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Order parameters in hydrocarbon chains
Chemical Physics 65 (1982) 259-270 North-Holland Pubii Company
OxRDER PARAIWERSIIV
CHAINS
HYDRWARRON
Ohe E&IOU4 Departrxent of TheoreticalPhysics. SIOO +# Stockholm 70, Sweden
Royal lrrsritute of Technology,
Received I6 January 198l;in fmal form 14 Oct0be11981
Order in lipid biiyers is often descriied by order parameters SCD, SHH and Scha which are rather directly obtained from deuterium NMR, proton NMR and ESR spectroscopy respectively. A relation is derived that makes one of these parameters redundant from purely geometrical seasons_ Further. the effect of conformntional disorder upon these order parameters is investigated theoretically using a tram-gauche model. Corrections from the kite width of the tram-gauche
we& and from non-fried valence angles are shown to be negligible.
1. Iutroduction Order parameters do oftenprovide a convenient way of describing order in liquid crystalline hydrocarbon chain systems. The order parameters are averages of functions of the angles characterizing the orientation of the chain or a segment of it. Some of them are rather easily accessible from spectroscopic experiments: s,,
= ;(3kos2eo)
sHJJ = :(3(cos26
- l),
(1)
HH) - 1)
(2)
and Fig. 1. The geometry in the text indicated.
The 0 are angles between respective vectors c FCH, r,H and Fcllain; seeFg1 . l)andthezaxis?the laboratory frame. S,, is obtained from deuterium nuclear magnetic resonance (NlUR) (see e.g. the review in ref. [ 1] ). Proton NMR, that gives SHH, has found some use 12-43. Fin&y electron spin resonance (ESR) by using spin labeled chains that are introduced into the bilayers gives Schain (see e.g. the review in ref. [S J ). These order parameters refer to different axes and are thus not dire&y comparable. The aim of this paper is to relate these order parameters to each other and to simple models for the chain disprder as the trans-gauche model.
of the CHz-gioup
with the vectors used
To make a comparison possible an order parameter tensor is introduced in section 2_ In section 3 these order parameters are given in terms of elements of this order parameter tensor and it is shown that there is a relation making one of them redundant. In section 4 conformational andmolecular disorder is discussed. In section 5 the conformational order parameters are expressed as functions of averages over new variables, the valence angles and the torsional rotation angles of the &bon-carbon bonds alor@ the chain. A transgauche model is introduced in section 6; and correcSons to the trans-gauche model are discussed and-
0. EdholmjOrder pammeters in h_vdrocerbon c&aim
260
shown to be negligible in section 7. As an illustration to disorder in the form of correlated rotational states along rhe chain, a -kink model is considered in section 8. Section 9 is devoted to a dis:ussion of possible conclusions from av2ilabIe experimental data.
2. Order parameters The orientation of a CHz-group may be described by giving three Euler angles R = (cr, .B.7) that rotate a coordinate system Axed to the CHZ-group into the l&oratory frame. In this work the frame f=ed to the CHJ-group has been chosen with the z-axis parallel to ‘ctin (see Eg. 1) and the x-2xis pointing in the direction bisecting the angle between the two CRbonds. A complete description of the ordering is given by the probability distribution P(a) over different orientations .Q for the CHZ-group. Equivalently, order parameters can be introduced 2; averages over rotation matrix elements, D$n(‘,(a: p, 7) (see e.g., Doane [61), s$jlI = Q$#,
8, r),,
(4)
where m, n = 0, +I, 5, . .. . rcl a?d I= 0; 1,2, . . . . -. A description of the properties of rotation matrix elements can be found e.g. in rtf. [7] _ The order parameters contain the same information as the probability distribution since this can be expanded after sotation matrix elements with coefficients that essentially are the or.der parameters
Now, the experiments considered here c%MR and ESR) are only sensitive to the orientation with respect to one axis (thzt of the applied magnetic field). This is chosen as z-axis of the laboratory frame. The x- and y-axis are not def-med and thus not the Euler angle 7 either. Then the only orde; parameters are s$;& = @&,
p, -)> = [4rr/(Z + l)] ‘RY~JjT,
a)). (6)
where Yl,,#, a) are spherical harmonics and the Euler angles 0, Q become the spherical coordinates that desc!%bc the orientation of the z-axis of thy laboratory
frame in the local coordinate system. Even now, inftitely many order parameters (m = 0, *l,+2, . . . . ti and I = 0, 1,2, .. . . -) are needed to give a complete description. Often in practice, however, it suffices with rather few. In liquid crys.taIs, the shape of the molecules is often close to that of a circular cylinder. Then there is rotational symmetry around the long axis of the molecule, which makes all order parameters with m diierent from zero equal to zero. Further, the two ends of the molecule are equivalent, thus making the order parameters with odd I equal to zero. Since S($, is constant, $2, .Si$, #, ___are the only interesting ones. Often, the higher order parameters are small and a single one remains: $3
= (4sr/S)‘Q
Y,,@)> = ;(3(cos+3>
- 1).
(7)
In reality the situation is not always 2s simple 2s this. For hydrocarbon chains, there is not a single molecular axis, the average orientation of which could describe the order, but a flexible chain. Therefore separate order parameters are needed for each chain segment. Since the hydrocarbon chtis are not shaped like circular cylinders, it is not necessary that the order parameters with fast lower index non-zero are zero. Finally the order parameters with higher I than 2 may very well be important. However, since it is second order order parameters that are detected through the spectroscopic experiments described in section 1, the attention will be focused upon them here. There 2re five of them namely, S&g, Si’a),S”&,S$ and SF&. An alternative des&-iption c&be given by the order matrix of Saupe [8], that has the elements: Sfj =
;tCOSBi COS 9j>
-
f
6ij
7
(8)
with i, j =x, y, z and tJi the angle between the Cartesian axis i and the z-axis of the laboratory system. Since (sii> is symmetric and traceless only Eve independent componen?s remain. This description is equivalent to the one above with the matrix elements related through the equations
0. EdholmfOrderparametersin hydiocnrbonchains Subtracting
S,, = -(3/8)1/2(S$;) - Sf&)
(gel
and S, = i(3/8)1/2(Sgj
t SF&).
(9f-j
The treatment using spherical instead of Cartesian order parameters seems preferable of two reasons. First it is easy to generalize to I different from 2. Secondly it becomes simpler if the distribution around the polar angle is nearly isotropic leaving only one order parameter distinctly different from zero, whiIe there are three related Cartesian ones.
261
eq. (11) from eq. (12) gives
SCD - SHH = $(3/8)1/2(S$;j
+ @_,).
(15)
Thus it is seen that the anisotropy in the distribution around the polar angle, reflected by a non-zero (S$$ f Sp&), is immediately related to the difference between the deuterium and proton NMR order parameters. Beside the order parameters considered until now, further informatiun could be gained since there are two hydrogens (or deuteriums) bonded to each carbon. ScD is taken as the average over these. If the surroundings are strongly anisotropic, it might be possible to resolve two CDorder parameters. Then, in addition to SCD, one might define;
3. Relating order parameters referring to different axes S,,-
The experimentally detectable order parameters of eqs. (lj-(3j can be expressed in terms of the order parameters Sgb : (loj
Schain = Skdy SHR = sru = -$S@
- (3/S)l/2(S$~
i-SF&),
(11)
and ScD, taken as the average over the two CD-bonds, SCD = ;s,,
+ :srr
= - +S#
- +(3/8)1/*(S$$
+ Sp$). (12)
Eqs. (IO)-(12) contain only two independent quantities S@ and (S@ + S(&) of which the second one does contain information about anisotropy of the distribution around the polar angle, Thus, analyzing experimentai order parameters in terms of those two quantities, one of the three experimental order parameters can be eliminated between eqs. (1 O)-(12) giving the relation Schain = sHH
- 3sCD-
(13)
This relation must be fulfilled provided the averages in eqs. (l)-(3) are averages over the same probability distribution. When the distribution around the polar angle is isotropic, (S$ij + SF&,) is zero, and the commonly used formu!a (see Seehg and Niederberger [9]) is regaiued: Schain = -2scD
’ -2sHH.
