Bending Lipid Bilayers: A Closed-Form Collective Variable for

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Biomolecular Systems

Bending lipid bilayers: a closed-form collective variable for effective free-energy landscapes in quantitative biology Diego Masone, Marina Uhart, and Diego Bustos J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00012 • Publication Date (Web): 05 Mar 2018 Downloaded from http://pubs.acs.org on March 7, 2018

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Journal of Chemical Theory and Computation

Bending lipid bilayers: a closed-form collective variable for effective free-energy landscapes in quantitative biology Diego Masone,†,¶ Marina Uhart,† and Diego M. Bustos∗,†,‡ †Instituto de Histología y Embriología (IHEM) - Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Cuyo (UNCuyo), 5500, Mendoza, Argentina ‡Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo (UNCuyo), Mendoza, Argentina ¶Facultad de Ingeniería, Universidad Nacional de Cuyo (UNCuyo), Mendoza, Argentina E-mail: [email protected] Phone: +54 261 449 4117. Fax: +54 261 449 4117 Abstract Curvature-related processes are of major importance during protein-membrane interactions. The illusive simplicity of membrane reshaping masks a complex molecular process crucial for a wide range of biological functions like fusion, endo and exocytosis, cell division, cytokinesis and autophagy. To date, no functional expression of a reaction coordinate capable of biasing molecular dynamics simulations to produce membrane curvature has been reported. This represents a major drawback given that the adequate identification of proper collective variables to enhance sampling is fundamental for restrained dynamics techniques. In this work, we present a closed-form equation of a collective variable that induces bending in lipid bilayers in a controlled

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manner, allowing for straightforward calculation of free energy landscapes of important curvature-related events, using standard methods such as umbrella sampling and metadynamics. As a direct application of the collective variable, we calculate the bending free energies of a ternary lipid bilayer in the presence and the absence of a Bin/Amphiphysin/Rvs domain with an N-terminal amphipathic helix (N-BAR), a well-known peripheral membrane protein known to induce curvature.

Introduction Free energy landscapes are a fundamental thermodynamic quantification of variables, describing significant events in soft matter and biological systems. Though, an interest alternative to study low probability events using computer simulations is restrained molecular dynamics, 1 where a collective reaction coordinate (some function of the coordinates of the system) is used to bias the simulation and adequately describe the macroscopic phenomena. With a properly chosen collective variable, restrained molecular dynamics may surpass intrinsic limitations of the physical model allowing for more efficient statistical sampling. 2–4 However, the adequate identification of proper collective variables to enhance sampling, remains a challenging problem. Biological membranes are not flat and they exist in many dynamic shapes. Their curvature is an ubiquitous biological feature, from the ATP biosynthesis in the mitochondria cristae, 5 to endo/exocitosis, fertilization 6 and pathogenesis, all requiring a correct membrane bending to maximize molecular contacts and minimize the energy of the system. Moreover, diverse methods have been proposed to induce and/or study membrane curvature in computer simulations, 7,8 calculating the bending moduli 9–12 and estimating remodeling energies, 13,14 but until now no closed analytical form of a collective variable has been found. Here, we present Ψ, a collective variable to be used in restrained molecular dynamics techniques that induces bending, exceeding existing methods in simplicity, applicability and versatility. This new reaction coordinate opens the door to study curvature-associated cellular processes 2


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applying widely used enhanced sampling methods, whose success rely on the proper choice of the collective variable. As a proof of concept, we calculate the free-energy landscape of a model membrane with cholesterol with and without one N-BAR domain. This is an α-helical protein domain that contains an N-terminal amphipathic helix which spontaneously inserts itself into the membrane and induces curvature. 15,16 Then, the membrane curvature is stabilized by the high density of positively charged residues of the BAR dimer, which acts as a scaffold. 17 We test our collective variable on this system proving that, as expected, the necessary free energy for membrane bending decreases in the presence of the N-BAR domain.

z

z

x Ψ=0

Ψ=-0.1

Ψ=-0.2

Ψ=-0.3

y Figure 1: Lipid bilayer simultaneous bending in the X and Y directions in bead representation. Spanning the reaction coordinate Ψ from 0 (for the planar bilayer) to -0.3 (for the maximum curvature). Values beyond Ψ = −0.3 produce excessive bending (see supporting information). For clarity, water molecules are not shown.

