Pulse velocities in initially stressed arteries

In this short paper, treating the artery as a thick walled cylindrical shell and employing the large deformation theory, an analytical expression for the pulse wave speed is obtained and the effect of lumen pressure and the axial stretch on the wave speed is discussed. Numerical results indicate that although the wave speed increases with inner pressure, it decreases with increasing axial stretch.     

The effect of heterogeneously-distributed RyR channels on calcium dynamics in cardiac myocytes

Calcium plays an essential role in excitation-contraction coupling in muscle, and derangements in calcium handling can produce a variety of potentially harmful conditions, especially in cardiac muscle. In cardiac tissue specialized invaginations of the sarcolemma, called T-tubules, penetrate deep into each sarcomere, and depolarization of the SL leads to an influx of calcium through voltage-sensitive channels in the T-tubules that in turn triggers further calcium release from the sarcoplasmic reticulum via ryanodine-sensitive calcium channels. Under certain conditions, such as elevated external Ca²⁺, cardiac cells can release calcium from the sarcoplasmic reticulum spontaneously, producing a calcium ’spark’ and propagating traveling waves of elevated Ca²⁺ concentration, without depolarization of the SL (Wier and Blatter, 1991a, Cell Calcium 12, 241–254; Williams, 1993, Cell Calcium 14, 724–735; Cheng et al., 1993a, Science 262, 740–744). However, under normal resting conditions these potentially harmful waves seldom occur. In this paper we investigate the role of the periodic distribution of ryanodine-sensitive channels in determining whether a spark can trigger a wave, using a modification of the kinetic model proposed by Tang and Othmer, 1994b, Biophys. J. 67, 2223–2235, for calcium-induced calcium release. We show that the spatial localization of these channels near the T-tubules has a significant effect on both wave propagation and the onset of oscillations in this system. Spatial localization provides a possible explanation for the differing effects of various experimental protocols on the system’s ability to propagate a traveling wave. Supported in part by NIH Grant GM29123.     

Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64]. An erratum to this article is available at .     

Gender-dependent effects of selenite on the perfused rat heart

Gender differences are related to the manner in which the heart responds to chronic and acute stress conditions of physiological and pathological nature. Depending on dose, sodium selenite acts as an antioxidant proven to have beneficial effects in several pathological conditions G. Drasch, J. Schopfer, and G. N. Schrauzer, Selenium/cadmium ratios in human prostates: indicators of prostate cancer risk of smokers and non-smokers, and relevance to the cancer protective effects of selenium,Biol. Trace Element Res. 103(2), 103–107 (2005); R. G. Kasseroller and G. N. Schrauzer, Treatment of secondary lymphedema of the arm with physical decongestive therapy and sodium selenite: a review,Am. J. Ther. 7(4), 273–279 (2000); G. N. Schrauzer, Anticarcinogenic effects of selenium,Cell. Mol. Life Sci. 57(13–14), 1864–1873 (2000); I. S. Palmer and O. E. Olson, Relative toxicities of selenite and selenate in the drinking water of rats,J. Nutr. 104(3), 306–314 (1974). To date, little is known about the gender-dependent direct effects of toxic doses of selenite on electrophysiology of the cardiovascular system H. A. Schroeder and M. Mitchener, Selenium and tellurium in rats: effect on growth, survival and tumors,J. Nutr. 101(11), 1531–1540 (1971); G. N. Schrauzer, The nutritional significance, metabolism and toxicology of selenomethionine,Adv. Food Nutr. Res. 47, 73–112 (2003). In the present study, the effects of in vitro toxic concentrations of sodium selenite ranging from 10^-6 M to 10^-3 M were tested on both male and female rat heart preparations. The toxic effects seen in an electrocardiogram and left ventricular pressure were dose and sex dependent at most of the tested concentrations. The present study clearly shows that at toxic doses, stress conditions are induced by selenite, resulting in gender-dependent modifications of the heart function. This modification is more pronounced in the contraction cascade of female rats. Males, on the other hand, had been much more affected in excitation-related parameters.     

