Resolving Internal Motional Correlations to Complete theConformational Entropy MeterIztok Urbancic, Ajasja Ljubetic, and Janez Strancar*
Laboratory of Biophysics, Condensed Matter Physics Department, “Jozef Stefan” Institute, Jamova cesta 39, SI-1000 Ljubljana,Slovenia
*S Supporting Information
ABSTRACT: Conformational entropy (SΩ) has long been used to theoreticallycharacterize the dynamics of proteins, DNA, and other polymers. Though recent advancesenabled its calculation also from simulations and nuclear magnetic resonance (NMR)relaxation experiments, correlated molecular motion has hitherto greatly hindered bothnumerical and experimental determination, requiring demanding empirical and computa-tional calibrations. Herein, we show that these motional correlations can be estimateddirectly from the temperature-dependent SΩ series that reveal effective persistence lengths ofthe polymers, which we demonstrate by measuring SΩ of amphiphilic molecules in modellipid systems by spin-labeling electron paramagnetic resonance (EPR) spectroscopy. Wevalidate our correlation-corrected SΩ meter against the basic biophysical interactionsunderlying biomembrane formation and stability, against the changes in enthalpy anddiffusion coefficients upon phase transitions, and against the energetics of fatty aciddissociation. As the method can be directly applied to conformational analysis of proteinsand other polymers, as well as adapted to NMR or polarized fluorescence techniques, we believe that the approach can greatlyenrich the scope of experimentally available statistical thermodynamics, offering new physical insights into the behavior ofbiomolecules.
SECTION: Biophysical Chemistry and Biomolecules
Molecular internal motion, effectively described byconformational (also termed configurational) entropy
(SΩ), greatly affects the function of biological molecules andsupramolecular structures, balancing the energy (enthalpy) ofthe driving forces. Although its roles have been recognized inprotein structure and activity,1−5 ligand binding,6,7 DNAstructure,8−10 polysaccharide interactions,11,12 alkane aggrega-tion,13 lipid arrangement,14,15 and membrane organization,16−18
determination of SΩ from molecular dynamics (MD)simulations19−23 or experiments10,13 remains considerablychallenging. Only recently has a rigorous procedure to measureSΩ of proteins by nuclear magnetic resonance (NMR) beendeveloped,24−30 employing the relaxation-determined orderparameter (O) as a proxy for SΩ. The predicted SΩ-to-Otransformation24−26 has been found to be significantly affectedby motional correlations though, which necessitate demandingcase-by-case empirical calibrations.29,30
In this work, we show that motional correlations along thestudied polymers, expressed by their persistence lengths, can bedirectly discerned from temperature-dependent measurementsof SΩ and further used to correct the measured values (Scheme1), avoiding the need for additional calibrations. Wedemonstrate the importance of the SΩ correction on modelbiomembrane systems, where motional correlations along thealkyl chains can be effectively tuned by temperature, phasetransitions, and chain length. In our case, electron para-magnetic/spin resonance (EPR/ESR) spectroscopy was
employed, detecting molecular motion at a similar nanosecondtime scale as relaxation NMR. EPR is therefore equally suitableto characterize fast molecular motion of protein sidechains31−34 or lipid alkyl chains,35−39 but has only once beenqualitatively interpreted in terms of SΩ.
