One-dimensional elastic continuum model of enterocyte layer migration

Necrotizing enterocolitis is the leading cause of death from gastrointestinal disease in preterm infants. It results from an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin into the circulation. Intestinal mucosal defects are repaired by the process of intestinal restitution, during which enterocytes migrate from healthy areas to sites of injury. In this article, we develop a mathematical model of migration of enterocytes during experimental necrotizing enterocolitis. The model is based on a novel assumption of elastic deformation of the cell layer and incorporates the following effects: i), mobility promoting force due to lamellipod formation, ii), mobility impeding adhesion to the cell matrix, and iii), enterocyte proliferation. Our model successfully reproduces the behavior observed for enterocyte migration on glass coverslips, namely the dependence of migration speed on the distance from the wound edge, and the finite propagation distance in the absence of proliferation that results in an occasional failure to close the wound. It also qualitatively reproduces the dependence of migration speed on integrin concentration. The model is applicable to the closure of a wound with a linear edge and, after calibration with experimental data, could be used to predict the effect of chemical agents on mobility, adhesion, and proliferation of enterocytes.     

A continuum model for root growth

This paper presents a study of a simple one-dimensional continuum model for growth of the plant root. A fundamental constitutive equation is derived. The model is studied by means of various special cases of increasing complexity. Asymptotic expansions are used to derive approximate solutions to the equation of the model under the fundamental assumption that cell wall thickness is small in comparison with the diameter of the cell. The basic results of the study may be summarized as follows. The observed growth pattern of the root cannot be modelled by a mechanical system whose properties are independent of position on the root. The observed pattern can be modelled by a simple mechanical system in which, for example, cell wall yield stress first decreases and then increases. Two fundamental observations are made based on the modelling study. The first is that any mechanical model must take into account the convective displacement from the tip of points along the root. The second is that in describing growth, data on cell wall mechanical properties are meaningless without corresponding data on cell water potential, and vice versa.     

A continuum model for coupled cells

A continuum model of diffusion-coupled cells that more accurately reflects the presence of low-permeability gap junctions between cells is analyzed. It is shown by a multi-scale analysis that to lowest order the slow evolution of the mean concentration is described by the usual ordinary differential equations for a discrete model. Furthermore, stable non-uniform steady solutions are shown to exist in the continuum model of a one component system, whereas this is impossible for the standard reaction-diffusion model of this system. It is also shown how to average the equations in this continuum model to obtain a system of reaction-diffusion equations with constant coefficients.     

Formamidinium-induced dimer stabilization and flicker block behavior in homo- and heterodimer channels formed by gramicidin A and N-acetyl gramicidin A.

Compared to the N-formyl gramicidin A (GA), the N-acetyl gramicidin A (NAG) channel has unchanged conductance in 1 M NH4+ (gamma NN/gamma GG = 1, conductance ratio) but reduced conductance in 1 M K+ (gamma NN/gamma GG = 0.6) methylammonium (gamma NN/gamma GG = 0.3), and formamidinium (gamma NN/gamma GG = 0.1) solutions. Except with formamidinium, "flicker blocks" are evident even at low cutoff frequencies. For all cations studied, channel lifetimes of N-acetyl homodimers (NN) are approximately 50-fold shorter than those of the GA homodimer (GG). The novel properties of GA channels in formamidinium solution (supralinear current-voltage relations and dimer stabilization (Seoh and Busath, 1993)) also appear in NN channels. The average single channel lifetime in 1 M formamidinium solution at 100 mV is 6-7-fold longer than in K+ and methylammonium solutions and, like in the GA channel, significantly decreases with increasing membrane potential. Experiments with mixtures of the two peptides, GA and NAG, showed three main conductance peaks. Oriented hybrids were formed utilizing the principle that monomers remain in one leaflet of the bilayer (O'Connell et al., 1990). With GA at the polarized side and NAG at the grounded side, at positive potentials (in which case hybrids were designated GN) and at negative potentials (in which case hybrids were designated NG), channels had the same conductances and channel properties at all potentials studied. Flicker blocks were not evident in the hybrid channels, which suggests that both N-acetyl methyl groups at the junction of the dimer are required to cause flickers. Channel lifetimes in hybrids are only approximately threefold shorter than those of the GG channels, and channel conductances are similar to those of GG rather than NN channels. We suggest that acetyl-acetyl crowding at the dimeric junction in NN channels cause dimer destabilization, flickers, and increased selectivity in N-acetyl gramicidin channels.     

