J. Phys. Chem. 1984,88, 3441-3446 restriction in geometry may also serve to select a certain mode of reaction. So far, only one size of each type of micelle has been considered, however, the size may change even in the same type of micelle. The increase in the ET efficiency observed in the region of 0.05-0.5 M SDS can relate to the increase in the micelle size as follows: If the micelle size increases in the fashion that the length of the cylinder increases with the SDS concentration, the ET efficiency will increase with the portion of the cylindrical region in a micelle. If this explanation is correct, the assumption of eq 1 in the above SDS concentration region should be reconsidered. It should be remarked that participation of quencher exchange has been proposed on the basis of the analysis assuming eq 1 As discussed
.’
3441
in the preceding section, the quenching kinetics can be explained, without invoking quencher exchange, in terms of micelle-size change, if certain conditions regarding the size and the first-order rate constant are met. The large micelle formed in the presence of sodium chloride may also be spherical or disklike as suggested by small aggregation numbers reported by a majority of aut h o r ~ . ’ ~If, surfactant ~~ molecules are also very densely packed in such a large micelle, reactions will be similar to those in the cylindrical region of the rodlike micelle, viz., the ET efficiency will be enhanced. Registry No. SDS,151-21-3; CU(DS)~, 7016-47-9; Cu2+, 15158-11-9; pyrene, 129-00-0; pyrene cation, 34506-93-9; sodium chloride, 7647-14-5.
Molecular Mechanics of Kink Formation in Lipid Monolayers Scott H. Northrup Department of Chemistry, Tennessee Technological University, Cookeville, Tennessee 38505 (Received: December 12, 1983)
The energetics of formation of kink disorders in lipid monolayers is studied by using empirical energy functions and a detailed atomic model of the monolayer. Hydrocarbon chains are induced to undergo transitions from an all-trans conformation to kink conformations at various chain positions. The surrounding layers of chains are allowed to relax adiabaticallyin response to the induced disorders, and potential energy profiles and other structural aspects of kink disorders are determined. The lowering of activation barriers to kink formation and increased stability of the kink state are found for kink formation in the presence of existing kinks in the same and in adjacent chains.
This “crankshaft” rotation preserves the overall direction of the chain, allowing it to remain packed essentially parallel to neighboring chains and perpendicular to the plane of the bilayer. Isolated gauche rotations may occur a t the chain terminus. A number of molecular mechanics studies have provided information on the energetics of various conformations available to lipid as~emblies.~ Several theoretical studies have calculated
interaction energies between packed all-trans hydrocarbon chains.6-8 Others have explored the interaction between hydrocarbon chain packing and the polar This paper provides a more detailed look at the hydrocarbon region. By using empirical energy functions, we study the energetics of stacking of hydrocarbon chains in a monolayer, including not only the preferred all-trans conformation but a number of specific disordered states. These include isolated kinks at various chain locations and kinks occurring in pairs in the same and in adjacent chains. This is carried out by selective adiabatic deformation of a monolayer of hexagonally packed hydrocarbon chains. Potential energy profiles are calculated in this fashion for the formation of isolated kinks and the cooperative formation of neighboring kinks in the same and in adjacent chains. Several important and useful quantities are thus obtained. For example, information is obtained on the interaction energy between neighboring chains which are respectively trans-trans, trans-kink, and kink-kink, energy differences between all-trans and kink states for isolated and dovetailed kinks, the internal degrees of freedom to which the disorder strain is disposed, adherence to crankshaft cooperative behavior, and the spatial distances over which disorder perturbations occur. The above information, particularly the energy data, may be immediately incorporated into explicit empirical biomembrane phase transition model^.^-^ Perhaps more importantly, the potential energy profiles yield activation barriers to the elementary mechanistic steps associated with the nucleation and propagation kinetics of phase transitions in and
(1) J. F. Wagle, Ann. Rev. Phys. Chem., 31, 157 (1980). (2) H. L. Scott, Jr., J . Theor. Biol., 46, 241 (1974); J . Chem. Phys., 62, 1347 (1975). ( 3 ) J. F. Nagle, J . Chem. Phys., 58, 252 (1973). (4) J. A. McCamrnon and J. M. Deutch, J . Am. Chem. SOC.,97,6675 (1975) (5) J. Belle and P. Bothorel, Biochem. Biophys. Res. Commun., 58, 433 (1974).
