Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Approximation method for electrolyte diffusion through a fixed charge medium

The diffusion of a binary salt through media containing uniformly dispersed immobile ions is formally equivalent to the diffusion of a nonelectrolyte with a concentration-dependent diffusion coefficient. This permits the application of a quasi-steady-state approximation, developed in a previous paper (Bull. Math. Biophysics,21, 19–32, 1959), to problems of transient diffusion. Free diffusion across an interface involving a discontinuity in immobile ion concentration is considered and expressions for the salt concentration and electrical potential at any point are developed. The dependence of electrical potential on the fixed charge density, ionic mobilities, as well as on the boundary concentrations is illustrated by a numerical example. This particular example is suggested by experiments on connective tissue.     

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.     

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.     

Functional characterization of a LAHC sucrose transporter isolated from grape berries in yeast

Large amounts of sugar are imported into grape berries from source leaves during ripening, and sucrose transporters play a key role during this process. In this study, a putative grape sucrose transporter gene VvSUC27, primarily expressed in sink tissue, was transformed into a yeast strain to characterize its function as a sucrose transporter. Sucrose was taken up by yeast transformed with VvSUC27 at an optimum pH of 4.0–5.0 and a K _m of 8.0–10.5 mM, indicating VvSUC27 is a LAHC (low-affinity/high-capacity) sucrose transporter. The ability of sucrose uptake in transformed yeast was activated by monosaccharides and inhibited by maltose and DEPC. Ya Li Zhang and Qing Yong Meng contributed equally to the paper.     

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 .     

Mathematical theory of biological periodicities II. Effect of active transport

A previous study (Bull. Math. Biophysics,30, 735–749) is generalized to the case of active transport, which acts together in general with ordinary diffusion. The basic results obtained are the same except for an additional important conclusion. In principle it is possible to obtain sustained oscillations even when the secretions of the different glands do not affect the rates of formation or decay of each other at all, but affect the “molecular pumps,” which are responsible for the active transports in various parts of the system. Thus no biochemical interactions need necessarily take place between then-metabolites to make sustained oscillations possible in principle. This is an addition to a previous finding (Bull. Math. Biophysics,30, 751–760) that due to effects of the secreted hormones on target organs, non-linearity of biochemical interactions is not needed for production of sustained oscillations.     

Phytoplankton blooms and fish recruitment rate: Effects of spatial distribution

We consider the spatio-temporal dynamics of a spatially-structured generalization of the phytoplankton-zooplankton-fish larvae model system proposed earlier (Biktashev et al., 2003, J. Plankton Res. 5, 21–33; James et al., 2003, Ecol. Model. 160, 77–90). In contrast to Pitchford and Brindley (2001, Bull. Math. Biol. 63, 527–546), who were concerned with small scale patchiness (i.e., 1–10m), on which the (stochastic) raptorial behaviour of individual larvae is important, we address here the much larger scale ‘patchy’ problems (i.e., 10–100 km), on which both larvae and plankton may be regarded as passive tracers of the fluid motion, dispersed and mixed by the turbulent diffusion processes. In particular, we study the dependence of the fish recruitment on carrying capacities of the plankton subsystem and on spatio-temporal evolution of that subsystem with respect to the larvae hatching site(s). It is shown that the main features found both in the nonstructured and age-structured spatially uniform models are observed in the spatially structured case, but that spatial effects can significantly modify the overall quantitative outcome. Spatial patterns in the metamorphosed fish distribution are a consequence of quasi-local interaction of larvae with plankton, in which the dispersion of larvae by large scale turbulent eddies plays little part due to the relatively short timescale of the larvae development. As a result, in a strong phyto/zooplankton subsystem, with fast reproduction rate and large carrying capacity of phytoplankton and high conversion ratio of zooplankton, recruitment success depends only on the localization and timing of the hatching with respect to the plankton patches. In a weak phyto/zooplankton system, with slow reproduction rate and small carrying capacity of phytoplankton and low conversion ratio of zooplankton, the larvae may significantly influence the evolution of the plankton patches, which may lead to nontrivial cooperative effects between different patches of larvae. However, in this case, recruitment is very low.     

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.     

Determination of <Superscript>13</Superscript>C CSA Tensors: Extension of the Model-independent Approach to an RNA Kissing Complex Undergoing Anisotropic Rotational Diffusion in Solution

Chemical shift anisotropy (CSA) tensor parameters have been determined for the protonated carbons of the purine bases in an RNA kissing complex in solution by extending the model-independent approach [Fushman, D., Cowburn, D. (1998) J. Am. Chem. Soc. 120, 7109–7110]. A strategy for determining CSA tensor parameters of heteronuclei in isolated X–H two-spin systems (X = ¹³C or ¹⁵N) in molecules undergoing anisotropic rotational diffusion is presented. The original method relies on the fact that the ratio κ₂=R₂^auto/R₂^cross of the transverse auto- and cross-correlated relaxation rates involving the X CSA and the X–H dipolar interaction is independent of parameters related to molecular motion, provided rotational diffusion is isotropic. However, if the overall motion is anisotropic κ₂ depends on the anisotropy D_||/D_⊥ of rotational diffusion. In this paper, the field dependence of both κ₂ and its longitudinal counterpart κ₁=R₁^auto/R₁^cross are determined. For anisotropic rotational diffusion, our calculations show that the average κ_av = 1/2 (κ₁+κ₂), of the ratios is largely independent of the anisotropy parameter D_||/D_⊥. The field dependence of the average ratio κ_av may thus be utilized to determine CSA tensor parameters by a generalized model-independent approach in the case of molecules with an overall motion described by an axially symmetric rotational diffusion tensor.     

