ALBERT

Notes: Abstract A somewhat different approach to the principle of biotopological mapping, discussed in previous publications, is given. The organism is considered as a set of properties, each of which is in its turn a set of numerous subproperties which are logically included in the corresponding properties. Topology is introduced by an appropriate definition of neighborhoods, and four postulates are stated which concern the mapping of the spaces corresponding to higher organisms on those of lower ones. A number of conclusions are drawn from the postulates. Some of them correspond to well-known facts. For example, in man and some higher organisms appropriate emotional stimuli should produce gastrointestinal or cardiovascular disturbances; or some microorganisms should produce substances harmful to other microorganisms (antibiotics). Some other conclusions are still awaiting verification. One of them is, for example, that there must exist unicellular organisms which produce antibodies to appropriate antigens.

Notes: Zusammenfassung Für die Praxis der Pflanzenernährung ist es wichtig, zu wissen, in welcher Weise die Ertragsbildung von der Konzentration eines mineralischen Nährstoffes in der Umgebung der Pflanze abhängt. Da nur diejenigen Nährstoffmengen das physiologische Geschehen in der Pflanze unmittelbar zu beeinflussen vermögen, die sich in der Pflanze befinden, wird angenommen, dass das Wachstum zum Zeitpunktt, d.h. die Geschwindigkeit der Trockensubstanzzunahme zu diesem Zeitpunkt, eine Funktion der zur Zeitt in der Pflanze enthaltenen Nährstoffmenge ist. Diese Nährstoffmenge wird natürlich im Intervall vor dem Zeitpunktt aufgenommen. Deshalb und auch noch aus anderen Gründen hängt das Wachstum zur Zeitt davon ab, wie die in der Umgebung der Pflanze herrschende Konzentration des betrachteten Nährstoffes in demjenigen Zeitintervall verläuft, das sich von der Aussaat bis zum Zeitpunktt erstreckt. Die angegebene Annahme fürhrt zusammen mit einigen weiteren naheliegenden Annahmen zu einem Ansatz, der Ergebnisse liefert, die in verschiedener Hinsicht gut mit der Erfahrung übereinstimmen. Jedoch gibt es auch noch Widersprüche zwischen Theorie und Erfahrung. Durch weitere Ausgestaltung der Theorie lassen sich diese Widersprüche beseitigen. Es wird angeregt, Versuche durchzuführen, deren Resultate Hinweise für die weitere Ausgestaltung der Theorie liefern.

Notes: Abstract The dynamics of cell multiplication and differentiation in tissues in asteady state and the kinetics of isotope incorporation into the DNA have been theoretically analyzed. Equations have been derived, with the aid of which thegeneration time, thelife span, and the distribution or rate of death of the cells can be obtained if the tissue is in asteady state, i.e., if the number of cells is maintained constant by constant, equal rates of cell division and cell death and if the mean DNA content per cell is also constant. An equation has also been derived which gives thegeneration time in the case of logarithmic multiplication of cells. Two special cases have been analyzed: InCase 1, the isotope is considered as being introduced into the metabolic system at zero time only; inCase 2, the specific activity of the DNA precursor is considered as being maintained constant. The use of the method has been illustrated by an example in which thegeneration time and themean, themedian, and themode life span, as well as the curve of the rate of death of leukocytes in a patient with chronic leukemic granulocytic leukemia, have been obtained from the rate of P32 incorporation into the DNA. The merits and the limitations of the method are discussed.

Notes: Abstract By means of the Laplace transform, the behavior of a simplified model of the cardiovascular system is mathematically formulated. This formulation allows mathematical expression of the periodicity of the cardiac output and the systemic response. With the cardiac output represented as half of a sine function cycle, the systolic aortic pressure becomes the sum of a sine term and exponential terms, while the sum of the exponential terms alone represents the diastolic pressure. The characteristics of the mathematical expressions for systole and diastole are analyzed, and some relationships of potentially practical value are derived. Variation in the parameters of the system yields mathematical results consistent with the expected physical ones.

Notes: Abstract A generalization of Landahl's approximation method (H. D. Landahl,Bull. Math. Biophysics,15, 49–61, 1953) for non-linear diffusion problems is suggested. The method is applied to sorption, desorption, and free diffusion problems involving concentration-dependent diffusion coefficients. With some limitations, the results compare favorably with those obtained by numerical methods.

Notes: Abstract Recently a theorem for representing current generators in a volume conductor by the superposition of a central dipole, quadrupole, octopole, etc., has been established by G. C. K. Yeh, J. Martinek, and H. de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958). This theorem makes possible the representation of any discrete or line, surface- or volume-distributed current source by a unique model which can be determined for each given case by surface potential measurements and closed form analysis. In this paper the multipole representations of an eccentric dipole and an eccentric double-layer are obtained in terms of the various parameters of the assumed singularities, and the contributions to surface potentials due to each of the multipoles are compared. Certain numerical results corresponding to those of E. Frank (Amer. Heart J.,46, 364–78, 1953) are carried out and compared. Furthermore, the multipole representation of a partially damaged double-layer is also determined and compared with that of an undamaged one. It is concluded that within the range of parameters corresponding to human subjects the higher-order multipoles can contribute significantly to the surface potentials compared with the dipole.

Notes: Abstract In line with a recent suggestion by the author (Bull. Math. Biophysics,20, 267–73, September, 1958) that not only does the organism as a whole map on the primordial, but that each organ can also be thus mapped, it is shown that the previously introduced abstract spaces, which represent an organism, contain subspaces which map continuously on the space of the primordial. Several theorems about those subspaces are proven.

Notes: Abstract The DNA-protein coding problem is given a general algebraic formulation, the consequences of which are then explored by standard mathematical methods. To keep the treatment self-contained, the mathematical techniques to be used are explained in detail. It is demonstrated that there exista priori a countably infinite number of different abstract DNA-protein codes, thereby showing that inductive attempts to construct such a code will most likely be fruitless. A notion of ergodicity is then introduced, which imposes a number of restrictions on the admissible codes, and, in fact, these considerations enable us toderive a small portion of a code which, if our hypothesis of ergodicity is correct, must occur in nature. Finally, we discuss briefly the problem as to whether there can exist more than one DNA-protein code in nature.

Notes: Abstract 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.

Notes: Abstract In a preceding paper (Bull. Math. Biophysics 20, 71–93, 1958) the principle of biotopological mapping was formulated in terms of a continuous mapping of an abstract space, made from the set of biological properties which characterize the organism, by an appropriate definition of neighborhoods. In this paper it is shown that we may consider directly the mappings of the different sets of properties which characterize different organisms without taking recourse to abstract spaces. All the verificable conclusions made in the preceding paper remain valid. A serious difficulty mentioned previously is, however, avoided and the possibility of more general predictions is established.

Notes: Abstract The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.

Notes: Abstract The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme. R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order of approximation.

Notes: Abstract An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.

