Systematic coarse graining from structure using internal states: Application to...

Systematic coarse graining from structure using internal states:Application to phospholipid/cholesterol bilayer

Teemu Murtola,1,a� Mikko Karttunen,2 and Ilpo Vattulainen1,3,4

1Department of Applied Physics and Helsinki Institute of Physics, Helsinki University of Technology,P.O. Box 1100, FI-02015 Espoo, Finland2Department of Applied Mathematics, The University of Western Ontario, 1151 Richmond St. N., London,Ontario N6A 5B7, Canada3Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland4Department of Physics, Memphys–Center for Biomembrane Physics, University of Southern Denmark,Campusvej 55, DK-5230 Odense M, Denmark

�Received 26 January 2009; accepted 10 June 2009; published online 5 August 2009�

We present a two-dimensional coarse-grained �CG� model for a lipid membrane composed ofphospholipids and cholesterol. The effective CG interactions are determined using radial distributionfunctions �RDFs� from atom-scale molecular dynamics simulations using the inverse Monte Carlo�IMC� technique, based on our earlier work �T. Murtola et al., J. Chem. Phys. 121, 9156 �2004�; J.Chem. Phys. 126, 075101 �2007��. Here, the original model is improved by including an internaldiscrete degree of freedom for the phospholipid tails to describe chain ordering. We also discuss theproblem of RDF inversion in the presence of internal states, in general, and present a modified IMCmethod for their inclusion. The new model agrees with the original models on large-scale structuralfeatures such as density fluctuations in pure dipalmitoylphosphocholine and cholesterol domainformation at intermediate concentrations and also indicates that ordered and disordered domainsform at all cholesterol concentrations, even if the global density remains uniform. The inclusion ofordering also improves transferability of the interactions between different concentrations, but doesnot eliminate transferability problems completely. We also present a general discussion of problemsrelated to RDF inversion. © 2009 American Institute of Physics. �DOI: 10.1063/1.3167405�

I. INTRODUCTION

With advances in computer power, atomistic simulationsusing classical molecular dynamics �MD� have become astandard tool in studies of complex biomolecular systemssuch as lipid bilayers. However, even with present-day com-puters and software, atom-scale simulations can only accesslength scales up to perhaps 20 nm and time scales of theorder of microseconds or less.1,2 Hence, simplified coarse-grained �CG� models are needed for studies of many phe-nomena occurring at larger scales,3 examples being phaseseparation and other structural rearrangements.

A large amount of effort has been devoted to accurateparametrization of atomistic potential energy functions,4–9

but following similar routes for CG models is less straight-forward. Because of the lower level of detail, quantitativereproduction of �some� properties for some specific systemcan typically only be achieved by a parametrization for thatspecific system. In such parametrization efforts, it is oftenadvantageous to use information from atom-scale models ina systematic fashion. Other approaches are also possible: Forexample, the popular MARTINI model10–12 has been param-etrized similarly to atomistic potentials, using densities andpartition coefficients between different substances. However,such an approach becomes more and more difficult as thelevel of detail decreases.

Systematic CG approaches calculate a set of target quan-tities from the atom-scale simulation, and then attempt toparametrize the CG model such that these quantities are re-produced as accurately as possible. Research on these meth-ods has focused on two different approaches, using eitherstructural information13–15 or forces.16–19 In the first, the ra-dial distribution functions �RDFs� are calculated for the CGparticles from the atomistic simulation, and pairwise interac-tions are then constructed such that the original RDFs arereproduced. In the second, a similar procedure is carried outfor pairwise forces using a set of atomistic configurations:The forces on each CG particle are calculated for each con-figuration, and a least-squares fit is performed to obtain theCG forces. Although at first sight very dissimilar, the force-matching approach can also be related to the structural prop-erties of the original system.20

The approaches above focus on determining CG interac-tions from atom-scale simulations. Another problem in con-structing CG models is choosing good degrees of freedom: Itis difficult to determine a priori whether some choice is bet-ter than another in describing the underlying system, givensome constraints on the complexity of the model. A few dif-ferent approaches have been proposed for this problem,mostly in the field of protein simulations.21–25 These are ei-ther based on analysis of a single structure21–23 or on preser-vation of dynamical information.24,25 Use of clustering algo-rithms or self-organizing maps in this context has also beendiscussed.26,27a�Electronic mail: [email protected].

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http://dx.doi.org/10.1063/1.3167405

http://dx.doi.org/10.1063/1.3167405

http://dx.doi.org/10.1063/1.3167405

This paper focuses on the problem of determining thepairwise interactions, given the degrees of freedom and theRDFs. This is the so-called inverse problem in classical sta-tistical mechanics, and it is known that such interactions areunique and they exist whenever such RDFs can beproduced.28–30 However, there are often practical problems indetermining this unique interaction; one aim of this paper isto critically discuss these issues.

As a model system, we use a lipid bilayer composed ofdipalmitoylphosphocholine �DPPC� and cholesterol, a sys-tem which we have also previously studied both at atomisticlevel31 and using coarse graining.32,33 Already such a two-component system shows interesting phase behavior, the mi-croscopic details of which are still elusive, in particular, atintermediate cholesterol concentrations �see, e.g., Ref. 34and references therein for recent experiments, and Ref. 35for theoretical work�. Computer simulations would, in prin-ciple, be ideal to study their detailed behavior. However,both the length and time scales needed to see phase separa-tion in multicomponent lipid bilayers are currently beyondthe reach of atomistic simulations. Hence, CG models areneeded, and here we focus on a very simple model, with theidea that even such simple models could be able to giveinsight into the phase behavior if carefully parametrized.Also, a high degree of coarse graining, i.e., going for assimple models as possible, provides a highly nontrivial testcase for development and assessment of CG models andmethodology.

In our previous CG studies, we have constructed simpletwo-dimensional �2D� models for the large-scale structure ofDPPC/cholesterol mixtures.32,33 In the first model,32 eachmolecule was replaced by a single pointlike particle, and thesecond model33 extended this description to include threeparticles for DPPC, one for the headgroup and one for eachtail. The particles were thought of as describing the center ofmass �CM� of �the part of� the molecule, and effective inter-actions were constructed to reproduce RDFs calculated fromatomistic simulations. In this paper, we take one step furtherand include the ordering of the DPPC tails in the modelusing a discrete, internal degree of freedom: Each chain hasa two-state degree of freedom that labels the chain as eitherordered or disordered, similarly to the phenomenologicalmodels of Nielsen et al.36,37 Hence, this paper completes theprocess of systematically constructing a CG model that issimilar to these phenomenological models; one of the origi-nal motivations was the success of these models in reproduc-ing the phase behavior of lipid/sterol mixtures. As the inter-actions in the above phenomenological models do notdepend on the thermodynamic state,36,37 it is also interestingto study the effect of the new degrees of freedom on thetransferability of the CG interactions: In the earlier modelswithout internal states, transferability was quite poor, in par-ticular, across phase boundaries. It should then be expectedthat including the internal states would improve transfer-ability.

In this paper, we will first discuss the theoretical back-ground of systematic coarse graining and RDF inversion inSec. II. Next, Sec. III discusses the inverse Monte Carlo�IMC� method,13 which has been used in this work to

achieve RDF inversion. Improvements and changes to theIMC algorithm, necessary for construction of the presentmodel, are also presented. The construction of the new CGmodel is presented in Sec. IV, and associated problems arediscussed. Monte Carlo �MC� simulations of larger systemsusing the model are described and discussed in Sec. V. Fi-nally, Sec. VI contains a thorough discussion of the resultsand also discusses the RDF-based coarse graining approachin general.

II. COARSE GRAINING AND CLASSICALINVERSE PROBLEM

The aim of systematic coarse graining is to reproducethe properties of the microscopic model as well as possible.If the microscopic degrees of freedom R� and the CG degreesof freedom r� are connected by a mapping function r�=M� �R� �,we can define an effective Hamiltonian Heff as

e−�Heff�r�� =� e−�H�R� ��M�R� � − r��dR� , �1�

where H�R� � is the configurational part of the microscopicHamiltonian and ��x� is the Dirac delta function. Althoughthis equation is typically written such that r� contains only theCG bead positions, it is also valid more generally. In particu-lar, we can include discrete internal degrees of freedom intor�; in this case, the mapping function should also give valuesfor the internal degrees of freedom. For clarity, we do notwrite these internal degrees of freedom explicitly, but it is tobe understood that they are included in r�.

