Diffusion and simultaneous chemical reactions: II. The equations of those systems in which transport occurs from one region to an adjoining region

A method is described for obtaining the solutions to the equations of systems in which a reacting substance diffuses between two adjacent regions. This substance may be produced or removed in the two regions at rates expressible as polynomial functions of time and may be removed throughout both regions by a first-order reaction. The solutions are obtained from simpler results, many of which are available and more of which are listed in this paper. Possible application in the study of the validity of cytochemical staining procedures is discussed.     

Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures

This work was motivated by the problems of analysing detailed 3D models of vascular districts with complex anatomy. It suggests an approach to prescribing realistic boundary conditions to use in order to obtain information on local as well as global haemodynamics. A method was developed which simultaneously solves Navier-Stokes equations for local information and a non-linear system of ordinary differential equations for global information. This is based on the principle that an anatomically detailed 3D model of a cardiovascular district can be achieved by using the finite element method. In turn the finite element method requires a specific boundary condition set. The approach outlined in this work is to include the system of ordinary differential equations in the boundary condition set. Such a multiscale approach was first applied to two controls: (i) a 3D model of a straight tube in a simple hydraulic network and (ii) a 3D model of a straight coronary vessel in a lumped-parameter model of the cardiovascular system. The results obtained are very close to the solutions available for the pipe geometry. This paper also presents preliminary results from the application of the methodology to a particular haemodynamic problem: namely the fluid dynamics of a systemic-to-pulmonary shunt in paediatric cardiac surgery.     

Unsteady Boundary Layer Flow and Heat Transfer of a Casson Fluid past an Oscillating Vertical Plate with Newtonian Heating

In this paper, the heat transfer effect on the unsteady boundary layer flow of a Casson fluid past an infinite oscillating vertical plate with Newtonian heating is investigated. The governing equations are transformed to a systems of linear partial differential equations using appropriate non-dimensional variables. The resulting equations are solved analytically by using the Laplace transform method and the expressions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and reduce to some well-known solutions for Newtonian fluids. Numerical results for velocity, temperature, skin friction and Nusselt number are shown in various graphs and discussed for embedded flow parameters. It is found that velocity decreases as Casson parameters increases and thermal boundary layer thickness increases with increasing Newtonian heating parameter.     

Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible,partly injectable multicompartment systems

An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.     

On the pressure and flow-rate distributions in tree-like and arterial-venous networks

A solution algorithm yielding the pressure and flow-rate distributions for steady flow in an arbitrary, tree-like network is provided. Given the tree topology, the conductance of each segment and the pressure distribution at the boundary nodes, the solution is obtained from a simple recursion based on perfect Gauss elimination. An iterative solution method using this algorithm is suggested to solve for the pressure and flow-rate distributions in an arbitrary diverging-converging (arterial-venous) network consisting of two tree-like networks which are connected to each other at the capillary nodes. A number of special solutions for tree-like networks are obtained for which the general algorithm is either simplified or can be replaced by closed form solutions of the pressure and flow-rate distributions. These special solutions can also be obtained for each tree of diverging-converging networks having particular topologies and conductance distributions. Sample numerical results are provided.     

Hierarchical and Joint Site‐Edge Methods for Medicare Hospice Service Region Boundary Analysis

Summary Hospice service offers a convenient and ethically preferable health‐care option for terminally ill patients. However, this option is unavailable to patients in remote areas not served by any hospice system. In this article, we seek to determine the service areas of two particular cancer hospice systems in northeastern Minnesota based only on death counts abstracted from Medicare billing records. The problem is one of spatial boundary analysis, a field that appears statistically underdeveloped for irregular areal (lattice) data, even though most publicly available human health data are of this type. In this article, we suggest a variety of hierarchical models for areal boundary analysis that hierarchically or jointly parameterize both the areas and the edge segments. This leads to conceptually appealing solutions for our data that remain computationally feasible. While our approaches parallel similar developments in statistical image restoration using Markov random fields, important differences arise due to the irregular nature of our lattices, the sparseness and high variability of our data, the existence of important covariate information, and most importantly, our desire for full posterior inference on the boundary. Our results successfully delineate service areas for our two Minnesota hospice systems that sometimes conflict with the hospices' self‐reported service areas. We also obtain boundaries for the spatial residuals from our fits, separating regions that differ for reasons yet unaccounted for by our model.     

