Externally driven countercurrent multiplication in a mathematical model of the urinary concentrating mechanism of the renal inner medulla

Substitution of measured permeabilities into mathematical models of the concentrating mechanism of the renal inner medulla yields less than the known urine osmolalities. To gain a better understanding of the mechanism we analyse a model in which a force of unspecified origin [expressed as fraction, ɛ, of entering descending thin limb (DTL) concentration] drives fluid from DTL to interstitial vascular space (CORE), thus concentrating the solution in DTL. When flow in the DTL reverses at the hairpin bend of the loop of Henle, the high solute permeability of ascending thin limb (ATL) permits solute to diffuse into the CORE thus permitting ɛ to be multiplied many-fold. Behavior of the model is described by two non-linear differential equations. In the limit for infinite salt permeability of ATL the two equations reduce to a single equation that is formally identical with that for the Hargitay and Kuhn multiplier, which assumes fluid transport directly from DTL to ATL (Z. Electrochem. Angew. Phys. Chem. 55, 539, 1951). Solutions of the equations describing the model with parameters taken from perfused thin limbs show that urine osmolalities of the order of 5000 mosm L⁻¹ can be generated by forces of the order of 20 mosm L⁻¹. It seems probable that mammals including desert rodents use some variant of this basic mechanism for inner medullary concentration.     

Relative amounts of sialic acid and fucose of amniotic fluid glycoconjugates in relation to pregnancy age

The present knowledge concerning the glycan structures and role of glycoconjugates derived from amniotic fluid is fragmentary and mainly focuses on the individual glycoproteins. The question has arisen as whether the general glycosylation pattern of amniotic fluid glycoconjugates can change with the progression of a normal pregnancy. In the present work we have described the dynamic, quantitative alterations in relative amounts of sialic acid and fucose linked by a variety of anomeric linkages to subterminal oligosaccharide structures of amniotic fluid glycoconjugates in relation to pregnancy age. The analysis was performed in the following groups of amniotic fluids derived from normal pregnancy by lectin dotting method: “2nd trimester” (14–19 weeks), “3rd trimester” (29–37 weeks), “perinatal period” (38–40 weeks) , “delivery at term” (39–41 weeks) and “post date pregnancy” (41–43 weeks). In the “3rd trimester” the amniotic fluid glycoconjugates contained higher relative amounts of glycans terminated by α2-6-linked sialic acid (p < 0.00002) and by α1-6 innermost fucose (p < 0.000001) than those in the 2nd trimester. In contrast, they showed the lower relative amount of fucose linked α1-3 (p < 0.02). At the perinatal period the relative amount of α2-6-linked sialic acid increased (p < 0.03), and it then decreased during delivery (p < 0.02) to the level found in the “3rd trimester” group. In the post date pregnancy all parameters studied increased. The sialyl- and fucosyl-glycotopes of the amniotic fluid glycoconjugates may play an critical role in growth and tissue remodeling of the foetus, as well as may might reflect maturation of a foetus. Additionally, a determination of the glycotope expressions might be helpful in prenatal diagnosis as predictor factors for well being of mother and child.     

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.     

Erythrocyte distribution in arterial blood flow: II. The hypothesis of minimal energy dissipation

In a previous paper (Bull. Math. Biophysics,29, 549–563, 1967) the author derived equations to represent the flow of blood in an artery. It was pointed out that these did not completely characterize the system and that an additional hypothesis was required. The hypothesis of minimal energy dissipation had been thought to imply a central tendency on the part of suspended particles (erythrocytes). It is here shown that if the fluid is non-Newtonian this may not be so.     