(14)
= f (SC& - &s)-
(16)
ScD can be expressed in terms of elements of the order parameter tensor SCD- = -i( 1/3)1/2(S$ij - SF&) = (1/2)1i2(sin2flsin
2a)= s&S
_XJJ .
(17)
In most cases only one CD order parameter can be resolved corresponding to SCD- being zero. SCD- different from zero means that the probability distribution for the orientation of the CH,-group contains a part that is antisymmetric in the polar angle. Thus the experimental order parameters give only three of the five independent second order order parameters. S@ and SF&, or equivalently S,, and SXZ are not accessible through the experiments described in section I_ Eq. (13) gives a relation between three order parameters. Two of them are independent, but they cannot vary totally freely- The simultaneous variation of the order parameters is limited since they are obtained by taking averages over the same probability distribution, that for the orientation of the CH2group. By eqs. (6), (11) and (12) and formulae for the spherical harmonics ScD and SFiN can be expressed as integrals over the same probability density P(x, CZ) (withx = cos p) for the orientation of the CHZ-group: SCD= i ~/(x,u)P(x,c)dxdcr, -10
(18)
262
0. Edkolm/Ordw
prometers
(1%
in kyiroaubon
ckuins
Schain
with f(x, CY) = -$(3xX’ - 1) - $ (1 -x”)
cos 2u
and g(X, ff) = -$(3x’
- I) - $ (I - 9)
X6 20.
(21)
Now, the probability dens&l is varied to find the maximum and mkimum of SHY A under the constraint that SCD is kept fixed. The optima are reached when alI the probability density is concentrated to the point where g(x, a) is extreme under the constraint that fix, a) has a given value. This gives after some algebra the following restrictions upon the simultaneous vaiation of Scr, and SHH: ‘HH,
s,,
b-1
for -$ 4ScD
29
2 3&-
- :,
< 0;
(22)
forO<~CD+
(23)
and SHHG$ISCD+~,
for -+
S$.
(24)
Thus :he order parameters S,, and S,, have to keep within the triangle shown in fig_ 2. By eq. (13) SHH may be replaced by Schain in eqs. (E)-(X) giving similar restrictions to the simultaneous variation of ScD and Schain- This is shown in fig_ 3. There are sirrilar restrictions upon the simultaneous variation ofS,, and Schnin. These restrictions are consequences of the geometry
‘HH
Fig. 2. Thr simultaneous vti~ation of ?he order parameters SC* and SHH is restricted to the triangle in this figure.
Fi_e 3. The simultaneous variation of the order parameters sCD and %hain is restricted to the triangle in this figure.
(tetrahedral) of the hydrocarbon group. Since ichain, form futed angles with one another, the & and &H _ reorientation of the CH2-group cannot make the order parameters vary totally freely of one another. Further, it should be noted that SCD is restricted to always be less than OS. This is a consequence of the fact that it is the average over two order parameters that both cannot be one at the same time.
4. Conformational and mokcular disorder Disorder in these systems may be of two types, either conformationa! disorder caused by the hydrocarbon chains being out of their ordered all trans state or molecular disorder caused by different orientations of the whoIe molecules. Order parameters reflect both these types of disorder and it is a far from easy problem to find out what type of disorder is reflected by experimental order parameters. Bumell and de Lange [IO] have considered the joint effect of molecular and conformational disorder under qtiite general circumstances. For the most general case when the two types of disorder are closely intermingled this becomes a formidable problem. It does, however, simplify under some assumptions. First they consider the weak collision approximation, that is they assume that a conformational chage does not cause any immediate change of molectilar order. Secondly they consider the two limiting time scales of slow and rapid interconversion.
0. Edholm/Order
In the first exe the system comes to an equilibrium over the different molecular orientations much faster than over the different conformations. Then a separate set of order parameters describing the molecular order is necessar- for each conformation. These sets may of course be very close to each other if the properties of the different conformations as regards interactions with neighbouring chains are not very different. Even if the conformational order parameters are different for at least some conformations, this might be of less importance if those conformations are uncommon. In the opposite limit (fast interconversion) one comes to an equilibrium over the conformations much faster than over the different orientations of the whole molecule. Then only one set of molecular order parameters is necessary. However, it seems to me that one then in principle would need a separate set of conformational order parameters for each orientation of the molecule. This is not discussed in ref. [lo]. Depending upon the precise properties 6f the system, most of these might cf course be redundant of sirr&u reasons as those given for the molecular order parameters above in the slow interconversion case. It is not clear which, if any, of these Iimits apply to the lipid bilayer case. In ref. [IO] it is stated that the fast interconversion limit cannot be completely satisfied for these systems and that the slow interconversion limit might be better approximation. To go further into this business one needs a detailed description of interchain and head group interactions, which hardly is available today. Here, fust in section 5 the case of conformational changes of a free chain will be considered, while in section 8 a kink model is treated. The latter is a very simple way of roughly including some of the effects of interchain interactions. When such simple models are used for the conformational order, it is not meaningful to do the comp!icated work necessary to take into account the coupling between conformational and molecular order in a proper way. I have therefore assumed, that the conformational order parameters are the average ones and that they are not affected by such things as molecular order or the actual (not average) interchain distance which might as weU be impor-
tant. Here the main concern will be to investigate the effect of conformational disorder. Therefore the order parameters
263
parameters in hyrlrocarbon chains
are defined with respect to a frame having
its z-axis parallel to the ali tram chain, thus giving&’ equal to one and the other four second order order
parameters zero for all the CHz-groups of the a tram chain. To include the effect of molecular disorder it is necessary to perform another transformation to a laboratory fmed coordinate system. This involves the averaging over a probability distribution over the Euler angles defining the orientation of the all-tram chain with respect to this frame. The order parameters LZ& are then obtained in terms of conformational and molecular order parameters
In this general case all components (m, n = 0,&l, 22, . .. . 9~ of the order parameter tensorSgronf are needed. Specializing to molecular disorder which is hotropic around the polar angle, this simplifies to
5. Conformational
disorder
Since the order parameters defined in section 1 are related to three corn onents (M = 0, ?2) of the crder parameter tensor Sli t: through eqs. (lo)--(12), it suffices to study these. They are, as discussed in section 2, related to an Euler angle description of the orientation of the CHz-group. For studying conformational disorder there are, however, better variables than the
Euler angles. I: is natural to use, for each CHxgroup, the valence angle 8 (i.e. the angle between the studied carbon-carbon bond and the preceding one) and the torsional angle Q (the angle of rotation around the carbcn-carbon bond). A coordinate system for each CHa-group is introduced as in section 2, with the and the x-axis bisecting the Z-axiS parallel t0 ‘chain angle between the two CH-bonds (see fig. 1). Then some space geometrical considerations give the Euler angles for the transformation between the coordinate systems adhering to two consecutive CHZ-groups in terms of these two angles: fY=~=arccos
1
sin(6/2)(1 - cos @)
[sit& + sir@/2)(1
- cos #I
U21 (27)
204
a EdtiolmfOrder pmmeters ix hydrocarbon ciztzins
Table 1 Decomposition of the Wiier a+diion maim according 10 eq. (19)
A3 =
0. Edholtn jO&r
parame:ers in hydrocarbon chains
and B = arccos[l
- cosqe/2)(r
- cos $)>I‘.
26.5
the rotational states of different carbon-carbon bonds upon the chain. At the free end of the ch&r this approximation is probably closer to reality thr?n in the upper and centra1 parts.
