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.     

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.     

Influence of magnetostimulation therapy on rheological properties of blood in neurological patients

ABSTRACT

The aim of the study is to test the influence of in vivo magnetostimulation on the rheological properties of blood in neurological patients. Blood circulation in the body depends both on the mechanical properties of the circulatory system and on the physical and physicochemical properties of blood. The main factors influencing the rheological properties of blood are as follows: hematocrit, plasma viscosity, whole-blood viscosity, red cells aggregability, deformability, and the ability of red cells to orient in the flow. The blood samples were collected from neurological patients with pain. Blood samples were collected twice from each patient, that is, before the magnetostimulation and immediately after the therapy. For each blood sample, the hematocrit value was measured using the standard method. Plasma viscosity and whole-blood viscosity were measured by means of a rotary-oscillating rheometer Contraves LS40. Magnetic field was generated by the instrument Viofor JPS® and the magnetostimulation treatments were performed using M1P2 and M1P3 programs. The analysis of the results included estimation of the hematocrit value (Hct), plasma viscosity (η_p), whole-blood viscosity and rheological parameters of Quemada’s model: k₀, k_∞, γ′_c. Plasma viscosity values were obtained from the shear rate dependence of shear stress using the linear regression method. The results obtained in the study suggest that the blood rheological properties change in accord with applied magnetostimulation program.     

Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.     

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.     

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.     

A derivation of a rote learning curve from the total uncertainty of a task

The derivation of H. D. Landahl’s learning curve (1941,Bull. Math. Biophysics,3, 71–77) from a single information-theoretical assumption obtained previously (Rapoport, 1956,Bull. Math. Biophysics,18, 317–21) is extended to obtain the entire family of such curves with the number of stimuliM (to each of which one ofN responses is to be associated) as a parameter. No additional assumptions are required. The entire family thus appears as a function of a single free parameter,k, all other parameters being experimentally determined. The theory is compared with a set of experiments involving the learning of artificial languages. An alternative quasi-neurological model leading to the same equation is offered.     

Hematocrit is related to age but not to nutritional condition in greater flamingo chicks

We measured the hematocrit from greater flamingo chicks Phoenicopterus roseus over 4 years to test whether this blood parameter was related to the nutritional condition of chicks, as there are controversial results on whether hematocrit may be used as an index of body condition. We also tested whether hematocrit increased with chick age, as there would be an age-related increase of oxygen demand due to exercising. We found no evidences that hematocrit was related neither to the nutritional nor to the body condition of chicks. Hematocrit increased with chick age, which may be related to the increased requirements of chicks for oxygen delivery during development.     

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.     

Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters

A mathematical technique is described that relates detection model parameters to stimulus magnitude and experimental probability of detection. The normalizing transform is used to make the response statistics approximately Gaussian. Conventional probit analysis is then applied. From measurements at M stimulus levels, a system of M equations is solved and estimates of M unknown parameters of the detection model are obtained. The technique is applied to a threshold vision model based on additive and multiplicative Poisson noise. Results are obtained for the parameter estimates for individual subjects, and for the standard deviation of the estimates, for various values of the stimulus energy and number of trials. A frequency-of-seeing experiment is performed using a point-source stimulus that randomly assumes 3 energy levels with 200 trials per level. With a central efficiency of 50%, the estimated ocular quantum efficiency for our four subjects lies between 12% and 23%, the average dark count at the retina lies between 8 and 36 counts, and the threshold count for our (low falsereport rate) data lies between 11 and 32. The theoretical results reduce to those obtained by Barlow (J. Physiol. London 160, 155–168, 1962), in the absence of dark light and multiplication noise.This work was supported by the National Science Foundation     

On multipole theory in electrocardiography

This paper continues a comparison of the Taylor series and spherical harmonic forms of multipole representations initiated by Yeh (Bull. Math. Biophysics,24, 197–207, 1962). It is shown that while transformations from Taylor series form into spherical harmonic form is always possible, the inverse cannot be accomplished as suggested by Yeh; corrected transformation equations are given. It is also shown that direct measurement of Taylor coefficients, as outlined in Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), is actually not possible. Accordingly, only the spherical harmonic coefficients can be determined by measurement of surface potentials, as in electrocardiography.     

Length‐weight relationships of three cyprinid fishes from headwater of the Nujiang River,China

Length‐weight relationships (LWR) for three cyprinid fish species collected from the headwaters of the Nujiang River in Tibet, China, were determined. The values of parameter b in the LWR equations were estimated as 2.54 for Schizopygopsis thermalis, 3.3 for Ptychobarbus kaznakovi, and 2.74 for Schizothorax nukiangensis, respectively. These are the first LWR records for the three species.     

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.     

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.     

