Molecular Simulation of the DPPE Lipid Bilayer Gel Phase: Coupling

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Molecular Simulation of the DPPE Lipid Bilayer Gel Phase: Coupling Between Molecular Packing Order and Tail Tilt Angle Karthik Uppulury, Patrick S. Coppock, and James T. Kindt J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b05720 • Publication Date (Web): 25 Jun 2015 Downloaded from http://pubs.acs.org on July 6, 2015

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The Journal of Physical Chemistry

Molecular Simulation of the DPPE Lipid Bilayer Gel Phase: Coupling Between Molecular Packing Order and Tail Tilt Angle Karthik Uppulury, Patrick S. Coppock†, and James T. Kindt*. Department of Chemistry, Emory University, Atlanta, GA, 30322, USA

ABSTRACT:

The structural

properties

and

thermal

stability of

dipalmitoyl

phosphatidylethanolamine (DPPE) in the ordered gel phase have been studied by molecular dynamics simulation using two force fields, the Berger united-atom model and the CHARMM C36 atomistic model.

As is widely known, structural features are

sensitive to the initial preparation of the gel phase structure, as some degrees of freedom are slow to equilibrate on the simulation timescale of 100’s of nanoseconds. In particular we find that the degree of alignment of the lipids’ glycerol backbones, which join the two hydrocarbon tails of each molecule, strongly affects the tilt angle of the tails in the resulting structures. Disorder in the backbone correlates with lower tilt angles: bilayer configurations initiated with aligned backbones produced tilt angles near 21° and 29° for the Berger and C36 force fields respectively, while structures initiated with randomized backbone orientations showed average tilt angles of 7° and 18°, in closer agreement to 1 ACS Paragon Plus Environment


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the untilted structure observed experimentally. The transition temperature for the Berger force field gel bilayer has been determined by monitoring changes in width of gel phase stripe domains as a function of temperature, and is 12 +/- 5 degrees K lower than the experimental value.

INTRODUCTION Studying the behavior of simulation models of single-component lipid bilayers in ordered phases has several benefits. The ability of a force field to represent the structure of the ordered phase (e.g. the gel phase) is a measure of the force field’s accuracy. Knowledge of the temperature at which the simulation model spontaneously changes phase – which may not match experiment – is essential to avoid unexpected effects in simulations performed near the transition temperature. Understanding the structure and properties of the gel phase itself is useful as its distinct properties influence the behaviors of membrane mimetic systems used in biophysical analysis and encapsulation technologies, e. g. in bicelles1 and thermosensitive liposomes.2 Members of the phosphatidylcholine (PC) series of glycero-phospholipids have been the most extensively studied via simulation.3-5 Many PC lipids undergo transitions upon cooling from the fluid (Lα) phase first into a ripple phase (Pβ') (characterized by a spatial modulation of wavelength ~10-20 nm) and then into a tilted gel (Lβ’) phase where the extended hydrophobic tails tilt uniformly away from the bilayer normal. In contrast, the ripple phase is not observed experimentally in the phosphatidylethanolamine (PE) lipids DPPE or DMPE (dipalmitoyl- and dimyristoyl-PE).6-8 As first demonstrated by McIntosh,9 DPPE undergoes a transition directly from the fluid phase into an untilted gel 2 ACS Paragon Plus Environment

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(Lβ) phase upon cooling. The qualitative explanation for the difference in tilt, discussed by McIntosh9 and earlier by Nagle10 is that the PC headgroup (featuring a quaternary ammonium group) is more bulky than the PE headgroup (featuring a primary ammonium group), and so that packing at the same volume density of chains but a larger area per lipid can be achieved by tilting. McIntosh also suggested that the tendency for tails to tilt in the gel phase was correlated with the existence of the ripple phase.9 Given that the intervention of the ripple phase complicates experimental, computational, and theoretical studies of the gel-fluid transition, the lack of a ripple phase is a welcome simplification that makes DPPE an attractive model for exploring the physical behavior of membranes undergoing phase transitions. Accurate modeling of PE lipids is also important in the biological context as they are found at high levels in some bacterial membranes, and as a link between PE content and susceptibility to certain antimicrobial agents has been proposed.11 Two prior simulation studies on DPPE in the gel phase12,13 have produced tilt angles of 25° or higher; one employed the CHARMM C27 force field14 while the other used the GAFF (general Amber force field)15 parameters. A third study16 using the Berger force field17 produced structures with little tilt but (as apparent from the snapshots presented in Figure 4 of ref.