(28)
Thus the Wigrier rotation matrix can be expressed as
a function of these angles instead of the Euler angles: 6. The kans-gauche
model
(29) where Al, A,, Ax, 61 and 6, are real 5 X 5 matrices that only depend upon the &ence angle. They are given in fable 1 as a function of the parameters ,$= cos@/2) 2nd n = sin(6/2)The conformational order parameter tensor of CH2-group number (IV+ 1) is the average of the rotation matrix transforming between the local frame of that CH1-group and that of the f&t CHz-group. (The Zaxis of the latter frame is parallel to the all trans chain.) This rotation matrix can be factorized into N rotation matrices transforming between the frames of the intervening CH2-groups. In the general case, the average is to be taken over a probability distnbution ofNvaIence angles and N torsional angles. Now the simplifying approximation is made-that the orientation of each CHz-group with respect to the one closest above is independent of the configuration of the rest of the chain. Then it is possible to factorize the probability distribution and one gets
As a further speciahzation, the trans-gauche model (see e.g. Glory [ 12:) is now introduced. This t?xes the valence angles to be at the most probable value, 0, (approximately equal to the tetrahedral angle). Further is restricts the torsional angles to three discrete values corresponding to the tram and gauche* states. Both these assumptions will be abandoned in section 7 and it will there be shown (keeping the assumption of independent rotations) that these approximations iutroduce only negligib!e errors. In the tram-gauche model, the matrices A and B of eq. (29) and table 1 are constant. The avemges over the torsional angle, @,can be expressed as functions of the trans probability, pt. ‘Withthe tram state at torsional angle zero and the two gauche states symmetrically at ?120’, the averages become: &n9)=!sin2@,)=0
(31)
and (cos (6) = (cos 2@) = $(3p, - 1).
pt cornbe expressed in terms of the difference in energy between the gauche and tram states, Em: _
(30)
pt = [ 1 + ?exp(-Ets/liBT)]
This is a questionable approximation for systems with as important interchain interactions as those of a lipid bilayer. As further discussed in section 8, the interchain interactions will force correlations between
Table 2 The different o:der parameters probability @$
after having transformed
(32)
-I.
(33)
With Etg = 05 kca!/mol, pt becomes about 0.5. Now, inserting eqs. (3 1) and (32j into eq. (29) the average of the rotation matrix for one transformation along the chain becomes linear inpt. Since by eq. (30j the
a number of steps (N = 0, 1, 2) along the chain as a function
N
s chain
‘CD
‘HH
0
1
&-I +9pt)
-$
--$
1 2
+e(-5
1
+ 64 + 15~;) -
+Zp*)
-TPt
-f(l
$(I - 2Pt - P:,
$5(7 - 18Pt + 3Pf)
:
of the trans
.5 0~
1
2
a 5
Fig. 6. S$’ as a function of tmns probability for different chain positions (iindicated). Compare with tig. 4. Y z, function of chair position for difFig. 4. &) = L&n ferent tmns probabilities azwming independent rotations
order parameter tensor of CH2-group number (IV+ 1) is obtained as the matrix product N averaged rotation matrices, the order parameters are polynomials of degree Nin pt. In table 2 these are given for some order parameters forN less or e ual to two. For higherNthe order parametersS~~ and@ =S(2& have been evaluated numerically as a function o:fiv and pt. See figs. 3-7. An important featttre of the results is that for reasonable values of Pr (around OS), the order parameters decrease rapidly with chain position. Such a rapid decrease is not observed for NMR order param-
15
eters in lipid bilayers in the upper or middle parts of the chain but only close to the free end. This shows that an independent rotation model does not work in the upper and central part of lipid chains, but might be a better description at the free end. Further, there is an even-odd effect for smahN and pt not too large. Something like that can be observed in experimental NMR order parameters (see e.g. fig 3 of ref. [17]). For completeness the remaining order parameter S# = -Splo is given for the trans.-gauche model making one transformation along the chain (although this order parameter is not accessible through the experiments described in section 1)
10 15 chain position
Fig. 5. S$,’ = ‘&/378(S CD - S,,) as a function of chain ~osieon for different inns probabilities assumingindependent roktions
Fig. 7. S$’ as a function of tram probability for different chtin positions (indiated). Compare with tis 5.
0. EdholmjOrder parameters in hydrombm chains 7. Corrections to the kans-gawhe
267
mode!
A better descriptid;? than the trans-gauche model can be given by restricting the torsional angles by a potential having minima at the.trans and gauche states. Such a potential is shown in fig. 8. An often used expression is (see e.g. Volkenstein [13]): U(@)=U*/2[1 --x cos @-(I
-x)cos3l$].
(35)
Here x is a measure of the difference in energy, Etg, between the gauche and trans states. U, is the height of the barrier between the gauche wells. Reasonable values are here for UC 12 kJ/mol and for x 0.22 corresponding to Etg equal to about $ of UO. The potential is symmetric around Q equal to zero so kin @>and (sin 29) remain zero. If U, tends to infinity with _!?,a (not x> fjed, the trans-gauche model is recovered (although, of course the transition probabilities between the states become infinite in this manner). The tram probability is now defmed as the probability for being in the central well: (36) with k& being the angle at which the barrier between the trans and gauche wells has its maximum. Ev2luatin, the integrals in eq. (36) numerically for different values of the parameters of the potential,x and Uo/kBT, givesp, as a function of those two parameters. pt can thus replacex as one of the parameters of the poten&I. In this way kos 4s) and kos 21$) are calculated as functions of Pt and Uo/ksT. The result is shown in
Fig. 9. The variation of kos @)and tcos 3) with the trzns probability @t)_The different cmves refer to different potential barrier heights U,-,/kgT(4,6 and 8). The tram-gauche mode! gives the limiting straight line. fig. 9 with pt on the abscissa for Uo/kBT equal to 46 and 8. The straight line is given by eq. (32) and corresponds to infinite Uo, that is the tram-gauche model. The difference between the tram-gauche value of the averages and their actual v&e increases with decreasing Uo/kBT and increasingpt. The difference is larger for kos 29) thm for kos 0). Realistic values for the parameters are around 0.5 for pt and 4.5 for Uo/kB T. If one so wishes, two new order parameters, varying between -0.5 and 1 and 0 and 1 rsspectively, can be introduced: S t,=%,-
(37)
f,
and (38) St9 does then describe the disorder due to different populations in the trans and gauche states, while St,, describes disorder caused by the fact that the torsional angles are spread a bit around the minimum values of the wells. Further, the valence angle is allowed to vary, which makes the matrices A and 6 that appear in eq. (29) iuto variables. This variation is restricted by a potential Fig. 8. Schematic view of the potential hindering the torsional zotations. Ihe analytic form ke.g. that of eq. (35).
Y(8) = y&cos
0 - cos 6”)Z )
(39)
0. EdholmjOrderparamete.sin
268
discussed. Then, the only deviations from the straight aI tram chain that are allowed are kinks, that is
with 7e around 200 kf/mol (see e.g. Heifand et al. 2141) and 8, the most probable value (which could be put approximately equal to the tetrahedral angle, 109.5”). Thus, as a third order parameter one might introduce S val
=
re!(re
gauche’ trans gauche’ conformations. Such a conformation does only make a small excursion from the main direction of the chain and does then continue in the same direction as before. Thisminimates interchain interactions. Transforming the coordinate system through 2 kink is made by use of the rotation matrix
(43)
’ 2kg q-
Realistic values upon the three thus introduced order parameters are S,, = 0.25, St,, = 0.80 and Sval = 0.98 The influence of these order parameters upon spherical or NMR order parameters between different positions in the chain is not directly cIear but has to be investigated by numerical calculations The result is shown in table 3. There, it can be seen that the corrections are negligible and that one is indeed safe using 2 tram-gauche model. However, it must be stressed that these results are only valid under the assumption that the disorder is uncorrela?ed at different positions along the chain. Strondy correlated small scale disorder may very well give a large total result when added together.