Notes: Abstract Diffusion through a flat pore into a large open region is proportional to the linear dimension of the pore and not to its area. This was first explained by Brown and Escombe (1900) for a circular pore and is here generalized, by means of a dimensional argument, to include any type of regular opening. The problem is further generalized to include diffusion through pores of finite thickness, finite distance apart, and into finite regions. Since this problem cannot be solved exactly, an approximation method is introduced. Reasons for the credibility of the approximation are presented. It is then shown, by means of the approximation method, that the diffusive flow through a pore is equal to the total concentration difference divided by the resistance of the system. The resistance, in turn, is the sum of the resistances of all portions of the system, each of which is calculated. The result is compared with results which have been calculated exactly for limiting cases and found to agree very well. The results are then applied to a standard method of computing pore size in membranes, and it is shown that the correction factor is negligible.

Notes: Abstract V. S. Ivlev [Experimental Ecology of Nutrition of Fishes, 1955, Moscow (in Russia)] has shown that the food uptake by fishes during a fixed interval of time is an exponential function of the concentration of food. Ivlev's equation is derived here, and it is shown that it can hold only for non-stationary conditions, such as prevailed in Ivlev's experiments. For a stationary state, the rate of food uptake should tend asymptotically to a limiting value as the concentration increases, but the variation is not exponential. Different other aspects of the problem are investigated, and definite new experimental procedures suggested. The implications of Ivlev's findings on the effect of non-uniformity of food distribution upon the rate of food consumption are studied from a mathematical point of view. The conclusion is reached that whereas a fish does not, in the process of eating, move directly to an individual food particle which it perceives, it does move more or less directly to large aggregates of particles, if the latter are distributed nonuniformly.

Notes: Abstract Some relational aspects of the property of self-reproduction of biological systems are studied. If in addition to the requirement of the property of self-reproduction we add also the requirement of adaptability of the organism to changing environment, this imposes certain conditions on the topology of the graphs which represent such systems. A further study of the relational properties of such systems seems to offer the possibility of deriving the principle of biological mapping from the requirement of self-reproduction and adaptability. An examination of the problem of the original formation of such self-reproducing systems in connection with the established fact of impossibility of spontaneous generation leads to the conclusion that an organism must inhibit such processes which, in the absence of organisms, would lead to spontaneous generation.

Notes: Abstract In the human, the antagonistic, extensor-flexor system of the leg is an example of a common type of neurophysiological feedback system. After a brief introduction to the neuroanatomy and physiology of this feedback system, the paper formulates transfer functions from temporal response data available in the literature. A feedback stability analysis, based on the extension of Nyquist's stability criteria to multiple-loop systems and utilizing flow-graph techniques, demonstrates the stable behavior of the system. Expressions are given relating the sensitivity of the system to variations in muscle response and Golgi tendon organ (tension receptor) response. By considering the events taking place at synapses and end-plates during “isometric tension-small knee angle excursion” conditions as stationary stochastic processes, an external “noise” input to the system is given, whose spectrum is derived from the statistics of a shot-process representation of these events. The paper concludes with some correlations between the analytical results and clinical syndromes.

Notes: Abstract In this continuation of a previous report it is shown how the Volterra population dynamics, which underlies the statistical theory, can be based on a variational principle; how the dynamics can be generalized as regards both the behavior of total populations and migration phenomena; and how many directly observable data, such as amplitudes and frequencies of oscillation of a population, fit into the statistical theory and can test it. Such a test is carried out in some detail using the fox-catch data of Elton, with a clear indication that the theory is capable of comprehending the major statistical properties of population-time curves. A final section sketches an extension of the theory to cover secular variations of external conditions such as temperature of the environment.

Notes: Abstract In the framework of the neural network theory effects similar to hypnotic displays are constructed. They are based on the associative paradigm involving non-linear interaction of excitatory and inhibitory channels with synaptic memory. The non-linearity of long-term memorizing processes may cause effects exhibited by blind spots, which are interpreted as the first stage of hypnosis. More complicated phenomena are discussed in terms of a two-layer network.

Notes: Abstract Mutation is introduced into autocatalytic reaction networks. The differential equations obtained are neither of repliator-type nor can they be transformed straightway into a linear equation. Examples of low dimensional dynamical systems —n=2, 3 and 4 — are discussed and complete qualitative analysis is presented. Error thresholds known from simple replication-mutation kinetics with frequency independent replication rates occur here as well. Instead of cooperative transitions or higher order phase transitions the thresholds appear here as supercritical or subcritical bifurcations being analogous to first-order phase transitions.

Notes: Abstract The non-linear behavior of a differential equations-based predator-prey model, incorporating a spatial refuge protecting a consant proportion of prey and with temperature-dependent parameters chosen appropriately for a mite interaction on fruit trees, is examined using the numerical bifurcation code AUTO 86. The most significant result of this analysis is the existence of a temperature interval in which increasing the amount of refuge dynamically destabilizes the system; and on part of this interval the interaction is less likely to persist in that predator and prey minimum population densities are lower than when no refuge is available. It is also shown that increasing the amount of refuge can lead to population outbreaks due to the presence of multiple stable states. The ecological implications of a refuge are discussed with respect to the biological control of mite pests.

Notes: Abstract In many applications of control theory on plant growth models biomass maximization is postulated to avoid analytically unsolvable problems while fruit maximization is commonly considered to be a more realistic criterion. In a special case, we are able to compare these criteria. Iwasa and Roughgarden (1984,Theor. Pop. Biol. 25, 78–105) have investigated a certain class of plant growth models using a fruit maximization criterion. They proved that, in the vegetative growth period, the organs follow a certain path of balanced growth. We show that this path remains optimal when biomass maximization is postulated. This underlines the importance of the balanced growth path found by Iwasa and Roughgarden. Furthermore, our result suggests that in the vegetative growth period the biomass maximization criterion is a good approximation of fruit maximization. In another theoretical control investigation, Schultzeet al. (1983,Oecologia 58, 169–177) derived a different type of balanced growth path. We apply the theory of Iwasa and Roughgarden to an improved version of the model of Schulzeet al. This leads to a new description of balanced growth between root and shoot that reflects non-linearities in the water uptake process and constitutes an interesting hypothesis for further experimental testing.

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

Notes: Abstract Many models of immune networks have been proposed since the original work of Jerne [1974,Ann. Immun. (Inst. Pasteur) 125C, 373–389]. Recently, a limited class of models (Weisbuchet al., 1990,J. theor. Biol. 146, 483–499) have been shown to maintain immunological memory by idiotypic network interactions. We examine generalizations of these models when the networks are both large and highly connected to study their memory capacity, i.e. their ability to account for immunization to a large number of random antigens. Our calculations show that in these minimal models, random connectivities with continuously distributed affinities reduce the memory capacity to essentially nil.

