Generation of dynamic structures in nonequilibrium reactive bilayers

Spatial organization of active particles with field-mediated interactions

We consider a system of independent pointlike particles performing a Brownian motion while interacting with a Gaussian fluctuating background. These particles are in addition endowed with a discrete two-state internal degree of freedom that is subjected to a nonequilibrium source of noise, which affects their coupling with the background field. We explore the phase diagram of the system and pinpoint the role of the nonequilibrium drive in producing a nontrivial patterned spatial organization. We are able, by means of a weakly nonlinear analysis, to account for the parameter-dependence of the boundaries of the phase and pattern diagram in the stationary state.

Fluid transport by active elastic membranes

A flexible membrane deforming its shape in time can self-propel in a viscous fluid. Alternatively, if the membrane is anchored, its deformation will lead to fluid transport. Past work in this area focused on situations where the deformation kinematics of the membrane were prescribed. Here we consider models where the deformation of the membrane is not prescribed, but instead the membrane is internally forced. Both the time-varying membrane shape and the resulting fluid motion result then from a balance between prescribed internal active stresses, internal passive resistance, and external viscous stresses. We introduce two specific models for such active internal forcing: one where a distribution of active bending moments is prescribed, and one where active inclusions exert normal stresses on the membrane by pumping fluid through it. In each case, we asymptotically calculate the membrane shape and the fluid transport velocities for small forcing amplitudes, and recover our results using scaling analysis.

Phase separation in biological membranes: integration of theory and experiment

Lipid bilayer model membranes that contain a single lipid species can undergo transitions between ordered and disordered phases, and membranes that contain a mixture of lipid species can undergo phase separations. Studies of these transformations are of interest for what they can tell us about the interaction energies of lipid molecules of different species and conformations. Nanoscopic phases (<200 nm) can provide a model for membrane rafts, specialized membrane domains enriched in cholesterol and sphingomyelin, which are believed to have essential biological functions in cell membranes. Crucial questions are whether lipid nanodomains can exist in stable equilibrium in membranes and what is the distribution of their sizes and lifetimes in membranes of different composition. Theoretical methods have supplied much information on these questions, but better experimental methods are needed to detect and characterize nanodomains under normal membrane conditions. This review summarizes linkages between theoretical and experimental studies of phase separation in lipid bilayer model membranes.

Calcium-actin waves and oscillations of cellular membranes

We propose a mechanism for the formation of membrane oscillations and traveling waves, which arise due to the coupling between the actin cytoskeleton and the calcium flux through the membrane. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature, which act as nucleators of actin polymerization and adhesion. Such a continuum model couples the forces of cell-substrate adhesion, actin polymerization, membrane curvature, and the flux of calcium through the membrane. Linear stability analysis shows that sufficiently strong coupling among the calcium, membrane, and protein dynamics may induce robust traveling waves on the membrane. This result was checked for a reduced feedback scheme and is compared to the results without the effects of calcium, where permanent phase separation without waves or oscillations is obtained. The model results are compared to the published observations of calcium waves in cell membranes, and a number of testable predictions are proposed.

Nonequilibrium patterns in phase-separating ternary membranes

We present a nonequilibrium approach for the study of a two-dimensional phase-separating ternary mixture. When the component that promotes phase separation is dynamically exchanged with the medium, the separation process is halted and actively maintained finite-size segregation domains appear in the system. In addition to this effect, already reported in our earlier work [J. Gómez, F. Sagués, and R. Reigada, Phys. Rev. E 77, 021907 (2008)], the use of a generic Ginzburg-Landau formalism and the inclusion of thermal fluctuations provide a more dynamic description of the resulting domain organization. Its size, shape, and stability properties are studied. Larger and more circular and stable domains are formed when decreasing the recycling rate, increasing the mobility of the exchanged component, and the mixture is quenched deeper. We expect this outcome to be of applicability in raft phenomenology in plasmatic cell membranes.