A continuum model for surface contraction waves

The periodic travelling waves which appear on some animal eggs after fertilization are considered here. These are thought to be caused by a calcium initiated calcium release on the surface, causing calcium waves. A continuum model is developed where the cell is treated as a small viscous droplet with a surface contamination. When a periodic source of surfactant acts at one pole and propagates down the cell surface to the opposite pole, the drop responds by forming constriction rings which move from pole to pole.     

A new model for the continuum concept

A reformulation of the continuum concept is presented after considering the implications of the community/continuum controversy and current niche theory. Community is a spatial concept dependent on landscape pattern while the continuum is an environmental concept referring to an abstract space. When applying niche theory to plants, the mechanisms of competition are ill-defined and the assumption of bell-shaped response curves for species unrealistic.Eight testable propositions on the pattern of response of vegetation to environmental gradients are presented 1. Environmental gradients are of two types. a) resource gradients or b) direct physiological gradients. 2. The fundamental niche response of species to resource gradients is a series of similar nested response curves. 3. The fundamental niche response of species to direct gradients is a series of separate, independent, overlapping response curves. 4. Species fundamental response curves are such that they have a relative performance advantage in some part of the environmental space. 5. The shape of the realized niche is variable even bimodal but predictable from the fundamental response given the other species present. Propositions 6–8 describe the response shapes of emergent community properties to environmental gradient; species richness is bimodal, dominance trimodal and standing crop unimodal. Detailed comparisons of these propositions are made with the alternative theories of Ellenberg, Gauch and Whittaker, Grime, and Tilman. These theories are incomplete lacking several generally accepted properties of plants and vegetation.     

Proton conduction in gramicidin A and in its dioxolane-linked dimer in different lipid bilayers.

Gramicidin A (gA) molecules were covalently linked with a dioxolane ring. Dioxolane-linked gA dimers formed ion channels, selective for monovalent cations, in planar lipid bilayers. The main goal of this study was to compare the functional single ion channel properties of natural gA and its covalently linked dimer in two different lipid bilayers and HCl concentrations (10-8000 mM). Two ion channels with different gating and conductance properties were identified in bilayers from the product of dimerization reaction. The most commonly observed and most stable gramicidin A dimer is the main object of this study. This gramicidin dimer remained in the open state most of the time, with brief closing flickers (tau(closed) approximately 30 micros). The frequency of closing flickers increased with transmembrane potential, making the mean open time moderately voltage dependent (tau(open) changed approximately 1.43-fold/100 mV). Such gating behavior is markedly different from what is seen in natural gA channels. In PEPC (phosphatidylethanolamine-phosphatidylcholine) bilayers, single-channel current-voltage relationships had an ohmic behavior at low voltages, and a marked sublinearity at relatively higher voltages. This behavior contrasts with what was previously described in GMO (glycerylmonooleate) bilayers. In PEPC bilayers, the linear conductance of single-channel proton currents at different proton concentrations was essentially the same for both natural and gA dimers. g(max) and K(D), obtained from fitting experimental points to a Langmuir adsorption isotherm, were approximately 1500 pS and 300 mM, respectively, for both the natural gA and its dimer. In GMO bilayers, however, proton affinities of gA and the dioxolane-dimer were significantly lower (K(D) of approximately 1 and 1.5 M, respectively), and the g(max) higher (approximately 1750 and 2150 pS, respectively) than in PEPC bilayers. Furthermore, the relationship between single-channel conductance and proton concentration was linear at low bulk concentrations of H+ (0.01-2 M) and saturated at concentrations of more than 3 M. It is concluded that 1) The mobility of protons in gramicidin A channels in different lipid bilayers is remarkably similar to proton mobilities in aqueous solutions. In particular, at high concentrations of HCl, proton mobilities in gramicidin A channel and in solution differ by only 25%. 2) Differences between proton conductances in gramicidin A channels in GMO and PEPC cannot be explained by surface charge effects on PEPC membranes. It is proposed that protonated phospholipids adjacent to the mouth of the pore act as an additional source of protons for conduction through gA channels in relation to GMO bilayers. 3) Some experimental results cannot be reconciled with simple alterations in access resistance to proton flow in gA channels. Said differences could be explained if the structure and/or dynamics of water molecules inside gramicidin A channels is modulated by the lipid environment and by modifications in the structure of gA channels. 4) The dioxolane ring is probably responsible for the closing flickers seen in the dimer channel. However, other factors can also influence closing flickers.     

