Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t=±. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.Supported in part by NSF Grant MCS8301724 and by a Biomedical Research Support Grant 2SO7RR0706618 from NIH     

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.Supported in part by NSF Grant DMS-8704369Supported in part by NSF Grant BSR-8512844     

Non-linear phenomena in oxygen transport to tissue

A non-linear partial differential equation is analyzed using multiple scale techniques and similarity transformations in order to examine the role of hemoglobin and myoglobin in facilitating oxygen transport to tissue.Supported by NSF Grant DCB 8902472     

The tanycyte of the rat median eminence

Summary The present ultrastructural study proves the existence of nerve terminals closely apposed to the plasmalemmata of tanycytes in the rat median eminence. Several of these axo-tanycytic endings display remarkable accumulations of agranular endoplasmic reticulum in the form of pleomorphic vesicles which are closely apposed on either side of the plasma membrane of each cell compartment. Some of these vesicular profiles give the impression of structural continuity across both membrane systems. This phenomenon is discussed in the context of being a potential substratum for communication between both cell compartments.Supported by NIH Grant NS13717 and NSF Grant BNS 78-11820     

The evolution of dispersal

A non-local model for dispersal with continuous time and space is carefully justified and discussed. The necessary mathematical background is developed and we point out some interesting and challenging problems. While the basic model is not new, a spread parameter (effectively the width of the dispersal kernel) has been introduced along with a conventional rate paramter, and we compare their competitive advantages and disadvantages in a spatially heterogeneous environment. We show that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal. However, fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results. For example, in a fairly general setting given two phenotypes with different, but small spread, the smaller spread is selected while in the case of large spread the larger spread is selected. S. Martinez was partially supported by Fondecyt 1020126 and Fondecyt Lineas Complementarias 8000010. K. Mischaikow was supported in part by NSF Grant DMS 0107396. Key words or phases:Non-local dispersal – Integral kernel – Evolution of dispersal     

Associations between isozyme phenotypes and environment in the slender wild oat (Avena barbata) in Israel

Summary Collections from 31 populations of A. barbata from diverse habitats in Israel were assayed electrophoretically for seven enzyme systems. Phenotype frequencies were scored in nine enzyme zones, probably representing 27 loci, to determine isozyme variability within and among populations. Many different isozyme phenotypes were found in all of the populations; also the array of isozyme phenotypes found in each population differed distinctly from that found in each other population. Overlays of phenotypic frequencies on map locations showed that isozyme variability is distributed in mosaic patterns not related to geographical distance. Principal-component and multiple-regression analyses revealed that temperature and moisture-related variables are significantly correlated with particular isozyme phenotypes. Further, the mosaic patterns of isozyme variation were found to correspond closely to mosaic patterns of the habitat. This structuring of the genetic variability into multilocus combinations was attributed to the combined effects of directional and diversifying selection. Comparisons of patterns and extent of genetic variation in Israel and California led to the conclusion that the evolution of ecotypes, each adapted to a specific habitat and marked by a particular set of enzyme alleles, has proceeded further in Israel, where A. barbata is endemic, than in California, where it is a recent introduction.This study was supported in part by NSF Grant BMS-01113-A01. Seed collections were supported by a United States-Israel Binational Science Foundation Grant     

Convergence of a structured metapopulation model to Levins’s model

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting régime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Steins method.Supported in part by Schweizer Nationalfonds Projekt Nrs 20–61753.00 and 20–67909.02Supported in part by CNR of Italy under Grant n. 00.0142.ST74     

A recovery-relapse epidemic model with spatial diffusion

An epidemic model of an infectious phenomenon is analyzed. The model allows for an age-dependency to describe the phases of incubation, recovery, and relapse, and for a spatial dependency to describe diffusion of the population in geographical space.Supported in part by the National Science Foundation Grant NSF MCS 7903047     

The Freter model: A simple model of biofilm formation

A simple, conceptual model of biofilm formation, due to R. Freter et al. (1983), is studied analytically and numerically in both CSTR and PFR. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid. One of these is stable for any particular set of parameter values and sharp and explicit conditions are given for the stability of each. The effects of adding an anti-microbial agent to the CSTR are examined.Supported by NSF Grant DMS 0107439 and UTA Grant REP 14748717Supported by NSF Grant DMS 0107160     

Entrainment of oscillatory neural activity in the cat's lateral geniculate nucleus

Oscillatory neural activity in the frequency range 7–12 Hz is observed in the lateral geniculate nucleus (LGN) of the lightly anesthetized cat. This paper describes a series of experiments in which the interactions between ongoing oscillatory potentials and periodic photic and electrical stimuli are analyzed using frequency domain techniques. The principal results of these experiments are consistent with a model of the neural system as an entrainable oscillator in which ongoing oscillations are suppressed by stimulation at nearby frequencies, but coexist with stimulus frequencies farther away. The physiological interpretation of these results may be closely tied to the role of the LGN as a gating mechanism between retina and cortex.Supported by NIH Training Grant #GM01455 and NSF Grant #ENG-7515736     

Identification and genetic control of two rabbit low-density lipoprotein allotypes

Rabbits were immunized with a low-density lipoprotein (LDL) preparation isolated from rabbit serum by ultracentrifugation. This elicited precipitin isoantibodies which distinguished two antigenically different genetic variants, i.e., allotypes of serum LDL. Both allotypes were identified as LDL by the following criteria: (1) the precipitin lines stained intensely with the lipid stain Sudan black B; (2) the antigens were found in the low-density but not the high-density lipoprotein fraction; (3) the antigens migrated electrophoretically on Agarose in the ₂ to region. That the inheritance of these two allotypes is controlled by a pair of allelic genes at an autosomal locus is based on allotypes present in 323 progeny from six possible mating combinations. This LDL locus designated Lpqwas shown not to be linked to the light-chain or heavy-chain loci of immunoglobulins. This investigation was supported (in part) by NSF Grant GB-5536 and USPHS Grant AI07043-03.Supported by USPHS predoctoral fellowship (1-F1-GM-37, 211-01) from the National Institute of General Medical Sciences.     