40
Molecular motion of spin-labeled chains, discerned throughadvanced spectral simulations, has been most commonlyparametrized by empirical O,37,38 as in NMR, or by freerotational space (Ω),31,39 which is a compact representation ofthe asymmetric wobbling cone41 of the paramagnetic nitroxide(NO) moiety. In this work, we redefined Ω so that its valuedirectly represents the normalized solid angle of the cone
ϑ φπ
Ω =−(1 cos )
20 0
(1)
where ϑ0 and φ0 represent the polar and azimuthal angles of thecone, respectively (Scheme 1). As Ω is now proportional to thenumber of possible NO orientations within the cone, whichdirectly reflect possible local orientations of the chain due torather rigid sp3 orbital configuration at the joint C atom, theconformational entropy of the chain at the site of NO (SΩ) canbe calculated through the Boltzmann relation
Received: September 9, 2014Accepted: October 1, 2014
where kB represents Boltzmann’s constant. Note that a rigorousthermodynamic derivation yields an identical expression.24 Theterm S0 = kB ln(2π) comes from the normalization of an openrotational space (Ω = 1) describing isotropic rotational motionand is constant across the studied systems within the scope ofthe applied model of molecular motion. We chose the Ωparametrization, which can be directly converted to O,24,42 onlyfor the sake of convenience in Ω-to-SΩ transformation;analytical relation between O and SΩ is only available for thesame wobbling model if φ0 = 2π (eq S6 and Figure S7 in the
Supporting Information) but requires numerical evaluationotherwise.24
Because the spectral model used to determine Ω relies onfast motion approximation,43 the method is highly suitable todescribe fast dynamics of small biomolecules, such as alkylchains of fatty acids and lipids39 or protein side chains,32,33 atphysiological temperatures. Its application to severely restrictedwobbling of the probes (i.e., Ω < 0.05 and SΩ − S0 < −3 kB)that could possibly result from a slow motion regime(compared to a nanosecond EPR time scale) should thereforebe avoided, which is further discussed in the SupportingInformation (section S1.6).To demonstrate the proposed approach and expose the need
for the correction of correlated motion, described below, wechose a well-characterized, yet nontrivial model system; weanalyzed temperature-dependent EPR spectra (Figures S1−S5,Supporting Information) of several spin labels (SLs; Table 1) indifferent model membranes, composed of saturated lipids ofvarious chain lengths, namely, dimyristoyl phosphocholine(DMPC, 14 C), dipalmitoyl phosphocholine (DPPC, 16 C),and distearoyl phosphocholine (DSPC, 18 C). Because theintroduction of SLs at low concentrations does not affectmembrane properties significantly,44,45 any differences reportedby various labels from the same biochemical environment resultfrom the probes’ intrinsic conformational preferences due tothe minimization of their own free energy, which can varybetween the probes due to differences in chemical structure.Both similarities and differences of their response served us toevaluate our entropy meter.In general, a common entropic behavior was observed for all
SLs, following the expected temperature dynamics of lipidmembranes, as reported previously by EPR, NMR, dynamicscanning calorimetry (DSC), and other methods44−52 (Figure1); heating the sample toward the temperature of gel-to-ripplepretransition (Tp), SΩ rose, showing a gradual loss of structuralorder. Between Tp and the temperature of the ripple-to-liquiddisordered main phase transition (Tm), SΩ increased steeply,approaching the plateau value, characteristic for the liquiddisordered phase at high temperature.Before quantitatively examining the results in detail, the first
visual inspection already reveals a paramount feature; thechanges of SΩ with T are nearly linear within broad T ranges.The derivative of entropy with respect to temperature (i.e., theslope dSΩ/dT) is proportional to c/T, where T stands forabsolute temperature and c for specific heat capacity. As hasbeen recognized before,26 the applied model of diffusion in asquare well potential does not itself predict any temperaturedependence at each wobbling bond. However, taking intoaccount that motion at each site along the chain is affected alsoby neighboring bonds, the measured c reflects the number ofdegrees of freedom (ν) that contribute half of the Boltzmannconstant each (c = νkB/2), giving
ν= =Ω ΩS
TcT
kT
dd 2
B(3)
As SΩ is measured at the site of NO, the corresponding cΩrepresents a partial heat capacity due to conformational changesof a fragment of the SL molecule that affects NO orientation.The length of such a segment, which represents the persistencelength (lp) of the SL chain, can therefore be estimated from theslopes of SΩ(T) profiles, considering that each bond of the alkylchain contributes two degrees of freedom to the measured cΩ(ν = 2lp).
53 Expressed in the units of C−C bonds
Scheme 1. Overview of the Methodologya
aThe acquired temperature (T) series of EPR spectra were modeledby multiple motional patterns, characterized by two wobbling coneangles (ϑ0 and φ0). Solutions were presented in terms of freerotational space (Ω), where bubble areas represent relative spectralportions of the coexisting motional patterns, or averaged into the meanconformational entropy (SΩ). The rates of SΩ changes withtemperature yielded persistence lengths (lp) of alkyl chains, whichwere finally used to correct the results for motional correlations (SΩ
cc).The approach could be generalized also to other experimentaltechniques (NMR, fluorescence anisotropy, etc.) or parameterizations(order parameter). For details, see the Supporting Information.
