^{1}, Hirsh Nanda

^{1,3}, Mathias Lösche

^{1,2,3,a)}and Frank Heinrich

^{1,3}

### Abstract

Biological membranes are composed of a thermally disordered lipid matrix and therefore require non-crystallographic scattering approaches for structural characterization with x-rays or neutrons. Here we develop a continuous distribution (CD) model to refine neutron or x-ray reflectivity data from complex architectures of organic molecules. The new model is a flexible implementation of the composition-space refinement of interfacial structures to constrain the resulting scattering length density profiles. We show this model increases the precision with which molecular components may be localized within a sample, with a minimal use of free model parameters. We validate the new model by parameterizing all-atom molecular dynamics (MD) simulations of bilayers and by evaluating the neutronreflectivity of a phospholipid bilayer physisorbed to a solid support. The determination of the structural arrangement of a sparsely-tethered bilayer lipidmembrane (stBLM) comprised of a multi-component phospholipid bilayer anchored to a gold substrate by a thiolated oligo(ethylene oxide) linker is also demonstrated. From the model we extract the bilayer composition and density of tether points, information which was previously inaccessible for stBLM systems. The new modeling strategy has been implemented into the *ga_refl*reflectivity data evaluation suite, available through the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR).

We thank Dr. Norbert Kucerka, Dr. Charles F. Majkrzak, Dr. John F. Nagle, and Dr. David Worcester for fruitful discussions, and Dr. Joseph Dura for the preparation of the thermally oxidized silicon wafer. We acknowledge Dr. Paul Kienzle for expert help regarding the data analysis software and Dr. Peter Yaron and Dr. Duncan McGillivray for critically reading the manuscript. This work was supported by the U.S. Department of Commerce through the MSE program under Grant No. 70NANB8H8009 and by the NIH (1P01 AG032131). This research was partially performed at the NIST Center for Nanoscale Science and Technology.

I. INTRODUCTION

A. Composition-space refinement of interfacial structures

II. DISTRIBUTION OF MOLECULAR COMPONENTS ACROSS AN INTERFACE:MODEL IMPLEMENTATION

A. Design of an error function-based model and comparison to Gaussian distributions

B. Error functions ensure volume filling intrinsically

C. Negative areas

III. VALIDATION OF THE CD MODEL WITH MD SIMULATIONS OF DMPC AND DOPC BILAYERS

IV. DISCUSSION OF THE CD MODEL

V. APPLICATIONS OF THE CD MODEL IN NR MEASUREMENTS (REF. 38)

A. Neutron reflection from a solid-supported lipid bilayer

B. Neutron reflection from a sparsely tethered bilayer lipidmembrane

VI. CONCLUSIONS

### Key Topics

- Lipids
- 49.0
- X-ray scattering
- 15.0
- Neutron scattering
- 14.0
- Molecular dynamics
- 13.0
- Neutrons
- 13.0

## Figures

(Color online) Distribution of sub-molecular components along the membrane normal in a DMPC bilayer, as obtained from a MD simulation (Ref. 52). The overlaid DMPC molecules are drawn to scale.

(Color online) Distribution of sub-molecular components along the membrane normal in a DMPC bilayer, as obtained from a MD simulation (Ref. 52). The overlaid DMPC molecules are drawn to scale.

Possibilities of accounting for atomic positions along the normal direction *z* at an interface for reflectivity modeling. (A) Average positions of the five atoms of a lipid phosphate group. (B) Lumping together atomic positions into one average position as a basis of describing the phosphate distribution by a Gaussian function. (C) Approximating the associated volume *V* of the component atoms by a boxcar function of width *l* and height *A* _{ b } = *V*/*l* that represents the cross-sectional area of the component. This approximation is the basis of the model developed in this work.

Possibilities of accounting for atomic positions along the normal direction *z* at an interface for reflectivity modeling. (A) Average positions of the five atoms of a lipid phosphate group. (B) Lumping together atomic positions into one average position as a basis of describing the phosphate distribution by a Gaussian function. (C) Approximating the associated volume *V* of the component atoms by a boxcar function of width *l* and height *A* _{ b } = *V*/*l* that represents the cross-sectional area of the component. This approximation is the basis of the model developed in this work.

(Color online) (A) The sum of two error functions represents an area profile *A*(*z*) of a sub-molecular component within a layered interface structure with the following parameters: *l* = 4 Å, *V* = 8 Å^{3}, σ_{1} = σ_{2} = 1 Å. The overlaid Gaussian function has the same volume *V* and a width parameter σ_{Gauss} ≈ 1.63 σ_{1}. (B) Adding a second sub-molecular component with the same *l*, *V*/*l*, and σ_{1} at the common interface results in ideal volume filling, even if σ_{1} ≠ σ_{2} for the second component. For details, see text.