A continuum model of elephant trunks

A continuum model is presented that relates the trunk parameters of loading, geometry, and muscle structure to the necessary conditions of static equilibrium. Linear theory for stress-strain behavior is used to describe an elephant trunk for an incremental displacement as the animal slowly lifts a weight at the trunk tip. With this analysis and experimental values for the trunk parameters, the apparent trunk stiffness Ea is estimated for the living animal. For an Asian elephant with a maximum compression strain of 33 percent, Ea is of the order of 10(6) N/m2. The continuum model is quite general and may be applied to similar nonskeletal appendages and bodies of other animals.     

Framework model for single proton conduction through gramicidin

This paper describes a framework model for proton conduction through gramicidin; a model designed to incorporate information from molecular dynamics and use this to predict conductance properties. The state diagram describes both motion of an excess proton within the pore as well as the reorientation of waters within the pore in the absence of an excess proton. The model is constructed as the diffusion limit of a random walk, allowing control over the boundary behavior of trajectories. Simple assumptions about the boundary behavior are made, which allow an analytical solution for the proton current and conductance. This is compared with corresponding expressions from statistical mechanics. The random walk construction allows diffusing trajectories underlying the model to be simulated in a simple way. Details of the numerical algorithm are described.     

The gramicidin ion channel: A model membrane protein

The linear peptide gramicidin forms prototypical ion channels specific for monovalent cations and has been extensively used to study the organization, dynamics and function of membrane-spanning channels. In recent times, the availability of crystal structures of complex ion channels has challenged the role of gramicidin as a model membrane protein and ion channel. This review focuses on the suitability of gramicidin as a model membrane protein in general, and the information gained from gramicidin to understand lipid-protein interactions in particular. Special emphasis is given to the role and orientation of tryptophan residues in channel structure and function and recent spectroscopic approaches that have highlighted the organization and dynamics of the channel in membrane and membrane-mimetic media.     

A continuum approach to blood vessel growth: Axisymmetric elastic structures

In this paper we discuss the morphogenesis of small blood vessels (venules) as the nautral consequence of physical forces prevailing during endothelial cell division. A physical model is developed in which the blood vessel is treated as a growing, thin elastic shell embedded in a viscous fluid (i.e. the surrounding tissue). It is explained how a pre-existing cylindrical vessel, induced to grow by some promoter, can buckle and thereby develop a spatially periodic structure displaying varicosity, sinuosity, and/or helicity. Growth manifests itself dynamically in terms of a “growth pressure” which disturbs any pre-existing force balance. The governing set of non-linear partial differential equations are derived, and solutions corresponding to uniform dilation are obtained. The buckled structure emerges as an instability of this time dependent basic state of uniform dilation. A linear stability analysis yields the dominant wavelength of the varicose mode; these results compare favorably with crude measurements made from the experimental literature. In the hope of uncovering the mechanism which underlies the selection of sprouting sites along a parent vessel, it is suggested that reaction and diffusion processes (between growth promoting and inhibiting substances) on buckled surfaces be coupled to the dynamical force balances discussed here.     

A continuum model for pressure-flow relationship in human pulmonary circulation

A continuum model was introduced to analyze the pressure-flow relationship for steady flow in human pulmonary circulation. The continuum approach was based on the principles of continuum mechanics in conjunction with detailed measurement of vascular geometry, vascular elasticity and blood rheology. The pulmonary arteries and veins were considered as elastic tubes and the "fifth-power law" was used to describe the pressure-flow relationship. For pulmonary capillaries, the "sheet-flow" theory was employed and the pressure-flow relationship was represented by the "fourth-power law". In this paper, the pressure-flow relationship for the whole pulmonary circulation and the longitudinal pressure distribution along the streamlines were studied. Our computed data showed general agreement with the experimental data for the normal subjects and the patients with mitral stenosis and chronic bronchitis in the literature. In conclusion, our continuum model can be used to predict the changes of steady flow in human pulmonary circulation.     

