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Phase segregation of membranal components, such as proteins, lipids, and cholesterols, leads to the formation of aggregates or domains that are rich in specific constituents. This process is important in the interaction of the cell with its surroundings and in determining the cell’s behavior and fate. Motivated by published experiments on curvature-modulated phase separation in lipid membranes, we formulate a mathematical model aiming at studying the spatial ordering of composition in a two-component biomembrane that is subjected to a prescribed (imposed) geometry. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively explore their influence. We reproduce published experimental observations and extend them to surfaces with geometric features (imposed geometry) and lipid phases beyond those used in the experiments. In addition, we demonstrate the possibility for curvature-modulated phase separation above the critical temperature and propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition.

The biological lipid bilayer membrane, or in short “biomembrane,” is a fundamental building block of the cell. It forms the barrier that separates the interior of the cell from its surroundings but is also responsible for almost all interaction of the cell with its environment, including transport, adhesion, regulation, transduction, and signaling [

The fluidity of the biomembrane combined with its spatial heterogeneity brings about a unique coupling between shape (geometry) and composition. For example, lipid phases that possess high bending stiffness highly favor regions with small (magnitude) curvature [

In the last two decades, much effort has been invested into understanding the consequences of the coupling between shape and composition in biomembranes. Theoretical models have generalized uniform composition models [

In a recent work, Parthasarathy et al. [

In the current paper we analyze this type of experiment by means of a mathematical model combined with numerical simulations. The main goal is to reproduce the experimental observations mentioned above but also to generalize them and motivate new experiments. Accordingly, the structure of the paper is as follows: Section

Consider a biomembrane composed of two components, for example, two different lipid molecules or two different lipid phases, that lies on a smooth nonflat surface (in their experiment, Parthasarathy et al. [

Composition field,

It is convenient to define

Equilibrium configurations correspond to local minima of the free energy, subjected to the relevant constraints. In our case, the system is closed so the total number of molecules of each type is preserved:

In what follows, we formulate the equations that govern the evolution of the concentration field

Next, we rewrite the governing equation in a nondimensional form. Besides the convenient formulation, this procedure enables us to identify of the nondimensional quantities that govern the behavior. To this end, we consider the characteristic scales of energy, length, and time.

The coefficient of the mixing energy

Applying a similar procedure to the third term in (

Nondimensional spatial coordinates (location),

In this section, we present numerical results focusing on the influence of the nondimensional quantities that were identified in the previous section on the response of a biomembrane with imposed geometry.

In our numerical simulations we calculate the evolution in time of the composition field,

Snapshots of typical simulations. (a)

All simulations start from random noncorrelated values near

In the experiments of Parthasarathy et al. [

The main purpose of the numerical investigation is to study the interplay between the imposed geometry (curvature) and composition of the biomembrane. Hence, we focus our attention on the influence of the nondimensional quantities

Motivated by the experiments of Parthasarathy et al. [

In our model, the magnitude of the surface curvature is accounted for by the nondimensional quantity

Figure

Final (steady state) composition field for different values of

Recall that

The parameter

The effect of overall (average) composition,

Similarly to Figure

Behavior above critical temperature. (a)

A comparison between the definitions of

Following the discussion above, we focus our attention below on demonstrating the consequences of the fact that the sign of the third term in (

The effect of spontaneous curvature for different values of

We formulated a simple mathematical model to study the spatial ordering of composition in a two-component biomembrane that is subjected to prescribed (imposed) geometry. The mathematical model does not account for possible anisotropy of the membrane constituents or for possible interaction between lipids in the two leaflets of the bilayer membrane. In addition, the numerical scheme, which follows the steepest descent method, leads to metastable equilibrium configurations associated with local minima of the free energy. We note, however, that applying the numerical scheme to slightly different initial conditions or to a larger domain of solution did not change the essence (topology) of the solution. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively study their influence. Our numerical results show that the geometry-driven ordering of the biomembrane composition is largely governed by the difference between the nondimensional bending stiffness and spontaneous curvatures of each phase, while the magnitude of this phenomenon is proportional to the ratio between the bending energy and the (chemical) interaction energy of the phases. Roughly speaking, energy considerations favor configurations in which the phase that is stiffer and has smaller spontaneous curvature is located at regions having smaller curvature. The numerical simulations reproduced the experimental observation of Parthasarathy et al. [

An important advantage of a mathematical model is that it enables studying the behavior at various settings with minimal effort and resources, before entering the lab. The agreement of our model results with experimental observations strengthens our confidence in the model and numerical scheme and opens the door to examining new and different conditions than those used in the original experiments. For example, we have demonstrated that, if the surface geometry is properly designed, phase separation can occur above the critical temperature. Such curvature-induced phase separation above the critical temperature awaits experimental examination. Also, we propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition. The procedure is based on the observation that the mechanism associated with the difference in bending stiffness depends on the magnitude of the surface curvature but indifferent to the sign (direction) of the curvature. On the contrary, the mechanism related to the spontaneous curvatures strongly depends on both magnitude and sign (direction) of the surface curvature. The consequences of these differences have been demonstrated by a set of simulations.

The authors declare that there are no conflicts of interest regarding the publication of this paper.