Volume-energy correlations in the slow degrees of freedom of computer-simulated phospholipid membranes
Ulf R. Pedersen, Günther H. Peters, Thomas B. Schrøder, Jeppe C. Dyre
aa r X i v : . [ phy s i c s . b i o - ph ] S e p Volume-energy correlations in the slow degrees of freedom of computer-simulatedphospholipid membranes
Ulf R. Pedersen, ∗ G¨unther H. Peters, † Thomas B. Schrøder , ∗ and Jeppe C. Dyre ∗ ∗ DNRF Centre “Glass and Time,” IMFUFA, Department of Sciences,Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark † Center for Membrane Biophysics (MEMPHYS), Department of Chemistry,Technical University of Denmark, DK-2800 Lyngby, Denmark
Constant-pressure molecular-dynamics simulations of phospholipid membranes in the fluid L α phase reveal strong correlations between equilibrium fluctuations of volume and energy on thenanosecond time-scale. The existence of strong volume-energy correlations was previously deducedindirectly by Heimburg from experiments focusing on the phase transition between the L α andthe L β phases. The correlations, which are reported here for three different membranes (DMPC,DMPS-Na, and DMPSH), have volume-energy correlation coefficients ranging from 0.81 to 0.89.The DMPC membrane was studied at two temperatures showing that the correlation coefficientincreases as the phase transition is approached. Biological membranes are essential parts of living cells.They not only act as passive barriers between outside andinside, but also play an active role in various biologicalmechanisms. The major constituent of biological mem-branes are phospholipids. Pure phospholipid membranesoften serve as a models for the more complex biologicalmembranes. Close to physiological temperatures mem-branes undergo a transition from the high-temperaturefluid L α phase (often referred to as the “biologically rele-vant phase”) to a low-temperature ordered gel phase L β .In the melting regime response functions such as heat ca-pacity, volume-expansion coefficient, and area-expansioncoefficient increase dramatically. Also, the characteris-tic time for the collective degrees of freedom increasesand becomes longer than milliseconds. Some time agoHeimburg found that the slow, dominating parts of heatcapacity and volume-expansion coefficient of DMPC asa function of temperature can be superimposed close tothe melting temperature T m [1] (see also Refs. 2 and 3).Thus the response functions are related in such a waythat a single function describes the temperature depen-dence of both.The fluctuation-dissipation (FD) theorem connects(linear) response functions to equilibrium fluctuations.The isobaric heat capacity c p can be calculated from en-thalpy fluctuations as follows: c p = h (∆ H ) i / ( V k B T ),where h . . . i is an average in the constant temperatureand pressure ensemble and ∆ is deviation from the aver-age value. Similarly, volume fluctuations are connectedto the isothermal volume compressibility by the expres-sion κ T = h (∆ V ) i / ( V k B T ). If the response functionswere described by a single parameter, fluctuations arealso described by a single parameter [1, 4] and the mi-crostates were connected via the relation ∆ H i = γ vol ∆ V i .At constant pressure this relation applies if and only if∆ E i = γ vol ∆ V i (where E is energy), which is the relationinvestigated below. This situation is referred to as a thatof a single-parameter description [4]. A single-parameterdescription applies to a good approximation for several models of van der Waals bonded liquids as well as forexperimental super-critical argon [5].Unfortunately, molecular-dynamics simulations arenot possible for investigating “single parameter”-ness ofmembranes close to T m , because the relaxation time forthe collective modes by far exceeds possible simulationtimes. We show below, however, that a single-parameterdescription applies to a good approximation for the slowdegrees of freedom of the fluid L α phase, a descriptionthat applies better upon approaching T m . At the end ofthis note we briefly discuss how this property may betested in experiments monitoring frequency-dependentthermoviscoelastic