8. Correlated
D$&_
=D@ =0u, @=2;r/3jD@
= [Al - ;(A,
+ Aj) + i(fi/2)&
X [A, - ;(A2 t A,) - i(a/2)(B2 The matrix multiplications
1 2 3 4 5 6
(Dc2) g+rg _)llnl = &,a(-l)“,
a) pt = 0.5and yg/2kgT
= 40.
b)pt=O.jandL’
(41)
n, m = 0,+I,22. (42)
This matrix describes a coordinate transformation, where the direction of the z-axis is kept while ‘Jle direction of thex-axis is turned 180”. This is exactly the result of a coordinate transformation through three all trans bonds and thus the kink does not change the orientation of CH~_groups outside the kink. For the two CH?-groups between the trans bond and the gauche+ and gauche- bonds respectively, the orientation does, however, change and it changes in the same way as if there where a single gauche bond preceding
Coxections from nonconstant valence angles a)
0.0082 0.0085 0.0027 0.0027 0.0002 0.0001
- B&j.
give
to the order parameters
s$
- Bj)]
X(Al+A,+A31
disorder
Fosition
= 0, Q = 0)
XD@=e,,+=--2x/3)
As stated earlier there is reason to believe that the rotations about different bonds in the lipid hydrocarbon chain are correlated. Further arguments for this are given by Edholm and Blomberg [ 151. There, also a simple model for the correlations, introduced originally by Trluble and Haynes [IS], the kink model is Table 5 Corrections
hydrocarbonchains
s!;’
-0.0023 -0.0002 -0.0004 o.ooo-: -0.0001 0.0002 ‘k T=4 . O/B
Corrections due to that the torsional angIes do net stick to the transgauche vah~es b) S$)
s!‘,
-0.0025 -0.0119 -0.0083 -0.0031 -0.0036 0.0001
-0.0149 -0.0227 -CL0077 -0.0055 0.0017 -0.0001
it. Thus, the results of section 6 for the transgauche model with N= 1 can be immediately used. Since there are three C&-groups to take the average over in 2 kink, but only two of them have another orientation in the kink than in the all tram chain, pt has to be replaced by (I - $u). pbd is then the probability for a certain CHz-group to be part of a kink (+ or -). (This probability must vary sIowly along the chain for the result to be meaningful.) A kink energy of twice the gauche energy (2 X 0.5 kcal/mol) givespkink about 0.3. The kink energy could be even larger, due to interchain interactions. This would give an even smaller pEti. Thus (1 - $pkink) becomes much larger than what pt would become in the tram-gauche model by ordinary Boltzmann statistics.
9. Comparison
with experiments
and discussion
The difference between the order parameters S,--D and SHH does give immediate information about the anisotropy around the polar angle according to eq. (1.5). IJnfortunately the experimental information concerning this difference is ambiguous. Petersen and Chan [IS] give SCD = -0.20 t 0.01 and S,, = -0.17 i 0.04 (Seiter and Ghan [2]). This gives 2 difference in eq. (15) of -0.03 + 0.05. Higgs and McKay [4] compare SHH values they have measured to SCD values of Davies and Jeffrey f17] and have at 60°C (fig. 5 of ref. [43) SHH = -0.27 and SC, = -0.23, that is a difference of 0.04 having opposite sign. Vaz and Deane [ 111 do also find an anisotropy of order parameters around the polar angle. Their order parameters do, however, refer to 2 different coordinate system. The parameter they use as a measure of the anisotropy depends therefore also upon other order parameters than (S!$)+S~~a)(S$ among else) and is thus not immediatdy relevant fey the ccmparison here. Petersen and Ghan [ 181 try to interpret the order parameters by 2 tram-gauche model, transforming just one step along the chain. That is they take only into account the effect of the tram-gauche state of the immediately preceding bond. The effect of disorder at bonds higher up in the chain is accounted for by averaging over an isotropic disrribution over all directions of the all tram chain within a cone of half top angle A@_This corresponds to adding molecular disorder 2s described by eq. (26). With the conforma-
tional order parameters
of the second row (N = 1) of
table 2 and S&)mol = S,, this $ves SCD = -&J&
(43)
and
+D - &iH = i(’
- pt)s,fl’
(44)
with SAP =
$0, A@(1+ cos A@).
(45)
The difference between the order parameters in eq. (44) is always positive. This comes into conflict with the negative experimental value of ref. [ 181. However, it is possible to get within the experimental accuracy with a small positive (less than 0.02) difference. Since the product S,,p, is @en by eq. (43) to be about 0.40, eq. (44) ieads to a restriction upon S,e. S,e needs to be less than about 0.56 to have SCD - SHH less than 0.02 and SC0 about -0.20 at the same time. However, such a small S, gives by eq. (43) that pt must be larger than about 0.7. To get such a large pt there must be other restrictions making gauche conformations more uncommon than the internal chain tram-gauche energy of OS kcal/mol does. However, it should be noted that these arguments of Petersen and Char-r [ 1S] fit very well into a kink model. The high trans probability gets an explanation. Further, a reason is provided for doing just one coordinate transformation along the chain not as many as are needed to get to the first CH,-group. It should be noted that by do&g more coordinate transformations than one there is no problem of reducing the order parameter expressing the anisotro y P around the polar angle. The problem is, that also Soi) decreases, and that it does so far too much. Finally, the results of Higgs and McKay [4] give 2 larger positive difference between SOD and SHH, which is easier to reconcile with trans-gauche model without having a small SAP. The ESR order parameters are hard to reconcile with NMR order parameters. McFarland and McConnell [ 191 give Schain (for multilamellar bilayers) by ESR to be 0.68 for the 12th carbon atom from the end of the chain, 0.50 for the 9th, 0.33 for the 7th and 0.13 for the 3rd. Comparing this to NMR order parameters by eq. (13), it is seen that neither for the 12th or for the 9th carbon atom are the results consistent. In general ESR order parameters seem to vary much more
0. EdhohjOrder parametersin i@racarbo~ ciurins
270
with chain position than the NMR ones. The plateau re@on in the upper and middle parts cf the chain, where &!!lR order parameters are rather independent of chain position, seem often to be missing in the ESR CaSe.
eq.
(13) instead of the simpler eq_ (14) may change the possibility to get agreement a bit. With S,, - S,, less than zero Schain becomes larger than in the case when there is isotropy around the polar angle. For instance SCD = -0.30 and S - HK -_ _woH~~S-~~” ._ a. @veSchain = 0.70 wwe s,, -s s chllin only 0.50.1~ only one Mv!R order pararneter is avtiable, sayScD, and upper limit for Sc,, could anyhow be given by the restrictions of fig. 3 Using
However. the available experimentaIinformation does enough anisotropy or even anisotropy of the correct sign to explain ESR order parameters. The data of Petersen and Chan [ 181 give Schain = Q43 + 0.07 and the data of Higgs and McKay [4] give 0.43. By eq (46) SC, restricts Schain to be s less than 0.55 and 0.66 respectively. If there is not agreement, the only possible explanation is that the averaging occurs over different probability distributions in the NMR and ESR cases. This in turn can have two reasons that hav,o been connot give large
chain
=
sidenbly debated in tie literature. First, spin probes mi&t disturb the order in the ESR case. Secondly,
since the two techniques are sensitive on different time scales, the difference might be due to the fact that the order parameters reflect ordering on different time scales.
Acknowledgement The author wants to thank Dr. Clas Blomberg for helpful discussions.
References [ I]
1. SeeIig and A. Seclig, Quart 19.
Rev.
[2] CH.A. [3]
Biophys
Seiter and 8.1. C&m, J. Am Chea 0973) 7541. XA Hemming and H.J.C Berendsen, J.
S (1972) 133. [4] T.P. H&s and kL
McKay, Chem
13 (1980)
Sot
95
Mago. Reson.
Phys. Lipids 20
(1977) 105. [S] S Schreier, CF. Polnaszek
and LCP. Smith, Biochim. Biophys Acta 515 (1978) 375. [6] J.W. Deane, in: Magnetic resonance and phase tramitions, e&F-J. Owens, CP. Poole Jr. and H.A. Faxach
(Academic Press, New York, 1979) pp_ 171 ff. [7] D.&I.Brinck and G.R. Satchler, Angular momentum (Oxford University Press, London, 1972) [8] -4. Saupe, Z Naturforsch. 19a (1974) 161. [9] J. See& and W. Niederberger, J. Am Chem. Sot 96 (1974) 2069. [fO] EE. BurnelI and Ck de Lange, J. hfag. Reson. 39
(1,080) 461. Ill]
N.A.P. Vaz and J.W. Doanne, Phys Rev. A22 (1980)
2238. [12] P. Glory, Statistical mechanics of chain molecules Wiley, New York, 1969). [ 131 M.V. Volkenstein, Confi_gu&ional statistics of polymeric c&ins (Wiley, New York, 1963). [14]
E. Helfand, ZR. Wasserman and T.A. Weber, Macromolecules 13 (1980) 526. [ 151 0. Edhohn a;ld C. Blombers Chem. Phys. 53 (1980) 185. [16] H. TGuble and A. Haynes, Chem. Phys. Lipids 7 (1971) 324. [17] I.E. Davies and K.R. Jeffrey, Chem. Phys. Lipids 20 (1977) 87. [IS] N.O. Petersen and.S.1. Ghan, Biochemistry 16 (1977) 2657. [19] B.C. McFarland and H.M. McConneU. Proc. Natl. Acad. Sci. USA 68 (1971) 1274.