Models for reserpine effects on normal and brain-operated animals on a continuous reinforcement bar pressing schedule

Mathematical models are presented for the relationship between changes in the bar pressing rate of albino rats on a continuous reinforcement schedule and alterations of the brain amines norepinephrine and serotonin induced by electrolytic lesions in the hypothalamus and by the drug, reserpine. The discussion utilizes central concepts from some of the more prominent hypotheses on the relation of the amines to behavior and for the potentiation effects of the drug by the lesion (A. Carlsson, 1964. “Functional Significance of Drug-induced Changes in Brain Monoamine Levels.” InBiogenic Amines, eds. H. E. Himwich and W. A. Himwich, New York: Elsevier, 9–27; B. B. Brodie and E. Costa, 1962. “Some Current Views on Brain Monoamines.”Psychopharm. Service Centre Bull.,2, 1–25.) Laboratory data (F. Grabarits and J. Harvey, 1966. “The Effects of Reserpine on Behavior and on Brain Concentrations of Serotonin and Norepinephrine in Control Rats and Rats with Hypothalamic Lesions.”J. Pharm.,153, 403–411) are interpreted, and suggestions for further experimental and theoretical work are made.     

Residue Specific Ribose and Nucleobase Dynamics of the cUUCGg RNA Tetraloop Motif by MNMR 13C Relaxation

The dynamics of the nucleobase and the ribose moieties in a 14-nt RNA cUUCGg hairpin-loop uniformly labeled with ¹³C and ¹⁵N were studied by ¹³C spin relaxation experiments. R₁, R_1ρ and the ¹³C-{¹H} steady-state NOE of C₆ and C_1′ in pyrimidine and C₈ and C_1′ in purine residues were obtained at 298 K. The relaxation data were analyzed by the model-free formalism to yield dynamic information on timescales of pico-, nano- and milli-seconds. An axially symmetric diffusion tensor with an overall rotational correlation time τ_c of 2.31±0.13 ns and an axial ratio of 1.35±0.02 were determined. Both findings are in agreement with hydrodynamic calculations. For the nucleobase carbons, the validity of different reported ¹³C chemical shift anisotropy values (Stueber, D. and Grant, D. M., 2002 J. Am. Chem. Soc. 124, 10539–10551; Fiala et al., 2000 J. Biomol. NMR 16, 291–302; Sitkoff, D. and Case, D. A., 1998 Prog. NMR Spectroscopy 32, 165–190) is discussed. The resulting dynamics are in agreement with the structural features of the cUUCGg motif in that all residues are mostly rigid (0.82 < S² < 0.96) in both the nucleobase and the ribose moiety except for the nucleobase of U7, which is protruding into solution (S² = 0.76). In general, ribose mobility follows nucleobase dynamics, but is less pronounced. Nucleobase dynamics resulting from the analysis of ¹³C relaxation rates were found to be in agreement with ¹⁵N relaxation data derived dynamic information (Akke et al., 1997 RNA 3, 702–709). Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users.     

Two remarks on the mathematical biology of automobile driving

In connection with a series of previous papers by this author (Bulletin of Mathematical Biophysics,21, 299–308, 375–385;22, 257–262, 263–267;23, 19–29;24, 319–325) results obtained by A. Crawford (Economics 5, 417–428) on the effects of irrelevant lights on reaction times toward a given light stimulus are discussed. The conclusions from a previous paper of this author (Bulletin of Mathematical Biophysics,23, 19–29) are elaborated.     

On the quantitative relationship between survival rate of larvae and their food supply

The survival rate of fishes in their earlier stages of development and the influencing factors present one of the most fundamental problems of fish population dynamics. After I. Hjort's (Cons. L.'explor. Ner.,20, 3–228, 1914) work, there have been many investigators in this field and there is no doubt about the very important role of ova and larvae mortality in the fate of a given fish generation. Less clear are the ideas concerning factors determining the high mortality of fishes in their earlier stages of development; especially the factor of food supply of larvae during the period of transition to exogenic nutrition. The value of this factor has been estimated differently from different points of view. For example, R. J. H. Beverton and S. J. Holt (On the Dynamics of Exploited Fish Population, 1957) have given to the food supply factor its deserved importance. On the other hand, T. V. Dekhnik (Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960;Ibid.,14, 222–243, 1961) has proved in her investigations that at least for pelagic larvae of Black Sea fishes there is an excessive amount of food, and that therefore food cannot play an important role in larva survival. Not wanting to stop to review the literature of the problem (see Dekhnik,Trudy Sevastopolskoi Biologicheskoi Stantsii,13, 216–244, 1960), we will only remark that the problem as a whole needs further investigation. Not only new data are needed, but also methods for following up analysis have to be worked out.     

