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  • Springer  (85,424)
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  • 1986  (99,217)
  • 1967  (50,028)
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  • 2015-2019
  • 2010-2014
  • 1985-1989  (99,217)
  • 1965-1969  (50,028)
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  • 1
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    Bulletin of mathematical biology 29 (1967), S. 1-16 
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A vast number of biologically important processes are based upon bimolecular systems. In these systems intermediate complexes are formed. Bimolecular systems in which no complex-complex interactions occur are called linear systems of complexes. A definition and some characteristic properties of these systems are given here. There may exist a contradiction of Onsager's principle of detailed balancing in these systems; however, no principal differences are found between the steady state behavior of an open system and that of a closed system. It is shown that the steady state behavior of a linear system of complexes of arbitrary complexity has some similarities with the steady state behavior of a simple bimolecular system, e.g., Michaelis-Menten enzymatic reaction. Multiplicity of action of the substances participating in biomolecular processes may produce some qualitative differences in the steady state behavior of the system.
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  • 2
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    Bulletin of mathematical biology 29 (1967), S. 17-32 
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    Notes: Abstract A time-dependent DNA histogram is calculated for an irradiated population of cells under the limiting assumption that the cells cannot pass through prophase due to the effects of the radiation. The population is assumed to increase exponentially prior to irradiation, but after irradiation to neither gain nor lose cells. Chromosome-number dispersion is taken into account in the calculation. The qualitative behavior of the calculated and experimental histograms are in reasonable agreement. The quantitative agreement between the two is relatively good at short post-irradiation times but is poor at long post-irradiation times (say, greater than half the doubling time). This suggests that recovery phenomena cannot be neglected at long post-irradiation times.
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  • 3
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    Bulletin of mathematical biology 29 (1967), S. 187-188 
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    Notes: Abstract It is pointed out that the three different stimuli for a corrective turn, namely the distance from the edge of the lane, the rate of approach to the edge, and the angle between the direction of the car and the direction of the lane (Bull. Math. Biophysics,28, 645–654, 1966,29, 181–186, 1967) may act all three simultaneously. It is found that in that case the tracking curve of the car is stable below a critical speed and becomes unstable above it.
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  • 4
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    Bulletin of mathematical biology 29 (1967), S. 181-186 
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    Notes: Abstract Continuing a previous study (Bull. Math. Biophysics, 28, 645–654, 1966), the biophysical mechanism of a corrective turn is investigated for the case where the stimulus for the corrective turn is produced not only by the perception of the nearness of an edge of the lane, but also by the rate of approach of the car towards the edge. In that case it is found that the tracking curve of the car may consist of a series of damped sinusoids and safe driving would be possible at any speed if it were not for the endogenous fluctuation in the driver's central nervous system. If the effect of the rate of approach increases sufficiently rapidly as the distance to the edge of the lane decreases, then a stable undamped oscillating tracking curve is possible. The case is also studied where the driver makes a corrective turn in response to a direct perception of the angle between the direction of the lane and the longitudinal axis of the car.
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  • 5
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    Bulletin of mathematical biology 29 (1967), S. 245-259 
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    Notes: Abstract The principle of minimal work requires that the conducting airways of the human lung should have a maximum radius for minimal resistance to gas flow. At the same time there is a requirement that the airways should have a minimal volume for economy of space. These two opposing requirements have been investigated mathematically, and a method for calculating the angle of branching which produces minimal volume has been derived. The relationship of the radii of the parent and daughter branches to produce minimal resistance has been similarly defined. By measurement of a bronchial cast from a human lung the extent to which the predicted optimum structure is realized in practice has been shown. The change in structure associated with change of function at the transition from conducting airway to diffusion zone has been demonstrated.
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  • 6
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    Bulletin of mathematical biology 29 (1967), S. 191-206 
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    Notes: Abstract This paper considers a class of set-theoretical entities, calledn-rank Linnaean structures, which are intended as abstract models of the taxonomic classificatory systems of biology. In the first part, devoted to formalism, finite Linnaean structures are discussed in complete generality; but, in addition, eight distinct subclasses are noted and some of the properties of their elements are explored. In the second part, concerned with applications, it is shown that taxonomic systems may be recast in the form of finite Linnaean structures, and an effort is made to show that some undesirable features of earlier models are avoided without artificiality and without abandoning extensional mathematics.
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  • 7
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    Bulletin of mathematical biology 29 (1967), S. 207-216 
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    Notes: Abstract Generalizing an idea of M. Richardson (Fundamentals of Mathematics, New York: Macmillan Co., 1958), an APS on a given populationP is a non-empty collection of non-empty subsets ofP such that ifA is in the collection andA⊆B, thenB is in the collection. From a structure of this kind a partial ordering ofP, called therelated bumping order, is derived. The question is raised as to what kinds of partial orderings can be so obtained. For structures determined by voting weights of the members of the population, a complete characterization of all possible bumping orders is obtained.
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    Bulletin of mathematical biology 29 (1967), S. 217-226 
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    Notes: Abstract The “second method” of Liapunov is used to perform a stability analysis of a mathematical model of the neuron. This analysis is based on the hypothesis that the firing of the neuron coincides with a temporary state of instability of the system, and that the initiation of all-or-none process depends on the magnitude of membrane depolarization and its first time derivative. It is found that the stability (and hence the possibility of a second firing) is restored approximately when the rate of membrane repolarization is at a maximum. This result predicts that the duration of the period of absolute refractoriness in neurons would be about 75 per cent of the spike duration, and thus shorter than the value usually obtained from experimental measurements.
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  • 9
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    Bulletin of mathematical biology 29 (1967), S. 227-232 
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    Notes: Abstract Some aspects of masking phenomena are considered in terms of the simplest possible model of two-factor neural elements. The effect of a number of variables can be accounted for, but the introduction of an internuncial element results in a masking function which need not be symmetric about zero delay interval. As an illustration, the results for a special case are compared with available data. In general, such a model results in a masking function which depends on the intensity, area, and duration of the stimuli, as well as on the temporal and spatial separation between them.
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    Bulletin of mathematical biology 29 (1967), S. 377-388 
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    Notes: Abstract The general equations are discussed describing two species in competition or in symbiosis or feeding one on the other.
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    Bulletin of mathematical biology 29 (1967), S. 403-404 
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    Bulletin of mathematical biology 29 (1967), S. 389-393 
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    Notes: Abstract It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.