The right-hand side of Eq. �1� is the sum of the Boltz-mann factors of all microstates consistent with specified val-ues of the CG degrees of freedom r�; the left-hand side is justthe Boltzmann factor of a system with Hamiltonian Heff.Hence, if we could sample CG configurations according toHeff defined by Eq. �1�, all properties that depend only on theCG degrees of freedom r� could be calculated exactly. Notethat Heff is a free energy of a particular configuration r�, andhence contains entropic effects in addition to the potentialenergy. For the same reason, Heff depends on the thermody-namic state.

It is not practical to estimate Heff directly for severalreasons: �i� r� can be high-dimensional, making it impossibleto store Heff, e.g., on a grid, �ii� as such, Heff can only beapplied to a system identical in size to the microscopic one,and �iii� evaluation of any free energies from microscopicsimulations is costly. For problems �i� and �ii�, it is possibleto approximate Heff with

Heff = w0 + �i,j

w2�rij� + �i,j,k

w3�rij,rik,rjk� + ¯

� V0 + �i,j

V2�rij� , �2�

where we have assumed that the system is isotropic and ho-mogeneous, and wn is the direct n-particle interaction thancan be fixed by requiring that Eq. �2� should hold for all Nwith the same functions wn, where N is the number of par-ticles. The sum on the first line can in principle range up toan N-body term. The second line suggests a computationally

055101-2 Murtola, Karttunen, and Vattulainen J. Chem. Phys. 131, 055101 �2009�


convenient approximation: The sum is replaced by an effec-tive pair interaction V2�r�, which also contains effects fromthe higher-order terms. In this form, we only need to storethe pair interaction, and the form can also be extrapolated tolarger systems. It should be noted that the approximation �2�results in an additional state dependence of Heff: The bestV2�r� may have a different state dependence than the exactHeff. This state dependence can then lead to inconsistent ther-modynamics if the origins of the pair interaction are not keptin mind.38 Also, although the constant and the single-particlefield �absorbed into V0 in Eq. �2�� are independent of particlepositions, they may depend on the thermodynamic state andshould be taken into account to treat thermodynamicsaccurately.39

Approximation �2� does not solve problem �iii�; hence,some derived quantity, not Heff itself, needs to be used forobtaining V2�r�. Force matching uses derivatives of Eq. �1�with respect to r�;18 the approach discussed here is based onstructural quantities. The structural quantity most often usedin this context is the RDF g�r�: It gives the probability ofobserving a particle at distance r from the origin, given thatthere is one at the origin. The normalization is such thatg�r�=1 everywhere for a homogeneous distribution, i.e., inan ideal gas.

It is not immediately clear that taking the RDFs as thetarget would result in well-defined interactions. A partial an-swer to this problem was provided by Henderson in 1974,who proved that if two systems with pairwise interactionsgive the same RDFs, the Hamiltonians can only differ by aconstant, i.e., the pairwise interactions are unique.28 Tenyears later, a rigorous proof was given for the existence ofsuch interactions.29,30 It has also been shown that of all sys-tems that reproduce the given RDFs, the one with pairwiseinteractions has the highest entropy.40 More recently, RDFmatching has been shown to be equivalent to maximizing thelikelihood that the configurations produced by the CG modelproduce the atomistic probability distribution.41

The uniqueness proof is a straightforward application ofthe inequality

F2 − F1 � �H2 − H11, �3�

where H1,2 are the Hamiltonians of the two systems, F1,2 aretheir free energies, and the equality holds only if the Hamil-tonians differ by a constant.28 The existence proof is morecomplex, but the basic idea is to show that the functionalF�V� defined by

�F�V� = lnexp�− �V�x,x��2��x,x��dxdx��

exp�− ��W�x1, . . . ,xN� + �i,jV�xi,xj��dNx�4�

attains its maximum with a pair potential V, which is thenshown to give the correct RDF g �which is the normalizedversion of the target two-particle density ��2��.29,30

W�x1 , . . . ,xN� is an arbitrary fixed N-particle interaction thatacts in the system. The proof by Chayes et al.29,30 then showsthat for any such W �with some finiteness criteria� and any��2� that is a two-particle reduction of an admissibleN-particle density, there exists a unique pair potential thatreproduces ��2�. Here, admissibility just refers to finiteness ofcertain thermodynamical functionals.30 We also note that F

differs from the relative entropy defined in Ref. 41 only by aconstant that does not depend on V. It should be noted that inthe commonly used scenario where the system has a finitevolume, the existence proof is strictly valid only in a systemwith the same volume as in which the RDFs were deter-mined. In a system with a different volume, the pair interac-tions will also, in general, be different. Hence, when devisingmethods for the inversion, one should perform fitting forsystems with identical sizes, as also noted in the originalIMC paper.13

As discussed above, the RDFs calculated from any simu-lation define a unique set of effective pair interactions underquite general conditions. Thus, using them as a target incoarse graining results, at least theoretically, in a well-defined model. By construction, this model then reproducesthe short-range structure, although using the effective inter-actions in larger systems than in which they were determinedcan lead to minor changes in the RDFs.33 In principle, thethermodynamics of the system can also be recovered throughthe compressibility route38,42 because the equation only in-volves RDFs. Other thermodynamic equations, such as thevirial pressure, lead, in general, to different results.38 Also,three-particle correlation functions will, in general, be differ-ent from the atomistic ones.42,43

III. IMC METHOD

IMC is one practical method for constructing the uniquepair interactions that lead to given RDFs.13 For other ap-proaches, see, e.g., Ref, 15 and references therein. In thissection, we briefly describe the IMC method and our modi-fications and improvements.

A. Basic idea

The central idea of the IMC method is to write theHamiltonian of the system in the form13

H = ��

S�V�, �5�

where the sum goes over potential bins, V� is the value of thepotential within the bin �, and S� is the number of particlepairs that fall within this bin. Now, �S�MD, where the sub-script MD indicates that the average is calculated for theatomistic positions mapped to the CG degrees of freedom, isa grid approximation to the target RDFs that we want the CGmodel to reproduce. Using Eq. �5� for the Hamiltonian, onecan then write a linear approximation for the change in �S�in terms of changes to V� as13

��S� = ��

��S��V�

�V�,��S��V�

= −�S�S� − �S��S�

kBT.

�6�

IMC then proceeds as follows. First, we take an initial guessV0 for the potentials and evaluate �S� and the derivativesfrom simulations of the CG model that employ this initialpotential. Then, we set ��S�= �S�MD− �S�. This allows usto solve for �V from a linear set of equations and set V1

=V0+�V, resulting in a better approximation for the interac-

055101-3 Structural coarse graining using internal states J. Chem. Phys. 131, 055101 �2009�


tions. This procedure is then iterated until the RDFs matchsufficiently well. Any interaction �or any other quantitywhich appears as such a linear term in the Hamiltonian� forwhich a target value �S�MD �or an equivalent quantity� isknown can be included in the iteration, although for bondedinteractions, special care must be taken to make the solutionto the matrix equation unique: In this case, the derivativematrix has a zero eigenvalue for each bonded potential, cor-responding to addition of a constant to the potential.33

Here, we note an alternative approach for arriving at thesame equations: If the functional �4� is maximized withNewton’s method and the Hamiltonian �5� is used, the equa-tion for one iteration becomes exactly Eq. �6�. This alsogives some insight into the convergence of IMC: if the de-rivatives can be calculated exactly and the trial potential isclose to the correct one, Newton’s method convergesquadratically.44 However, the effect of noise on the deriva-tives requires more careful analysis, which is beyond thescope of the present discussion. Also, if the trial potential isnot close, there is no guarantee of convergence; the directionof the change is an ascent direction for Eq. �4�, but the stepmay be too long.

