Electrophoresis, a transport technology that transitioned from moving boundary method to zone method.

Gel electrophoresis, a transport technology, is one of the most widely used experimental methods in biochemical and pharmaceutical research and development. Transport technologies are used to determine hydrodynamic or electrophoretic properties of macromolecules. Gel electrophoresis is a zone technology, where a small volume of sample is applied to a large separation gel matrix. In contrast, a seldom-used electrophoresis technology is moving boundary electrophoresis, where the sample is present throughout the separation phase or gel matrix. While the zone method gives peaks of separating macromolecular solutes, the moving boundary method gives a boundary between solute-free and solute-containing phases. We will review electrophoresis as a transport technology of zone and moving boundary methods and describe its principles and applications.

Preparative separation of immunoglobulins from bovine colostrum by continuous divergent-flow electrophoresis.

Immunoglobulins in bovine colostrum were separated and fractionated from other proteins using the method and instrumentation developed in our laboratory. The proposed separation was based on bidirectional isotachophoresis/moving boundary electrophoresis with electrofocusing of the analytes in a pH gradient from 3.9 to 10.1. The preparative instrumentation included the trapezoidal non-woven fabric that served as separation space with divergent continuous flow. The defatted and casein precipitate-free colostrum supernatant was loaded directly into the instrument without any additional colostrum pre-preparation. Immunoglobulin G was fractionated from other immune proteins such as bovine serum albumin, ß-lactoglobulin, and α-lactalbumin, and was continuously collected in separated fractions over 3 h. The fractions were further processed, and isolated immunoglobulin G in the liquid fractions was confirmed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and by re-focusing in gel isoelectric focusing. Separated immunoglobulin G was detected in seven fractions by sodium dodecyl sulfate-polyacrylamide gel electrophoresis with a gradually decreased concentration in the fractions. Re-focusing of the proteins in the fractions by gel isoelectric focusing revealed multiple separated zones of immunoglobulin G with the isoelectric point values covering the range from 5.4 to 7.2. Each fraction contained distinct zones with gradually increased isoelectric point values and decreased concentrations from fraction to fraction.

System eigenmobilities in zone electrophoresis: A general moving-boundary approach.

System zones in capillary zone electrophoresis represent an important topic, very interesting from the theoretical point but also important for practice. This paper is aimed at contributing to the understanding of system zones as one of the very fundamental properties of electrophoretic systems, by developing an alternative approach to the so far used vector-matrix model of calculation of system mobilities (system eigenmobilities). The presented model is based on the solution of the differential form of the moving-boundary equation. The result for acid-base systems is a single algebraic equation valid universally for a zone comprising any number of constituents (mono- or polyhydric strong or weak acids or bases and/or amphoteric compounds). The value of the described solution against previous models consists in its explicit form, expressing the system eigenmobility of a homogeneous zone of given composition as a function of only known quantities. The obtained equation is shown to be the common source of various simplified equations obtained in the past for particular simple systems. The applicability of the simplified equations is discussed in terms of completeness of the results (number of output system eigenmobilities). For non-buffered systems, the occurrence of a previously unreported non-zero value of system eigenmobility is discussed that is equal to the arithmetic average of mobilities of the solvent ions.

The role of mechanical interactions in EMT.

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-ß, produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

Preparative continuous flow electrophoretic instrumentation for purification of biological samples.

We constructed a preparative instrumentation and developed the methods that are based on separation of the samples by bidirectional isotachophoresis/moving boundary electrophoresis in continuous divergent flow. The described instrumentation can be used for a variety of the samples, however, it can be easily optimized and tailored for the specific sample. The trapezoid separation bed from nonwoven textile exhibited minimum adsorption effect for sample and it can be used repeatedly. By the addition of different spacers via separation space inlets, the sections of pH gradient can be modified to enhance the separation. The liquid flow from two inlets positioned on each side of the sample inlet prevented the contact of the sample with anolyte and catholyte at the analysis beginning. One pair of thin electrodes (graphite and stainless-steel) was placed at the separation space output. The electrode products were washed out into drains without disturbing the focusing process. The influence of EOF was managed by tilting the separation bed in the direction from cathodic to anodic side. The components of spirulina supernatant and color pI markers were separated in the pH gradient from 3.9 to 10.1. pH gradient was stable for at least 4.5 h and spirulina supernatant from about 0.12 g of dry powder was processed. Compared to other preparative methods used for spirulina separation, the presented method/instrumentation working with a continuous divergent flow had essential advantages. The efficient separation was fast, and no intermediate steps were necessary to obtain liquid fractions with separated components compatible with further biological experiments.