An ant’s eye view of culture: propagation of new traditions through triggering dormant behavioural patterns

We consider a previously unknown way of propagation of behavioural traditions in animal communities using hunting in ants as an example. We experimentally revealed that common litter dwelling ants Myrmica rubra effectively hunt jumping prey and the way the hunting behavioural pattern is distributed within ant colonies is rather sophisticated. Comparison of our results with those obtained on vertebrates enables us to suggest that “distributed social learning” plays an important role in spreading new traditions in animal communities: initial performances by a few carriers of an “at once and entirely” available behavioural pattern propagate this pattern among specimens which have only dormant “sketches” of it. Spread of these behaviours in populations is based on relatively simple forms of social learning such as social facilitation which underlies species’ predisposition to learn certain sequences of behavioural acts. To be triggered, carriers of dormant “sketches” of a relevant behavioural pattern should encounter performances of this pattern with sufficient frequency. We call this strategy triggering of dormant behavioural patterns. Integration of behaviour thus takes place not only at the individual level but at the population level as well.     

Properties of New,Long-Wavelength,Voltage-sensitive Dyes in the Heart

Membrane potential measurements using voltage-sensitive dyes (VSDs) have made important contributions to our understanding of electrophysiological properties of multi-cellular systems. Here, we report the development of long wavelength VSDs designed to record cardiac action potentials (APs) from deeper layers in the heart. The emission spectrum of styryl VSDs was red-shifted by incorporating a thienyl group in the polymethine bridge to lengthen and retain the rigidity of the chromophore. Seven dyes, Pittsburgh I to IV and VI to VIII (PGH I-VIII) were synthesized and characterized with respect to their spectral properties in organic solvents and heart muscles. PGH VSDs exhibited 2 absorption, 2 excitation and 2 voltage-sensitive emission peaks, with large Stokes shifts (> 100 nm). Hearts (rabbit, guinea pig and Rana pipiens) and neurohypophyses (CD-1 mice) were effectively stained by injecting a bolus (10–50 μl) of stock solution of VSD (2–5 mM) dissolved in in dimethylsulfoxide plus low molecular weight Pluronic (16% of L64). Other preparations were better stained with a bolus of VSD (2–5 mM) Tyrode’s solution at pH 6.0. Action spectra measured with a fast CCD camera showed that PGH I exhibited an increase in fractional fluorescence, ΔF/F = 17.5 % per AP at 720 nm with 550 nm excitation and ΔF/F = − 6% per AP at 830 nm with 670 nm excitation. In frog hearts, PGH1 was stable with ∼30% decrease in fluorescence and AP amplitude during 3 h of intermittent excitation or 1 h of continuous high intensity excitation (300 W Xe-Hg Arc lamp), which was attributed to a combination of dye wash out > photobleaching > dynamic damage > run down of the preparation. The long wavelengths, large Stokes shifts, high ΔF/F and low baseline fluorescence make PGH dyes a valuable tool in optical mapping and for simultaneous mapping of APs and intracellular Ca²⁺.     

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.     

An Annular Plasmonic Lens Under Illumination of Circularly Polarized Light

In this paper, we consider a circular central aperture surrounded with annular depth-tuned grooves and investigate the beaming effect of the structure under illumination of a circularly polarized (CP) plane wave. As a CP plane wave is equivalent to the superposition of two linearly polarized plane waves (TM and TE) with a phase difference of π/2, the superposition of the electric field intensity, ( | E_x |² + | E_y |² ) \left( {{{\left| {E_x} \right|}^2} + {{\left| {E_y} \right|}^2}} \right) , is observed in the transmission field. In addition, two plasmonic modes are found at the resonant wavelengths λ ₁ and λ ₂ with each consisting of multiple wavelengths. At the wavelength λ ₁ = 420 nm, the significant near-field collimation is formed along the direction z, having a long propagation distance up to 1.75 μm (≈4λ) away from the exit plane of the new plasmonic lens.     

Erythrocyte distribution in arterial blood flow: I. Basic equations and anisotropic effects

Equations are derived for the flow of an anisotropic fluid in a tube. It is argued that these provide a model for arterial blood flow. Particular attention is paid to the effect of radial differences in hematocrit. Sequels to this paper (Bull. Math. Biophysics,29, 565–574; forthcoming, 1967) will respectively demonstrate possible wall-directed forces on the erythrocyte and enlarge on the physiological consequence of hematocrit variations. The present article develops the basic equations and explores the possible role of anisotropic effects in blood flow.     

