Three-dimensional simulation of anisotropic cell-driven collagen gel compaction

Tissue equivalents (TEs), formed by entrapping cells in a collagen gel, are an important model system for studying cell behavior. We have previously (Barocas and Tranquillo in J Biomech Eng 117:161–170, 1997a) developed an anisotropic biphasic theory of TE mechanics, which comprises five coupled partial differential equations describing interaction among cells and collagen fibers in the TE. The model equations, previously solved in one or two dimensions, were solved in three dimensions using an adaptive finite-element platform. The model was applied to three systems: a rectangular isometric cell traction assay, an otherwise- acellular gel containing two islands of cells, and an idealized tissue-engineered cardiac valve leaflet. In the first two cases, published experimental data were available for comparison, and the model results were consistent with the experimental observations. Fibers and cells aligned in the fixed direction in the isometric assay, and a region of strong fiber alignment arose between the two cell islands. For the valve problem, the alignment predicted by the model was generally similar to that observed experimentally, but an asymmetry in the experiment was not captured by the model.     

Hierarchical analysis of large-scale two-dimensional gel electrophoresis experiments

Large-scale two-dimensional gel experiments have the potential to identify proteins that play an important role in elucidating cell mechanisms and in various stages of drug discovery. Such experiments, typically including hundreds or even thousands of related gels, are notoriously difficult to perform, and analysis of the gel images has until recently been virtually impossible. In this paper we describe a scalable computational model that permits the organization and analysis of a large gel collection. The model is implemented in Compugen's Z4000 system. Gels are organized in a hierarchical, multidimensional data structure that allow the user to view a large-scale experiment as a tree of numerous simpler experiments, and carry out the analysis one step at a time. Analyzed sets of gels form processing units that can be combined into higher level units in an iterative framework. The different conditions at the core of the experiment design, termed the dimensions of the experiment, are transformed from a multidimensional structure to a single hierarchy. The higher level comparison is performed with the aid of a synthetic "adaptor" gel image, called a Raw Master Gel (RMG). The RMG allows the inclusion of data from an entire set of gels to be presented as a gel image, thereby enabling the iterative process. Our model includes a flexible experimental design approach that allows the researcher to choose the condition to be analyzed a posteriori. It also enables data reuse, the performing of several different analysis designs on the same experimental data. The stability and reproducibility of a protein can be analyzed by tracking it up or down the hierarchical dimensions of the experiment.     

Line defects in gel phase lipid monolayers

We investigate various models of the hydrolysis of gel-phase phosphatidylcholine monolayers by phospholipase A2 (Grainger et al. (1989) FEBS Lett. 252, 73-82). We assume that the probability of hydrolysis of a given lipid depends only upon how many of its nearest neighbour lipids have already been hydrolysed. We find that the experimental data are consistent with a model in which line defects exist in the gel phase and that lipids on such defects are more easily hydrolysed than the other gel-phase lipids. Based on this model, we calculate the course of hydrolysis of a gel-phase region possessing line defects, and we suggest how such a structure might be made and the model tested. An experiment, similar to that proposed by us, has been carried out by Grainger et al. (1990) Biochim. Biophys. Acta 1023, 365-379). We also calculate the fractal dimension, df, of the interface created by the hydrolytic process and show that a measurement of df might identify how this process proceeds.     

Proposal for a standard representation of two-dimensional gel electrophoresis data

The global analysis of proteins is now feasible due to improvements in techniques such as two-dimensional gel electrophoresis (2-DE), mass spectrometry, yeast two-hybrid systems and the development of bioinformatics applications. The experiments form the basis of proteomics, and present significant challenges in data analysis, storage and querying. We argue that a standard format for proteome data is required to enable the storage, exchange and subsequent re-analysis of large datasets. We describe the criteria that must be met for the development of a standard for proteomics. We have developed a model to represent data from 2-DE experiments, including difference gel electrophoresis along with image analysis and statistical analysis across multiple gels. This part of proteomics analysis is not represented in current proposals for proteomics standards. We are working with the Proteomics Standards Initiative to develop a model encompassing biological sample origin, experimental protocols, a number of separation techniques and mass spectrometry. The standard format will facilitate the development of central repositories of data, enabling results to be verified or re-analysed, and the correlation of results produced by different research groups using a variety of laboratory techniques.     