16

) with considerable disorder in the tails. Apart from the differences in

force field, there were notable differences in the set-ups of these simulation trajectories. In simulations of the gel phase, lipids are not observed to diffuse or rotate over the course of a trajectory, so initial conditions are critical to the final result. Tilted gel phases were produced when the initial configurations used were regularly ordered structures, either based on the crystal structure of DLPE18 (following a common procedure3,4) or from

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another regular array. In contrast, Leekumjorn and Sum16 annealed a fluid-phase DPPE structure to obtain an untilted gel. (A similar procedure, was followed by Qin et al.19 with similar results.) These previous works raise the question: to what extent does the difference in initial preparation (as opposed to differences in force field) affect the resulting structure – not only the disorder in the tails, but the tilt angle as well? Certain elements of the gel phase structure are known from experiment, but some details are unknown. X-ray scattering shows that tails are packed in a hexagonal lattice, but not how neighboring tails are connected to each other via the lipid's glycerol backbone. In the crystal structures, pairs of tails are organized in regular rows with neighboring molecules sharing the same backbone orientation, but a disordered backbone arrangement has been suggested for the gel phase based on indirect experimental evidence.20,21 In support of this arrangement, simulations by Coppock and Kindt22 found backbone disorder in growing regions of DPPC and DSPC gel phases even started from an ordered template. Schubert et al.5 set up DPPC gel systems using several different methods and observed a correlation between higher order and greater tilt angles, along with greater melting temperatures and phase transition enthalpies. In the present study we compare the effect of backbone order on tilt in the gel phase of DPPE simulated with the Berger united-atom force field17 and the CHARMM C36 force field.23 The backbone-ordered united-atom system displayed a definitely tilted gel phase with a tilt angle of 20.6 ± 0.9° while the backbone-disordered system displayed a net tilt averaging 7.0 ± 1.5° with fluctuations between 0 and 15°, consistent with the lack of tilt seen in experiment.

A backbone-ordered gel phase modeled using the C36

potential showed a mean tilt angle of 29.2 ± 0.4° while the averages tilts over three 4 ACS Paragon Plus Environment

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trajectories for the backbone-disordered system were 17.7±1.0°, 18.1±1.0° and 20.1±0.9°. Arguments based on a simple model are presented to explain for why backbone disorder is expected to produce lower tilt angles. The effects of disorder on area per lipid and hydrogen bonding were also studied. As the Berger model was more successful in reproducing the DPPE gel phase structure, we proceeded to evaluate its transition temperature by observing the rates of growth or loss of a gel phase stripe domain embedded in a fluid phase.22 From this study we estimate Tm at 325 ± 5 K, between 7 and 17 K lower than the experimental value. METHODS General simulation methods. The Gromacs 4.5 simulation package24 was used for all molecular dynamics calculations, with a time-step of 2 fs.

Two sets of force-field

parameters were used for comparison. The first, referred to here as the Berger force field, is based on the united-atom parameters of Berger et al.17 with modifications described by de Vries et al.,25 and was used in conjunction with the SPC water model.26 The second, referred to here as the C36 force field, is the CHARMM36 all-atom parameter set,23,27 used in conjunction with the TIP3P water model.28 Electrostatic forces were calculated using the particle-mesh Ewald method and standard Gromacs settings.29 The Berendsen temperature and pressure coupling algorithms30 were used, with a thermostat coupling time of 0.4 ps and a barostat coupling time of 5 ps (based on an assumed compressibility of 4.5 × 10-5 bar-1); for the C36 simulations the velocity-rescaling thermostat31 was used instead for temperature equilibration. Pressure coupling is always applied independently in the x and y dimensions to allow the spacings of rows and columns of tails in the gel



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phase to equilibrate independently, as is customary for gel-phase simulations.3 The VMD program was used for molecular graphics and visualization.32 Simulation set-up: ordered vs. disordered gel-phase structures. The arrangement of lipids in the initial bilayer patch was critical in determining the structure of the resulting gel-phase bilayer. The first step was to set up either an ordered or a disordered arrangement of pairs of sites on a periodically repeating 20 × 20 triangular lattice, as shown in Figure 1. Each site corresponds to the initial position of one lipid tail; each pair represents a two-tailed lipid.

Figure 1. Lattice site (open blue circles) connectivity patterns generated for initial placement of lipids in a pattern with hexagonal tail packing and random backbone orientation (left) versus ordered backbone pattern (right). Red segments represent the originally placed bond positions; magenta segments are those that migrated during the Monte Carlo annealing procedure; green segments are new bonds added during the


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annealing procedure; and black segments indicate the continuation of bonds across period boundaries.

