Estimating the initial reaction velocity of a soluble dehydrogenase in situ

Summary The initial reaction velocities (v _i ) of lactate dehydrogenase in single hepatocytes were determined, by microdensitometry or computer-assisted image analysis, in sections of unfixed mouse liver incubated at 37°C on substrate-containing agarose gel films. They were found to fit the equations v _i =2.82°A and v _i =v+2°A, where v and °A are, respectively, gradients (or steady-state linear velocities) and the intercepts on the absorbance axis of the linear regression lines of the absorbance (A) on incubation time plots for incubation times between 1 and 3 min. Both equations were independent of section thickness between 4 and 14 m. The observed and calculated values of v _i , agreed within 11.5% (n = 71). The validity of the equations for v _i was confirmed by showing that the calculated v _i was proportional to the thickness of the section and hence the amount of enzyme present. Thus, v _i can be determined from measurements of either °A alone or v and °A.     

Rotating wave solutions of the FitzHugh–Nagumo equations

This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

A singular dispersion relation arising in a caricature of a model for morphogenesis

In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value k _c from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value l _l ,the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l _n the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l>l _n the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u=0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.     

Solutions to a degenerate system of parabolic equations from marine biology

Summary A system of parabolic and ordinary differential equations u _t = a ² u _xx + F(u, v, w), v _t = a ² v _xx + G(u, v, w),w _x = – k(u)w is studied which has been proposed by Radach and Maier-Reimer for the dynamics of phytoplankton and nutrient in dependence of light intensity. It is shown that there is a unique solution to this system satisfying given initial and boundary conditions. The solution depends continuously on the data. For specific nonlinearities F, G, and k bounds for the solutions are given.     

The initial reaction velocities of lactate dehydrogenase in various cell types

Summary The initial reaction velocities (v _v ) of lactate dehydrogenase in hepatocytes, cardiac muscle fibres, skeletal (gastrocnemius) muscle fibres, gastric parietal cells, ductal epithelial and acinar cells of the parotid gland, and oocytes were determined, by computer-assisted image analysis, in unfixed sections of these tissues incubated at 37°C on substrate-containing agarose gel films. They were found to fit the equations v _i = a₁A (equation 1) and v _i – v = a₂A (equation 2) reported previously for mouse hepatocytes (Nakae & Stoward, 1993a, b), where v and A are, respectively, the gradients (or steady-state velocities) and the intercepts on the absorbance axis of the linear regression lines of the absorbance (A) of the finalreaction product on incubation times between 1 and 3 min, and a ₁ and a ₂ are constants. Both equations 1 and 2 fitted the observed v _i closely for mouse (a ₁ = 2.7, a ₂ = 2.2) and human (a ₁ = 3.0, a ₂ = 1.9) hepatocytes. However, equation 2 fitted the observed v _i better than equation 1 for mouse cardiac muscle fibres (a ₂ = 1.5), skeletal muscle fibres (a ₂ = 1.2), gastric parietal cells (a ₂ = 1.7), acinar (a ₂ = 1.4) and striated ductal (a ₂ = 2.2) epithelial cells of the parotid gland, and oocytes (a ₂ = 1.6). The values of v _i calculated from the two equations agreed with the observed v _i to within about 11%. They ranged from 105 mole hydrogen equivalents/cm³ cell/min units in hepatocytes to 24 units in parotid acinar cells, but for other cell types they were between 46 and 61 units. These are all considerably higher than values reported previously.     

Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.     

Kinetic analysis of lactate dehydrogenase in situ in mouse liver determined with a quantitative histochemical technique

Summary The kinetics of lactate dehydrogenase in situ were studied in sections of unfixed liver of the male mouse using a quantitative histochemical technique. The sections were incubated on substrate-containing gel films. The absorbance of the final reaction products deposited in a single hepatocyte was measured continuously during the incubation as a function of incubation time using a scanning microdensitometer. The absorbance increased non-linearly during the first minute of incubation, but linearly for at least the next 3 min afterwards. The initial velocity (v _i ) of the dehydrogenase was calculated from two equations proposed previously by us, v _i=2.82 °A and v _i =v+2°A, where v and °A are, respectively, the gradient and intercept o linear regression line of absorbance on time for incubation times between 1 and 3 min.The dependence of v _i on lactate concentration gave the following mean kinetic constants. For periportal hepatocytes, the apparent K _m =14 mM and V _max =80 moles hydrogen equivalents formed cm^–3 hepatocyte cytoplasm min^–1. For pericentral hepatocytes, K _m =12 mM and V _max =87 moles hydrogen equivalents cm^–3 min^–1. The K _m values are very similar to those determined previously from biochemical assays. The concentrations of the enzyme in single hepatocytes calculated from the V _max values are in good agreement with those obtained by another method. These data substantiate the validity of our equations.     