response functions.It is not a priori obvious that a single parameter maybe sufficient for describing slow thermodynamic fluctua-tions of a membrane. For instance, simulations of waterand methanol showed no “single parameter”-ness. Ap-parently, what happens here is that contributions to vol-ume and energy fluctuations from hard-core repulsioncompete with those from directional hydrogen bonds todestroy any significant correlation [5]. Membranes arecomplex anisotropic systems, and we cannot give anyobvious reason that volume and energy should correlatestrongly in their fluctuations.A membrane may be pertubated via three thermody-namic energy bonds. The change of enthalpy dH can bewritten as a sum of contributions from a thermal energybond, a mechanical volume energy bond, and a mechan-ical area energy bond, dH = dE + pdV + Π dA , where p is pressure, V volume, Π membrane surface tension,and A membrane area. The natural ensemble to con-sider is the constant T , p , and Π ensemble, since the sur-rounding water acts as a reservoir. If a single parametercontrols the microstates, for all states i one would have∆ E i = γ vol ∆ V i = γ area ∆ A i where the γ ’s are constants.In general, the microstates may of course be controlledby several parameters. An interesting question is howmany parameters are sufficient to describe the membranethermodynamics to a good approximation. This questionis addressed below by investigating molecular-dynamicssimulations of different phospholipid membranes.An overview of the simulated systems is found in tableI. The simulated systems include different head groups(both charged and zwitterions) and temperatures. Allsimulations was carried out in the constant pressure, tem-perature ensemble. The membranes are fully hydratedand in the fluid L α phase. The simulations were per-formed using the program NAMD [7] and a modifiedversion of CHARMM27 all hydrogen parameter set [6, 8].More simulation details are given in Ref. 6.The correlations between equilibrium time-averagedfluctuations of volume and energy on the nanosecondtime-scale of a DMPC membrane at 310 K are shownon Fig. 1. If E ( t ) and V ( t ) is the energy and volumeaveraged over 1 nanosecond, the figure shows that to agood approximation one has∆ E ( t ) ≃ γ vol ∆ V ( t ) (1)where γ vol = σ E /σ V is a constant (standard de-viation σ ) and ∆ is difference from the thermody-namical average value. Similar results were foundfor the other membranes studied. Table I showsthat volume-energy correlation coefficients ( R EV = h ∆ E ∆ V i / q h (∆ E ) ih (∆ V ) i ) range between 0.81 and0.89.The correlation depends on the time scales con-sidered. This can be investigated by evaluatingΓ( t ) = C EV ( t ) / p C EE ( t ) C V V ( t ) where C AB ( t ) = h ∆ A ( τ )∆ B ( τ + t ) i / p h (∆ A ) ih (∆ B ) i is a time corre-lation function. Γ( t ) = 0 implies that energy at time τ is uncorrelated with volume at time t + τ , whereas Γ( t )close to unity implies strong correlation. Γ( t ) is plottedin the inset of Fig. 2. At short time (picoseconds) Γ isaround 0.5 but approaches unity at t ≃ T [K] R EV A lip [˚A ] t [ns] t tot [ns]DMPC 310 0.885 53.1 60 114DMPC 330 0.806 59.0 50 87DMPS-Na 340 0.835 45.0 22 80DMPSH 340 0.826 45.0 40 77TABLE I: Data from simulations of fully hydrated phos-pholipid membranes of 1,2-Dimyristoyl-sn-Glycero-3-Phosphocholine (DMPC), 1,2-Dimyristoyl-sn-Glycero-3-Phospho-L-Serine with sodium as counter ion (DMPS-Na)and hydrated DMPS (DMPSH). The columns list temper-ature, volume-energy correlation coefficient, average lateralarea per lipid, simulation time in equilibrium (used in thedata analysis), and total simulation time. N o r m a li ze d f l u c t u a ti on s [ σ ] EnergyVolume R EV = 0.89 FIG. 1: Normalized fluctuations of energy ( × ) and volume ( ◦ )for a DMPC membrane at 310 K. Each data point representsa 1 ns average. Energy and volume correlate with correlationcoefficient R EV = 0 . functions approaching T m . Figure 2 shows the time-correlation functions of energy, volume, and area of aDMPC membrane at 330 K and 310 K. Time constantas well as magnitude of the slow fluctuations