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).     

Eccentric multipole representations of current generators in a spherical volume conductor

In Yeh, Martinek and de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958), a method is presented for determining successively better central multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. This paper generalizes the method such that the multipoles may be located at any chosen point in the conductor. The spherical harmonic expansion is advantageously used and the “interior sphere theorem” of Ludford, Martinek and Yeh (Proc. Cambridge Philos. Soc.,51, 389–93, 1955) makes possible disturbance potential expressions in closed forms. A method for approximate determination of the eccentricity is also presented. In the theory of electrocardiography, the eccentric multipoles can more accurately represent the heart as a current generator with fewer surface potential measurements than the central multipoles. This investigation was supported by The National Heart Institute under Research Grant H-2263(c-4).     

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.     

On multipole representation of current generators

In Yeh, Martinek, and de Beaumont (Bull. Math. Biophysics,20, 203–216, 1958), multipole representations of current generators in a volume conductor are used, based upon the Taylor series expansion of the potential function. In Yeh, (Bull. Math. Biophysics,23, 263–276, 1961) multipole representations of current generators in a spherical volume conductor are used, based upon the spherical harmonic expansion. This paper correlates these two systems of multipole representations so that formulations in terms of one system of the representations may be readily transformed into formulations in terms of the other system. Since the Taylor series representation is more graphic, whereas the spherical harmonic representation is more compact, such a transformation between these two systems of formulations can serve useful purposes in the application of the theory of electrocardiography. This investigation was supported by the National Heart Institute under Research Grant H-2263 (c-4).     

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.     

Multipole representations of current generators in a volume conductor

As far as the potential distribution outside the current generators is concerned, any current source distribution may be replaced by a suitable collection of multipoles. If these current generators lie close to the geometrical center of the volume conductor, a central dipole is a good approximation for potentials at surface points which are at considerable distances from the center. For better accuracy and for points close to the center, additional singularities such as a central quadrupole, a central octopole, etc., should be included. Potential expressions due to such multipoles in a spherical conductor can be obtained in closed forms by means of the “interior sphere theorem”. This paper presents a method for determining successively better multipole representations of the current generators in a homogeneous conducting sphere by measuring surface potentials at a successively increasing number of points. It is shown that Einthoven's triangle and Wilson's tetrahedron in the theory of electrocardiography are first and second approximations of this method. This concept also applies to conductors of other shapes. This investigation was supported by The National Heart Institute under a research grant H-2263(c).     

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.     

Predicting anti-HIV activity of 2,3-diaryl-1,3-thiazolidin-4-ones: computational approach using reformed eccentric connectivity index

The eccentric connectivity index, which has recently been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature, has been reformed to overcome its limitations caused by degeneracy and insensitivity towards heteroatoms. The reformed eccentric connectivity index, termed the eccentric connectivity topochemical index, overcomes the limitations of the eccentric connectivity index by exhibiting very low degeneracy and displaying sensitivity to both the presence and relative position of heteroatoms without compromizing the discriminating power of the eccentric connectivity index. The relationship of the eccentric connectivity topochemical index, eccentric connectivity index and Wieners index with regard to the anti-HIV activity of 2, 3-diaryl-1, 3-thiazolidin-4-one derivatives was subsequently investigated. The values of the eccentric connectivity topochemical index, the eccentric connectivity index and Wieners index of each of 31 analogues comprizing the data set were computed using in-house computer program. Resultant data was analyzed and suitable models developed after identification of active ranges. Subsequently, each derivative was assigned a biological activity using these models, which was then compared with the reported anti-HIV activity. The accuracy of prediction using these models was found to vary from 81 to 90%. The proposed index offers a vast potential for virtual screening of combinatorial libraries, structure property/activity studies and drug design.Figure Basic structure of 2,3-diaryl-1, 3-thiazoidin-4-ones.     