To obtain a disordered arrangement, starting with an empty triangular lattice, an unoccupied lattice site and one of its unoccupied neighbor sites were selected at random. A segment linking this pair of sites was drawn to indicate that they are occupied by tails attached to the same headgroup. This procedure was repeated until no more adjacent pairs of unoccupied sites remained on the lattice. Then, pairs were created using an algorithm that allows the remaining unoccupied sites to diffuse about the lattice until they can pair up and be filled. In this algorithm, an unoccupied site is first selected at random. One of its six neighboring (occupied) sites is then chosen to serve as a pivot, and that site’s paired partner is moved into the open site, moving the location of the unoccupied site. This procedure continues until the unoccupied site moves to a position adjacent to an existing unoccupied site, and the pair of sites is then treated as an occupied pair, annihilating both defects. Finally, after all sites are occupied, one site of each pair is selected at random as the sn-1 chain and the other to be the sn-2. For disordered bilayers, an independently created disordered lattice was used for each leaflet. An example is shown Figure 1 (left hand side).

An ordered gel phase DPPE bilayer system was

generated from the ordered lattice grid shown on the right hand side of Figure 1, with the backbone orientations are all aligned and oriented in the same direction. To convert the lattice structures into atomistic coordinates for the lipids, a lipid molecule configuration obtained from a preliminary calculation (selected for having a



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compact cross-section of both its tails and its headgroup) was replicated on the grid so that the center of mass of the glycerol backbone moiety of the lipid molecule coincides with the midpoint of the line segment specified in the grid, and the vector connecting the tail carbonyl carbons coincides with the direction of the sn-1 sn-2 vector on the grid. To avoid steric clashes between the DPPE lipids, molecules were initially placed on an expanded grid with an area per lipid of 0.82 nm2. Two leaflets were created with an intervening gap between them, as shown in Figure 2(a). After hydration with 43-45 water molecules per headgroup, the systems were then compressed laterally at a pressure of 50,000 bar for a duration of 2 ps at T=275 K with the x and y box dimensions allowed to respond independently to the applied pressure but with the z dimension fixed. The result was a closely packed array of lipids with a much smaller gap, little or no tilt in the lipid tails, and a high degree of extension as shown in Figure 2(b). The packing of the lipids is visualized by representing the C5 carbon atoms (highlighted as yellow spheres in Figure 2(c)) of each lipid chain along the bilayer normal. The packing of lipid tails produced showed regular hexagonal lattice ordering for both ordered and disordered backbone arrangements, as shown in Figure 2(d) and (e). Upon further equilibration at 1 bar pressure, the gap between the leaflets in the ordered and disordered Berger systems closed rapidly. In contrast, the C36 systems experienced increased separation between the leaflets once the pressure was reduced. To overcome the increased separation for the C36 ordered system, a pressure of 3000 bar in the direction of the bilayer normal was applied for 4 ps to close the excess gaps between leaflets before further equilibration. The C36 backbone disordered structures were prepared using three slightly different protocols, which we refer to as P1, P2 and P3. The 8 ACS Paragon Plus Environment

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starting structure (similar to Figure 2a) is the same for all P1, P2 and P3. After lateral compression of the starting structure, equilibration at 1 bar pressure tended to widen rather than close the gaps between the leaflets. To overcome this, a compressive force was applied in Z-dimension in a different manner in each of the protocols. The protocol P1 consists of: (i) Equilibration for 2 ns initially, allowing the separation of the leaflets and letting their tilt directions equilibrate independently, (ii) Compression of the structure obtained in Z-dimension at 15000 bar pressure (while the pressures in X and Y were set to 1 bar) for 6 ps at 320 K to close in the leaflets. The protocol P2 consists of: (i) Compression at a pressure of 50, 50 and 300 bar in X, Y and Z respectively for 220 ps at 320 K to close in the gaps between leaflets. The protocol P3 consists of: (i) Compression at 15000 bar in Z-dimension (whereas the pressures in X and Y were set to 1 bar) for 6 ps at 275 K to close the gaps. Following the initial preparation of untilted structures and (where necessary as described above) extra steps for elimination of the inter-leaflet gap, all systems were simulated under anisotropic pressure coupling conditions at 1 bar pressure, with coordinates saved every 10 ps. The first 50 ns of all trajectories were allowed for equilibration (except for the C36 ordered system, which stabilized very quickly and for which 20 ns was allowed for equilibration), and averaged values with standard deviations of tilt angles and areas per lipid taken from the remainders (between 40 and 100 ns) of the trajectories.