OxRDER PARAIWERSIIV
CHAINS
HYDRWARRON
Ohe E&IOU4 Departrxent of TheoreticalPhysics. SIOO +# Stockholm 70, Sweden
Royal lrrsritute of Technology,
Received I6 January 198l;in fmal form 14 Oct0be11981
Order in lipid biiyers is often descriied by order parameters SCD, SHH and Scha which are rather directly obtained from deuterium NMR, proton NMR and ESR spectroscopy respectively. A relation is derived that makes one of these parameters redundant from purely geometrical seasons_ Further. the effect of conformntional disorder upon these order parameters is investigated theoretically using a tram-gauche model. Corrections from the kite width of the tram-gauche
we& and from non-fried valence angles are shown to be negligible.
1. Iutroduction Order parameters do oftenprovide a convenient way of describing order in liquid crystalline hydrocarbon chain systems. The order parameters are averages of functions of the angles characterizing the orientation of the chain or a segment of it. Some of them are rather easily accessible from spectroscopic experiments: s,,
= ;(3kos2eo)
sHJJ = :(3(cos26
- l),
(1)
HH) - 1)
(2)
and Fig. 1. The geometry in the text indicated.
The 0 are angles between respective vectors c FCH, r,H and Fcllain; seeFg1 . l)andthezaxis?the laboratory frame. S,, is obtained from deuterium nuclear magnetic resonance (NlUR) (see e.g. the review in ref. [ 1] ). Proton NMR, that gives SHH, has found some use 12-43. Fin&y electron spin resonance (ESR) by using spin labeled chains that are introduced into the bilayers gives Schain (see e.g. the review in ref. [S J ). These order parameters refer to different axes and are thus not dire&y comparable. The aim of this paper is to relate these order parameters to each other and to simple models for the chain disprder as the trans-gauche model.
of the CHz-gioup
with the vectors used
To make a comparison possible an order parameter tensor is introduced in section 2_ In section 3 these order parameters are given in terms of elements of this order parameter tensor and it is shown that there is a relation making one of them redundant. In section 4 conformational andmolecular disorder is discussed. In section 5 the conformational order parameters are expressed as functions of averages over new variables, the valence angles and the torsional rotation angles of the &bon-carbon bonds alor@ the chain. A transgauche model is introduced in section 6; and correcSons to the trans-gauche model are discussed and-
0. EdholmjOrder pammeters in h_vdrocerbon c&aim
260
shown to be negligible in section 7. As an illustration to disorder in the form of correlated rotational states along rhe chain, a -kink model is considered in section 8. Section 9 is devoted to a dis:ussion of possible conclusions from av2ilabIe experimental data.
2. Order parameters The orientation of a CHz-group may be described by giving three Euler angles R = (cr, .B.7) that rotate a coordinate system Axed to the CHZ-group into the l&oratory frame. In this work the frame f=ed to the CHJ-group has been chosen with the z-axis parallel to ‘ctin (see Eg. 1) and the x-2xis pointing in the direction bisecting the angle between the two CRbonds. A complete description of the ordering is given by the probability distribution P(a) over different orientations .Q for the CHZ-group. Equivalently, order parameters can be introduced 2; averages over rotation matrix elements, D$n(‘,(a: p, 7) (see e.g., Doane [61), s$jlI = Q$#,
8, r),,
(4)
where m, n = 0, +I, 5, . .. . rcl a?d I= 0; 1,2, . . . . -. A description of the properties of rotation matrix elements can be found e.g. in rtf. [7] _ The order parameters contain the same information as the probability distribution since this can be expanded after sotation matrix elements with coefficients that essentially are the or.der parameters
Now, the experiments considered here c%MR and ESR) are only sensitive to the orientation with respect to one axis (thzt of the applied magnetic field). This is chosen as z-axis of the laboratory frame. The x- and y-axis are not def-med and thus not the Euler angle 7 either. Then the only orde; parameters are s$;& = @&,
p, -)> = [4rr/(Z + l)] ‘RY~JjT,
a)). (6)
where Yl,,#, a) are spherical harmonics and the Euler angles 0, Q become the spherical coordinates that desc!%bc the orientation of the z-axis of thy laboratory
frame in the local coordinate system. Even now, inftitely many order parameters (m = 0, *l,+2, . . . . ti and I = 0, 1,2, .. . . -) are needed to give a complete description. Often in practice, however, it suffices with rather few. In liquid crys.taIs, the shape of the molecules is often close to that of a circular cylinder. Then there is rotational symmetry around the long axis of the molecule, which makes all order parameters with m diierent from zero equal to zero. Further, the two ends of the molecule are equivalent, thus making the order parameters with odd I equal to zero. Since S($, is constant, $2, .Si$, #, ___are the only interesting ones. Often, the higher order parameters are small and a single one remains: $3
= (4sr/S)‘Q
Y,,@)> = ;(3(cos+3>
- 1).
(7)
In reality the situation is not always 2s simple 2s this. For hydrocarbon chains, there is not a single molecular axis, the average orientation of which could describe the order, but a flexible chain. Therefore separate order parameters are needed for each chain segment. Since the hydrocarbon chtis are not shaped like circular cylinders, it is not necessary that the order parameters with fast lower index non-zero are zero. Finally the order parameters with higher I than 2 may very well be important. However, since it is second order order parameters that are detected through the spectroscopic experiments described in section 1, the attention will be focused upon them here. There 2re five of them namely, S&g, Si’a),S”&,S$ and SF&. An alternative des&-iption c&be given by the order matrix of Saupe [8], that has the elements: Sfj =
;tCOSBi COS 9j>
-
f
6ij
7
(8)
with i, j =x, y, z and tJi the angle between the Cartesian axis i and the z-axis of the laboratory system. Since (sii> is symmetric and traceless only Eve independent componen?s remain. This description is equivalent to the one above with the matrix elements related through the equations
0. EdholmfOrderparametersin hydiocnrbonchains Subtracting
S,, = -(3/8)1/2(S$;) - Sf&)
(gel
and S, = i(3/8)1/2(Sgj
t SF&).
(9f-j
The treatment using spherical instead of Cartesian order parameters seems preferable of two reasons. First it is easy to generalize to I different from 2. Secondly it becomes simpler if the distribution around the polar angle is nearly isotropic leaving only one order parameter distinctly different from zero, whiIe there are three related Cartesian ones.
261
eq. (11) from eq. (12) gives
SCD - SHH = $(3/8)1/2(S$;j
+ @_,).
(15)
Thus it is seen that the anisotropy in the distribution around the polar angle, reflected by a non-zero (S$$ f Sp&), is immediately related to the difference between the deuterium and proton NMR order parameters. Beside the order parameters considered until now, further informatiun could be gained since there are two hydrogens (or deuteriums) bonded to each carbon. ScD is taken as the average over these. If the surroundings are strongly anisotropic, it might be possible to resolve two CDorder parameters. Then, in addition to SCD, one might define;
3. Relating order parameters referring to different axes S,,-
The experimentally detectable order parameters of eqs. (lj-(3j can be expressed in terms of the order parameters Sgb : (loj
Schain = Skdy SHR = sru = -$S@
- (3/S)l/2(S$~
i-SF&),
(11)
and ScD, taken as the average over the two CD-bonds, SCD = ;s,,
+ :srr
= - +S#
- +(3/8)1/*(S$$
+ Sp$). (12)
Eqs. (IO)-(12) contain only two independent quantities S@ and (S@ + S(&) of which the second one does contain information about anisotropy of the distribution around the polar angle, Thus, analyzing experimentai order parameters in terms of those two quantities, one of the three experimental order parameters can be eliminated between eqs. (1 O)-(12) giving the relation Schain = sHH
- 3sCD-
(13)
This relation must be fulfilled provided the averages in eqs. (l)-(3) are averages over the same probability distribution. When the distribution around the polar angle is isotropic, (S$ij + SF&,) is zero, and the commonly used formu!a (see Seehg and Niederberger [9]) is regaiued: Schain = -2scD
’ -2sHH.
(14)
= f (SC& - &s)-
(16)
ScD can be expressed in terms of elements of the order parameter tensor SCD- = -i( 1/3)1/2(S$ij - SF&) = (1/2)1i2(sin2flsin
2a)= s&S
_XJJ .
