Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

Abstract

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes-namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another-are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.

Figure 1

Snapshot of the simulation box created with the Martini force field. For better visualization, we do not illustrate water and ion molecules. We report (top) front view and (bottom) top view of the membrane-protein system. The protein is represented by the black domain, and the lipids are colored depending on the species. Cholesterol: red, POPC: blue, DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine): lime (lighter) green, PAPI (1-palmitoyl-2-arachidonoyl-sn-glycero-3-phosphoinositol): white, DGPE (1,2-digondoyl-sn-glycero-3-phosphoethanolamine): magenta, DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine): orange, POPS (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoserine): violet, DPSM (N-palmitoyl-D-sphingomyelin): yellow, PNSM (N-nervonoyl-D-sphingomyelin): pink, POSM (N-oleoyl-D-sphingomyelin): (darker) green. To see this figure in color, go online.

Figure 2

Cumulative distribution Π(r², Δ) with Δ = 1, 10, 100, 400, 1000, 5000, and 10,000 ns (from left to right). The dashed line corresponds to a theoretical curve proportional to r². To see this figure in color, go online.

Figure 3

One-dimensional and two-dimensional marginal distributions for model M2. The highlighted area below the one-dimensional marginal distributions corresponds to 95%. The ellipses in the two-dimensional marginal distributions correspond to 68 and 95% confidence regions. The reduced ${\tilde{\chi }}^2$ is 0.94. To see this figure in color, go online.

Figure 4

Comparison between the MSD data from MD simulations (blue dots) and GLE model (blue line). The long-time asymptotic MSD (black) and the characteristic times τ (green) and ${\omega }_s^{-1}$ (orange) and ${\omega }_0^{-1}$ (cyan) are also shown. To see this figure in color, go online.

Figure 5

(Top) Local MSD dependence on time of protein and NEAR and FAR lipids. α = 2 indicates the ballistic regime, α = 1 the Brownian regime, and 0 < α < 1 the subdiffusive regime. (Bottom) Comparison between protein’s time-dependent diffusion coefficient normalized to the value D_∞ = k_BT/(ξ_s + ξ_p) as derived from the model and the numerical counterpart derived from MD simulations. Numerical results for NEAR and FAR lipids as derived from MD simulations are shown as well. Gray lines help to visualize the onset of the Brownian dynamics. To see this figure in color, go online.

Figure 6

Distribution $p_{\lambda \nu }^{\delta ,\tau }$(f) of relaxation rates f underlying the memory kernel. The gray line corresponds to 1/τ, where τ = 13.4 ns. To see this figure in color, go online.

Figure 7

Comparison between the VACF data from MD simulations of protein and NEAR and FAR lipids. Dashed gray line indicates the point at which lipid negative correlations start. To see this figure in color, go online.

Figure 8

Illustration of the three-parameter Mittag-Leffler function (solid lines) for three different set of parameters (blue: as presented in Table 2; red: τ = 13.4 ns, λ = 0.7, ν = 0.98, δ = 1.4; green: τ = 13.4 ns, λ = 0.7, ν = 1.0, δ = 1.0) as well as their corresponding long-time asymptotic decay (dashed lines). Notice that the case τ = 13.4 ns, λ = 0.7, ν = 1.0, δ = 1.0 reduces to the one-parameter Mittag-Leffler function, whose short-time behavior corresponds to the stretched exponential exp(−t/τ)^λ (86) (black dashed line). To see this figure in color, go online.

Figure 9

One-dimensional and two-dimensional marginal distributions for model M1. The highlighted area below the one-dimensional marginal distributions corresponds to 95%. The ellipses in the two-dimensional marginal distributions correspond to 68 and 95% confidence regions. The reduced ${\tilde{\chi }}^2$ is 1.02. To see this figure in color, go online.