Results and discussion More than ten years ago Wim Briels and collaborators developed an efficient method to form hydrophilic pores in lipid bilayers using computer simulations. 18,19 Inspired by these works, we propose here a closed-form analytical expression of a collective variable, defined Ψ (see eq 1), that induces bending in a lipid bilayer. We apply Ψ to calculate the free energy 3



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for bending a model lipid bilayer of pure dipalmitoylphosphatidylcholine (DPPC) using the well-known restrained molecular dynamics technique umbrella sampling. We also use Ψ to study the free energy contribution during membrane remodeling by protein induced bending, in a complex ternary lipid mixture. PN Ψ=

1

tanh ( 12

p

d2x + d2y + d2z ) − N0 N − N0

(1)

In eq 1, the sum runs over the centers of mass of all lipid molecules (N = 1024, in all cases studied here). Variables dx , dx and dz are the components of the three-dimensional distance between the center of mass of each lipid molecule and the center of mass of the membrane. N0 is the average equilibrium value of the summation in the numerator, obtained from an unrestrained simulation of an intact membrane.

Ψ collective variable Conceptually, Ψ is a dimensionless order parameter that controls a normalized density at the geometric XYZ center of the bilayer. As defined by eq 1, Ψ takes practical values in the interval [-0.3,0] for the largest bending and the equilibrium flat state, respectively (see fig 1). Negative values of Ψ increase local lipid density to which the membrane responds by occupying positions along the normal direction to the XY plane (initially containing the planar bilayer) and hence inducing bending.

Saddle shaped bending The sum of hyperbolic tangents included in the definition of Ψ runs over all 3D distances between each membrane molecule center of mass and the center of mass of the membrane itself. Consequently, the collective response of the many-particle system forming the membrane ends up in a simultaneous bending in both X and Y directions, giving its complex saddle-like shape. Fig 2 shows this resemblance together with the free energy of the si-

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DPPC

300 250 200

ΔG [kB T]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


150 100 50 0

−0.25 −0.2 −0.15 −0.1 −0.05 Ψ

0

Figure 2: Free energy profile for bending a pure DPPC bilayer. Computed from umbrella sampling molecular dynamics (for convergence details, see supporting information). Inset membrane snapshots are colored along the positive Z axis direction from blue to red. multaneous bending. As expected, (and as predicted by the well-known Helfrich model of membrane bending 20 ) the free energy needed is considerably higher than values reported for bending along one dimension and in the lowest mode ('80kB T ), 14 but in full agreement with calculations made for more complex events, such as dividing a single spherical phospholipid bilayer into two smaller vesicles ('350kB T ) 21 and the generation of membrane curvature by fusion or fission catalyzed by specialized proteins ('450kB T ). 22 To check whether a non bending solution satisfying the collective variable condition existed, we simulated 1µs of the pure DPPC bilayer imposing the restraint Ψ = −0.3. We observed permanent curvature during all simulation time, suggesting that no other configuration of the membrane satisfies the condition and hence validating our method. Fig 3 shows lipid density profiles averaged over 500 ns, depicting invariable curvature. Importantly, when a large bending condition is applied (i.e. Ψ0 = −0.3, for this bilayer) projected XY area

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reveals fluctuations of ±10%, (see supporting information for more details on this issue). a)

c)

b) 1

13

14

11

0.9

12

0.8

10

y [nm]

9 8 7 6 5

0.6 8

0.5

6

0.4 0.3

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0.2

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3 2 0

0.7

10

Normalized density

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z [nm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4

6

8

y [nm]

10

12

14

0 0

0.1 2

4

6

8

x [nm]

10

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0

Figure 3: Membrane bending is an intricate roughly saddle-shaped surface. (a-b) Averaged density profiles exhibiting permanent curvature. Side and top views of the 2D density maps for lipid molecules only, averaged along 500 ns of simulation time with the maximum bending restraint applied (Ψ0 = −0.3). (c) Smoothed linear regression grid surface to the scatter plot. Black dots indicate PO4 beads under Ψ0 = −0.3 bending restraint.

Protein induced membrane curvature Functioning biological membrane structure, dynamics, and molecular organization are constantly under an exchange of energy and material with the environment 23 (or are modulated by associated active proteins and enzymes 24 ). In particular, a considerable amount of computational 25–29 and experimental 15,30–32 studies have shown the effects of proteins containing a Bin/Amphiphysin/Rvs (BAR) domain 33 in flat membranes sculpture. Klaus Schulten and collaborators have shown how different arrangements of N-BAR domains (BAR domains containing an N-terminal amphipathic helix) on the membrane surface lead to bending. 16,34 As a direct application of Ψ collective variable, we have studied here the contribution a single N-BAR domain adsorbed on the bilayer surface has on the bending energetics. Two membranes were then constructed (in addition to the pure DPPC one), following a mixture already used by George Khelashvili and collaborators to study the bending modulus for multicomponent lipid membranes in different thermodynamic phases. 35 The chosen lipid mixture (see table 1) takes into account a saturated component with dipalmitoylphosphatidylcholine (DPPC), an unsaturated one with dioleoylphosphatidylcholine (DOPC) and choles6


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a)

c)

b)

