ELECTROPHORESIS OF PEPSIN

1. A number of pepsin solutions containing several protein components have been studied by the electrophoresis method. All samples show a homogeneous boundary moving to the anode at pH 4.4. 2. The activity of this material may be higher than that of the original solution on the basis of total nitrogen but is the same as that of the original solution on the basis of protein nitrogen. 3. There is no separation of the various protein components under these conditions. 4. The apparent isoelectric point at pH 2.7, previously obtained by the collodion particle method is due to the presence of decomposition products. Pure crystalline pepsin, free from decomposition products, is always negatively charged.     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

Diffusion and simultaneous chemical reactions: I. A method for solving the equations of some systems in which a fixed concentration exists at a boundary

Many solutions are available to the differential equations for systems consisting of a space region with a boundary at which the concentration is fixed, diffusion occurring across this boundary. A method is described for readily transforming these solutions into results for similar systems in which the diffusing substance is removed by a first-order reaction and also removed or produced at a rate which is expressible as a polynomial in the time variable. Subsidiary transformations and steady-state conditions are also discussed. An indication is given of biological applications of the results made available by this method.     

Kinetics of biopolymerization on nucleic acid templates

The kinetics of biopolymerization on nucleic acid templates is discussed. The model introduced allows for the simultaneous synthesis of several chains, of a given type, on a common template, e.g., the polyribosome situation. Each growth center [growing chain end plus enzyme(s)] moves one template site at a time, but blocks L adjacent sites. Solutions are found for the probability n_j(t) that a template has a growing center that occupies the sites j — L + 1,…, j at time t. Two special sets of solutions are considered, the uniform-density solutions, for which n_j(t) = n, and the more general steady-state solutions, for which dn_j(t)/dt = 0. In the uniform-density case, there is an upper bound to the range of rates of polymerization that can occur. Corresponding to this maximum rate, there is one uniform solution. For a polymerization rate less than this maximum, there are two uniform solutions that give the same rate. In the steady-state case, only L = 1 is discussed. For a steady-state polymerization rate less than the maximum uniform-density rate, the steady-state solutions consist of either one or two regions of nearly uniform density, with the density value(s) assumed in the uniform region(s) being either or both of the uniform-density solutions corresponding to that polymerization rate. For a steady-state polymerization rate equal to or slightly larger than the maximum uniform-density rate, the steady-state solutions are nearly uniform to the single uniform-density solution for the maximum rate. The boundary conditions (rate of initiation and rate, of release of completed chains from the template) govern the choice among the possible solutions, i.e., determine the region(s) of uniformity and the value(s) assumed in the uniform region(s).     

Spatial structures in a reaction-diffusion system--detailed analysis of the "Brusselator"

Continuous dependence of spatially nonuniform concentration profiles for the 'Brussellator" reaction mechanims on the characteristic length of the system is given both for zero flux and fixed boundary conditions. Branches of solutions arising through primary bifurcation form closed curves. Secondary bifurcations giving rise to spatially asymmetric solutions exist for fixed boundary conditions. Results of a stability analysis of individual solutions are discussed. A method of composing complex spatial profiles for higher lengths from elementary solutions for smaller lenghts is suggested and tested in the case of zero flux boundary conditions. Emergence of subsequently more complex stable patterns in dependence on increasing length of the system suggests many similarities to gradual build up of complex morphogenetic patterns.     

A cryomicroscope for the analysis of solute polarization during freezing

A better understanding of the freezing process in the extracellular suspension medium implies the consideration of deviations from equilibrium, i.e., unsteady diffusion of heat and mass with a moving phase boundary. Such phenomena, especially solute redistribution in front of the advancing phase interface, can readily be investigated with a special cryomicroscope equipped with a spectrophotometer. A major advantage of this method is the combination of quantitative measurements in conjunction with visual observations, allowing a control of the solid-liquid interface morphology (planar-cellular-dendritic) which is crucial to the solidification process. The freezing stage designed for this purpose produces a temperature field in the sample layer resembling that within a large plate-shaped container, and hence well-defined thermal gradients (having a dominant effect on the shape of the interface). Aqueous solutions of NaMnO₄, exhibiting a maximum absorption at 525 nm and a phase diagram as well as diffusive properties very similar to NaCl in water, turned out to be a particularly suitable model for simulating of solidification of biological solutions. As long as freezing is unidimensional (planar), the concentration profiles can be scanned on-line, while multidimensional (cellular, dendritic) structures require off-line densitometric determination from photomicrographs. The experimental results agree quite well with mathematical models for both types of solidification. The observed transition points between planar freezing and higher-order structures correspond to those resulting from constitutional supercooling, a criterion roughly indicating the conditions for interface instability based on temperature and concentration gradients at the phase boundary.     