(Color online) (A) The sum of two error functions represents an area profile *A*(*z*) of a sub-molecular component within a layered interface structure with the following parameters: *l* = 4 Å, *V* = 8 Å^{3}, σ_{1} = σ_{2} = 1 Å. The overlaid Gaussian function has the same volume *V* and a width parameter σ_{Gauss} ≈ 1.63 σ_{1}. (B) Adding a second sub-molecular component with the same *l*, *V*/*l*, and σ_{1} at the common interface results in ideal volume filling, even if σ_{1} ≠ σ_{2} for the second component. For details, see text.

(Color online) Validation of the model by fitting to an NAMD simulation (Ref. 51) of a DMPC bilayer (72 lipids, 6 waters/lipid; *T* = 315 K, initial configurations from Ref. 55) that used the CHARMM27 force field (Refs. 52–54). For details, see Supplemental Information. (A) Best-fit of the error-function based continuous distribution model with constraints (see text) to component distributions obtained from an average of 1000 configuration snapshots of the MD simulation. The overall space filling of the lipid molecules is broken down into the area profiles of individual lipid components, as indicated, for both the model (dashed black lines) and the simulation (continuous lines). Residuals shown at the top of the panel quantify the agreement between the model and the simulation data. (B) Comparison of the interfacial positions of molecular components, defined in the text, and their interfacial roughness with mean atomic positions and standard deviations from the mean obtained in the simulation. Different background gray levels indicate the projected extensions of sub-molecular lipid components. Crosses (x) on the interfaces between two sub-molecular components indicate the interfacial roughness σ of adjacent error functions determined by the fit. These parameters describe the thermally disordered distributions of neighboring molecular components. Upright crosses (+) provide the averaged locations and standard deviations from the mean of non-hydrogen atomic positions observed in the MD snapshot. For the methyl, carboxyl-glycerol, phosphate and choline components, non-hydrogen atoms are labeled. Dashed lines are guides for the eye.

(Color online) Validation of the model by fitting to an NAMD simulation (Ref. 51) of a DMPC bilayer (72 lipids, 6 waters/lipid; *T* = 315 K, initial configurations from Ref. 55) that used the CHARMM27 force field (Refs. 52–54). For details, see Supplemental Information. (A) Best-fit of the error-function based continuous distribution model with constraints (see text) to component distributions obtained from an average of 1000 configuration snapshots of the MD simulation. The overall space filling of the lipid molecules is broken down into the area profiles of individual lipid components, as indicated, for both the model (dashed black lines) and the simulation (continuous lines). Residuals shown at the top of the panel quantify the agreement between the model and the simulation data. (B) Comparison of the interfacial positions of molecular components, defined in the text, and their interfacial roughness with mean atomic positions and standard deviations from the mean obtained in the simulation. Different background gray levels indicate the projected extensions of sub-molecular lipid components. Crosses (x) on the interfaces between two sub-molecular components indicate the interfacial roughness σ of adjacent error functions determined by the fit. These parameters describe the thermally disordered distributions of neighboring molecular components. Upright crosses (+) provide the averaged locations and standard deviations from the mean of non-hydrogen atomic positions observed in the MD snapshot. For the methyl, carboxyl-glycerol, phosphate and choline components, non-hydrogen atoms are labeled. Dashed lines are guides for the eye.

(Color online) Validation of the model by fitting to an NAMD simulation of a DOPC bilayer (72 lipids, 5.4 waters/lipid, *T* = 296 K; initial configuration from Ref. 33). Other details as given in Fig. 4 and the Supplemental Information. (A) Best-fit of the error-function based continuous distribution model with constraints to a simulation snapshot. (B) Comparison of the interfacial positions of molecular components and their interfacial roughness with mean atomic positions and standard deviations from the mean obtained in the simulation.

(Color online) Validation of the model by fitting to an NAMD simulation of a DOPC bilayer (72 lipids, 5.4 waters/lipid, *T* = 296 K; initial configuration from Ref. 33). Other details as given in Fig. 4 and the Supplemental Information. (A) Best-fit of the error-function based continuous distribution model with constraints to a simulation snapshot. (B) Comparison of the interfacial positions of molecular components and their interfacial roughness with mean atomic positions and standard deviations from the mean obtained in the simulation.

(Color online) NR at two different solvent contrasts and structural analysis of a bilayer, deposited from floating surface monolayers with a composition, DMPS:DMPC-d_{54} 30:70, on a thermally oxidized Si wafer. (A) Experimental data. Error bars represent 66% confidence intervals. Inset: Neutron SLD determined from the fit using the molecular distribution model. (B) Decomposition of the area profile into sub-molecular components. The PS:PC compositions in the two bilayer leaflets is distinctly different from that of the parent monolayer, particularly in the leaflet proximal to the solid substrate. Note the slight displacement of the PS with respect to the PC head groups toward the bilayer center, which illustrates the capabilities of the new model, as such shifts of molecular fragments against each other cannot be described within the conventional box model.