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ν= = ΩlS T T
k2(d /d )
pB (4)
Below Tp, the slopes, and thus lp, increased with the length ofSL (Figure 1, compare light and dark symbols of the samecolor) and decreased with the length of lipid chains in theenvironment (Figure 1, compare the same symbols in panelsA−C or D−F), indicating interleaflet interdigitation of theprobes’ alkyl chains due to the probe-to-lipid chain lengthmismatch (dm = dSL − dlip in Figure 2).47 In the gel phase, the
determined lp values at zero mismatch (dm = 0) were for mostSLs found to take values around 12−16 C−C bonds (Figure2A), suggesting that motional correlations extend almostthroughout the whole chain. At higher mismatch (dm > 0), lpeven exceeded the length of probe and lipid chains, indicatingthat the wobbling is significantly affected also by degrees offreedom from neighboring molecules.Above Tm, noninterdigitating FASLs showed nearly no T
dependence of SΩ (Figure 2B), which importantly signpostsalmost independent wobbling of individual bonds. The range of
Table 1. Chemical Structures of the SLs Used in the Study
Figure 1. Temperature (T) dependence of measured conformational entropy relative to unrestricted motion (SΩ − S0) for probes with NO at the(A−C) 5th and (D−F) 12th/13th C atom of the chain (see the color legend and Table 1) in different lipid environments (columns for DMPC,DPPC, and DSPC). Gray shaded areas indicate the expected temperatures of the pretransition (Tp) and main lipid phase transition (Tm).
46 Valuesbelow −2.5 kB, possibly affected by the slow motion regime, are omitted.
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obtained lp values for this type of SLs thus agrees with previousT-dependent estimations in nematic liquid crystals.54 Never-theless, a length mismatch with the environment (dm > 0) oranchoring to another alkyl chain (PCSL) can considerablyextend motional correlations even in the liquid disorderedphase.Due to such motional correlations, a simple summation of
measured SΩ values along the molecule would significantlyoverestimate the total SΩ of the system, which necessitatesdemanding empirical calibrations29,30 or computational correc-tions.55 However, the measured lp can directly serve to obtainthe thermodynamically relevant total conformational entropy ofthe molecular segment of interest (SΩ
seg), for which we proposea simple approximation
= =+Ω Ω
ΩS S lS l
l( 1)seg cc
segseg
p (5)
where lseg represents the number of C−C bonds in theobserved segment and SΩ
cc the correlation-corrected contribu-tion per bond. In the limit of independent wobbling (lp = 0),the expression reproduces additive entropy contributions alongthe chain (SΩ
seg = SΩ lseg), whereas for substantially correlatedmotion (lp = lseg − 1), the total entropy is already sampled atany site (SΩ
seg = SΩ).
Due to a lack of relevant literature data, a quantitative checkof the values, obtained by the introduced approach, was notstraightforward. The few directly comparable results ofnumerical simulations are obviously method-dependent.14,18
As SΩ is often undetermined to an additive constant, the onlyreliable comparison was obtained with detailed MD data byBaron et al. that presented depth-dependent SΩ of quarterlychain fragments of DPPC at 323 K.14 Between the segmentsthat correspond to our labeling depths, that is, the 5th and 12/13th C atoms, and neglecting any motional correlations, theyreported a 30 J K−1 mol−1 (0.9 kB/bond) increase of SΩ towardthe end of the chain.14 The value reasonably agrees with ournoncorrected result for the most DPPC-like SLs that showedan increase of around 0.7 kB (compare black and gray symbolsfor 5- and 12-doxyl PCSL in Figure 1B and E, respectively).However, our correction by the measured lp (eq 4) suggested anearly 3-fold reduction of this value (see section S2.1,Supporting Information), whereas their treatment of pairwisecorrelations yielded only a modest correction (−9%).14Nevertheless, their choice of a smaller segment size formolecular alignment prior to conformational analysis decreasedthe total SΩ almost by half, effectively accounting forcorrelations,14 which might call for evaluating also higher-order correlations, as indicated by our data.