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    Bulletin of mathematical biology 48 (1986), S. 29-57 
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    Notes: Abstract Approximate equations for epithelial solute and water transport have been combined with the relations of mass conservation to yield a single differential equation representing volume flow along the proximal tubule. This flow equation is first order, quasilinear and may be integrated directly. For the steady state, the result is an implicit relation between volume flow and distance along the tubule. For two time-dependent problems (step change of tubule inlet velocity or osmolality) the trajectories (distance as a function of transit time) of a fluid element starting at the inlet are obtained. Differentiation of the steady-state relation with respect to the inlet velocity yields a first-order differential equation relating inlet and outlet velocity. This equation is considered in detail, particularly with regard to the influence of solute-linked water reabsorption. Model calculations with parameters representing rat proximal tubule indicate that it will be difficult to discern coupled water flux in this epithelium from only outlet and inlet flows. Calculations using lower transport rates and lower permeabilities suggest that this equation may be useful in quantifying coupled water flow in proximal tubules from other species.
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  • 14
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    Bulletin of mathematical biology 29 (1967), S. 407-407 
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    Bulletin of mathematical biology 29 (1967), S. 409-409 
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    Bulletin of mathematical biology 48 (1986), S. 105-105 
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    Bulletin of mathematical biology 48 (1986), S. I 
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    Bulletin of mathematical biology 48 (1986), S. 97-103 
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    Notes: Abstract The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.
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    Bulletin of mathematical biology 48 (1986), S. 125-136 
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    Notes: Abstract Galerkin's finite element-Laplace transform technique (GAFELTTE) has been used to study transient temperature distribution in human skin and subcutaneous tissues. This study incorporates heat conduction, heat carried by perfusion of blood in the capillary beds and metabolic heat generation in the tissues. Different values of various quantities have been considered in all three parts, namely epidermis, dermis and subcutaneous tissues, depending on physiological considerations. The GAFELTTE provides interface temperatures for a wide range of the values of skin surface temperatures. These values have been used to obtain temperature profiles in the region considered. Steady-state temperature distribution has been deduced from the solution obtained by GAFELTTE and has been compared with the results obtained by using different methods.
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    Bulletin of mathematical biology 48 (1986), S. 137-148 
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    Notes: Abstract Necessary and sufficient conditions are given for three equilibria to occur in a predatorprey model and conditions are given for two of these to be stable. The existence of two stable equilibria requires predator intraspecific competition for either space or food, and the lower the prey growth rate the stronger this predator self-regulation must be. A prey growth rate that is skewed to the right, the ability of a few predators to survive at low prey densities, and predators with high searching effectiveness, long handling times, and large maximum per capita rate of increase all make two stable equilibria more likely.
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    Bulletin of mathematical biology 48 (1986), S. 107-124 
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    Notes: Abstract Drawing evidence from a variety of cardiovascular studies on the heart rate in homeothermic animals, the author establishes the following thesis. The servocontrol (i.e. the autonomic and reflex control) by the medulla oblongata of the heart (rate) is a negative feedback dynamic which is isomorphic (i.e. ‘diffeomorphic’) to the dyamic underlying the heat rate control in those animals (cf. Kuyk,Bull. math. Biol. 46, 81–102, 1984). In fact, unlike in the heat rate case, the qualitative evidence supporting this thesis can not be fully complemented by quantitative data stemming from experiments, because of a lack of pertinent experiments—which, indeed, should measuresimultaneously the heart rate state parameter and thefour control parameters at the input side of the medulla. The results of some of the existing experiments on animal preparations can nevertheless be adduced to recognize that this dynamic can be graphed by the five-dimensional butterfly catastrophe type. The theory leads to new ways of looking at experiments in the field and/or setting up such experiments in the future.
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  • 22
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    Notes: Abstract A model based upon minimization of surface energy is proposed as an explanation for compaction and internalization of cells during mammalian embryo development. The model is used to simulate and graphically display these phenomena on a computer.
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    Bulletin of mathematical biology 48 (1986), S. 197-211 
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    Notes: Abstract This paper describes a growth model for binary topological trees. The model defines the branching probability of all segments in the tree. The branching probability of a segment is formulated as a function of two variables, one indicating its type (intermediate or terminal), the other representing its order, i.e. the topological distance to the root segment. The function is determined by two parameters, namely the ratio of branching probabilities of intermediate and terminal segments and the strength of the order dependency, implemented in an exponential form. Expressions are derived for the calculation of symmetry properties of the partitions and it is indicated which part of the parameter domain results in predominantly symmetrical trees.
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    Bulletin of mathematical biology 48 (1986), S. 213-228 
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    Notes: Abstract The problem of cellular differentiation and consequent pattern generation during embryonic development has been mathematically investigated with the help of a reaction-diffusion model. It is by now a well-recognized fact that diffusion of micromolecules (through intercellular gap junctions), which is dependent on the spatial parameter (r), serve the purpose of ‘positional information’ for differentiation. Based on this principle the present model has been constructed by coupling the Goodwin-type equations for RNA and protein synthesis with the diffusion process. The homogeneous Goodwin system can exhibit stable periodic solution if the value of the cooperativity as measured by the Hill coefficient (ρ) is greater than 8, which is not biologically realistic. In the present work it has been observed that inclusion of a negative cross-diffusion can drive the system into local instability for any value of ρ and thus a time-periodic spatial solution is possible around the unstable local equilibrium, eventually leading to a definite pattern formation. Inclusion of a negative cross-diffusion thus makes the system biologically realistic. The cross-diffusion can also give rise to a stationary wave-like dissipative structure.
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    Bulletin of mathematical biology 48 (1986), S. I 
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    Bulletin of mathematical biology 29 (1967), S. 605-613 
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    Notes: Abstract This paper deals with bimolecular systems in which also complex-complex interactions occur. Because of the complexity of the problem, an approximation in a form of coupled linear systems of complexes (Bull. Math. Biophysics,29, 1–16, 1967) is considered. Two types of couplings, serial and parallel, are studied. In the serial coupling the nonlinear system of complexes has the same behavior as its subsystems. An entity, initial sensitivity, has interesting properties: in serial coupling it is at most equal to the product and in parallel coupling, at most equal to the sum of partial initial sensitivities.
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    Bulletin of mathematical biology 29 (1967), S. 615-623 
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    Notes: Abstract Amplification effect in the catalytic bimolecular systems is a consequence of the kinetic characteristic of the catalyst. Two types of the coefficient of amplification are defined. The applicability of these definitions is given by the type of the bimolecular system. In a simple example it is shown that the concept of amplification is meaningful in these systems. Furthermore, two rules, analogous to those for a coupling of amplifiers, are derived for the two basic modes of coupling of catalytic systems. Thus, in biological systems the catalytic reactions may be regarded as biologically effective amplifiers.
