On a Model for Phase Separation on Biological Membranes and its Relation to the Ohta-Kawasaki Equation

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, R\"atz, R\"oger and the second author. The model is an extended Cahn-Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta-Kawasaki equation as the limit for infinitely large affinity between membrane components.Comment: 41 page

Kink dynamics with oscillating forces

It is well known that the dynamics of a one-dimensional dissipative system driven by the Ginzburg-Landau free energy may be described in terms of interacting kinks: two neighbouring kinks at distance $\ell$ feel an attractive force $F(\ell)\approx\exp(-\ell)$. This result is typical of a bistable system whose inhomogeneities have an energy cost due to surface tension, but for some physical systems bending rigidity rather than surface tension plays a leading role. We show that a kink dynamics is still applicable, but the force $F(\ell)$ is now oscillating, therefore producing configurations which are locally stable. We also propose a new derivation of kink dynamics, which applies to a generalized Ginzburg-Landau free energy with an arbitrary combination of surface tension, bending energy, and higher-order terms. Our derivation is not based on a specific multikink approximation and the resulting kink dynamics reproduces correctly the full dynamics of the original model. This allows to use our derivation with confidence in place of the continuum dynamics, reducing simulation time by orders of magnitude.Comment: 15 pages in Revtex, one column style. Minor changes. To appear in J. Stat. Mec

Transient domain formation in membrane-bound organelles undergoing maturation

The membrane components of cellular organelles have been shown to segregate into domains as the result of biochemical maturation. We propose that the dynamical competition between maturation and lateral segregation of membrane components regulates domain formation. We study a two- component fluid membrane in which enzymatic reaction irreversibly converts one component into another, and phase separation triggers the formation of transient membrane domains. The maximum domains size is shown to depend on the maturation rate as a power-law similar to the one observed for domain growth with time in the absence of maturation, despite this time dependence not being verified in the case of irreversible maturation. This control of domain size by enzymatic activity could play a critical role in intra-organelle dynamics.Comment: 7 pages, 6 figure

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.Comment: 34 pages, 10 figure

Reaction-Driven Assembly and Diffusiophoresis: Mechanisms for Control and Organization of Life-like Systems

The interior of cells, a fundamental building block of biological systems, is a hetero- geneous environment comprised of a multitude of molecular species. Its organization in the form of aggregates and compartments is tightly bound to the cellular function and demands precise coordination, positioning, and transportation. To achieve this, bi- ological systems operate in non-equilibrium, dissipating energy and exporting entropy to the exterior. Driven chemical reactions, wherein molecules of high internal energy are consumed, are one way to achieve this. In such a scenario, the formation and shape of membraneless organelles, characterized as phase-separated fluid condensates, result from an interplay of phase separation and chemical reactions that drive the system out of equilibrium. In addition to this, the properties of membrane-bound organelles, com- partments that are surrounded by an amphiphilic bilayer, may be influenced away from equilibrium, granting cells control over their topologies. This dissertation is dedicated to the question how driven chemical reactions influence the dynamics of biologically inspired systems. In particular, what are the properties and dynamics of aggregates formed from amphiphilic molecules, which are dynami- cally produced within a chemical reaction cycle? And how do long-range concentra- tion gradients that emerge from locally and externally replenished reactants influence the positioning and growth of condensates, both with and without membranes? Theoretical concepts from polymer physics are employed to answer these questions. To this end, a software is developed that simulates the time evolution of concentra- tion fields within a continuum model. The implementation is kept flexible and allows the blending of arbitrary numbers of diblock copolymer and homopolymer species. It makes efficient use of modern GPUs, which facilitates the investigation of large length and time scales. As a complementary simulation scheme, particle-based simulations are employed, wherein the reactions are implemented either as type conversions of en- tire macromolecules or as the conversion of monomeric units. Firstly, the reaction-driven assembly of molecules, that are switched between a hy- drophilic and an amphiphilic state within a reaction cycle, is investigated. Such systems could be implemented synthetically or may have played a role prebiotically. Both the- oretical considerations within the continuum model and simulations are used to study the initial dynamics of structure formation and, assuming instantaneous fuel recov- ery in the system, nonequilibrium steady states are identified. Aggregates may stack in lattices and the reaction rates influence the lattice spacing. Since the amphiphiles’ architecture dictates the membrane thickness in equilibrium, there is an interplay of length scales – the reaction-dictated lattice spacing and the membrane thickness. To- gether, these determine the membrane topology in the steady state. Theoretically, the membrane thickness is found to be approximately unaltered by the chemical reactions. Nonetheless, we show in the simulations, that compartments that form in the process, closed vesicles, accumulate precursor material which imparts tension on the membrane and decreases the membrane thickness. For certain parameter regions, this may even stabilize the formation of pores in the vesicles, which are propelled by the efflux of pre- cursor in opposite direction of the pore. Chemical reactions hence serve as a means to alter and control membrane topologies. Secondly, a strategy for cells to organize intra-cellular condensates via reaction-driven diffusiophoresis is demonstrated. In its original context, diffusiophoresis refers to the movement of hard colloids in external concentration gradients. Here, it is shown that passive liquid droplets are also transported in an external concentration gradient, be- cause the external component maintains a finite flux inside the droplet, depending on the interactions. Incompressibility in turn dictates a droplet flux in the opposite direc- tion, which induces the motion. This effect occurs naturally in systems with a reac- tion cycle that is driven by the conversion of high-internal-energy molecules, termed fuel, into a state of low internal energy, termed waste. To continuously enable the re- action cycle, fuel needs to be locally replenished and waste locally drained. In this case, fuel and waste concentrations exhibit gradients, such that dynamically emerging droplets are driven to or away from sources and sinks, depending on their interactions with fuel and waste. Theoretical considerations reveal an additional contribution to the movement of droplets that stems from an asymmetry in production in the droplets’ surroundings. In addition, it is explicitly shown that droplets with different composi- tions and distinct interactions can be transported in opposite directions simultaneously. Therefore, the phenomenon serves as a precise tool for biological systems to selectively position and move aggregates. Finally, the concept of reaction-driven diffusiophoresis is applied to aggregates that are assembled from amphiphilic molecules. By systematically varying molecular ar- chitectures and interactions, different pathways to form vesicles are explored. These dynamics circumvent free-energy barriers, which would emerge in their equilibrium counterpart and hamper self-assembly. More generally, there are two concepts that are common threads of these investiga- tions: Driven chemical reactions that change molecular solubility introduce a length scale, which determines phase-domain morphologies in an interplay with the system’s other intrinsic length scales. Furthermore, the investigations highlight the possibility of transport processes that originate from the dissipation of chemical energy through reaction cycles. To this end, an accurate treatment of the incompressibility constraint of typical aqueous solutions is essential.2024-07-1

Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation

We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term $q$. For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term $q$. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term $q$. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as $t\to +\infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor

Pattern formation aspects of electrically charged tri-stable media with implications to bulk heterojunction in organic photovoltaics

A common thread in designing electrochemically-based renewable energy devices comprises materials that exploit nano-scale morphologies, e.g., supercapacitors, batteries, fuel cells, and bulk heterojunction organic photovoltaics. In these devices, however, Coulomb forces often influence the fine nano-details of the morphological structure of active layers leading to a notorious decrease in performance. By focusing on bulk heterojunction organic photovoltaics as a case model, a self-consistent mean-field framework that combines binary (bi-stable) and ternary (tri-stable) morphologies with electrokinetics is presented and analyzed, i.e., undertaking the coupling between the spatiotemporal evolution of the material and charge dynamics along with charge transfer at the device electrodes. Particularly, it is shown that tri-stable composition may stabilize stripe morphology that is ideal bulk heterojuction. Moreover, since the results rely on generic principles they are expected to be applicable to a broad range of electrically charged amphiphilic-type mixtures, such as emulsions, polyelectrolytes, and ionic liquids.Comment: 8 pages, 4 figure

Ohta–Kawasaki Energy and Its Phase-Field Simulation

Understanding pattern formation in nature is an important topic in applied mathematics. For more than three decades, the Ohta–Kawasaki energy has attracted considerable attention from applied mathematicians. This energy functional, which combines surface energy and electrostatic potential energy, captures the intricate patterns observed in various physical and biological systems. Despite its apparent simplicity, the Ohta–Kawasaki energy serves as a versatile framework for describing a wide range of pattern formation phenomena induced by competing interactions. In this dissertation, we aim to gain a better understanding of the important properties of the Ohta–Kawasaki energy, specifically its stationary points, global minimizers, and energy landscape. We explore these properties in the context of broad applications such as nuclear physics, block copolymers, and biological membranes. In order to investigate the complicated geometries in these applications, we utilize asymptotic analysis and numerical simulations. Firstly, we explore the stationary points of the Ohta–Kawasaki energy. Specifically, we study how a three-dimensional ball loses stability as the nonlocal coefficient increases in the binary case. Our approach combines numerical simulations and bifurcation analysis. We calculate the minimum energy path for the transition from a single ball to two separate balls, as well as the bifurcation branch orginating from the ball. In the context of nuclear physics, this bifurcation branch is known as the Bohr–Wheeler branch. Our simulations suggest that, unlike the previous understanding, all the stationary points on this bifurcation branch are unstable. Similar results are observed in two dimensions. This finding illustrates the unexpected mechanism governing the stability loss of balls and disks. Secondly, we explore the global minimizers of the Ohta–Kawasaki energy. We numerically compute the one-dimensional energy minimizers of relatively short patterns in the non-degenerate ternary case. Inspired by our numerical results, we propose an array of periodic candidates. We then show that our candidates can have lower energy than the previously conjectured global minimizer which is of the cyclic pattern. Our results are consistent with simulations based on other theories and physical experiments of triblock copolymers, in which noncyclic lamellar patterns have been found. This finding indicates that even in one dimension, the global minimizers of the Ohta–Kawasaki energy can exhibit unexpected richness. Lastly, we explore the energy landscape of the Ohta–Kawasaki energy. We propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model in the degenerate ternary case. Our phase-field simulations and asymptotic results suggest that the limit of the recovery sequence exhibits behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes. This finding reveals the intricate landscape of the Ohta–Kawasaki energy. In summary, this dissertation sheds light on three important aspects of the Ohta–Kawasaki energy: its stationary points, global minimizers, and energy landscape. Our findings are timely contributions to the ongoing research on pattern formation driven by energetic competition