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a

b

d

e

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c

Figure 2. (a) Uncompressed (disordered) structure (xz) (b) Compressed (disordered) structure (xz) (c) single lipid molecule with C5 carbons highlighted in yellow on the sn1 and sn2 chains. (d) Compressed (disordered) top view, showing C5 carbons of a single leaflet colored such that sites on the two tails of the same molecule are the same color (e) Compressed (ordered) top view of a single leaflet colored as in (d). Analysis of mean tilt angle and area per lipid. The tilt vector was defined by a vector that connects the C4 and the C14 carbons of each of the DPPE tail chains. The vectorial average over all chains in each leaflet was first calculated, and its angle with respect to the z-axis was plotted separately for each leaflet.

The area per lipid is

calculated as 2 × Lx × Ly/N, where Lx and Ly are the lengths of the simulation box in the x and y dimensions respectively, and N is the number of lipid molecules in the system. 10 ACS Paragon Plus Environment

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Set-up of alternating gel/fluid stripe domains for determination of Tm. The equilibrated backbone-disordered structure (Berger model) of the DPPE gel phase was used to create a box containing 619 DPPE lipids with lateral dimensions 13.2 nm × 9.7 nm.

To create alternating gel and fluid stripe domains, 50 DPPE molecules were

removed from each leaflet, from positions randomly selected within the bilayer but leaving intact a stripe of width 4 nm. The resulting structure, under periodic boundary conditions, had a central section with lipid packing characteristic of the gel phase flanked by regions in which the lateral density of lipids was about 24% lower. (Compared with the local temperature jump method previously used22 to set up alternating gel and fluid stripes, melting through depletion produced a disordered bilayer in a much shorter time.) The structure was solvated with 21093 waters, and upon brief (10 ps) initial equilibration at 320 K at 1 bar pressure, lipid tails in the low-density regions filled in the gaps, in the process losing their ordered gel-phase structure and contracting to leave an empty space between the leaflets. A subsequent 10 ps MD equilibration at high pressure (2000 bar) resulted in closing this gap. The structure was equilibrated at 320 K for 6 ns at 1 bar to produce a 519 DPPE structure with dimensions 12.3 × 9.6 × 10.4 nm3 showing a distinct zone of gel-like ordering embedded in a disordered bilayer structure. This was then used as the starting configuration for domain growth/contraction runs at intervals of 5 K between 315 and 335 K. In each case, the system was equilibrated for 100 ns (or until the phase transition was evidently completed) with the y dimension (parallel to the gel-fluid boundary) fixed and the x and z dimensions free to equilibrate independently at 1 bar pressure.



Calculating the hexagonal order parameter (S6).

To study melting or freezing

processes by characterizing lipid molecules in a stripe system as a gel/fluid lipid, the ‘hexagonal order parameter’ (S6) was calculated. The hexagonal order parameter ‘S6’ is defined for each of the chains of a lipid molecule. Each chain's structure is reduced to a point in two dimensions, calculated as the average of the x and y coordinates of the respective chain. The order parameter for a lattice site 'j' is calculated as follows:

, = (1)

where site 'j' has 'k' nearest neighbor sites and θjk is the angle between the vector connecting sites 'j' and 'k' and the x axis. Nearest neighbor sites are defined as those that reside within a circle centered at lattice site 'j' and with radius of 0.65 nm. A lipid is defined as a gel lipid if the order parameter S6 for each of the chains is at least 0.72. The fraction of gel-phase lipids according to this definition was determined in bands of width 1 nm extending along the y axis; typical profiles of the gel-phase fraction versus x are shown in Figure 3. The width of the gel stripe was defined as the portion over which the gel-phase fraction was greater than 65%. 1

Gel Fraction

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Lower,0ns Upper,0ns Lower,80ns Upper,80ns

0.8 0.6 0.4 0.2 0 0

3

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Position along x (nm)

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Figure 3. Profile of gel fraction of gel/fluid stripe system at 320 K (fraction of lipids within a 1 nm range of x values that satisfy the gel state criteria) vs. position along the x axis. The gel region is defined as having a gel fraction greater than 0.65.

RESULTS AND DISCUSSION Effect of backbone disorder on gel-phase structure. In simulations employing the Berger force field, gel phase lipid bilayers equilibrated from initial states that were backbone-ordered or backbone-disordered showed differences in their structural properties, as evident from snapshots taken following 100 ns equilibration at 320 K and shown in Figure 4. The backbone-ordered system adopted a distinct tilt in the lipid tails while the disordered structure maintained low tilt, but with some curvature in the bilayer. Looking at the evolution of the average collective tilt angles in the two systems (Figure 5) we see that the degree of tilting in the disordered system fluctuates on the 10 ns timescale from around 0 to 12°; the mean tilt angle was 7.0 ±1.5°. In the backboneordered system, after an initial transition from the untilted compressed structure to a tilted structure, the tilt angle averaged to 20.6±0.9°. The fluctuations in tilt and the dynamical undulations of the bilayer surface in the backbone disordered structure are evident from snapshots of characteristic “high-tilt” and “low-tilt” configurations sampled during the equilibration period (Supporting Information, Figure S1). The mean area per lipid of the ordered system was 0.425 ± 0.002 nm2 while for the disordered system it was 0.407 ± 0.002 nm2, as seen in Figure 6. Even though the area per headgroup is smaller for the disordered structure, the tilt-corrected area per chain (calculated as one-half the area per