Figure 2. Approximate persistence lengths (lp, right vertical axes) were calculated according to eq 4 from the rates of the SΩ change with temperature(dSΩ/dT) (A) down to 15 K below Tp and (B) up to 15 K above Tm. The values are plotted against the difference in alkyl chain lengths of probeswith respect to surrounding lipids (chain length mismatch: dm = dSL − dlip; the values were slightly shifted for the clarity of presentation). The insetsdemonstrate the determination for 13-doxyl MeFASL (18 C) in DPPC. The schemes below illustrate interleaflet interdigitation of probe tails at dm >0.
Figure 3. Reduction of correlation-corrected conformational entropy (SΩcc) with respect to completely isotropic rotational motion (S0) is shown
against the length of surrounding lipids in (A) gel and (B) liquid disordered phases. Colored stripes denote the estimated contributions ofhydrophobic (Ehp) and van der Waals (EvdW) interactions.
56 The insets demonstrate the determination for 13-doxyl MeFASL (18 C) in DPPC.
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We further assessed the correlation correction of the entropymeter through three basic biophysical phenomena. First,considering that SLs share equal free energy in water and ina bilayer but differ in conformational freedom, the measured SΩreduction with respect to isotropic rotational motion (S0) wasused to estimate the energy needed to restrict conformationalmotion into a bilayer: ΔE = T(SΩ
cc − S0). This ΔE is providedby hydrophobic interaction and van der Waals attractionbetween alkyl chains, both of which contribute down to about−0.9 kBT per C atom of the chain56 (see section S2.2,Supporting Information). In the gel phase, where bothinteractions are fully relevant, all of our data with very differentmeasured SΩ and lp yielded consistent ΔSΩcc of around −1.8 kB(Figure 3A), which perfectly agrees with the prediction above.Without the correlation correction, the obtained valuesscattered in the range between −0.7 and −3.5 kB (Figure S8,Supporting Information), which could not be reasonablyexplained. In addition, above Tm, where interchain attractionshould be largely released, monoalkyl probes indeed showedapproximately half of the previously described SΩ reduction(Figure 3B), which could be realized by hydrophobicinteraction alone. However, significant interdigitation of theprobes in thin membranes (DMPC) or the presence of another
chain with a fixed common anchor (PCSL) obviously enforcedvan der Waals interaction to some extent also at high T.Experimentally determined intermolecular interactions, illus-trated above, could further help to understand the organizationof biomembranes, for example, lipid rafts and nanodomains.57
The role of SΩ can be further investigated in lipid phasetransition, perhaps the most significant thermodynamicproperty of the bilayer. Considering different lp’s below Tp
and above Tm, we calculated the combined SΩ jump upon bothphase transitions for each SL (ΔSΩcc, Figure 4). The obtainedvalues for PCSL nicely agree with those derived from DSCenthalpy changes,46 from which we subtracted the contributionof translational entropy58,59 change due to markedly differentdiffusion coefficients in both phases60−63 that affects DSCmeasurements but not EPR (open squares in Figure 4A; seealso section S2.3, Supporting Information). It is again worthnoting that the agreement with DSC data would be lost ifcorrelation correction was not be applied (Figure S9,Supporting Information). FASLs faithfully mimicked PCSLsonly when strongly interdigitated (DMPC) but tended to showmuch greater ΔSΩcc otherwise (Figure 4B). The reason shouldbe sought in the much smaller change of translational diffusionfor monoalkyls in PC membranes60 and the larger per-bond
Figure 4. Differences in correlation-corrected conformational entropy (SΩcc) between the temperatures of the pretransition (Tp) and main phase
transition (Tm) (determined as indicated by the inset for 13-doxyl MeFASL (18 C) in DPPC) are shown against the length of surrounding lipids for(A) lipid-based PCSL probes and (B) monoalkyl H- and MeFASLs (solid symbols). Open squares represent the changes in total entropy of lipids, asdetermined from calorimetry (ΔSDSC),46 from which the contribution of translational entropy58−63 (ΔStr) was subtracted (see section S2.3 in theSupporting Information). Open circles denote the total entropy change for fatty acids of the corresponding chain length.64
Figure 5. (A) pH dependence of free rotational space (Ω; left vertical axis) and conformational entropy (SΩ − S0; right vertical axis) of coexistingspectral components for 5-doxyl H- and MeFASL (both 16 C) in DSPC at 334 K; the bars represent the components’ distribution, whereas bubbleareas denote their spectral weights (wi). Complete temperature series are presented in the Supporting Information (Figure S10). (B) Histograms ofwi from the data in panel A; lower and upper sets of bars correspond to spectral components with Ω below and above 0.5, respectively. The schemeon the right depicts the difference in vertical positioning between the protonated (O) and dissociated (−) forms of HFASL and its interaction withcations (+).
