Alternative mechanisms of structuring biomembranes: self-assembly versus self-organization

Time-reversal and parity-time symmetry breaking in non-Hermitian field theories

We study time-reversal symmetry breaking in non-Hermitian fluctuating field theories with conserved dynamics, comprising the mesoscopic descriptions of a wide range of nonequilibrium phenomena. They exhibit continuous parity-time (PT) symmetry-breaking phase transitions to dynamical phases. For two concrete transition scenarios, exclusive to non-Hermitian dynamics, namely, oscillatory instabilities and critical exceptional points, a low-noise expansion exposes a pretransitional surge of the mesoscale (informatic) entropy production rate, inside the static phases. Its scaling in the susceptibility contrasts conventional critical points (such as second-order phase transitions), where the susceptibility also diverges, but the entropy production generally remains finite. The difference can be attributed to active fluctuations in the wavelengths that become unstable. For critical exceptional points, we identify the coupling of eigenmodes as the entropy-generating mechanism, causing a drastic noise amplification in the Goldstone mode.

Sorting of multiple molecular species on cell membranes

Eukaryotic cells maintain their inner order by a hectic process of sorting and distillation of molecular factors taking place on their lipid membranes. A similar sorting process is implied in the assembly and budding of enveloped viruses. To understand the properties of this molecular sorting process, we have recently proposed a physical model [Zamparo et al., Phys. Rev. Lett. 126, 088101 (2021)]10.1103/PhysRevLett.126.088101, based on (1) the phase separation of a single, initially dispersed molecular species into spatially localized sorting domains on the lipid membrane and (2) domain-induced membrane bending leading to the nucleation of submicrometric lipid vesicles, naturally enriched in the molecules of the engulfed sorting domain. The analysis of the model showed the existence of an optimal region of parameter space where sorting is most efficient. Here the model is extended to account for the simultaneous distillation of a pool of distinct molecular species. We find that the mean time spent by sorted molecules on the membrane increases with the heterogeneity of the pool (i.e., the number of distinct molecular species sorted) according to a simple scaling law, and that a large number of distinct molecular species can in principle be sorted in parallel on cell membranes without significantly interfering with each other. Moreover, sorting is found to be most efficient when the distinct molecular species have comparable homotypic affinities. We also consider how valence (i.e., the average number of interacting neighbors of a molecule in a sorting domain) affects the sorting process, finding that higher-valence molecules can be sorted with greater efficiency than lower-valence molecules.

Non-reciprocity induces resonances in a two-field Cahn-Hilliard model

We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Active control of equilibrium, near-equilibrium, and far-from-equilibrium colloidal systems

The development of top-down active control over bottom-up colloidal assembly processes has the potential to produce materials, surfaces, and objects with applications in a wide range of fields spanning from computing to materials science to biomedical engineering. In this review, we summarize recent progress in the field using a taxonomy based on how active control is used to guide assembly. We find there are three distinct scenarios: (1) navigating kinetic pathways to reach a desirable equilibrium state, (2) the creation of a desirable metastable, kinetically trapped, or kinetically arrested state, and (3) the creation of a desirable far-from-equilibrium state through continuous energy input. We review seminal works within this framework, provide a summary of important application areas, and present a brief introduction to the fundamental concepts of control theory that are necessary for the soft materials community to understand this literature. In addition, we outline current and potential future applications of actively-controlled colloidal systems, and we highlight important open questions and future directions.

Steady states and coarsening in one-dimensional driven Allen-Cahn system

We study the steady states and the coarsening dynamics in a one-dimensional driven nonconserved system modeled by the so-called driven Allen-Cahn equation, which is the standard Allen-Cahn equation with an additional driving force. In particular, we derive equations of motion for the phase boundaries in a phase ordering system obeying this equation using a nearest-neighbor interaction approach. Using the equations of motion we explore kink binary and ternary interactions and analyze how the average domain size scale with respect to time. Further, we employ numerical techniques to perform a bifurcation analysis of the one-period stationary solutions of the equation. We then investigate the linear stability of the two-period solutions and thereby identify and study various coarsening modes.

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.

Exploring the interdependence between self-organization and functional morphology in cellular systems

All living matter is subject to continuous adaptation and functional optimization via natural selection. Consequentially, structures with close morphological resemblance repeatedly appear across the phylogenetic tree. How these designs emerge at the cellular level is not fully understood. Here, we explore core concepts of functional morphology and discuss its cause and consequences, with a specific focus on emerging properties of self-organizing systems as the potential driving force. We conclude with open questions and limitations that are present when studying shape-function interdependence in single cells and cellular ensembles.

