in 2005 , heimburg and jackson showed that biomembranes may carry solitonic sound waves whose maximum amplitude and minimum velocity are close to the propagation velocity in myelinated nerves.(1 ) their paper concluded :  it would be surprising if nature did not exploit these features . 
subsequent works by the same authors argue directly that nerve signals are not primarily electrical but solitonic sound waves carried by the nerve cell membrane .
the conventional wisdom is that nerve signals propagate via electrical current as formulated in the hodgkinhuxley theory.(4 ) the heimburgjackson theory explains anesthesia as a straightforward effect of melting - point depression of the orderdisorder transition .
this can in turn explain why biomembranes have transition temperatures close to physiological temperatures.(5 ) the heimburgjackson nerve - signal theory motivated this study .
one element of the theory is the assumption that volume and enthalpy correlate strongly in their thermal equilibrium fluctuations , at least as it regards the slow parts of these fluctuations .
this assumption was justified by experimental observations that membranes specific heat and compressibility are proportional in their temperature dependence across the phase transition .
also , a correlation between lipid area and enthalpy has been proposed(6 ) on the basis of experimental observations.(8 ) if so , both volume , membrane area , and enthalpy are controlled by a single parameter .
below , we present first results from extensive computer simulations of different phospholipid membranes performed in order to investigate whether strong volumeenergy and/or areaenergy correlations are observed and , if so , what causes them .
recently , we studied equilibrium thermodynamic fluctuations of much simpler systems , namely , various model liquids like the standard lennard - jones ( lj ) liquid and similar systems . in many such simple liquids ,
one finds a strong correlation between the equilibrium fluctuations of virial w and potential energy u , when fluctuations are studied at constant particle number n , constant volume v , and constant temperature t ( the so - called nvt ensemble(14 ) ) .
recall(15 ) that the virial w = w(t ) gives the nonideal contribution to the instantaneous pressure p = p(t ) via the defining equation p(t)v = nkbt(t ) + w(t ) , where t(t ) is the instantaneous temperature defined via the instantaneous kinetic energy .
the virial is a function of the particle positions.(15 ) for the lj liquid , as well as for a united - atom toluene model , a dumbbell model , the kobandersen binary lj liquid,(16 ) and other van der waals liquids , w and u correlate better than 90% in their equilibrium fluctuations.(9 ) this reflects an effective inverse power - law potential dominating fluctuations , as discussed in detail in refs ( 11 and 12 ) .
the correlations are ruined by the hydrogen bonds , which are conventionally modeled via coulomb forces ; the existence of competing interactions prevents strong wu correlation in hydrogen - bonded liquids . for liquids with time - scale separation , like highly viscous liquids ,
strong wu correlations are particularly significant : viscous liquids with strong wu correlations are close to being  single - order parameter liquids  in the classical prigoginedefay sense .
this implies that complex frequency - dependent thermoviscoelastic response functions like the isobaric / isochoric dynamic specific heat , the dynamic thermal expansion coefficient , the dynamic compressibility , etc .
, are all given by a single function.(19 ) in particular , these can not 
decouple(20 ) from one another ; they must all exhibit relaxations in the same frequency range .
it has also been shown that strongly correlating viscous liquids obey density scaling , i.e. , that if the relaxation time  is measured at different temperatures and density  ,  is a unique function(21 ) of /t , where the exponent  may be determined from studies of wu correlations at a single state point.(22 ) finally , it was recently found that strongly correlating viscous liquids have much simpler aging properties than viscous liquids in general.(23 ) fluctuations are ensemble dependent , of course , and one may ask what happens if fluctuations are studied instead in the ensemble of constant temperature and pressure ( npt ensemble ) . in this ensemble ,
virial fluctuations are not interesting , but there are strong correlations for simple liquids between the fluctuations of volume and potential energy.(10 ) this is the ensemble used below for studying biomembrane thermodynamic fluctuations . motivated by the findings for simple model liquids and the heimburgjackson theory , we decided to focus on phospholipid membranes in order to investigate whether the heimburgjackson assumption of strong volumeenthalpy correlations is confirmed .