Influence of model boundary conditions on blood flow patterns in a patient specific stenotic right coronary artery

Background

In literature, the effect of the inflow boundary condition was investigated by examining the impact of the waveform and the shape of the spatial profile of the inlet velocity on the cardiac hemodynamics. However, not much work has been reported on comparing the effect of the different combinations of the inlet/outlet boundary conditions on the quantification of the pressure field and flow distribution patterns in stenotic right coronary arteries.

Method

Non-Newtonian models were used to simulate blood flow in a patient-specific stenotic right coronary artery and investigate the influence of different boundary conditions on the phasic variation and the spatial distribution patterns of blood flow. The 3D geometry of a diseased artery segment was reconstructed from a series of IVUS slices. Five different combinations of the inlet and the outlet boundary conditions were tested and compared.

Results

The temporal distribution patterns and the magnitudes of the velocity, the wall shear stress (WSS), the pressure, the pressure drop (PD), and the spatial gradient of wall pressure (WPG) were different when boundary conditions were imposed using different pressure/velocity combinations at inlet/outlet. The maximum velocity magnitude in a cardiac cycle at the center of the inlet from models with imposed inlet pressure conditions was about 29% lower than that from models using fully developed inlet velocity data. Due to the fact that models with imposed pressure conditions led to blunt velocity profile, the maximum wall shear stress at inlet in a cardiac cycle from models with imposed inlet pressure conditions was about 29% higher than that from models with imposed inlet velocity boundary conditions. When the inlet boundary was imposed by a velocity waveform, the models with different outlet boundary conditions resulted in different temporal distribution patterns and magnitudes of the phasic variation of pressure. On the other hand, the type of different boundary conditions imposed at the inlet and the outlet did not have significant effect on the spatial distribution patterns of the PD, the WPG and the WSS on the lumen surface, regarding the locations of the maximum and the minimum of each quantity.

Conclusions

The observations from this study indicated that the ways how pressure and velocity boundary conditions are imposed in computational models have considerable impact on flow velocity and shear stress predictions. Accuracy of in vivo measurements of blood pressure and velocity is of great importance for reliable model predictions.

     

Transient distributions arising from initially sharp boundaries in the ultracentrifuge

An infinite series solution to the Mason-Weaver equation is presented for the case in which a synthetic boundary is formed originally between solution and solvent. Digital computations based on this series, and confirmed independently, have been made for a range of parameters. For given conditions, the maximum rate of change of concentration at the meniscus and the time at which it occurs can be easily estimated by means of the curves presented. In equilibrium experiments which commence with formation of a sharp boundary, this enables the fringes to be identified with certainty.     

Sedimentation Patterns of Rapidly Reversible Protein Interactions

The transport behavior of macromolecular mixtures with rapidly reversible complex formation is of great interest in the study of protein interactions by many different methods. Complicated transport patterns arise even for simple bimolecular reactions, when all species exhibit different migration velocities. Although partial differential equations are available to describe the spatial and temporal evolution of the interacting system given particular initial conditions, a general overview of the phase behavior of the systems in parameter space has not yet been reported. In the case of sedimentation of two-component mixtures, this study presents simple analytical solutions that solve the underlying equations in the diffusion-free limit previously subject to Gilbert-Jenkins theory. The new expressions describe, with high precision, the average sedimentation coefficients and composition of each boundary, which allow the examination of features of the whole parameter space at once, and may be used for experimental design and robust analysis of experimental boundary patterns to derive the stoichiometry and affinity of the complex. This study finds previously unrecognized features, including a phase transition between boundary patterns. The model reveals that the time-average velocities of all components in the reaction mixture must match—a condition that suggests an intuitive physical picture of an effective particle of the coupled cosedimentation of an interacting system. Adding to the existing numerical solutions of the relevant partial differential equations, the effective particle model provides physical insights into the relationships of the parameters that govern sedimentation patterns.     

带一个参数的四阶边值问题的一个变号解

讨论了带一个参数的非线性四阶狄利克雷边值问题变号解的存在性．在非线性项满足一定条件时,通过利用一些新的概念如O-有界锥和O-正算子,我们得到该边值问题至少存在一个变号解．     

Concentration polarization phenomenon in the case of mechanical pressure difference on the membrane

We analyzed the transport of KCl solutions through the bacterial cellulose membrane and concentration boundary layers (CBLs) near membrane with pressure differences on the membrane. The membrane was located in horizontal-plane between two chambers with different KCL solutions. The membrane was located in horizontal-plane between two chambers with different KCL solutions. As results from the elaborated model, gradient of KCL concentration in CBLs is maximal at membrane surfaces in the case when pressure difference on the membrane equals zero. The amplitude of this maximum decreases with time of CBLs buildup. Application of mechanical pressure gradient in the direction of gradient of osmotic pressure on the membrane causes a shift of this maximum into the chamber with lower concentration. In turn, application of mechanical pressure gradient directed opposite to the gradient of osmotic pressure causes the appearance of maximum of concentration gradient in chamber with higher concentration. Besides, the increase of time of CBLs buildup entails a decrease of peak height and shift of this peak further from the membrane. Similar behavior is observed for distribution of energy dissipation in CBLs but for pressure difference on the membrane equal to zero the maximum of energy dissipation is observed in the chamber with lower concentration. We also measured time characteristics of voltage in the membrane system with greater KCl concentrations over the membrane. We can state that mechanical pressure difference on the membrane can suppress or strengthen hydrodynamic instabilities visible as pulsations of measured voltage. Additionally, time of appearance of voltage pulsations, its amplitude, and frequency depend on mechanical pressure differences on the membrane and initial quotient of KCl concentrations in chambers.     