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    Bulletin of mathematical biology 29 (1967), S. 583-596 
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    Notes: Abstract It is postulated that cell hydration is governed by adsorption of water on cell proteins in accord with the Bradley adsorption isotherm, and that the action of a solute in the surrounding solution is to lower the vapor pressure of the solution so that cell water adsorption is decreased by moving down the Bradley isotherm. From these concepts, it is derived that cell volume (V) should be related to solute concentration (x) by the equationV=−E log10 x+F whereE andF are constants which are independent of type of solute. For a non-adsorbed solute this agrees well with experimental data. For solutes which are adsorbed by cell proteins, a correction in the above equation may be necessary at higher solute concentrations, which is shown to be compatible with various experimental data. The types of experiments which are generally used to support the osmotic pressure theory of cell hydration agree equally well with the adsorption theory. The virtue of the adsorption theory is that, unlike the osmotic pressure theory of cell swelling, it is compatible with permeability of the cell membrane to solutes, which has been experimentally observed for various solutes.
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    Bulletin of mathematical biology 29 (1967), S. 657-664 
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    Notes: Abstract Conditions under which a time varying electromagnetic field problem (such as arises in electrophysiology, electrocardiography, etc.) can be reduced to the conventional quasistatic problem are summarized. These conditions are discussed for typical physiological parameters.
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    Bulletin of mathematical biology 29 (1967), S. 711-718 
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    Notes: Abstract A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.
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    Bulletin of mathematical biology 29 (1967), S. 677-690 
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    Notes: Abstract A physical model that incorporates all the experimental information on the formation of the visual pigment rhodopsin is presented. The visual pigments consist of a chromophore bound to an appropriate protein. Thus rhodopsin (λm 505 mμ) is formed by a Schiff’s base linkage C19H27CH=NH+-opsin (λm 440 mμ) between 11-cis retinal (λm 380 mμ) and the protein opsin (λm 280 mμ). It is found that there exists a red shift in the spectrum of rhodopsin from the Schiff’s base. The model brings an explanation for this red shift. It is shown that such a shift may be due to a charge transfer process (R. S. Mulliken,J. Am. Chem. Soc.,74, 811–824, 1952) between an electron at the double bond of carbons C11−C12 and an atomic orbital of the sulphur present in cysteine. This provides an explanation of the presence of SH-groups in the protein after the absorption of light. A one-electron approximation is used and the dipole momentμ NV ; hence, the oscillator strengthf of the transitionNV is estimated and compared with the experimentally determined extinction coefficient ∈m by mixing 3.5×10−3 M of 11-cis retinal with 8.3×10−5 M of cysteine at pH ranges 6 through 8. Reasonable agreement is found. Solvent, concentration and temperature dependence are shown also.
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    Notes: Abstract A nes software system is described for building simulation programs on micro- and minicomputers. Model equations are written as C subroutines, compiled and linked to the SCoP package to produce a menu-driven, interactive program. The system maintains a database of names, values, and units for all model parameters and variables. Run-time options include several methods for interactive parameter modification and both graphic and tabular outputs, with output values presented as they are calculated. Simulation output values can be compared with experimental data graphically and a companion program SCoPFit is provided for formal optimization of parameter values.
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    Bulletin of mathematical biology 48 (1986), S. 455-468 
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    Notes: Abstract We consider the existence and global stability of aq-member equilibrium (1≤q≤n) in partially closed food-chains of lengthn having an abiotic component as resource. We observe that such existence demands bounds of resource supply rate and these bounds are weighted sums of interaction coefficients. Particular results of global sector-stability of partially feasible equilibria of simple food-chains obeying Lotka-Volterra dynamics are shown. Lastly the elasticity of such food-chains when a new species is introduced at the highest trophic level is investigated.
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    Bulletin of mathematical biology 48 (1986), S. 485-492 
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    Notes: Abstract Criteria are established for three classes of models of single-species dynamics with a single discrete delay to have a globally asymptotically stable positive equilibrium independent of the length of delay.
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    Bulletin of mathematical biology 48 (1986), S. 493-508 
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    Notes: Abstract The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically).
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    Bulletin of mathematical biology 48 (1986), S. 509-523 
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    Notes: Abstract In this paper a general class of semi-Markov compartmental systems is studied. Two models for different input processes are analysed. Attention has been paid to the recurrence times associated with each compartment and to the distribution of the number of particles in each compartment. As an example, a three-compartment system is discussed to study the movement between three health states of patients with chronic diseases.
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    Bulletin of mathematical biology 48 (1986), S. 569-583 
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    Notes: Abstract A strategy is presented for searching the gene and protein sequence data banks which combines the use of two previously described algorthms. The implementation of this strategy is thoroughly evaluated with respect to sensitivity, specificity and speed. The establishment of standard benchmarks for comparing programs that rearch the sequence data banks for homology is proposed.
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    Bulletin of mathematical biology 48 (1986), S. 545-567 
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    Notes: Abstract During functional linkage, ligand receptors are coupled to other receptors and to the cell's metabolic-transport apparatus. The linkage guides the cellular processing of matter, energy and information. Previous conceptions of functional linkage have used the ideas of classical physics appropriate to macroscopic objects. This study presents an initial quantum mechanical model of functional linkage in the case of ligands moving through lipid bilayers and hydrophilic transmembrane channels (‘pores’) of molecular dimensions. On the basis of permeability data, energy surfaces consisting of piecewise-constant potential regions are used to model the lipid bilayers and transmembrane channels. The centre-of-mass wavefunction for a ligand on such energy surfaces is analysed and the permeability coefficients calculated from the wavefunction's transmission characteristics. It is found that quasi-bound states in the several ligand-binding regions of a bilayer or pore system can functionally link to facilitate the passage of the molecule across the permeability barrier. Appearance of the linkage is a sensitive function of the ligand's energy. If the centre-of-mass energies are distributed as in a thermalized fluid, the flux via the quantum functional linkage can equal or exceed that of a classical flux for proton transport through rigid pores in which the intrasite barriers are relatively high (0.25–1 eV) and narrow (0.1–1 Å). The functional linkage plays a less important role in bilayer (rather than pore) energy surfaces and at higher molecular weights. If the ligand-receptor interaction is accompanied by energy transfer to or from ligands, the flux via the quantum functional linkage can equal or exceed the classically expected flux at all relevant ligand molecular weights. These findings are discussed in relation to earlier work and the limitations of the model emphasized.
