ALBERT

Notes: Abstract The entropy budget of a white-tailed deer (50kg) on a maintenance diet and a full-feed diet in a standing posture in an open field under clear nocturnal skies with an air temperature of −20°C is investigated based on the energetics given by Moen. Entropy inflow into a white-tailed deer due to infra-red radiation and entropy outflows from a deer due to infra-red radiation, convection, evaporation of water and conduction to ingested food are calculated. Also the entropy production due to metabolic heat production is estimated. Net entropy flow into a deer from its environment becomes negative. On the assumption that a white-tailed deer is in a steady state in entropy, the total entropy production in a deer on a maintenance diet becomes +0.46 J/sec/K. Positiveness of the entropy production shows that the Second Law of Thermodynamics certainly holds in a white-tailed deer. The entropy production per effective radiating surface area of a deer on a maintenance diet is 0.32×10−4 J/cm2/sec/K. On the other hand, the entropy production in a deer on a full-feed diet is 0.59 J/sec/K and that per effective surface area is 0.41×10−4 J/cm2/sec/K. Uptake of 1 g of food produces 22 J/K of entropy within the body of a white-tailed deer. Comparison is made with the results for entropy production in a lizard and in plant leaves.

Notes: Abstract Biological adaptability has been proved to be analysable by means of the Maximum Entropy Formalism (MAXENT) in some cases of non-interacting systems. This formalism is extended to the biomass statistical structures of populations exhibiting internal interactions (i.e. predatorprey effects).

Notes: Abstract The temporal behaviours of the nonlinear substructure of a self-organized compartmental model of calcium metabolism were investigated. The order-two autocatalytic process included in this simple two-dimensional model is compared to some secondary nucleation mechanisms which should take place at the extracellular fluid-bone interface. The model gives rise to complex dynamic behaviours, and multistability properties, involving up to two stable periodic regimes (birhythmicity), were established in different topological configurations. The bifurcations occurring on the boundaries between regions of different qualitative behaviour have been determined. These properties are discussed in relation to the dynamical behaviour of other two-variable models, especially those including the same nonlinearity.

Notes: Abstract A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.

Notes: Abstract A theoretical approach to the explanation of the structural design of metabolic pathway is presented. It is based on the hypothesis that due to natural selection during evolution the cellular metabolism of present-day organisms may be characterized by optimal properties. Two cardinal terms enter the theory: (i) the efficiency of a metabolic pathway and (ii) the evolutionary effort for the change of the kinetic parameters of enzymes by mutations of the corresponding genes. For both quantities simple mathematical expressions are proposed. While the efficiency is related to the reaction rates of the enzymes constituting the metabolic pathway, the evolutionary effort is considered to be a monotonically increasing function of the parameter values. By maximizing the efficiency under the constraint of a fixed evolutionary effort the theory allows the calculation of the optimal parameter distribution as the outcome of evolution processes. The methods developed are applied to the following systems: (a) linear reaction sequences with very low affinities of the enzymes towards substrates, (b) linear sequences consisting of saturable enzymatic reactions, (c) branched metabolic pathways consisting of segments of linear chains and (d) glycolysis of erythrocytes. The conclusion is derived that the optimal distribution of kinetic constants depends strongly on the equilibrium constants of the reactions as well as on the total osmolarity of the metabolic intermediates. Without osmotic constraints the evolutionary effort is mainly spent on the enzymes at the beginning of the chain. Using Michaelis-Menten equations the optimal state is characterized by a decrease of the maximal activities of the enzymes towards the end of the chain. These results are modified if osmotic constraints are taken into account. At the investigation of branched pathways the following results were obtained: firstly, if a certain end product may be synthesized along different pathways those which are thermodynamically more unfavourable (e.g. characterized by a small change of free energy) are eliminated in the course of evolution; secondly, if a branched pathway leads to several important end products those reaction segments which are thermodynamically unfavourable are characterized by a higher evolutionary effort. The application of the theory to a realistic model of glycolysis of erythrocytes leads to a correct description of various functionally important properties of the system, such as the ratio between fluxes through different branches and the ATP/ADP ratio, whereas the theory cannot predict the strong separation of time constants observed in the real glycolytic system. It is concluded that the improvement of the predictive power of the theory necessitates the use of more complex functionals for the efficiency which take into account not only the fluxes but also other system properties such as the stability of the pathway or homoeostatic effects.

Notes: Abstract We present a new symmetric model of the idiotypic immune network. The model specifies clones of B-lymphocytes and incorporates: (1) influx and decay of cells; (2) symmetric stimulatory and inhibitory idiotypic interactions; (3) an explicit affinity parameter (matrix); (4) external (i.e. non-idiotypic) antigens. Suppression is the dominant interaction, i.e. strong idiotypic interactions are always suppressive. This precludes reciprocal stimulation of large clones and thus infinite proliferation. Idiotypic interactions first evoke proliferation, this enlarges the clones, and may in turn evoke suppression. We investigate the effect of idiotypic interactions on normal proliferative immune responses to antigens (e.g. viruses). A 2-D, i.e. two clone, network has a maximum of three stable equilibria: the virgin state and two asymmetric immune states. The immune states only exist if the affinity of the idiotypic interaction is high enough. Stimulation with antigen leads to a switch from the virgin state to the corresponding immune state. The network therefore remembers antigens, i.e. it accounts for immunity/memory by switching beteen multiple stable states. 3-D systems have, depending on the affinities, 9 qualitatively different states. Most of these also account for memory by state switching. Our idiotypic network however fails to account for the control of proliferation, e.g. suppression of excessive proliferation. In symmetric networks, the proliferating clones suppress their anti-idiotypic suppressors long before the latter can suppress the former. The absence of proliferation control violates the general assumption that idiotypic interactions play an important role in immune regulation. We therefore test the robustness of these results by abandoning our assumption that proliferation occurs before suppression. We thus define an “escape from suppression” model, i.e. in the “virgin” state idiotypic interactions are now suppressive. This system erratically accounts for memory and never for suppression. We conclude that our “absence of suppression from idiotypic interactions” does not hinge upon our “proliferation before suppression” assumption.