Nonequilibrium microstructures in reactive monolayers as soft matter systems

Chemical systems provide classical examples of nonequilibrium pattern formation. Reactions in weak aqueous solutions, such as the extensively investigated Belousov-Zhabotinsky reaction, demonstrate a rich variety of patterns, ranging from travelling fronts to rotating spiral waves and chemical turbulence. Pattern formation in such systems is based on interplay between the reactions and diffusion. Intrinsically, this puts a restriction on the minimum length scale of the developing structures, which cannot be shorter than the diffusion length of the reactants. However, much smaller nonequilibrium structures, with characteristic lengths reaching down to nanoscales, are also possible. They are found in reactive soft matter, where energetic interactions between molecules are present as well. In these systems, chemical reactions and diffusion interfere with phase transitions, yielding active, stationary or dynamic microstructures. Nonequilibrium soft-matter microstructures are of fundamental importance for biological cells and may have interesting engineering applications. In this Minireview, we focus on the microstructures found in reactive soft-matter monolayers at solid surfaces or liquid-air interfaces.

Gradient sensing in reactive, ternary membranes

Using computer simulations, we investigate the behavior of reactive ternary ABC membranes that are subjected to an external, spatially nonuniform stimulus, which controls the rate of interconversion between the A and B components. We assume that A and B have different spontaneous curvatures. Furthermore, the C component is taken to be nonreactive and incompatible with both A and B. We find that a gradient in the applied stimulus causes the dynamic reconstruction of the membrane, with a preferential reorientation of the reactive AB domains along the gradient. In addition, the external gradient effectively controls the transport of the nonreactive C component within the membrane. The latter effect could potentially be exploited for cleaning the membrane of the nonreactive C "impurities" or for the targeted delivery of the C component to specific locations.

Nonlinear dynamics of a double bilipid membrane

The nonlinear dynamics of a biological double membrane that consists of two coupled lipid bilayers, typical of some intracellular organelles such as mitochondria or nuclei, is studied. A phenomenological free-energy functional is formulated in which the curvatures of the two parts of the double membrane and the distance between them are coupled to the lipid chemical composition. The derived nonlinear evolution equations for the double-membrane dynamics are studied analytically and numerically. A linear stability analysis is performed, and the domains of parameters are found in which the double membrane is stable. For the parameter values corresponding to an unstable membrane, numerical simulations are performed that reveal various types of complex dynamics, including the formation of stationary, spatially periodic patterns.

Phase transitions of the coupled membrane-cytoskeleton modify cellular shape

Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of testable predictions are proposed.

Modeling multicomponent reactive membranes

Using analytical calculations and computer simulations, we study binary AB and ternary ABC membranes that respond to an external stimulus by interconverting A and B components. The C component is assumed to be nonreactive and is incompatible with both A and B . We also assume that A and B have different spontaneous curvatures. The dynamics of the ternary system is described in terms of three order parameters: two specify the local composition and a third characterizes the local height of the membrane. Our description of the two-component membrane is based on a recent model proposed by Reigada [Phys. Rev. E. 72, 051921 (2005)]; we extend the latter approach by explicitly including the effects of the membrane's surface tension on the phase behavior of the system. By performing a linear stability analysis, we determine the behavior of the reactive AB membrane for a given bending elasticity and surface tension at different values of the reaction rate coefficients. We also numerically integrate the governing dynamic equations, and the results of these simulations are in agreement with the analytical predictions. For the two-component membranes, we calculate two critical values of the reaction rate coefficients, which define the behavior of the system, and plot the phase diagrams in terms of different parameters. We illustrate that the surface tension of the membrane strongly affects these critical values of the reaction rate coefficients and therefore the location of the phase boundaries. We also pinpoint the regions on the phase diagram where the late-time behavior is affected by the initial fluctuations, i.e., where such a reactive system has some "memory" of its prior state. In the case of the three-component system, we show that the presence of the nonreactive C component strongly affects the composition and topology of the membrane, as well as critically altering the propagation of the traveling waves within the system.

Finite-size domains in membranes with active two-state inclusions

The distribution of inclusion-rich domains in membranes with active two-state inclusions is studied by simulations. Our study shows that typical size of inclusion-rich domains (L) can be controlled by inclusion activities in several ways. When there is effective attraction between state-1 inclusions, we find: (i) Small domains with only several inclusions are observed for inclusions with time scales (approximately 10(-3)s) and interaction energy [approximately Omicron(kBT)] comparable to motor proteins. (ii) L scales as 13 power of the lifetime of state-1 for a wide range of parameters. (iii) L shows a switch-like dependence on state-2 lifetime k12(-1). That is, L depends weakly on k12 when k12<k12* but increases rapidly with k12 when k12>k12*, the crossover k12* occurs when the diffusion length of a typical state-2 inclusion within its lifetime is comparable to L. (iv) Inclusion-curvature coupling provides another length scale that competes with the effects of transition rates.