Accurate kinetic parameter estimation during progress curve analysis of systems with endogenous substrate production

We have identified an error in the published integral form of the modified Michaelis–Menten equation that accounts for endogenous substrate production. The correct solution is presented and the error in both the substrate concentration, S, and the kinetic parameters V_m, K_m, and R resulting from the incorrect solution was characterized. The incorrect integral form resulted in substrate concentration errors as high as 50% resulting in 7–50% error in kinetic parameter estimates. To better reflect experimental scenarios, noise containing substrate depletion data were analyzed by both the incorrect and correct integral equations. While both equations resulted in identical fits to substrate depletion data, the final estimates of V_m, K_m, and R were different and K_m and R estimates from the incorrect integral equation deviated substantially from the actual values. Another observation was that at R = 0, the incorrect integral equation reduced to the correct form of the Michaelis–Menten equation. We believe this combination of excellent fits to experimental data, albeit with incorrect kinetic parameter estimates, and the reduction to the Michaelis–Menten equation at R = 0 is primarily responsible for the incorrectness to go unnoticed. However, the resulting error in kinetic parameter estimates will lead to incorrect biological interpretation and we urge the use of the correct integral form presented in this study. Biotechnol. Bioeng. 2011;108: 2499–2503. © 2011 Wiley Periodicals, Inc.     

Window automata analysis of population dynamics in the immune system

A formalism based on window automata is proposed as a method to analyse complex population dynamics. The method is applied to a model of the immune network (Weisbuch, G.et al., 1990.J. theor. Biol. 146, 483–499), and used to predict which attractor the system reaches after antigenic stimulation, as a function of the parameters. The attractors of the dynamics are interpreted in terms of immune conditions such as vaccination or tolerance. Scaling laws that define the regimes in the parameter space corresponding to the specific attractor reached under antigenic stimulation are derived.     

Stenogrammitis, a new genus of grammitid ferns segregated from Lellingeria (Polypodiaceae)

Stenogrammitis , a new genus of grammitid ferns, is segregated from Lellingeria based on morphological and molecular evidence. It differs from Lellingeria by linear leaves usually less than 5 mm wide, clathrate iridescent rhizome scales that are glabrous except for a single apical cilium, veins unbranched and only one per segment, fertile veins usually with the dark sclerenchyma visible beneath the sporangia, and x = 33. In contrast, Lellingeria has broader laminae, veins pinnate within the segments, and fertile veins not visible beneath the sporangia. Melpomene, which is sister to Stenogrammitis and Lellingeria, differs from those two genera by reddish setae on the leaves and rhizome scales papillate at the apex. Some species of Stenogrammitis are also distinctive by hemidimorphic laminae that have the fertile portion less dissected than the sterile. Stenogrammitis is pantropical and currently comprises 24 species, 12 of which occur in the Neotropics, six in Africa, four in Madagascar, and two on Pacific Islands. New combinations are made for Stenogrammitis aethiopica, S. anamorphosa, S. ascensionensis, S. boivinii, S. delitescens, S. jamesonii, S. hartii, S. hellwigii, S. hildebrandtii, S. limula, S. luetzelburgii, S. myosuroides, S. nutata, S. oosora, S. paucipinnata, S. prionodes, S. pumila, S. ruglessii, S. rupestris, S. saffordii, S. strangeana, S. tomensis, S. subcoriacea, and S. wittigiana. Lectotypifications are made for Grammitis muscosa, Polypodium itatiayense, P. oosorum var. micropecten, P. serrulatum forma major, P. serrulatum forma minor, S. luetzelburgii, S. myosuroides, and S. wittigiana. Illustrations are presented for the diagnostic characters of the genus, as well as a map with the geographical distribution.     

Some remarks on the central nervous system

It is pointed out that the successes obtained in the mathematical biology of the central nervous system are based mostly on a number of more or less complicated neuronic circuit models, each inventedad hoc for the purpose of explaining a given phenomenon. The individual models remain disconnected from each other, however, and the unity of the CNS is not apparent. (Rashevsky,Mathematical Biophysics, 3rd Edition, Vol. II, 1960. New York, Dover Publications, Inc.) Some “field theories” of the CNS, as for example that of Griffith (Bull. Math. Biophysics,25, 111–120, 1963;27, 187–195, 1965), give more expression to this unity but lose in the explanation of specific phenomena. The present paper starts with the picture thatevery neuron in the brain isdirectly or indirectly affected to some extent byevery other neuron. This leads to a system of equations with a very large number of variables. Such a system can be replaced in the limiting case by an integral equation of the first kind. At least two specific results can be obtained with this approach and suggestions for further improvement are made.     

Some possible quantitative aspects of a neurophysiological model of schizophrenias

In continuation of a previous paper (Bull. Math. Biophysics,26, 167–185, 1964) simple equations are derived for the rate of development of schizophrenia as a function of some psychobiological parameters of the individual and of an index which characterizes the frequency of traumatic experiences of the individual. A clue to the understanding of why schizophrenia is more likely to develop at an early adult age is thus provided. Formerly of the Committee on Mathematical Biology, The University of Chicago.     

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.