lipid multiplied by the cosine of the tilt angle) is about 4% higher for the disordered case; as one would expect, the disordered system packs less densely.

Disordered-XZ

Disordered-YZ

Ordered-XZ

Ordered-YZ

Figure 4. Cross-sectional views from two sides of final structures from Berger force field DPPE bilayer equilibrated at 320 K from backbone-disordered (top) and backboneordered (bottom) initial structures.

Chain Molecular Tilt

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30

Lower,Dis

ordered Upper,Dis

25

Lower,Ord Upper,Ord

20 15

disordered

10 5 0 0

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40

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80

time (ns)

100

120

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Figure 5. Collective tail tilt angle in backbone-ordered and –disordered Berger force field DPPE gel phase systems along MD trajectories for upper (red) and lower (blue) leaflets.

Area Per Lipid (nm2)

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0.46

Disordered Ordered

0.44 0.42 0.4 0.38 0.36 0.34

0

20

40

60

time (ns)

80

100

120

Figure 6. Evolution of average area per lipid in backbone-disordered and -ordered Berger force field DPPE gel phase systems (blue and red curves respectively). Qualitatively, the backbone-disordered structure shown in Figure 4 displays greater curvature undulations than does the backbone-ordered structure. This trend appears to be true on average throughout the simulation, and suggests that the degree of backbone order influences the bending rigidity of the bilayer, either directly or via the effect on tilt. The fluctuations in the average tilt are greater for the disordered system as well.

The

curvature of the structure is not static but fluctuates (along with the tilt) over time, as can be observed in snapshots taken throughout the trajectory (examples of which are given in Figure S1). The undulations and tilt fluctuations might be related; recent work has shown that a broad distribution of molecular tilt angles in a simulated bilayer is predictive of a low bending modulus,33,34 although those studies considered fluid phases and fluctuations



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of single-tail tilt angles, as opposed to the fluctuations of net tilt angle of the entire leaflet calculated here. In simulations using the C36 atomistic force field, as in the results described above, the DPPE tail tilt angle (Figure 7) was also greater in the backbone-ordered system than in the backbone-disordered case. In both cases, however, the angle was much greater than that observed experimentally; the mean angle was 29.2±0.4° for the ordered case and ranged from 17.7±1.0° for trajectory P2 to 20.1 ±0.9° for trajectory P3 with the backbone-disordered system. Area per lipid, shown in Figure 8, is consistently higher for the C36 force field than the Berger force field, and higher for ordered than disordered structures.

As evident in Figure 9, the backbone-disordered bilayer structure from

protocol P2 differs from P1 and P3 in that it develops a cross-tilted structure or herringbone structure. In protocol P1 the tilt directions of the two leaflets were allowed to develop independently while there was a gap between the leaflets, so one would predict equal probabilities of aligned and crossed tilt vectors for the two leaflets. Even in P2 and P3, where the gap was eliminated through pressure applied to the bilayer normal before the tilt directions were locked in, both aligned and crossed tail tilts are seen. Therefore, we conclude that contact between the leaflets during the evolution of the tilt does not reproducibly determine the relative tilts of the leaflets. It is interesting to note, however, that the rows of tails observed along the X direction line up between the two leaflets in P1 and P3, whereas the rows are staggered in the cross-tilted P2 structure. The ordered C36 structure shown in Figure 9 also displays a cross-tilt, with opposing tilt directions in the two leaflets, directed largely along the X direction. Irrespective of the tilt direction, the three disordered systems prepared according to slightly different initial


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conditions converge on very similar tilt angles and areas per lipid, consistently lower than the backbone-ordered system. The reproducibility of the trend is further underlined in a pair of trajectories that were first mistakenly equilibrated using a combination of the C36 lipid force field and the SPC water model, then switched over to the correct TIP3P water model; as shown in Figure S2 (Supporting Information), these reach values for area per lipid and tilt angle that are consistent with the results shown in Figures 7 and 8 for systems equilibrated from the start with the TIP3P model.