Analytical solutions for transient thermal behavior in biological systems

Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.     

Analysis of a model of membrane potential for a skin receptor

A linear model for the membrane potential of a nerve ending associated with a sense-of-touch skin receptor is formulated and analyzed. Because the nerve has cytoplasmic extensions and is stimulated through a membrane strain mechanism, the model is a generalized cable equation with linear oblique boundary conditions (i.e. boundary conditions involving both time and space derivatives). The model is stimulated through a certain conductance parameter. We establish solutions for the problem and we consider a particular physiological question, both analytically and computationally, of determining conditions on parameters and stimulus which give threshold level potential values at a particular boundary point.     

Proton flux measurements from tissues in buffered solution

Proton movement across plant cell membranes is part of many important physiological processes. The net proton flux to or from tissues can be determined non-invasively by measuring the proton electrochemical potential gradient in the adjacent solution. In buffered solution, some of the protons crossing the tissue boundary diffuse as proto-nated buffer whose flux is not included in the flux calculated from the proton (hydrogen ion) electrochemical gradient. In this theoretical paper, it is shown how experimenters can calculate the protonated buffer flux from the measured proton flux in solution. The ratio of these two components of total proton flux depends on the pH of the solution and on the concentration and pK of the buffer. For a given concentration of a buffer which has a single pK, the flux ratio rises with pH when the solution pH is lower than the buffer pK. The slope is about 2 on a log₁₀ scale. As the pH increases above the pK, the flux ratio levels off to approach its maximum. With mixed buffers, or one having two or more pK values, the flux ratios are additive: each buffer acts independently based on its concentration and its pK value. Unbuffered solutions always have the buffering effects of water itself and also of carbonates due to carbon dioxide dissolved from the atmosphere. In unbuffered solutions at pH 6, the flux carried by water and carbonate is about 1 % of the measured proton flux. This validates measurements of proton flux from tissues, made by a number of workers, in unbuffered solutions below pH 6.     

Effects of transients on pulsatile flow in arteries

Analytical solutions to the model problem of unsteady Newtonian fluid flow in straight, elastic-walled vessels can provide: theoretical insights into the flow of blood in arteries; a theoretical basis for clinical measurements in diagnoses of arterial flow rates; and guidance for boundary conditions in numerical simulations of flow in finite computational domains. However, while Womersley's analyses of blood flow assume solution forms that treat the flow as periodic and continuously unsteady, many flow variables in the smaller arteries are not continuously unsteady at all. They are characterized more accurately as rapid transient motions followed by a period of recovery to a stationary state, repeated in successive cycles. These flows are not continually unsteady ones described by Womersley's solutions but unsteady flows restarted from rest in each cycle, characterized as initial-boundary value problems. In this paper, we compare the Womersley and initial-boundary value solutions for model transients that stop then restart, explain these previously unreported limitations of Womersley's solutions, and demonstrate how the initial-boundary value solutions provide excellent agreement with measurements of blood flow in the anterior tibial and popliteal arteries of patients. Some consequences of these findings for understanding and interpreting measurements of blood flow, and for prescribing boundary conditions in computer simulations of arterial blood flow are discussed.     

带存放率的周期竞争扩散系统的稳定共存

应用上、下解方法和抛物型方程的极值原理,研究了带存放率的周期竞争系统ut-D1Δu=u（α-bu-cv）＋h,vt-D2Δv=v）d-eu-fv）＋k 在齐次Neumann边界条件下解的渐近性态,得到了该系统的全局渐近性．     

A theoretical model for peripheral tissue heat transfer using the bioheat equation of Weinbaum and Jiji

In this paper the new bioheat equation derived in Weinbaum and Jiji is applied to the three layer conceptual model of microvascular surface tissue organization proposed in. A simplified one-dimensional quantitative model of peripheral tissue energy exchange is then developed for application in limb and whole body heat transfer studies. A representative vasculature is constructed for each layer and the enhancement in the local tensor conductivity of the tissue as a function of vascular geometry and blood flow is examined. Numerical solutions for the boundary value problem coupling the three layers are presented and these results used to study the thermal behavior of peripheral tissue for a wide variety of physiological conditions from supine resting state to maximum exercise.     

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595     

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.