Notes: Abstract A kinetic model is proposed to delineate the factors that determine the coronary reactive hyperemic response (RHR) to transient ischemia. The model comprises of myocardial-interstitial (M) and vascular (V) compartments. Vasodilator metabolites (VM) are produced in the M compartment during the interval of coronary occlusion. The rate of VM production is dependent on the flow rate during the ischemic period, the ratio of excess flow above the control level (R) to the loss of flow during occlusion period (D), the amount of oxygen stored and the degree of vasodilation in the V compartment prior to occlusion. Following a complete release of occlusion, VM are transported from the M to V compartment and are washed out or degraded with time. The time course of RHR is determined by the coronary patency which is proportional to VM concentration in the V compartment. Based on a set of numerical constants, the model is tested by simulating RHR to the various occlusion manoeuvres: a pair of 10 sec occlusions separated by brief release, a 15 sec release followed by a second brief occlusion, a brief release of an occlusion followed by restriced inflow and a period of restricted inflow after occlusion. The simulated results fit the experimental R/D and RH durations data of canine hearts. Factors that determine the impairment of RH capacity in coronary stenosis are suggested in terms of the model scheme.

Notes: Abstract In the present paper a kinetic study is made of the behaviour of a Michaelis-Menten enzyme-catalysed reaction in the presence of irreversible inhibitors rendered unstable in the medium by their reaction with the product of enzymatic catalysis. A general mechanism involving competitive, non-competitive, uncompetitive and mixed irreversible inhibition with one or two steps has been analysed. The differential equation that describes the kinetics of the reaction is non-linear and computer simulations of its dynamic behaviour are presented. The results obtained show that the systems studied here present kinetic co-operativity for a target enzyme that follows the simple Michaelis-Menten mechanism in its action on the substrate, except in the case of an uncompetitive-type inhibitor.

Notes: Abstract The relative contributions of mitochondrial β-oxidation and peroxisomal β-oxidation and peroxisomal ω-oxidation to the oxidation of a given fatty acidin vivo can be quantitated by an isotopic method. The approach requires infusion of a fatty acid labelled on two specific carbon atoms (e.g. [1-14C] and [11-14C] palmitate) to an isotopic steady state, with subsequent isolation and degradation of an acetylated conjugate as a product of the liver cytosolic acetyl CoA pool and of ketone bodies as a product of the liver mitochondrial acetyl CoA pool.

Notes: Abstract Pancreatic β-cells in intact islets of Langerhans perfused with various glucose concentrations exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The fraction of the time spent in the active phase is called the plateau fraction and appears to be strongly correlated with the rate of release of insulin from islets as glucose concentration is varied. Here this correlation is quantified and a theoretical development is presented in detail. Experimental rates of insulin release are correlated with “effective” plateau fractions over a range of glucose concentrations. There are a number of different models for BEA in pancreatic β-cells and a method is developed here to quantify the dependence of a glucose dependent parameter on glucose concentration. As an example, the plateau fractions computed from the Sherman-Rinzel-Keizer model are matched with experimental plateau fractions to obtain a relationship between the model's glucose-dependent parameter, β, and glucose concentration. Knowledge of the relationships between β and glucose concentration and between experimental measurements of rates of insulin release and plateau fractions permits the determination of theoretical rates of insulin release from the model.

Notes: Abstract When a suspension of bacterial cells of the speciesBacillus subtilis is placed in a chamber with its upper surface open to the atmosphere complex bioconvection patterns are observed. These arise because the cells: (1) are denser than water; and (2) usually swim upwards, so that the density of an initially uniform suspension becomes greater at the top than the bottom. When the vertical density gradient becomes large enough, an overturning instability occurs which ultimately evolves into the observed patterns. The reason that the cells swim upwards is that they are aerotactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell (N) and oxygen (C) concentrations, and these are solved in the pre-instability phase of development whenN andC depend only on the vertical coordinate and time. Numerical results are obtained for both shallow- and deep-layer chambers, which are intrinsically different and require different mathematical and numerical treatments. It is found that, for both shallow and deep chambers, a thin boundary layer, densely packed with cells, forms near the surface. Beneath this layer the suspension becomes severely depleted of cells. Furthermore, in the deep chamber cases, a discontinuity in the cell concentration arises between this cell-depleted region and a cell-rich region further below, where no significant oxygen concentration gradients develop before the oxygen is fully consumed. The results obtained from the model are in good qualitative agreement with the experimental observations.

Notes: Abstract We describe a classification scheme for bursting oscillations which encompasses many of those found in the literature on bursting in excitable media. This is an extension of the scheme of Rinzel (inMathematical Topics in Population Biology, Springer, Berlin, 1987), put in the context of a sequence of horizontal cuts through a two-parameter bifurcation diagram. We use this to describe the phenomenological character of different types of bursting, addressing the issue of how well the bursting can be characterized given the limited amount of information often available in experimental settings.

Notes: Abstract To ensure its sustained growth, a tumour may secrete chemical compounds which cause neighbouring capillaries to form sprouts which then migrate towards it, furnishing the tumour with an increased supply of nutrients. In this paper a mathematical model is presented which describes the migration of capillary sprouts in response to a chemoattractant field set up by a tumour-released angiogenic factor, sometimes termed a tumour angiogenesis factor (TAF). The resulting model admits travelling wave solutions which correspond either to successful neovascularization of the tumour or failure of the tumour to secure a vascular network, and which exhibit many of the characteristic features of angiogenesis. For example, the increasing speed of the vascular front, and the evolution of an increasingly developed vascular network behind the leading capillary tip front (the brush-border effect) are both discernible from the numerical simulations. Through the development and analysis of a simplified caricature model, valuable insight is gained into how the balance between chemotaxis, tip proliferation and tip death affects the tumour's ability to induce a vascular response from neighbouring blood vessels. In particular, it is possible to define the success of angiogenesis in terms of known parameters, thereby providing a potential framework for assessing the viability of tumour neovascularization in terms of measurable quantities.

Notes: Abstract The parameter domain for which the quasi-steady state assumption is valid can be considerably extended merely by a simple change of variable. This is demonstrated for a variety of biologically significant examples taken from enzyme kinetics, immunology and ecology.

Notes: Abstract The ability of random fluctuations in selection to maintain genetic diversity is greatly increased when generations overlap. This result has been derived previously using genetic models with very special assumptions about the population age structure. Here we explore its robustness in more realistic population models, with very general age structure or physiological structure. For a range of genetic models (haploid, diploid, single and multilocus) we find that the condition for maintaining genetic diversity generalizes almost without change. Genetic diversity is maintained by selection if a product of the form (generation overlap)×(selection intensity)×(variability in the selection regime) is sufficiently large, where the generation overlap is measured in units of Fisher's reproductive value. This conclusion is based on a local evolutionary stability analysis, which differs from the standard “protected polymorphism” criterion for the maintenance of genetic diversity. Simulation results match the predictions from the local stability analysis, but not those from the protected polymorphism criterion. The condition obtained here for maintaining genetic diversity requires fitness fluctuations that are substantial but well within the range observed in many studies of natural populations.

Notes: Abstract Premised on relatively simple assumptions, mathematical models like those of Monod, Pirt or Droop inadequately explain the complex transient behavior of microbial populations. In particular, these models fail to explain many aspects of the dynamics of aTetrahymena pyriformis-Escherichia coli community. In this study an alternative approach, an individual-based model, is employed to investigate the growth and interactions ofTetrahymena pyriformis andE. coli in a batch culture. Due to improved representation of physiological processes, the model provides a better agreement with experimental data of bacterial density and ciliate biomass than previous modeling studies. It predicts a much larger coexistence domain than rudimentary models, dependence of biomass dynamics on initial conditions (bacteria to ciliate biomasses ratio) and appropriate timing of minimal bacteria density. Moreover, it is found that accumulation ofE. coli sized particles andE. coli toxic metabolites has a stabilizing effect on the system.