35 P2 P1 P3 Ordered

Chain Tilt

30 25 20 15 10 5 0 0

50

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150

Time (ns)

Figure 7. Collective average tail tilt angle in CHARMM C36 backbone-ordered (red dotted curves) and –disordered (solid curves – blue (P1), red (P2) and black (P3)) DPPE gel phase systems along MD trajectories.

Area Per Lipid (nm2)

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Figure 8. Evolution of average area per lipid in CHARMM C36 backbone-disordered (solid curves – blue (P1), red (P2) and black (P3)) and -ordered (red dotted curve) DPPE gel

phase

systems.

Backbone-ordered

Disordered – P1

Disordered –P2

Disordered – P3

Figure 9. Cross-sectional views in two planes (xz at left and yz at right) of final structures from DPPE bilayers equilibrated at 320 K from backbone-ordered and backbonedisordered structures using CHARMM C36 force-field.


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Our primary conclusion is that the backbone order is coupled to the degree of tilt in the lipid tails, with a disordered backbone arrangement producing less strongly tilted tails in both force fields examined. A similar correlation was noted by Schubert et al. in the case of DPPC.5 In the present case, considering both the tilt and the area per lipid, the backbone-disordered structure is closer to the experimental structure for both force-fields used. X-ray diffraction studies indicate that the gel phase of DPPE is approximately untilted.9 To our knowledge, the area per lipid of DPPE bilayers in the gel phase has not been measured, but for DLPE (four carbons shorter in each chain) an area of 0.41 nm2 at 293 K has been obtained,35 consistent with the simulation results for the backbonedisordered Berger system as shown in Figure 6.

Although it is possible that a

cancellation of errors between the configuration and some fault in the force field is at play, the backbone-disordered structure's ability to better reproduce experimental properties agreement adds to previous evidence22 that it is more representative of the experimental structure than is a backbone-ordered structure.

The implication for

modeling and simulation is that a fully-ordered crystal-like structure is not the best jumping-off point for creating a gel-phase bilayer, but that introducing disorder in the glycerol backbone orientations is likely to yield a more realistic structure upon equilibration. The observation that backbone disorder reduces tail tilt can be rationalized on the basis of symmetry considerations.

Qualitatively, assuming that lipid structure dictates a

direction within the plane of the bilayer that is most favorable for a molecule to tilt, an orientationally disordered array of molecules cannot satisfy that preference for all molecules and so the degree of tilting in the "compromise" direction is likely to be



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weakened. To illustrate this idea, we consider a 1-d simplification of the actual geometry of the gel phase (depicted in Figure 10), consisting of a single row of molecules on the X axis.

An internal vector representing the backbone of each molecule can point either

towards the positive or negative X direction; the normalized projection of this vector of molecule on the X axis we will call α and give a value of +/- 1.

θ=0 θ = 16° θ = -16° X

Figure 10.

Cartoon representation of backbone-ordered (left, all α=1 lipids) and

backbone-disordered (right, mixture of α=1 and α=-1 lipids) arrangements of lipids in 1-d model at three tilt angles θ. Small red arrows represent backbone orientation vectors. The primary effect of tilt angle on the stability of the system is by its effect on the area per head group, given the constraints of (approximately) fixed length and volume of the extended tail groups. This will produce a contribution to the system's free energy that we will denote F0(θ). Because this effect does not depend on tilt direction, it will not depend on the backbone orientations. F0 must therefore be an even function of θ, and may either have its global minimum at θ0=0 or symmetric minima at ±θ0. When θ=0, symmetry dictates that reversing all molecule backbone orientations will not affect the energy. Similarly, reversing all molecular backbone orientations in a system with equal numbers of molecules in the α = +1 and α = -1 states (that is, a state 20 ACS Paragon Plus Environment

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with =0) will not affect the energy. Finally, symmetry requires that the energy to tilt an α =+1 lipid to an angle +θ be equal to the energy needed to tilt an α =-1 lipid through the same angle in the opposite direction, -θ. Therefore, we may expect a term f1 in the free energy, consistent with these properties, to account for the sum of the tilt preferences associated with the backbone anisotropy of each lipid: = () + (−) = (1 + 〈!〉) () + (1 − 〈!〉) (−)# (2)

where f1(0) = 0. Although not required by symmetry, is reasonable to assume that f1 is a monotonically increasing or decreasing function over moderate range of tilt angles. Here we will assume that the backbone orientation vector is defined so that f1 is a decreasing function of θ, i.e. that lipids oriented with α=+1 prefer to be tilted with θ > 0. If U0 has its minimum at θ=0, then a system of N lipids with =0 retains a minimum in the untilted state under the influence of both U0 and U1: %

