Diffusion of the Reaction Boundary of Rapidly Interacting Macromolecules in Sedimentation Velocity

Sedimentation velocity analytical ultracentrifugation combines relatively high hydrodynamic resolution of macromolecular species with the ability to study macromolecular interactions, which has great potential for studying dynamically assembled multiprotein complexes. Complicated sedimentation boundary shapes appear in multicomponent mixtures when the timescale of the chemical reaction is short relative to the timescale of sedimentation. Although the Lamm partial differential equation rigorously predicts the evolution of concentration profiles for given reaction schemes and parameter sets, this approach is often not directly applicable to data analysis due to experimental and sample imperfections, and/or due to unknown reaction pathways. Recently, we have introduced the effective particle theory, which explains quantitatively and in a simple physical picture the sedimentation boundary patterns arising in the sedimentation of rapidly interacting systems. However, it does not address the diffusional spread of the reaction boundary from the cosedimentation of interacting macromolecules, which also has been of long-standing interest in the theory of sedimentation velocity analytical ultracentrifugation. Here, effective particle theory is exploited to approximate the concentration gradients during the sedimentation process, and to predict the overall, gradient-average diffusion coefficient of the reaction boundary. The analysis of the heterogeneity of the sedimentation and diffusion coefficients across the reaction boundary shows that both are relatively uniform. These results support the application of diffusion-deconvoluting sedimentation coefficient distributions c(s) to the analysis of rapidly interacting systems, and provide a framework for the quantitative interpretation of the diffusional broadening and the apparent molar mass values of the effective sedimenting particle in dynamically associating systems.     

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.     

Transport theory for growing cell populations

The partial differential equation that describes the growth of cell populations whose maturation rate is random is developed. The equation resembles that used in classical transport theory but mitotic boundary conditions and the restriction of the maturation rate to non-negative values brings out new features and new problems. This is a generalization of a previously published formulation in which cells could make transitions at random between only two maturation velocities: a characteristic velocity and zero. Growth rates, cycle time distributions and pulsed labeled mitotic curves are calculated for a simple choice of parameters. A numerical algorithm that is suited to the solution of the transport equation is given.     

Flow Past a Permeable Stretching/Shrinking Sheet in a Nanofluid Using Two-Phase Model

The steady two-dimensional flow and heat transfer over a stretching/shrinking sheet in a nanofluid is investigated using Buongiorno’s nanofluid model. Different from the previously published papers, in the present study we consider the case when the nanofluid particle fraction on the boundary is passively rather than actively controlled, which make the model more physically realistic. The governing partial differential equations are transformed into nonlinear ordinary differential equations by a similarity transformation, before being solved numerically by a shooting method. The effects of some governing parameters on the fluid flow and heat transfer characteristics are graphically presented and discussed. Dual solutions are found to exist in a certain range of the suction and stretching/shrinking parameters. Results also indicate that both the skin friction coefficient and the local Nusselt number increase with increasing values of the suction parameter.     

A parameter identification problem arising from a model of canalicular bile formation

We develop a simple mathematical model for bile formation and analyze some features of the model that suggest the design for future physiological experiments. The mathematical model results in a boundary value problem for a system of functional differential equations depending on several physical parameters. From the observability of the boundary values we can identify, both qualitatively and quantitatively, some of these physical parameters. This identification then suggests physical experiments from which one could infer some of the bile transport phenomena that are not, at present, directly observable. The mathematical parameter identification problem is solved by converting the boundary value problem to a transition time problem for a quadratic system of ordinary differential equations on the plane where we are able to employ some special properties of quadratic systems in order to obtain a solution.The author was supported by the Air Force Office of Scientific Research and the National Science Foundation under the grants AF-AFOSR-89-0078 and DMS-9022621The author was supported by National Institutes of Health under grant number R37 DK-27623     

ESS analysis of mechanistic models for territoriality: the value of scent marks in spatial resource partitioning

In this paper elements of game theory are used to analyse a spatially explicit home range model for interacting wolf packs. The model consists of a system of nonlinear partial differential equations whose parameters reflect the movement behavior of individuals within each pack and whose solutions describe the patterns of space-use by each pack. By modifying the behavioral parameters, packs adjust their patterns of movement so as to maximize their reproductive output. This involves a tradeoff between maximizing prey intake and minimizing conflict with neighbors. Evolutionarily stable choices of the behavioral parameters yields territories that are immune to invasion by groups with alternate behaviors.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Differential transform semi-numerical analysis of biofluid-particle suspension flow and heat transfer in non-Darcian porous media