Notes: Abstract Analytical bounding functions for diffusion problems with Michaelis-Menten kinetics were recently presented by Anderson and Arthurs, 1985 (Bull. math. Biol. 47, 145–153). Their methods, successful to some extent for a small range of parameters, has the disadvantage of providing a weak upper bound. The optimal approach for the use of one-line bounding kinetics is presented. The use of two-line bounding kinetics is also shown, in order to give, sufficient accuracy in those cases where the one-line approach does not provide satisfactory results. The bounding functions provide excellent upper and lower bounds on the true solution for the entire range of kinetic and transport parameters.

Notes: Abstract Thresholds for survival and extinction are important for assessing the risk of mortality in systems exposed to exogeneous stress. For generic, rudimentary population models and the classical resource-consumer models of Leslie and Gallopin, we demonstrate the existence of a survival threshold for situations where demographic parameters are fluctuating, generally, in a nonperiodic manner. The fluctuations are assumed, to be generated by exogenous, anthropogenic stresses such as toxic chemical exposures. In general, the survival threshold is determined by a relationship between mean stress measure in organisms to the ratio of the population intrinsic growth rate and stress response rate.

Notes: Abstract A stochastic analog to a deterministic model describing subpopulation emergence in heterogeneous tumors is developed. The resulting system is described by the Fokker-Planck or forward Kolmogorov equation. A finite element approach for the numerical solution to this equation is described. Four biological and clinical scenarios are simulated (emergence of heterogeneity, exclusion of a subpopulation, and induction of drug resistance in both pure and heterogeneous tumors). The results of the simulations show that the stochastic model describes the same basic dynamics as its deterministic counterpart via a convective component, but that for each simulation a distribution of tumor sizes and mixes can also be derived from a diffusive component in the model. These distributions yield estimates for subpopulation extinction probabilities. The biological and clinical relevance of these results are discussed.

Notes: Abstract Bayesian image processing formalisms which incorporatea priori information about valued-uncorrelated and valued-correlated (patterned) source distributions are introduced and the corresponding iterative algorithms are derived using the EM technique. Striking improvement in image processing is demonstrated when applying these algorithms to Poisson and Gaussian randomized data in one-dimensional cases.

Notes: Abstract Two algorithms for the efficient identification of segment neighborhoods are presented. A segment neighborhood is a set of contiguous residues that share common features. Two procedures are developed to efficiently find estimates for the parameters of the model that describe these features and for the residues that define the boundaries of each segment neighborhood. The algorithms can accept nearly any model of segment neighborhood, and can be applied with a broad class of best fit functions including least squares and maximum likelihood. The algorithms successively identify the most important features of the sequence. The application of one of these methods to the haemagglutinin protein of influenza virus reveals a possible mechanism for conformational change through the finding of a break in a strong heptad repeat structure.

Notes: Abstract Given a sequenceA and regular expressionR, theapproximate regular expression matching problem is to find a sequence matchingR whose optimal alignment withA is the highest scoring of all such sequences. This paper develops an algorithm to solve the problem in timeO(MN), whereM andN are the lengths ofA andR. Thus, the time requirement is asymptotically no worse than for the simpler problem of aligning two fixed sequences. Our method is superior to an earlier algorithm by Wagner and Seiferas in several ways. First, it treats real-valued costs, in addition to integer costs, with no loss of asymptotic efficiency. Second, it requires onlyO(N) space to deliver just the score of the best alignment. Finally, its structure permits implementation techniques that make it extremely fast in practice. We extend the method to accommodate gap penalties, as required for typical applications in molecular biology, and further refine it to search for substrings ofA that strongly align with a sequence inR, as required for typical data base searches. We also show how to deliver an optimal alignment betweenA andR in onlyO(N+logM) space usingO(MN logM) time. Finally, anO(MN(M+N)+N 2logN) time algorithm is presented for alignment scoring schemes where the cost of a gap is an arbitrary increasing function of its length.

Notes: Abstract The stochastic complexity of a data base of 365 protein-coding regions is analysed. When the primary sequence is modeled as a spatially homogeneous Markov source, the fit to observed codon preference is very poor. The situation improves substantially when a non-homogeneous model is used. Some implications for the estimation of species phylogeny and substitution rates are discussed.

Notes: Abstract We present, in an easy to use form, the large deviation theory of the binomial distribution: how to approximate the probability ofk or more successes inn independent trials, each with success probabilityp, when the specified fraction of successes,a≡k/n, satisfies 0〈p〈a〈1.

Notes: Abstract A linear segment in which a number of pairs of intervals of equal length are identified as potential stems is the subject of a folding problem analogous to inference of RNA secondary structure. A quantity of free energy (or equivalently, energy per unit length) is associated with each stem, and the various types of loops are assigned energy costs as a function of their lengths. Inference of stable structures can then be carried out in the same way as in RNA folding. More important, perturbation of stem lengths and energy densities (modelling various mutational processes affecting nucleotide sequences) allows the delineation of domains of stability of various foldings, through the explicit calculation of their boundaries, in a low-dimensional parameter space.

Notes: Abstract In sensory physiology, various System Identification methods are implemented to formalized stimulus-response relationships. We applied the Volterra approach for characterizing input-output relationships of cells in the medial geniculate body (MGB) of an awake squirrel monkey. Intraspecific communication calls comprised the inputs and the corresponding cellular evoked responses—the outputs. A set of vocalization was used to calculate the kernels of the transformation, and these kernels subserved to predict the responses of the cell to a different set of vocalizations. It was found that it is possible to predict the response (PSTH) of MGB cells to natural vocalizations, based on envelopes of the spectral components of the vocalization. Some of the responses could be predicted by assuming a linear transformation function, whereas other responses could be predicted by non-linear (second order) kernels. These two modes of transformation, which are also reflected by a distinct spatial distribution of the linearvis-à-vis non-linear responding cells, apparently representa new revelation of parallel processing of auditory information.