Determination of the effective diffusion coefficient of oxygen in gel materials in relation to gel concentration

Summary The effective diffusion coefficient of oxygen, ID_e, was determined in different gel support materials (calcium alginate, -carrageenan, gellan gum, agar and agarose) which are generally used for immobilization of cells. The method used was based upon fitting Crank's model on the experimental data. The model describes the solute diffusion from a well-stirred solution into gel beads which are initially free of solute. The effect of the gel concentration on ID_e of oxygen in the gel was investigated. The results showed a decreasing ID_e for both agar and agarose at increasing gel concentration. In case of calcium alginate and gellan gum, a maximum in ID_e at the intermediate gel concentration was observed. It is hypothesized that this phenomenon is due to a changing gelpore structure at increasing gel concentrations. The ID_e of oxygen in calcium alginate, -carrageenan and gellan gum varied from 1.5^*10^–9 to 2.1^*10^–9 m²s^–1 in the gel concentration range of 0.5 to 5% (w/v).     

A thermodynamic analysis of calcium–alginate gel formation in the presence of inert electrolyte

The effect of mixed salt (1:1 and 2:1 electrolyte) on alginate (a charged polysaccharide) gel formation is analyzed within the Poisson-Boltzmann cell model utilizing experimental data from the literature. The concentration of calcium ions needed to induce gelation of alginate goes through a minimum as 1 : 1 electrolyte is added. The theoretical model can account for this in a qualitative manner. According to the theoretical model, however, it is only in terms of concentrations that the minimum exists. In terms of chemical potentials for the ions (or salt) the curve is monotonic. The effect is due to the highly nonideal interactions in polyelectrolyte solutions when the total salt content is low. © 1992 John Wiley & Sons, Inc.     

Normalization and analysis of residual variation in two-dimensional gel electrophoresis for quantitative differential proteomics

Although two-dimensional gel electrophoresis (2-DE) has long been a favorite experimental method to screen proteomes, its reproducibility is seldom analyzed with the assistance of quantitative error models. The lack of models of residual distributions that can be used to assign likelihood to differential expression reflects the difficulty in tackling the combined effect of variability in spot intensity and uncertain recognition of the same spot in different gels. In this report we have analyzed a series of four triplicate two-dimensional gels of chicken embryo heart samples at two distinct development stages to produce such a model of residual distribution. In order to achieve this reference error model, a nonparametric procedure for consistent spot intensity normalization had to be established, and is also reported here. In addition to variability in normalized intensity due to various sources, the residual variation between replicates was observed to be compounded by failure to identify the spot itself (gel alignment). The mixed effect is reflected by variably skewed bimodal density distributions of residuals. The extraction of a global error model that accommodated such distribution was achieved empirically by machine learning, specifically by bootstrapped artificial neural networks. The model described is being used to assign confidence values to observed variations in arbitrary 2-DE gels in order to quantify the degree of over-expression and under-expression of protein spots.     

A theoretical analysis of the use of zonal gel filtration in the detection and purification of protein-ligand complexes.

A simple model of zonal gel filtration is analysed numerically to establish semiquantitatively how the capacity of a protein-ligand complex to survive gel filtration intact depends on several experimental variables and on the kinetic characteristics of the complex. The design of gel-filtration experiments aimed at detecting and partially purifying binding proteins is discussed.     

Immunoassay principle based on exchange reactions: investigations on agarose gel and silanized quartz.

Preliminary studies of a new immunoassay principle based on exchange reactions is reported. Exchange of 125I-labelled insulin with unlabelled insulin from immobilized monoclonal antibodies was investigated. From antibody immobilized on a gel substrate the tagged insulin was exchanged according to a first-order process. With antibody immobilized on a quartz substrate by two different methods, the kinetics was changed dramatically, probably because of the non-specific interaction between the ligand and the surface. The recorded adsorption isotherms could not be described by the Langmuir adsorption equation, and a model allowing for non-specific adsorption of the ligand was developed. This model gave a satisfactory fit to the experimental data, allowing computation of adsorption parameters. It is concluded that even the best method used to immobilize receptors on quartz is not adequate for an exchange assay to be made. However, this coating method may lead to more sensitive receptor-based assays of more conventional type.     