Figure 4: Protein-induced curvature. (a) Coarse-grained model of the 244 residue N-BAR domain from Drosophila (PDB ID: 1URU). (b) Top view of the 1024 DPPC/DOPC/CHOL (blue/red/yellow) lipid bilayer in beads representation with a single the N-BAR domain (orange). Unrestrained molecular dynamics snapshot at t = 1µs. (c) Side view of the N-BAR domain adsorbed on the bilayer surface showing protein induced local bending. Water molecules are not shown. terol (CHOL). One of these membranes included a single N-BAR domain from Drosophila melanogaster (PDB ID: 1URU) adsorbed on the surface (see fig 4), which already has shown to induce curvature in all-atom and coarse-grained molecular dynamics simulations. 16,34 Supporting information includes a group of other 10 bilayers also tested as a benchmark for Ψ. Table 1: Lipid membranes Lipids DPPC DPPC/DOPC/CHOL DPPC/DOPC/CHOL

Mixture 1 0.58:0.12:0.3 0.58:0.12:0.3

N-BAR no no yes

min(Ψ0 ) -0.3 -0.5 -0.5

N0 998.255 991.771 991.771

Generalization to heterogeneous membranes When applying Ψ to multi-component lipid membranes, the center of mass of the whole bilayer is no longer a useful reference point from which to calculate distances towards the 7



center of mass of each lipid molecule. This is due to the fact that in an heterogeneous bilayer different lipids with a variety of lengths will contribute differently to the three dimensional distances needed by the collective variable. In particular, the membrane might find a solution to satisfy the reaction coordinate condition without bending itself but rearranging the lipids: the shortest molecules nearer the XY center of the bilayer and the longest near the borders. To avoid this artifact, we have defined more than one center of mass: one for each lipid species. This correction makes no changes in the general equation (1), but generalizes Ψ to use distances from each lipid molecule towards a center of mass defined particularly for each lipid type, giving the method the versatility needed for any kind of heterogeneous and/or asymmetric membrane. However, by doing this, Ψ becomes a function of the lipid composition and its bending range is now dependent on it. For example, to produce an equivalent amount of bending in the pure DPPC and the DPPC/DOPC/CHOL bilayer (as defined in table 1), Ψ must take equilibrium values of Ψ0 − 0.3 and Ψ0 = −0.5, respectively. Hence, free energy profiles are only comparable between bilayers of the same composition. As expected, the presence of the adsorbed N-BAR domain lowers the energetic cost for bending the bilayer in the Ψ space, (see fig 5), where free energy profiles for bending the same bilayer have been calculated with and without the N-BAR domain (free energy differences for maximum induced curvature are ' 100kB T ). This result is crucial to demonstrate that Ψ allows for the membrane to bend and reorganize as it would do under unrestrained conditions, cooperating with N-BAR as it induces bending. Consequently, we suggest that Ψ collective variable correctly captures the mechanisms that take place while inducing curvature in biological lipid bilayers, and hence we propose it as a tool to study bilayer curvature-related events. Given that Ψ controls the local lipid density, the overall pattern of the curved bilayer emerges as the result of this density adjustment. The procedure is analogous to the one applied by proteins inducing curvature, such as N-BAR domains. Hence, Ψ collective variable mimics the behavior of the bending phenomena and does not produce a simple geometrical

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DPPC/DOPC/CHOL DPPC/DOPC/CHOL/NBAR

500 400 ΔG [kB T]

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300 200 100 0 −0.5

−0.4

−0.3

Ψ

−0.2

−0.1

0

Figure 5: Three component bilayer (DPPC/DOPC/CHOL) free energy calculations during bending with and without the N-BAR domain. Computed from umbrella sampling molecular dynamics using Ψ collective variable. Snapshots show N-BAR position and the curvature induced. For clarity, water molecules are not shown. See supporting information for convergence details. surface (following the lowest bending modes) but a complex reshaped many-body object that spontaneously reorganizes itself in a saddle-like shape. This arrangement (see fig 1) becomes the most stable configuration satisfying each reaction coordinate equilibrium condition, as studied here, for Ψ in [-0.3 0] in the pure DPPC bilayer and Ψ in [-0.5 0] for the rest. In spite of the general application of our method and as an indication of the complexity and roughness of the energetic landscape being explored here, around 3% of all simulations performed in this work got trapped in pathological membrane shapes, depicting excessive aberrant bending or increased abnormal thickness. These undesired effects were verified by running many short simulations under a particular Ψ restraint and randomizing velocities at each restart. We observed that a few amount of metastable states of unsatisfactory curvature exist in the space of Ψ, but if spotted and discarded, do not represent a problem

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from a central limit theorem point of view, which molecular dynamics simulations adhere to. In all calculations performed in this work, these incorrect configurations were identified by visual inspection using Visual Molecular Dynamics (VMD), 36 deleted and repeated from the beginning. A full molecular description of membrane remodeling processes together with the calculation of free energy landscapes, is a step forward in understanding how membranes modulate signaling pathways. We expect that the simplicity of our approach will open up a broad range of applications in cellular transport functions where membrane bending is essential.

