Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996     

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.     

A consistent model for thermal fluctuations and protein-induced deformations in lipid bilayers

We present an elastic Hamiltonian for membrane energetics that captures bilayer undulation and peristaltic deformations over all wavelengths, including the short wavelength protrusion regime. The model implies continuous functional forms for thermal undulation and peristaltic amplitudes as a function of wavelength and predicts previously overlooked relationships between these curves. Undulation and peristaltic spectra display excellent agreement with data from both atomistic and coarse-grained models over all simulated length scales. Additionally, the model accurately predicts the bilayer's response to a cylindrical protein inclusion as observed in coarse-grained simulation. This elastic response provides an explanation for gramicidin ion channel lifetime versus membrane thickness data that requires no fit constants. The physical parameters inherent to this picture may be expressed in terms of familiar material properties associated with lipid monolayers. Inclusion of a finite monolayer spontaneous curvature is essential to obtain fully consistent agreement between theory and the full range of available simulation/experimental data.     

Role of unsaturated lipid and ergosterol in ethanol tolerance of model yeast biomembranes

We present a combined atomic force microscopy and fluorescence microscopy study of the behavior of a ternary supported lipid bilayer system containing a saturated lipid (DPPC), an unsaturated lipid (DOPC), and ergosterol in the presence of high ethanol (20 vol %). We find that the fluorescent probe Texas Red DHPE preferentially partitions into the ethanol-induced interdigitated phase, which allows the use of fluorescence imaging to investigate the phase behavior of the system. Atomic force microscopy and fluorescence images of samples with the same lipid mixture show good agreement in sample morphology and area fractions of the observed phases. Using area fractions obtained from fluorescence images over a broad range of compositions, we constructed a phase diagram of the DPPC/DOPC/ergosterol system at 20 vol % ethanol. The phase diagram clearly shows that increasing unsaturated lipid and/or ergosterol protects the membrane by preventing the formation of the interdigitated phase. This result supports the hypothesis that yeast cells increase ergosterol and unsaturated lipid content to prevent interdigitation and maintain an optimal membrane thickness as ethanol concentration increases during anaerobic fermentations. Changes in plasma membrane composition provide an important survival factor for yeast cells to deter ethanol toxicity.     

Applications of a model for scale-invariant pattern formation in developing systems

A fundamental problem in developmental biology concerns the proportioning of the developing tissue of a morphallactic system into different cell types in a way that is independent of the overall size of the tissue. The two main models for positional information in pattern formation, the source-sink models and the Turing reaction-diffusion models, have shortcomings that limit their applicability. In a previous paper, we described a model that can produce perfectly scale-invariant spatial patterns and analyzed some of its mathematical properties. In the present paper, we demonstrate some of the shortcomings of the standard reaction-diffusion models and discuss the applicability of our model to developmental systems.     

Travelling wave patterns in a model of the spinal pattern generator using spiking neurons

The aim of this study is to produce travelling waves in a planar net of artificial spiking neurons. Provided that the parameters of the waves – frequency, wavelength and orientation – can be sufficiently controlled, such a network can serve as a model of the spinal pattern generator for swimming and terrestrial quadruped locomotion. A previous implementation using non-spiking, sigmoid neurons lacked the physiological plausibility that can only be attained using more realistic spiking neurons. Simulations were conducted using three types of spiking neuronal models. First, leaky integrate-and-fire neurons were used. Second, we introduced a phenomenological bursting neuron. And third, a canonical model neuron was implemented which could reproduce the full dynamics of the Hodgkin–Huxley neuron. The conditions necessary to produce appropriate travelling waves corresponded largely to the known anatomy and physiology of the spinal cord. Especially important features for the generation of travelling waves were the topology of the local connections – so-called off-centre connectivity – the availability of dynamic synapses and, to some extent, the availability of bursting cell types. The latter were necessary to produce stable waves at the low frequencies observed in quadruped locomotion. In general, the phenomenon of travelling waves was very robust and largely independent of the network parameters and emulated cell types.     