Libraries of atomic multipole moments for precise modeling of electrostatic properties of amino acids

Summary Contemporary theoretical models used in describing electrostatic properties of amino acids in polypeptides rely usually on atomic point charges. Recently noted defects of such models in reproducing protein folding originate from the inadequate representation of the electrostatic term, in particular inability of atomic charges to account for local anisotropy of molecular charge distribution. Such defects could be corrected by multicenter multipole moments derived directly from any high quality quantum chemical wavefunctions. This is illustrated by comparison of monopole and multipole electrostatic interactions between some amino acids within glutathione S-transferase.High quality Point Charge Models (PCM) can be derived analytically from multipole moment databases. Preliminary results suggest that torsional potentials are controlled by electrostatic interactions of atomic multipoles.Examples illustrating various uses of multicenter multipole moment databases of protein building blocks in modeling various properties of amino acids and polypeptides have been described, including calculation of molecular electrostatic potentials, electric fields, interactions between amino acid residues, estimates of pK_a shifts and changes in catalytic activity induced by amino acid substitutions in mutated enzymes.     

Organismic supercategories and qualitative dynamics of systems

The representation of biological systems by means of organismic supercategories, developed in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–71;32, 539–561), is further discussed. The different approaches to relational biology, developed by Rashevsky, Rosen and by Băianu and Marinescu, are compared with Qualitative Dynamics of Systems which was initiated by Henri Poincaré (1881). On the basis of this comparison some concrete result concerning dynamics of genetic system, development, fertilization, regeneration, analogies, and oncogenesis are derived.     

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 .     

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.     

An evolution of pulse speed in arteries

In this work, treating the artery as a thick-walled cylindrical shell made of an incompressible, isotropic and elastic solid, utilizing the large deformation theory and the stress-strain relation proposed by Demiray (1976b,Trans. ASME Ser. E, J. Appl. Mech.,98, 194–197), an explicit expression for the pulse speed is obtained and the effect of lumen pressure and the axial stretch on wave speed is discussed. Numerical results indicate that the wave speed increases with lumen pressure but decreases with the axial stretch. The results of the present model are compared with our previous work (Demiray, 1988,J. Biomech. 21, 55–58) on the same subject.     

Indices of skeletal muscle damage and connective tissue breakdown following eccentric muscle contractions

 Indirect indices of exercise-induced human skeletal muscle damage and connective tissue breakdown were studied following a single bout of voluntary eccentric muscle contractions. Subjects (six female, two male), mean (SD) age 22 (2) years performed a bout of 50 maximum voluntary eccentric contractions of the knee extensors of a single leg. The eccentric exercise protocol induced muscle soreness (P < 0.05 Wilcoxon test), chronic force loss, and a decline in the 20:100 Hz percutaneous electrical myostimulation force ratio [P < 0.01, repeated measures analysis of variance (ANOVA)]. Serum creatine kinase (CK) and lactate dehydrogenase (LDH) activities were elevated (P < 0.01, repeated measures ANOVA) following the bout. The mean (SD) CK and LDH levels recorded 3 days post-exercise were 2815 (4144) IU · l^–1 and 375 (198) IU · l^–1, respectively. Serum alkaline phosphatase activity showed no changes throughout the study, and a non-significant increase (P = 0.058, repeated measures ANOVA) in pyridinoline was recorded following the bout. Urinary hydroxyproline (HP) and hydroxylysine (HL) excretion, expressed in terms of creatinine (Cr) concentration, increased after exercise (P < 0.05 and P < 0.01, respectively, repeated measures ANOVA). An increased HP:Cr was recorded 2 days post-exercise and HL:Cr was increased above baseline on days 2, 5, and 9 post-exercise. This indirect evidence of exercise-induced muscle damage suggests that myofibre disruption was caused by the eccentric muscle contractions. Elevated urine concentrations of indirect indices of collagen breakdown following eccentric muscle contractions suggests an increased breakdown of connective tissue, possibly due to a localised inflammatory response. Accepted: 9 October 1996     

Light-initiated responses of retinula and eccentric cells in the Limulus lateral eye

The relationship between retinula and eccentric cells in the lateral eye of Limulus polyphemus was studied using a double electrode technique which permitted simultaneous recording of light-initiated responses in two sense cells and the labeling of the cells for subsequent histological examination and identification. The following results were obtained: (a) light-initiated slow responses with and without superimposed spike potentials were recorded from retinula cells and from eccentric cells (only one eccentric cell yielded responses without superimposed spike potentials); (b) spike potentials recorded in different cells within the same ommatidium were always synchronous; (c) a complete absence of spike potentials was observed in two experiments in which no eccentric cells could be found in the ommatidia containing the labeled retinula cells; (d) the greatest differences in the characteristics of responses recorded simultaneously occurred in those recorded from retinula-eccentric combinations. The results indicate that there is only one source of spike potential activity within an ommatidium (presumably the eccentric cell) and that the light-initiated response of retinula cells may be independent of the eccentric cell response. The suggestion is advanced that the response of the retinula cell may "trigger" the eccentric cell response.     