(6) L. Salem, J . Chem. Phys., 37, 2100 (1962). (7) R. Zwanzig, J . Chem. Phys., 39, 2251 (1963). (8) E. Shapiro and S . Ohki, J . Colloid Interface Sci., 47, 38 (1974). (9) M. Kreissler and P. Bothorel, Chem. Phys. Lipids, 22, 261 (1978). (10) S. P. Gupta, G. Govil, and R. K. Mishra, J . Theor. Biol., 51, 13 (1975). (1 1) J. McAlister, N. Yathindra, and M. Sundaralingarn, Biochemistry, 12, 1189 (1973).
Introduction Lipid bilayers, along with proteins, are the primary constituents of cell membranes. A large body of evidence exists which indicates that lipid assemblies exhibit a fairly pronounced phase transition from a highly ordered gel state, with hydrocarbon chains in an all-trans conformation, to a more disordered liquid crystalline state, in which many chain bonds are rotated into gauche conformations.’ The role of this transition in biological function and control is under widespread investigation. A complete understanding of both structural and dynamical aspects of lipid bilayers requires knowledge on the detailed molecular level. A number of theoretical descriptions of lipid bilayer phase transitions recognize the existence of kinks or related phenomena as a fundamental manifestation of disorder in the hydrocarbon chain^.^-^ In the transition of a hydrocarbon chain from an all-trans (t) conformation to kink conformation, two gauche (g+) rotations occur in the opposite direction, symbolized by the usual notation ...ttt... ...g+tg-... (1)
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0022-365418412088-3441$01.5010
0 1984 American Chemical Society
3442 The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
Northrup
7
0
i I
looo
, 210°
NO0
,,_,.... ."' . ' 2m"
,'.
..
300"
a I
Figure 1. Initial hexagonal packing arrangement of the 61 hydrocarbon chains of the monolayer section used in this study. The interchain spacing is 4.6 A.
demonstrate the basis for dynamical cooperativity of kink formation. Another manifestation of disorder which has been proposed is the large-scale cooperative tilting of hydrocarbon chain axes relative to the bilayer n0rma1.l~ This type of behavior is not accounted for in our model and is probably more characteristic of an assembly in a highly fluid state. Our model, on the other hand, is more appropriate for the initial stages of departure from a completely ordered gel state. It is thus most useful in connection with the nucleation of order-disorder transitions and, to some extent, the growth of ordered or disorder clusters at phase boundaries.
Model Lipid Monolayer In this section we describe the model lipid monolayer employed in this study. Our model is deemed applicable to bilayers as well, in that the two layers have been shown to behave independently.20 Only the hydrocarbon chains of the monolayer are modeled explicitly. The presence of the relatively immobile hydrophilic head groups is accounted for by anchoring the first carbon atom in each hydrocarbon chain to a plane in a hexagonal array.g An isotropic harmonic potential function with a force constant equal to 800 kcal/(mol A2) was employed to constrain the head atoms. A total of 61 saturated hydrocarbon chains, each containing 15 carbon atoms, was initially oriented perpendicular to the head group plane in an all-trans configuration. This is consistent with experimental structure observations of the solid phase at a temperature just below the main transition.21 The hydrocarbon chain planes are assumed to be coparallel for simplicity, although there is some evidence which suggests mutual tilting of chain planes in some phospholipids.ii*22 A view perpendicular to the lateral plane of (12) T. Y. Tsong and M. 1. Kanehisa, Biochemistry, 16, 2674 (1977). (13) B. Gruenewald, Biochim. Biophys. Acta, 687, 71 (1982). (14) P. Yager and W. L. Peticolas, Biochim. Biophys. Acfa, 688, 775 (19821. (15) R. C. Gamble and P. R. Schimmel, Proc. Natl. Acad. Sci. U.S.A., 75, 3011 (1978). (16) T. Sano, J. Tanaka, T. Yasunaga, and Y. Toyoshima, J. Phys. Chem., 86, 3013 (1982). (17) J. E. Harkness and R. D. White, Biochim. Biophys. Acta, 551, 450 (1979). (18) K. Elamrani and A. Blume, Biochemistry, 22, 3305 (1983). (19) P. van der Ploeg and H. J. C. Berendsen, J . Chem. Phys., 76,3271 (1982). (20) L. 0. Sillerud and R. E. Barnet, Biochemistry, 21, 1756 (1982). (21) B. Cater, D. Chapman, S. Hawkes, and J. Saville, Biochim. Biophys. Acta, 363, 54 (1974).