a phospholipid has van der waals interactions between its acyl chains and hydrogen bonds in the head region .
similarities to simple van der waals liquids are not at all obvious , and the microscopic origin of the volumeenthalpy correlation tentatively derived from experiments is not trivial .
thermodynamic fluctuations of membranes have been studied in the past , however , not with a focus on volumeenergy correlations .
because of the large amount of water in the system and the hydrophilic head groups , one does not expect strong instantaneous correlations of phospholipid membrane thermodynamic fluctuations .
the heimburgjackson theory , however , relates to strong correlations of slow degrees of freedom of the biomembrane ( on millisecond time scales ) , and so do the experiments they quote .
this is analogous to the situation for highly viscous liquids where time - scale separation between the fast , vibrational degrees of freedom and the much slower configurational ones is also , e.g. , essential for viscous flow or visco - elastic responses .
phospholipids are the major constituent of biological membranes.(28 ) close to physiological temperature , these membranes undergo a transition from an ordered phase ( l ) to a disordered phase ( l ) .
below , we evaluate the strength of vu correlations for both the disordered and ordered phases with a main focus on the disordered phase .
we show that vu fluctuations correlate strongly but only on long time scales.(32 ) we also investigate how well membrane area as well as chain order parameter fluctuations correlate with v and u fluctuations ; such correlations are generally weak .
finally , the cause of the correlations is identified by splitting volume and energy into contributions from the tail , head , and water regions : the origin of the slow , strongly correlating vu fluctuations are located in the tail region which are dominated by van der waals interactions .
thus , establishing a conceptual link to strongly correlating fluctuations of simple van der waals liquids.(9 )
dmpc - f and dmpc - g : fully hydrated dimyristoyl - phosphatidyl - choline membrane in the fluid and ordered phases , respectively.(33 ) figure 1 shows configurations of these membranes .
dppc - f and dppc - g : fully hydrated dipalmitoyl - phosphatidyl - choline membrane in the fluid and ordered phases , respectively.(34 ) dppg : a fully hydrated dipalmitoyl - phosphatidyl - glycerol membrane in the fluid phase with calcium counterions.(35 ) dpps : a fully hydrated dimyristoyl - phosphatidyl - serine membrane in the fluid phase with calcium counterions.(35 ) dmpsh : a fully hydrated and protonated dimyristoyl - phosphatidyl - serine membrane in the fluid phase.(35 ) phospholipids ( and counterions ) were modeled using the all - atom charmm27 force field(36 ) with modified charges of the headgroup as described in detail elsewhere.(33 ) membranes were hydrated using explicit water represented by the tip3 model.(37 ) tsim , total simulation time in nanoseconds ; tprod , length of production run in nanoseconds ( only membranes in quasi - equilibrium , i.e. , with no detectable drift in the area per molecule , were included in the data analysis ) ; nlipid , number of lipid molecules ; t , thermostat temperature in kelvin ; tactual , average temperature and standard deviation of fluctuations ; nwater / nlipid , number of water molecules per lipid molecule .
the uncertainty is estimated from the dmpc - f and dppc - f trajectories as described in the text ( 67% confidence interval ) ;  , energyvolume scaling factor in units of 10 ml / j ( see eq 11 ) ; ruut , energyenergy of tail region  correlation coefficient ; rua , energyarea correlation coefficient ; rva , volumearea correlation coefficient ; rascd , areachain order parameter 
correlation coefficient ; ruscd , energychain order parameter  correlation coefficient ; rvscd , volumechain order parameter  correlation coefficient .
snapshots of dmpc membranes in the fluid phase ( dmpc - f ; panel a ) and in the ordered phase ( dmpc - g ; panel b ) .
the red atoms are the oxygen atoms of water molecules ; hydrogen atoms were removed for visual clarity ( but included in the simulations ) .
simulations were carried out using the namd software package.(38 ) in all simulations , a time step of 1.0 fs was used .
temperature and pressure were controlled by a langevin thermostat ( damping coefficient : 5 ps ) and a noshoover langevin barostat ( anisotropic regulation ; piston oscillation time of 100 fs ; damping time of 50 fs).(14 ) electrostatic interactions were evaluated using the particle - mesh - ewald method with a grid spacing of about 1  and updated every 4 fs .
