^{1,a)}, Yonggang Peng

^{2,3}, Yujun Zheng

^{3}and Frank L. H. Brown

^{1,2,a)}

### Abstract

A numerical scheme based upon established hydrodynamic and elastic considerations is introduced and used to predict the intermediate scattering function for lipid bilayer membranes. The predictions span multiple wavelength regimes, including those studied by dynamic light scattering (DLS; microns) and neutron spin-echo (NSE) spectroscopy (10–100 nm). The results validate a recent theory specific to the NSE regime and expose slight inaccuracies associated with the theoretical results available in the DLS regime. The assumptions that underlie both our numerical methods and the related theoretical predictions are reviewed in detail to explain when certain results can be applied to experiment and where caution must be exercised.

We thank R. Granek, J.-H. Lee, J. Nagle, P. Pincus, and A. Zilman for helpful discussions. This work was supported in part by the National Science foundation (NSF) (CHE-0848809). Financial support was also provided by the Los Alamos National Laboratory Institute for Multiscale Materials Studies, operated under the auspices of the National Nuclear Security Administration of the U. S. Department of Energy (DOE) under contract DE-AC52-06NA25396. F.L.H.B. is a Camille Dreyfus Teacher-Scholar.

I. INTRODUCTION

II. MEMBRANE DYNAMICS

III. INTERMEDIATE SCATTERING FUNCTION

IV. ANALYTIC EXPRESSIONS FOR

A. Zilman and Granek theory: Structureless membrane sheet with no internal dissipation

B. Bilayer dissipation included within NSE regime

V. NUMERICAL RESULTS

A. No dissipation: DLS regime

B. Dissipation included: DLS regime

C. Dissipation included: NSE regime

D. Geometric considerations

VI. CONCLUSION

## Figures

Membrane fluctuations have been studied across several length scales using a wide variety of techniques. The ranges of applicability for several techniques are shown above. Using the constants from Fig. 5, we also show the bilayer thickness and cross-over wavelengths associated with DMPC (see Sec. II).

Membrane fluctuations have been studied across several length scales using a wide variety of techniques. The ranges of applicability for several techniques are shown above. Using the constants from Fig. 5, we also show the bilayer thickness and cross-over wavelengths associated with DMPC (see Sec. II).

The lipid bilayer is treated as a periodic thin sheet. Its shape may be described by *h*(*x*, *y*), the local *z*-direction deviation of the sheet from a planar *L* × *L* fiducial state (pale green). So long as there are no overhangs, the height function *h*(*x*, *y*) is single valued for all points in the *xy* plane and provides a complete description of membrane shape. Although the dual leaflet structure of the bilayer may be considered for the purposes of calculating membrane dynamics, *S*(*k*, *t*) is calculated assuming that scattering occurs only from the single surface *h*(*x*, *y*).

The lipid bilayer is treated as a periodic thin sheet. Its shape may be described by *h*(*x*, *y*), the local *z*-direction deviation of the sheet from a planar *L* × *L* fiducial state (pale green). So long as there are no overhangs, the height function *h*(*x*, *y*) is single valued for all points in the *xy* plane and provides a complete description of membrane shape. Although the dual leaflet structure of the bilayer may be considered for the purposes of calculating membrane dynamics, *S*(*k*, *t*) is calculated assuming that scattering occurs only from the single surface *h*(*x*, *y*).

Schematic of a bilayer as described in the Seifert and Langer theory (Refs. 50 and 51). The neutral surface of each monolayer (dashed) lies parallel to the bilayer midplane (dotted). These surfaces are separated by a constant distance *d*. When the membrane is curved, the lipid number densities at the neutral surfaces ϕ^{±} are necessarily different from those at the bilayer midplane ψ^{±}.

Schematic of a bilayer as described in the Seifert and Langer theory (Refs. 50 and 51). The neutral surface of each monolayer (dashed) lies parallel to the bilayer midplane (dotted). These surfaces are separated by a constant distance *d*. When the membrane is curved, the lipid number densities at the neutral surfaces ϕ^{±} are necessarily different from those at the bilayer midplane ψ^{±}.

(A) An undeformed bilayer, where the density at every surface is ϕ_{0}. Each rectangle represents a lipid. (B) When the membrane is suddenly bent, dissipative forces at the midplane (dotted) hinder the lipids from rearranging. Hence, the number densities at the midplane remain unchanged. (C) Over time, the lipids rearrange so that the minimal energy in the bent geometry is achieved, in which ϕ^{±} = ϕ_{0} (dashed). Bilayer bending over short timescales is significantly more energetically costly than over longer timescales.