.
Kink Formation in Lipid Monolayers
i
at room temperature,22y2628which suggest spacings in the range 4.6-4.8 A. At this spacing a total stabilization of 18.9 kcal/mol of chains is obtained by using eq 2 and looking at the central chain. The maximum stabilization energy is 29.3 kcal/mol of chains at an optimum spacing of 4.0 A. However, the latter spacing would most closely correspond to a monolayer at 0 K, and thus the 4.6 %, value was deemed more appropriate for representing a roomtemperature monolayer. The van der Waals interaction energy W,, between a pair of all-trans chains a t 4.6 A is -6.3 kcal/mol in our model, which corresponds to the value predicted by Salem6 for a 15-carbon chain at 4.94 A. The empirical energy function was used in conjunction with a steepest descents energy minimization procedure to obtain adiabatic energy profiles for inducing various disorder transitions in the monolayer. The all-trans monolayer was prepared for subsequent deformation studies by first performing 100 cycles of steepest descents energy minimization on the initial ordered coordinate set. This resulted in a reduction of only 0.3 kcal/mol of chains and had little effect on the geometry of the monolayer. A series of runs was performed to determine the energetics of the formation of an isolated kink disorder at various chain positions in the central chain of the section. In each run an external stiff harmonic constraint was applied to one selected torsion angle 4L to induce it to undergo a trans -,gauche transition. This rotation was performed in 15’ increments, starting from the 180’ trans angle and carrying out 10 cycles of steepest descent energy minimization after each increment of rotation. This generated a set of unrelaxed coordinate sets having angles 4, = 180, 195, ..., 300, and 315’. Each of these coordinate sets was subsequently relaxed by 200 cycles of minimization to allow the surrounding monolayer to accommodate the externally induced disorder, during which the angle I#J~ continued to be constrained at the designated positions. By this procedure adiabatic energy profiles were generated for disorder formation at various chain positions. The procedure was carried out for angles &, #3, +g, 411,and $q2, where angle q512 is the torsion angle at the end of the 15-carbon chain furthest from the anchored head atom. In a second study, the above procedure was performed adjacent to existing kinks. A kink was first induced by rotation of 45 in the central chain from 180 to 300’ by the above procedure. Using this as a starting coordinate set, we studied energetics of formation of a second kink by inducing a trans -,gauche transition at 49 in the same chain, producing a kink of the same rotation at the ( q 5 g , ~ l l )position. This procedure was also performed using a trans --* gauche rotation of angle 4g in the same chain to produce a second kink in the chain with an opposite rotation. In a third study, a kink was induced in the (45,47)position in an adjacent chain, forming a dovetailed pair of kinks. During these deformations and minimizations, the original kink remained in its conformation without imposing an external potential. Results and Discussion We first present the results of the study of isolated kink formation in the monolayer. The term “kink” has come to refer to the result of a “crankshaft” transition of two torsion angles 4i and 4r+2in opposite directions and has been symbolized as the transition
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ttttttt ttg+tg-tt (3) Here, t stands for trans (4 = 180’) and g+ and g stand for gauche (4 = 300’) or antigauche (4 = 60°), respectively. On the other hand, the external force used in this study to drive the formation of the kink acts only upon one individual torsion angle 4,. The chain exhibited the crankshaft cooperativity in angle 4i+2 of its own accord, in order to minimize nonbonded contacts with adjacent chains and preserve the overall directionality of the chain (26) P. B. Hitchcock, R. Mason, K. M. Thomas, and G. G. Shipley, Proc. Natl. Acad. Sci. U.S.A.,71, 3036 (1974). (27) Y.K. Levine, “X-ray Diffraction Studies of Membranes”, Pergamon Press, Oxford, 1973. (28) Y. K.Levine and M. H. F. Wilkins, Nature (London) New Bioi., 230, 69 (1971).