Self-Propulsion Enhances Polymerization

Self-assembly is a spontaneous process through which macroscopic structures are formed from basic microscopic constituents (e.g., molecules or colloids). By contrast, the formation of large biological molecules inside the cell (such as proteins or nucleic acids) is a process more akin to self-organization than to self-assembly, as it requires a constant supply of external energy. Recent studies have tried to merge self-assembly with self-organization by analyzing the assembly of self-propelled (or active) colloid-like particles whose motion is driven by a permanent source of energy. Here we present evidence that points to the fact that self-propulsion considerably enhances the assembly of polymers: self-propelled molecules are found to assemble faster into polymer-like structures than non self-propelled ones. The average polymer length increases towards a maximum as the self-propulsion force increases. Beyond this maximum, the average polymer length decreases due to the competition between bonding energy and disruptive forces that result from collisions. The assembly of active molecules might have promoted the formation of large pre-biotic polymers that could be the precursors of the informational polymers we observe nowadays.

Pattern Formation in Precipitation Reactions: The Liesegang Phenomenon

Pattern formation is a frequent phenomenon in physics, chemistry, biology, and materials science. Bottom-up pattern formation usually occurs in the interaction of the transport phenomena of chemical species with their chemical reaction. The oldest pattern formation is the Liesegang phenomenon (or periodic precipitation), which was discovered and described in 1896 by Raphael Edward Liesegang, who was a German chemist and photographer who was born 150 years ago. The purpose of this feature article is to provide a comprehensive overview of this type of pattern formation. Liesegang banding occurs because of the coupling of the diffusion process of the reagents with their chemical reactions in solid hydrogels. We will discuss several phenomena observed and discovered in the past century, including reverse patterns, precipitation patterns with dissolution (due to complex formation), helicoidal patterns, and precipitation waves. Additionally, we will review all existing models of the Liesegang phenomenon including pre- and postnucleation scenarios. Finally, we will highlight several applications of periodic precipitation.

Cytoskeletal actin patterns shape mast cell activation

Activation of immune cells relies on a dynamic actin cytoskeleton. Despite detailed knowledge of molecular actin assembly, the exact processes governing actin organization during activation remain elusive. Using advanced microscopy, we here show that Rat Basophilic Leukemia (RBL) cells, a model mast cell line, employ an orchestrated series of reorganization events within the cortical actin network during activation. In response to IgE antigen-stimulation of FCε receptors (FCεR) at the RBL cell surface, we observed symmetry breaking of the F-actin network and subsequent rapid disassembly of the actin cortex. This was followed by a reassembly process that may be driven by the coordinated transformation of distinct nanoscale F-actin architectures, reminiscent of self-organizing actin patterns. Actin patterns co-localized with zones of Arp2/3 nucleation, while network reassembly was accompanied by myosin-II activity. Strikingly, cortical actin disassembly coincided with zones of granule secretion, suggesting that cytoskeletal actin patterns contribute to orchestrate RBL cell activation.

Huw Colin-York et al. use advanced microscopy techniques to show that the cortical actin network within a model mast cell line undergoes a series of reorganizational events at the basal interface during activation. They find that actin patterns co-localize with zones of Arp2/3 nucleation and myosin-II activity accompanies network reassembly.

Self-organizing actin patterns shape cytoskeletal cortex organization

Living systems rely, for biological function, on the spatiotemporal organization of their structures. Cellular order naturally emerges by dissipation of energy. Consequently, energy-consuming processes operating far from thermodynamic equilibrium are a necessary condition to enable biological systems to respond to environmental cues that allow their transitions between different steady-states. Such self-organization was predicted for the actin cytoskeleton in theoretical considerations and has repeatedly been observed in cell-free systems. We now demonstrate in our recent work how self-organizing actin patterns such as vortices, stars, and asters may allow cells to adjust their membrane architecture without affecting their cell mechanical properties.

When Is "Enough" Enough?

Computational simulations of polymerizing actin filaments indicate that competition for a limiting pool of building blocks is not sufficient to control their length.