B. Regularization and thermodynamic constraints

As described previously, it is advantageous to change thelinear equation to a quadratic minimization problem:33 In-stead of solving a matrix equation of the type Ax=b, weminimize Ax−b . In this form, it is possible to apply effi-cient regularization to better deal with noise in the input data.Further, it is also possible to add a thermodynamic constraintto constrain the surface tension of the system to avoid un-physical interactions.33 That is, the potentials can be deter-mined under the constraint that the surface tension � is con-strained to a predetermined value. We define � using thestandard virial formula,

�V = �Ekin +1

2��i�j

f ijrij� , �7�

where �Ekin=NkBT is the average kinetic energy. Here, V isthe area of the system and f ij and rij are the force and thedistance, respectively, between particles i and j. The internaldegrees of freedom only enter through the forces f ij, whichdepend on the internal states of the involved particles. Theconstraint can be treated identically to Ref. 33: The con-straint that � should remain at a particular value is addedthrough a Lagrange multiplier. One should note that thevalue of the surface tension defined by Eq. �7� is not thethermodynamic surface tension of the system because theinteractions depend on the thermodynamic state �see Sec. II�.Because of this, the value needs to be fixed using some aux-iliary quantity. In Ref. 33, the area compressibility was quali-tatively fitted to experimental values, and the same surfacetensions are also used in this study to facilitate comparison.It is also worth to note that the exact value of the surfacetension does not alter the qualitative results obtained fromthe model.33

C. Internal degrees of freedom

For the present model, we also have discrete degrees offreedom that describe the chain ordering. In practice, thismeans that the degrees of freedom for a single chain are nowxi= �r�i ,si�, where si� �0,1�. Here, and also in the rest of themanuscript, we refer to particle types when all internal statesare considered as a single type and particle kinds when eachinternal state is separated. The states si enter the descriptionthrough two ways: �i� each internal state is treated as a sepa-rate particle kind, determining the pair interactions throughwhich the particle interacts, and �ii� additional terms areneeded in the Hamiltonian to describe the internal �free� en-ergies of the states. Full treatment of the issue is given inAppendix A; here we just present the end result that theHamiltonian to be parametrized becomes

H = ��

S�V� + �i

Eini + �i,j

Eij�ni�nj . �8�

Here, Ei and Eij are parameters and ni are the numbers ofparticle of each kind. In Eq. �8�, �ni=ni−ni

ave measures thefluctuations from an arbitrary constant reference numberni

ave. Since the total number of particles of each type remainsconstant, the ni are not all independent of each other. Tomake the Hamiltonian unique, Eq. �8� should be interpretedsuch that the sums only include an independent set of the ni.In the case of the present model, this means that the totalnumber of tails is a constant. Hence, only the number ofdisordered tails �denoted as nd� is required �or equivalentlyno, the number of ordered tails�. The Hamiltonian then be-comes

H = ��

S�V� + �End + Efluct�nd2, �9�

where we have used �E to denote the internal energy differ-ence between the ordered and disordered kinds and Efluct forthe second-order term.

Full justification for Eq. �8� is given in Appendix A.Here it suffices to note that the second-order term in ni,which distinguishes the system from a standard semigrandcanonical ensemble, is needed for precise control of the num-ber of particle pairs in the system: ninj terms appear in thenumber of pairs in the system, and a certain number of pairsis required for exact reproduction of the two-particledensities.

The nonlinearity in ni makes it nontrivial to generalizethe model to a larger system as the Hamiltonian as such is nolonger an extensive quantity. If the particle number increasesby a factor M, ni

ave should be transformed to Mniave to pre-

serve �ni /N, where N is the total particle number. However,the Eij terms can be treated in two different ways: �i� Eij arekept constant or �ii� Eij are scaled to Eij /M. However, neitherworks well if Eij �0, as �ii� does not preserve the energeticsof flipping a single state, while �i� can make the cases ni=0and ni=N exceedingly favorable. As accurate matching ofthe RDFs seems to lead to such problems in the present case,we have only used interactions with Eij 0 for larger sys-tems, and the other cases are only studied in a system iden-tical in size to the underlying atomistic simulations.



It is also possible to apply the IMC in such a way that Eij

is set to zero and not included in the iteration, also neglectingthe value of ��ni�nj during the inversion. This correspondsto the standard semi-grand canonical ensemble. This can becombined with any of the implementations discussed belowand is applied in some of the calculations discussed in Secs.IV C and V.

D. Straightforward implementation

IMC can be generalized straightforwardly to Hamilto-nians of the form �8� since all the parameters appear linearlyin the Hamiltonian. The calculation of the derivatives can becarried out as before, and the linearization �6� still holds withadditional terms for the derivatives with respect to Ei and Eij.A similar linearization can also be written for the ��ni and��ni�nj, which results in an extended set of linear equa-tions that can be solved for changes in the potentials and theinternal energies. Subsequently, this approach is referred toas method A. In Sec. IV C, we also briefly employ a variantwhere Ei are fixed and the linearization �6� for each ��ni areused as a constraint to keep �ni at its target values. Thisvariant is referred to as A�.

E. Alternative implementations

There are issues with method A that may result in poorgeneralizability of the determined interactions to larger sys-tems; these are discussed in more detail in Secs. IV C andVI. Hence, we have implemented several alternative ap-proaches, collectively referred to as method B. The basicprinciple in this approach is to optimize the RDFs andthe energy parameters separately. This is achieved by calcu-lating the target pair counts for each iteration as S�

t

= �S�MD�N� / �N�MD, where N� is the number of pairs inter-acting through the interaction associated with S�. This effec-tively separates the changes in the structure from changes inthe number of particle pairs. This is in contrast with methodA, where the target pair counts are calculated directly fromthe target two-particle density �S�

t = �S�MD� and remain thesame for every iteration. For consistency, we also need to useEq. �6� for �S� without the terms coming from Ei or Eij inmethod B. Essentially, method B then solves �V as if therewere no internal states, only scaling the target pair countsappropriately.

Several different subimplementations of method B arepossible, differing in the way the derivative matrix��S� /�V� is evaluated.

�1� Use Eq. �6� directly �without terms from Ei or Eij�.�2� Do an ad hoc modification to factor out the effects of

state occupancy changes on the pair counts, using areplacement S�→S��N� /N� inside all the expectationvalues in Eq. �6�.

�3� Calculate the derivative for �S� / �N� instead of �S�.

Of these, �i� is the simplest, but it does not necessarilyconverge to the correct solution because the derivative ma-trix still contains effects from changes in the number ofpairs. The concrete effect is that Ei remain close to their

initial values because any change in the interactions thatwould change the pair counts is disfavored. This limits thespace of interactions that the method searches, but withinthis space, it still finds a good match for the RDFs. Approach�ii� cannot be easily justified theoretically, but solves theabove problem. �iii� leads to a nonsymmetric matrix and hasstability problems and was only studied briefly in this work.The first two approaches are referred to as B� �for �i�� and B,respectively. The reason for this labeling becomes more ap-parent when comparing with method A, since A and B, and,respectively, A� and B� behave similarly. The last approachis referred to as B�iii�.

The implementation so far discussed does not considerhow Ei and Eij should change. For all the subimplementa-tions, we have used the same approach to adjust these pa-rameters. First, after the potential change has been evaluated,Ei are adjusted using the linearizations �6� for ��ni to keep�ni constant. An additional adjustment is then made to ob-tain the desired ��ni and ��ni�nj, using the iteration for-mulas derived in Appendix B. Here, let us only quote thefinal formulas,

�E�2� =1

2��−1� , �10�

��E�1� − 2�E�2�nave� = − ��n + ��−1�n� . �11�

Here, E�1� and E�2� are the vector and matrix formed from Ei

and Eij, respectively, �n is a vector with the average stateoccupancies from the CG simulation, and is a matrix withthe fluctuations in the state occupancies from the CG simu-lation. In our implementation, we have neglected the secondterm in the latter equation because � is typically small.

IV. MODEL CONSTRUCTION

A. General

As before, we construct a simple model with the follow-ing assumptions.32,33 First, the interaction between the twomonolayers is assumed to be weak, allowing us to constructa model for a single monolayer only. The effect of undula-tions is also assumed weak. Together, these assumptions al-low a 2D model to be constructed. Also, we assume that thesystem can be adequately described with pairwise and iso-tropic effective potentials.

Our new model uses three particles per each DPPC mol-ecule, similarly to Ref. 33. One particle is used for the head-group and the glycerol parts and one for each of the tails.One particle is used for each cholesterol molecule. Theseparticles are thought to describe the CM positions of thecorresponding groups of atoms. Each tail particle has a two-state internal degree of freedom that describes ordering. Thisstate can change during the simulation, and it selects thenonbonded interaction tables that the particle uses. Althoughthe tails of a DPPC molecule are not completely equivalent,we assume that the nonbonded interactions of these particlescan be treated identically.