(A) An undeformed bilayer, where the density at every surface is ϕ_{0}. Each rectangle represents a lipid. (B) When the membrane is suddenly bent, dissipative forces at the midplane (dotted) hinder the lipids from rearranging. Hence, the number densities at the midplane remain unchanged. (C) Over time, the lipids rearrange so that the minimal energy in the bent geometry is achieved, in which ϕ^{±} = ϕ_{0} (dashed). Bilayer bending over short timescales is significantly more energetically costly than over longer timescales.

The scattering wave vector is defined as the difference between the incident and outgoing wave vectors . In order to calculate *S*(*k*, *t*), we integrate over all solid angles Ω. The membrane (blue) is shown with wavenumber *q* = 2π/λ.

The scattering wave vector is defined as the difference between the incident and outgoing wave vectors . In order to calculate *S*(*k*, *t*), we integrate over all solid angles Ω. The membrane (blue) is shown with wavenumber *q* = 2π/λ.

Behavior of the two relaxation modes for a DMPC bilayer over typical NSE time and length scales. Within this length regime, the decay rate of the modified bending mode (solid) has reached its asymptotic short wavelength behavior (red circles). The mode damped by the monolayer surface viscosity relaxes on a slower timescale than τ_{ NSE } and remains effectively constant during the course of the measurements.

Behavior of the two relaxation modes for a DMPC bilayer over typical NSE time and length scales. Within this length regime, the decay rate of the modified bending mode (solid) has reached its asymptotic short wavelength behavior (red circles). The mode damped by the monolayer surface viscosity relaxes on a slower timescale than τ_{ NSE } and remains effectively constant during the course of the measurements.

The normalized scattering functions at *k* = 0.02 nm^{−1} when κ = 12*k* _{ B } *T*. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm, which were obtained numerically. As the patch size increases, the intermediate scattering function converges to the ZG result (solid).

The normalized scattering functions at *k* = 0.02 nm^{−1} when κ = 12*k* _{ B } *T*. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm, which were obtained numerically. As the patch size increases, the intermediate scattering function converges to the ZG result (solid).

The normalized scattering functions at *k* = 0.02 nm^{−1} when κ = 2*k* _{ B } *T*. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm, which were calculated numerically. Due to the amplified effect of orientational averaging, the ZG result (solid) for *S*(*k*, *t*) is no longer accurate. The scattering functions are less sensitive to patch size than in Fig. 8 since η*L* ^{3}/κ has increased by a factor of 6.

The normalized scattering functions at *k* = 0.02 nm^{−1} when κ = 2*k* _{ B } *T*. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm, which were calculated numerically. Due to the amplified effect of orientational averaging, the ZG result (solid) for *S*(*k*, *t*) is no longer accurate. The scattering functions are less sensitive to patch size than in Fig. 8 since η*L* ^{3}/κ has increased by a factor of 6.

The normalized scattering functions at *k* = 0.02 nm^{−1} using the material parameters listed in Fig. 5 with *T* = 27 °C. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm. The numerically generated curves do not converge to the ZG result (solid black). The underlying assumptions of our analytic result (Eqs. (35) and (36)) are not valid within this regime (solid grey).

The normalized scattering functions at *k* = 0.02 nm^{−1} using the material parameters listed in Fig. 5 with *T* = 27 °C. The non-solid lines correspond to *L* = 6.4 μm, *L* = 9.6 μm, and *L* = 12.8 μm. The numerically generated curves do not converge to the ZG result (solid black). The underlying assumptions of our analytic result (Eqs. (35) and (36)) are not valid within this regime (solid grey).

For high scattering wave vectors such as *k* = 1 nm^{−1}, the effects of hydrodynamic dissipation within the bilayer may easily be observed. The numerical results for *L* = 200 nm, *L* = 400 nm, *L* = 800 nm, and *L* = 1600 nm, (non-solid) agree with analytic approximation (solid grey) of Eqs. (35) and (36). The ZG theory, which does not include internal dissipative forces, is no longer valid (solid black). The values of the material parameters given in Fig. 5 with *T* = 27 °C were used for these calculations.