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3443
60
I
I
180”
210°
,
I
270’
240’
‘\
3 0”
Figure 3. Variation of torsion angles b4-& in a chain undergoing a crankshaft transition to kink state at the (&&) position. The dashed line represents an idealized crankshaft motion of 6,.
\
\
I
180’
I
1
210’
240’
270’
300‘
4 Figure 4. Adherence to ideal crankshaft motion (---) for n = 5 (0) n = 9 ( O ) , and n = 9 when a kink already exists at position (&,&) (X).
within the monolayer. This provides another piece of direct evidence for the expected crankshaft cooperativity in formation of kink disorders in monolayers. In Figure 3 the various angles are monitored in a chain undergoing a kink formation at interior angles and d7,clearly demonstrating the cooperativity expected. An idealized crankshaft motion of qb7 is indicated by the dashed line which is followed closely by the observed behavior of d7. The perturbation of other torsion angles is remarkably small. In Figure 4 the adherence to crankshaft motion is demonstrated when forming kinks at various chain positions. Note that the idealized crankshaft correlation is not followed quite as strictly near the chain ends as near the chain head but is nevertheless necessary except for angle
3444
The Journal of Physical Chemistry, Vol. 88, No. 16, I984
Northrup
71
Figure 5. Potential energy vs. q55 for kink formation at the ($+) position of three cases: (---) pure crankshaft rotation in rigid chain surroundings;).( pure crankshaft rotation of a chain in a vacuum, (0) adiabatic trans kink transition in a relaxing environment of chains.
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dll or dI2transitions. As we will discuss later, the ability of the chain to deviate from an idealized crankshaft motion has the effect of reducing the energy barrier for kink formation. The steepest descents procedure allows one to calculate the potential energy profile for the adiabatic transition of a chain from all-trans to kink conformation. First, a comparison is made of a 15-carbon chain executing a pure crankshaft motion in angles & and 4, in a vacuum, in a rigid monolayer environment (4.6 A spacing), and finally in a monolayer which is allowed to relax during the twist. The latter transition is not a purely enforced crankshaft motion but was carried out by driving 41~and allowing 47to follow of its own accord as in the previous discussion. The results are summarized in Figure 5 . The kink transition of a chain in a vacuum is simply the superposition of two simultaneous trans gauche transitions, each contributing 3 kcal/mol to the total energy barrier and each contributing a trans gauche energy difference E = Eg - Et = 1.5 kcal/mol in our model. This gives a total vacuum energy barrier of 6.0 kcal/mol and a difference of 3.0 kcal/mol between trans and kink states. The kink formation is impossible in the presence of rigid surroundings. However, if the surrounding chains are capable of undergoing relaxation, the surroundings have little effect on the net barrier to kink formation, indicating the ease at which surrounding chains may move to accommodate the induced disorder. Almost all of the energy barrier to kink formation is accounted for by the intrachain torsion energy terms. However, adherence to the crankshaft-type motion is obviously still necessitated by interchain excluded volume interactions. The chain must still cross two 3.0 kcal/mol intrachain barriers in an approximately concerted fashion to avoid developing repulsive nonbonded contacts with neighboring chains. The kink-state minimum occurs at 45 = 285’ and 47 = 75O and is less stable than the vacuum kink state by about 2 kcal/mol due to development of stress in the surroundings. The resulting barrier for the dekinking transition thus becomes only about 1 kcal/mol, rather than 3 kcal/mol for the chain in a vacuum. In Figure 6 one may observe the disposition of the strain energy into various energy components resulting from a kink disorder at position (&,&). In midtransition, virtually all of the strain energy is manifested in the soft & and d7intrachain torsion angle components and none in the much more steeply varying bond, bond angle, and excluded volume components. In the kink state, most of the energy is disposed of by bond angle deformation in the kinked chain, while almost no excess energy is present in interchain nonbonded interactions (the excluded volume portion of which
-
-
Figure 6. Disposition of strain energy into energy components as a kink is formed adiabatically at the (&,&) position: (0) torsion angle energy; (X) van der Waals energy; (e) bond angle energy.