Modelling cell guidance and curvature control in evolving biological tissues.

Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue's moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.

Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging.

Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential cancer drug therapies. Standard experiments involve growing and imaging spheroids to explore how different conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI) has enabled real-time in situ visualisation of the cell cycle progression. Observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work, we adapt the mathematical framework originally proposed by Ward and King (Math Med Biol 14:39-69, 1997. https://doi.org/10.1093/imammb/14.1.39 ) to produce a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and (iii) dead cells that are not fluorescent. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can qualitatively capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to qualitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software programs required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub. at https://github.com/wang-jin-mathbio/Jin2021.

Invading and Receding Sharp-Fronted Travelling Waves.

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.

Spots, stripes, and spiral waves in models for static and motile cells : GTPase patterns in cells.

The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.

A free boundary model of epithelial dynamics.

In this work we analyse a one-dimensional, cell-based model of an epithelial sheet. In the model, cells interact with their nearest neighbouring cells and move deterministically. Cells also proliferate stochastically, with the rate of proliferation specified as a function of the cell length. This mechanical model of cell dynamics gives rise to a free boundary problem. We construct a corresponding continuum-limit description where the variables in the continuum limit description are expanded in powers of the small parameter 1/N, where N is the number of cells in the population. By carefully constructing the continuum limit description we obtain a free boundary partial differential equation description governing the density of the cells within the evolving domain, as well as a free boundary condition that governs the evolution of the domain. We show that care must be taken to arrive at a free boundary condition that conserves mass. By comparing averaged realisations of the cell-based model with the numerical solution of the free boundary partial differential equation, we show that the new mass-conserving boundary condition enables the coarse-grained partial differential equation model to provide very accurate predictions of the behaviour of the cell-based model, including both evolution of the cell density, and the position of the free boundary, across a range of interaction potentials and proliferation functions in the cell based model.

On the theory of the unsteady-state growth of spherical crystals in metastable liquids.

Motivated by a large number of applications, we consider the process of non-stationary growth of spherical crystals in a supercooled binary melt. The moving-boundary problem describing the unsteady-state distributions of temperature and impurity concentration around the growing crystal as well as the dynamics of its radius and growth rate is solved by means of the methods of small-parameter expansion and Laplace-Carson integral transform. We show that the growth rate of crystals contains the main contribution (which is proportional to the supercooling degree Δ) and the first correction (which is proportional to Δ2 t, where t is time). The second correction is also found. The non-stationary temperature and concentration fields are determined as power functions of Δ and t. We demonstrate that the first corrections to the dynamics of crystal radius R( t) and its growth rate V ( t) play an important role. It is shown that R( t) and V ( t) can change more than twice in comparison with the previously known steady-state solution with the course of time. Such a behaviour will significantly modify the dynamics of a polydisperse ensemble of crystals evolving in a metastable liquid. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Nonlinear studies of tumor morphological stability using a two-fluid flow model.

We consider the nonlinear dynamics of an avascular tumor at the tissue scale using a two-fluid flow Stokes model, where the viscosity of the tumor and host microenvironment may be different. The viscosities reflect the combined properties of cell and extracellular matrix mixtures. We perform a linear morphological stability analysis of the tumors, and we investigate the role of nonlinearity using boundary-integral simulations in two dimensions. The tumor is non-necrotic, although cell death may occur through apoptosis. We demonstrate that tumor evolution is regulated by a reduced set of nondimensional parameters that characterize apoptosis, cell-cell/cell-extracellular matrix adhesion, vascularization and the ratio of tumor and host viscosities. A novel reformulation of the equations enables the use of standard boundary integral techniques to solve the equations numerically. Nonlinear simulation results are consistent with linear predictions for nearly circular tumors. As perturbations develop and grow, the linear and nonlinear results deviate and linear theory tends to underpredict the growth of perturbations. Simulations reveal two basic types of tumor shapes, depending on the viscosities of the tumor and microenvironment. When the tumor is more viscous than its environment, the tumors tend to develop invasive fingers and a branched-like structure. As the relative ratio of the tumor and host viscosities decreases, the tumors tend to grow with a more compact shape and develop complex invaginations of healthy regions that may become encapsulated in the tumor interior. Although our model utilizes a simplified description of the tumor and host biomechanics, our results are consistent with experiments in a variety of tumor types that suggest that there is a positive correlation between tumor stiffness and tumor aggressiveness.

Skin hydration analysis by experiment and computer simulations and its implications for diapered skin.