For bonded interactions, there are two separate head-tailbond interactions �one for each tail� and one tail-tail interac-tion, but no angle potentials. A standard practice in many



simulations is to use exclusions for bonded interactions, i.e.,nonbonded interactions are ignored between particles con-nected by bonds. However, the derivation for Eq. �8� isstrictly valid only when exclusions are not used. Hence,when considering models with the full Hamiltonian �Efluct

�0�, nonbonded interactions are also present betweenbonded pairs, in contrast to Ref. 33. If Efluct is zero, thetreatment of exclusions is theoretically not as important, andin this case we did use exclusions. For simplicity, we assumethat the bonded interactions do not depend on the orderingstate of the particles.

With the above assumptions, the model consists of fourkinds of particles �headgroup, ordered and disordered tails,and cholesterol� that interact through ten different non-bonded interactions �one for each pair� and three differentbonded interactions. There are also two internal energy pa-rameters �see Eq. �9��. We set the reference point for thefluctuations as nd

ave= �ndMD.As with the previous models,32,33 the IMC process re-

quires the RDFs between all pairs of CG particles and theareas per molecule from the atomistic simulations. In addi-tion, �ndMD and ��nd�ndMD are needed. For method A, weneed the two-particle density �S�MD and not the RDFs di-rectly; hence, we also need the expected number of particlepairs for each RDF �or we can calculate the two-particledensity directly�.

In order to calculate the above quantities from atomisticsimulations, we need a definition that allows us to character-ize any conformation of a DPPC tail as either ordered ordisordered. For this purpose, we have decided to use thechain order parameter that is commonly used in characteriz-ing the ordering of the tails, even when considering instan-taneous order.35,45,46 Here, we have averaged the order pa-rameter over all carbon atoms in the chain,

Szz =1

Nc − 2 �i=2

Nc−11

2�3 cos2 �i − 1� , �12�

where Nc=16 is the number of carbon atoms in a chain and�i is the angle between the bilayer normal and the orientationof the chain at the ith carbon. We have taken the z axis as thebilayer normal, and the orientation �i is calculated using thevector connecting the carbons i−1 and i+1. A chain wasidentified as being ordered if Szz�C, where C is a cutoffvalue whose selection will be discussed below.

B. MD analysis

The MD simulations used as the basis for the modelconstruction were the same as with the previous models inRefs. 32 and 33, and details of the simulations can be foundin Ref. 31. Briefly, the simulated systems consisted of 128fully hydrated lipid molecules, i.e., DPPCs and cholesterols.The Berger set of parameters was used for DPPC,47 and cho-lesterol parameters were taken from Höltje et al.48 The simu-lations were performed with 0, 3, 8, 13, and 19 cholesterols/ML, corresponding to cholesterol molar concentrations 0%,5%, 13%, 20%, and 30%. Each simulation was runfor 100 ns in the NPT ensemble at T=323 K using the

GROMACS simulation package,49 and first 20 ns was treatedas equilibration.

Figure 1 shows the distribution of instantaneous chainSzz values for the different cholesterol concentrations. Thedistribution is smooth with a single maximum at all concen-trations, showing that there is no clear distinction betweenordered and disordered chains. Hence, the selection of the Szz

cutoff for our two-state model is somewhat arbitrary and isonly limited by the condition that at all concentrations, thereshould be a significant fraction of both ordered and disor-dered chains to give reasonable statistics for all the RDFs.Also, a priori it is reasonable to take the cutoff at the samevalue for each concentration. Based on these considerations,a value of 0.6 was selected. However, it should be noted thatother cutoffs could also be used without any clear advantageor disadvantage. More than two states could also be justified,but would lead to a significant increase in the number ofinteractions and would also increase the statistical noise inthe RDFs. This makes it a substantial amount of work toconstruct such models. As there are no clear a priori advan-tages over two-state models, such models were not studied inthis work in any detail.

Figure 2 shows the RDFs calculated with an orderedchain defined by Szz�0.6. The fraction of ordered chains asa function of cholesterol concentration is also shown, to-gether with the fluctuation around this average value.Bonded RDFs are not shown, as they are identical to thosepublished previously33 and are not important for the presentdiscussion. The RDFs were calculated from the 2D projec-tions of the CM positions corresponding to the CG particles,considering each monolayer separately. Note that bondedpairs were not excluded in the calculation, i.e., the resultscannot be directly compared to those in Ref. 33, where ex-clusions were used. This treatment is in line with the selectedform of the Hamiltonian, i.e., exactly those pairs that wouldinteract through a certain CG pair potential were included inthe calculation of the corresponding RDF.

Qualitatively, the RDFs behave as in Ref. 33, i.e., ordergenerally increases with increasing cholesterol concentra-tion, while still maintaining a liquidlike structure. This effectis most clearly seen in the RDFs in the tail region, i.e., thetwo bottom rows. However, the division of the tails intoordered and disordered shows that the heights of the RDFpeaks increase the most for the RDFs involving the disor-

0

1

2

3

Prob

abili

ty

-0.5 0.0 0.5 1.0

Szz

0%5%13%20%30%

FIG. 1. Order parameter distributions for different cholesterol concentra-tions calculated from MD data. The vertical dotted line shows the cutoffemployed for classifying a chain as ordered �see text for details�.



dered tails. In contrast, the nearest-neighbor peak of orderedtails actually decreases when cholesterol concentration in-creases. Nevertheless, the minima become deeper for all theRDFs. The fact that the ordering of disordered tails also in-creases indicates that the presence of cholesterol has an ef-fect that is not only mediated by tail ordering, at least whenordering is measured using Szz only �the RDFs behave simi-larly for other Szz cutoffs as well�. Also, the ordered-disordered RDF shows much smaller changes than the oth-ers, the peaks remaining small. This indicates that tails preferto be next to other tails which are in the same state, i.e., thatordered and disordered tails tend to occupy different regions.The ordered fraction behaves as expected: With increasingcholesterol concentration, more and more tails become or-dered. Fluctuations around the average are quite small, witha maximum at 20% concentration. This maximum is easilyexplained by the fact that at this concentration, the orderedfraction is closest to 0.5, i.e., entropic effects from the dif-ferent numbers of microstates for different fractions limit thefluctuations least at this concentration.

Areas per molecule were also obtained for each choles-terol concentration by dividing the average total box area bythe number of molecules in one monolayer �i.e., by 64�. Thevalues are identical to those used for the previousmodels31–33 and are omitted here. It is sufficient to note thatthe area decreases monotonically with increasing cholesterolconcentration, starting from 65 Å2 for the 0% concentrationand ending at 42 Å2 at the 30% concentration.31 The linearsizes of the MD systems hence decreased from 6.5 to 5.2 nm.

C. Determination of interactions

Several sets of effective interactions for the CG particleswere constructed using the different IMC approaches de-

scribed in Sec. III, using the RDFs in Fig. 2 �or ones withexclusions�. The details of the calculations are given in TableI and discussed in more detail below. The names in the firstcolumn of the table are used to refer to the calculationsthroughout the text.

For all calculations, a separate interaction was used foreach pair used for RDF calculation �Fig. 2 plus three bondedinteractions33�. The cutoff for all interactions was 2.5 nm, alimit imposed by the fact that the RDFs can only be calcu-lated up to half the linear size of the system. The input RDFsand the effective potentials were pre- and postprocessedidentically to the previous model �Ref. 33�, i.e., a spline-fitting algorithm50 was used to smooth the input RDFs andthe final potentials, and power-law forms were used for thepotentials in regions where the RDFs were zero.

0.0

0.5

1.0

g(r)

head - head head - o head - d head - chol

0.0

1.0

2.0

3.0

g(r)

o - o o - d d - d

0.0

1.0

2.0

3.0

g(r)

0.0 0.5 1.0 1.5 2.0

r [nm]

o - chol

0.0 0.5 1.0 1.5 2.0

r [nm]

d - chol

0.0 0.5 1.0 1.5 2.0 2.5

r [nm]

chol - chol

0.0

0.5

1.0

Ord

ered

frac

tion

0.00

0.05

0.10 Standarddev.

0 10 20 30

Chol. conc. [%]

0%5%13%20%30%

FIG. 2. RDFs for pairs of CG beads calculated from atomistic MD simulations calculated without exclusions. The fraction of ordered chains �squares� and thefluctuations around this average �circles� are shown in the small figure. A chain was identified as ordered if Szz�0.6. The bonded RDFs are identical to thosepublished previously �Ref. 33� and are not shown here. RDFs not involving tail particles are identical to those in Ref. 33.

TABLE I. List of IMC calculations performed. All the calculations wereperformed only for the 30% cholesterol concentration; only calculations A3,B1, B2, and B3 were performed for all concentrations.