Notes: Abstract At the core of contemporarymorphometrics—the quantitative study of biological shape variation—is a synthesis of two originally divergent methodological styles. One contributory tradition is the multivariate analysis of covariance matrices originally developed as biometrics and now dominant across a broad expanse of applied statistics. This approach, couched solely in the linear geometry of covariance structures, ignores biomathematical aspects of the original measurements. The other tributary emphasizes the direct visualization of changes in biological form. However, making objective the biological meaning of the features seen in those diagrams was always problematical; also, the representation of variation, as distinct from pairwise difference, proved infeasible. To combine these two variants of biomathematical modeling into a valid praxis for quantitative studies of biological shape was a goal earnestly sought though most of this century. That goal was finally achieved in the 1980s when techniques from mathematical statistics, multivariate biometrics, non-Euclidean geometry and computer graphics were combined in a coherent new system of tools for the complete regionalized quantitative analysis oflandmark points together with the biomedical images in which they are seen. In this morphometric synthesis, correspondence of landmarks (biologically labeled geometric points, like “bridge of the nose”) across specimens is taken as a biomathematical primitive. The shapes of configurations of landmarks are defined as equivalence classes with respect to the Euclidean similarity group and then represented as single points in David Kendall'sshape space, a Riemannian manifold with Procrustes distance as metric. All conventional multivariate strategies carry over to the study of shape variation and covariation when shapes are interpreted in the tangent space to the shape manifold at an average shape. For biomathematical interpretation of such analyses, one needs a basis for the tangent space compatible with the reality of local biotheoretical processes and explanations at many different geometric scales, and one needs graphics for visualizing average shape differences and other statistical contrasts there. Both of these needs are managed by thethin-plate spline, a deformation function that has an unusually helpful linear algebra. The spline also links the biometrics of landmarks to deformation analysis of the images from which the landmarks originally arose. This article reviews the history and principal tools of this synthesis in their biomathematical and biometrical context and demonstrates their usefulness in a study of focal neuroanatomical anomalies in schizophrenia.

Notes: Abstract A competition model describing tumor-normal cell interaction with the added effects of periodically pulsed chemotherapy is discussed. The model describes parameter conditions needed to prevent relapse following attempts to remove the tumor or tumor metastasis. The effects of resistant tumor subpopulations are also investigated and recurrence prevention strategies are explored.

Notes: Abstract Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in the embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.

Notes: Abstract The quasi-stationary distribution of a population within a system of interacting populations is approximated by a stochastic logistic process. The parameters of this process can be expressed in the parameters of the full system. Using the diffusion approximation, an expression for the expected extinction time is derived from this logistic process. Since the expected extinction time is expressed in the parameters of the full system, the effect of these parameters on the extinction risk can be easily evaluated, which may be of use for studies in ecology, conservation biology and epidemiology. The outcome is compared with simulation results for the case of a prey-predator system.

Notes: Abstract The cytokines are the information superhighway of the immune system. They are an important component of the integrated behavior of the system. In order to be able to have a good understanding of the immune system, we must be able to model the effect of cytokines and their combined effect. This work is a step in that direction. We study the combined effect of two cytokines: interleukin-2 (IL-2) and interleukin-4 (IL-4) on some cells of the immune system. Interleukin-2 and interleukin-4 are important growth and differentiation factors for B and T cells. Interleukin-4 antagonizes the effect of interleukin-2 on B cells and some T cells while it synergizes with interleukin-2 on other T cells. We build a mathematical model of the interaction of both cytokines on T and B cells as a building block toward a model of the Th1/Th2 cross-regulation. The response of a given cell to the combination of interleukin-2 and interleukin-4 is shown to involve competing dynamical effects which can lead to either antagnostic or synergistic combined effect.

Notes: Abstract We propose a mathematical approach to the modelling of self-organizing hierarchies in animal societies. This approach relies on a basic positive feedback mechanism that reinforces the ability of a given individual to win or to lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. Motivated by experiments carried out on primitively eusocial waspsPolistes, the model, is based on coupled differential equations supplemented with a small stochastic term. Numerical integrations allow many different hierarchical profiles to be obtained depending on the model parameters: (1) the particular form of the probability for an individual to win or lose a fight given its history, (2) the probability of interaction between two individuals, (3) the forgetting strength, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual and (4) two individual recognition parameters, which set the contribution of individual recognition in the process of hierarchical genesis. We compare the results, expressed in terms of a hierarchical index or of the Landau number that describes the degree of linearity of the hierarchy, with various experimental results.

Notes: Abstract The normal process of dermal wound healing fails in some cases, due to fibro-proliferative disorders such as keloid and hypertrophic scars. These types of abnormal healing may be regarded as pathologically excessive responses to wounding in terms of fibroblastic cell profiles and their inflammatory growth-factor mediators. Biologically, these conditions are poorly understood and current medical treatments are thus unreliable. In this paper, the authors apply an existing deterministic mathematical model for fibroplasia and wound contraction in adult mammalian dermis (Olsenet al., J. theor. Biol. 177, 113–128, 1995) to investigate key clinical problems concerning these healing disorders. A caricature model is proposed which retains the fundamental cellular and chemical components of the full model, in order to analyse the spatiotemporal dynamics of the initiation, progression, cessation and regression of fibro-contractive diseases in relation to normal healing. This model accounts for fibroblastic cell migration, proliferation and death and growth-factor diffusion, production by cells and tissue removal/decay. Explicit results are obtained in terms of the model processes and parameters. The rate of cellular production of the chemical is shown to be critical to the development of a stable pathological state. Further, cessation and/or regression of the disease depend on appropriate spatiotemporally varying forms for this production rate, which can be understood in terms of the bistability of the normal dermal and pathological steady states—a central property of the model, which is evident from stability and bifurcation analyses. The work predicts novel, biologically realistic and testable pathogenic and control mechanisms, the understanding of which will lead toward more effective strategies for clinical therapy of fibro-proliferative disorders.

Notes: Abstract Populations often exhibit abrupt changes in abundance associated with a smooth, continuous change in some component of their environment, with the abruptness usually attributed to inter-specific interactions or physical extremes. This paper presents a spatially explicit single-species population model in which intra-specific interactions alone are responsible for such an abrupt change. The essential mechanism involves cooperation in both colonization (through enhanced recruitment near other individuals) and mortality (protection through a “safety-in-numbers” interaction). Large fluctuations in population density would likely be observable near the transition region.

Notes: Abstract A mathematical model of the nitrogen transformation cycle in an aquatic environment is studied. Using Pontryagin's maximum principle, a preferential utilization of ammonium to nitrate by phytoplankton is explained and verified by experimental data. A multiparameter bifurcation is given. The model was found to have four types of equilibrium sets. It is shown that a Hopf bifurcation may occur.