Notes: Abstract The time-dependent surface coverage of antigen-antibody complexes for a sensor in which antigens are bound to surface immobilized antibodies is determined analytically. Assuming a reversible first order reaction between the antigens and antibodies, a model is derived describing the dynamical response of the sensor. The surface coverage is related explicitly to the antigen concentration which is of special interest in experimental situations. The stationary state and short time behaviour are determined explicitly. Several illustrations of the full solution are provided.

Notes: Abstract Simple reaction time is the minimum time required to respond to a signal such as a steady light or tone. Such a reaction time is taken to be the time required for transmission of a fixed quantity of information, ΔH, from stimulus to subject. That is, information summation replaces energy summation. This information is calculated from consideration of the quantum nature of the stimulus. The theoretically derived equation for reaction time is fitted to experimental data. Piéron's empirical law for reaction time is obtained as an approximation from a proposed informational equation. The exponent in Piéron's law is found to be the same as the exponent in the power law of sensation. Threshold appears to be the smallest stimulus capable of transmitting the quantity of information ΔH.

Notes: Abstract The performance of a symmetric nonrecursive filter can be improved by multiple use of the same filter. The method is based on an Amplitude Change Function (ACF). An approach to the design of nonrecursive filters using an ACF is discussed in this paper. The prototype filter chosen is a Recursive Running Sum (RRS) filter which does not require any multipliers for its implementation. The required filter specifications are met by multiple use of the RRS filters. The overall filter requires a much smaller number of multiplications and adders than the one designed using the conventional method. It is shown that this method provides reduced noise due to coefficient quantization and product quantization compared with the conventional design technique.

Notes: Abstract This paper establishes the large-sample accuracy properties of two nonlinear least-squares estimators (NLSE) of sine-wave parameters: the basic NLSE, which ignores the possible correlation of the noise, and the optimal NLSE, which, besides the sine-wave parameters, also estimates the noise correlation (appropriately parametrized). It is shown that these two NLS estimators have thesame accuracy in large samples. This result provides complete justification for preferring the computationally less expensive basic NLSE over the “optimal” NLSE. Both estimators are shown to achieve the Cramér-Rao Bound (CRB) as the sample size increases. A simple explicit expression for the CRB matrix is provided, which should be useful in studying the performance of sine-wave parameter estimators designed to work in the colored-noise case.

Notes: Abstract Multidimensional lossless networks are of special interest for use as reference structures for multidimensional wave digital filters [l]–[3]. The starting point of the presented synthesis procedure for two-dimensional representatives of the networks mentioned is a scattering matrix description of the desired multiport. This given matrix is assumed to have those properties which have turned out to be necessary [9], [10] for any scattering matrix of a multidimensional lossless network. The method presented for the synthesis of 2-D reactancem-ports is based mainly on known properties of block-companion matrices and the factorization of a univariable rational matrix which is discrete para-Hermitian and nonnegative definite on the unit circle. The resulting network always contains only a minimal number of frequency-dependent building elements. No restrictions are made concerning the coefficients of the rational entries of the scattering matrix; they may be either real or complex, so as to include even complex networks which are of special interest for multi-dimensional wave digital filters [3].

Notes: Abstract Examples are given concerning the range of applicability of recent representation results that provide a means of studying the input-output properties of nonlinear systems in terms of the familiar impulse-response concept, and which extend the concept of integral transformation to nonlinear maps. We show that such representations, which we call “g-” and “h-representations,” exist for important classes of systems governed by nonlinear integral equations. In particular, it is proved that a large class of maps that have Volterra series representations also have these representations.

Notes: Abstract For a broad class of interconnected nonlinear systems, this paper develops a complete design methodology for decentralized variable structure control. Specifically, the paper sets forth design schemes for local switching surfaces and the related local switched feedback gains which together force the original nonlinear interconnected system to behave as a reduced order interconnected equivalent system having a desired response such as stability, tracking, or prespecified eigenvalues. Also developed is a numerical algorithm for constructing the switched local feedback gains. A simple nonlinear example illustrates the control strategy.

Notes: Abstract A new stabilization method of a large-scale dynamic system, consisting of a set of interconnected subsystems, is presented in this paper. The topology of the interconnected subsystems is given as a network containing nodes with only one ingoing link, and none, one, or more outgoing links. Here, when the notion “node” is used a subsystem is assumed, and the links stand for the subsystem interconnections. The stabilization method is made only by the use of local linear state feedback around each subsystem, in order to satisfy constraints given in the problem. The interconnections among the subsystems are assumed to be nonlinear, time-varying. According to the topology of the large-scale system, the method of stabilization is hierarchic, one proceeds from node to node, and is applicable from a computer standpoint. A design algorithm follows directly, and can be made using the Generate and Test method for each subsystem independently, thus enabling designers to use a computer which has a video terminal as a peripheral unit and providing a possibility for interactive applications.

Notes: Abstract The estimation of covariance matrices which are structured, for example, of Toeplitz type, from measurement data is considered. The problem is considered in the context of array beamforming, and various methods of estimation are derived and compared, such comparison including consideration of the behavior of the estimate in beamforming applications.

Notes: Abstract The planar least-squares inverse (PLSI) polynomials are used for stabilization of two-dimensional unstable recursive filters. In order to obtain the PLSI polynomials, the main work involved consists in forming a set of linear equations and then solving them. In this paper we present an efficient and simple method to form the necessary set of linear equations (i.e., the required coefficient matrix) for a chosen pattern and order of the desired PLSI polynomial, starting from the denominator polynomial of a two-dimensional unstable recursive filter.

