Travelling lipid domains in a dynamic model for protein-induced pattern formation in biomembranes

Protein lipidation in health and disease: molecular basis, physiological function and pathological implication

Posttranslational modifications increase the complexity and functional diversity of proteins in response to complex external stimuli and internal changes. Among these, protein lipidations which refer to lipid attachment to proteins are prominent, which primarily encompassing five types including S-palmitoylation, N-myristoylation, S-prenylation, glycosylphosphatidylinositol (GPI) anchor and cholesterylation. Lipid attachment to proteins plays an essential role in the regulation of protein trafficking, localisation, stability, conformation, interactions and signal transduction by enhancing hydrophobicity. Accumulating evidence from genetic, structural, and biomedical studies has consistently shown that protein lipidation is pivotal in the regulation of broad physiological functions and is inextricably linked to a variety of diseases. Decades of dedicated research have driven the development of a wide range of drugs targeting protein lipidation, and several agents have been developed and tested in preclinical and clinical studies, some of which, such as asciminib and lonafarnib are FDA-approved for therapeutic use, indicating that targeting protein lipidations represents a promising therapeutic strategy. Here, we comprehensively review the known regulatory enzymes and catalytic mechanisms of various protein lipidation types, outline the impact of protein lipidations on physiology and disease, and highlight potential therapeutic targets and clinical research progress, aiming to provide a comprehensive reference for future protein lipidation research.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

Nonreciprocal Cahn-Hilliard Model Emerges as a Universal Amplitude Equation

Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatiotemporal pattern formation. Starting from a linear large-scale oscillatory instability-a conserved-Hopf instability-that naturally occurs in many active systems with two conservation laws, we derive a corresponding amplitude equation. It belongs to a hierarchy of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s). The derived universal equation generalizes a phenomenological model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard model, and may be of a similar relevance for the classification of pattern forming systems as the complex Ginzburg-Landau equation.

Influence of physical interactions on spatiotemporal patterns

Spatiotemporal patterns are often modeled using reaction-diffusion equations, which combine complex reactions between constituents with ideal diffusive motion. Such descriptions neglect physical interactions between constituents, which might affect resulting patterns. To overcome this, we study how physical interactions affect cyclic dominant reactions, like the seminal rock-paper-scissors game, which exhibits spiral waves for ideal diffusion. Generalizing diffusion to incorporate physical interactions, we find that weak interactions change the length- and time scales of spiral waves, consistent with a mapping to the complex Ginzburg-Landau equation. In contrast, strong repulsive interactions typically generate oscillating lattices, and strong attraction leads to an interplay of phase separation and chemical oscillations, like droplets co-locating with cores of spiral waves. Our work suggests that physical interactions are relevant for forming spatiotemporal patterns in nature, and it might shed light on how biodiversity is maintained in ecological settings.

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'.

Revealing local molecular distribution, orientation, phase separation, and formation of domains in artificial lipid layers: Towards comprehensive characterization of biological membranes

Lipids, together with molecules such as DNA and proteins, are one of the most relevant systems responsible for the existence of life. Selected lipids are able to assembly into various organized structures, such as lipid membranes. The unique properties of lipid membranes determine their complex functions, not only to separate biological environments, but also to participate in regulatory functions, absorption of nutrients, cell-cell communication, endocytosis, cell signaling, and many others. Despite numerous scientific efforts, still little is known about the reason underlying the variability within lipid membranes, and its biochemical significance. In this review, we discuss the structural complexity of lipid membranes, as well as the importance to simplify studied systems in order to understand phenomena occurring in natural, complex membranes. Such systems require a model interface to be analyzed. Therefore, here we focused on analytical studies of artificial systems at various interfaces. The molecular structure of lipid membranes, specifically the nanometric thickens of molecular bilayer, limits in a major extent the choice of highly sensitive methods suitable to study such structures. Therefore, we focused on methods that combine high sensitivity, and/or chemical selectivity, and/or nanometric spatial resolution, such as atomic force microscopy, nanospectroscopy (tip-enhanced Raman spectroscopy, infrared nanospectroscopy), phase modulation infrared reflection-absorption spectroscopy, sum-frequency generation spectroscopy. We summarized experimental and theoretical approaches providing information about molecular structure and composition, lipid spatial distribution (phase separation), organization (domain shape, molecular orientation) of lipid membranes, and real-time visualization of the influence of various molecules (proteins, drugs) on their integrity. An integral part of this review discusses the latest achievements in the field of lipid layer-based biosensors.