lennard - jones potentials were brought smoothly to zero by multiplying a switching function s(rij ) ; s(rij ) = 1 for rij < ron , s(rij ) = ( roff2  rij2)(roff2 + 2rij2  3ron2)/(roff2  ron2 ) for ron < rij < roff , and s(rij ) = 0 for rij > roff with ron = 10  and roff = 12  .
initial configurations of dmpc - f , dppc - f , dppg , dpps , dmps , and dmpsh are taken from refs ( 3335 ) .
initial configurations of the ordered membranes , dmpc - g and dppc - g , were built from a membrane studied by venable and co - workers.(41 ) throughout the simulation , the acyl chains remain in an ordered structure , as shown in figure 1b.(31 ) thermal equilibrium was ensured by monitoring the membrane area ; only trajectories with no drift ( compared to the thermal fluctuations ) were used in the data analysis .
lengths of equilibrium trajectories are listed in the top section of table 1 in the column under tprod .
the importance of firmly establishing that the system is in equilibrium should be emphasized , since an apparent strong correlation would appear if volume and energy relax from some ( arbitrary ) out - of - equilibrium state .
the following collective quantities were evaluated every 0.5 ps : potential energy u , volume of simulations box v = xyz ( where x , y , and z are the box dimensions ) , projected membrane area a = xy , box thickness z , and average chain order parameter scdch .
the latter quantity characterizes the overall order of acyl chains(43 ) and is defined by scdch = |(/2 )  cos(cd )  ( /2)ch| , where cd is the angle between the membrane normal ( z ) and the ch bond of a given methylene group .  ...
first , we consider the fluid dmpc - f membrane shown in figure 1a . instantaneous fluctuations of volume and energy do not show any significant correlation ( data not shown ) .
this is not surprising , since a significant part of the simulation box is water , and water is known not to show strong correlation.(9 ) however , if fluctuations are averaged over time windows of 0.5 ns , volume and energy correlate strongly , as shown in the top panel of figure 2 .
this is quantified by the correlation coefficient 

 where the bar here and henceforth indicates a 0.5 ns average .
r = 0 corresponds to no correlation , whereas |r| = 1 corresponds to perfect correlation . we will refer to r  0.75 as strong correlation . for comparison ,
correlations in the slow thermal equilibrium fluctuations of volume and energy ( top ) and correlation matrix for the dmpc - f membrane ( bottom ) .
the normalized fluctuations of volume and potential energy shown are averaged over time intervals of 0.5 ns .
data are shifted and scaled such that the average value is zero and the standard deviation is unity .
a significant correlation is observed and quantified by the correlation coefficient , ruv = 0.77 .
this strong correlation can be associated with the tail region , as seen by the similarities with figure 5 .
the bottom panel represents the absolute values of the elements of the correlation matrix of energy , volume , membrane area , thickness , and the average chain order parameter , where dark red illustrates strong correlation .
membrane area , thickness , and average chain order parameter are strongly correlated , but these quantities only correlate weakly with energy and volume .
similar results are found for the other fluid membranes ( except for dmpc - g and dmps where the energyvolume correlation is only 0.47 and 0.59 , respectively ) ; see table 1 .
it is an appealing idea to establish a direct connection between thermodynamic properties and microscopic structures , and several studies have focused on the ordering of the acyl chains as the important microscopic structure .
following this philosophy , one possible explanation for the observed strong correlation is that the order of the acyl chains is the single parameter controlling the fluctuations : if chains , as a result of a thermal fluctuation , become more ordered , one expects a decrease of energy u , volume v , and area a , but an increase of thickness z , and chain order parameter scdch .
thus , the correlation has the expected sign , but it is significantly weaker than ruv. the same is the case for z and scdch .
the bottom panel of figure 2 shows the full correlation matrix . clearly , a single parameter description is not sufficient .
we need two parameters to describe the fluctuations : one parameter controlling v and u and one  geometrical  parameter controlling a , z , and scdch .