Notes: Abstract This paper shows how to use orthogonal functions to invert singular (i.e., generalized state-space) systems. The approach is to express the inverse system itself as a singular system, and then to apply the theory of orthogonal functions to convert that differential-algebraic system to a purely algebraicgeneralized Lyapunov equation whose solution yields the input of the original system given its output. Both left and right inversion are treated. Necessary and sufficient conditions for the existence and uniqueness of the generalized Lyapunov equation are derived, and a generalizedQZ algorithm is given for its efficient solution. It is also shown that the coefficients in the Walsh function expansion may be approximately found using an FFT-type butterfly network. These results provide both an extension in theory, by investigating the properties of a new Lyapunov equation, and an extension in the implementation of system inversion, by providing a scheme which applies to generalized state-space systems and uses an unconventional approach which may prove to be a useful contribution.

Notes: Abstract In digital communication networks, a special class of complex biquad recursive digital filters called orthogonal filters is increasingly being used. The separate effects of the overflow and quantization nonlinearities on these orthogonal filters' responses have been investigated [5], [6]. In this paper we examine the zero-input stability properties of the actual orthogonal filter having both overflow and quantization nonlinearities. The overflow nonlinearities considered include saturation, bit-by-bit inversion, zeroing, and modulo 2 arithmetic. The quantization techniques used may be roundoff, magnitude, or value truncation. An example demonstrates the adverse coupling effect between the overflow and quantization nonlinearities. Two criteria are therefore derived to ensure asymptotic overflowstability of the filter in the presence of quantization. These criteria have been translated to the coefficient plane; various regions corresponding to different minimum wordlengths required to ensure decoupling of the overflow and quantization phenomena have been derived.

Notes: Abstract The identification problem for electromagnetic objects excited by transients is discussed. Several classes of models are reviewed, and an output error model is selected. An algorithm for solving the transient identification problem using this model is presented, and some of the issues connected with its use are considered. Examples of the application of this algorithm to electromagnetic data are given.

Notes: Abstract A stochastic model for the population regulated by logistic growth and spreading in a given region of two-or three-dimensional space has been introduced. For many-species population the interactions among the species have also been icorporated in this model. From the random variables that describe stochastic processes of a Wiener type the space-dependent random population densities have been formed and shown to satisfy the Langevin equations. The Fokker-Planck equation corresponding to these Langevin equations has been approximately solved for the transition probability of the population spreading and it has been found that such approximate expressions of the transition probability depend on the solutions of the deterministic equations of the diffusion model with logistic growth and interactions. Also, the stationary or equilibrium solutions of the Fokker-Planck equation together with the special discussion on the pattern of single-species population spreading have been made.

Notes: Abstract The notion of an evolutive hierarchical system proposed in this paper is a mathematical model for systems, like organisms, with more or less complex objects. This model, based on category theory, retains the following characteristics of natural systems: they have an internal organization consisting of components with interrelations; they maintain their organization in time though their components are changing; their components are divided into several levels corresponding to the increasing complexity of their own organization, and the system may be studied at any of these levels (e.g. molecular, cellular...). The state of the system at a given instant is modeled by a category whose objects are its components, the state transition by a functor, a complex object by the (direct) limit of a pattern of linked objects (which describes its internal organization). The properties of limits in a category make it possible to ‘measure’ the emergence of properties for a complex object with respect to its components, and to reduce the study of a hierarchical system to that of its components of the lowest degree and their links. Categorical constructions describe the formation of a hierarchical evolutive system stepwise, by means of the operations: absorption of external objects, destruction of some components, formation of new complex objects.

Notes: Abstract An algorithmic formulation is presented for the inference procedure concerning lineage models. The problem is to find lineage rules from observed sequences of tree structures under the assumption that no interactions take place in the course of development and that sufficiently frequent observations are available at equal time intervals. The underlying structural pattern is taken to be a OL system, and the goal is to find propagating and deterministic OL schemes with minimal properties satifsying certain biological reliance criteria. Upper bounds have been found for the complexity of the inference algorithms.

Notes: Abstract A statistical theory of non-equilibrium fluctuation in Volterra-Lotka systems has been presented on the basis of the technique of statistical linearization of non-linear coupled stochastic differential equations.

Notes: Abstract Under certain basic assumptions the branching pattern of dendrites can be modeled as a Galton-Watson process in varying environment. Using results from graph theory we compute the probability distributions, expectations and variances for biologically significant variables such as the number of (intermediate and terminal) branches, the maximum number of orders, etc., together with the limit behavior of these quantities. Furthermore, the probability measure induced by the Galton-Watson process on the set of all trees is calculated. The measure assigns to any set of branching patterns the probability that it is realized by a certain process, which is completely described through the bifurcation probabilities.

Notes: Abstract Theoretical models for DNA repair mechanisms are constructed. Reliability studies considering the living cell as a repairable system are done. The DNA repair process is discussed along with applications and comparison with available experimental data.

Notes: Abstract A mathematical model has been developed to simulatein vivo transmural accumulation of an intravenously injected tracer in the aortic wall of experimental animals. Parameters have been included to represent the following processes that affect tracer distribution: permeation of the blood-tissue interface, diffusion through the layers of the artery wall,convective solute drag through the same, and degradation. Of particular interest for thein vivo situation situation is the inclusion of boundary conditions that account for the variation in the plasma concentration of injected tracer as a function of time. Two analytical solutions are presented. The first describes a system in which two boundaries must be delineated; it pertains if the tracer is allowed to circulate until it enters the avascular media of the artery wall both across its luminal boundary and from the capillaries in its outer layer. The second applies to shorter duration experiments in which entry across only the luminal boundary is considered. A limiting case of the solution for short circulation times is presented, compared with a previously published solution, and examined for its potential utility in parameter estimation. Because of its treatment of time-dependent boundary conditions, the model has unique application toin vivo experiments related to macromolecular transport in atherosclerosis that may otherwise elude proper interpretation.