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    Bulletin of mathematical biology 48 (1986), S. 617-632 
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    Notes: Abstract A new measure of subalignment similarity is introduced. Specifically, similaritys(l,c) is defined as the logarithm to the basep of the probability of findingc or fewer mismatches in a subalignment of lengthl, wherep is the probability of a match. Previous algorithms can not use this measure to find locally optimal subalignments because, unlike Needleman-Wunsch and Sellers similarities, this measure is nonlinear. A new pattern recognition algorithm is described for finding all locally optimal subalignments of two nucleotide sequences. The DD algorithm can uses(l, c) or any other reasonable similarity function to assess the relative interest of subalignments. The DD algorithm searches only the diagonal graph, which lacks insertions and deletions. This search strategy greatly decreases the computation time and does not require an arbitrary choice of gap cost. The paths of the resulting DD graph usually draw attention to likely locations for insertions and deletions. A heuristic formula is derived for estimating significance levels fors(l, c) in the context of the lengths of the two aligned sequences. The DD algorithm has been used to find interesting subalignments between the nucleotide sequences for human and murine interleukin 2.
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    Bulletin of mathematical biology 29 (1967), S. 841-862 
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    Notes: Abstract By assigning coordinates to the environmental function space comprising all physical and mental stimuli, mathematical interpretations can be based on such terms as adaptability, and reactivity which relate to individuals interacting with their environment within a society. These psychometric concepts are incorporated into a framework of functional analysis, which permits the optimization of social change by maximizing the satisfaction integral through the use of variational or dynamic programming methods in conjunction with some optimal social policy. The approach provides a mathematical connection between psychology and sociology, and further demonstrates that existing forms of government are simulated by differential equations belonging to the same general class. The synthesis of new classes of functional equations describing social progress is visualized as a legitimate objective for abstract mathematical sociology.
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    Bulletin of mathematical biology 29 (1967), S. 863-877 
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    Notes: Abstract The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.
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    Bulletin of mathematical biology 48 (1986), S. 633-660 
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    Notes: Abstract Nonlinear similarity functions are often better than linear functions at distinguishing interesting subalignments from those due to chance. Nonlinear similarity functions useful for comparing biological sequences are developed. Several new algorithms are presented for finding locally optimal subalignments of two sequences. Unlike previous algorithms, they may use any reasonable similarity function as a selection criterion. Among these algorithms are VV-1, which finds all and only the locally optimal subalignments of two sequences, and CC-1, which finds all and only the weakly locally optimal subalignments of two sequences. The VV-1 algorithm is slow and interesting only for theoretical reasons. In contrast, the CC-1 algorithm has average time complexityO(MN) when used to find only very good subalignments.
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    Bulletin of mathematical biology 48 (1986), S. 701-703 
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    Bulletin of mathematical biology 48 (1986), S. 681-699 
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    Notes: Abstract A resource-based competition model of two consumer species and one resource species is formulated in the form of a Lotka-Volterra system. The competition involves both exploitation and interference. By a method of asymptotic estimates, sufficient conditions are derived for the three species system to converge ast→∞ to an equilibrium point with all three species present; a generalization of the result forn≥2 and single resource species is indicated. The strong form of equilibrium perisistence of the three species consumer-resource system is achieved by the ability of each of the consumer species to exploit the resource and interfere with others in such a way which will avoid exclusion by the other.
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    Circuits, systems and signal processing 5 (1986), S. 3-36 
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    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.
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    Bulletin of mathematical biology 29 (1967), S. 233-243 
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    Notes: Abstract A neuron is assumed to receive synaptic input of both excitatory and inhibitory natures from a large number of neighboring neurons; it is also assumed that a large number of such impulses are required to raise the neuron’s transmembrane potential to its threshold potential, at which it “fires” or “spikes”. The model is similar to one of Gerstein and Mandelbrot, except that in the absence of input an exponential decay of potential toward a resting level is introduced. Computational methods of determining the spike timeinterval distribution are discussed, along with the inverse problem of estimating the parameters of the system from observed spike time-interval data.
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    Bulletin of mathematical biology 29 (1967), S. 311-318 
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    Notes: Abstract Moens-Korteweg relations are developed for the velocity of wave propagation through an orthotropic elastic tube based on the three dimensional equations of elasticity. Numerical examples are presented for the femoral artery of a dog and several other orthotropic materials. These results are compared with those obtained from the equations of motion for an orthotropic elastic medium.
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    Bulletin of mathematical biology 29 (1967), S. 335-341 
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    Notes: Abstract Up to the present time, the following property of the product component in the reversible one substrate-one intermediate-one product enzymic mechanism has been taken only as anassumption, viz., during the course of the reaction, the time-rate of change of product concentration is never negative and the product concentration never exceeds its equilibrium value. Applying the methods of the geometric theory of ordinary differential equations it is shown that this result follows as a direct deduction from the differential equations governing the mechanism together with the initial conditions. Further, the nature of the equilibrium point as a stable node is established.
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    Bulletin of mathematical biology 29 (1967), S. 319-333 
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    Notes: Abstract An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.
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    Bulletin of mathematical biology 29 (1967), S. 405-405 
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    Bulletin of mathematical biology 29 (1967), S. 395-401 
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    Notes: Abstract In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.
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    Bulletin of mathematical biology 48 (1986), S. 59-75 
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    Notes: Abstract An understanding of the comparative statics of biological communities is important both as a means of explaining the long-term effects of changes in environmental conditions, and as a framework for viewing community time trajectories. A general formulation of community dynamics is presented here which, given full information about a particular community's dynamic behavior, describes the impact of a change in environmental conditions on the community steady state. However, since such full information is often lacking in studies of biological communities, various approaches to partial information analysis of comparative statics are presented and compared, including a generalized protocol for isocline analysis. The suggested isocline protocol is shown to be a useful tool for both full and partial information analyses, as well as for both general and partial equilibrium studies.
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    Bulletin of mathematical biology 48 (1986), S. 77-86 
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    Notes: Abstract Fujita's diagrams in phyllotaxis, showing the frequencies of divergence angles as a function of these angles for low phyllotactic patterns such as (2, 1) and (3, 2), which are approximately normal curves centered at the limitdivergence angle of 137.51°, are shown to be puzzling when compared to results and observations in the field. An analysis of these diagrams is proposed, in the context of Fujita's methodology, of data from other sources, of a mathematical theorem on lattices, and of the contact pressure theory of phyllotaxis.