Notes: Abstract Parallel computation employing a domain decomposition method was used to calculate precisely without approximations the spatio-temporal distribution of Ca2+ in nerve terminals. The results showed, contrary to expectations, that for equal admitted Ca2+ currents at low (one channel open) and high (four channels open) depolarization, the average Ca2+ concentration at the release area is higher at the low depolarization. These calculations provide additional support for the Ca2+-voltage hypothesis for neurotransmitter release.

Notes: Abstract Type I hypersensitivity, which functions to protect the organism from parasites, is caused by binding of antigen to IgE antibodies pre-attached to the cell surface of tissue mast cells and their circulating counterparts, the basophils. In “allergy,” type I hypersensitivity is inappropriately induced by protein-based foreign substances (such as pollen) or protein components of insect stings, which in the normal course of events would be cleared from the organism without causing any damage. Paradoxically, a successful clinical treatment of allergy involves repeated immunization of allergic persons with low doses of the allergen—immunotherapy. Investigation of the available experimental evidence leads to the conclusion that the phenomena of immunotherapy are best addressed in terms of the interplay among the mechanism(s) of immune memory—Th1/Th2 cross-regulation—and the physical compart-mentalization of the immune system. These conclusions are illustrated with a numerical simulation.

Notes: Abstract We consider a host-solitary parasitoid system with three categories of individuals: parasitoids, healthy hosts and parasitized hosts. Parasitoids are assumed to discriminate perfectly between the two kinds of hosts and they can reject those which are already parasitized. If parasitoids systematically accept or reject superparasitism or behave randomly, the system is always unstable. Using an optimal foraging model, we determine the behavior of parasitoids which leads to maximization of the instantaneous reproductive rate. When following this adaptive decision rule, parasitoids accept or refuse superparasitism according to the densities of both healthy and parasitized hosts. We study the dynamics of the system when parasitoids follow the optimal rule and show that under certain conditions it possesses a locally stable equilibrium point. In addition, our model predicts that at equilibrium parasitoids show partial preferences for superparasitism.

Notes: Abstract A system of differential equations for the control of tumor growth cells in a cycle nonspecific chemotherapy is analyzed. Spontaneously acquired drug resistance is taken into account, and a criterion for the selection of chemotherapeutic treatment is used. This criterion purports to describe the possibility of improvement of the patient's health when treatment is discontinued. Contrary to our early results which also take drug resistance into account, in this context strategies of continuous chemotherapy in which rest periods take part may be better than maximum drug concentration throughout the treatment (which appears to be in accordance with clinical practice). This bears out our previous conjecture that when drug resistance is accounted for, the imperfections in the usual modelling of treatment criteria, which in general do not allow for patient recuperation, ruled out the possibility of rest periods in optimal continuous chemotherapy.

Notes: Abstract A logistic density-dependent matrix model is developed in which the matrices contain only parameters and recruitment is a function of adult population density. The model was applied to simulate introductions of white-tailed deer into an area; the fitted model predicted a carrying capacity of 215 deer, which was close to the observed carrying capacity of 220 deer. The rate of population increase depends on the dominant eigenvalue of the Leslie matrix, and the age structure of the simulated population approaches a stable age distribution at the carrying capacity, which was similar to that generated by the Leslie matrix. The logistic equation has been applied to study many phenomena, and the matrix model can be applied to these same processes. For example, random variation can be added to life history parameters, and population abundances generated with random effects on fecundity show both the affect of annual variation in fecundity and a longer-term pattern resulting from the age structure.

Notes: Abstract In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of “regular” (neutrally stable orbits and tori) and chaotic motion typical of non-integrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.

Notes: Abstract The standard method for measuringin vitro antibiotic efficacy is based on a point observation of bacterial activity 18 hours after inoculation. The method, while simple, forgoes significant information by ignoring the dynamics of the interations between antibiotic and bacteria. This paper proposes a simple dynamic model describing these interactions. The model consists of two non-linear differential equations of the S-system type. Its parameter values are estimated, through the minimization of residual errors, from data on the effect of the carbapenem antibiotic imipenem onPseudomonas aeruginosa. The model adequately describes the dynamic behavior of the bacterial populations in the presence of the antibiotic: beginning with drug administration, then through the decline of the bacterial population and possibly ending with bacterial resurgence.

Notes: Abstract We have studied an ecological system of two species, which we denotestrong andweak, respectively, that compete for a single food resource. This system is modelled as a three component reaction-diffusion process. In the presence of a solitary pulse of increased resources, the weaker competitor can diffuse toward this surplus, gaining a competitive advantage and hence persisting in contraposition with the classical Lotka-Volterra result. An exact analytical solution has been found through a quantum mechanical analogy. A stability analysis of this solution against changes in different parameters has been carried out.

Notes: Abstract Collapsible-tube flow with self-excited oscillations has been extensively investigated. Though physiologically relevant, forced oscillation coupled with self-excited oscillation has received little attention in this context. Based on an ODE model of collapsible-tube flow, the present study applies modern dynamics methods to investigate numerically the responses of forced oscillation to a limit-cycle oscillation which has topological characteristics discovered in previous unforced experiments. A devil's staircase and period-doubling cascades are presented with forcing frequency and amplitude as control parameters. In both cases, details are provided in a bifurcation diagram. Poincaré sections, a frequency spectrum and the largest Lyapunov exponents verify the existence of chaos in some circumstances. The thin fractal structure found in the strange attractors is believed to be a result of high damping and low stiffness in such systems.

Notes: Abstract In this work, we show that a one-dimensional model of the blood flow across the lungs can reproduce the evolution of a bolus versus the time. Solving the differential equation governing the bolus concentration in the framework of this model, we determine the solution which fulfills Gaussian initial boundary conditions. An effective parameter related to the ratio of a diffusion coefficient to the square of the mean speed of the flow is defined. The determination of its numerical values following a semi-empirical approach enables us to know accurately the mean transit time and the cardiac output. The results have been compared to other methods, and were found in good agreement. Such an approach could be of interest in all studies where the knowledge of flow—including micro-circulation—is needed.

Notes: Abstract How two species interact during and after colonization influences which of them will be present in each stage of succession. In the tolerance model of ecological succession in a patchy environment, empty patches can be colonized by any species, but the ability to tolerate reduced resource levels determines which species will exclude the other. Here, we analyze a meta-population model of the possible roles of competition in colonization and succession, using non-linear Markov chains as a mathematical framework. Different kinds of competition affect the final equilibrial, abundances of the species involved in qualitatively different ways. An explicit criterion is given to determine which interactions have stronger effects on the final equilibrial levels of the weaker, species. Precise conditions are stated for the co-existence of both species. Both species are more likely to co-exist in the presence of an intermediate disturbance frequency.

Notes: Abstract A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Spontaneously acquired drug resistance is accounted for, as well as the evolution in time of normal cells. In addition, optimization of conflicting objectives forms the aim of the chemotherapeutic treatment. For general cell growth, some results are given, whereas for the special case of Malthusian (exponential) growth of tumor cells and rather general growth rate for normal cells, the optimal strategy is worked out. The latter, from the clinical standpoint, corresponds to maximum drug concentration throughout the treatment.

Notes: Abstract This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.