&(' + ) & & ) = 0+ , − , = 0 (3) &( 2 & *' 2 & *' *'

In contrast, within a backbone-ordered state with α = 1 for all lipids, the derivative of the free energy with respect to tilt angle is N (df1/dθ)θ=0, which by our assumption will be negative and will produce an equilibrium tilt angle greater than zero. So, by symmetry, it is quite plausible that a backbone-ordered system should in general have a non-zero tilt angle while a system of the same lipids with randomized backbone directions is untilted. If U0 has degenerate minima at ± θ0 , then in a system of N lipids with uniform α =+1, the sum of U0 and U1 will be lower at θ =+ θ0 than at θ =- θ0. The shift in the position of



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the minimum will be proportional to the of the derivative of the sum U0 + U1 with respect to θ at this angle. The value of this slope in the backbone-ordered system: &(' + ) % ) &( **

/ ; 〈〉1

= 0+,

& (4) & **/

is larger in magnitude than its value the backbone-disordered system: &(' + ) % ) &( **

/ ; 〈〉'

=0+

& & , − , (5) 2 & **/ 2 & **/

given the assumption that df1/dθ < 0 over all θ of interest. Therefore, the magnitude of the tilt is expected to be greater for the backbone-ordered than the backbone-disordered system even when a mismatch between the cross-sectional areas of the head- and tailgroups promotes tilting. The above discussion does not take into account the fact that interaction between neighboring lipids with different orientations may have a different energy than between lipids with the same orientation.

If these interactions become more favorable with

increasing tilt angle θ, then this higher-order effect could counteract the trends described above for systems with non-zero equilibrium tilt. Overall, however, the most probable trend is that increasing backbone order would produce a greater equilibrium tilt angle. The structural properties (tilt angle and area per lipid) obtained using the Berger simulation model (with backbone disorder) are closer to experiment than those obtained from the C36 model. Hence, the Berger model is used to in further exploration of


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hydrogen-bonding statistics and the order-disorder phase transition temperature for the DPPE bilayer system. Hydrogen bonding between DPPE headgroup sites.

The degree of hydrogen

bonding between the amine hydrogens and the oxygen sites of the phosphoester and acyl ester groups of DPPE is shown in Table I for both gel-phase structures and for a fluidphase bilayer. The density of the hydrogen bond network depends strongly on the definition used to define a hydrogen bond.

Using the default definition from the

Gromacs g_hbond routine (distance between N and O within 3.5 Å and H-N-O bond angle less than 30°) there is about one H-bond for every two lipids in the gel phase, slightly more in the backbone-disordered case than in the backbone-ordered structure. The small excess in the backbone-disordered case comes from an increase in NH3+phosphate oxygen hydrogen bonds as opposed to the bonding to the acyl ester oxygen atoms. Hydrogen bonding within the gel phase at 320 K is more prevalent than in the fluid phase at 340 K, where we found an average of 0.35 hydrogen bonds per lipid. All these trends are maintained when more relaxed criteria are used to identify hydrogen bonds (distance between H and O within 2.5 Å and H-N-O bond angle less than 60°), but with total hydrogen bond counts increasing to about 2 for the gel phases and 1.5 in the fluid phase. This definition is comparable to that of ref. 24, where 1.2 PE-PE hydrogen bonds per lipid group was reported for pure fluid-phase DOPE.



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Table I. Hydrogen bonding statistics. Average Number of DPPE-DPPE Hydrogen bonds per lipid

NH3+ - phosphate NH3+ - ester

Gel, 320 K

Gel, 320 K

Fluid, 340 K

backbone-disordered;

backbone-ordered

0.25a / 1.05b

0.20 / 0.94

0.18 / 0.87

0.27 / 0.96

0.28 / 0.95

0.17 / 0.59

a. Using 3.5 Å N-O bond distance and 30° H-N-O bond angle cutoffs b. Using 2.5 Å H-O bond distance and 60° H-N-O bond angle cutoffs

Estimation of the transition temperature for the Berger force field the simulation model. To determine the limit of thermal stability of the gel phase, a system containing alternating stripe-shaped domains of fluid and gel phase bilayers was prepared as described in the Methods section. The Berger simulation model was used as it yielded an untilted gel phase, consistent with experiment. Equilibration at five temperatures evenly spaced between 315 K and 335 K showed definite growth of the gel phase at temperatures 320 K and below, and definite melting of the gel phase at temperatures 330 K and above, as evident in final snapshots shown in Figure 11. (Final snapshots for all systems are given in the Supporting Information, Figure S3.)