the bottom section of table 1 shows that dmpc - f , dppc - g , dppc - g , dmpg , dmps , and dmpsh all have strong volumeenergy correlation ( ruv  0.75 ) .
volume fluctuations of the dmpc - g and dmps membranes , however , are only weakly correlated with energy fluctuations . to show that this finding is not due to uncertainty from random noise
, we estimated the error bar as follows : two of the simulations , dmpc - f and dppc - f , are about 3 times longer than the remaining five simulations .
we used these two long simulations to estimate the error bar of the calculated correlation coefficients by dividing them into three blocks of 40 ns ( regarded as uncorrelated blocks).(44 ) the standard deviation of the correlation coefficient is 0.03 and 0.10 for dmpc - f and dppc - f , respectively , or about 0.07 on average .
each of the five short runs corresponds to a single block , and thus , we estimate the error bar of these to be about 0.07 ( within a 67% confidence interval ) . for
the two longer runs , we use the n rule and estimate an error bar of 0.07/3 = 0.04 .
thus , the uncertainty is smaller than the spread among ruv s , and the weak vu correlation of the dmpc - g and dmps membranes is genuine , a point returned to later . the term  slow fluctuations  has so far been defined via the 0.5 ns averaging time window ( indicated with a bar ) .
a more general approach is to investigate the following time - dependent correlation coefficient 

 similarly , one defines the time - dependent energyarea correlation coefficient ua(t ) .
in contrast to ua(t ) , we observe strong correlation on long time scales for uv(t ) ( uv(t )  0.75 ) for the five membranes where slow fluctuations of v and u correlate strongly consistent with ruv  0.75 ( table 1 , bottom section ) . in the following section ,
the slow parts of the volume and energy fluctuations are investigated via the auto - correlation functions v(0)v(t) and u(0)u(t). time - dependent correlation coefficient of volumeenergy uv(t ) ( left ) and areaenergy ua(t ) ( right ) .
< 10 ns , only correlate weakly , whereas the slow fluctuations , t  1 ns , correlate strongly .
a membrane is a highly heterogeneous system,(28 ) and it is reasonable to divide it into regions.(27 ) let the three regions t , h , and w be defined as follows : t ( tail ) refers to the hydrophobic acyl - chain atoms ( i.e. , atoms of methylene and methyl groups in the acyl chains ) , h ( head ) refers to the hydrophilic lipid atoms ( the remaining lipid atoms ) , and w refers to the water atoms ( and counterions ) . to identify the origin of the slow volume fluctuations , we construct voronoi polyhedra(45 ) of heavy atoms ( i.e. , ignoring hydrogen ) and sum the voronoi volumes for the regions t , h , and w. in this way
, the total volume of the simulation box is divided into three terms : 

 the auto - correlation function of the volume in eq 2 can now be split into a sum of three auto- and three cross - correlation functions : 

 figure 4a shows these six functions for the dmpc - f membrane . the only nonvanishing function at long times ( responsible for the slow fluctuations ) is the auto - correlation function of the hydrophobic ( tail ) part of the membrane , vt(0)vt(t). this is verified by rvvt = 0.94 being close to unity .
time - dependent auto- and cross - correlation function of volume and energy terms of the dmpc - f membrane .
panel a : volume correlation functions of the tail , head , and water regions ( eq 4 ) .
the shot - time tailwater anticorrelation is probably an artifact of the way the voronoi polyhedron is constructed ( related to the tailwater interface , see text for details ) .
panel c : energy correlations split into intramolecular , lennard - jones ( lj ) , and coulombic energies ( eq 7 with x = tail ) .
the short time cross - correlation between water and the tail region is significant and negative .
this is , however , probably a spurious consequence of the voronoi construction between neighboring groups of different size : when a water molecule approaches the boundary of the tail region , it gains an ( unfair ) increase of the voronoi volume , due to methylene groups being larger than water molecules .
this shows up as a negative correlation , since the water molecule  steals  volume from the methyl groups , however , only at short times ( t < 0.1 ns ) . in the simulation ,
the potential energy of the system consists of a sum of lennard - jones terms , coulomb pair energy terms , and intramolecular energies : 

 where uijlj = 4ij((ij / rij )  ( ij / rij ) ) are lennard - jones terms ( van der waals interactions ) , uijcoul = qiqj/(40rij ) are columbic terms , and u is a sum of the reminder terms ( which are intramolecular ; harmonic bonds , angles , dihedrals , and improper dihedrals ) .