The differential transform method (DTM) is semi-numerical method which is used to study the steady, laminar buoyancy-driven convection heat transfer of a particulate biofluid suspension in a channel containing a porous material. A two-phase continuum model is used. A set of variables is implemented to reduce the ordinary differential equations for momentum and energy conservation (for both phases) to a dimensionless system. DTM solutions are obtained for the dimensionless system under appropriate boundary conditions. We examine the influence of momentum inverse Stokes number (Sk_m), Darcy number (Da), Forchheimer number (Fs), particle loading parameter (p_L), particle-phase wall slip parameter (Ω) and buoyancy parameter (B) on the fluid-phase velocity (U) and particle-phase velocity (U_p). Padé approximants are also employed to achieve satisfaction of boundary conditions. Excellent correlation is obtained between the DTM and numerical quadrature solutions. The results indicate that there is a strong decrease in fluid-phase velocities with increasing Darcian (first-order) drag and the second-order Forchheimer drag, and a weaker reduction in particle-phase velocity field. Fluid and particle-phase velocities are also strongly affected with inverse momentum Stokes number. DTM is shown to be a powerful tool providing engineers with an alternative simulation approach to other traditional methods for multi-phase computational biofluid mechanics. The model finds applications in haemotological separation and biotechnological processing.     

Density contrast sedimentation velocity for the determination of protein partial-specific volumes

The partial-specific volume of proteins is an important thermodynamic parameter required for the interpretation of data in several biophysical disciplines. Building on recent advances in the use of density variation sedimentation velocity analytical ultracentrifugation for the determination of macromolecular partial-specific volumes, we have explored a direct global modeling approach describing the sedimentation boundaries in different solvents with a joint differential sedimentation coefficient distribution. This takes full advantage of the influence of different macromolecular buoyancy on both the spread and the velocity of the sedimentation boundary. It should lend itself well to the study of interacting macromolecules and/or heterogeneous samples in microgram quantities. Model applications to three protein samples studied in either H(2)O, or isotopically enriched H(2) (18)O mixtures, indicate that partial-specific volumes can be determined with a statistical precision of better than 0.5%, provided signal/noise ratios of 50-100 can be achieved in the measurement of the macromolecular sedimentation velocity profiles. The approach is implemented in the global modeling software SEDPHAT.     

Three-Dimensional Flow of an Oldroyd-B Nanofluid towards Stretching Surface with Heat Generation/Absorption

This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are then solved analytically by using the homotpy analysis method (HAM). Graphically results are presented and discussed for various parameters, namely, Deborah numbers and , heat generation/absorption parameter Prandtl parameter , Brownian motion parameters, thermophoresis parameter and Lewis number . We have seen that the increasing values of the Brownian motion parameter and thermophoresis parameter leads to an increase in the temperature field and thermal boundary layer thickness while the opposite behavior is observed for concentration field and concentration boundary layer thickness. To see the validity of the present work, the numerical results are compared with the analytical solutions obtained by Homotopy analysis method and noted an excellent agreement for the limiting cases.     

Hydromagnetic Flow and Heat Transfer over a Porous Oscillating Stretching Surface in a Viscoelastic Fluid with Porous Medium

An analysis is carried out to study the heat transfer in unsteady two-dimensional boundary layer flow of a magnetohydrodynamics (MHD) second grade fluid over a porous oscillating stretching surface embedded in porous medium. The flow is induced due to infinite elastic sheet which is stretched periodically. With the help of dimensionless variables, the governing flow equations are reduced to a system of non-linear partial differential equations. This system has been solved numerically using the finite difference scheme, in which a coordinate transformation is used to transform the semi-infinite physical space to a bounded computational domain. The influence of the involved parameters on the flow, the temperature distribution, the skin-friction coefficient and the local Nusselt number is shown and discussed in detail. The study reveals that an oscillatory sheet embedded in a fluid-saturated porous medium generates oscillatory motion in the fluid. The amplitude and phase of oscillations depends on the rheology of the fluid as well as on the other parameters coming through imposed boundary conditions, inclusion of body force term and permeability of the porous medium. It is found that amplitude of flow velocity increases with increasing viscoelastic and mass suction/injection parameters. However, it decreases with increasing the strength of the applied magnetic field. Moreover, the temperature of fluid is a decreasing function of viscoelastic parameter, mass suction/injection parameter and Prandtl number.     