MEPSi: A tool for simulating tomograms of membrane-embedded proteins

The throughput and fidelity of cryogenic cellular electron tomography (cryo-ET) is constantly increasing through advances in cryogenic electron microscope hardware, direct electron detection devices, and powerful image processing algorithms. However, the need for careful optimization of sample preparations and for access to expensive, high-end equipment, make cryo-ET a costly and time-consuming technique. Generally, only after the last step of the cryo-ET workflow, when reconstructed tomograms are available, it becomes clear whether the chosen imaging parameters were suitable for a specific type of sample in order to answer a specific biological question. Tools for a-priory assessment of the feasibility of samples to answer biological questions and how to optimize imaging parameters to do so would be a major advantage. Here we describe MEPSi (Membrane Embedded Protein Simulator), a simulation tool aimed at rapid and convenient evaluation and optimization of cryo-ET data acquisition parameters for studies of transmembrane proteins in their native environment. We demonstrate the utility of MEPSi by showing how to detangle the influence of different data collection parameters and different orientations in respect to tilt axis and electron beam for two examples: (1) simulated plasma membranes with embedded single-pass transmembrane αIIbβ3 integrin receptors and (2) simulated virus membranes with embedded SARS-CoV-2 spike proteins.     

A tubule-sheet continuum model for the mechanism of nuclear envelope assembly

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A parameterized, continuum electrostatic model for predicting protein pKa values

Recognizing the limits of trying to achieve chemical accuracy for pK(a) calculations with a purely electrostatic model, we include empirical corrections into the Poisson-Boltzmann solver macroscopic electrostatics with atomic detail (Bashford, Biochemistry 1990;29:10219-10225), to improve the reliability and accuracy of the model. The total number of parameters is kept to a minimum to maximize the robustness of the model for compounds outside of the fitting dataset. The parameters are based on: (a) the electrostatic interaction between functional groups close to the titratable site, (b) the electrostatic work required to desolvate the residue, and (c) the site-to-site interactions. These interactions are straightforward to calculate once the electrostatic field has been solved for each residue using the linearized Poisson-Boltzmann equation and are assumed to be linearly related to the intrinsic pK(a). Two hundred and eighty-six residues from 30 proteins are used to determine the empirical parameters, which result in a root mean square error (RMSE) of 0.70 for the entire set. Eight proteins with 46 experimentally known values were excluded from the parameterization to test the model. This test set had a RMSE of 1.08. We show that the parameterized model improves the results over other models, although like other models the error is strongly correlated with the degree to which a residue is buried. The parameters themselves indicate that local effects are most important for determining the pK(a), whereas site-to-site interactions are found to be less significant.     

A continuum model for skeletal muscle contraction at homogeneous finite deformations

The contractile force in skeletal muscle models is commonly postulated to be the isometric force multiplied by a set of experimentally motivated functions which account for the muscle’s active properties. Although both flexible and simple, this approach does not automatically guarantee a thermodynamically consistent behavior. In contrast, the continuum mechanical model proposed herein is derived from fundamental principles in mechanics and guarantees a dissipative behavior. Further, the contractile force is associated with a friction clutch which provides a simple and well-defined macroscopic model for cycling cross-bridges. To show the performance of the model, it is specialized to standard experiments for rabbit tibialis anterior muscle. The results show that the model is able to capture important characteristics of skeletal muscle.     

Free energy calculations on dimer stability of the HIV protease using molecular dynamics and a continuum solvent model

Dimerization of HIV-I protease (HIV PR) monomers is an essential prerequisite for viral proteolytic activity and the subsequent generation of infectious virus particles. Disrupting dimerization of the enzyme can inhibit its activity. We have calculated the relative binding free energies between different dimers of the HIV protease using molecular dynamics and a continuum model, which we call MM/PBSA. We examined the dominant negative inhibition of the HIV PR by a mutated form of the protease and found relative dimerization free energies of homo- and hetero-dimerization consistent with experimental data. We also developed a rapid screening method, which was called the virtual mutagenesis method to consider other mutations which might stabilize non-wild-type heterodimers. Using this approach, we considered the mutations near the dimer interface which might cause dominant negative inhibition of the HIV PR. The rapid method we developed can be used in studying any ligand-protein and protein-protein interaction, in order to identify mutations that can enhance the binding affinities of the complex.