Notes: Abstract Game theory has had remarkable success as a framework for the discussion of animal behaviour and evolution. It suggested new interpretations and prompted new observational studies. Most of this work has been done with 2-player games. That is the individuals of a population compete in pairwise interactions. While this is often the case in nature, it is not exclusively so. Here we introduce a class of models for situations in which more than two (possibly very many) individuals compete simultaneously. It is shown that the solutions (i.e. the behaviour which may be expected to be observable for long periods) are more complex than for 2-player games. The concluding section lists some of the new phenomena which can occur.

Notes: Abstract A method allowing to measure the inhomogeneous distribution of purines/pyrimidines in nucleotide sequences is developed. We show that this measure relates to the coding or non-coding character of the considered sequence. Coding sequences present a near to the random Pu or Py distribution. This property is shared by both protein-coding DNA and functional RNA-coding DNA. Non-coding sequences present a highly clustered inhomogeneity. We propose the hypothesis, corroborated with appropriate computer simulations, that this is due to the action of various transposition events accumulated for long time periods.

Notes: Abstract The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.

Notes: Abstract Parabolic growth invariably results in the survival of all competing types. Under the constraint of constant total concentration, there is a unique equilibrium in the simplex interior, which is asymptotically stable inside the whole simplex. The appropriate Lyapunov function is obtained in terms of the excess productivity which is shown to be maximized for the competitive system with fractional order kinetics. Claims to the contrary are refuted.

Notes: Abstract Due to the increasing importance of the extracellular matrix in many biological problems, in this paper we develop a model for fibroblast and collagen orientation with the ultimate objective of understanding how fibroblasts form and remodel the extracellular matrix, in particular its collagen component. The model uses integrodifferential equations to describe the interaction between the cells and fibers at a point in space with various orientations. The equations are studied both analytically and numerically to discover different types of solutions and their behavior. In particular we examine solutions where all the fibroblasts and collagen have discrete orientations, a localized continuum of orientations and a continuous distribution of orientations with several maxima. The effect of altering the parameters in the system is explored, including the angular diffusion coefficient for the fibroblasts, as well as the strength and range of the interaction between fibroblasts and collagen. We find the initial conditions and the range of influence between the collagen and the fibroblasts are the two factors which determine the behavior of the solutions. The implications of this for wound healing and cancer are discussed including the conclusion that the major factor in determining the degree of scarring is the initial deposition of collagen.

Notes: Abstract This paper considers the time to extinction for a stochastic epidemic model of SEIR form without replacement of susceptibles. It first shows how previous rigorous results can be heuristically explained in terms of the more transparent dynamics of an approximating deterministic system. The model is then extended to include a host population structured into patches, with weak nearest-neighbour mixing of infection. It is shown, by considering the approximating deterministic system, that the expected time to extinction in a population of n + 1 patches each of size N is of the form a log N + bn, provided that N 〉 N c where N c is a critical patch size below which transits are unlikely to occur. This corresponds to the simple decomposition of the time of an epidemic into the time it takes to spread through one patch plus the time it takes to transit to each of n successive patches. Expressions for this threshold and the coefficients of the time to extinction are given in terms of the transmission parameters of infection and the coupling strength between patches. These expressions are compared with numerical results using parameters relevant to a study of phocine distemper virus in North Sea seals, and the agreement is found to be good for large and small N. In the region when N ≈ N c , where transits may or may not occur, interesting transitional behaviour is seen, leading to a non-monotonicity of the extinction time as a function of N.

Notes: Abstract When directly transmitted infectious diseases are modeled assuming an everlasting induced immunity (and constant contact rate), there are well-established formulas to deal with, which is not true if we include the loss of induced immunity. In general, the immunity induced by the disease is everlasting. We propose a model considering the loss of immunity and present methods for the estimation of two epidemiological parameters: the force of infection and the basic reproduction ratio. We also analyze the effects of the loss of immunity on these parameters. Based on these results, we conclude that reinfection can play an important role in highly vaccinated populations.

Notes: Abstract We studied mathematical models for the length distributions of actin filaments under the effects of polymerization/depolymerization, and fragmentation. In this paper, we emphasize the effects of these two processes acting alone. In this case, simple discrete and continuous models can be derived and solved explicitly (in several special cases), making the problem interesting from a modeling and pedagogical point of view. In a companion paper (Ermentrout and Edelstein-Keshet, 1998, Bull. Math. Biol. 60, 477–503) we investigate what happens when the processes act together, with particular attention to fragmentation by gelsolin, and with a greater level of biological detail.

Notes: Abstract A possible experimental design for combination experiments is to compare the doseresponse curve of a single agent with the corresponding curve of the same agent using either a fixed amount of a second one or a fixed dose ratio. No interaction is then often defined by a parallel shift of these curves. We have performed a systematic study for various types of doseresponse relations both for the dose-additivity (Loewe additivity) and for the independence (Bliss independence) criteria for defining zero interaction. Parallelism between doseresponse curves of a single agent and those of the same agent in the presence of a fixed amount of another one is found for the Loewe-additivity criterion for linear doseresponse relations. For nonlinear relations, one has to differentiate between effect parallelism (parallel shift on the effect scale) and dose parallelism (parallel shift on the dose scale). In the case of Loewe additivity, zero-interaction dose parallelism is found for power, Weibull, median-effect and logistic doseresponse relations, given that special parameter relationships are fulfilled. The mechanistic model of competitive interaction exhibits dose parallelism but not effect parallelism for Loewe additivity. Bliss independence and Loewe additivity lead to identical results for exponential doseresponse curves. This is the only case for which dose parallelism was found for Bliss independence. Parallelism between single-agent doseresponse relations and Loewe additivity mixture relations is found for examples with a fixed doseratio design. However, this is again not a general property of the design adopted but holds only if special conditions are fulfilled. The comparison of combination doseresponse curves with single-agent relations has to be performed taking into account both potency and shape parameters. The results of this analysis lead to the conclusion that parallelism between zero interaction combination and single-agent doseresponse relations is found only for special cases and cannot be used as a general criterion for defining zero-interaction in combined-action assessment even if the correct potency shift is taken into account.

Notes: Abstract The purpose of this study was to investigate strategies in the monotherapy treatment of HIV infection in the presence of drug-resistant (mutant) strains. A mathematical system is developed to model resistance in HIV chemotherapy. It includes the key players in the immune response to HIV infection: virus and both uninfected CD4+ and infected CD4+ T-cell populations. We model the latent and progressive stages of the disease, and then introduce monotherapy treatment. The model is a system of differential equations describing the interaction of two distinct classes of HIV—drug-sensitive (wild type) and drug-resistant (mutant)—with lymphocytes in the peripheral blood. We then introduce chemotherapy effects. In the absence of treatment, the model produces the three types of qualitative clinical behavior—anuninfected steady state, andinfected steady state (latency), andprogression to AIDS. Simulation of treatment is provided for monotherapy, during theprogression to AIDS state, in the consideration of resistance effects. Treatment benefit is based on an increase or retention in CD4+ T-cell counts together with a low viral titer. We explore the following treatment approaches: an antiviral drug which reduces viral infectivity that is administered early—when the CD4+ T-cell count is ≥300/mm3, and late—when the CD4+ T-cell count is less than 300/mm3. We compare all results with data. When treatment is initiated during the progression to AIDS state, treatment prevents T-cell collapse, but gradually loses effectiveness due to drug resistance. We hypothesize that it is the careful balance of mutant and wild-type HIV strains which provides the greatest prolonged benefit from treatment. This is best achieved when treatment is initiated when the CD4+ T-cell counts are greater than 250/mm3, but less than 400/mm3 in this model (i.e. not too early, not too late). These results are supported by clinical data. The work is novel in that it is the first model to accurately simultate data before, during and after monotherapy treatment. Our model also provides insight into recent clinical results, as well as suggests plausible guidelines for clinical testing in the monotherapy of HIV infection.