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    Bulletin of mathematical biology 48 (1986), S. 87-95 
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    Notes: Abstract It is shown that a representative Fisher-Wright model withn(≥3) diallelic loci admits a necessary condition for existence of a time-independent steady-state probability distribution. This necessary condition states that a global integral depending on the phenotype fitness functions of natural selection must be larger than a certain quantity depending on the parameters associated with genetic drift.
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    Bulletin of mathematical biology 29 (1967), S. 437-449 
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    Notes: Abstract A general model of a large 2m-ploid breeding population, withr loci ands h alleles at the h th locus is considered. It is assumed that the population is bisexual, non-overlapping and breeds by random mating. The genotypic structure of the population is presented as a bilinear form in the gametic output vectors where the genotype distribution is in the matrix form. Using the concept of the segregation distribution, the genotype proportions in the (n+1)st generation are given. An equilibrium condition for random chromosome segregation is obtained in terms of gene frequencies.
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    Bulletin of mathematical biology 48 (1986), S. 149-166 
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    Notes: Abstract Equations for the time-dependent concentrations of all species involved in the general mechanism of human plasminogen activation proposed by Wohlet al. (J. biol. Chem. 255, 2005–2013, 1980) have been derived. These equations are valid for the whole course of the reaction: for both the transient phase and the steady state. In addition, we compare our results with the ones obtained by the above-mentioned authors for the steady state assuming rapid equilibrium conditions. Finally, we propose a method for the determination of all velocity constants.
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    Bulletin of mathematical biology 48 (1986), S. 189-195 
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    Notes: Abstract A special class of interval graphs is defined and characterized, and an algorithm is given for their construction. These graphs are motivated by an important representation of DNA called restriction maps by molecular biologists. Circular restriction maps are easily included.
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    Bulletin of mathematical biology 29 (1967), S. 541-548 
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    Notes: Abstract The rigidity of the skull and the inertial characteristics and incompressibility of its contents cause the elastic cerebral arteries and veins to act over brief periods of time like rigid tubes of relatively small diameter. Poiseuille's law is applicable to their behavior. The use of this law, in combination with the fact that, during brief intervals, the total volume of the cerebral arteries and veins remains constant, permits derivation of a mathematical expression for the average arterial flow in terms of an average arterial radius. The differentiated equation has five positive roots which represent maxima and minima of the average flow in terms of the average arterial radius. The theoretical results have physiological implications and potential clinical usefulness, which are discussed.
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    Bulletin of mathematical biology 29 (1967), S. 549-563 
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    Notes: Abstract 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.
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    Bulletin of mathematical biology 29 (1967), S. 575-581 
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    Notes: Abstract Concentric circular lines are present on X-ray diffraction patterns of DNA. They cannot be accounted for by the double helix model. It is suggested that they may indicate the presence of double helical side chains.
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    Bulletin of mathematical biology 48 (1986), S. 253-278 
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    Notes: Abstract Over the past 25 years stepwise improvement in the cure of disseminated cancers has been good, fair or very poor depending on the particular cancer one is discussing. “Cancer chemotherapy provides variably effective treatment for the majority of forms of human cancer and curative treatment for some 12 categories.” We have been slow to gain and learn how to apply quantitative information on the biologic phenomena that underlie the responsiveness, or lack of responsiveness, of many different cancers to single drugs and combinations of drugs delivered in different ways. I am of the opinion that continuing development and integration of rational biomathematical models based on principles already identified, and testing them for compatibility with much already available experimental and clinical data, will lead to models that will help in planning more effective treatment regimens for cancers now classified as moderately refractory or very refractory to chemotherapy. Some of the critical variables are considered briefly. My advice, for what it is worth, is “try to be sure that the biologic concepts that you use in modeling are almost as good as the arithmetic.”
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    Bulletin of mathematical biology 48 (1986), S. 309-322 
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    Notes: Abstract A discussion of the bases of physiological pharmacokinetics is followed by a brief review of the fundamental mass balance equations of the models. Some examples are outlined, together with a listing of published reviews which give many more references and detailed examples. Finally, some thoughts on future research directions are presented.
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    Bulletin of mathematical biology 48 (1986), S. 323-336 
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    Notes: Abstract The use of stochastic simulation languages in cell kinetics research is discussed. Two special purpose simulation languages; CELLSIM and CELLGROW are described and example problems are presented.
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    Bulletin of mathematical biology 48 (1986), S. 293-307 
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    Notes: Abstract Mathematical models predicting tissue doses of chemical toxicants can be either highly complex or simple, depending upon the end results needed. As an example of a highly complex mathematical model, the Miller Model of the distribution of reactive gases in human and animal lungs is described. The Miller Model accounts for the convection, the radial and axial diffusion, and the chemical reactions of gases as an inhaled breath passes down the airways. The geometry and physiology of human and animal lungs are used to calculate the convection and diffusion likely in each generation or bifurcating series of airways commencing with the trachea and extending 24 generations in humans. The chemical reactivity of ozone, an air pollutant, is accounted for by simulating second-order chemical reactions with the fluid lining materials of the lung and tissue biological molecules. The flux of ozone into three compartments (pulmonary tissue, overlying liquid layer and capillary blood) in each generation of the lung is calculated to provide molecular doses of ozone reaching each region of the lung. These results of calculated molecular dose are then used to construct dose-response curves for a variety of biological endpoints. A much simpler model is also described which recognizes the saturable or Michaelis-Menten type of kinetics controlling the removal of nickelous ion (nickel) from the lung. This model is used to calculate the chronic lung burden of the human lung for occupational, environmental and cigarette smoking exposure scenarios. In both the complex Miller Model and the simpler nickel lung burden model, the results can be used to calculate molecular doses at the potential site of action of these environmental chemicals and to unify a wide variety of studies. The predictions made are more likely to be valid since multiple investigators using a variety of animal species have participated in generation of the primary data. As a methodology, mathematical modeling based on physiological, physicochemical and anatomical principles provides a means of eliminating non-scientific considerations from the important process of regulating and recognizing toxic or cancer causing chemicals in the human environment.
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    Bulletin of mathematical biology 48 (1986), S. 337-351 
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    Notes: Abstract This paper presents a brief review of applications of kinetic simulation of multi-enzyme networks to the study of antimetabolite drugs used as anticancer agents. Kinetic models consist of systems of nonlinear differential equations that describe changes in concentrations of cellular metabolites with respect to time. Such models have been used to predict the effect of changes in activity of enzymes, or changes in enzyme kinetic parameters, on sensitivity to inhibition. Kinetic simulation has provided insight into several aspects of the biochemical pharmacology of antimetabolites, including drug sensitivity and resistance, and drug-drug interactions. Two specific studies are described in detail. The first concerns the importance of the ratio of competing enzymes in determining the selectivity of inhibitors of one of the competing enzymes, studied by a simple model. The second case study examines the effect of alternative biosynthetic pathways, thede novo and salvage pathways of pyrimidine nucleotide biosynthesis, on the selectivity of antipyrimidine drugs, as studied by a detailed model of 27 reactions of pyrimidine metabolism.