(17)
In most cases only one CD order parameter can be resolved corresponding to SCD- being zero. SCD- different from zero means that the probability distribution for the orientation of the CH,-group contains a part that is antisymmetric in the polar angle. Thus the experimental order parameters give only three of the five independent second order order parameters. S@ and SF&, or equivalently S,, and SXZ are not accessible through the experiments described in section I_ Eq. (13) gives a relation between three order parameters. Two of them are independent, but they cannot vary totally freely- The simultaneous variation of the order parameters is limited since they are obtained by taking averages over the same probability distribution, that for the orientation of the CH2group. By eqs. (6), (11) and (12) and formulae for the spherical harmonics ScD and SFiN can be expressed as integrals over the same probability density P(x, CZ) (withx = cos p) for the orientation of the CHZ-group: SCD= i ~/(x,u)P(x,c)dxdcr, -10
(18)
262
0. Edkolm/Ordw
prometers
(1%
in kyiroaubon
ckuins
Schain
with f(x, CY) = -$(3xX’ - 1) - $ (1 -x”)
cos 2u
and g(X, ff) = -$(3x’
- I) - $ (I - 9)
X6 20.
(21)
Now, the probability dens&l is varied to find the maximum and mkimum of SHY A under the constraint that SCD is kept fixed. The optima are reached when alI the probability density is concentrated to the point where g(x, a) is extreme under the constraint that fix, a) has a given value. This gives after some algebra the following restrictions upon the simultaneous vaiation of Scr, and SHH: ‘HH,
s,,
b-1
for -$ 4ScD
29
2 3&-
- :,
< 0;
(22)
forO<~CD+
(23)
and SHHG$ISCD+~,
for -+
S$.
(24)
Thus :he order parameters S,, and S,, have to keep within the triangle shown in fig_ 2. By eq. (13) SHH may be replaced by Schain in eqs. (E)-(X) giving similar restrictions to the simultaneous variation of ScD and Schain- This is shown in fig_ 3. There are sirrilar restrictions upon the simultaneous variation ofS,, and Schnin. These restrictions are consequences of the geometry
‘HH
Fig. 2. Thr simultaneous vti~ation of ?he order parameters SC* and SHH is restricted to the triangle in this figure.
Fi_e 3. The simultaneous variation of the order parameters sCD and %hain is restricted to the triangle in this figure.
(tetrahedral) of the hydrocarbon group. Since ichain, form futed angles with one another, the & and &H _ reorientation of the CH2-group cannot make the order parameters vary totally freely of one another. Further, it should be noted that SCD is restricted to always be less than OS. This is a consequence of the fact that it is the average over two order parameters that both cannot be one at the same time.
4. Conformational and mokcular disorder Disorder in these systems may be of two types, either conformationa! disorder caused by the hydrocarbon chains being out of their ordered all trans state or molecular disorder caused by different orientations of the whoIe molecules. Order parameters reflect both these types of disorder and it is a far from easy problem to find out what type of disorder is reflected by experimental order parameters. Bumell and de Lange [IO] have considered the joint effect of molecular and conformational disorder under qtiite general circumstances. For the most general case when the two types of disorder are closely intermingled this becomes a formidable problem. It does, however, simplify under some assumptions. First they consider the weak collision approximation, that is they assume that a conformational chage does not cause any immediate change of molectilar order. Secondly they consider the two limiting time scales of slow and rapid interconversion.
0. Edholm/Order
In the first exe the system comes to an equilibrium over the different molecular orientations much faster than over the different conformations. Then a separate set of order parameters describing the molecular order is necessar- for each conformation. These sets may of course be very close to each other if the properties of the different conformations as regards interactions with neighbouring chains are not very different. Even if the conformational order parameters are different for at least some conformations, this might be of less importance if those conformations are uncommon. In the opposite limit (fast interconversion) one comes to an equilibrium over the conformations much faster than over the different orientations of the whole molecule. Then only one set of molecular order parameters is necessary. However, it seems to me that one then in principle would need a separate set of conformational order parameters for each orientation of the molecule. This is not discussed in ref. [lo]. Depending upon the precise properties 6f the system, most of these might cf course be redundant of sirr&u reasons as those given for the molecular order parameters above in the slow interconversion case. It is not clear which, if any, of these Iimits apply to the lipid bilayer case. In ref. [IO] it is stated that the fast interconversion limit cannot be completely satisfied for these systems and that the slow interconversion limit might be better approximation. To go further into this business one needs a detailed description of interchain and head group interactions, which hardly is available today. Here, fust in section 5 the case of conformational changes of a free chain will be considered, while in section 8 a kink model is treated. The latter is a very simple way of roughly including some of the effects of interchain interactions. When such simple models are used for the conformational order, it is not meaningful to do the comp!icated work necessary to take into account the coupling between conformational and molecular order in a proper way. I have therefore assumed, that the conformational order parameters are the average ones and that they are not affected by such things as molecular order or the actual (not average) interchain distance which might as weU be impor-
tant. Here the main concern will be to investigate the effect of conformational disorder. Therefore the order parameters
263
parameters in hyrlrocarbon chains
are defined with respect to a frame having
its z-axis parallel to the ali tram chain, thus giving&’ equal to one and the other four second order order
parameters zero for all the CHz-groups of the a tram chain. To include the effect of molecular disorder it is necessary to perform another transformation to a laboratory fmed coordinate system. This involves the averaging over a probability distribution over the Euler angles defining the orientation of the all-tram chain with respect to this frame. The order parameters LZ& are then obtained in terms of conformational and molecular order parameters
In this general case all components (m, n = 0,&l, 22, . .. . 9~ of the order parameter tensorSgronf are needed. Specializing to molecular disorder which is hotropic around the polar angle, this simplifies to
5. Conformational
disorder
Since the order parameters defined in section 1 are related to three corn onents (M = 0, ?2) of the crder parameter tensor Sli t: through eqs. (lo)--(12), it suffices to study these. They are, as discussed in section 2, related to an Euler angle description of the orientation of the CHz-group. For studying conformational disorder there are, however, better variables than the
Euler angles. I: is natural to use, for each CHxgroup, the valence angle 8 (i.e. the angle between the studied carbon-carbon bond and the preceding one) and the torsional angle Q (the angle of rotation around the carbcn-carbon bond). A coordinate system for each CHa-group is introduced as in section 2, with the and the x-axis bisecting the Z-axiS parallel t0 ‘chain angle between the two CH-bonds (see fig. 1). Then some space geometrical considerations give the Euler angles for the transformation between the coordinate systems adhering to two consecutive CHZ-groups in terms of these two angles: fY=~=arccos
1
sin(6/2)(1 - cos @)
[sit& + sir@/2)(1
- cos #I
U21 (27)
204
a EdtiolmfOrder pmmeters ix hydrocarbon ciztzins
Table 1 Decomposition of the Wiier a+diion maim according 10 eq. (19)
A3 =
0. Edholtn jO&r
parame:ers in hydrocarbon chains
and B = arccos[l
- cosqe/2)(r
- cos $)>I‘.
26.5
the rotational states of different carbon-carbon bonds upon the chain. At the free end of the ch&r this approximation is probably closer to reality thr?n in the upper and centra1 parts.