Evolutionary tradeoff and equilibrium in an aquatic predator-prey system

Due to the conventional distinction between ecological (rapid) and evolutionary (slow) timescales, ecological and population models have typically ignored the effects of evolution. Yet the potential for rapid evolutionary change has been recently established and may be critical to understanding how populations persist in changing environments. In this paper we examine the relationship between ecological and evolutionary dynamics, focusing on a well-studied experimental aquatic predator-prey system (Fussmann et al., 2000, Science, 290, 1358–1360; Shertzer et al., 2002, J. Anim. Ecol., 71, 802–815; Yoshida et al., 2003, Nature, 424, 303–306). Major properties of predator-prey cycles in this system are determined by ongoing evolutionary dynamics in the prey population. Under some conditions, however, the populations tend to apparently stable steady-state densities. These are the subject of the present paper. We examine a previously developed model for the system, to determine how evolution shapes properties of the equilibria, in particular the number and identity of coexisting prey genotypes. We then apply these results to explore how evolutionary dynamics can shape the responses of the system to ‘management’: externally imposed alterations in conditions. Specifically, we compare the behavior of the system including evolutionary dynamics, with predictions that would be made if the potential for rapid evolutionary change is neglected. Finally, we posit some simple experiments to verify our prediction that evolution can have significant qualitative effects on observed population-level responses to changing conditions.     

Examine the characterization of biofilm formation and inhibition by targeting SrtA mechanism in Bacillus subtilis: a combined experimental and theoretical study

A unified coarse-grained model of three major classes of biological molecules—proteins, nucleic acids, and polysaccharides—has been developed. It is based on the observations that the repeated units of biopolymers (peptide groups, nucleic acid bases, sugar rings) are highly polar and their charge distributions can be represented crudely as point multipoles. The model is an extension of the united residue (UNRES) coarse-grained model of proteins developed previously in our laboratory. The respective force fields are defined as the potentials of mean force of biomacromolecules immersed in water, where all degrees of freedom not considered in the model have been averaged out. Reducing the representation to one center per polar interaction site leads to the representation of average site–site interactions as mean-field dipole–dipole interactions. Further expansion of the potentials of mean force of biopolymer chains into Kubo’s cluster-cumulant series leads to the appearance of mean-field dipole–dipole interactions, averaged in the context of local interactions within a biopolymer unit. These mean-field interactions account for the formation of regular structures encountered in biomacromolecules, e.g., α-helices and β-sheets in proteins, double helices in nucleic acids, and helicoidally packed structures in polysaccharides, which enables us to use a greatly reduced number of interacting sites without sacrificing the ability to reproduce the correct architecture. This reduction results in an extension of the simulation timescale by more than four orders of magnitude compared to the all-atom representation. Examples of the performance of the model are presented.

Molecular Dynamics on an Excel Spreadsheet

Abstract

We discuss some of the problems that have frustrated the development of reliable model intermolecular potentials for polyatomic molecules. In particular, the usual assumption of an isotropic atom-atom model potential is analysed, and evidence for its inadequacies is presented. A new approach to designing model potentials, an anisotropic site—site model, is introduced by describing several applications to both small and organic molecules, including molecular dynamics and Monte Carlo simulations. The anisotropy required in an atom—atom potential can be directly linked to the non-spherical features in the valence electron distribution, such as lone pairs and π electrons. An accurate electrostatic model for these effects can be constructed from a distributed multipole analysis of the ab initio wavefunction. The empirically required forms of anisotropy in the repulsion potential can also be qualitatively linked to the molecular electron density difference map. Thus, consideration of the molecular bonding can be a useful indication of how to construct adequate model intermolecular pair potentials.     

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.