Name Method�E

�kBT� aEfluct

�10−2kBT� a Figure 4 line style

A1 A 22 2.5 Gray solidA2 A� 1.4 0.7 Gray dashedA3 A�

b 1.48 0 Gray dash dottedB1 B 21 2.1 Black solidB2c B� 1.4 1.2 Black dashedB3d B�

b 1.7 0 Black dash-dottedB4 B�

b,e 3.9 0 Black dotted

aEnergy values given for 30% cholesterol concentration.bIn standard semigrand canonical ensemble with nonbonded interactions ex-cluded between bonded particles.cUsed for large-scale studies and shown in Fig. 3.dUsed for transferability studies and shown in Fig. S1 �Ref. 51�.eWith a manual intervention to change �E.



The sampling for the IMC runs, as well as the simula-tions reported in Sec. V, were done using standard Metropo-lis MC. Two types of moves were used: �i� a randomly se-lected particle was displaced or �ii� the internal state of arandomly selected chain was flipped.

Using the potential of mean forces �V�r�=−kBT ln g�r��as the initial potentials resulted in very unstable iteration.Hence, for each concentration, we first performed an IMCrun where the internal states were treated as identical. Thefinal potentials from this run were then used as the initialpotentials for the full IMC inversion.

As with our previous model,33 we observed that the sur-face tension needs to be constrained during the IMC processto obtain positive virial pressures. The target values for thesurface tension constraints were the same as with the previ-ous model to facilitate comparison, and the obtained areacompressibilities were within 10% of those obtained withinthe previous model. The constraint was applied for all thereported calculations: For pure DPPC systems, the surfacetension was constrained to ��90 dyn /cm, while forcholesterol-containing systems a value of ��220 dyn /cmwas used.33

Figure 3 shows the set of potentials for calculation B2,which is also the set used in Sec. V to study large-scalebehavior of the system �the potentials used for transferabilitystudies are shown in supplementary material, Fig. S1 �Ref.51��. The internal energy terms are also shown as a functionof cholesterol concentration. The B2 potentials were deter-mined using IMC approach B� from the RDFs in Fig. 2. Thepotentials are, in general, very soft due to the high degree ofcoarse graining.32,33 Note also that the internal energies �Eare very close to the potential of mean force result

�E=−kBT ln��ndMD / �noMD� used as the initial guess for �E�shown in triangles in the figure�, as expected for approachB�.

The behavior of the effective potentials as a function ofcholesterol concentration is very similar to the earlier three-particle model in Ref. 33. In the tail region, higher choles-terol concentrations imply stronger repulsion at short ranges,while at intermediate ranges the 20% potentials are the mostattractive, with monotonic reduction in attractiveness whenthe concentration is changed either way. The main differencebetween the potentials involving ordered and disordered tailsseems to be the increase in detail at short length scales forthe potentials involving ordered tails. Interestingly, the inclu-sion of the ordering in the model has very little effect in thestrength of the concentration dependence, in contrast to oura priori assumption that the new degree of freedom wouldimprove transferability.33 However, despite the concentrationdependence, the transferability properties actually improve;this is discussed in more detail in Sec. V.

The B2 potentials in Fig. 3 do not result in perfect RDFagreement. This is because the method B� restricts �E to anincorrect value. However, one should note that the root-mean-square difference between the CG and MD RDFs is ofthe order of 0.02 or less, i.e., the differences are mostly ofthe order of the linewidth in Fig. 2 �the differences are shownin Figs. S2 and S3 in the supplementary material51�. Never-theless, these minor differences prompted us to constructseveral different sets of interactions to understand the RDFinversion in more detail. Largest differences between differ-ent approaches were observed in the highest cholesterol con-centrations, and hence we focus here on only the 30% case.It turned out that not all of the potentials generalized well tolarger systems, as will be discussed below in more detail. For

0

5

V(r

)[k

BT

]

head - head o - o o - d d - d

0

2

4

V(r

)[k

BT

]

head - o head - d head - chol

0

5

V(r

)[k

BT

]

0.0 0.5 1.0 1.5 2.0

r [nm]

o - chol

0.0 0.5 1.0 1.5 2.0

r [nm]

d - chol

0.0 0.5 1.0 1.5 2.0 2.5

r [nm]

chol - chol

-2

-1

0

1

2

E[k

BT

]

0.00

0.01

0.02

Efluct [k

B T]

0 10 20 30

Chol. conc. [%]

0%4.7%12.5%20.3%29.7%

FIG. 3. Effective pairwise interactions determined from RDFs in Fig. 2 using IMC approach B� �calculation B2�. The small figure shows the internal energydifference between the disordered and ordered tails �squares�, as well as the magnitude of the fluctuation energy term �circles�. The initial �E values in theiteration are shown as triangles.



the pure DPPC, all potentials worked, but the problems be-came larger and larger as cholesterol concentrationincreased.

Figure 4 shows all the different potential sets from TableI for the 30% cholesterol concentration. Only head-ordered,ordered-ordered, and ordered-cholesterol potentials areshown for brevity; the full figure is provided as supplemen-tary material �Fig. S4�.51 Each set of potentials is identifiedwith a different line style as described in Table I. The poten-tials from Fig. 3 are shown in dashed black. The color of theline determines the IMC approach: Black lines used ap-proach B, gray lines approach A. The solid lines �A1 and B1�imposed no restrictions on the internal energy terms and usedeither A or B. Dashed lines �A2 and B2� used either A� or B�and had �E�−kBT ln��ndMD / �noMD�. The dash-dotted lines�A3 and B3� are otherwise similar, but have, in addition,Efluct=0. The dotted black line �B4� had �E�−4kBT andEfluct=0, i.e., �E was in between those of B1 and B2 �ap-proach B� was used with some manual intervention tomodify the internal energies�.

The best RDF agreement is achieved, as expected, withthe full Newton-type method in A1, i.e., the solid gray line inFig. 4 �see Figs. S2 and S3 in the supplementary material51�.Methods B and B�iii� also lead to similar potentials and in-ternal energy values �calculation B1�, although B�iii� seemsto have stability problems and was only briefly studied. The

behavior of the internal energies for the A1 potentials isshown in Fig. 5; the full set of potentials can be found in thesupplementary material �Fig. S5�.51 The difference to Fig. 3is quite large, as �E is now of the order of 10 to 20 kBT.The lower end is of the same order of magnitude as the freeenergy difference between the ordered and disordered statesin the phenomenological models of Nielsen et al.36,37 Theenergy terms �E and Efluct also show nonmonotonic behavioras a function of cholesterol concentration. Their magnitudeseems to be closely coupled, with a larger −�E resulting in alarger −Efluct. It is not straightforward to give a physicalmeaning to these parameters; they are mainly mathematicalconstructs that are used in combination with the effectivepair interactions to estimate the effective Hamiltonian, whichincludes both energetic and entropic contributions. Neverthe-less, the qualitative match between the phenomenologicalmodels and the present model at most concentrations isencouraging.

By construction, each of the interaction sets should re-produce the RDFs when a system identical in size to theoriginal MD system is simulated. This is indeed the case,with the root-mean-square difference between the CG andMD RDFs being of the order of 0.02 or less in all cases. Allof the sets seem to have minor systematic deviations fromthe target RDFs �see Figs. S2 and S3 in the supplementarymaterial51�, but it is very difficult to judge whether they aresignificant or not, in particular a priori without comparisonto other potentials. However, we performed a test that showsthat the differences, although almost invisible to the eye,actually are important: If we take the A2 potentials for the30% concentration and use them to produce the target RDFsfor an IMC run that is otherwise identical to the B2 run, theA2 potentials are recovered. The same holds true if the rolesof the two potential sets are reversed. Similar observationshave also been reported by Bolhuis and Louis in the contextof semidilute polymer systems.52

Although the differences between the potentials in Fig. 4were minor in a system identical in size to the atomisticsimulation, major differences were found when linear systemsize was increased eightfold �we still focus on the 30% case,but similar effects could also be seen in lower concentra-

-2

0

2

4

V(r

)[k

BT

]

head - o

-2

0

2

V(r

)[k

BT

]

o - o

-2

0

2

V(r

)[k

BT

]

0.0 0.5 1.0 1.5 2.0

r [nm]

o - chol

FIG. 4. Effective pairwise interactions for 30% cholesterol concentrationfrom different calculations in Table I. All calculations give essentially thesame RDFs and virial pressure for a small system. Each calculation is iden-tified by a different line style, as described in Table I. The dashed black linecorresponds to Fig. 3. A figure of the omitted potentials is provided assupplementary material �Fig. S4� �Ref. 51�.