(28)
Thus the Wigrier rotation matrix can be expressed as
a function of these angles instead of the Euler angles: 6. The kans-gauche
model
(29) where Al, A,, Ax, 61 and 6, are real 5 X 5 matrices that only depend upon the &ence angle. They are given in fable 1 as a function of the parameters ,$= cos@/2) 2nd n = sin(6/2)The conformational order parameter tensor of CH2-group number (IV+ 1) is the average of the rotation matrix transforming between the local frame of that CH1-group and that of the f&t CHz-group. (The Zaxis of the latter frame is parallel to the all trans chain.) This rotation matrix can be factorized into N rotation matrices transforming between the frames of the intervening CH2-groups. In the general case, the average is to be taken over a probability distnbution ofNvaIence angles and N torsional angles. Now the simplifying approximation is made-that the orientation of each CHz-group with respect to the one closest above is independent of the configuration of the rest of the chain. Then it is possible to factorize the probability distribution and one gets
As a further speciahzation, the trans-gauche model (see e.g. Glory [ 12:) is now introduced. This t?xes the valence angles to be at the most probable value, 0, (approximately equal to the tetrahedral angle). Further is restricts the torsional angles to three discrete values corresponding to the tram and gauche* states. Both these assumptions will be abandoned in section 7 and it will there be shown (keeping the assumption of independent rotations) that these approximations iutroduce only negligib!e errors. In the tram-gauche model, the matrices A and B of eq. (29) and table 1 are constant. The avemges over the torsional angle, @,can be expressed as functions of the trans probability, pt. ‘Withthe tram state at torsional angle zero and the two gauche states symmetrically at ?120’, the averages become: &n9)=!sin2@,)=0
(31)
and (cos (6) = (cos 2@) = $(3p, - 1).
pt cornbe expressed in terms of the difference in energy between the gauche and tram states, Em: _
(30)
pt = [ 1 + ?exp(-Ets/liBT)]
This is a questionable approximation for systems with as important interchain interactions as those of a lipid bilayer. As further discussed in section 8, the interchain interactions will force correlations between
Table 2 The different o:der parameters probability @$
after having transformed
(32)
-I.
(33)
With Etg = 05 kca!/mol, pt becomes about 0.5. Now, inserting eqs. (3 1) and (32j into eq. (29) the average of the rotation matrix for one transformation along the chain becomes linear inpt. Since by eq. (30j the
a number of steps (N = 0, 1, 2) along the chain as a function
N
s chain
‘CD
‘HH
0
1
&-I +9pt)
-$
--$
1 2
+e(-5
1
+ 64 + 15~;) -
+Zp*)
-TPt
-f(l
$(I - 2Pt - P:,
$5(7 - 18Pt + 3Pf)
:
of the trans
.5 0~
1
2
a 5
Fig. 6. S$’ as a function of tmns probability for different chain positions (iindicated). Compare with tig. 4. Y z, function of chair position for difFig. 4. &) = L&n ferent tmns probabilities azwming independent rotations
order parameter tensor of CH2-group number (IV+ 1) is obtained as the matrix product N averaged rotation matrices, the order parameters are polynomials of degree Nin pt. In table 2 these are given for some order parameters forN less or e ual to two. For higherNthe order parametersS~~ and@ =S(2& have been evaluated numerically as a function o:fiv and pt. See figs. 3-7. An important featttre of the results is that for reasonable values of Pr (around OS), the order parameters decrease rapidly with chain position. Such a rapid decrease is not observed for NMR order param-
15
eters in lipid bilayers in the upper or middle parts of the chain but only close to the free end. This shows that an independent rotation model does not work in the upper and central part of lipid chains, but might be a better description at the free end. Further, there is an even-odd effect for smahN and pt not too large. Something like that can be observed in experimental NMR order parameters (see e.g. fig 3 of ref. [17]). For completeness the remaining order parameter S# = -Splo is given for the trans.-gauche model making one transformation along the chain (although this order parameter is not accessible through the experiments described in section 1)
10 15 chain position
Fig. 5. S$,’ = ‘&/378(S CD - S,,) as a function of chain ~osieon for different inns probabilities assumingindependent roktions
Fig. 7. S$’ as a function of tram probability for different chtin positions (indiated). Compare with tis 5.
0. EdholmjOrder parameters in hydrombm chains 7. Corrections to the kans-gawhe
267
mode!
A better descriptid;? than the trans-gauche model can be given by restricting the torsional angles by a potential having minima at the.trans and gauche states. Such a potential is shown in fig. 8. An often used expression is (see e.g. Volkenstein [13]): U(@)=U*/2[1 --x cos @-(I
-x)cos3l$].
(35)
Here x is a measure of the difference in energy, Etg, between the gauche and trans states. U, is the height of the barrier between the gauche wells. Reasonable values are here for UC 12 kJ/mol and for x 0.22 corresponding to Etg equal to about $ of UO. The potential is symmetric around Q equal to zero so kin @>and (sin 29) remain zero. If U, tends to infinity with _!?,a (not x> fjed, the trans-gauche model is recovered (although, of course the transition probabilities between the states become infinite in this manner). The tram probability is now defmed as the probability for being in the central well: (36) with k& being the angle at which the barrier between the trans and gauche wells has its maximum. Ev2luatin, the integrals in eq. (36) numerically for different values of the parameters of the potential,x and Uo/kBT, givesp, as a function of those two parameters. pt can thus replacex as one of the parameters of the poten&I. In this way kos 4s) and kos 21$) are calculated as functions of Pt and Uo/ksT. The result is shown in
Fig. 9. The variation of kos @)and tcos 3) with the trzns probability @t)_The different cmves refer to different potential barrier heights U,-,/kgT(4,6 and 8). The tram-gauche mode! gives the limiting straight line. fig. 9 with pt on the abscissa for Uo/kBT equal to 46 and 8. The straight line is given by eq. (32) and corresponds to infinite Uo, that is the tram-gauche model. The difference between the tram-gauche value of the averages and their actual v&e increases with decreasing Uo/kBT and increasingpt. The difference is larger for kos 29) thm for kos 0). Realistic values for the parameters are around 0.5 for pt and 4.5 for Uo/kB T. If one so wishes, two new order parameters, varying between -0.5 and 1 and 0 and 1 rsspectively, can be introduced: S t,=%,-
(37)
f,
and (38) St9 does then describe the disorder due to different populations in the trans and gauche states, while St,, describes disorder caused by the fact that the torsional angles are spread a bit around the minimum values of the wells. Further, the valence angle is allowed to vary, which makes the matrices A and 6 that appear in eq. (29) iuto variables. This variation is restricted by a potential Fig. 8. Schematic view of the potential hindering the torsional zotations. Ihe analytic form ke.g. that of eq. (35).
Y(8) = y&cos
0 - cos 6”)Z )
(39)
0. EdholmjOrderparamete.sin
268
discussed. Then, the only deviations from the straight aI tram chain that are allowed are kinks, that is
with 7e around 200 kf/mol (see e.g. Heifand et al. 2141) and 8, the most probable value (which could be put approximately equal to the tetrahedral angle, 109.5”). Thus, as a third order parameter one might introduce S val
=
re!(re
gauche’ trans gauche’ conformations. Such a conformation does only make a small excursion from the main direction of the chain and does then continue in the same direction as before. Thisminimates interchain interactions. Transforming the coordinate system through 2 kink is made by use of the rotation matrix
(43)
’ 2kg q-
Realistic values upon the three thus introduced order parameters are S,, = 0.25, St,, = 0.80 and Sval = 0.98 The influence of these order parameters upon spherical or NMR order parameters between different positions in the chain is not directly cIear but has to be investigated by numerical calculations The result is shown in table 3. There, it can be seen that the corrections are negligible and that one is indeed safe using 2 tram-gauche model. However, it must be stressed that these results are only valid under the assumption that the disorder is uncorrela?ed at different positions along the chain. Strondy correlated small scale disorder may very well give a large total result when added together.
8. Correlated
D$&_
=D@ =0u, @=2;r/3jD@
= [Al - ;(A,
+ Aj) + i(fi/2)&
X [A, - ;(A2 t A,) - i(a/2)(B2 The matrix multiplications
1 2 3 4 5 6
(Dc2) g+rg _)llnl = &,a(-l)“,
a) pt = 0.5and yg/2kgT
= 40.
b)pt=O.jandL’
(41)
n, m = 0,+I,22. (42)
This matrix describes a coordinate transformation, where the direction of the z-axis is kept while ‘Jle direction of thex-axis is turned 180”. This is exactly the result of a coordinate transformation through three all trans bonds and thus the kink does not change the orientation of CH~_groups outside the kink. For the two CH?-groups between the trans bond and the gauche+ and gauche- bonds respectively, the orientation does, however, change and it changes in the same way as if there where a single gauche bond preceding
Coxections from nonconstant valence angles a)
0.0082 0.0085 0.0027 0.0027 0.0002 0.0001
- B&j.