Ion induced deformation of soft tissue

In this paper the effects of changing the ion concentration in and around a sample of soft tissue are investigated. The triphasic theory developed by Laiet al. (1990,Biomechanics of Diarthrodial Joints, Vol. 1, Berlin, Springer-Verlag) is reduced to two coupled partial differential equations involving fluid ion concentration and tissue solid deformation. These equations are given in general form for Cartesian, cylindrical and spherical geometries. After solving the two equations quantities such as fluid velocity, fluid pressure, chemical potentials and chemical expansion stress may be easily calculated. In the Cartesian geometry comparison is made with the experimental and theoretical work of Myerset al. (1984,ASME J. biomech. Engng,106, 151–158). This dealt with changing the ion concentration of a salt shower on a strip of bovine articular cartilage. Results were obtained in both free swelling and isometric tension states, using an empirical formula to acount for ion induced deformation. The present theory predicts lower ion concentrations inside the tissue than this earlier work. A spherical sample of tissue subjected to a change in salt bath ion concentration is also considered. Numerical results are obtained for both hypertonic and hypotonic bathing solutions. Of particular interest is the finding that tissue may contract internally before reaching a final swollen equilibrium state or swell internally before finally contracting. By considering the relative magnitude, and also variation throughout the time course of terms in the governing equations, an even simpler system is deduced. As well as being linear the concentration equation in the new system is uncoupled. Results obtained from the linear system compare well with those from the spherical section. Thus, biological swelling situations may be modelled by a simple system of equations with the possibility, of approximate analytic solutions in certain cases.     

Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model

More than 20 years after its proposal, Keller and Segel's model (1971,J. theor. Biol.,30, 235–248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherrattet al. (1993,J. theor. Biol.,162, 23–40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

A mathematical model for the inhibition of the multidrug resistance-associated P-glycoprotein pump

An extension of an earlier model of the p170 glycoprotein pump is presented. In an earlier work (Michelson and Slate,Bull. math. Biol. 54, 1023–1038, 1992), the pump was modeled using an energy-dependent model of facilitated diffusion. In this paper we add an inhibitor to the model. New equations are derived which represent either competitive or non-competitive inhibition in the pumping action of the glycoprotein. Numerical simulations were run which provide a response surface (initial loading concentration of inhibitor and its ability to compete with an ideal anti-cancer drug vs a summary measure of cytoplasmic exposure) for each scenario. The importance of the exposure profile, how it is related to ultimate tumor cell survival, and the binding requirements for developing multidrug resistance inhibitors are discussed.     

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 .     

Inferring the Pattern of Spontaneous Mutation from the Pattern of Substitution in Unitary Pseudogenes of Mycobacterium leprae and a Comparison of Mutation Patterns Among Distantly Related Organisms