is the most steeply varying energy term). The energy disposal in bond angle deformations may be explained as follows. A chain rotated into a kink state in a vacuum and having no other degrees of freedom distorted would be approximately 1.25 A shorter than an all-trans chain. This is accompanied by a lateral displacement of atoms which, in the presence of surrounding chains, leads to development of stress perpendicular to the chain direction. This stress is alleviated by bond angle stretching throughout the chain, causing the chain to stretch back toward its original length and decreasing lateral motion. The shift of atomic positions caused by the kink disorder at position (&,4,) is surprisingly small. The largest atom shift is in atom 8 in the kinked chain, which shifts 1.25 A largely perpendicularly to the chain direction. Atoms 7, 9, and 10 shift by 0.5-0.7 A, while all others in the main chain shift by only 0.1-0.2 A. Thus, due to the bond angle stretching mentioned earlier, the chain is not shortened significantly by kinking. This result seems to contrast with eaperimental findings of Lagaly and Weiss on related sy~tems,2~ which demonstrate well-defined chain shortening on 1.2-1.3 A decrements as the temperature is increased. This decrement is just what one would expect from kink formation. Our model, on the other hand, looks a t isolated kinks in a wellordered region which is not expanding laterally as layer width decreases. Presumably, decreases in chain length do not occur substantially for isolated kinks but when kinks occur in clusters. The atom shifts in the six neighbor chains are virtually insignificant, amounting to 0.1 A at most. These results imply that the stress in the monolayer is fully alleviated by very small atom shifts over a large number of atoms but still localized to the main chain and, to a much smaller extent, the six adjacent chains. No effects are communicated to next-nearest-neighbor chains. The potential energy profiles for kink transitions as a function of chain position are shown in Figure 7. The energy barriers for kinks from position (43,4s) to (49,411)are somewhat comparable, showing a moderate decline from 6.3 to 5.1 kcal/mol toward the free-chain end. This plateau behavior is followed by an abrupt decline to 3 kcal/mol for the trans gauche transition of (which has no corresponding cooperative partner in a chain with 12 torsion angles). This is comparable to the plateau behavior as a function of chain position observed in NMR relaxation exp e r i m e n t ~ , ~although *~~ it has been pointed out r e c e r ~ t l y ~that ~,’~
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(29) G. Lagaly and A. Weiss, Angew. Chem., Int. Ed. Engl., 10, 558 (1971). (30) M.F. Brown, J . Magn. Reson., 35, 203 (1979). (31) M. F.Brown, J. Seelig, and UbHaberlen, J . Chem. Phys., 70, 5045 (1979).
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3445
Kink Formation in Lipid Monolayers 8
n ~ 3 - 0 - 5
-D-
7:n...x...
't
9 11
6 -
a
*
u 3-
pe W
w 2-
Figure 7. Potential energy profiles for isolated kink formation at various chain locations (&,&+2).