give
to the order parameters
s$
- Bj)]
X(Al+A,+A31
disorder
Fosition
= 0, Q = 0)
XD@=e,,+=--2x/3)
As stated earlier there is reason to believe that the rotations about different bonds in the lipid hydrocarbon chain are correlated. Further arguments for this are given by Edholm and Blomberg [ 151. There, also a simple model for the correlations, introduced originally by Trluble and Haynes [IS], the kink model is Table 5 Corrections
hydrocarbonchains
s!;’
-0.0023 -0.0002 -0.0004 o.ooo-: -0.0001 0.0002 ‘k T=4 . O/B
Corrections due to that the torsional angIes do net stick to the transgauche vah~es b) S$)
s!‘,
-0.0025 -0.0119 -0.0083 -0.0031 -0.0036 0.0001
-0.0149 -0.0227 -CL0077 -0.0055 0.0017 -0.0001
it. Thus, the results of section 6 for the transgauche model with N= 1 can be immediately used. Since there are three C&-groups to take the average over in 2 kink, but only two of them have another orientation in the kink than in the all tram chain, pt has to be replaced by (I - $u). pbd is then the probability for a certain CHz-group to be part of a kink (+ or -). (This probability must vary sIowly along the chain for the result to be meaningful.) A kink energy of twice the gauche energy (2 X 0.5 kcal/mol) givespkink about 0.3. The kink energy could be even larger, due to interchain interactions. This would give an even smaller pEti. Thus (1 - $pkink) becomes much larger than what pt would become in the tram-gauche model by ordinary Boltzmann statistics.
9. Comparison
with experiments
and discussion
The difference between the order parameters S,--D and SHH does give immediate information about the anisotropy around the polar angle according to eq. (1.5). IJnfortunately the experimental information concerning this difference is ambiguous. Petersen and Chan [IS] give SCD = -0.20 t 0.01 and S,, = -0.17 i 0.04 (Seiter and Ghan [2]). This gives 2 difference in eq. (15) of -0.03 + 0.05. Higgs and McKay [4] compare SHH values they have measured to SCD values of Davies and Jeffrey f17] and have at 60°C (fig. 5 of ref. [43) SHH = -0.27 and SC, = -0.23, that is a difference of 0.04 having opposite sign. Vaz and Deane [ 111 do also find an anisotropy of order parameters around the polar angle. Their order parameters do, however, refer to 2 different coordinate system. The parameter they use as a measure of the anisotropy depends therefore also upon other order parameters than (S!$)+S~~a)(S$ among else) and is thus not immediatdy relevant fey the ccmparison here. Petersen and Ghan [ 181 try to interpret the order parameters by 2 tram-gauche model, transforming just one step along the chain. That is they take only into account the effect of the tram-gauche state of the immediately preceding bond. The effect of disorder at bonds higher up in the chain is accounted for by averaging over an isotropic disrribution over all directions of the all tram chain within a cone of half top angle A@_This corresponds to adding molecular disorder 2s described by eq. (26). With the conforma-
tional order parameters
of the second row (N = 1) of
table 2 and S&)mol = S,, this $ves SCD = -&J&
(43)
and
+D - &iH = i(’
- pt)s,fl’
(44)
with SAP =
$0, A@(1+ cos A@).
(45)
The difference between the order parameters in eq. (44) is always positive. This comes into conflict with the negative experimental value of ref. [ 181. However, it is possible to get within the experimental accuracy with a small positive (less than 0.02) difference. Since the product S,,p, is @en by eq. (43) to be about 0.40, eq. (44) ieads to a restriction upon S,e. S,e needs to be less than about 0.56 to have SCD - SHH less than 0.02 and SC0 about -0.20 at the same time. However, such a small S, gives by eq. (43) that pt must be larger than about 0.7. To get such a large pt there must be other restrictions making gauche conformations more uncommon than the internal chain tram-gauche energy of OS kcal/mol does. However, it should be noted that these arguments of Petersen and Char-r [ 1S] fit very well into a kink model. The high trans probability gets an explanation. Further, a reason is provided for doing just one coordinate transformation along the chain not as many as are needed to get to the first CH,-group. It should be noted that by do&g more coordinate transformations than one there is no problem of reducing the order parameter expressing the anisotro y P around the polar angle. The problem is, that also Soi) decreases, and that it does so far too much. Finally, the results of Higgs and McKay [4] give 2 larger positive difference between SOD and SHH, which is easier to reconcile with trans-gauche model without having a small SAP. The ESR order parameters are hard to reconcile with NMR order parameters. McFarland and McConnell [ 191 give Schain (for multilamellar bilayers) by ESR to be 0.68 for the 12th carbon atom from the end of the chain, 0.50 for the 9th, 0.33 for the 7th and 0.13 for the 3rd. Comparing this to NMR order parameters by eq. (13), it is seen that neither for the 12th or for the 9th carbon atom are the results consistent. In general ESR order parameters seem to vary much more
0. EdhohjOrder parametersin i@racarbo~ ciurins
270
with chain position than the NMR ones. The plateau re@on in the upper and middle parts cf the chain, where &!!lR order parameters are rather independent of chain position, seem often to be missing in the ESR CaSe.
eq.
(13) instead of the simpler eq_ (14) may change the possibility to get agreement a bit. With S,, - S,, less than zero Schain becomes larger than in the case when there is isotropy around the polar angle. For instance SCD = -0.30 and S - HK -_ _woH~~S-~~” ._ a. @veSchain = 0.70 wwe s,, -s s chllin only 0.50.1~ only one Mv!R order pararneter is avtiable, sayScD, and upper limit for Sc,, could anyhow be given by the restrictions of fig. 3 Using
However. the available experimentaIinformation does enough anisotropy or even anisotropy of the correct sign to explain ESR order parameters. The data of Petersen and Chan [ 181 give Schain = Q43 + 0.07 and the data of Higgs and McKay [4] give 0.43. By eq (46) SC, restricts Schain to be s less than 0.55 and 0.66 respectively. If there is not agreement, the only possible explanation is that the averaging occurs over different probability distributions in the NMR and ESR cases. This in turn can have two reasons that hav,o been connot give large
chain
=
sidenbly debated in tie literature. First, spin probes mi&t disturb the order in the ESR case. Secondly,
since the two techniques are sensitive on different time scales, the difference might be due to the fact that the order parameters reflect ordering on different time scales.
Acknowledgement The author wants to thank Dr. Clas Blomberg for helpful discussions.
References [ I]
1. SeeIig and A. Seclig, Quart 19.
Rev.
[2] CH.A. [3]
Biophys
Seiter and 8.1. C&m, J. Am Chea 0973) 7541. XA Hemming and H.J.C Berendsen, J.
S (1972) 133. [4] T.P. H&s and kL
McKay, Chem
13 (1980)
Sot
95
Mago. Reson.
Phys. Lipids 20
(1977) 105. [S] S Schreier, CF. Polnaszek
and LCP. Smith, Biochim. Biophys Acta 515 (1978) 375. [6] J.W. Deane, in: Magnetic resonance and phase tramitions, e&F-J. Owens, CP. Poole Jr. and H.A. Faxach
(Academic Press, New York, 1979) pp_ 171 ff. [7] D.&I.Brinck and G.R. Satchler, Angular momentum (Oxford University Press, London, 1972) [8] -4. Saupe, Z Naturforsch. 19a (1974) 161. [9] J. See& and W. Niederberger, J. Am Chem. Sot 96 (1974) 2069. [fO] EE. BurnelI and Ck de Lange, J. hfag. Reson. 39
(1,080) 461. Ill]
N.A.P. Vaz and J.W. Doanne, Phys Rev. A22 (1980)
2238. [12] P. Glory, Statistical mechanics of chain molecules Wiley, New York, 1969). [ 131 M.V. Volkenstein, Confi_gu&ional statistics of polymeric c&ins (Wiley, New York, 1963). [14]
E. Helfand, ZR. Wasserman and T.A. Weber, Macromolecules 13 (1980) 526. [ 151 0. Edhohn a;ld C. Blombers Chem. Phys. 53 (1980) 185. [16] H. TGuble and A. Haynes, Chem. Phys. Lipids 7 (1971) 324. [17] I.E. Davies and K.R. Jeffrey, Chem. Phys. Lipids 20 (1977) 87. [IS] N.O. Petersen and.S.1. Ghan, Biochemistry 16 (1977) 2657. [19] B.C. McFarland and H.M. McConneU. Proc. Natl. Acad. Sci. USA 68 (1971) 1274.