(Color online) NR at two different solvent contrasts and structural analysis of a bilayer, deposited from floating surface monolayers with a composition, DMPS:DMPC-d_{54} 30:70, on a thermally oxidized Si wafer. (A) Experimental data. Error bars represent 66% confidence intervals. Inset: Neutron SLD determined from the fit using the molecular distribution model. (B) Decomposition of the area profile into sub-molecular components. The PS:PC compositions in the two bilayer leaflets is distinctly different from that of the parent monolayer, particularly in the leaflet proximal to the solid substrate. Note the slight displacement of the PS with respect to the PC head groups toward the bilayer center, which illustrates the capabilities of the new model, as such shifts of molecular fragments against each other cannot be described within the conventional box model.

(Color online) NR at three different solvent contrasts and structural analysis of a stBLM based upon WC14:βME 3:7 completed with DMPC-d_{54} (Ref. 8; details, see Supplemental Information). (A) Experimental data. Error bars represent 66% confidence intervals. Inset: Neutron SLD determined from the fit using the molecular distribution model. (B) Decomposition of the area profile into sub-molecular components. Note the high density of hydrocarbons in the proximal bilayer leaflet compared to the distal leaflet which comes about by the large content (≈ 63%) of myristyl chains associated with WC14 that pack more densely than the myristoyl chains of DMPC. The decomposition of the sub-membrane space between the bilayer and the substrate into its components, βME, oligo(ethyleneoxide) of the WC14 and water, is a unique capability of the new model and has not been possible within the limitations of conventional box models. This allows, for the first time, quantitative estimates of the area density ratio between βME and WC14 at the interface and of WC14 and free DMPC in the proximal bilayer leaflet within the self-organized tBLM structure.

(Color online) NR at three different solvent contrasts and structural analysis of a stBLM based upon WC14:βME 3:7 completed with DMPC-d_{54} (Ref. 8; details, see Supplemental Information). (A) Experimental data. Error bars represent 66% confidence intervals. Inset: Neutron SLD determined from the fit using the molecular distribution model. (B) Decomposition of the area profile into sub-molecular components. Note the high density of hydrocarbons in the proximal bilayer leaflet compared to the distal leaflet which comes about by the large content (≈ 63%) of myristyl chains associated with WC14 that pack more densely than the myristoyl chains of DMPC. The decomposition of the sub-membrane space between the bilayer and the substrate into its components, βME, oligo(ethyleneoxide) of the WC14 and water, is a unique capability of the new model and has not been possible within the limitations of conventional box models. This allows, for the first time, quantitative estimates of the area density ratio between βME and WC14 at the interface and of WC14 and free DMPC in the proximal bilayer leaflet within the self-organized tBLM structure.

## Tables

Parameters of the continuous distribution model fitted to the area profiles obtained from MD simulation of a DMPC bilayer (6 waters per lipid, Fig. 4) and equivalent quantities directly determined from the MD snapshot (see text). The parameters σ refer to the interface of the respective sub-molecular component with that on the preceding row. *A* ^{lipid} determined from the fit was (59.50 ± 0.04) Å^{2} and 59.43 Å^{2} in the simulation.

Parameters of the continuous distribution model fitted to the area profiles obtained from MD simulation of a DMPC bilayer (6 waters per lipid, Fig. 4) and equivalent quantities directly determined from the MD snapshot (see text). The parameters σ refer to the interface of the respective sub-molecular component with that on the preceding row. *A* ^{lipid} determined from the fit was (59.50 ± 0.04) Å^{2} and 59.43 Å^{2} in the simulation.

Parameters of the continuous distribution model fitted to the area profiles obtained from MD simulation of a DOPC bilayer (5.4 water per lipid, Fig. 5) and equivalent quantities directly determined from the MD snapshot. *A* ^{lipid} determined from the fit was (56.53 ± 0.04) Å^{2} and 57.06 Å^{2} in the simulation.

Parameters of the continuous distribution model fitted to the area profiles obtained from MD simulation of a DOPC bilayer (5.4 water per lipid, Fig. 5) and equivalent quantities directly determined from the MD snapshot. *A* ^{lipid} determined from the fit was (56.53 ± 0.04) Å^{2} and 57.06 Å^{2} in the simulation.

Fit parameters for a solid supported bilayer derived using the continuous distribution model. 95.4% confidence intervals were computed using a Monte Carlo resampling technique (Ref. 9).

Fit parameters for a solid supported bilayer derived using the continuous distribution model. 95.4% confidence intervals were computed using a Monte Carlo resampling technique (Ref. 9).

Fit parameters for the sparsely tethered lipid bilayer membrane derived using the continuous distribution model. 95.4% confidence intervals were computed using a Monte Carlo resampling technique (Ref. 9).

Fit parameters for the sparsely tethered lipid bilayer membrane derived using the continuous distribution model. 95.4% confidence intervals were computed using a Monte Carlo resampling technique (Ref. 9).

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