The pattern of spontaneous mutation can be inferred from the pattern of substitution in pseudogenes, which are known to be under very weak or no selective constraint. We modified an existing method (Gojobori T, et al., J Mol Evol 18:360, 1982) to infer the pattern of mutation in bacteria by using 569 pseudogenes from Mycobacterium leprae. In Gojobori et al.’s method, the pattern is inferred by using comparisons involving a pseudogene, a conspecific functional paralog, and an outgroup functional ortholog. Because pseudogenes in M. leprae are unitary, we replaced the missing paralogs by functional orthologs from M. tuberculosis. Functional orthologs from Streptomyces coelicolor served as outgroups. We compiled a database consisting of 69,378 inferred mutations. Transitional mutations were found to constitute more than 56% of all mutations. The transitional bias was mainly due to C→T and G→A, which were also the most frequent mutations on the leading strand and the only ones that were significantly more frequent than the random expectation. The least frequent mutations on the leading strand were A→T and T→A, each with a relative frequency of less than 3%. The mutation pattern was found to differ between the leading and the lagging strands. This asymmetry is thought to be the cause for the typical chirochoric structure of bacterial genomes. The physical distance of the pseudogene from the origin of replication (ori) was found to have almost no effect on the pattern of mutation. A surprising similarity was found between the mutation pattern in M. leprae and previously inferred patterns for such distant taxa as human and Drosophila. The mutation pattern on the leading strand of M. leprae was also found to share some common features with the pattern inferred for the heavy strand of the human mitochondrial genome. These findings indicate that taxon-specific factors may only play secondary roles in determining patterns of mutation. Electronic Supplementary Material Electronic Supplementary material is available for this article at and accessible for authorised users. [Reviewing Editor:Dr. Dmitri Petrov]     

MCMC for hidden Markov models incorporating aggregation of states and filtering

This paper is concerned with the statistical analysis of single ion channel records. Single channels are modelled by using hidden Markov models and a combination of Bayesian statistics and Markov chain Monte Carlo methods. The techniques presented here provide a straightforward generalization to those in Rosales et al. (2001, Biophys. J., 80, 1088–1103), allowing to consider constraints imposed by a gating mechanism such as the aggregation of states into classes. This paper also presents an extension that allows to consider correlated background noise and filtered data, extending the scope of the analysis toward real experimental conditions. The methods described here are based on a solid probabilistic basis and are less computationally intensive than alternative Bayesian treatments or frequentist approaches that consider correlated data.     

Nonquasineutral current equilibria as elementary structures of plasma dynamics

A study is made of the fundamental features of current filaments with a nonzero electron vorticity Ω _e ≡ B − (c/e) ▿ × p _ee ≠ 0 and the corresponding Lagrangian invariant I _e . Such current structures can exist on spatial scales of up to ω _pi ⁻¹. It is shown that the dissipative stage of the plasma evolution and the violation of Thomson’s theorem on vorticity conservation in an electron fluid are of fundamental importance for the onset of electron current structures. A key role of the screening of electric and magnetic fields at distances on the order of the magnetic Debye radius r _B = B/(4πen _e )—the main property of such current structures in a Hall medium with σB/(en _e c) ≫ 1—is stressed. Since the minimum size of a vortex structure is the London length c/ω _pe , the structures under consideration correspond to the condition r _B > c/ω _pe or B ² > 4πn _e m _e c ², which leads to strong charge separation in the filament and relativistic electron drift. It is demonstrated that the specific energy content in current structures is high at a filament current of 10–15 kA: from 100 J/cm³ at a plasma density of 10¹⁴ cm⁻³ (the parameters of a lightning leader) to 10⁷ J/cm³ for a fully ionized atmospheric-pressure air. Estimates are presented showing that the Earth’s ionosphere, circumsolar space, and interstellar space are all Hall media in which current vortex structures can occur. A localized cylindrical equilibrium with a magnetic field reversal is constructed—an equilibrium that correlates with the magnetic structures observed in intergalactic space. It is shown that a magnetized plasma can be studied by using evolutionary equations for the electron and ion Lagrangian invariants I _e and I _i . An investigation is carried out of the evolution of a current-carrying plasma in a cylinder with a strong external magnetic field and with a longitudinal electron current turned on in the initial stage—an object that can serve as the simplest electrodynamic model of a tokamak. In this case, it is assumed that the plasma conductivity is low in the initial stage and then increases substantially with time. Based on the conservation of the integral momentum of the charged particles and electromagnetic field in a plasma cylinder within a perfectly conducting wall impenetrable by particles, arguments are presented in support of the generation of a radial electric field in a plasma cylinder and the production of drift ion fluxes along the cylinder axis. A hypothesis is proposed that the ionized intergalactic gas expands under the action of electromagnetic forces.