THE EFFECT OF THE H ION CONCENTRATION ON THE AVAILABILITY OF IRON FOR CHLORELLA SP

The data obtained in these experiments indicate clearly that unless the necessary precautions are taken to keep the iron of the culture medium in solution the results obtained by varying the H ion concentration will not represent the true effect of this factor on growth. The availability of iron in nutrient solutions has been the subject of numerous recent investigations and it is now known that iron is precipitated at the lower hydrogen ion concentrations, that the iron of certain iron salts is less likely to be precipitated than that of others, and that certain salts of organic acids tend to keep the iron in solution. In general, ferric citrate seems to be the most favorable source of iron. In addition to chemical precipitation, however, it is also possible for the iron to be removed by adsorption on an amorphous precipitate such as calcium phosphate. As this precipitate is frequently formed when nutrient solutions are made alkaline, this may account for the discordant results reported in the literature as to the availability of certain forms of iron. By omitting calcium from the culture solution iron can be maintained in a form available for growth in alkaline solutions by the addition of sodium citrate. In such solutions the maximum growth of Chlorella occurred at pH 7.5. The alkaline limit for growth has not been established as yet. In investigating the availability of iron at varying concentrations of the hydrogen ion, changes in the pH value of the solution during the course of an experiment should also be taken into account. This is especially important in unbuffered solutions. The differential absorption of the ions of ammonium salts may cause a marked increase in the hydrogen ion concentration, which in turn will cause an increase in the solubility of iron. In strongly buffered solutions as used in these experiments this effect is slight.     

Phase equilibria of rodlike molecules in binary solvent systems

Phase relationships in solutions of rodlike, molecules were investigated with solutions of poly(γ-benzyl L -glutamate) having degrees of polymerization of 1500 and 3600, in N,N′-dimethylformamide–methanol and in N,N′-dimethylformamide–water at 30°C. With these systems, corresponding boundary curves for isotropic and anisotropic solution were obtained as a function of composition. Phase diagrams for these ternary systems were analysed on the basis of a theoretical treatment by Flory for a binary system consisting of a rodlike polymer and a solvent and found to be in good agreement with that predicted theoretically. For example, the polymer concentration above which the isotropic phase cannot exist and that of anisotropic conjugate phase agree with the values calculated. Furthermore, viscosities were measured as a function of polymer concentration by the falling-sphere method to confirm the boundary composition between the isotropic solution and heterogeneous region. These results were also found to coincide with those on phase equilibrium.     

Diffusion through a membrane: Approach to equilibrium

The transfer of solute through a membrane separating two aqueous solutions is studied with the time-dependent diffusion equation for composite media. By introducing new independent and dependent variables it is shown that the differential equations and boundary conditions can be transformed into a dimensionless form which does not explicitly depend on the diffusivities of the media. Laplace transforms are used to derive explicit solutions for the solute concentration as a function of position and time. It is shown that at large time the concentration approaches the equilibrium distribution exponentially. Explicit results are given for the decay time as a function of the parameters of the system. In addition, an accurate and simplified expression is derived for the decay time for the case of small membrane permeability. The accuracy of the analytic solutions for the concentration profiles is tested by comparing them with numerical results obtained by solving the diffusion equations by the method of finite differences. Excellent agreement is found. Research supported in part by a grant from the National Science Foundation.     

Sedimentation of Clusters of Spheres

We describe a numerical method for calculating hydrodynamic interactions between spherical particles efficiently and accurately, both for particles immersed in an infinite liquid and for systems with periodic boundary conditions. Our method is based on a multipole expansion in Cartesian tensors. We then show how to solve the equations of motion for translational and rotational motion of suspended particles at large Peclet numbers. As an example we study the sedimentation of an array of spheres with and without periodic boundary conditions. We also study the effect of perturbations on the stability of the trajectories.Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1007/s0089460020227     

Theory of a rapid method for determining molecular weights of giant molecules

A new method is proposed for finding molecular weights from the boundary condition of the ultracentrifuge. The method is more rapid than the conventional Archibald method by at least one order of magnitude and should be especially useful for giant molecules with small diffusion coefficients. It employs a preset concentration gradient. A theoretical study of the method has been made by obtaining solutions to the Lamm equation analogous to the “mirror image” solution (for early times) and the Fourier expansion solution (for late times) of Mason and Weaver. Numerical examples of theoretical results are given, and the errors are discussed. Centrifugation in a preset linear gradient can also be used to reduce the time needed to reach equilibrium.     

Bifurcation analysis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations

The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.     

Analytical ultracentrifugation as a tool for studying protein interactions

The last two decades have led to significant progress in the field of analytical ultracentrifugation driven by instrumental, theoretical, and computational methods. This review will highlight key developments in sedimentation equilibrium (SE) and sedimentation velocity (SV) analysis. For SE, this includes the analysis of tracer sedimentation equilibrium at high concentrations with strong thermodynamic non-ideality, and for ideally interacting systems, the development of strategies for the analysis of heterogeneous interactions towards global multi-signal and multi-speed SE analysis with implicit mass conservation. For SV, this includes the development and applications of numerical solutions of the Lamm equation, noise decomposition techniques enabling direct boundary fitting, diffusion deconvoluted sedimentation coefficient distributions, and multi-signal sedimentation coefficient distributions. Recently, effective particle theory has uncovered simple physical rules for the co-migration of rapidly exchanging systems of interacting components in SV. This has opened new possibilities for the robust interpretation of the boundary patterns of heterogeneous interacting systems. Together, these SE and SV techniques have led to new approaches to study macromolecular interactions across the entire spectrum of affinities, including both attractive and repulsive interactions, in both dilute and highly concentrated solutions, which can be applied to single-component solutions of self-associating proteins as well as the study of multi-protein complex formation in multi-component solutions.     

