Model studies of lipid flip-flop in membranes

Abstract

Biomembranes, which are made of a lipid bilayer matrix where proteins are embedded or attached, constitute a physical barrier for cell and its internal organelles. With regard to the distribution of their molecular components, biomembranes are both laterally heterogeneous and transversally asymmetric, and because of this they are sites of vital biochemical activities. Lipids may translocate from one leaflet of the bilayer to the opposite either spontaneously or facilitated by proteins, hence they contribute to the regulation of membrane asymmetry on which cell functioning, differentiation, and growth heavily depend. Such transverse motion—commonly called flip-flop—has been studied both experimentally and computationally. Experimental investigations face difficulties related to time-scales and probe-induced membrane perturbation issues. Molecular dynamics simulations play an important role for the molecular-level understanding of flip-flop. In this review we present a summary of the state of the art of computational studies of spontaneous flip-flop of phospholipids, sterols and fatty acids. Also, we highlight critical issues and strategies that have been developed to solve them, and what remains to be solved.

Appendix: Diffusion model of flip-flop

If other molecular and collective degrees of freedom have a negligible effect, as occurs for sterols and some fatty acids, flip-flop can be described as a diffusive process on a free energy landscape defined by the Z-position (normal to the bilayer) and the tilt angle of the lipid with respect to the bilayer normal. The existence of high energy barriers between the free energy minima allows the reduction of the problem to a kinetic description, according to the multi-dimensional extension [96, 108] of Kramers theory for activated processes [94]. If the free energy presents only two minima, corresponding to the preferred position and orientation of the molecule in either leaflet, the flip-flop process can be simply described in terms of jumps from one to the other minimum. The rate constant for the transition connecting the free energy minima j′ and j through the saddle point can be expressed as:

$$ k_{{j \leftarrow j^{\prime } }} = - \frac{{\lambda_{s} }}{{2\pi k_{B} T}}\sqrt {\frac{{\left| {\det {{\textbf{\"{U}}}}_{{j^{\prime } }} } \right|}}{{\left| {\det {{\textbf{\"{U}}}}_{s} } \right|}}} \exp \left[ { - \frac{{\left( {U_{s} - U_{{j^{\prime } }} } \right)}}{{k_{B} T}}} \right] $$

(A1)

where k _B is the Boltzmann constant, T is the temperature, U is the free energy, Ü is the matrix of the second derivatives of the free energy, and λ is the single negative eigenvalue of the product matrix DÜ, with D being the roto-translational diffusion tensor. The subscript indicates whether quantities are calculated in the minima (j, j′) or in the saddle point (s).

More generally, if in the free energy there are also metastable minima, the process can be pictured as a molecule first leaving the absolute minimum in one leaflet, then visiting other metastable sites within the bilayer, and finally reaching the absolute minimum in the other leaflet. In this case, the time evolution of the system is described in terms of the transitions between the free energy minima, according to the Master Equation [95]:

$$ \frac{\partial }{\partial t}p_{j} (t) = - \sum\limits_{{j^{{\prime }} }} {W_{{jj^{{\prime }} }} p_{{j^{{\prime }} }} (t)} $$

(A2)

where p _j(t) is the time dependent probability of the j state, $W_{jj^{\prime}} = - k_{{j \leftarrow j^{{\prime }} }} (j \ne j^{{\prime }} )$, and $W_{jj} = \sum\nolimits_{q} {k_{q \leftarrow j} }$. The time dependent probabilities are calculated by solving Eq. (A2) for a given initial condition, under the assumption that they approach the equilibrium probabilities at infinite time, p _j (t → ∞) = p _j,eq , with:

$$ p_{j,\;eq} = \frac{{\exp \left[ { - E_{j} /k_{B} T} \right]}}{{\sum\limits_{m} {\exp \left[ { - E_{m} /k_{B} T} \right]} }} $$

(A3)

where

$$ E_{j} = U_{j} + \frac{{k_{B} T}}{2}\ln \left| {\det \frac{{{{\ddot{\rm U}}}_{j} }}{{2\pi k_{B} T}}} \right|. $$

(A4)

Let us consider now the flip-flop process from the minimum A to the minimum B. By solving the master equation, Eq. (A2), with initial conditions p _A (t = 0) = 1, p _j≠A (t = 0) = 0, we calculate how the probability is transferred from A to any other minimum on the free energy surface. The flip-flop rate constant can then be obtained from the time evolution of the probability for the other absolute minimum, p _B (t), according to the expression:

$$ k_{flip - flip}^{ - 1} = \int\limits_{0}^{\infty } {\frac{{p_{B} (t) - p_{B} (\infty )}}{{ - p_{B} (\infty )}}dt} $$

(A5)

In the case of only two minima, p _B (t) follows a mono-exponential growth with rate constant k _B←A , and the integral in Eq. (A5) becomes equal to 1/k _B←A .

By performing the Boltzmann average of the free energy over the angular variable, the resulting 1D-PMF is a function of the single Z variable only. In this case the flip-flop rate constant can be evaluated as the rate of escape from the minimum over the barrier in the free energy profile. Thus the Kramers expression, which is the mono-dimensional analogue of Eq. (A5), can be used [94]:

$$ k_{flip - flop} = \frac{{D_{max} }}{{2\pi k_{B} T}}\sqrt {\frac{{{\ddot{U}}_{max} }}{{{\ddot{U}}_{min} }}} exp\left[ { - \frac{{U_{max} - U_{min} }}{{k_{B} T}}} \right] $$

(A6)

where U is the free energy and Ü is the curvature of the free energy profile, in the minimum (min) and in the maximum (max), respectively, with the diffusion coefficient along the single coordinate of interest evaluated on the top of the barrier, D _max .