BACKGROUND: Experimental work on skin hydration is technologically challenging, and mostly limited to observations where environmental conditions are constant. In some cases, like diapered baby skin, such work is practically unfeasible, yet it is important to understand potential effects of diapering on skin condition. To overcome this challenge, in part, we developed a computer simulation model of reversible transient skin hydration effects. METHODS: Skin hydration model by Li et al. (Chem Eng Sci, 138, 2015, 164) was further developed to simulate transient exposure conditions where relative humidity (RH), wind velocity, air, and skin temperature can be any function of time. Computer simulations of evaporative water loss (EWL) decay after different occlusion times were compared with experimental data to calibrate the model. Next, we used the model to investigate EWL and SC thickness in different diapering scenarios. RESULTS: Key results from the experimental work were: (1) For occlusions by RH=100% and free water longer than 30 minutes the absorbed amount of water is almost the same; (2) Longer occlusion times result in higher water absorption by the SC. The EWL decay and skin water content predictions were in agreement with experimental data. Simulations also revealed that skin under occlusion hydrates mainly because the outflux is blocked, not because it absorbs water from the environment. Further, simulations demonstrated that hydration level is sensitive to time, RH and/or free water on skin. In simulated diapering scenarios, skin maintained hydration content very close to the baseline conditions without a diaper for the entire duration of a 24 hours period. CONCLUSION: Different diapers/diaper technologies are known to have different profiles in terms of their ability to provide wetness protection, which can result in consumer-noticeable differences in wetness. Simulation results based on published literature using data from a number of different diapers suggest that diapered skin hydrates within ranges considered reversible.

Electrolyte system for fast preparative focusing in wide pH range based on bidirectional isotachophoresis.

In this paper, we suggest new electrolyte system for fast preparative electrofocusing in wide pH range. It is based on bidirectional ITP with multiple counterions and spacers created by commercially available defined simple buffers. The migration course of proposed focusing model can be simulated in advance by using separation conditions and electrolyte components that are consequently applied during the experiments. The suggested electrolyte system allows high current densities at the initial stages of focusing without danger of local overheating, which strongly reduces the time needed for analysis completion. The performance of the electrolyte system is demonstrated by the focusing of synthetic colored low molecular weight indicators and proteins in the arrangements with both linear narrow strip and nonwoven fabric sheet with continuous flow.

Electrolyte system strategies for anionic ITP with ESI-MS detection. 3. The ITP spacer technique in moving-boundary systems and configurations with two self-maintained ITP subsystems.

This contribution is the third part of the project on strategies used in the selection and tuning of electrolyte systems for anionic ITP with ESI-MS detection. The strategy presented here is based on the creation of self-maintained ITP subsystems in moving-boundary systems and describes two new principal approaches offering physical separation of analyte zones from their common ITP stack and/or simultaneous selective stacking of two different analyte groups. Both strategic directions are based on extending the number of components forming the electrolyte system by adding a third suitable anion. The first method is the application of the spacer technique to moving-boundary anionic ITP systems, the second method is a technique utilizing a moving-boundary ITP system in which two ITP subsystems exist and move with mutually different velocities. It is essential for ESI detection that both methods can be based on electrolyte systems containing only several simple chemicals, such as simple volatile organic acids (formic and acetic) and their ammonium salts. The properties of both techniques are defined theoretically and discussed from the viewpoint of their applicability to trace analysis by ITP-ESI-MS. Examples of system design for selected model separations of preservatives and pharmaceuticals illustrate the validity of the theoretical model and application potential of the proposed techniques by both computer simulations and experiments. Both new methods enhance the application range of ITP-MS and may be beneficial particularly for complex multicomponent samples or for analytes with identical molecular mass.

Total protein quantitation using the bicinchoninic acid assay and gradient elution moving boundary electrophoresis.

We investigated the ability of gradient elution moving boundary electrophoresis (GEMBE) with capacitively coupled contactless conductivity detection (C(4) D) to assay total protein concentration using the bicinchoninic acid (BCA) reaction. We chose this format because GEMBE-C(4) D behaves as a concentration dependent detection system, unlike optical methods that also rely on pathlength (due to Beer's law). This system tolerates proteins well compared with other capillary electrophoretic methods, allowing the capillary to be reused without coatings or additional hydroxide wash steps. The typical reaction protocol was modified by reducing the pH slightly from 11.25 to 9.4, which enabled elimination of tartrate from the reagents. We estimated that copper (I) could be detected at approximately 3.0 µmol/L, which agrees with similar GEMBE and CZE systems utilizing C(4) D. Under conditions similar to the BCA "micro method" assay, we determined the LOD for three common proteins (insulin, BSA, and bovine gamma globulin) and found that they agree well with the existing spectroscopic detection methods. Further, we investigated how long reaction times impact the LOD and found that the conversion was proportional to log(time). This indicated that little sensitivity is gained by extending the reaction past 1 h. Hence, GEMBE provides an alternative platform for total protein assays while maintaining the excellent sensitivity of the optical-based methods.