Model and computer simulations of the motion of DNA molecules during pulse field gel electrophoresis

A model is presented for the motion of individual molecules of DNA undergoing pulse field gel electrophoresis (PFGE). The molecule is represented by a chain of charged beads connected by entropic springs, and the gel is represented by a segmented tube surrounding the beads. This model differs from earlier reptation/tube models in that the tube is allowed to leak in certain places and the chain can double over and flow out of the side of the tube in kinks. It is found that these kinks often lead to the formation of U shapes, which are a major source of retardation in PFGE. The results of computer simulations using this model are compared with real DNA experimental results for the following cases: steady field motion as seen in fluorescence microscopy, mobility in steady fields, mobility in transverse field alternation gel electrophoresis (TFAGE), mobility in field inversion gel electrophoresis (FIGE), and linear dichroism (LD) of DNA in agarose gels during PFGE. Good agreement between the simulations and the experimental results is obtained.     

A theoretical method to predict DNA permutation gel electrophoresis from the sequence.

The gel electrophoretic permutation assays of DNA fragments experimentally investigated by different authors were theoretically reproduced using our theoretical model of sequence-dependent curvature. The general pattern of agreement obtained suggests that our method can be usefully adopted as an alternative to the experimental assay, in particular where the lack of a sufficient number of unique restriction sites in the fragment prevents the correct localization of the main bend site.     

Simulation of gel phase formation and melting in lipid bilayers using a coarse grained model

The transformation between a gel and a fluid phase in dipalmitoyl-phosphatidylcholine (DPPC) bilayers has been simulated using a coarse grained (CG) model by cooling bilayer patches composed of up to 8000 lipids. The critical step in the transformation process is the nucleation of a gel cluster consisting of 20-80 lipids, spanning both monolayers. After the formation of the critical cluster, a fast growth regime is entered. Growth slows when multiple gel domains start interacting, forming a percolating network. Long-lived fluid domains remain trapped and can be metastable on a microsecond time scale. From the temperature dependence of the rate of cluster growth, the line tension of the fluid-gel interface was estimated to be 3+/-2 pN. The reverse process is observed when heating the gel phase. No evidence is found for a hexatic phase as an intermediate stage of melting. The hysteresis observed in the freezing and melting transformation is found to depend both on the system size and on the time scale of the simulation. Extrapolating to macroscopic length and time scales, the transition temperature for heating and cooling converges to 295+/-5 K, in semi-quantitative agreement with the experimental value for DPPC (315 K). The phase transformation is associated with a drop in lateral mobility of the lipids by two orders of magnitude, and an increase in the rotational correlation time of the same order of magnitude. The lipid headgroups, however, remain fluid. These observations are in agreement with experimental findings, and show that the nature of the ordered phase obtained with the CG model is indeed a gel rather than a crystalline phase. Simulations performed at different levels of hydration furthermore show that the gel phase is stabilized at low hydration. A simulation of a small DPPC vesicle reveals that curvature has the opposite effect.     

Diffusion of Cu2+ in calcium alginate gel beads: further analyses

The diffusivity of Cu(2+), as determined by previous authors from analysis of experimental data in terms of the shrinking core (SCM) and linear absorption (LAM) models, is examined in light of the ability of the models to curve fit all the data. It is concluded from this further analysis that previous conclusions depicting the LAM to have an advantage over the SCM for predictive value are not justified. It is also shown that equally good curve fits can be obtained with a recent absorption/desorption model of diffusion which considers directly, through distribution theory, the effect of heterogeneity of material properties on the rate of diffusion. (c) 1995 John Wiley & Sons, Inc.     