Phosphorylase kinase deficiency

Phosphorylase kinase deficiency in I strain mice and in humans both show X-chromosomal inheritance. Neither deficient adult humans nor deficient mice show any sign of disease. Thus the two conditions resemble each other. However, there are differences. The enzyme is only partially deficient in human patients in liver, muscle, and blood cells; in mice the deficiency is complete and seems to be confined to the muscles.This paper was presented at a symposium entitled Genetic Control of Mammalian Metabolism held at The Jackson Laboratory, Bar Harbor, Maine, June 30–July 2, 1969. The symposium was supported in part by an allocation from NIH General Research Support Grant FR 05545 from the Division of Research Resources to The Jackson Laboratory.Supported by grant AM 13359 of the National Institutes of Health.     

The dynamics of density dependent population models

Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a strange attractor is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).Supported by NSF Grant No. BMS 74-21240.     

Calculation of the inbreeding coefficient

Wright's rule for calculating the inbreeding coefficient for an arbitrary pedigree is proven for both autosomal and X-linked loci.Supported by NSF Grant PHY-84-16691     

An eikonal-curvature equation for action potential propagation in myocardium

We derive an eikonal-curvature equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential ellipses from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.Research was supported in part by NSF Grant DMS-8801446     

Changes in competible IAA binding in relation to bud development in pea seedlings

An insoluble (particulate) ³H-IAA-binding system similar tothat reported by Hertel et al. (3) is described in buds frompea seedlings. The binding is competed by NAA as well as IAA.Auxin-competible binding is optimal at 25?C and pH6.5; Ca⁺²increases binding as does a 1 hr preincubation at 4?C. Releaseof apical dominance produces outgrowth and a large decreasein the NAA-competible ³H-IAA-binding activity in the axillarybuds; this correlates with a decreased ability of auxin to inhibitthe buds. Both the antiauxin triiodobenzoic acid and the cytokininbenzyladenine also compete with the bound IAA. ¹Supported in part by a Special Research Assistance Grant fromthe College of Literature, Science and the Arts, Universityof Michigan and by USPHS Grant no. ES-000634. ²Present address: Universitet u Pritini, Prirodno-matemat-facultet,38000 Pritina, Yugoslavia. (Received May 16, 1974; )     

A gene apparently determining the extent of sialylation of lysosomal α-mannosidase in mouse liver

Organ-specific electrophoretic heterogeneity of lysosomal -mannosidase has been observed within individual strains of inbred mice. Polymorphism between C57BL/6J and CBA/J for liver lysosomal -mannosidase is determined by a single genetic locus on chromosome 5 and appears to be the result of differences in sialylation of the lysosomal enzyme. Two different patterns of expression of development of the liver electrophoretic forms have been observed.Supported in part by Grant GM-19521 from the U.S. Public Health Service. One of the authors (M.D.) was supported in part from USPHS Grant TAO-CA05016.     

Quantitation of sulfhydryl groups on erythrocytes in polycythemia vera

Summary The number of sulfhydryl groups on the surface of intact erythrocytes from patients with polycythemia vera is significantly less (34%) than those from control subjects. Sulfhydryl groups were measured indirectly by reversibly blocking-SH groups with the thiol reagent, 6,6-dithiodinicotinic acid, which forms stable mixed disulfides on the surface of erythrocytes. Glutathione was used to break the disulfides and release thione into the supernatant. Thione was then quantitated spectrophotometrically.Supported by NIH Biomedical Research Development Grant 1-S08-RR-09171-01     

Regulation of population cycles by genetic feedback: Existence of periodic solutions of a mathematical model

Populations of voles, and lemmings of the Northern hemisphere exhibit cyclic fluctuations with a cycle of three to four years. Krebs et al. presented evidence that the cycles are driven by changes in the genotypic structure of the population [9]. Incorporating some of their hypotheses we present a mathematical model of a one locus two allele population with density dependent selection and assuming a slow selection hypothesis, the existence of periodic solutions is proved. These solutions arise by Hopf bifurcation in ₁/¦₁¦, the ratio of the residual death and birth rates of the density sensitive homozygote.Partially supported by NSF Grant # MCS-8005777     

On the optimal sex-ratio: A stability analysis based on a characterization for one-locus multiallele viability models

Theoretical one-locus multiallele sex-determination models are found to admit even sex ratio equilibrium surfaces besides the equilibria for corresponding one-locus multiallele viability models. Both types of equilibria can be defined in terms of a single spectral radius function, the former corresponding to level surfaces and the latter to critical points. The stable equilibria in the corresponding viability models are associated with the local maxima, and the equilibrium structures for the sex-determination models can be fully described. Several optimality properties of the even-sex-ratio equilibrium surfaces can be deduced.Supported in part by NIH Grant 5R01 GM10452-20 and NSF Grant MCS79-24310