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    Bulletin of mathematical biology 48 (1986), S. 381-404 
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    Notes: Abstract General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.
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    Bulletin of mathematical biology 48 (1986), S. 353-380 
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    Notes: Abstract Complex networks of biological processes are analogous to electrical circuits. For each step in a biological or electrical network, flow is dependent on the driving force and the conductivity of the step. The relationship between biological flows and their driving forces can therefore be expressed as relationships between analogous currents and voltages. The time dependence of approach to equilibrium or a steady state is determined by the rates of depletion of material in various compartments. Electrical capacitance is therefore analogous to compartment volume. Once these generalized concepts of flow, force and capacitance are recognized, it becomes clear that computer programs designed for analysis of electrical circuits may be used for simulation of biological networks. A set of simple mathematical descriptions of the individual steps and a diagram showing how the steps are arranged with respect to each other are all that is necessary to perform a simulation; there is no need for computer programming skills or differential equations. The use of SPICE2 for simulation of the cellular and plasma pharmacokinetics of cytosine arabinoside (araC) is described as an example. A network model is developed which considers cellular pharmacokinetics (membrane transport, intracellular phosphorylation and dephosphorylation), and plasma pharmacokinetics following infusions of araC.
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    Bulletin of mathematical biology 48 (1986), S. 405-415 
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    Notes: Abstract Developments in computer hardware and software are making significant improvements in the availability of simulation for biomedical researchers. This paper reviews past and present techniques for digital computer simulation and looks at improvements likely in the near future. In the area of hardware, personal computers are making computing and simulation more widely available and at the same time, supercomputers and special-purpose numerical processors are making it possible to solve larger problems. Software developments for simulation are reducing the time, effort and special skills required to produce a simulation program. A new hierarchical linker is proposed to make it easy to synthesize a global model by combining existing submodels. In the more distant future, computer models may be constructed graphically and with the assistance of intelligent programs capable of analysis and information retrieval.
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    Bulletin of mathematical biology 48 (1986), S. 417-426 
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    Notes: Abstract Modeling is a ubiquitous and often misunderstood enterprise in which data from diverse disciplines are analyzed by techniques from other diverse disciplines in an attempt to confirm or falsify a set of hypotheses about the real world. Guidelines are offered for designing models to match the goals of modeling biological systems. Techniques for the construction and interpretation of models are discussed. The requirements for credibility of models are detailed, and tests are suggested for their validation.
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    Bulletin of mathematical biology 48 (1986), S. 453-453 
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    Bulletin of mathematical biology 48 (1986), S. 443-452 
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    Notes: Abstract Presentations and discussions at the symposium illustrate some general issues in biomedical modeling for cancer research. Given the motivations for modeling and assumptions concerning who should be involved in the modeling process, one can identify some basic needs to be met in supports to modelers. These concern both the models themselves and ways of presenting them to users. In conclusion, some thoughts are offered on economic and educational issues that may affect the infusion of modeling into biomedical research.
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    Bulletin of mathematical biology 48 (1986), S. I 
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    Bulletin of mathematical biology 29 (1967), S. 781-791 
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    Notes: Abstract An extension of an earlier model simulating the effects of light on the drosophila eclosion rhythm is presented. The effects of variable light intensity are described. This allows not only the simulation of certain experiments not covered by the earlier model, but also it permits an extension of the model to other organisms. By changing only its sensitivity to light the model simulates the phase response curves of certain mammals as well as Aschoff’srule.
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    Bulletin of mathematical biology 29 (1967), S. 827-829 
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    Notes: Abstract Previous derivations of the Stevens Power Law relating loudness to intensity assume, in addition to level invariance, that the relation must be analytic or at least differentiable. This last condition is replaced here by the weaker one of requiring only continuity.
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    Circuits, systems and signal processing 5 (1986), S. 109-123 
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    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract Certain properties of solutions similar to set invariance, set attractivity, boundedness, BIBO stability, etc. are investigated for the semistate model $$P(t)\dot x = M(t,x)x + D(t,x)u,y = q(t,x,u).$$ For systems considered, it is assumed that the reduction to a normal form of lower order is not possible. Using the direct method of Liapunov, the properties of solutions are investigated without actual knowledge of solutions.
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    Circuits, systems and signal processing 5 (1986), S. 153-169 
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    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract In this paper we extend the results on the multiple time-scale structure for linear autonomous systems of the form $$\dot x = A( \in )x$$ (cf. Coderchet al. [1]) to nonlinear autonomous systems. Our main result is in obtaining conditions under which the linearized system and the nonlinear system around an equilibrium point have the same time-scale structure.
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    Bulletin of mathematical biology 29 (1967), S. 33-40 
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    Notes: Abstract A method of analyzing thymidine labeling in a population of cells is formulated. The formulation establishes a unique relation between a specific set of labeling data and a specific set of cells in the population, viz. that set of cells having a particular chromosome number. The analysis employs a cell-state variable, i.e., a quantity which specifies the progress of a cell through its lifecycle. This variable is defined in terms of the nucleo-protein content and configuration of the chromosomes. The relation mentioned above leads immediately to an expression for the number of cells present at a particular time following labeling which have a given amount of label per cell and a given chromosome number.
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    Bulletin of mathematical biology 29 (1967), S. 41-56 
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    Notes: Abstract An equation relating radiation-induced metaphase delay to the dose-rate and duration of irradiation is obtained. The equation is derived from a model specifying the effects of radiation on the normal chromosome coiling process. The basic assumptions of the model are (1) that normal coiling proceeds by contractile protein acting on segements of a viscoelastic chromosomal fiber; (2) that radiation causes cross-linking of adjacent chromosomal fibers which hinders the coiling process.
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    Bulletin of mathematical biology 29 (1967), S. 57-65 
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    Notes: Abstract Normal micturition is controlled primarily by a neural system. Certain physical effects become evident when neural control is destroyed, and the automatic or autonomous bladder phenomena occur. It is shown in this paper that a physical system simulating the alternating periods of continence and voiding of the automatic bladder may comprise only passive elastic components, and that periodic voiding does not per se imply neural control.