Traveling wave solutions from microscopic to macroscopic chemotaxis models

In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044–1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller–Segel model, (Keller and Segel in J Theor Biol 30:225–234; 377–380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: F(v) = lnv{\Phi(v) = \ln v} and F(v) = ln[v/(1-v)]{\Phi(v) = \ln[v/(1-v)]}. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on \mathbb R{\mathbb R}. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

Evolution of discontinuous solutions to transport equations in stellarators

It was shown earlier that, in the range of rare collisions, transport equations for stellarators allow steady discontinuous solutions for the ambipolar electric field and for the plasma density and temperature gradients. Moreover, such solutions are non-single-valued; that is, their explicit form depends on the initial values of the ambipolar electric field. The time-independent transport equations are derived under the conventional quasineutrality condition; i.e., it is assumed that the electron and ion densities, N _e and N _i , are related by the relationship N _e = ZN _i (where Z is the ion charge number). In other words, the plasma charge density is assumed to be much less than the product e _i N _i . Under typical conditions, the corresponding inequality is satisfied by a large margin. However, if the electric field E has discontinuities, then it can be seen from the equation ▿·E = 4πρ that, at the discontinuity points, the charge density becomes infinite and the relationship N _e = ZN _i fails to hold, so it is necessary to replace it with N _e = ZN _i + ρ/e _e . In the transport equations, this latter replacement produces additional terms, proportional to the second radial derivative of the field E. With these additional terms, the steady solutions are modified substantially. First, the ambipolar field and the derivatives of the density and temperatures all become continuous functions of the coordinates, a result that seems to be quite obvious. The second, not-so-obvious result is that the steady solutions become single-valued, i.e., independent of the initial values of the ambipolar electric field. It turns out that, in this case, two regimes are possible, depending on the values of the plasma parameters. In the first regime, the solution is unique and is independent of the initial conditions. In the second regime, two steady solutions can exist, depending on the initial conditions. One of the solution is similar to that obtained in the first regime, and the other differs from the first one both in the ambipolar field profile and in the dependence of the density and temperatures on the minor plasma radius. It cannot be excluded that different plasma confinement modes revealed in experiments are associated with the existence of such solutions.     

Preferential Targeting of Nav1.6 Voltage-Gated Na+ Channels to the Axon Initial Segment during Development

During axonal maturation, voltage-gated sodium (Na_v) channels accumulate at the axon initial segment (AIS) at high concentrations. This localization is necessary for the efficient initiation of action potentials. The mechanisms underlying channel trafficking to the AIS during axonal development have remained elusive due to a lack of Na_v reagents suitable for high resolution imaging of channels located specifically on the cell surface. Using an optical pulse-chase approach in combination with a novel Na_v1.6 construct containing an extracellular biotinylation domain we demonstrate that Na_v1.6 channels are preferentially inserted into the AIS membrane during neuronal development via direct vesicular trafficking. Single-molecule tracking illustrates that axonal channels are immediately immobilized following delivery, while channels delivered to the soma are often mobile. Neither a Na_v1.6 channel lacking the ankyrin-binding motif nor a chimeric K_v2.1 channel containing the Na_v ankyrinG-binding domain show preferential AIS insertion. Together these data support a model where ankyrinG-binding is required for preferential Na_v1.6 insertion into the AIS plasma membrane. In contrast, ankyrinG-binding alone does not confer the preferential delivery of proteins to the AIS.     

Spatiotemporal dynamics of two generic predator–prey models

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ^∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L ^p -estimate, uniform in time, for all p≥1, implies L ^∞-uniform bounds, given any nonnegative L ^∞-initial data. The applicability of the L ^∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.     

An enzyme rate equation for the overall rate of reaction of gel-immobilized glucose oxidase particles under buffered conditions. II. Two limiting substrates

The overall rate of reaction of gel-immobilized glucose oxidase particles in buffered media has been investigated theoretically under two substrate diffusion limited conditions by the numerical solution of the diffusion equations. It has been found that the Enzyme Rate Equation (Atkinson and Lester), Biotechnol. Bioeng., 16 , 1299 (1974), together with an analytical solution which describes the asymptotic conditions associated with a large particle size, provides an adequate estimation of the values resulting from the numerical solution outside the region of the parameter space defined by 0.4 < M_g′, M₀′ < 10. When the dimensionless parameter (B₀′/B_g′)(M_g′²/M₀′²) is greater than unity the overall rate of reaction is limited principally by the external concentration and when the parameter has a value less than unity, by the external oxygen concentration. The results are generally applicable to enzymes whose kinetics are similar to those of glucose oxidase or for which the equation describing glucose oxidase kinetics provides an adequate curve-fit of experimental data.     

Identification of random amplified polymorphic DNA (RAPD) markers linked to the v locus in barley, Hordeum vulgare L.