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.

Reconstruction of Par-dependent polarity in apolar cells reveals a dynamic process of cortical polarization

Cellular polarization is fundamental for various biological processes. The Par network system is conserved for cellular polarization. Its core complex consists of Par3, Par6, and aPKC. However, the general dynamic processes that occur during polarization are not well understood. Here, we reconstructed Par-dependent polarity using non-polarized Drosophila S2 cells expressing all three components endogenously in the cytoplasm. The results indicated that elevated Par3 expression induces cortical localization of the Par-complex at the interphase. Its asymmetric distribution goes through three steps: emergence of cortical dots, development of island-like structures with dynamic amorphous shapes, repeating fusion and fission, and polarized clustering of the islands. Our findings also showed that these islands contain a meshwork of unit-like segments. Furthermore, Par-complex patches resembling Par-islands exist in Drosophila mitotic neuroblasts. Thus, this reconstruction system provides an experimental paradigm to study features of the assembly process and structure of Par-dependent cell-autonomous polarity.

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.

To forge a solid immune recognition

Phagocytosis and innate immune responses to solid structures are topics of interest and debate. Alum, monosodium urate, calcium pyrophosphate dehydrate, silica and by extension all solid entities draw varying degrees of attention from phagocytes, such as antigen presenting cells. For some, innocuous soluble metabolites turn into fierce irritants upon crystallization, pointing to divergent signaling mechanisms of a given substance in its soluble and solid states. Over the years, many mechanisms have been proposed, including phagocytic receptors, toll like receptors, and NACHT-LRRs (NLRs), as well as several other protein structure mediated recognition of the solids. Is there a more general mechanism for sensing solids? In this perspective, I present an alternative view on the topic that membrane lipids can engage solid surfaces, and the binding intensity leads to cellular activation. I argue from the stands of evolution and biological necessity, as well as the progression of our understanding of cellular membranes and phagocytosis. The effort is to invite debate of the topic from a less familiar yet equally thrilling viewing angle.

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.

Onset of unidirectional pulse propagation in an excitable medium with asymmetric heterogeneity

Heterogeneity is one of the most important and ubiquitous types of external perturbations in dissipative systems. To know the behaviors of pulse waves in such media is closely related to studying the collision process between the pulse and the heterogeneity-induced-ordered pattern. In particular, we focus on unidirectional propagation of pulses in a medium with an asymmetric bump heterogeneity. This topic has attracted much interest recently with respect to potential computational aspects of chemical pulse propagation as well as with respect to pulse propagation in biological signal processing. We employ a three-component reaction-diffusion system with one activator and two inhibitor species to illustrate these issues. The reduced dynamics near a drift bifurcation describes the phenomena in the full partial differential equations by ordinary differential equations. Such a reduced dynamics is able to capture unidirectional propagation properties of pulses near an asymmetric heterogeneity in a qualitatively correct way. A remarkable feature is that such unidirectional behavior is linked to the imperfection of global bifurcation structure and the resulting asymmetric locations of critical points.

Numerical study of the phase separation in binary lipid membrane containing protein inclusions under stationary shear flow

The phase separation of lipids is believed to be responsible for the formation of lipid rafts in biological cell membrane. In the present work, a continuum model and a particle model are constructed to study the phase separation in binary lipid membrane containing inclusions under stationary shear flow. In each model, employing the cell dynamical system (CDS) approach, the kinetic equations of the confusion-advection process are numerically solved. Snapshot figures of the phase morphology are performed to intuitively display such phase evolving process. Considering the effects from both the inclusions and the shear flow, the time growth law of the characteristic domain size is discussed.

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.

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

We study two mechanisms for the formation of protein patterns near membranes of living cells by mathematical modelling. Self-assembly of protein domains by electrostatic lipid-protein interactions is contrasted with self-organization due to a nonequilibrium biochemical reaction cycle of proteins near the membrane. While both processes lead eventually to quite similar patterns, their evolution occurs on very different length and time scales. Self-assembly produces periodic protein patterns on a spatial scale below 0.1 microm in a few seconds followed by extremely slow coarsening, whereas self-organization results in a pattern wavelength comparable to the typical cell size of 100 microm within a few minutes suggesting different biological functions for the two processes.