Notes: Abstract Several current reaction-diffusion mechanisms have been proposed as models for morphogenesis in the Turing (1952,Phil. Trans. R. Soc. Lond. B 237, 37–72) sense. We introduce and exploit a quantity, we have termed heterogeneity, which allows us to elaborate the differences between the various models with regard to spatial pattern formation. It is shown that this quantity provides a concise view for the comparison of theoretical models with experimental observations. Two model mechanisms are treated explicitly both for linear and for biased diffusion.

Notes: Abstract The optimum number of total capillaries in the whole human body was estimated from the analysis of the efficiency for oxygen (O2) transport in the vascular-tissue system. We used a tissue model composed of uniform spheres in which O2 diffuses from the capillary located at the centre of each sphere towards the surrounding tissue consuming O2 at a constant rate. The tissue mass supplied by a single capillary was estimated as the area of positive O2 concentration under a given condition of capillary flow and O2 consumption rate. Total tissue mass was determined as the function of the capillary numbern and the total blood flow. On the other hand, the energy cost required to maintain the vascular system withn terminals was assessed by using the minimum volume model (Kamiya and Togawa,Bull. math. Biophys. 34, 431–438, 1972). The efficiency of the entire vascular-tissue system was evaluated by calculating the ratio of total tissue mass/cost function. The result of the calculation using physiological data of cardiac output and O2 consumption for an average human adult during a heavy exercise revealed the maximum efficiency occurring at the capillary number 3.7×1010 which coincided well with its normal range of physiological estimates (3.2×1010–4.2×1010). We concluded that the entire vascular-tissue system is constructed so as to attain the highest efficiency in O2 transport at its maximum activity.

Notes: Abstract A kinetic model involving synthesis of proinsulin in the rough endoplasmic reticulum, maturation through the Golgi apparatus and granules, with conversion to insulin, is proposed to account for data on the amount of insulin and of proinsulin both secreted during various time intervals and remaining in islets. Introducing three compartments for granules makes it possible to account for the measurement of both hot (pulse labeled with tritiated leucine) and cold proinsulin and insulin over a period of 21/2 hr under constant glucose. Data from islets from animals pretreated with tolbutamide are also presented and modeled. The model is then expanded so that it can be successfully applied to available data on the effects of a period of glucose deprivation on secretion of both hot and cold hormone. Parameters have essentially the same values, where they overlap, as were obtained (Landahl and Grodsky, 1982Bull. math. Biol. 44, 399–410) from insulin secretion by perfused rat pancreas stimulated by a variety of temporal patterns of glucose concentration.

Notes: Abstract Two models of binary tree growth are examined in terms of the Strahler order branching ratio (Rb) and the types of vertex produced during growth, and their inter-relationship. The sequential growth model is that described by Van Pelt and Verwer (1985,Bull. math. Biol. 47, 323–336) in which random growth occurs according to attributed probabilities on terminal or internal segments, one branch at a time. This model generates values ofRb≥3. The synchronous growth model is new and permits more than one segment to branch at a time, again randomly with attributed probabilities. This model generates values ofRb≥2 and in particular, when only terminal branching is permitted, gives 2≤Rb〈3. Such a model might explain the branching in the human bronchial tree, in which 2.5≤Rb≤2.8. Our synchronous model is an alternative to the centrifugal-order-dependent sequential model of Van Pelt and Verwer.

Notes: Abstract From an energy budget of a deciduous plant leaf in moderate conditions, entropy fluxes into or out of the leaf due to solar radiation, infrared radiation, evaporation of water and heat conduction are calculated. Net entropy flow into the leaf is negative. On the assumption that the entropy in the leaf is in a steady state, the entropy production in the typical deciduous leaf in moderate conditions [the solar energy absorbed by both sides of the leaf isE solar=0.0602 (J cm−2 s−1)] becomesS prod=1.8×10−4 (J cm−2 s−1 K−1). The positiveness of the entropy production shows that the Second Law of Thermodynamics certainly holds in the plant leaf. Entropy productions in other conditions are also calculated. The entropy production in the leafS prod becomes a linear function of the solar energy absorbed by the leafE solar:S prod≈-(29.5E solar)×10−4. A theorem is presented: the entropy production in plant leaves oscillates during the period of one day, paralleling the daily solar energy absorbed by leaves.

Notes: Abstract Molecular biologists strive to infer evolutionary relationships from quantitative macromolecular comparisons obtained by immunological, DNA hybridization, electrophoretic or amino acid sequencing techniques. The problem is to find unrooted phylogenies that best approximate a given dissimilarity matrix according to a goodness-of-fit measure, for example the least-squares-fit criterion or Farris'sf statistic. Computational costs of known algorithms guaranteeing optimal solutions to these problems increase exponentially with problem size; practical computational considerations limit the algorithms to analyzing small problems. It is established here that problems of phylogenetic inference based on the least-squares-fit criterion and thef statistic are NP-complete and thus are so difficult computationally that efficient optimal algorithms are unlikely to exist for them.

Notes: Abstract This article extends the use of dynamic programming algorithms in molecular sequence comparison to the alignment of the α-carbon (Cα-) coordinates of two protein structures in three dimensions. The algorithm is described in detail and is applied to the comparison of α-lactalbumin with both hen egg white lysozyme and T4 lysozyme. In the first case, the structures are similar, while the second comparison is between two distantly related molecules. References are made to the usual sequence alignments. A variety of complementary methods are introduced to display the results.

Notes: Abstract The composition of naturally occurring DNA sequences is often strikingly heterogeneous. In this paper, the DNA sequence is viewed as a stochastic process with local compositional properties determined by the states of a hidden Markov chain. The model used is a discrete-state, discreteoutcome version of a general model for non-stationary time series proposed by Kitagawa (1987). A smoothing algorithm is described which can be used to reconstruct the hidden process and produce graphic displays of the compositional structure of a sequence. The problem of parameter estimation is approached using likelihood methods and an EM algorithm for approximating the maximum likelihood estimate is derived. The methods are applied to sequences from yeast mitochondrial DNA, human and mouse mitochondrial DNAs, a human X chromosomal fragment and the complete genome of bacteriophage lambda.