 Recombinant backcross lines of barley were produced from a cross between Kanto Nakate Gold (KNG; two-rowed) and Azumamugi (AZ; six-rowed) after backcrosses of F₁ plants with AZ as the recurrent parent. Each of these lines had an introgressed segment from chromosome 2 of KNG. Two recombinant backcross lines, L1 and M3-13, were used for an initial screening of polymorphism. After screening a total of 888 oligonucleotides as arbitrary primers, we identified eight random amplified polymorphic DNAs (RAPDs) between backcross lines and AZ. Among the RAPD fragments, CMNA-38₇₀₀ was linked to the v locus with a recombination frequency of zero, while OPJ-09₈₅₀ and OPP-02₇₀₀ were linked to the v locus at a map distance of 1.4 cM. Thus, the three RAPD markers were clustered around the v locus since the lengths of introgressed chromosomal segments in the L1 and M3-13 lines were no less than 38 cM. The other five RAPD fragments that we identified were not linked to the v locus. Received: 14 January 1997 / Accepted: 14 February 1997     

Light-scattering and depolarization in biopolymers: Distinction between flexible chain models

The general formulas describing the low-angular dependence of polarized (V_v) and depolarized (H_v) light-scattering intensity by a chain macromolecule of arbitrary form with anisotropic polarizability are derived. It is shown that the value dH_v/dtH_vo (where t = [〈R²〉S/L]², S being the scattering vector) is an indication of the chain-flexibility mechanism. This permits one to distinguish between the models of a wormlike chain, regular zigzag, or statistical zigzag. The results of numerical calculations useful for the interpretation of experimental data are presented.     

Molecular configuration of amylose and its complexes in aqueous solutions. Part IV. Determination of DP of amylose by measuring the concentration of free iodine in solution of amylose–iodine complex

In aqueous solutions of the amylase–iodine complex the concentration of free iodine [I_f]_v after reaching equilibrium (or closely approximating it) is determined by the following factors: temperature, pH, concentration of iodide ions and amylose, and DP of amylose. In the present paper the role of temperature, amylose concentration, and DP has been investigated. At half-saturation of amylose by iodine, the reciprocal value of free iodine defines the equilibrium constant: 1/[I_f]_v = K. The relation between [I_f]_v, in normality and temperature is the following: 5 + log [I_f]_v = ?(2.132/T) + 8.52, for DP _n = 1290, 0.4 mg. amylose in 100 ml. 0.1N HCl. The value of the energy of activation E_a between 2 and 52°C. is 9.72 kcal./mole. The influence of amylose concentration [Am] on photometrically determined [I_f]_v, at 20°C, in the range of 0.1–1.2 mg./100 ml. 0.1 N HCl for DP _n = 1290 is: 5 + log [I_f]_v = 0.209 ? 0.047 log [Am]. At [Am] = 0.6 mg. amylose/ 100 ml. 0.1 N HCl and 20°C, the value of [I_f]_v depends on DP _n as follows: 5 + log [I_f]_v = 0.085 = + 0.222 log (10⁴/DP _n). These above equations are summarized by the relation: [I_f]_v = exp {16.865 ? (E_a/RT)}[Am]^0.047(10⁴/DP _n)^0.222 ×10^?5 Considering that the determination of [I_f]_v by automatic photometric titration can be performed quickly and with appropriate reproducibility, this method is convenient for a rapid empirical and approximate determination of DP of amylose on a microscale. The iodine-binding capacity [IBC] as well as the value of λ_max, have been also investigated as functions of DP _n, by photometric and by amperometric titration.     

An unstable gene controlling developmental variegation in okra (Abelmoschus esculentus (L.) Moench)

Summary Genetic studies on radiation-induced chlorina and variegated mutants of okra (Abelmoschus esculentus (L.) Moench) revealed the existence of an unstable gene. The normal green color of the leaves is controlled by duplicate genes C₁ and C₂, either of which produces the green colour. The chlorina plants are C ₁ C ₁ C ₂ C ₂. The allele c ₁ ^v is dominant to both C ₁ and C ₂ but is unstable. The homozygote c ₁ ^v c ₁ ^v c ₂ c ₂ is a normal green while the heterozygote c _i ^v c ₁ c ₂ c ₂ has a variegated phenotype as a result of the mutation of c ₁ ^v to c ₁ during development. In green plants with a c ₁ ^v c{sh1/v}c ₂ c ₂ genotype, the autonomous mutation of one of the c ₁ ^v alleles to c ₁ may take place at the pre-meiotic stage. In the variegated genotype (c ₁ ^v c ₁ c ₂ c ₂), the mutation of c ₁ to c ₁ ^v may take place in early ontogeny, thus producing green plants. The allele C ₁, when associated with c ₁ ^v in a heterozygous condition, mutates to c ₁ at the pre-meiotic stage even in the presence of the allele C ₂.