Thermal and Concentration Stratifications Effects in Radiative Flow of Jeffrey Fluid over a Stretching Sheet

In this article we investigate the heat and mass transfer analysis in mixed convective radiative flow of Jeffrey fluid over a moving surface. The effects of thermal and concentration stratifications are also taken into consideration. Rosseland''s approximations are utilized for thermal radiation. The nonlinear boundary layer partial differential equations are converted into nonlinear ordinary differential equations via suitable dimensionless variables. The solutions of nonlinear ordinary differential equations are developed by homotopic procedure. Convergence of homotopic solutions is examined graphically and numerically. Graphical results of dimensionless velocity, temperature and concentration are presented and discussed in detail. Values of the skin-friction coefficient, the local Nusselt and the local Sherwood numbers are analyzed numerically. Temperature and concentration profiles are decreased when the values of thermal and concentration stratifications parameters increase. Larger values of radiation parameter lead to the higher temperature and thicker thermal boundary layer thickness.     

Mixed Convection Boundary Layer Flow over a Moving Vertical Flat Plate in an External Fluid Flow with Viscous Dissipation Effect

The steady boundary layer flow of a viscous and incompressible fluid over a moving vertical flat plate in an external moving fluid with viscous dissipation is theoretically investigated. Using appropriate similarity variables, the governing system of partial differential equations is transformed into a system of ordinary (similarity) differential equations, which is then solved numerically using a Maple software. Results for the skin friction or shear stress coefficient, local Nusselt number, velocity and temperature profiles are presented for different values of the governing parameters. It is found that the set of the similarity equations has unique solutions, dual solutions or no solutions, depending on the values of the mixed convection parameter, the velocity ratio parameter and the Eckert number. The Eckert number significantly affects the surface shear stress as well as the heat transfer rate at the surface.     

Toxicokinetics of Metals in Terrestrial Invertebrates: Making Things Straight with the One-Compartment Principle

In this analysis, we first performed a critical review of one-compartment models used to describe metal toxicokinetics in invertebrates and found mathematical or conceptual errors in almost all published studies. In some publications, the models used do not represent the exact solution of the underlying one-compartment differential equations; others use unrealistic assumptions about constant background metal concentration and/or zero metal concentration in uncontaminated medium. Herein we present exact solutions of two differential-equation models, one describing simple two-stage toxicokinetics (metal toxicokinetic follows the experimental phases: the uptake phase and the decontamination phase) and another that can be applied for more complex three-stage patterns (toxicokinetic pattern does not follow two phases determined by an experimenter). Using two case studies for carabids exposed via food, based on previously published data, we discuss and compare our models to those originally used to analyze the data. Our conclusion is that when metal toxicokinetic follows a one-compartment model, the exact solution of a set of differential equations should be used. The proposed models allow assimilation and elimination rates to change between toxicokinetic stages, and the three-stage model is flexible enough to fit patterns that are more complex than the classic two-stage model can handle.     

Differential interferometry in the analytical ultracentrifuge. II. General applications

Various ways of applying differential interferometry to ultracentrifugal analyses are examined and several analytical techniques are established. In transport and moving boundary methods, the sedimentation coefficient is more precisely determined in the differential interference system than in the schlieren optical system because fringe measurement accuracy is much higher in the former system. Compared to interference and absorption optics, the differential interferometer provides a more exact s value in the transport method since an accurate calculation procedure can be adopted. Moreover, the following advantages of differential interferometry are noted. Determination of the initial solute concentration, which must be done in the usual interference method, is unnecessary in this sedimentation equilibrium method. Regardless of the partial loss of solute from the observed system due to rapid precipitation or adsorption to the cell wall during centrifugation, the molecular weight of the rest of the solute can be determined exactly. The diffusion coefficient can be determined accurately by fringe displacement analysis at the hinge point during the transient state. Together with the molecular weight and diffusion coefficient, the partial specific volume and sedimentation coefficient of a solute can be obtained from the result of a single low-speed centrifugation when the sample solutions in H₂O and D₂O are compared.     

A differential game theoretical analysis of mechanistic models for territoriality

In this paper, elements of differential game theory are used to analyze a spatially explicit home range model for interacting wolf packs when movement behavior is uncertain. The model consists of a system of partial differential equations whose parameters reflect the movement behavior of individuals within each pack and whose steady-state solutions describe the patterns of space-use associated to each pack. By controlling the behavioral parameters in a spatially-dynamic fashion, packs adjust their patterns of movement so as to find a Nash-optimal balance between spreading their territory and avoiding conflict with hostile neighbors. On the mathematical side, we show that solving a nonzero-sum differential game corresponds to finding a non-invasible function-valued trait. From the ecological standpoint, when movement behavior is uncertain, the resulting evolutionarily stable equilibrium gives rise to a buffer-zone, or a no-wolf’s land where deer are known to find refuge.