Notes: Abstract A new and apparently rather useful and natural concept in cluster analysis is studied: given a similarity measure on a set of objects, a sub-set is regarded as a cluster if any two objectsa, b inside this sub-set have greater similarity than any third object outside has to at least one ofa, b. These clusters then form a closure system which can be described as a hypergraph without triangles. Conversely, given such a system, one may attach some weight to each cluster and then compose a similarity measure additively, by letting the similarity of a pair be the sum of weights of the clusters containing that particular pair. The original clusters can be reconstructed from the obtained similarity measure. This clustering model is thus located between the general additive clustering model of Shepard and Arabie (1979) and the standard hierarchical model. Potential applications include fitting dendrograms with few additional nonnested clusters and simultaneous representation of some families of multiple dendrograms (in particular, two-dendrogram solutions), as well as assisting the search for phylogenetic relationships by proposing a somewhat larger system of possibly relevant “family groups”, from which an appropriate choice (based on additional insight or individual preferences) remains to be made.

Notes: Abstract An important component of computer programs for determining the solution conformation of proteins and other flexible molecules from nuclear magnetic resonance data are the so-called “bound smoothing algorithms”, which compute lower and upper limits on the values of all the interatomic distances from the relatively sparse set which can usually be measured experimentally. To date, the only methods efficient enough for use in large problems take account of only the triangle inequality, but an appreciable improvement in the precision of the limits is possible if the algebraic relations between the distances among each quadruple of atoms are also considered. The goal of this paper is to use a recently improved algorithm for computing these “tetrangle inequality limits” to determine just how much improvement really is possible, given the types of experimental data that are usually available.

Notes: Abstract We apply the concept of marginal stability hypothesis, which has been proposed for solving the problem of dendritic crystal growth, to the pattern selection problem in the Gierer-Meinhardt models. In the case of a large system, the system selects a definite wavelength of the ultimate spatial pattern when the unstable homogeneous steady state is locally disturbed. The numerical results are analyzed theoretically by means of the marginal stability hypothesis, and they are in good agreement with it. Biologically, these results imply why for large systems the Gierer-Meinhardt model (and presumably other reaction-diffusion schemes) have the ability to explain the observation that pattern-generating mechanisms are remarkably insensitive to a wide range of environmental and experimental conditions.

Notes: Abstract Small networks of threshold automata are used to model complex interactions between populations of regulatory cells (helpers and suppressors, antigen specific and anti-idiotypic) which participate in the immune response. The models, being discrete and semiquantitative, are well adapted to the situation of incomplete information often encounteredin vivo. However, the dynamics of many different network structures usually end up in the same attractor set. Thus, many different theories are equivalent in their explicative power for the same facts. This property, known as underdetermination of the theories by the facts, is given a quantitative estimate. It appears that such an underdetermination, as a kind of irreductible complexity, can be expected in manyin vivo biological processes, even when the number of interacting and functionally coupled elements is relatively small.

Notes: Abstract The “paradox of enrichment” predicts that increasing the growth rate of the resource in a resource-consumer dynamic system, by nutrient enrichment, for example, can lead to local instability of the system—that is, to a Hopf bifurcation. The approach to the Hopf bifurcation is accompanied by a decrease in resilience (rate of return to equilibrium). On the other hand, studies of nutrient cycling in food webs indicate that an increase in the nutrient input rate usually results in increased resilience. Here these two apparently conflicting theoretical results are reconciled with a model of a nutrient-limited resource-consumer system in which the tightly recycled limiting nutrient is explicitly modelled. It is shown that increasing nutrient input may at first lead to increased resilience and that resilience decreases sharply only immediately before the Hopf bifurcation is reached.

Notes: Abstract Statistical properties of topological binary trees are studied on the basis of the distribution of segments in relation to centrifugal order. Special attention is paid to the mean of this distribution in a tree as it will be used as a measure of tree topology. It will be shown how the expectation of the mean centrifugal order depends both on the size of the tree and on the mode of growth in the context of modelling the growth of tree structures. Observed trees can be characterized by their mean orders and procedures are described to find the growth mode that optimally corresponds to these data. The variance structure of the mean-order measure appears to be a crucial factor in these fitting procedures. Examples indicate that mean-order analysis is an accurate alternative to partition analysis that is based on the partitioning of segments over sub-tree pairs at branching points.

Notes: Abstract We propose certain general conditions that we believe are reasonable for any pattern recognition algorithm. We find that these conditions give rise to paradoxical identification. The algorithms are incapable of distinguishing composite patterns and must be able to distinguish patterns at an atomistic level.

Notes: Abstract A stratigraphically oriented series of the Miocene foraminiferal speciesBrizalina mandoroveensis from Ikang, Cameroon, was analyzed both by conventional multivariate morphometric procedures and by the tensor biometric method of Bookstein (1986;Statist. Sci. 1, 181–142), a method which analyzes sets of landmark points rather than specific variables of shape or size. The conventional analysis used five size-measures upon 170 specimens from five stratigraphic levels; the tensor analysis encompassed six landmarks (12 coordinates) upon 50 specimens. Whereas certain features appeared in both analyses, such as the separation between levels one and five, the techniques did not always agree with respect to the interpretation of those findings or about most details in the sequence of mean phenotypes. The canonical variate analysis bases its ordination upon a general size factor (the meaning of which is obscured by the foreshortening of within-group variation which is built into the technique). The tensor analysis locates a similar ordination using mainly features of shape.

Notes: Abstract In a separate paper, we developed a mathematical model describing HIV infection and used it to suggest experiments for quantifying characteristic viral parameters. In this paper we generalize the model to any well-mixed assay system. We also present complete and rigorous derivations of fundamental results needed for the design and analysis of HIV infectivity assays. The model is applicable to infectious agents with multiple receptors for their target cell (e.g. HIV, Epstein-Barr virus and Plasmodium), and to blockers (both reversible and irreversible), as long as blocker and target cells are the same diffusion compartment.