A note on the development of organismic sets

It is suggested that the development of organismic sets is governed not by the maximalization of the integral survival value, as suggested previously (Bull. Math. Biophysics,28, 283–308, 1966;29, 139–152, 1967;30, 163–174, 1968), but by maximizing the number of new relations which appear as an organismic set develops.     

Linearized oscillations in population dynamics

A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.     

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.     

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.     

Mathematical theory of biological periodicities: Formulation of then-body case

In a series of papers, L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956;19, 9–18, 1957) showed that a non-linear biochemical interaction between the anterior pituary gland and the thyroid gland may result under certain conditions in sustained periodical oscillations of the rates of production and of the blood level of the thyrotropic and of the thyroid hormone. They treated the systems, however, as a homogeneous one. N. Rashevsky (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1965;Bull. Math. Biophysics,29, 395–401, 1967) generalized the above results by taking into account the histological structures of the two glands as well as the diffusion coefficients and permeabilities of cells involved. The present paper is the first step toward the theory of interaction of any numbern of glands or, more generally,n components. The differential equations which govern the behavior of such a system represent a system of2n ²+n non-linear first order ordinary equations and involve a total of 7 n ²+3n parameters of partly histological, partly biochemical nature. The requirements of the existence of sustained oscillations demand 4n ²+2n+2 inequalities between those 7n ²+3n parameters.     

Threonine diffusion and threonine transport in Corynebacterium glutamicum and their role in threonine production

Transmembrane threonine fluxes (i.e., uptake, diffusion, and carrier-mediated excretion) all contribut-ing to threonine production by a recombinant strain of Corynebacterium glutamicum, were analyzed and quantitated. A threonine-uptake carrier that transports threonine in symport with sodium ions was identified. Under production conditions (i.e., when internal threonine is high), this uptake system catalyzed predominantly threonine/threonine exchange. Threonine export via the uptake system was excluded. Threonine efflux from the cells was shown to comprise both carrier-mediated excretion and passive diffusion. The latter process was analyzed after inhibition of all carrier-mediated fluxes. Threonine diffusion was found to proceed with a first-order rate constant of 0.003 min^–1 or 0.004 μl min^–1 (mg dry wt.)^–1, which corresponds to a permeability of 8 × 10^–10 cm s^–1. According to this permeability, less than 10% of the efflux observed under optimal conditions takes place via diffusion, and more than 90% must result from the activity of the excretion carrier. In addition, the excretion carrier was identified by (1) inhibition of its activity by amino acid modifying reagents and (2) its dependence on metabolic energy in the form of the membrane potential. Activity of the excretion system depended on the membrane potential, but not on the presence of sodium ions. Threonine export in antiport against protons is proposed. Received: 25 August 1995 / Accepted: 18 October 1995     

The inconsistencies of present-day mathematical-physical methods pertaining to current generators in volume conductors used in electrocardiography

The present-day practices of electrocardiography and vectorardiography are based upon the theory that the surface potential differences can be assumed to be due to a single dipole inside the body. It is shown in this paper that a dipole cannot account for all the surface potentials due to realistic current generators, and hence the determination of the current generator from surface potential measurements based upon such a theory will lead to inconsistent representations of the heart for one and the same subject. To demonstrate this point two eccentric dipoles of different strengths and locations representing two muscle fibers are taken to be the current generator in a homogeneous spherical conductor. The exact surface potentials are then expressed by means of the “interior sphere theorem” of the authors. With these expressions the magnitude, direction, and location of the resultant dipole are determined by the method of D. Gabor and C. V. Nelson (J. App. Physics,25, 413–16, 1954). The surface potentials due to this resultant dipole are again exactly expressed by means of the “interior sphere theorem” and compared with those due to the eccentric dipoles assumed. It can be seen that the differences can be considerable. It is suggested that the multipole model of the authors (Bull. Math. Biophysics,20, 203–16, 1958) be used as a more accurate and the only unique representation of the heart. This investigation was supported by the National Heart Institute under a research grant H-2263(c).     

Urea uptake and urease activity in Corynebacterium glutamicum

When Corynebacterium glutamicum is grown with a sufficient nitrogen supply, urea crosses the cytoplasmic membrane by passive diffusion. A permeability coefficient for urea diffusion of 9 × 10^–7 cm s^–1 was determined. Under conditions of nitrogen starvation, an energy-dependent urea uptake system was synthesized. Carrier-mediated urea transport was catalyzed by a secondary transport system linked with proton motive force. With a K _m for urea of 9 μM, the affinity of this uptake system was much higher than the affinity of urease towards its substrate (K _m approximately 55 mM urea). The maximum uptake velocity depended on the expression level and was relatively low [2–3.5 nmol min^–1 (mg dry wt.)^–1]. Received: 11 August 1997 / Accepted: 2 December 1997     

Contributions to relational biology

The principle of biotopological mapping (Rashevsky, 1954,Bull. Math. Biophysics,16, 317–48) is given a generalized formulation, as the principle of relational epimorphism in biology. The connection between this principle and Robert Rosen’s representation of organisms by means of categories (1958,Bull. Math. Biophysics,20, 317–41) is studied. Rosen’s theory of (M,R)-systems, (1958,Bull. Math. Biophysics,20, 245–60) is generalized by dropping the assumption that only terminalM _i components are sending inputs into theR _i components. It is shown that, if the primordial organism is an (M,R)-system, then the higher organisms, obtained by a construction well discussed previously (1958,Bull. Math. Biophysics,20, 71–93), are also (M,R)-systems. Several theorems about such derived (M,R)-systems are demonstrated. It is shown that Rosen’s concept of an organism as a set of mappings throws light on phenomena of synesthesia and also leads to the conclusion that Gestalt phenomena must occur not only in the fields of visual and auditory perception but in perceptions of any modality.