Notes: Abstract A mathematical model which describes adhesion of bacteria to host cell lines is presented. The model is flexible enough to account for the following situations: extracellular bacteria are either in exponential or in stationary phase. Adhesion is described as a reversible binding process in which the bacteria attach to or detach from specific receptors uniformly distributed on the cell surface. In turn, attached bacteria can either replicate or, conversely, they are restrained to remain in stationary phase. In the first case, however, we must consider the problem of whether the decrease of unoccupied receptors as adhesion progresses imposes a limit to the replicating capacity of the attached bacteria. The effect exerted by the multiplicity of infection (MOI), i.e. the ratio of the number of bacteria to the number of host cells, on the process of adhesion is also contemplated by the model. This has revealed that experiments performed at the same values of MOI can show completely different levels of adhered bacteria, depending on the number of host cells in the assays. This finding demonstrates that the report of the MOI values is insufficient to characterize comparative studies of bacterial adhesion since it could lead to a misunderstanding of the corresponding data. Simplified models based on the steady-state approximation and in equilibrium analysis by means of a Lagmuir adsorption isotherm for the attached bacteria are also discussed. This allows us to define the adhesion coefficient (β) in a given bacterium-cell system so that, with the exception of those systems where these coefficients cannot be defined, larger values of β are related to a greater adhesion capacity. An overview of the procedures to perform quantitative adhesion data analysis is outlined. Finally, theoretical predictions are compared with experimental results from the literature.

Notes: Abstract A new measure of toxicity based on stochastic modelling of single photon-counting processes, representing time-resolved phagocyte luminescence of xenobiotic-perturbed human neutrophils, has been constructed. The stochastic measure of toxicity has been verified by the QSAR method, and then compared and contrasted with the traditional toxicity measure used in bio- and chemiluminescent research. Phenol and benzene homologues were chosen as perturbers due to their importance from the viewpoint of ecotoxicology and occupational medicine.

Notes: Abstract We describe a mathematical model of the flow and deformation in a human teat. Our aim is to compare the theoretical milk yield during infant breast feeding with that obtained through the use of a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk, whereas breast pumps use a cyclic pattern of suction only. Our model is based on quasi-linear poroelasticity whereby the teat is modelled as a cylindrical porous elastic material saturated with fluid. We impose a cyclic axial suction pressure difference across the teat and impose a radial compressive force moving along the teat which mimics infant suckling. This is compared to the case of cyclic and steady pumping only which models the action of breast pumps. The results illustrate that there is an optimal time to apply the compressive force during the suction cycle that will increase the flow rate in our theoretical teat. The model and results may be of use in the future design of effective breast pumps.

Notes: Abstract In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.

Notes: Abstract Adult dermal wounds, in contrast to fetal wounds, heal with the formation of scar tissue. A crucial factor in determining the degree of scarring is the ratio of types I and III collagen, which regulates the diameter of the combined fibers. We developed a reaction-diffusion model which focuses on the control of collagen synthesis by different isoforms of the polypeptide transforming growth factor-β (TGFβ). We used the model to investigate the current controversy as to whether the fibroblasts migrate into the wound from the surrounding unwounded dermis or from the underlying subcutaneous tissue. Numerical simulations of a spatially independent, temporal model led to a value of the collagen ratio consistent with that of healthy tissue for the fetus, but corresponding to scarring in the adult. We investigated the effect of topical application of TGFβ and show that addition of isoform 3 reduces scar tissue formation, in agreement with the experiment. However, numerical solutions of the reaction-diffusion system do not exhibit this sensitivity to growth factor application. Mathematically, this corresponds to the observation that behind healing wavefront solutions, a particular healed state is always selected independent of transients, even though there is a continuum of possible positive steady states. We explain this phenomenon using a caricature system of equations, which reflects the key qualitative features of the full model but has a much simpler mathematical form. Biologically, our results suggest that the migration into a wound of fibroblasts and TGFβ from the surrounding dermis alone cannot account for the essential features of the healing process, and that fibroblasts entering from the underlying subcutaneous tissue are crucial to the healing process.

Notes: Abstract Oscillations in cytosolic Ca2+ concentrations in living cells are often a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1, 4, 5-trisphosphate (IP3)-induced Ca2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP3 receptor that includes the influence of Ca2+ buffers. For concentrations of IP3 in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca2+. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca2+. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.

Notes: Abstract The robustness of patterning events in development is a key feature that must be accounted for in proposed models of these events. When considering explicitly cellular systems, robustness can be exhibited at different levels of organization. Consideration of two widespread patterning mechanisms suggests that robustness at the level of cell communities can result from variable development at the level of individual cells; models of these mechanisms show how interactions between participating cells guarantee community-level robustness. Cooperative interactions enhance homogeneity within communities of like cells and the sharpness of boundaries between communities of distinct cells, while competitive interactions amplify small inhomogeneities within communities of initially equivalent cells, resulting in fine-grained patterns of cell specialization.

Notes: Abstract Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate, numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.

Notes: Abstract A model, based on the principles of continuum mechanics, is presented for the analysis of cell-velocity fields within wool follicles. The model requires specification of three follicle characteristics in the form of spatially varying fields: viscosity, cell density and cell production rate. The viscosity is introduced as an attempt to model both complex intercellular interactions and individual cell deformation as the cells move. It is demonstrated that the distribution of cell production is more important than axial variation in viscosity in determining the overall flow pattern.

Notes: Abstract A microbial trichome extracts nutrients from its immediate surroundings. It may also oxidize electron donors, reduce electron acceptors, and exude the ‘waste’ products of endogenous redox metabolism. Finally, it may effect light harvesting. These exchange fluxes are summed up in a generic model, which covers photoautotrophs as well as chemoheterotrophs. The focus is on endogenous metabolism and the cellular homeostasis of both reducing and phosphorylating equivalents. A novel result is the formulation of four ‘rules’, akin to the Pasteur effect, which govern the compatibility of endogenous metabolism with various assimilatory processes.

Notes: Abstract In this paper we present a deterministic, discrete-time model for a two-patch predator-prey metapopulation. We study optimal harvesting for the metapopulation using dynamic programming. Some rules are established as generalizations of rules for a single-species metapopulation harvesting theory. We also establish rules to harvest relatively more (or less) vulnerable prey subpopulations and more (or less) efficient predator subpopulations.