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    Bulletin of mathematical biology 29 (1967), S. 91-94 
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    Notes: Abstract A number of inaccuracies in previous papers are pointed out and amended, and some implications of the correct situation are outlined.
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    Bulletin of mathematical biology 29 (1967), S. 67-89 
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    Notes: Abstract We investigate a model of the renal medulla in which active NaCl transport is restricted to the thick ascending limb of Henle's loop. The model contains a vas rectum, a loop of Henle, salt, and water. The model generates interstitial osmolality curves consonant with the known functioning of the kidney in water diuresis. Using data from the white rat and the curves generated by the model, one can predict the permeability of the thin limb of Henle's loop to NaCl and the percentage of total renal blood flow entering the inner medulla. In this model interstitial osmolality at the papilla can be about twice plasma osmolality, so that NaCl transport restricted to the outer medulla can contribute significantly to the work required in producing a hypertonic urine. However, the interstitial osmolality monotonically decreases proceeding from the junction of the outer and inner medulla to the papilla, and the maximum interstitial osmolality in the outer medulla is greater than the maximum interstitial osmolality in the inner medulla. Thus we infer that a source of active transport located in the inner medulla is needed to explain the high osmolalities observed in hydropenia. A sketch of an alternative model, a “lineal multiplication mechanism”, for the renal concentrating process is presented in which active transport in the inner medulla is restricted to active salt transport by the collecting duct. The lineal multiplication mechanism makes no use of counter-current multipliers in the inner medulla.
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    Bulletin of mathematical biology 29 (1967), S. 95-121 
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    Notes: Abstract Starting from the basic flux equation, it is possible to obtain an integral form relating the current componentsI i at an arbitrary pointr 2 to the distribution of mobilities and concentrationsc i, potential forces $$\bar \mu $$ , and chemical productivityp i without any restrictive assumptions such as constant mobilities, constant field, steady state, or electrical neutrality. The equation is $$\begin{gathered} I_i (r_2 ) = G_i (r_2 )\left[ {\Delta \bar \mu _i - \int_{r_1 }^{r_2 } {z_i } FA\left( {p_i - dc_i /dt} \right)\left( {\frac{1}{{G_i (r)}}} \right)dr} \right]; \hfill \\ G_i (r) = 1/\int_{r_1 }^r {\frac{{dr}}{{z_i^2 F^2 c_i u_i }}.} \hfill \\ \end{gathered} $$ On the basis of this equation, it is possible to give a more general and systematic development of the basic equations of electrophysiology which clarifies a number of questions concerning the physical interpretation of and the necessary and sufficient conditions for the applicability of some of the standard equations and gives their proper extensions to more general conditions. It is found that the relation between the current components and chemical reactions present arises in a very natural way via the continuity equation and enables one to discuss the incorporation of the metabolic and active transport parameters by assuming a very general physical condition. On the basis of this general integration technique one may then compare the physical interpretation of the differential conductance, the chord conductance and the integral conductance.
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    Notes: Abstract Previous papers by F. M. Snell (Jour. Theor. Biol.,8, 469–479, 1965) and M. A. Fox and H. D. Landahl (Bull. Math. Biophysics,27, Spec. Issue, 183–190, 1965) have found that the formulation by previous authors for the oxygen flow rates through hemoglobin solution as a function of pressure determined by E. Hemmingsen and P. F. Scholander (Science,132, 1379–1381, 1960) did not give a satisfactory quantitative fit of the curve for constant pressure difference. The suggestion of Fox and Landahl that the Bohr effect involving the shift in acidity accompanying the oxidation of Hb should give rise to voltage and pH differences in oxyhemoglobin transport is examined in more detail. In this paper, the previous expressions for the total oxygen flow rate in terms of the end point concentrations are extended to include the effects of the electrical field. Estimates of the potential difference shows it to be negligible. A derivation of a voltage-pH relation shows that the Nernst relation does not apply and a negligible voltage difference does not preclude a pH shift which is the more probable explanation of the discrepancies observed. Several other predictions suitable for experimental testing are made.
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    Bulletin of mathematical biology 29 (1967), S. 153-174 
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    Notes: Abstract A model for the human eye-movement mechanism is derived. The derivation is based on a literature search directed toward identifying and mathematically describing each component through physiological and anatomical considerations. It is felt that although certain parameter values may not be exactly correct (for the data were taken from a wide variety of animals), we can place a great deal of confidence in the configuration.
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    Bulletin of mathematical biology 29 (1967), S. 175-179 
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    Notes: Abstract The urethra as seen on X-ray films may show alternate regions of constriction and distension. That these regions do not necessarily correspond to high and low tensions in the circumferential muscle sheath is shown by calculated stable configurations under uniform tension.
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    Bulletin of mathematical biology 29 (1967), S. 139-152 
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    Notes: Abstract The discussion as to whether societies are organisms andvice versa has been going on for a long time. The question is meaningless unless a clear definition of the term “organism” is made. Once such a definition is made, the question may be answered by studying whether there exists any relational isomorphism between what the biologist calls an organism and what the sociologist calls society. Such a study should also include animal societies studied by ecologists. Both human and animal societies are sets of individuals together with certain other objects which are the products of their activities. A multicellular organism is a set of cells together with some products of their activities. A cell itself may be regarded as a set of genes together with the products of their activities because every component of the cell is either directly or indirectly the result of the activities of the genes. Thus it is natural to define both biological and social organisms as special kinds of sets. A number of definitions are given in this paper which define what we call here organismic sets. Postulates are introduced which characterize such sets, and a number of conclusions are drawn. It is shown that an organismic set, as defined here, does represent some basic relational aspects of both biological organisms and societies. In particular a clarification and a sharpening of the Postulate of Relational Forces given previously (Bull. Math. Biophysics,28, 283–308, 1966) is presented. It is shown that from the basic definitions and postulates of the theory of organismic sets, it folows that only such elements of those sets will aggregate spontaneously, which are not completely “specialized” in the performance of only one activity. It is further shown that such “non-specialized” elements undergo a process of specialization, and as a result of it their spontaneous aggregation into organismic sets becomes impossible. This throws light on the problem of the origin of life on Earth and the present absence of the appearance of life by spontaneous generation. Some applications to problems of ontogenesis and philogenesis are made. Finally the relation between physics, biology, and sociology is discussed in the light of the theory of organismic sets.