For high scattering wave vectors such as *k* = 1 nm^{−1}, the effects of hydrodynamic dissipation within the bilayer may easily be observed. The numerical results for *L* = 200 nm, *L* = 400 nm, *L* = 800 nm, and *L* = 1600 nm, (non-solid) agree with analytic approximation (solid grey) of Eqs. (35) and (36). The ZG theory, which does not include internal dissipative forces, is no longer valid (solid black). The values of the material parameters given in Fig. 5 with *T* = 27 °C were used for these calculations.

κ_{ fit } is the best fit value of the bending modulus when fitting Eqs. (31), (32), and (39) to our numerical results when the normalized intermediate scattering function is greater than 0.7 (○) and greater than 0.2 (◻). κ is the actual value of the bending modulus used in Eq. (15). Top: the fits to *S*(*k*, *t*) using Eqs. (32) and (39). Bottom: the fits to *S*(*k* _{ z }, *t*) using Eqs. (31) and (39). The values of the material parameters given in Fig. 5 with *T* = 27 °C were used for these calculations.

κ_{ fit } is the best fit value of the bending modulus when fitting Eqs. (31), (32), and (39) to our numerical results when the normalized intermediate scattering function is greater than 0.7 (○) and greater than 0.2 (◻). κ is the actual value of the bending modulus used in Eq. (15). Top: the fits to *S*(*k*, *t*) using Eqs. (32) and (39). Bottom: the fits to *S*(*k* _{ z }, *t*) using Eqs. (31) and (39). The values of the material parameters given in Fig. 5 with *T* = 27 °C were used for these calculations.

The best fit exponent α, acquired by fitting the equation *S*(*k*, *t*)/*S*(*k*, 0) = Exp[ − (Γ*t*)^{α}] to our numerically generated curves. When including the effects of internal dissipation, α was obtained over values for *t* for which *S*(*k*, *t*)/*S*(*k*, 0) > 0.7 (⋆) and *S*(*k*, *t*)/*S*(*k*, 0) > 0.2 (□). When the effect of internal dissipation is not included, α is nearly constant (○). Since only one timescale is involved in this case, the value of α is nearly independent of how long the decay is measured. The values of the material parameters given in Fig. 5 with *T* = 27°C were used for these calculations.

The best fit exponent α, acquired by fitting the equation *S*(*k*, *t*)/*S*(*k*, 0) = Exp[ − (Γ*t*)^{α}] to our numerically generated curves. When including the effects of internal dissipation, α was obtained over values for *t* for which *S*(*k*, *t*)/*S*(*k*, 0) > 0.7 (⋆) and *S*(*k*, *t*)/*S*(*k*, 0) > 0.2 (□). When the effect of internal dissipation is not included, α is nearly constant (○). Since only one timescale is involved in this case, the value of α is nearly independent of how long the decay is measured. The values of the material parameters given in Fig. 5 with *T* = 27°C were used for these calculations.

The points are data from Yi *et al.* (Ref. 20) for DMPC vesicles at *T* = 60 °C. From the top down, the various data sets correspond to 2π/*k* = (12.8, 10, 7.2, 6.9, 6.0, 5.2, 4.7) nm. The solid curves are the fits using the numerical procedure. The patch size was set equal to the mean radius *R* = 200 nm. The dashed lines are fits to Eqs. (32) and (36). In both situations, the only fitting parameter is *d*. The rest of the material parameters were taken from Fig. 5.

The points are data from Yi *et al.* (Ref. 20) for DMPC vesicles at *T* = 60 °C. From the top down, the various data sets correspond to 2π/*k* = (12.8, 10, 7.2, 6.9, 6.0, 5.2, 4.7) nm. The solid curves are the fits using the numerical procedure. The patch size was set equal to the mean radius *R* = 200 nm. The dashed lines are fits to Eqs. (32) and (36). In both situations, the only fitting parameter is *d*. The rest of the material parameters were taken from Fig. 5.

*S*(*k*, *t*) at *k* = 1 nm^{−1} and *L* = 200 nm for a 2*L* × 2*L* system consisting of four identical patches (solid), and a single “floating” *L* × *L* patch (dashed) placed within a 2*L* × 2*L* system (see text).

*S*(*k*, *t*) at *k* = 1 nm^{−1} and *L* = 200 nm for a 2*L* × 2*L* system consisting of four identical patches (solid), and a single “floating” *L* × *L* patch (dashed) placed within a 2*L* × 2*L* system (see text).

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