collective order fluctuations rather than the fast local segmental rotations considered in this work may provide the dominant contribution to spin-lattice relaxation. Regardless, it is interesting to note the semiquantitative agreement with the activation energy profile vs. chain position determined by temperature studies of deuterated potassium palmitate 70% aqueous dispersions above the transition temperaturee3* There the activation energies for relaxation of carbons 2-14 in the hydrocarbon chains are essentially constant (;=5 kcal/mol), followed by an abrupt decline to 1.5 kcal/mol at the chain end (carbon 16). The primary factor determining the activation barrier to kink formation appears to be the degree of adherence to crankshaft behavior enforced by the surrounding chains. The more concerted the coupled t g+ and t g rotations are, the higher the barrier. This, in turn, depends on the volume available to the chain undergoing transition and the facility with which neighboring chains adjust their packing positions. One might imagine that in a highly disordered liquid-crystalline layer where large-scale cooperative tilting of chains is occurring, transient packing holes may allow g+ rotation followed in nonconcerted fashion an individual t by a second t g rotation to form a kink. Such transitions could thereby occur with enthalpy barriers as small as 3 kcal/mol. Such is not observed in our model, which is more applicable to transitions in a well-ordered system. We also observe the kink conformations are kinetically more stable near the chain end than near the chain head. The barrier to the dekinking transition is about 1.5 kcal/mol at positions (43,45) and (45,471 and is 2.5 kcal/mol at ('$9,411). In a second aspect of this study, we observed the effect of existing kinks on the formation of new kinks. We first discuss position by enforcing the results for kink formation at the (49,411) a d9 twist in a chain already having a kink in the (&,4~~)position. For comparison, the @9 twist was performed in the same and in the opposite sense as the existing kink. These two cases are denoted by transitions I and 11, symbolized as
Figure 8. Potential energy profiles for kink formation at the position with an existing kink at the (&,c$~)position in the same chain: ( 0 ) transition I (same rotation as first kink); transition I1 (opposite rotation as first kink); ( 0 ) isolated kink at the (q59,&1) position for comparison.
7t
- -
--
ttttg+tg-ttttt
I
ttttg+tg-tg+tg-t
I1
ttttg+tg-ttttt
ttttg+tg-tg-tg+t
(4)
Figure 8 illustrates the energy profiles for formation of the second kink and compares those with the isolated (49,4ll) kink formation. (32) J. H. Davis, K. R. Jeffrey, and M. Bloom, J . Magn. Reson., 29, 191 (1978).
(33) M. F. Brown, J . Chem. Phys., 77, 1576 (1982). (34) M. F. Brown, A. A. Ribeiro, and G. D. Williams, Proc. Natl. Acad. Sci. U.S.A.,80, 4325 (1983).
R Figure 9. Potential energy profiles for (0)isolated kink formation at the ($5r@7) position and ( 0 ) dovetailed kink formation at the same chain position adjacent to an existing kink.
By transition I the energy barrier in the presence of a kink becomes broader and lower than for an isolated kink by 1.0 kcal/mol. The broadness is due to the fact that the trans gauche transitions do not occur in concerted fashion (which would give a 6 kcal/mol barrier). This explains the somewhat bimodal nature of curve I. The crankshaft plot in Figure 4 shows the transition of lagging as much as 40' behind the pure crankshaft behavior. This is presumably due to the larger volume available for the main chain, since the adjacent chains have already been laterally displaced by the formation of the kink at (45,47).Thus, we observe an intrachain cooperativity effect in the energetics of kink formation in monolayers. The transition of type I1 has a broader barrier than the corresponding isolated kink transition but is of nearly the same height. An energetic cooperativity effect may also be observed in the formation of dovetailed kinks, i.e., kink formation in chains adjacent to existing kinks. A kink is first produced in the (45,47) position of the central chain. Then, an adiabatic minimization study is performed on the t g+ rotation in chain 2 (cf. Figure
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The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
1) on the fifth dihedral angle $J~,inducing a kink in the (&,4,) position in chain 2. Rotation of angle & in chain 2 is in the same direction as in chain 1. In Figure 9 we compare the adiabatic energy profile for isolated kink formation to kink formation which dovetails with an existing kink. There one clearly sees the interchain cooperativity effect in kink formation adjacent to existing kinks. The barrier height is lower only by about 0.5 kcal/mol, but more significantly, the kink conformation is stabilized by 1.7 kcal/mol compared to the case for the isolated kink. It requires about 5 kcal to form a mole of isolated kinks, while about 4 kcal is required if the kinks occur in dovetailed pairs. In the limit that a perfectly dovetailed cluster of kinks occurs, only 3 kcal is required to form a mole of kinks. Interestingly, this is the amount required for forming a kink in a vacuum. Note by comparison with Figure 5 that the dovetail kink conformation adjacent to an existing kink is nearly as stable as an isolated kink in a vacuum. This stabilization is presumably due to the relief of strain when the neighbor chain dovetails with the original kink. This is not an enhanced volume effect as in the intrachain case which would lead to a broader and lower bimodal barrier. Here, the shape and height of the barrier are consistent with a nearly concerted transition of chain 2 angles & and 4, in crankshaft manner. Recent ultrasonic relaxation experiments on phospholipid bi1 a y e r ~ ’ observe ~ J ~ a fast relaxation process with a time constant in the 10-20-11s range. This process reaches a maximum amplitude a t the main gel-liquid crystalline transition temperature, so is thought to be intimately associated with that transition. Furthermore, it is speculated that this fast relaxation process is the kink isomerization of lipid chains. By analyzing the temperature dependence of the relaxation time data in ref 15 using an Arrhenius plot, we obtain an activation barrier in the range 4-5 kcal/mol for these transitions. This value is greater than that expected for an isolated t g transition (3 kcal/mol) and less than the calculated maximum (6.3 kcal/mol) for isolated kinks near the polar head. If one assumes that many kinking transitions occur in cooperative pairs and clusters or near the chain ends, the activation barriers for kink formation calculated here fall in the range consistent with this experimental value and are thus consistent with the kink isomerizations.