-30

-25

-20

-15

-10

-5

E[k

BT

]

-0.03

-0.02

-0.01

0.0

Efluct [k

B T]

0 10 20 30

Cholesterol concentration [%]

FIG. 5. Internal energy terms for interactions determined with IMC ap-proach A without any constraints �calculation A1�. �E is shown in squaresand Efluct in circles. Full set of potentials is provided as supplementarymaterial �Fig. S5� �Ref. 51�.



tions, although not as clearly�. This is in contrast to our ear-lier model, where the value of the virial pressure could beconstrained to different values without any qualitativechanges in the behavior.33

In the present case, potentials A2 and A3 result in clearlyunphysical situations, having dense clusters of particles sepa-rated by empty space. For A1 and B1, the failure is moresubtle and is not unphysical per se: These potentials result invery strong density variations, caused by phase separation ofcholesterol and ordered tails in one phase and disorderedtails in another phase. In principle, such a behavior couldoccur also for the 30% case, but the number �nd /N actuallydeviates significantly from the value used in the parametri-zation, casting some doubt on the validity. Also, this does nothappen for other interactions, nor for our earlier models, sowe have opted to use interactions for which this does nothappen. Representative configurations from both these casesare included as supplementary material �Fig. S6�.51

The potentials determined with IMC approach B�, i.e.,B2, B3, and B4, did not have the above problems �a repre-sentative configuration of this case is also shown in thesupplementary material51�. All of these gave similar results.For B2 �Fig. 3�, where Efluct was not zero, the interactionsworked only when the Efluct were not scaled with the numberof particles; if scaling was used, similar problems as abovewere observed. The reason for these differences is returnedto in Sec. VI, after discussing the large-scale behavior of themodel with the potentials B2 and B3. For these interactions,the RDFs change only little when the system size is in-creased �see Figs. S7– S9 in the supplementary material51�.However, there is an effect that comes from segregation ofthe ordered and disordered tails into separate regions �seeSec. V�.

V. BEHAVIOR AT LARGE LENGTH SCALES

Let us now briefly analyze the behavior of the new CGmodel at larger scales. The results presented here were ob-tained using the potentials B2 from Fig. 3 �transferabilitystudies were done with B3 that had Efluct=0, shown in Fig.S1 in the supplementary material51�. The MC simulationswere mostly conducted for systems with 16 times the linearsize of the original MD simulation �transferability studieswere done with half this size�. That is, the linear system sizewas in the range of 80–110 nm, depending on the cholesterolconcentration. Reasonable statistics, good for qualitativeconclusions, could be obtained within a few days on a stan-dard desktop computer.

We use the static structure factor S�k�� to characterize thelong-range order in the system. S�k�� is defined by

S�k�� =1

N��i=1

N

�j=1

N

exp�− ik� · �r�i − r� j�� , �13�

where N is the number of particles, r�i are the positions of theparticles, and k� is a reciprocal space vector. It is also possibleto calculate the structure factor for a subset of the particlesby restricting the summations to these particles. Although inprinciple, S�k�� and the corresponding RDF carry the same

information �they are related by a Fourier transform�, thestatic structure factor is more convenient to characterize thelong-range structure �small k� values�.

Figure 6 shows the total static structure factor for eachcholesterol concentration. The behavior is very similar to theearlier models: At 13% and 20% concentrations, there issome large-scale order, as shown by an increasing S�k� whenk is reduced. However, this effect is weaker than for thethree-particle model without ordering33 and comparable tothe one-particle model.32 For 20% concentration, it is evenslightly weaker than for either of the previous models. Nev-ertheless, decomposition of the structure factors into partialstructure factors shows that the cholesterol-cholesterol struc-ture factor has a similar peak at small k. In the presentmodel, the ordered and disordered tails give an additionalcontribution to the peak �see below�. We also note that the0% and 5% systems have a peak around k�1 nm−1 simi-larly to the earlier three-particle model; this peak was inter-preted as indicating strong density fluctuations in the fluidphase of pure DPPC systems and was later confirmedthrough large-scale atomistic simulations.46

The new feature of the present model is chain ordering,and it is interesting to see how the different chain states takepart in the behavior discussed above. This is shown in Fig. 7:It shows the partial static structure factors for the tail-tailpairs for the 0%, 13%, and 30% concentrations. Both the o-oand d-d structure factors increase as k→0 at all concentra-

0.0

0.5

1.0

1.5

2.0

2.5

S(k)

0 5 10 15

k [2 /nm]

0.0

0.2

0.4

0.6

0 1 2 3

FIG. 6. Total static structure factors for different cholesterol concentrations.The inset shows the small k region magnified. Different line styles corre-spond to different cholesterol concentrations as in Fig. 3. The structurefactors were obtained from systems with 16 times the linear size of theoriginal MD simulations, with the interactions from Fig. 3.

-0.2

-0.1

0.0

0.1

0.2

S(k)

0 1 2 3

k [2 /nm]

0%

0 1 2

k [2 /nm]

13%-4

-2

0

2

4

S(k)

0 1 2 3

k [2 /nm]

30%

FIG. 7. Partial structure factors at three cholesterol concentration for o–o�solid black�, d–d �dashed�, and o–d �gray� pairs. The structure factors wereobtained from systems with 16 times the linear size of the original MDsimulations, with the interactions from Fig. 3. Note that the large oppositepeaks in 30% cancel more or less exactly to give the total structure factor inFig. 6 �there is also a small contribution from the other partial structurefactors�.



tions, while the cross-S�k� is negative, and becomes more soat the k→0 limit. This shows that at all concentrations, theordered and disordered tails tend to segregate and formlarger domains; however, at the 30% concentration the totaldensity remains homogeneous, as seen in the total structurefactor �Fig. 6�. At 0% and 5% concentrations, it is seen thatexactly this segregation is the reason for the peak at k�1 nm−1. This is in agreement with the atomic-scale simu-lations, which showed that the chains within the denser do-mains are generally more ordered.46 The RDF comparisonshown in the supplementary material �Figs. S7–S9�51 alsoshow that such segregation occurs when system size isincreased.

One of the main motivations for including the chain or-dering was an a priori assumption that it would improve thetransferability of the effective interactions between the dif-ferent concentrations.33 As before, we evaluate the transfer-ability by using the interactions determined at a certain con-centration to study the neighboring concentrations andcompare the results to those obtained with the correct effec-tive potential. This addresses the consistency of the effectiveinteractions evaluated from different atomistic simulationsand assesses whether it could be possible to evaluate, e.g.,phase boundaries with the current model. However, it doesnot provide any direct measure for the quality of the CGmodel.

In the general case, the form �8� for the Hamiltonianmakes transferability studies more difficult. Namely, it is notclear how the final term should be treated, as it includesinformation from the thermodynamic state in the values ni

ave.In the present case, we only studied the transferability ofpotentials B3, which had Efluct=0, and hence these problemswere avoided. A figure of these potentials is included assupplementary material �Fig. S1�.51

The transferability results are summarized in Fig. 8�similar figures, but with RDFs instead of structure factors,can be found in the supplementary material, Figs. S10 andS11 �Ref. 51��. Between all concentrations, the results arequalitatively correct for the nearest-neighbor peak �k�13 nm−1�. This is demonstrated in the middle figure,which shows the transferability between 5% and 13% con-centrations: Although the change in the nearest-neighborpeak is smaller than the difference between the correct struc-ture factors, it is in the correct direction. Between 20% and30%, as well as from 13% to 20%, the transferability is evenbetter in this respect: Also the height of the nearest-neighborpeak is reproduced �the bottom figure�. For the small k re-gion, the results are not as good: The interactions determinedat 5% or 30% do not give the marked increase at 13% or20% concentration. However, the opposite direction worksbetter: Very little remains of the small k peak when the in-teractions from 13% and 20% are used in 5% and 30%,respectively. Between 13% and 20% the transferability isreasonable in both directions, although from 13% to 20% thesmall k peak increases slightly instead of decreasing.

The transferability is mostly similar to the earlier three-particle model without the internal degrees of freedom:33

Also the earlier model showed qualitatively correct behaviorof the nearest-neighbor peak. However, there are two im-

provements: �i� the behavior of the nearest-neighbor peak isalso quantitatively correct in some cases, and �ii� the small kregion shows a degree of transferability also across the phaseboundaries, i.e., from the inhomogeneous concentrations of13% and 20% to the homogeneous ones of 5% and 30%.However, the latter does not work in the opposite direction,i.e., from the homogeneous to the inhomogeneous. Hence,the transferability did improve, but only slightly.

