Lescanec-Muthukumar procedure. The details are given in ref 8. The molecules are represented as strings of beads of diameter d = 1 and of step length 1.2. Each spacer contains P steps. Lescanec and Muthukumar considered the cases P = 1, 3, 5, 7, 9, and 1 1 , whereas we have done computations only for P = 1, 2, or 3. Exact calculation of the intrinsic viscosity of flexible macromolecules is a formidable task. Fortunately, a number of approximate formulas are available. Most of these are known to be either upper or lower bounds, so that we can bracket the exact result. The formulas usually require us to replace segments of the molecule with hydrodynamic beads (simple enough for a bead model such as this). The following expression, due to Zimm,I2
is rigorously valid only for rigid molecules, and was shown by Wilemski and TanakaI3 to be an upper bound to the intrinsic viscosity of flexible molecules. Here H is the hydrodynamic interaction tensor, U is a vector to be defined momentarily, d is the bead diameter, and M is the molecular weight. The angular (8) Lescanec, R. L.; Muthukumar, M. Macromolecules 1990,23, 2280. (9) Caminati, G.; T w o , N. J.; Tomalia, D. A. J. Am. Chem. SOC.1990, 112, 8515. (10) Moreno-Bondi, M. C.; Orellana, G.; Turro, N. J.; Tomalia, D. A. Macromolecules 1990, 23, 912. (1 1) Gopidas, K.R.; Leheny, A. R.; Caminati, G.; Turro, N. J.; Tomalia, D. A. J . Am. Chem. SOC.,in press. (12) Zimm, B. H.Macromolecules 1980, 13, 592. (1 3) Wilemski, G.; Tanaka, G. Macromolecules 1981, 14, 1531.
3996 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992
brackets denote an ensemble average. An adaptation of this formula replaces the hydrodynamic interaction tensor with its preaveraged form, Ha:
3rd [e] = -(UT*H,-'-U) M
(3)
Here Ha is preaveraged over all internal degrees of freedom and over all orientations. Equations 2 and 3 have the advantage over eq 1 that only the inverse of Ha is needed, not that of H. We have used the Rotne-Prageri6 hydrodynamic interaction tensor:
Hip8 = q(Rij)6a, + D:(Rij)Rij&i~ D;f=O
9 R = 1--@.(R) lJ 16 d
D;(R) =
3 16dR
if R
m
R.
Rsn
radii R,(lower bound)
R.(uDDer bound)
1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3
1 2 3 4
0.95 1.48 1.99 2.51 3.04 3.63 1.34 2.13 2.88 3.62 4.36 5.15 6.11 1.65 2.65 3.58 4.48 5.38 6.31 7.44
1.20 2.26 2.99 3.64 4.34 5.08 2.21 3.36 4.39 5.43 6.43 7.46 8.55 2.69 4.24 5.64 6.84 8.06 9.31 10.60
0.91 1.46 2.06 2.72 3.43 4.21 1.26 2.06 2.92 3.87 4.88 5.98 7.23 1.54 2.54 3.60 4.75 6.00 7.34 a
0.92 1.53 2.25 3.14 4.24 5.61 1.30 2.20 3.24 4.52 6.10 8.08 10.62 1.60 2.72 4.02 5.59 7.54 10.00 a
5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7
"This starburst is very large (1 144 beads), so R, was not calculated.
(4b)
beads having the same familial relationship. This means, for example, that all pairs of first cousins in a given generation are combined, but they are not combined with any pairs of second cousins, nor are they combined with any pairs of first cousins in any other generation. H is also averaged over orientations, which provides
d
(6b)
Therefore, Ha is defined as if R
>d
(40
In eq 4, italic indexes refer to bead numbers and Greek to Cartesian coordinates. R , is the distance between beads i and j ; Rija is one component of the vector between beads i and j . Excluded volume repulsions were strictly enforced in this calculation, meaning that eq 4c and 4e are never required. The vector U is given by uix
P
(4a)
ifR