Cable analysis of a motor-nerve terminal branch in a volume conductor

An equivalent electrical circuit is given for a branch of an amphibian motor-nerve terminal in a volume conductor. The circuit allows for longitudinal current flow inside the axon as well as between the axon and its Schwann cell sheath, and also for the radial leakage of current through the Schwann cell sheath. Analytical and numerical solutions are found for the spatial and time dependence of the membrane potential resulting from the injection of depolarizing current pulses by external electrodes at one or two separate locations on the terminal. These solutions show that the depolarization at an injection site can cause a hyperpolarization at sites a short distance away. This effect becomes more pronounced in a short terminal with sealed-end boundary conditions. The hyperpolarization provides a possible explanation for recent experimental results, which show that the average quantal release due to a test depolarizing current pulse delivered by an electrode at one site on a nerve terminal is reduced by the application of an identical conditioning pulse at a neighbouring site.     

Initial/boundary-value problems of tumor growth within a host tissue

This paper concerns multiphase models of tumor growth in interaction with a surrounding tissue, taking into account also the interplay with diffusible nutrients feeding the cells. Models specialize in nonlinear systems of possibly degenerate parabolic equations, which include phenomenological terms related to specific cell functions. The paper discusses general modeling guidelines for such terms, as well as for initial and boundary conditions, aiming at both biological consistency and mathematical robustness of the resulting problems. Particularly, it addresses some qualitative properties such as a priori non-negativity, boundedness, and uniqueness of the solutions. Existence of the solutions is studied in the one-dimensional time-independent case.     

NRSub: a non-redundant database for Bacillus subtilis.

In the context of the international project aimed at sequencing the whole genome of Bacillus subtilis we have developed a non-redundant, fully annotated database of sequences from this organism. Starting from the B.subtilis sequences available in the EMBL, GenBank and DDBJ collections we have removed all encountered duplications and then added extra annotations to the sequences (e.g. accession numbers for the genes, locations on the genetic map, codon usage, etc.) We have also added cross-references to the EMBL, MEDLINE, SWISS-PROT and ENZYME data banks. The present system results from merging of the NRSub and SubtiList databases and the sequence contigs used in the two systems are identical. NRSub is distributed as a flatfile in EMBL format (which is supported by most sequence analysis software packages) and as an ACNUC database, while SubtiList is distributed as a relational database under 4th Dimension. It is possible to access the data through two dedicated World Wide Web servers located in France and Japan.     

An improved function for fitting sedimentation velocity data for low-molecular-weight solutes.

Many traditional approaches to the analysis of sedimentation velocity data work poorly with data for low-molecular-weight solutes, which have sedimentation boundaries that are severely broadened by diffusion. An approach that has previously had some success is to directly fit these broad boundaries to approximate solutions of the Lamm equation that directly account for the high diffusion. However, none of the available approximate solutions work well at times both early and late in the run, or give boundary shapes that are highly accurate, especially for species of molecular weight < 10,000. An improved fitting function has been developed to overcome some of these limitations. The new function adds two correction terms to the Fujita-MacCosham solution. The optimum coefficients for these new correction terms were determined by a least-squares approach. The accuracy and limitations of fitting with this new function were tested against synthetic data sets obtained by finite-element methods, for analysis of samples containing either single species or several noninteracting species. We also compare the strengths and weaknesses of this method of analysis, and its ability to work with noisy data, relative to recently developed time-derivative methodologies.     

Analysis of a mathematical model for transport of I-albumin

The hypotheses and results given are motivated by the study of the distribution of albumin in man which represents a class of delay-differential systems. The approach used is to study the behavior of the solutions of nonlinear delay-differential systems with variable coefficients under the assumptions of continuity and boundedness of coefficients. The criterions are conditions on the roots of a certain “quasi-polynomial”, i.e., a polynomial in a variable and exponential of that variable. These criterions bear a resemblance to the ones in the case of constant coefficients and retardations and are applicable to this case also. The method is based on Lyapunov type functional with appropriate properties.     

Determination of the quaternary phase diagram of the water-ethylene glycol-sucrose-NaCl system and a comparison between two theoretical methods for synthetic phase diagrams

Characterization of the thermodynamic properties of multi-solute aqueous solutions is of critical importance for biological and biochemical research. For example, the phase diagrams of aqueous systems, containing salts, saccharides, and plasma membrane permeating solutes, are indispensible in the field of cryobiology and pharmacology. However, only a few ternary phase diagrams are currently available for these systems. In this study, an auto-sampler differential scanning calorimeter (DSC) was used to determine the quaternary phase diagram of the water-ethylene glycol-sucrose-NaCl system. To improve the accuracy of melting point measurement, a “mass-redemption” method was also applied for the DSC technique. Base on the analyses of these experimental data, a comparison was made between the two practical approaches to generate phase diagrams of multi-solute solutions from those of single-solute solutions: the summation of cubic polynomial melting point equations versus the use of osmotic virial equations with cross coefficients. The calculated values of the model standard deviations suggested that both methods are satisfactory for characterizing this quaternary system.