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entropy change upon their intrinsic phase transition,64 whichsets an upper limit to our data (open circles in Figure 4B). Theability to experimentally quantify such differences could beimportant to understand various biological roles of fatty acidsor other monoalkyls in membranes.65−67
Finally, from the observed −0.6 kB difference in SΩ betweenthe protonated and dissociated form of 5-doxyl HFASL,revealed by individual spectral components during a pH scan(Figure 5), the difference in vertical anchoring interactions dueto an additional charge was assessed. As the influence ofdissociation was greatest near the head group (5-doxyl) andvanished toward the end of the chain (13-doxyl; compareorange and blue symbols in Figure 1D−F for H- and MeFASLs,respectively), we roughly estimated that the conformationaldifferences affect approximately half of the chain, adding up toΔSΩchain ≈ −5 kB (lp ≈ 0 from Figure 2B). Because the thermalequilibrium of both HFASL forms again imposes their equalfree energies, the loss of SΩ for the dissociated probes should becompensated for by an interaction of ΔE = TΔSΩ ≈ −5 kBT.The value plausibly corresponds to Coulomb’s attractionbetween a counterion pair in water at a distance of 0.1−0.2nm (see section S2.4, Supporting Information) that can beformed by the charged dissociated head group with eithercholines or dissolved ions, which confirms the reliability of ourentropy meter even to study coexisting molecular structures.To conclude, we thoroughly validated the proposed
procedure to experimentally determine SΩ from EPR spectra,which yielded quantitatively compelling clues about the mostbasic properties of lipid bilayers and about the underlyinginteractions. The introduced correction for motional correla-tions, directly discerned from persistence lengths throughtemperature-dependent data, was shown to be essential forconsistent agreement with known data. First, hydrophobic andvan der Waals interactions were shown to provide for themeasured entropy reduction involved in membrane formationand stabilization in different lipid phases. Second, thedetermined contributions of conformational and translationalentropy of the lipid SL upon phase transitions were in line withDSC and diffusion data for lipids, whereas conformationalentropy change was found to increase significantly formembrane-incorporated monoalkyl molecules. Finally, themeasured entropy reduction upon head group dissociationwas explained by electrostatic interaction.Generalizations of the introduced approach are manifold.
Namely, the same spectral and/or conformational analysis canbe applied to site-directed spin labeling of intrinsicallydisordered and membrane proteins31−34 or other polymers.68
Next, the transformation from a wobbling cone or orderparameter to entropy could be exploited in polarizedfluorescence micro(spectro)scopy.69−71 Furthermore, thecorrelation correction could be adopted by the establishedorder-parameter-based NMR entropy meter (Figure S7,Supporting Information),29,30 alleviating the need for empiricalcalibrations, that would greatly simplify the measurements andenhance the interpretation. We thus expect that theexperimental correlation-corrected conformational entropymeter can greatly enrich our understanding of physicalinteractions involved in (supra)molecular structures, theirtransitions, and coexistence, which should importantly comple-ment the emerging computational studies.
■ ASSOCIATED CONTENT*S Supporting InformationDetailed description of the experimental methodology andanalysis, figures of EPR spectra, and results of simulations. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.
NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTSThe authors thank the Slovenian Research Agency (ARRS) forfinancial support (project No. P1-0060), Profs. S. Pecar and J.Mravljak (Faculty of Pharmacy, University of Ljubljana) forsynthesis of the spin labels, M. Nemec, S. Kure (“J. Stefan”Institute), and Drs. S. Pajk and N. Zidar (Faculty of Pharmacy,University of Ljubljana) for their help with EPR measurements,and Drs. A. L. Bozic and M. Sentjurc (“J. Stefan” Institute) foruseful discussions and careful reading of the manuscript.
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