increasewhen temperature decreases and the phase transition isapproached. γ vol in Eq. (1) is 9 . × − cm /J. Thisis of the same order of magnitude as γ vol = 7 . × − cm /J calculated from the experimental data of C p ( T )and κ volT ( T ) at T m [1].Both volume and energy time-correlation functionsshow a two-step relaxation at 310 K for DMPC (Fig. 2B).As temperature is lowered towards T m , the separation isexpected to become more significant. It makes sense todivide the dynamics into two separated processes, a fastand a slow collective process. Our simulations suggestthat the slow degrees of freedom can be described by asingle parameter, but not the fast degrees of freedom.To see the “single-parameter”-ness of the L α phase,the fast degrees of freedom must be filtered out. Ex-periments deal with macroscopic samples where fluctu-ations are small and difficult to measure (the relativemagnitude of fluctuations goes as 1 / √ N where N is the T i m e C o rr e l a ti on EnergyVolumeEnr.-vol. Cross. Corr.Membrane area
Time [ns] T i m e C o rr e l a ti on Time [ns] Γ A: T = 330 KB: T = 310 K C
FIG. 2: Time-correlation functions for a DMPC membrane ofpotential energy C EE ( × ), total volume C V V ( ◦ ), membranearea C AA ( △ ), and cross correlation between energy and vol-ume C EV ( ⋄ ). Time correlation for membrane areas are scaledby a factor 0.2. Panel A shows data at 330 K, panel B at 310K. The inset (C) displays Γ( t ) = C EV ( t ) / p C EE ( t ) C V V ( t )at 310 K ( ◦ ) and 330 K ( × ). Γ approaches unity at t ≃ number of molecules). It is therefore difficult in exper-iment to perform the same analysis as we have donehere; it is easier to measure response functions. Fastfluctuations can be filtered out by measuring frequency-dependent response functions. The slow collective de-grees of freedom give rise to a separate “loss” peaks inthe imaginary parts. A frequency-dependent Prigogine-Defay ratio Λ T p ( ω ) was recently suggested as a test quan-tity for single-parameter-ness in a paper focusing on theproperties of glass-forming liquids [4]. If c ′′ p ( ω ), κ ′′ T ( ω ),and α ′′ p ( ω ) are the imaginary parts of the frequency-dependent isobaric specific heat (per volume), isother-mal compressibility, and isobaric expansion coefficient, respectively, by definitionΛ T p ( ω ) = c ′′ p ( ω ) κ ′′ T ( ω ) T [ α ′′ p ( ω )] . (2)In general Λ T p ( ω ) ≥
1, and Λ
T p ( ω ) = 1 if and only ifa single parameter describes the fluctuations [4]. Thequantity 1 / p Λ T p may be interpreted as a correlationscoefficient.In summary, we found strong volume-energy corre-lations of the slow degrees of freedom in molecular-dynamics simulations of different phospholipid mem-branes in the L α phase. An experimental test was sug-gested. ACKNOWLEDGMENTS
The authors would like to thank Nick P. Bailey,Thomas Heimburg, and Søren Toxværd for fruitful dis-cussions and useful comments. This work was supportedby the Danish National Research Foundation Centre forViscous Liquid Dynamics “Glass and Time.“ GHP ac-knowledges financial support from the Danish NationalResearch Foundation via a grant to the MEMPHYS-Center for Biomembrane Physics. Simulations were per-formed at the Danish Center for Scientific Computing atthe University of Southern Denmark. [1] T. Heimburg, Biochim. Biophys. Acta (1998) 147.[2] T. Heimburg and A. D. Jackson, Proc. Natl. Acad. Sci. (2005) 9790; T. Heimburg, Biochim. Biophys. Acta (1998) 147.[3] H. Ebel, P. Grabitz, and T. Heimburg, J. Phys. Chem. B (2001) 7353.[4] N. L. Ellegaard, T. Christensen, P. V. Christiansen, N. B.Olsen, U. R. Pedersen, T. B. Schrøder, and J. C. Dyre, J.Chem. Phys. (2007) 074502.[5] U. R. Pedersen, N. P. Bailey, T. B. Schrøder, and J. C.Dyre, ArXiv: cond-mat/0702146 (2007).[6] U. R. Pedersen, C. Leidy, P. Westh, and G. H. Peters,Biochim. Biophys. Acta (2006) 573; U. R. Pedersen,G. H. Peters, and P. Westh, Biophysical Chemistry (2007) 104.[7] J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E.Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, L. Kale,and K. Schulten, J. Comput. Chem. (2005) 1781.[8] N. Foloppe and A. D. MacKerell, J. Comput. Chem.21