Phenomenological theory of gel electrophoresis of protein-nucleic acid complexes

A phenomenological theory of gel electrophoresis is elaborated for protein-DNA complexes involving one, two, or three binding sites on the DNA molecule. The computed electrophoretic patterns simulate experimental patterns shown by both prokaryotic and eukaryotic systems. The mechanism whereby the electrophoretic protein-DNA ladder is generated upon titration of the operator with repressor is embodied in theory of mass transport coupled to reversible interactions under chemical kinetic control. In contrast to strong interactions (association constant greater than 10(12) M-1), patterns observed with weak complexes (K less than 10(10) M-1) could be simulated only by applying the cage effect, a model of which is formulated. Theoretical underpinning is provided for the electrophoretic estimation of equilibrium association constants, and requisite chemical kinetic conditions are elucidated for direct estimation of the rate constant for dissociation of the protein-DNA complex from gel patterns. The theory thus affords an experimenter with a means for determining the conditions required to render the gel retardation method a valid procedure for evaluating equilibrium constants and/or kinetic parameters for the particular protein-nucleic acid system under investigation. These several considerations apply not only to interactions of proteins with nucleic acids (DNA or RNA) but also to a wide range of macromolecular interactions involving peptides, drugs, and other ligands as well as large assemblies such as multienzyme complexes.     

An update on data standards for gel electrophoresis

The use of gel electrophoresis to separate and, in some instances, to quantify the abundance of large numbers of proteins from complex mixtures, has been well established for several decades. The quantity of publicly available data is still relatively modest due to a lack of community accepted data standards, tools to facilitate the data sharing process and controlled vocabularies to ensure that consistent terminology is used to describe the experimental methodology. It is becoming widely recognised that there are significant benefits in data sharing for proteomics, allowing results to be verified and new findings to be generated by re-analysis of published studies. We report on standards development by the Gel Analysis Workgroup of the Proteomics Standards Initiative. The workgroup develops reporting requirements, data formats and controlled vocabularies for experimental gel electrophoresis, and informatics performed on gel images. We present a tutorial on how such resources can be used and how the community should get involved with the on-going projects. Finally, we present a roadmap for future developments in this area.     

Glucose oxidase release from calcium alginate gel capsules

Diffusion of glucose oxidase within calcium alginate gel capsules has been assayed and the experimental data fitted to a simple semi-empirical power equation, which is used to analyse the solute release from polymeric devices. It was found that an increase in the concentration of sodium alginate and calcium chloride gives rise to a reduction in the enzyme leakage. This was verified when glucose oxidase (GOD) diffusion percentages were compared in capsules with thicknesses of the same order of magnitude but obtained under different experimental conditions. So, the use of sodium alginate and calcium chloride solutions of concentrations 0.5% w/v and 2.6% w/v, respectively, lead to a diffusion percentage of 25 +/- 2. This percentage was reduced to 8 +/- 3 when sodium alginate and calcium chloride concentrations were fixed at 1% w/v and 4% w/v, respectively, even though the thicknesses of the capsules were of the same order of magnitude.     

Stress field in actin gel growing on spherical substrate

Polymerization of actin to form an elastic gel is one of the main mechanisms responsible for cellular motility. The particular problem addressed here stems from the need to model theoretically the growth of actin gel under controlled conditions, as observed in experiments. A biomimetic in vitro system which consists of a spherical latex bead, coated by the enzymatic protein ActA, and a reconstituted cytoplasm within which such beads are placed, induces polymerization of actin on the surface of the bead in the form of successive elastic thin spherical layers. Each newly formed layer pushes outward, and is pushed inward by, the already formed spherical layers which altogether constitute an elastic spherical shell of thickness h varying with time. Thus, a stress field is created in the shell which in turn affects the rate of polymerization as well as that of dissociation of actin gel. Given this bio-chemo-mechanical coupling, the accurate determination of the stress field becomes a subject of great importance for the understanding of the process, and it is the main objective of this work. The problem is addressed by first assuming appropriate constitutive laws for the actin gel elastic material, and then solving the only non-trivial stress equilibrium differential equation along the radial direction assuming spherical symmetry. A linear and a non-linear constitutive model for isotropic elasticity is used, appropriate for small and finite strains, respectively, and the solution is found in closed analytical forms in both cases. Two important conclusions are reached. First, the stress field depends strongly on the compressibility of the actin gel medium via the value of the Poisson ratio, for both linear and non-linear analysis. Second, the linear and non-linear solutions are very close for small strains, but they diverge progressively as the strains increase from small to large. Guided by available experimental data on the observed strain levels, the analytical results are illustrated by selected graphs of stress variation along the radial direction. At the end some comments and suggestions on the bio-chemo-mechanical coupling of actin gel growth and resorption are presented, where the role of properly defined joint isotropic invariants of stress and a unit vector along the predominant direction of free ends of actin filaments at the polymerization site is introduced.