Dynamic membrane patterning, signal localization and polarity in living cells

We review the molecular and physical aspects of the dynamic localization of signaling molecules on the plasma membranes of living cells. At the nanoscale, clusters of receptors and signaling proteins play an essential role in the processing of extracellular signals. At the microscale, "soft" and highly dynamic signaling domains control the interaction of individual cells with their environment. At the multicellular scale, individual polarity patterns control the forces that shape multicellular aggregates and tissues.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

Traveling waves and global oscillations triggered by attractive molecular interactions in an excitable system

During pattern formation in spatially extended systems, different mechanisms with different characteristic length scales, e.g., reaction-diffusion processes or molecular interactions, can be active. Such multiscale effects may generate new phenomena, which are not observed in systems where pattern formation occurs on a single scale. Here, we derive and analyze a reaction-diffusion model of the FitzHugh-Nagumo type with short-range attractive molecular interactions of the activator species. The model exhibits a wave instability. Simulations in one and two dimensions show traveling waves with a wavelength set by the parameters of the molecular interaction in the model. In two dimensions, simulations reveal a labyrinthine arrangement of the waves in systems with isotropic diffusion, whereas parallel bands of counterpropagating waves are formed in simulations of a model with anisotropic diffusion. The latter findings are in good qualitative agreement with experimental observation in the catalytic NO+H_{2} reaction on an anisotropic Rh(110) surface. In addition we have identified a transition regime in the simulations, where a short scale instability triggers global oscillations in an excitable regime.

Systems biology: a biologist's viewpoint

The debate over reductionism and antireductionism in biology is very old. Even the systems approach in biology is more than five decades old. However, mainstream biology, particularly experimental biology, has broadly sidestepped those debates and ideas. Post-genome data explosion and development of high-throughput techniques led to resurfacing of those ideas and debates as a new incarnation called Systems Biology. Though experimental biologists have co-opted systems biology and hailed it as a paradigm shift, it is practiced in different shades and understood with divergent meanings. Biology has certain questions linked with organization of multiple components and processes. Often such questions involve multilevel systems. Here in this essay we argue that systems theory provides required framework and abstractions to explore those questions. We argue that systems biology should follow the logical and mathematical approach of systems theory and transmogrification of systems biology to mere collection of higher dimensional data must be avoided. Therefore, the questions that we ask and the priority of those questions should also change. Systems biology should focus on system-level properties and investigate complexity without shying away from it.

Self-organization and the self-assembling process in tissue engineering

In recent years, the tissue engineering paradigm has shifted to include a new and growing subfield of scaffoldless techniques that generate self-organizing and self-assembling tissues. This review aims to cogently describe this relatively new research area, with special focus on applications toward clinical use and research models. Particular emphasis is placed on providing clear definitions of self-organization and the self-assembling process, as delineated from other scaffoldless techniques in tissue engineering and regenerative medicine. Significantly, during formation, self-organizing and self-assembling tissues display biological processes similar to those that occur in vivo. These processes help lead to the recapitulation of native tissue morphological structure and organization. Notably, functional properties of these engineered tissues, some of which are already in clinical trials, also approach native tissue values. This review endeavors to provide a cohesive summary of work in this field and to highlight the potential of self-organization and the self-assembling process for providing cogent solutions to currently intractable problems in tissue engineering.

Aggregation on a membrane of particles undergoing active exchange with a reservoir

We investigate the dynamics of clusters made of aggregating particles on a membrane which exchanges particles with a reservoir. Exchanges are driven by chemical reactions which supply energy to the system, leading to the establishment of a non-equilibrium steady state. We predict the distribution of cluster size at steady state. We show in particular that in a regime, that cannot exist at equilibrium, the distribution is bimodal: the membrane is mainly populated of single particles and finite-size clusters. This work is motivated by the observations that have revealed the existence of submicrometric clusters of proteins in biological membranes.

Dynamic as well as stable protein interactions contribute to genome function and maintenance

The cell nucleus is responsible for the storage, expression, propagation, and maintenance of the genetic material it contains. Highly organized macromolecular complexes are required for these processes to occur faithfully in an extremely crowded nuclear environment. In addition to chromosome territories, the nucleus is characterized by the presence of nuclear substructures, such as the nuclear envelope, the nucleolus, and other nuclear bodies. Other smaller structural entities assemble on chromatin in response to required functions including RNA transcription, DNA replication, and DNA repair. Experiments in living cells over the last decade have revealed that many DNA binding proteins have very short residence times on chromatin. These observations have led to a model in which the assembly of nuclear macromolecular complexes is based on the transient binding of their components. While indeed most nuclear proteins are highly dynamic, we found after an extensive survey of the FRAP literature that an important subset of nuclear proteins shows either very slow turnover or complete immobility. These examples provide compelling evidence for the establishment of stable protein complexes in the nucleus over significant fractions of the cell cycle. Stable interactions in the nucleus may, therefore, contribute to the maintenance of genome integrity. Based on our compilation of FRAP data, we propose an extension of the existing model for nuclear organization which now incorporates stable interactions. Our new “induced stability” model suggests that self-organization, self-assembly, and assisted assembly contribute to nuclear architecture and function.