Human trachea primary epithelial cells express both sialyl(α2-3)Gal receptor for human parainfluenza virus type 1 and avian influenza viruses, and sialyl(α2-6)Gal receptor for human influenza viruses

We reported previously that the dominant receptors of influenza A and B viruses, and human and murine respiroviruses, were sialylglycoproteins and gangliosides containing monosialo-lactosamine type I-and II-residues, such as sialic acid-α2-3(6)-Galβ1-3(4)-GlcNAcβ1-. In addition, the Siaα2-3Gal linkage was predominantly recognized by avian and horse influenza viruses, and human parainfluenza virus type 1 (hPIV-1), whereas the Siaα2-6Gal linkage was mainly recognized by human influenza viruses (Paulson JC in “The Receptors' [Conn M Ed] 2, 131–219 (1985); Suzuki Y, Prog Lipid Res 33, 429–57 (1994); Ito T, J Virol 73, 6743–51 (2000); Suzuki Y, J Virol 74, 11825–31 (2000); Suzuki T, J. Virol 75, 4604–4613 (2001); Suzuki Y, Biol. Pharm. Bull. 28, 399–408 (2005)). To clarify the distribution of influenza virus receptors on the human bronchial epithelium cell surface, we investigated a primary culture of normal human bronchial epithelial (NHBE) cells using two types of lectin (MAA and SNA), which recognize sialyl linkages (α2-3 and α2-6), using fluorescence-activated cell-sorting analysis. The results showed that both α2-3- and α2-6-linked Sias were expressed on the surface of primary human bronchial epithelial cells. The cells infected by hPIV-1 bound to MAA, confirming that cells targeted by hPIV-1 have α2-3-linked oligosaccharides. We also compared the ability of hPIV-1 and human influenza A virus to infect primary human bronchial epithelial cells pre-treated with Siaα2-3Gal-specific sialidase from Salmonella typhimurium. No difference was observed in the number of sialidase pre-treated and non-treated cells infected with human influenza A virus, which binds to Siaα2-6Gal-linked oligosaccharides. By contrast, the number of cells infected with hPIV-1 decreased significantly upon sialidase treatment. Thus, cultured NHBE cells showed both α2-3-linked Sias recognized by hPIV-1 and avian influenza virus receptors, and α2-6-linked Sias recognized by human influenza virus receptors.     

Memory capacity in large idiotypic networks

Many models of immune networks have been proposed since the original work of Jerne [1974,Ann. Immun. (Inst. Pasteur) 125C, 373–389]. Recently, a limited class of models (Weisbuchet al., 1990,J. theor. Biol. 146, 483–499) have been shown to maintain immunological memory by idiotypic network interactions. We examine generalizations of these models when the networks are both large and highly connected to study their memory capacity, i.e. their ability to account for immunization to a large number of random antigens. Our calculations show that in these minimal models, random connectivities with continuously distributed affinities reduce the memory capacity to essentially nil.     

On the number of balanced signed graphs

The classical enumeration theorem of Pólya (Acta Math.,68, 145–254, 1937) is applied to a modified version of Harary’s (Pacific J. Math.,8, 743–755, 1958) generating functions for counting bicolored graphs to derive a counting function for the number of balanced signed graphs. Methods for computing these counting polynomial functions are discussed.     

Structure and spin density of ferric low-spin heme complexes determined with high-resolution ESEEM experiments at 35 GHz

The wide use of the heme group by nature is a consequence of its unusual “electronic flexibility.” Major changes in the electronic structure of this molecule can result from small perturbations in its environment. To understand the way the electronic distribution is dictated by the structure of the heme site, it is extremely important to have methods to reliably determine both of them. In this work we propose a way to obtain this information in ferric low-spin heme centers via the determination of g, A, and Q tensors of the coordinated nitrogens using electron spin echo envelope modulation experiments at Q-band microwave frequencies. The results for two bisimidazole heme model complexes, namely, PPIX(Im)₂ and CPIII(Im)₂, where PPIX is protoporphyrin IX, CPIII is coproporphyrin III, and Im is imidazole, selectively labeled with ¹⁵N on the heme or imidazole nitrogens are presented. The planes of the axial ligands were found to be parallel and oriented approximately along one of the N–Fe–N directions of the slightly ruffled porphyrin ring (approximately 10°). The spin density was determined to reside in an iron d orbital perpendicular to the heme plane and oriented along the other porphyrin N–Fe–N direction, perpendicular to the axial imidazoles. The benefit of the method presented here lies in the use of Q-band microwave frequencies, which improves the orientation selection, results in no/fewer combination lines in the spectra, and allows separation of the contributions of hyperfine and quadrupole interactions due to the fulfillment of the exact cancellation condition at g _Z and the possibility of performing hyperfine decoupling experiments at the g _X observer position. These experimental advantages make the interpretation of the spectra straightforward, which results in precise and reliable determination of the structure and spin distribution.     

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.     