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Summary In this paper we have described the employment of an adiabatic minimization technique to study the energetics of inducing discrete disorders in an orderly all-trans monolayer of parallel hydrocarbon chains. An external potential is used to induce a trans gauche transition in a chain. Of its own accord a correlated crankshaft transition occurs, resulting in the formation of a kink disorder. The barrier height to kink formation depends on how rigidly the chain adheres to pure crankshaft behavior. Less concerted t g, and t g.. transitions result in lower barriers. This in turn depends on volume available to the chain in a layer of other chains. Crankshaft behavior is adhered to less near the chain ends than near the polar head group, as expected. This study has determined the potential energy changes associated with kink formation in the same chain as an existing kink.
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The second kink forms with a lower, broader activation barrier and is more stable than an isolated kink. The existing kink creates a larger volume in which the remainder of the chain may form a second kink. Because of this volume availability, the latter transition departs substantially from a concerted crankshaft mode, giving rise to a lower, bimodal activation barrier. Additionally, the energetics of transition is determined for a kink which dovetails with an existing kink in a neighboring chain. The barrier to formation of the second kink is only slightly lower than that of an isolated kink, but the dovetail kink conformation is substantially more stable (by 1.7 kcal/mol) than an isolated kink. There are a number of limitations and deficiencies in this study. First, this study confines its attention to one type of chain defect. This is primarily because of the efficacy of inducing a g+tgdisorder in a chain, which is aided by the dynamical anticooperativity between torsions separated by one bond. Other defects not considered include g-tg- conformations and “jogs” (coupled defects with more than one intervening trans bond). These also preserve the overall chain direction but are not attainable by inducing a single t g, rotation, requiring rather the imposition of additional constraints. Second, the static deformation method is able to determine the energy only along the steepest descents pathway of a kink transition. The actual mechanism may occur along a more complicated collective reaction coordinate which, in fact, may be considered a “gated” mechanism,35 that is, one in which a transient packing defect occurs by chain tilting, followed g, and t g- transitions in the transient by nonconcerted t packing “hole”. Furthermore, even if the steepest descent pathway does give an accurate representation of the mechanism of kink formation, it does not provide information on the free energy change associated with such a transition. For this information, other calculational techniques must be employed such as umbrella sampling method^.^^,^^ Other limitations of this study include structural features. No account is taken of possible changes in the glycerol or polar head group conformation upon disordering of the aliphatic chains. Furthermore, the chains themselves are of unequal lengths in a typical phospholipid, and the chain planes may be mutually tilted relative to one another because of their attachment to the glycerol backbone. Finally, this analysis is limited to disorder inductions in an initially highly ordered layer of chains. In a certain sense, however, the intra- and interchain cooperativity study does anticipate features of a more disorderly system.
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Acknowledgment. I wish to express my appreciation to Michael Brown for pointing out several important references pertaining to this proposal and to Andy McCammon for useful discussions. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the Research Corporation for the partial support of this research. (35) J. A. McCarnrnon, C. Y . Lee, and S . H. Northrup, J . Am. Chem. SOC.,105, 2232 (1983). (36) S. H.Northrup, in preparation.