It could be possible to improve the transferability byimproving the mapping of the atomistic simulations to theCG degrees of freedom, in particular, for the ordering states.This could be done either by having more than two internalstates or with a different definition of the ordered-disordereddivision. However, the Szz order parameter distributions inFig. 1 seem to indicate that the order changes quite continu-ously, so there might not be any good way of dividing thetails into two �or more� well-defined, separate groups. In-creasing the number of states is also problematic from thetechnical point of view, as discussed in Sec. IV B. It is alsopossible that the system is simply too complex for goodtransferability. The fact that all the different models we haveconstructed actually have similar cholesterol-cholesterol in-teractions indicates that the concentration dependence mightbe an intrinsic feature of the chosen particle description. The

0.0

0.5

1.0

1.5

S(k)

5% 13%

0.2

0.4

0 1 2

0.0

0.5

1.0

1.5

2.0

S(k)

0 5 10 15

k [2 /nm]

20% 30%

0.0

0.1

0.2

0 1 2 3

FIG. 8. Potential transferability properties between different concentrationsusing interactions B3 �potentials shown in the supplementary material �Ref.51�, Fig. S1�. The diagram on top summarizes the transferability propertiesbetween the concentrations: The first + /− stands for �qualitative� reproduc-tion of the small k behavior of S�k�, the second + for qualitative reproduc-tion of the nearest-neighbor peak in S�k�, and the possible third + for quan-titatively nearly correct S�k� away from the small k region. The figures onthe bottom show total static structure factors for two cases: In both figures,the correct S�k� is shown as a solid line, and the S�k� given by the trans-ferred interactions as a dashed line. Each figure shows the transferability inboth directions between adjacent concentrations, and the color of the linesshows the simulation concentration.



transferability properties may also be adversely affected bysampling issues in the underlying MD; these are briefly dis-cussed in Sec. VI.

VI. DISCUSSION AND CONCLUSIONS

Although theoretically the solution to the inverse prob-lem is unique, we have demonstrated that widely differentpotentials can give rise to essentially the same RDFs �seeFig. 4�, in line with observations by Bolhuis and Louis.52

Further, in our earlier study the virial pressure of the systemcould be constrained to widely different values with onlyminimal changes in the RDFs.33 However, not all of the in-teractions perform well in larger systems. In Ref. 33, thevirial pressure was used as an additional constrained quantityto obtain workable interactions, but in the present case alsothat fails to distinguish “good” interactions from “bad” ones:All the interactions discussed in this paper give essentiallythe same value for the virial pressure in the small systemwhere they were determined. This demonstrates that it isnontrivial to evaluate the quality of effective interactionseven if the target RDFs are well reproduced.

The above problems also reflect in the sensitivity of thepotentials to the RDFs. However, although the differencesbetween the different interactions may not even be visible tothe eye without magnification, they are typically enough todistinguish the underlying interactions in the inversion: If apairwise interaction is used to create the RDFs, exactly thispairwise interaction is recovered with algorithms such asIMC if no additional constraints are imposed. Because of theabove issues, one needs to be careful when using RDFs de-termined from complex systems to determine just a single setof interactions because there may be subtle effects from in-complete sampling that may result in large changes in theinteractions.

Why do the interactions from some IMC approachesthen perform worse than those from others? One reasonseems to be that the “natural” internal energy values for thesystem have Efluct�0, which make it impossible to general-ize the interactions to a larger system �see Sec. III�. Theinteractions that worked had Efluct0 and did not suffer fromthis problem. However, not all of the interactions withEfluct0 worked, and most notably so those from IMC ap-proach A. A clue to this difference is provided by the obser-vation that the problems become larger as cholesterol con-centration increases. With increasing cholesterolconcentration, the size of the simulation box becomes closerand closer to twice the cutoff. This means that the number ofparticle pairs that, on average, are beyond the cutoff is muchsmaller for the 30% system. This makes it possible to betterand better compensate for incorrect values of �E and Efluct

with changes in the potential without other major changes �ifall the particles were within the cutoff, �E and Efluct param-eters would not be needed because their effect could be in-cluded in the potentials �see Appendix A��. Hence, one pos-sible reason for the poor performance of approach A is itstendency to compensate any inaccuracies in the energy pa-rameters with changes in the potential, which only works forthe original system size. In approach B, the target counts are

rescaled such that these inaccuracies are more or less invis-ible in the potential equation. However, this separation is notperfect, and it is possible that some of the variants sufferfrom similar problems as approach A. In particular, con-straining Efluct to zero in approach B does not solve the prob-lem, although Efluct no longer is negative.

It is possible that the underlying reason for the problemslies in incomplete sampling of the underlying MD simula-tions. As our model is more complex than the previous ones,the statistics on the RDFs are worse. The spline fitting usedto smooth the RDFs can then induce small errors. Also, theone-particle densities calculated from the underlying MDsimulations are not homogeneous, with the largest differ-ences seen at higher cholesterol concentrations in theordered/disordered tail densities. Further, autocorrelationfunctions of the chain states �calculated such that the twostates correspond to 0 and 1� show that the timescale forconverting a chain from one state to the other also dependsstrongly on the cholesterol concentration: For pure DPPC,the chains lose the memory of their previous state within 200ps, while for the highest cholesterol concentrations thisreaches 10 ns. In the highest concentrations, different chainsalso sample the different states quite differently, i.e., the frac-tion of time the chain is ordered depends on the chain muchmore than in pure DPPC. Together, these observations sug-gest that the complete phase space of the system may not besampled for the highest concentrations. This can then alterthe RDFs even to the point that the pair interactions thatexactly reproduce them no longer exist. However, with thepresent data, it is difficult to ascertain if this is the case.Studies on simpler model systems where the sampling is notan issue should provide additional insight into this problemand the suitability of RDF inversion for models with internalstates, in general, but is beyond the scope of the presentarticle.

It is also interesting to note that the problems seen inRef. 33 with the virial pressure follow the same pattern asthose here: The good virial pressure and the virial pressureobtained without any constraint in the IMC differ more andmore as the cholesterol concentration increases. Also in thatcase, the pure DPPC system worked quite well without thepressure constraint. It is thus possible that the origins ofthese problems are linked, although no concrete evidence canbe provided.

Let us now briefly turn to the different algorithms usedin inverting the RDFs. The most common of these are itera-tive Boltzmann �IB� inversion14,53 and IMC;13 for a review ofother possible approaches, see Ref. 15. In this paper, wenoted that IMC corresponds to Newton’s method for maxi-mizing the functional �4�, while Soper has provided an argu-ment for the global convergence of IB.14 In our experience,IMC requires careful regularization during the initial steps ofthe iteration to remain stable. Hence, IB might be more suit-able for use during the first steps in the optimization, whileIMC could be used for the final optimization to take advan-tage of the quadratic convergence of Newton’s method �as-suming exact sampling of the required quantities�. This isalso supported by the observation that the convergence of thelong-range part of the interaction is slow in IB, as measured



by the virial pressure.54 In contrast, our experience indicatesthat IMC converges to the “native” virial pressure within afew iterations if the RDFs are close to the target ones. As afinal note on the convergence of the different approxima-tions, we observe that the second derivative of the functional�4� is exactly the cross-correlation matrix that also appears inIMC. As such, it is positive definite, and hence the functionalis convex. This rules out the possibility of local maxima andensures that an iterative method to refine the RDFs cannotget stuck in a local optimum.