Persistence and global stability in a predator-prey model consisting of three prey genotypes with fertility differences

A model of a predator-prey interaction, where the prey population consists of three genotypes with random mating and continuous, nonlinear birth and death processes with fertility differences, is proposed. Sufficiency conditions giving the existence of a globally stable equilibrium on one of the coordinate planes are given. This extends results of Freedman and Waltman [J. Math. Biol. 6, 367–374 (1978) andRocky Mountain J. Math. 12, 779–784 (1982)]. In addition, conditions are derived which guarantee the persistence of all components of the populations. Research in part is from a Ph.D. thesis submitted to the Faculty of Graduate Studies and Research of the University of Alberta. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, grant no. NSERC A4813.     

Comparison of surface potentials due to several singularity representations of the human heart

In electrocardiography the electrical potentials due to the heart actions can be analyzed by assuming the human body to be a conductor of homogeneous medium and the heart to be a combination of singularities within it. For a spherical conductor the “interior sphere theorem” of G. Ludford, J. Martinek, and G. Yeh (Proc. Cambridge Phil. Soc.,51, 389–93, 1955) renders potential expressions due to any singularity. For a conductor of prolate spheroidal shape the potential expressions due to a source-sink pair and a general dipole have been given by J. R. Wait (Jour. App. Physics,24, 496–97, 1953) and the authors (paper at the Conference on the Electrophysiology of the Heart, Feb. 16–17, 1956, in New York, to appear in theAnn. N. Y. Acad. Sciences) respectively. (A theorem which applies to any singularity inside a prolate or oblate spheroid will be published by the authors soon). This paper presents numerical and graphical results of potentials on the surfaces of a prolate spheroid and a sphere due to source-sink pairs and dipoles of several locations and directions and compares the various representations. A discussion on the judicious choice of heart models concludes the paper. This investigation was supported by The National Heart Institute under a research grant H-2263.     

Mathematical biology of automobile driving: III

In a previous paper (Bull. Math. Biophysics,22, 257–262, 1960), an expression for the probability that a car jumps off a road as a function of the speed and the size of the car was derived mostly from geometric and kinematic considerations, introducing only the reaction time as a biological parameter. In subsequent papers (Bull. Math. Biophysics,29, 181–186, 187–188, 1967) a more detailed study was made of the exact shape of the tracking curve of the car which involved several biological parameters of the driver. In the present paper the results of the previous studies are combined, and a more general equation for the probability of jumping off the road is obtained. This probability, as in the earlier study, increases with the speedv, widths _o and lengthl _o of the car, and decreases with widths of the lane. However, this probability also depends on several parameters which characterize the psychobiological constitution of the driver. Unpublished experiments by Ehrlich, which corroborate the general conclusions, are briefly described.     

The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release

The fire-diffuse-fire model provides an idealized model of Ca²⁺ release within living cells. The effect of calcium pumps, which drive Ca²⁺ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer et al. [Keizer, J. E., G. D. Smith, S. Ponce Dawson and J. Pearson (1998). Saltatory propagation of Ca²⁺ waves by Ca²⁺ sparks. Biophys. J. 75, 595–600.]. For travelling waves, in which release events occur sequentially, we construct the speed of waves in terms of the time-scale at which pumps operate. An immediate consequence of this analysis is that the inclusion of calcium pumps leads to multiple solutions. A linear stability analysis determines those solution branches in parameter space which are stable. Numerical continuation is used to provide explicit examples of the bifurcation diagrams of the speed of waves as a function of physiologically significant system parameters.     

Some comments on re-estalishability

The present note consists of two separate but related parts. In the first, a new graphtheoretic proof is presented that an (ℳ,R)-system must always contain a nonreestablishable component. The second considers some questions concerning the relation between re-establishability and the time-lag structure in (ℳ,R)-systems. It is supposed that the reader is familiar with the terminology of the author's previous work on (ℳ,R)-systems, particularly R. Rosen,Bull. Math. Biophysics,20, 245–260, 1958.     

Two definitions of stability of equilibria

In classical physics the stability of an equilibrium requires that any, even infinitesimal, displacement from the configuration of equilibrium results in forces which tend to restore the original equilibrium configuration. In case of several stable equilibrium configurations, the height of the threshold, which must be exceeded by the deviarion from the stable equilibrium in order to bring the configuration into another stable equilibrium is taken as a measure of stability of the first configuration. In quantum mechanics, and in the recent work of I. Baianu, S. Comorosan and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968;31, 59–70, 1969;32, 539–561, 1970) on organismic supercategories, preference is given to take, as ameasure of the degree of stability of a configuration, or of a “state”, the length of time during which the system remains in that configuration. It is shown that under rather general conditions the two criteria are equivalent.