Numerical simulation of flow behavior in basilar bifurcation computed tomography angiography.

In this study, a moving boundary deformation model based on four-dimensional computed tomography angiography (4D-CTA) with high temporal resolution is constructed, and blood flow dynamics of cerebral aneurysms are investigated by numerical simulation. A realistic moving boundary deformation model of a cerebral aneurysm was constructed based on 4D-CTA in each phase. Four hemodynamic factors (wall shear stress [WSS], wall shear stress divergence [WSSD], oscillatory shear index [OSI], and residual residence time [RRT]) were obtained from numerical simulations, and these factors were evaluated in basilar artery aneurysms. Comparison of the rigid body condition and the moving boundary condition investigating the relationship between wall displacement and hemodynamic factors clarified that the spatial-averaged WSS and maximum WSSD considering only the aneurysmal dome has a large difference between conditions during the peak systole, and there were also significant differences in OSI and RRT.

Electrolyte system strategies for anionic isotachophoresis with electrospray-ionization mass-spectrometric detection. 2. Isotachophoresis in moving-boundary systems.

This contribution is the second part of the project on strategies used in the selection of electrolyte systems for anionic ITP with ESI-mass spectrometric detection. It presents ITP as a powerful tool for selective stacking of anionic analytes, performed in a nonconventional way in moving-boundary systems where two co-anions are present in both the leading and terminating zones. The theoretical background is given to substantiate the conditions for the existence and migration of ITP boundaries in moving-boundary systems and stacking of analytes at these boundaries. The practical aspects of the theory are shown in form of stacking-window diagrams that bring immediate information about which analytes are stacked in a given system. The presented theory and strategy are illustrated and verified on the example of analysis of a model mixture of salicylic acid, ibuprofen and diclofenac, and comparison of regular and free-acid ITP with moving-boundary ITP systems formed by formic and propionic acids and ammonium as counterion.

Pattern formation and travelling waves in a multiphase moving boundary model of tumour growth.

We employ the multiphase, moving boundary model of Byrne et al. (2003, Appl. Math. Lett., 16, 567-573) that describes the evolution of a motile, viscous tumour cell phase and an inviscid extracellular liquid phase. This model comprises two partial differential equations that govern the cell volume fraction and the cell velocity, together with a moving boundary condition for the tumour edge, and here we characterize and analyse its travelling-wave and pattern-forming behaviour. Numerical simulations of the model indicate that patterned solutions can be obtained, which correspond to multiple regions of high cell density separated by regions of low cell density. In other parameter regimes, solutions of the model can develop into a forward- or backward-moving travelling wave, corresponding to tumour growth or extinction, respectively. A travelling-wave analysis allows us to find the corresponding wave speed, as well as criteria for the growth or extinction of the tumour. Furthermore, a stability analysis of these travelling-wave solutions provides us with criteria for the occurrence of patterned solutions. Finally, we discuss how the initial cell distribution, as well as parameters related to cellular motion and cell-liquid drag, control the qualitative features of patterned solutions.

Computational Modeling of Diffusion-Based Delamination for Active Implantable Medical Devices.

Delamination at heterogeneous material interfaces is one of the most prominent failure modes in active implantable medical devices (AIMDs). A well-known example of an AIMD is the cochlear implant (CI). In mechanical engineering, a multitude of testing procedures are known whose data can be used for detailed modeling with respect to digital twins. Detailed, complex models for digital twins are still lacking in bioengineering since body fluid infiltration occurs both into the polymer substrate and along the metal-polymer interfaces. For a newly developed test for an AIMD or CI composed of silicone rubber and metal wiring or electrodes, a mathematical model of these mechanisms is presented. It provides a better understanding of the failure mechanisms in such devices and their validation against real-life data. The implementation utilizes COMSOL Multiphysics®, consisting of a volume diffusion part and models for interface diffusion (and delamination). For a set of experimental data, the necessary diffusion coefficient could be derived. A subsequent comparison of experimental and modeling results showed a good qualitative and functional match. The delamination model follows a mechanical approach. The results of the interface diffusion model, which follows a substance transport-based approach, show a very good approximation to the results of previous experiments.