Protein diffusion on charged membranes: a dynamic mean-field model describes time evolution and lipid reorganization

Abstract

As charged macromolecules adsorb and diffuse on cell membranes in a large variety of cell signaling processes, they can attract or repel oppositely charged lipids. This results in lateral membrane rearrangement and affects the dynamics of protein function. To address such processes quantitatively we introduce a dynamic mean-field scheme that allows self-consistent calculations of the equilibrium state of membrane-protein complexes after such lateral reorganization of the membrane components, and serves to probe kinetic details of the process. Applicable to membranes with heterogeneous compositions containing several types of lipids, this comprehensive method accounts for mobile salt ions and charged macromolecules in three dimensions, as well as for lateral demixing of charged and net-neutral lipids in the membrane plane. In our model, the mobility of membrane components is governed by the diffusion-like Cahn-Hilliard equation, while the local electrochemical potential is based on nonlinear Poisson-Boltzmann theory. We illustrate the method by applying it to the adsorption of the anionic polypeptide poly-Lysine on negatively charged lipid membranes composed of binary mixtures of neutral and monovalent lipids, or onto ternary mixtures of neutral, monovalent, and multivalent lipids. Consistent with previous calculations and experiments, our results show that at steady-state multivalent lipids (such as PIP(2)), but not monovalent lipid (such as phosphatidylserine), will segregate near the adsorbing macromolecules. To address the corresponding diffusion of the adsorbing protein in the membrane plane, we couple lipid mobility with the propagation of the adsorbing protein through a dynamic Monte Carlo scheme. We find that due to their higher mobility dictated by the electrochemical potential, multivalent lipids such as PIP(2) more quickly segregate near oppositely charged proteins than do monovalent lipids, even though their diffusion constants may be similar. The segregation, in turn, slows protein diffusion, as lipids introduce an effective drag on the motion of the adsorbate. In contrast, monovalent lipids such as phosphatidylserine only weakly segregate, and the diffusions of protein and lipid remain largely uncorrelated.

FIGURE 1

Schematic view of a simulated unit cell containing a protein (or peptide) adsorbed on a membrane. For illustration, we use the basic poly-peptide Lysine, 13 residues long (Lys¹³). The membrane is represented by a rectangular-shaped slab of thickness d. Charges on the lipid headgroups are represented by a continuous surface charge density. The distance of nearest approach between protein and membrane surfaces is h. In all our calculations the dielectric constant for the membrane interior as well as protein is ε_m = 2, the dielectric constant of the aqueous environment is ε_m = 80, and the Debye length in the electrolyte solution is λ_D = 10 Å.

FIGURE 2

Adsorption of a spherical macroion (R_p = 10 Å) onto binary (PC/PS) lipid membranes with (a) Steady-state (equilibrium) adsorption free energy (in k_BT units) as a function of macroion-membrane separation calculated using our method (solid line) and by the method used in May et al. (20) (dashed line). (b) Time sequence of radial profiles of the local fraction of PS lipids, for macroion-membrane separation of h = 3 Å. The steady-state distribution of PS lipids from the calculations by May et al. is shown for comparison (dotted-dashed line).

FIGURE 3

Adsorption of spherical macroion (R_p = 10 Å) onto ternary (PC/PS/PIP₂) lipid membrane with composition. (a) Normalized fraction φ* of PIP₂ lipids (upper panel) and PS lipids (lower panel) as a function of the radial distance from the macroion, r, at different times from the initial macroion binding. (b) Time sequence for the electrochemical potentials (μ – μ°), reported in k_BT units, of PIP₂ (upper panel) and PS (lower panel) lipids upon macroion binding.

FIGURE 4

Adsorption of Lys¹³ onto ternary (PC/PS/PIP₂) lipid membrane with (a and b), and onto binary (PC/PS) lipid membrane with (c). (a) Normalized local fraction of PIP₂ lipids in the ternary system. (b) Local PS lipid fractions in the ternary system after 0.5 μs. (c) Local PS lipid fraction in the binary mixture.

FIGURE 5

Electrostatic potential isosurfaces for Lys¹³ adsorbed on a ternary (PC/PS/PIP₂) lipid membrane. (a and b) Side and top views, respectively, of the system in the initial configuration. (c and d) Similar views for the final state of the system, after 500 ns. Lys¹³ van der Waals surfaces are colored in gray. The red surface represents Φ = −1.5 k_BT/e (−37.5 mV) equipotential contour, and the blue mesh depicts the Φ = +1.5 k_BT/e (+37.5 mV) equipotential contour. For clarity, the lipid membrane is not shown.

FIGURE 6

Diffusion of charged spherical macroion on mixed membranes. The panels show the local surface charge densities after 0.6 μs of simulations (color scale) and the entire macroion trajectories in that time (connected black lines) for binary (PC/PS) mixture, D′ = 10 (a); for binary (PC/PS) mixture, D′ = 2 (b); for ternary (PC/PS/PIP₂) mixture, D′ = 10 (c); and for ternary (PC/PS/PIP₂) mixture, D′ = 2 (d). The red-dashed circles on each panel represent the projected size of the macroion with black arrows indicating the starting position of macroion center of mass. For clarity, the figures zoom in on the relevant membrane surface region explored by the macroion.

FIGURE 7

Mean-square-displacement (MSD) as a function of time for D′ = 10 (upper panel) and D′ = 2 (lower panel). The table inset shows the apparent macroion and lipid diffusion coefficient ratios for free and for membrane-bound macroion, the latter calculated from linear regression analysis of the MSD plots.

FIGURE 8

Value of the instantaneous adsorption free-energy functional ΔF (in k_BT units) as a function of time in the CHDMC simulations for (a) D′ = 2 systems and for (b) D′ = 10. The horizontal lines show the calculated equilibrium adsorption free energies for the respective systems when the macroion is stationary (see text for details).