Oscillations in the lateral pressure of lipid monolayers induced by nonlinear chemical dynamics of the second messengers MARCKS and protein kinase C

The binding of the MARCKS peptide to the lipid monolayer containing PIP(2) increases the lateral pressure of the monolayer. The unbinding dynamics modulated by protein kinase C leads to oscillations in lateral pressure of lipid monolayers. These periodic dynamics can be attributed to changes in the crystalline lipid domain size. We have developed a mathematical model to explain these observations based on the changes in the physical structure of the monolayer by the translocation of MARCKS peptide. The model indicates that changes in lipid domain size drives these oscillations. The model is extended to an open system that sustains chemical oscillations.

Phase separation and bistability in a three-dimensional model for protein domain formation at biomembranes

Proteins in living cells interact with membranes. They may bind to or unbind from the membrane to the cytosol depending on the lipid composition of the membrane and their interaction with cytosolic enzymes. Moreover, proteins can accumulate at the membrane and assemble in spatial domains. Here, a simple model of protein cycling at biomembranes is studied, when the total number of proteins is conserved. Specifically, we consider the spatio-temporal dynamics of MARCKS proteins and their interactions with enzymes facilitating translocation from and rebinding to the membrane. The model exhibits two qualitatively different mechanisms of protein domain formation: phase separation related to a long-wave instability of a membrane state with homogeneous protein coverage and stable coexistence of two states with different homogeneous protein coverage in bistable media. We evaluate the impact of the cytosolic volume on the occurrence of protein pattern formation by simulations in a three-dimensional model. We show that the explicit treatment of the volume in the model leads to an effective rescaling of the reaction rates. For a simplified model of protein cycling, we can derive analytical expressions for the rescaling coefficients and verify them by direct simulations with the complete three-dimensional model.

Domain-driven morphogenesis of cellular membranes

Cellular membrane systems delimit and organize the intracellular space. Most of the morphological rearrangements in cells involve the coordinated remodeling of the lipid bilayer, the core of the membranes. This process is generally thought to be initiated and coordinated by specialized protein machineries. Nevertheless, it has become increasingly evident that the most essential part of the geometric information and energy required for membrane remodeling is supplied via the cooperative and synergistic action of proteins and lipids, as cellular shapes are constructed using the intrinsic dynamics, plasticity and self-organizing capabilities provided by the lipid bilayer. Here, we analyze the essential role of proteo-lipid membrane domains in conducting and coordinating morphological remodeling in cells.

Actively maintained lipid nanodomains in biomembranes

Lipid rafts, defined as domains rich in cholesterol and sphingolipids, are involved in many important plasma membrane functions. Recent studies suggest that the way cells handle membrane cholesterol is fundamental in the formation of such lateral heterogeneities. We propose to model the plasma membrane as a nonequilibrium phase-separating system where cholesterol is dynamically incorporated and released. The model shows how cellular regulation of membrane cholesterol may determine the nanoscale lipid organization when the lipid mixture is close to a phase separation boundary, providing a plausible mechanism for raft formation in vivo.

Molecular architectonic on metal surfaces

The engineering of highly organized systems from instructed molecular building blocks opens up new vistas for the control of matter and the exploration of nanodevice concepts. Recent investigations demonstrate that well-defined surfaces provide versatile platforms for steering and monitoring the assembly of molecular nanoarchitectures in exquisite detail. This review delineates the principles of noncovalent synthesis on metal substrates under ultrahigh vacuum conditions and briefly assesses the pertaining terminology-self-assembly, self-organization, and self-organized growth. It presents exemplary scanning-tunneling-microscopy observations, providing atomistic insight into the self-assembly of organic clusters, chains, and superlattices, and the metal-directed assembly of low-dimensional coordination architectures. This review also describes hierarchic-assembly protocols leading to intricate multilevel order. Molecular architectonic on metal surfaces represents a versatile rationale to realize structurally complex nanosystems with specific shape, composition, and functional properties, which bear promise for technological applications.

Coerced mechanical coarsening of nanoparticle assemblies

Coarsening is a ubiquitous phenomenon that underpins countless processes in nature, including epitaxial growth, the phase separation of alloys, polymers and binary fluids, the growth of bubbles in foams, and pattern formation in biomembranes. Here we show, in the first real-time experimental study of the evolution of an adsorbed colloidal nanoparticle array, that tapping-mode atomic force microscopy (TM-AFM) can drive the coarsening of Au nanoparticle assemblies on silicon surfaces. Although the growth exponent has a strong dependence on the initial sample morphology, our observations are largely consistent with modified Ostwald ripening processes. To date, ripening processes have been exclusively considered to be thermally activated, but we show that nanoparticle assemblies can be mechanically coerced towards equilibrium, representing a new approach to directed coarsening. This strategy enables precise control over the evolution of micro- and nanostructures.