Notes: Abstract An equivalent electrical circuit is given for a branch of an amphibian motor-nerve terminal in a volume conductor. The circuit allows for longitudinal current flow inside the axon as well as between the axon and its Schwann cell sheath, and also for the radial leakage of current through the Schwann cell sheath. Analytical and numerical solutions are found for the spatial and time dependence of the membrane potential resulting from the injection of depolarizing current pulses by external electrodes at one or two separate locations on the terminal. These solutions show that the depolarization at an injection site can cause a hyperpolarization at sites a short distance away. This effect becomes more pronounced in a short terminal with sealed-end boundary conditions. The hyperpolarization provides a possible explanation for recent experimental results, which show that the average quantal release due to a test depolarizing current pulse delivered by an electrode at one site on a nerve terminal is reduced by the application of an identical conditioning pulse at a neighbouring site.

Notes: Abstract Synthetic barriers such as gloves, condoms and masks are widely used in efforts to prevent disease transmission. Due to manufacturing defects, tears arising during use, or material porosity, there is inevitably a risk associated with use of these barriers. An understanding of virus transport through the relevant passageways would be valuable in quantifying the risk. However, experimental investigations involving such passageways are difficult to perform, owing to the small dimensions involved. This paper presents a mathematical model for analyzing and predicting virus transport through barriers. The model incorporates a mathematical description of the mechanisms of virus transport, which include carrier-fluid flow, Brownian motion, and attraction or repulsion via virus-barrier interaction forces. The critical element of the model is the empirically determined rate constant characterizing the interaction force between the virus and the barrier. Once the model has been calibrated through specification of the rate constant, it can predict virus concentration under a wide variety of conditions. The experiments used to calibrate the model are described, and the rate constants are given for four bacterial viruses interacting with a latex membrane in saline. Rate constants were also determined for different carrier-fluid salinities, and the salt concentration was found to have a pronounced effect. Validation experiments employing laser-drilled pores in condoms were also performed to test the calibrated model. Model predictions of amount of transmitted virus through the drilled holes agreed well with measured values. Calculations using determined rate constants show that the model can help identify situations where barrier-integrity tests could significantly underestimate the risk associated with barrier use.

Notes: Abstract Evaluation of the fluid flow pattern in a non-pregnant uterus is important for understanding embryo transport in the uterus. Fertilization occurs in the fallopian tube and the embryo (fertilized ovum) enters the uterine cavity within 3 days of ovulation. In the uterus, the embryo is conveyed by the uterine fluid for another 3 to 4 days to a successful implantation site at the upper part of the uterus. Fluid movements within the uterus may be induced by several mechanisms, but they seem to be dominated by myometrial contractions. Intra-uterine fluid transport in a sagittal cross-section of the uterus was simulated by a model of wall-induced fluid motion within a two-dimensional channel. The time-dependent fluid pattern was studied by employing the lubrication theory. A comprehensive analysis of peristaltic transport resulting from symmetric and asymmetric contractions is presented for various displacement waves on the channel walls. The results provide information on the flow field and possible trajectories by which an embryo may be transported before implantation at the uterine wall.

Notes: Abstract A mechanistically based mathematical model is used to investigate some of the important factors in priming hepatocytes to enter the G1 phase of the cell cycle. The model considers all of the relevant biochemical mechanisms from signal-receptor binding to the elevation of AP-1(activation protein transcription factor) levels. Focus is centered on the chain of biochemical events governing the sequential activation of protein kinase C (PKC), mitogen-activated protein kinase (MAPK) and AP-1. Factors such as amplitude and duration of growth factors signals, the kinetics of guanosine diphosphate (GDP) to guanosine triphosphate (GTP) conversion, and the negative feedback control mechanisms governing initial steps in cellular replication were theoretically examined. The results of our theoretical assessments support the finding that specific mutations along the PKC-AP1 pathways can have a critical effect on the rate at which cells enter the division cycle.

Notes: Abstract Normal cardiac muscle contraction occurs in response to a rapid rise followed by a slower decay in intracellular calcium concentration. When cardiac muscle cells are loaded with calcium, an intracellular store releases calcium into the cytosol by the process of calcium-induced calcium release (CICR). This release contributes to the rise in intracellular calcium which in turn triggers contraction. We use two qualitative piecewise linear reaction-diffusion models of this behaviour to investigate the speed, stability and waveform of plane waves using singular perturbation techniques.

Notes: Abstract Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1–22 were investigated. The models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. The reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. The Kolmogorov-Smirnov test revealed that even for 1000 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions.

Notes: Abstract To analyze signals measured from human blood flow in the time-frequency domain, we used the wavelet transform which gives good time resolution for high-frequency components and good frequency resolution for low-frequency components. Five characteristic frequency peaks, corresponding to five almost periodic rhythmic activities, were found on the time scale of minutes. These oscillations were characterized by time and spatial invariant measures. The potential of this approach in studying the blood-flow dynamics was illustrated by revealing differences between the groups of control subjects and athletes.

Notes: Abstract The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity.

Notes: Abstract Hemodynamic forces affect endothelial cell morphology and function. In particular, circumferential cyclic stretch of blood vessels, due to pressure changes during the cardiac cycle, is known to affect the endothelial cell shape, mediating the alignment of the cells in the direction perpendicular to stretch. This change in cell shape proceeds a drastic reorganization at the internal level. The cellular scaffolding, mainly composed of actin filaments, reorganize in the direction which later becomes the cell’s long axis. How this external mechanical stimulus is ’sensed’ and transduced into the cell is still unknown. Here, we develop a mathematical model depicting the dynamics of actin filaments, and the influence of the cyclic stretch of the substratum based on the experimental evidence that external stimuli may be transduced inside the cell via transmembrane proteins which are coupled with actin filaments on the cytoplasmic side. Based on this view, we investigate two approaches describing the formulation of the transduction mechanisms involving the coupling between filaments and the membrane proteins. As a result, we find that the mechanical stimulus could cause the experimentally observed reorganization of the entire cytoskeleton simply by altering the dynamics of the filaments connected with the integral membrane proteins, as described in our model. Comparison of our results with previous studies of cytoskeletal dynamics reveals that the cytoskeleton, which, in the absence of the effect of stretch would maintain its isotropic distribution, slowly aligns with the precise direction set by the external stimulus. It is found that even a feeble stimulus, coupled with a strong internal dynamics, is sufficient to align actin filaments perpendicular to the direction of stretch.

Notes: Abstract We study a general predator—prey system in a spatially heterogeneous environment. The predation process, which occurs on a behavioural time-scale, is much faster than the other processes (reproduction, natural mortality and migrations) occurring on the population dynamics time-scale. We show that, taking account of this difference in time-scales, and assuming that the prey have a refuge, the dynamics of the system on a slow time-scale become donor-controlled. Even though predators may control the prey density locally and on a behavioural fast time-scale, nevertheless, both globally and on a slow time-scale, the prey dynamics are independent of predator density: the presence of predators generates a constant prey mortality. In other words, in heterogeneous environments, the prey population dynamics depend in a switch-like manner on the presence or absence of predators, not on their actual density.