RDF inversion is only one of the approaches for con-structing CG models from detailed simulations. It is possibleto devise other approaches based on the matching offorces17,18 or potential energy.42,55 All these different ap-proaches lead to different effective interactions, each of theinteractions performing better for some quantities than forothers. In general, there is no single effective interaction thatcan reproduce all quantities of interest.38,42 To some extent,this problem of representability is related to the fact that thesame quantities can have a different meaning in the differentmodels: For example, if we coarse grain the water away froma solution of ions, the virial pressure in the CG model isactually the osmotic pressure in the detailed picture.40,56

However, it may be difficult to find such relationships for allquantities, and also the approximation of Heff by an effectivepairwise potential results in correction terms in such equa-tions that may be difficult to evaluate. Hence, care is neededwhen applying models with effective interactions, and theorigins of the interactions should be kept in mind.38

In conclusion, we have extended the IMC method tohandle models with internal degrees of freedom and appliedit to extend our earlier CG model for a DPPC/cholesterolbilayer to include the tail ordering in a discrete fashion. Ourseries of CG models of increasing complexity contributes tounderstanding of RDF inversion and its limitations as a CGtechnique, in general, and this issue has been discussed inthis paper at length. Despite the limitations, interesting re-sults can be obtained already with the simple models wehave constructed. Examples include the cholesterol-rich andcholesterol-poor domains at intermediate concentrations, taildensity fluctuations at low cholesterol concentrations, andthe effect of the internal degrees of freedom on the transfer-ability properties. However, detailed analysis of the behaviorof the models, e.g., in terms of domain sizes, still does notseem reasonable as there are uncertainties in the model con-struction process �such as the selection of the cutoff,33 pos-sibility of insufficient sampling, and definition of theordered/disordered boundary� that can have a significant ef-fect on the quantitative results.

It has proven to require a substantial amount of work toimprove our original one-particle CG model constructed inRef. 32, but each improvement has yielded some additionalinsight. The first three-particle model uncovered the tail den-sity fluctuations at low cholesterol concentrations. Thepresent work provided some additional information on order-ing of the tails, but the main contribution in our view is thegeneralization of RDF inversion to models with internalstates and demonstration of the problems that may result.

If the uncertainties in the model construction can be

solved, systematic construction of the interactions can givesubstantial added value to the phenomenological models byNielsen et al. In particular, the interactions could then belinked to a particular lipid system, and different mixtures,e.g., with different sterols could be studied to understand theeffects of the microscopic features on the general behavior.However, a substantial amount of work may be needed. Fur-ther, easy-to-interpret results may prove difficult to obtainbecause transferability problems are expected, even whenchain ordering is included as in the present work. Under-standing simpler model systems might help alleviate theproblems, and detailed studies in this respect are an interest-ing direction for future research.

It would also be interesting to study the extent of theproblems seen in the present study in semiatomistic modelsfor lipid bilayers,57,58 where the degree of coarse graining isnot as high. Again, systematic approach to the interactionscould provide a link to atom-scale chemical detail. Compari-son of the resulting interactions with the simple Lennard-Jones form used in, e.g., the MARTINI force field,10,11 couldalso yield insight into the strengths and weaknesses of gen-eral �e.g., MARTINI� versus specific �RDF and force match-ing� approaches to coarse graining. Finally, comparing dif-ferent systematic CG approaches, both on simple modelsystems and more realistic cases as studied here, could pro-vide better understanding of their differences, as well as thesource of the problems seen in the present study.

ACKNOWLEDGMENTS

This work was supported by the Academy of Finlandthrough its Center of Excellence Program �T.M. and I.V.�, theAcademy of Finland �I.V.�, the National Graduate School inNanoscience �T.M.�, and National Sciences and EngineeringResearch Council of Canada �M.K.�. We would also like tothank the Finnish IT Center for Science and the HorseShoe�DCSC� supercluster computing facility at the University ofSouthern Denmark for computer resources.

APPENDIX A: MOST GENERAL HAMILTONIANWITH INTERNAL STATES

In this appendix, we justify Eq. �8�, i.e., the form of theHamiltonian used for our CG model. The uniqueness andexistence proofs make no explicit reference to the form ofthe space,29,30 so they apply equally well for the present casewhere xi= �r�i ,si�. Now, the pair potential�s� V�x ,x�� can bethought of as consisting of several distinct pair potentials,each of which is a function of the positions r� only. There isone such pair potential for each combination of s values, andhence an equivalent description can be achieved for having aseparate potential for each combination of internal states.However, the number of particles of each kind is no longerconstant, and hence the two-particle density should be usedinstead of RDF as the target quantity: RDFs do not containthe information necessary to determine the occurrence prob-abilities of different s values. From now on, we will use thesecond picture unless explicitly noted, i.e., the pair potentialswill be functions of positions only.



The pairwise Hamiltonian is unique up to a constant.Without internal states, this statement is equivalent to eachpair interaction being unique up to a constant. With internalstates, this holds only for each V�x ,x��, i.e., not for eachindividual pair potential if they are considered as functionsof positions only. Hence, in the present case, we cannot re-quire that each potential approach zero when r→�.

When we introduce an approximation that each pair po-tential is zero beyond some cutoff, we limit the class ofHamiltonians available. In order to recover the full class ofHamiltonians on which the existence proof holds, we need tobe able to add additive constants to each individual pair po-tential also beyond the cutoff. To calculate the minimumHamiltonian that allows this, we calculate how the Hamil-tonian changes when a different constant is added to eachpair potential.

Let us consider a system with Nk particle kinds and12Nk�Nk+1� pairwise potentials Vab�r�. If we add constantsCab to respective potentials, the Hamiltonian becomes

H − H0 =1

2�i�j

Ckikj=

1

2�k=1

Nk

�l=1

Nk

�nknl − �klnk�Ckl

= −1

2�k=1

Nk

Ckknk +1

2�k=1

Nk

�l=1

Nk

Cklnknl,

�A1�

where ki is the kind of particle i, H0 is the Hamiltonianbefore the addition, and nk is the number of particles of kindk. Since the total number of particles of each type is constant�in a canonical ensemble�, not all nk are independent and wecan choose Ns=Nk−Nt independent nk, where Nt is the num-ber of particle types. In terms of these independent nk num-bers, the Hamiltonian becomes

H = H0 + �i=1

Ns

�Eini + �i=1

Ns

�j=i

Ns

�Eij�ni�nj + C0, �A2�

where �Ei, �Eij, and C0 are linear combinations of the Cab.Hence, if we take the original Hamiltonian as �8�, we cannow add �Ei and �Eij to the corresponding Ei and Eij, andthe new Hamiltonian has the same functional form as the oldone.

The last term in the Hamiltonian �8� can be written eitheras �ni�nj or ninj, and these forms can be interchanged usinga suitable modification of Ei �also, any value of ni

ave can beselected�. However, the one with �ni�nj �Eq. �8�� and ni

ave

= �niMD is more suitable for the IMC iteration because themagnitude of �ni�nj is much smaller than ninj. The formselected for Eq. �8� also perhaps shows more clearly thephysical significance of the parameters Eij in controlling themagnitude of fluctuations.

APPENDIX B: ITERATION FORMULASFOR INTERNAL ENERGIES

In this appendix, we derive the iteration formulas �10�and �11� used for the internal energies in the IMC approachB.

We start by writing the partition function of the systemand integrating out the positional degrees of freedom to ar-rive at

Z = Tr exp�− ��i

Eini + �i,j

Eij�ni�nj + f�n�� , �B1�

where f is some unknown function of the vector n formedfrom the ni, and the trace is over all possible combinations ofni. f�n� is essentially the free energy of such a combinationof ni �of which the two internal energy sums have been fac-tored out�. Switching to matrix notation and approximating fwith a quadratic form around �n as f�n�=C0+ �D�1��Tn+nTD�2�n, the partition function becomes

Z = Tr exp�− ��C̃0 + �D�1� + E�1� − 2E�2�nave�Tn

+ nT�D�2� + E�2��n�� , �B2�

where E�1� and E�2� are a vector and a matrix formed from Ei

and Eij, respectively. Setting v=D�1�+E�1�−2E�2�nave and A=D�2�+E�2� and replacing the trace with an integral over thefull real axis for each ni, the partition function can be evalu-ated exactly as a Gaussian integral,

ln Z = 14�vTA−1v + C , �B3�

where C does not depend on E�1�. Differentiating both sideswith respect to Ei and canceling a common factor −� gives

�n = − 12A−1v , �B4�

and a second differentiation gives

�ninj − �ni�nj =1

2��A−1�ij . �B5�

Denoting ij = �ninj− �ni�nj and inverting both sides of Eq.�B5�, we can then solve the change required in A to bring thefluctuations to the correct value as

��−1� = 2��A . �B6�

Substituting Eq. �B5� into Eq. �B4� gives

�n = − �v , �B7�

which can be linearized to yield

��n = − ��v − ��v . �B8�

Solving �v from Eq. �B7�, substituting this into the aboveequation and rearranging gives

��v = − ��n + ��−1�n� . �B9�

Assuming that D�1� and D�2� are constants, Eqs. �B6� and�B9� immediately give Eqs. �10� and �11�.

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