On a simplified model for pattern formation in honey bee colonies

We present a simplified version of a previously presented model (Camazine et al. (1990)) that generates the characteristic pattern of honey, pollen and brood which develops on combs in honey bee colonies. We demonstrate that the formation of a band of pollen surrounding the brood area is dependent on the assumed form of the honey and pollen removal terms, and that a significant pollen band arises as the parameter controlling the rate of pollen input passes through a bifurcation value. The persistence of the pollen band after a temporary increase in pollen input can be predicted from the model. We also determine conditions on the parameters which ensure the accumulation of honey in the periphery and demonstrate that, although there is an important qualitative difference between the simplified and complete models, an analysis of the simplified version helps us understand many biological aspects of the more complex complete model. Corresponding author     

Lipid modulation of protein-induced membrane domains as a mechanism for controlling signal transduction

The reason for the enormous lipid variety present in eukaryotic membranes remains largely an enigma. We suggest that its role is to provide an on-off switch for a signaling event at the membrane level. This is achieved through lipid-lipid interactions that convert membrane protein binding and association events into very cooperative processes while maintaining reversibility. We have previously shown [Hinderliter, A., at al. (2001) Biochemistry 40, 4181-4191] that thermodynamic linkage between an intrinsic tendency for lipid demixing and a preferential interaction of a protein with a specific lipid within the mixture leads to dramatic changes in lipid and protein domain formation. Here, we tested the hypothesis that small alterations in lipid chemical structure alter the magnitude of the net interaction free energy (omega(AB)) between unlike lipids in a predictable manner, and that even very small changes in omega(AB) lead to dramatic changes in bilayer organization when coupled with protein binding. We systematically varied the chemical structure of phosphatidylcholine (PC), in mixtures with a fixed phosphatidylserine (PS), by changing the PC acyl chain length and the degree of unsaturation, and examined domain formation upon addition of a peripheral protein, the synaptotagmin I C2A motif. Experimental excimer/monomer ratios (E/M) of pyrene-substituted lipids mimicking the PS were interpreted using Monte Carlo computer simulations. E/M is larger if the PC melting temperature is lower, suggesting that domain formation is a thermodynamic consequence of weak interactions between PC and PS. Consistent with our hypothesis, only very small changes in omega(AB) were required for prediction of large changes in lipid and protein domain formation.     

Markers for detergent-resistant lipid rafts occupy distinct and dynamic domains in native membranes

Lipid rafts isolated by detergent extraction and sucrose gradient fractionation from mast cells are enriched for the glycosylphosphatidylinositol-linked protein Thy-1, the ganglioside GM1, palmitoylated LAT, and cross-linked IgE receptors, FcepsilonRI. This study addresses the relationship of fractionation data to the organization of raft markers in native membranes. Immunogold labeling and electron microscopy shows there is little or no colocalization of the raft markers Thy-1, GM1, and LAT with each other or with FcepsilonRI on native membrane sheets prepared from unstimulated cells. External cross-linking of Thy-1 promotes coclustering of Thy-1 with LAT, but not with GM1. Thy-1 and LAT clusters occur on membrane regions without distinctive features. In contrast, external cross-linking of FcepsilonRI and GM1 causes their redistribution to electron-dense membrane patches independently of each other and of Thy-1. The distinctive patches that accumulate cross-linked FcepsilonRI and GM1 also accumulate osmium, a stain for unsaturated lipids, and are sites for coated vesicle budding. Electron microscopy reveals a more complex and dynamic topographical organization of membrane microdomains than is predicted by biochemical analysis of detergent-resistant membranes.     

A new model for biological pattern formation

Various non-equilibrium growth models have been used to explore the development of morphology in biological systems. Here we review a class of biological growth models which exhibit fractal structures and discuss the relationship of these models to a variety of other phenomena.     