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initial structure

final structure, T=320 K

final structure, T=330 K

Figure 11. Cross-sectional view of alternating gel/fluid phase stripe systems in initial configuration (same used for all temperature points) and at T=320 and T=330 K. A portion of the solvent is cropped out of each snapshot. The rate of growth was assessed from the slope of the changing width of the gel phase domain, assessed by a linear fit to width vs. time, as shown in Figure 12 for each temperature's MD run. At both 315 and 320 K, there is a plateau at a width of 6 nm, even though the total width of the system remains well over 11 nm. Periodic boundary effects appear to be responsible; lateral growth of the gel domain is not uniform, and a gel-phase bridge can form across the fluid phase (see Figure S4). Once the gel phase has percolated across the x dimension of the simulation box, compression of the box along the x dimension is effectively prevented by the solid-like nature of the continuous gel 25 ACS Paragon Plus Environment


structure. Without compression, the total area cannot decrease and further gelation is blocked. (A similar kinetic trap is likely responsible for the presence of regions of disorder in gel phase structures generated by annealing.16) Therefore, at these temperatures the rate of gel phase growth was calculated only using the initial portion of the trajectory, up to the time that a width of 6 nm was reached (30 ns at T=315 and 50 ns at T=320 K). The rate of gel phase width increase vs. temperature is shown in Figure 13. We may conclude that the melting temperature for this model is near 325 K, and lies between 320 K and 330 K in any case.

Width of the gel front (nm)

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315K 320K 325K 330K 335K

8 6 4 2 0 0

20

40

60

80

time (ns)

100

120

Figure 12. Time evolution of the width of the simulated gel phase domain at temperatures between 315K and 335 K.


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0.1

Rate (nm/ns)

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0

-0.1 -0.2 315

320

325

330

Temperature (K)

335

Figure 13. Rate of growth of gel-phase domain width vs. temperature from simulations. Error bars were determined as in reference 22. The experimental transition temperature Tm for DPPE is 337 K.8 The melting point for the present model gel phase therefore falls short of the experimental value by 12 ± 5 K, a greater factor than discrepancy of 5-6 K between simulation and experiment found previously for DPPC and DSPC bilayers, simulated with the same force field.22 A likely explanation is that the inter-headgroup hydrogen bonding may be too weak in the present model. Following reference 25, the partial charges of +0.248 on the ethanolamine group H sites were taken from the GROMOS96 parameters for protonated lysine. For comparison the charge of +0.33 for the same site is used in the CHARMM force fields.14 CONCLUSIONS Investigation of ordered DPPE structures by simulation indicates that molecular tilt in the gel phase is sensitive to the ordering of glycerol backbones, and that disordered arrangements tend to produce lower tail tilt angles. A simple explanation for this trend based on symmetry considerations is proposed based on a phenomenological model. Whatever its origin, the improved agreement with experiment in tail tilt angle and in area



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per lipid obtained when a backbone-disordered arrangement of lipids is used in simulations is evidence that backbone disorder is a feature of experimental gel-phase structures as well. Computational modeling of gel phase structures is therefore better initiated from a disordered structure than by partial melting of a crystalline packing of lipids, given that the rotation of lipids about their long axis is hindered in the gel phase and cannot be assumed to equilibrate over the course of a simulation. Simulations with the Berger united-atom model reproduced the untilted gel phase structure when backbone disorder was introduced, while simulations with the CHARMM C36 all-atom force field retained a significant tail tilt. The Berger force field parameter set used displays a transition temperature of 325 ±5 K for DPPE, between 7 and 17 degrees below the experimental value. These characteristics of a model lipid gel are essential to establish so that it may be used appropriately in investigation of the kinetic, thermodynamic, and mechanical features of the gel phase and the gel-fluid transition; knowledge of these characteristics for a model of a lipid like DPPE that lacks the complicating interjection of the ripple phase is likely to be valuable in future studies. ASSOCIATED CONTENT Supporting Information. Snapshots of bilayer structures depicting the influence of tilt on the curvature, the evolution of tail tilt angle and area per lipid for the C36 systems initially equilibrated with the SPC water model, the starting structure of the stripe system (320 K) and the final structures from 315-335 K, and a snapshot illustrating the structure of gel and fluid domains after the gel domain has made a bridge to its periodic image are provided in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org. 28 ACS Paragon Plus Environment

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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Present Addresses †School of Science and Technology, Georgia Gwinnett College, Lawrenceville, GA, 30043 Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources This material is based upon work supported by the National Science Foundation under Grant No. CHE-1213904. ACKNOWLEDGMENT This material is based upon work supported by the National Science Foundation under Grant No. CHE-1213904.

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Molecular Simulation of the DPPE Lipid Bilayer Gel Phase: Coupling

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