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    Bulletin of mathematical biology 29 (1967), S. 189-190 
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    Bulletin of mathematical biology 29 (1967), S. 261-266 
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    Notes: Abstract Ureteral obstruction has been known to reduce the renal concentration gradients of chloride, sodium, urea, and osmolality. A quantitative analysis of the factors responsible for the decrease in the gradients has been performed. By applying the equation of conservation of matter to data obtained in this laboratory it is concluded that:a. at least, 62.5 per cent of the decrease in concentration gradients is due to drainage by circulatory vessels;b. at most, 25 per cent of the decrease is due to increase in water content of medulla; andc. at most, 12.5 per cent of the decrease is due to diffusion of solutes through the interstitium.
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    Bulletin of mathematical biology 29 (1967), S. 267-289 
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    Notes: Abstract The vector equation for the general motion of a body in an inertial system is used to analyze the accelerations in the semicircular canals of the cat when the head undergoes rotation about a vertical axis only, rotation about the naso-occipital axis only, and both rotations simultaneously. The corresponding mean forces and mean pressures in the endolymph are calculated by means of a closed line integral along each canal circumference. The importance of the area of the semicircular canal and of its orientation in space become evident. One can see through this mathematical analysis that the input pattern received by the labyrinthine system depends on a set of well-specified geometrical and mechanical conditions, which must be precisely evaluated in order to interpret the nystagmic outputs.
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    Bulletin of mathematical biology 29 (1967), S. 291-310 
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    Notes: Abstract This paper describes some analytical models for the system which regulates the daily eclosion rhythm of the drosophila. A general topological model is described which can simulate practically all the known experimental results about the behavior of the system under various light stimuli. From that a more specific model is proposed which can shortly be described as follows: The system contains a basic oscillator whose output is a substances. This is produced in the presence of an enzymer. During part of the cycler is deactivated ands dissipates until it reaches a lower level whenr is reactivated again. Light has the effect of deactivatingr immediately. The substances causes the production of a second substanceq which triggers a series of reactions leading to eclosion when it exceeds a threshold. The main oscillator (s—r) is temperature-compensated, but the production ofq is accelerated in the presence of light or higher temperature.
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    Bulletin of mathematical biology 29 (1967), S. 343-348 
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    Notes: Abstract Some theoretical results obtained in a previous publication (Bull. Math. Biophysics,28: 25–50, 1966) are studied from the numerical point of view. Possible medical interpretations are suggested.
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    Bulletin of mathematical biology 29 (1967), S. 349-361 
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    Notes: Abstract The findings of molecular biology concerning biosynthesis of macromolecules are applied to the deduction of the kinetics of mass and volume growth in individual cells between divisions. The time course of increase of all macromolecules and of the total dry mass is found to be linear, in agreement with the available data; the corresponding volume growth curves are either quadratic, or exponential with a linear asymptote, depending on the relative contributions of metabolism and transport to cell water. A self-limiting mass and volume kinetics is derived by including repression among the other molecular mechanisms.
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    Bulletin of mathematical biology 29 (1967), S. 363-371 
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    Notes: Abstract An error in a previous analysis of a simple model for myelinated nerve propagation is pointed out and rectified. The model previously chosen does not lend itself to analysis because it has a region ofinfinite negative conductivity which leads to an instability in steady propagation. A model which assumes afinite negative conductivity is examined in detail and a more general results is derived.
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    Bulletin of mathematical biology 29 (1967), S. 373-376 
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    Notes: Abstract In a deterministic model for the spread of news in a closed homogeneously mixing population of individuals who never forget (or, of an epidemic without recovery), it is shown that the fractions πi of the population first hearing the news (contracting the disease)i th hand are given by the terms of a truncated Poisson distribution.
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    Bulletin of mathematical biology 48 (1986), S. 21-27 
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    Notes: Abstract The process of cooperative binding of ligands to DNA has been classified into different modes. An additional mode of cooperative interaction amongst ligands binding at sites on complementary strands has been emphasised. A statistical mechanical method has been applied to obtain an analytical expression for the fraction of nucleotide sites bound. Theoretical Scatchard plots have been drawn and analysed.
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    Bulletin of mathematical biology 48 (1986), S. 1-19 
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    Notes: Abstract Suspensions of chemotactic bacteria develop spatial and temporal structures in response to an initial inhomogeneity in the medium. A theoretical model is presented for the analysis of spatial and temporal evolution of bacterial bands in response to several attractants. Applications of the model to various experimental cases give good agreement between theory and observation. The theoretical analysis provides further insight to the mechanisms governing band formation and band migration.
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    Bulletin of mathematical biology 29 (1967), S. 465-471 
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    Notes: Abstract A mathematical model of the left ventricle is constructed. This model is a spherical thinwalled chamber with a radius and wall density approximately that of the left ventricle and enclosing a fluid whose pressure is that noted when heart sounds occur. The solution for the natural frequency of vibration is of the same value as that noted physiologically. Substitution of the values for the right ventricle yields similar results. The lowest natural frequencies of the idealized cardiac chamber possessing physical properties similar to a real cardiac chamber are of the same order of magnitude as the lowest significant observed cardiac frequencies. Such observation reinforces the likelihood that cardiac sounds are generated by the vibrating cardiac walls.
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    Bulletin of mathematical biology 29 (1967), S. 411-418 
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    Notes: Abstract From a quantum mechanical standpoint, electron tunneling may occur in a nerve axon because the nerve membrane (60 to 100 angstroms) is thin enough for the tunnel effect and because the external solution and the axoplasma are redox systems which can provide electrons. Calculations and diagrams of the density-of-states are given to predict theN-shaped current-voltage characteristics which have been observed in the studies of squid giant axons, of the reconstituted liquid membranes and of the marine algae.
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    Bulletin of mathematical biology 29 (1967), S. 429-436 
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    Notes: Abstract A theory of growth of a cell which takes up nutrients by diffusion or active transport is discussed. The main conclusion is that the volume should grow at least as fast as the third power of the time. Existing experimental evidence is not a conclusive test of the theory, and further experiments to test it are proposed.
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    Bulletin of mathematical biology 29 (1967), S. 419-428 
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    Notes: Abstract This is an analytic study of mucous flow caused by ciliary motion. The computed flow data may be compared with that already found exprimentally. The effects of mucous density, viscosity and layer depth on flow phenomena are investigated. The effects of cilia diameter, length, spacing and oscillation frequency are determined from the equations governing the flow of the mucous blanket. A pertinent finding of the analysis is that the mucous flow in the airway tubes can satisfy physical constraints only through the assumption of a variable viscosity in the covering mucous blanket. The mucous viscosity must increase considerably from the low value at the cillium layer to a much higher value at the air-mucus interface.
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