Notes: Abstract A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical valuer=1 of the inherent net reproductive rater. Stability is studied in this general setting near the bifurcation point and conditions are derived that determine which of these two bifurcating branches is the stable branch. These general results are supplemented by numerical studies of the asymptotic dynamics over wider parameter ranges where various other bifurcations and stable attractors are found. The implications of these results are discussed with respect to the effects on stability that age class competition within a population can have and whether such competition is stabilizing or destabilizing.

Notes: Abstract Phototransduction is a process which links the absorption of photons by a rod or cone to the modulation of voltage across the cell membrane. An important feature of many vertebrate photoreceptors is a mechanism that adjusts the sensitivity and dynamics of the response to light according to the level of illumination. We construct a system of ordinary differential equations that models what are currently thought to be the important molecule mechanisms involved in phototransduction: this includes consideration of both intracellular enzyme kinetics and the properties of light-insensitive and light-sensitive conductances in the cone membrane. The system contains negative feedback whose functional form is determined by constraining the steady-state behaviour of the system. Despite the highly nonlinear nature of the system of ordinary differential equations, our methods permit us to derive an analytic expression for the first-order frequency response parametric in the steady-state value of only one dynamic variable, the light input. Various unknown kinetic parameters are found by fitting the model to experimental data on the first-order frequency response of cones measured at several mean light levels spanning a range of four log units. Good fits are obtained to the data, and the computed shape of the feedback function agrees qualitatively with recent experiment. Moreover, the model accounts for the dramatic speeding up of the response kinetics and the decrease in response gain with increasing light level.

Notes: Abstract In order to understand the abnormal flow conditions of blood in a locally constricted blood vessel, the analytical results are obtained for the oscillatory flow of blood which behaves as a Newtonian fluid. It is here assumed that the surface roughness is cosine-shaped and the maximum height of the roughness is very small compared with the radius of the unconstricted tube. Numerical solutions are presented for the instantaneous flow rate, resistive impedance, wall shear stress and phase lag.

Notes: Abstract In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.

Notes: Abstract We first analyse a simple symmetric model of the idiotypic network. In the model idiotypic interactions regulate B cell proliferation. Three non-idiotypic processes are incorporated: (1) influx of newborn cells; (2) turnover of cells: (3) antigen. Antigen also regulates proliferation. A model of 2 B cell populations has 3 stable equilibria: one virgin, two immune. The twodimensional system thus remembers antigens, i.e. accounts for immunity. By contrast, if an idiotypic clone proliferates (in response to antigen), its anti-idiotypic partner is unable to control this. Symmetric idiotypic networks thus fail to account for proliferation regulation. In high-D networks we run into two problems. Firstly, if the network accounts for memory, idiotypic activation always propagates very deeply into the network. This is very unrealistic, but is an implication of the “realistic” assumption that it should be easier to activate all cells of a small virgin clone than to maintain the activation of all cells of a large (immune) clone. Secondly, graph theory teaches us that if the (random) network connectance exceeds a threshold level of one interaction per clone, most clones are interconnected. We show that this theory is also applicable to immune networks based on complementary matching idiotypes. The combination of the first “percolation” result with the “interconnectancr” result means that the first stimulation of the network with antigen should eventually affect most of the clones. We think this is unreasonable. Another threshold property of the network connectivity is the existence of a virgin state. A gradual increase in network connectance eliminates the virgin state and thus causes an abrupt change in network behaviour. In contrast to weakly connected systems, highly connected networks display autonomous activity and are unresponsive to external antigens. Similar differences between neonatal and adult networks have been described by experimentalists. The robustness of these results is tested with a network in which idiotypic inactivation of a clone occurs more generally than activation. Such “long-range inhibition” is known to promote pattern formation. However, in our model it fails to reduce the percolation, and additionally, generates semi-chaotic behaviour. In our network, the inhibition of a clone that is inhibiting can alter this clone into a clone that is activating. Hence “long-range inhibition” implies “long-range activation”, and idiotypic activation fails to remain localized. We next complicate this model by incorporating antibody production. Although this “antibody” model statically accounts for the same set of equilibrium points, it dynamically fails to account for state switching (i.e. memory). The switching behaviour is disturbed by the autonomous slow decay of the (long-lived) antibodies. After antigenic triggering the system now performs complex cyclic behaviour. Finally, it is suggested that (idiotypic) formation of antibody complexes can play only a secondary role in the network. In conclusion, our results cast doubt on the functional role of a profound idiotypic network. The network fails to account for proliferation regulation, and if it accounts for memory phenomena, it “explodes” upon the first encounter with antigen due to extensive percolation.

Notes: Abstract A new biomathematical description is given for the shape of the birch leaf roller's (Deporaus betulae) incisions. These incisions are investigated for different leaves. The theoretical patterns agree well with the real ones, and the presented mathematical expressions describe well the shape of the real incisions.

Notes: Abstract Nucleotide sequences carry genetic information of many different kinds, not just instructions for protein synthesis (triplet code). Several codes of nucleotide sequences are discussed including: (1) the translation framing code, responsible for correct triplet counting by the ribosome during protein synthesis; (2) the chromatin code, which provides instructions on appropriate placement of nucleosomes along the DNA molecules and their spatial arrangement; (3) a putative loop code for single-stranded RNA-protein interactions. The codes are degenerate and corresponding messages are not only interspersed but actually overlap, so that some nucleotides belong to several messages simultaneously. Tandemly repeated sequences frequently considered as functionless “junk” are found to be grouped into certain classes of repeat unit lengths. This indicates some functional involvement of these sequences. A hypothesis is formulated according to which the tandem repeats are given the role of weak enhancer-silencers that modulate, in a copy number-dependent way, the expression of proximal genes. Fast amplification and elimination of the repeats provides an attractive mechanism of species adaptation to a rapidly changing environment.