Materials and methods General setup All simulations were modeled under the MARTINI force-field 37 using the polarizable water model, 38 which has shown good results recently. 39–41 The MARTINI model groups 4 to 6 nonhydrogen atoms into a single particle. Then, reduced resolution particles interact according to a parametrization that well reproduced experimental data. A water layer was added above and below each 1024 lipid bilayer, verifying the ample water condition for MARTINI (>15 CG waters per lipid, which corresponds to 60 real waters per lipid 42 ). Molecular Dynamics simulations were performed with GROMACS-5.1.4 43–45 patched with Plumed 2.3.1, 46 a plug-in for free energy calculations and complex collective variable implementation. The calculation of Ψ was implemented in Plumed-2.3.1 46 compiled with matheval library, which is able to parse and evaluate symbolic mathematical expressions, such as the hyperbolic tangent used here as switching function. All simulations were run in the semi-isotropic NPT ensemble, at T = 303.15 K. 47–49 The temperature was controlled by a V-rescale thermostat 50 using a coupling constant of 1 ps. The pressure was maintained at 1.0 bar using the Berendsen barostat 51 with a 5 ps coupling constant with compressibility of 3x10−4 bar−1 . Long-range electrostatic interactions were computed with the fourth-order 10


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Particle-Mesh-Ewald method (PME). 52 This simulation scheme lets the membrane to reach a tensionless equilibrium state after a reasonably long period of equilibration. 53

Free energy calculations The Potential of Mean Force (PMF) for membrane bending was computed by umbrella sampling 54,55 using a set of 31 windows to span the reaction coordinate Ψ in the interval [-0.3 0] for the DPPC bilayer and 51 windows for spanning Ψ in the interval [-0.5 0] for the DPPCDOPC-CHOL bilayer. All membranes were minimized and equilibrated for more than 100 ns to generate properly relaxed initial configurations, with the exception of the membrane containing the N-BAR domain which was equilibrated for 1.4 µs to allow for the protein domain to be fully adsorbed on the bilayer surface. In all cases, each umbrella sampling window ran for 100 ns. Free energy profiles were recovered with the Weighted Histogram Analysis Method (WHAM). 56 Convergence was assessed by repeating these calculations on consecutive trajectory blocks of 10 ns (see supporting information).

Acknowledgement This work was supported by grants from MinCyT (Iniciativa de Proyectos Acelerados de Cálculo IPAC 2016: SACT 017-00721036), PICT-2014-1659, CONICET (PIP-13CO01), SeCTyP-UNCuyo (J051) and the Roemmers Foundation. Supercomputing time granted by the Sistema Nacional de Computación de Alto Desempeño (SNCAD-MinCyT) in clusters Mendieta (CCAD-UNC), Piluso (UNR), Coyote (CIMEC) and Pirayú (CIMEC) is gratefully acknowledged as well as GPU hardware granted by the NVIDIA Corporation. The authors thank L. Mayorga, C. Tomes and A. Torrano for invaluable scientific discussions and J. Giunta and N. Wolovick for their technical assistance while using the supercomputer clusters.

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Supporting Information Available Supporting information includes convergence details for all umbrella sampling simulations. The persistent curvature analysis along 1 µs depicting the area fluctuations involved and the N-BAR domain displacements and orientations when adsorbed on the bilayer surface, are described as well. Figures depicting maximum possible bending, center of mass shifts during umbrella sampling and a benchmark of other 10 bilayers where Ψ was tested are also included. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Graphical TOC Entry

19


z 1 2 3 z 4 5 6 7

Journal of Chemical Theory and Computation Page 20 of 24

x Ψ=0

Ψ=-0.1

Ψ=-0.2


y

Ψ=-0.3

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a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

b)



c)

300

DPPC

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250 200

ΔG [kB T]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27


150 100 50 0


−0.25 −0.2 −0.15 −0.1 −0.05

0

a)

b)

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1

13

14

11

0.9

12

0.8

10 9 8 7 6 5

0.6 8

0.5

6

0.4 0.3

4

0.2

4

2

3 2 0

0.7

10

2

4

6

8

y [nm]

10

12

14

0 0

0.1 2

4

6


8

x [nm]

10

12

14

0

Normalized density

12

y [nm]

z [nm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

c)


DPPC/DOPC/CHOL DPPC/DOPC/CHOL/NBAR


500 400 ΔG [kB T]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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300 200 100 0 −0.5

−0.4

−0.3


−0.2

−0.1

0

Bending Lipid Bilayers: A Closed-Form Collective Variable for

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