again , we split the total potential energy into contributions from regions of tails , heads , and water : 

 where 

 where x represents either tails , heads , or water .
as for the volume , the auto - correlation function of the energy fluctuations in eq 2 is also split into a sum of three auto- and three cross - correlation functions : 
  figure 4b shows the six auto- and cross - correlation functions of ut , uh , and uw .
again , the slow fluctuations are dominated by the tail region . it should be noted , though , that the slow tailtail correlation is not quite as dominating , as for the volume fluctuations . this is quantified by ruut = 0.82 ( table 1 , bottom section ) not being as close to unity as rvvt = 0.94 ; thus , slow energy fluctuations of the head uh and water region uw are significant ( in contrast to slow volume fluctuations of the head and water region ) .
figure 4c shows the auto- and cross - correlation functions corresponding to an additional splitting of the tail energy into  intramolecular interactions  ,  coulombic interactions  , and 
in the previous section , it was demonstrated that slow energyvolume correlation ( of some membranes ) originates from van der waals interactions in the tail region ; thus , 

 where 

 this is consistent with our findings for simple strongly correlating liquids .
it should be remembered , however , that eq 9 neglects the energy terms uh and uw with some significance . if we correlate the slow fluctuations ( defined as a 0.5 ns average ) of the voronoi volume of the tail region and the van der waals energy of the tail region , we find a correlation coefficient of rvtutlj = 0.87 , as shown in figure 5 .
the same number for the fluctuations for the whole simulation box was rvu = 0.77 .
consistent with this , the bottom section of table 1 shows a general correlation between rvu and ruut ; e.g. , for the weakly correlated membranes , dmps and dmpc - g , both rvu and ruut have low values .
a better understanding of how this loss of correlation is related to various lipid head groups deserves future study .
normalized fluctuations of voronoi volume and van der waals energy of the tail region of the dmpc - f membrane , vt(t ) and utlj(t ) .
vt(t ) and utlj(t ) are shifted and scaled so the average value is zero and the standard deviation is unity .
the correlation is strong with a correlation coefficient of 0.87 ( justifying eq 9 ) .
note the similarity with figure 2a ; thus , slow vu fluctuations are dominated by the tail region .
it is convenient to define a volumeenergy scaling factor via 

 so that if ruv = 1 then v = u. for the seven membranes , given in table 1 , we find that  ranges from 6.1  10 ml / j ( dmpc - g ) to 13.2  10 ml / j ( dmpsh ) .
this scaling factor is indirectly accessible though experiments : close to the phase transition temperature , curves of excess enthalpy h(t ) and volume v(t ) can be superimposed.(7 ) heimburg showed that such a relation is trivial if excess volume and enthalpy of relevant microstates are proportional:(6 ) v = vhh .
the inverse vh scaling factor can be written as ( vh ) = h/v = u/v + p. since p = 1 atm = 10 j / ml is much smaller than u/v  10 j / ml ( a typical value as seen in table 1 ) , then h/v  u/v and the pv contribution to enthalpy is negligible .
moreover , the excess volume and enthalpy of the phase transition is expected to be related to the slow equilibrium fluctuations close to the transition .
thus , the scaling factor calculated from equilibrium simulations ( eq 11 ) is closely related to the ( experimental )  phase transition  scaling factor vh , and we expect them to have the same value . in agreement with this ,
ebel , grabitz , and heimburg(7 ) report a scaling factor of ( 9.01  0.71 )  10 and ( 8.14  0.67 )  10 ml / j for large unilamellar vesicles of dmpc and dppc , respectively , only slightly smaller than our findings of 9.6  10 and 10.2  10 ml / j , respectively ( table 1 , bottom section ) .
this supports that we have identified the microscopic origin of the scaling of the phase transition .
the proportionality constant  , however , is not the same in both phases ( table 1 , bottom section ) but is about /3 of the fluid value in the ordered phase .