Mutations in multiple domains activate paramyxovirus F protein-induced fusion

SER virus, a paramyxovirus that is closely related to simian virus 5 (SV5), is unusual in that it fails to induce syncytium formation. The SER virus F protein has an unusually long cytoplasmic tail (CT), and it was previously observed that truncations or specific mutations of this domain result in enhanced syncytium formation. In addition to the long CT, the SER F protein has nine amino acid differences from the F protein of SV5. We previously observed only a partial suppression of fusion in a chimeric SV5 F protein with a CT derived from SER virus, indicating that these other amino acid differences between the SER and SV5 F proteins also play a role in regulating the fusion phenotype. To examine the effects of individual amino acid differences, we mutated the nine SER residues individually to the respective residues of the SV5 F protein. We found that most of the mutants were expressed well and were transported to the cell surface at levels comparable to that of the wild-type SER F protein. Many of the mutants showed enhanced lipid mixing, calcein transfer, and syncytium formation even in the presence of the long SER F protein CT. Some mutants, such as the I310 M, T438S, M489I, T516V, and N529K mutants, also showed fusion at lower temperatures of 32, 25, and 18 degrees C. The residue Asn529 plays a critical role in the suppression of fusion activity, as the mutation of this residue to lysine caused a marked enhancement of fusion. The effect of the N529K mutation on the enhancement of fusion by a previously described mutant, L539,548A, as well as by chimeric SV5/SER F proteins was also dramatic. These results indicate that activation to a fusogenic conformation is dependent on the interplay of residues in the ectodomain, the transmembrane domain, and the CT domain of paramyxovirus F proteins.     

Lowering line tension with high cholesterol content induces a transition from macroscopic to nanoscopic phase domains in model biomembranes

Chemically simplified lipid mixtures are used here as models of the cell plasma membrane exoplasmic leaflet. In such models, phase separation and morphology transitions controlled by line tension in the liquid-disordered (Ld)?+?liquid-ordered (Lo) coexistence regime have been described [1]. Here, we study two four-component lipid mixtures at different cholesterol fractions: brain sphingomyelin (BSM) or 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC)/1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC)/1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC)/cholesterol (Chol). On giant unilamellar vesicles (GUVs) display a nanoscopic-to-macroscopic transition of Ld?+?Lo phase domains as POPC is replaced by DOPC, and this transition also depends on the cholesterol fraction. Line tension decreases with increasing cholesterol mole fractions in both lipid mixtures. For the ternary BSM/DOPC/Chol mixture, the published phase diagram [19] requires a modification to show that when cholesterol mole fraction is >~0.33, coexisting phase domains become nanoscopic.     

Physical mechanism of ATP formation in biomembranes

Membrane-bound and isolated H+ ATPases of various origin are able to synthesize ATP from ADP and Pi after a jump-like pH increase. In the course of this increase the pH of solution (or suspension) must cross a value corresponding to pK of certain acid groups in the catalytic component of ATPase. In the case of isolated soluble enzymes it is possible to obtain up to 10 ATP molecules per one pH jump per one enzyme molecule. A physical mechanism of this phenomenon as well as of oxidative and photosynthetic phosphorylation is suggested.     

Fluorescence correlation studies of lipid domains in model membranes

Advances in optical microscopy techniques and single-molecule detection have paved the way to exploring new approaches for investigating membrane dynamics and organization, thereby revealing details on the processing of signals, complex association/dissociation, chemical reactions and transport at and around the membrane. These events rely on a tight regulation of lipid-protein and protein-protein interactions in space and time. Fluorescence Correlation Spectroscopy (FCS) provides exquisite sensitivity in measuring local concentrations, association/dissociation constants, chemical rate constants and, in general, in probing the chemical environment of the species of interest and its interactions with potential partners. Here, we review some applications of FCS to lipid and protein organization in biomimetic membranes with lateral heterogeneities, which share some physico-chemical properties with cellular rafts. What we learn from investigations of lipid-lipid and lipid-protein interactions in simple model membranes can be regarded as an essential basic lecture for studies in more complex cellular membranes.     

Reaction and diffusion on growing domains: Scenarios for robust pattern formation

We investigate the sequence of patterns generated by a reaction—diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction—diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.     

The formation of polymeric model biomembranes from diacetylenic fatty acids and phospholipids

Diacetylenic fatty acid monolayers at the air/water interface and multilayers on suitable supports polymerise when exposed to ultraviolet radiation. It has been found that polymerisation still occurs when monolayers are diluted with cholesterol or gramicidin. The rigid, crystalline nature of the films formed makes them useful biomembrane models. Phospholipids made from the fatty acids were less reactive. Multilayers deposited on hydrophobic supports would polymerise but not monolayers on water.