ebel , grabitz , and heimburg(7 ) reported a similar decrease ( 65% ; multilamellar vesicles of dmpc or dppc ) when comparing scaling factors of the pretransition with the main transition .
this suggests that the configuration of the ordered membranes ( dmpc - g and dppc - g ) is related to the pretransition , and that the nature of the correlations changes when passing the phase transition .
interestingly , this is in contrast to our earlier findings for crystallization of the standard lennard - jones liquid .
more work is needed to clarify the cause of this difference between biomembranes and simple liquids .
biologically relevant membranes are mixtures of many sorts of amphiphilic molecules and are far more complex than the membranes studied here .
the origin of the strong correlations is , however , not specific , but reflects the van der waals bonded nature of the core of the membranes .
how solutes in membranes affect the volumeenergy correlation is an interesting subject for a future study .
another concern could be the finite size of the simulated membranes , since fluctuations on length scales larger than the length of the simulation box can not be represented.(26 ) again , since the origin of strong vu correlations is intrinsic to van der waals interactions , the correlations are not expected to be affected by the size .
one caveat is that , although we do observe some correlation between lipid area and energy fluctuations , the correlation is weaker than the volumeenergy correlation .
their conjecture depends on the membrane being close to the phase transition , and it is possible that such strong au correlation appears only when approaching the phase transition .
this can be investigated by evaluating the slow au correlation coefficient as a function of temperature .
also , it is possible that such a correlation only appears on even slower time scales ( e.g. , s ) .
this paper reports a study of thermodynamic equilibrium fluctuations of phospholipid membranes . on long ( nanosecond ) time scales , we identify strong volumeenergy correlations of a kind that were previously only observed for simple model liquids .
these correlations are mainly documented for the fluid phase , but we also show that they exist in the ordered phase .
it may seem surprising that a complex system like a biomembrane exhibits such strong thermodynamic correlations .
the identification of the origin of the correlations as deriving from the van der waals interactions of the hydrophobic part of the membrane , however , points to a common origin of strong thermodynamic correlations in simple van der waals liquids and biomembranes .
this is consistent with the finding that there are strong energyvolume correlations in both fluid and ordered phases : the correlations do not depend on the degree of chain order , just as for simple liquids where the strong correlations survive crystallization .
we find weaker correlation between energy / volume and  geometrical  order parameters such as membrane area , thickness , and scd ( characterizing ordering of acyl chains ) .
thus , one parameter is not sufficient to describe thermodynamic fluctuations . regarding the heimburgjackson nerve signal theory
, our findings largely confirm one crucial assumption of this theory , namely , that volume and energy ( enthalpy ) correlate strongly for microstates .
we find strong correlations only on the nanosecond and longer time scales , which are , however , precisely the relevant times for nerve signals .
 this paper reports all - atom computer simulations of five phospholipid membranes ( dmpc , dppc , dmpg , dmps , and dmpsh ) with focus on the thermal equilibrium fluctuations of volume , energy , area , thickness , and chain order . at constant temperature and pressure 
 , volume and energy exhibit strong correlations of their slow fluctuations ( defined by averaging over 0.5 ns ) . these quantities , on the other hand , do not correlate significantly with area , thickness , or chain order . 
 the correlations are mainly reported for the fluid phase , but we also give some results for the ordered ( gel ) phase of two membranes , showing a similar picture . 
 the cause of the observed strong correlations is identified by splitting volume and energy into contributions from tails , heads , and water , and showing that the slow volumeenergy fluctuations derive from van der waals interactions of the tail region ; they are thus analogous to the similar strong correlations recently observed in computer simulations of the lennard - jones and other simple van der waals type liquids ( u. r. pedersen et al . , 
 phys . 
 rev . 
 lett.2008 , 100 , 015701 ) . 
 the strong correlations reported here confirm one crucial assumption of a recent theory for nerve signal propagation proposed by heimburg and jackson ( t. heimburg and a. d. jackson , proc . 
 natl . 
 acad . 
 sci.2005 , 102 , 97909795 ) . 
Introduction
Simulation Details
Results
Discussion
Conclusions