Notes: Abstract The kinematics of an area-conserving tank-treading disk-shaped red blood cell membrane is studied using the stream function method suggested by Secomb and Skalak (Q. Jl Mech. appl. Math. 35, Pt 2, 233–247, 1982). Two simple area-conserving velocity fields are superimposed to satisfy the continuity condition at the curved edges of the disk. A differential equation for the trajectory of any material point of the membrane is derived. The requirement of synchrony of the cycle for all membrane points leads to an integral equation which determines a magnitude function. An approximate solution is made possible by assuming small trajectory deflections.

Notes: Abstract The probability of becoming infected with HIV is formulated in terms of the total number of sexual contacts (N), the probability that a sexual act is infectious (r) and the prevalence (p). Using the appropriate equations we studied the effect of reducing each of the risk factors on lowering the probability of infection. We show that for many realistic situations the probability of becoming infected by multiple partners is equal to the probability of becoming infected by one partner in a monogamous relationship given that the prevalence is the same in both cases; however if the multiple partners are chosen over time from a pool of a growing prevalence, then one is better off in a monogamous relationship where that partner is chosen early in the epidemic.

Notes: Abstract In this article the question of reconstructing a phylogeny from additive distance data is addressed. Previous algorithms used the complete distance matrix of then OTUs (Operational Taxonomic Unit), that corresponds to the tips of the tree. This usedO(n 2) computing time. It is shown that this is wasteful for biologically reasonable trees. If the tree has internal nodes with degrees that are bounded onO(n*log(n)) algorithm is possible. It is also shown if the nodes can have unbounded degrees the problem hasn 2 as lower bound.

Notes: Abstract In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatica) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimmensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species. Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.

Notes: Abstract A series of Bayesian image processing algorithms which incorporate various classes ofa priori source information in treating data which obeys Poisson and Gaussian statistics is derived using maximum entropy considerations. The standard maximum likelihood equations are shown to be a special case of Bayesian image processing when thea priori information about a source distribution φ j is solely that a non-vanishing probability for each element value φ j exists only in some finite interval,a j ≤φ j ≤φ j . Bayesian image processing equations for thea priori source information that all φ j are finite -∞〈φ j 〈∞ and each φ j distribution has a defined mean φ j and a defined variance σ j are derived. The Bayesian image processing equations are also derived when thea priori source information is that all φ j ≥0 and that each φ j distribution has a defined mean φ j and a defined variance σ j . The a priori source distribution constraint that a correlation exists among nearby elements is also considered. The results indicate improvement over standard methods.

Notes: Abstract The superficial capillary network of the gastric mucosa can be monitored for red blood cell velocity measurements by a microscopic technique. This network, however, reflects the blood flow in capillaries of more physiological interest, namely those passing by the acid-producing cells and emptying into the superficial network. It is, however, not possible to study these capillaries directly and therefore the problem is to determine in what way and to what degree blood flow measurements in the superficial network reflect the capillary flow of interest. A probabilistic approach where the movements of the red blood cells have been analysed, gives indications of determinable relations between observations on the superficial network flow and the flow passing the acid-producing cells.

Notes: Abstract The present paper describes a calculation technique to determine a value of Young's modulus for the cornea when the intraocular pressure, the shape of the cornea and the thickness variation along the cornea are known fromin vivo measurements. Twenty-eight persons were examined in both eyes and a mean value of Young's modulus for the cornea was calculated to 2.45×104±0.57×104 N/m2 (S.D.).

Notes: Abstract It is pointed out that the asymptotic general solution to the ϕ-model equation for a periodic carrying capacityK(t) andt≳r −1 is identical in form to the generalized logistic equation solution with a built-in developmental time delay τ(≲r −1) and associated parameter ranges of primary biological interest. In the case of the ϕ-model equation, the time delay is a purely dynamical consequence of the nonlinear form featured by the population growth rate.

Notes: Abstract The pulsatile flow of blood through arteries is investigated in this paper by treating the blood vessel as a thin-walled anisotropic, non-linearly viscoelastic, incompressible circular cylindrical shell; nonlinearities of the flow of blood are also paid due consideration. The displacement components of the vessel wall are obtained from the equations of equilibrium which have been linearized by employing the principle of superimposition of a small deformation on a state of known finite deformation. The influence of the wall deformation on the flow properties of blood, has been accounted for by considering suitably formulated continuity conditions. A finitedifference scheme is employed for solving the flow equations together with the boundary and initial conditions by using the locally measured values of pressure and pressure gradient. Numerical results obtained for the velocity profile of blood flowing in a canine middle descending thoracic aorta have been presented through figures.

Notes: Abstract The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.

Notes: Abstract To describe stationary-state kinetics of solute clearance from an interstitial space with an initial uniform concentration, an explicit solution of the Sangren-Sheppard model and a stratagem for an explicit solution of the Johnson-Wilson model are presented. In both cases expressions for computing the exact intravascular and extravascular concentrations at any time and location are complicated and inconvenient. Simpler and far more convenient formulae for determining upper and lower bounds on solute concentrations and on the pseudo-first-order clearance rate ‘constant’ (k) are derived. For the Johnson-Wilson model, the bounds are so tight thatk, for example, can be estimated with considerable accuracy whenever the capillary blood flow exceeds the permeability-surface area product.

Notes: Abstract Different types of random binary topological trees (like neuronal processes and rivers) occur with relative frequencies that can be explained in terms of growth models. It will be shown how the model parameter determining the mode of growth can be estimated with the maximum likelihood procedure from observed data. Monte Carlo simulations were used to study the distributional properties of this estimator which appeared to have a negligible bias. It is shown that the minimum chi-square procedure yields an estimate that is very close to the maximum likelihood estimate. Moreover, the goodness-of-fit of the growth model can be inferred directly from the chi-square statistic. To illustrate the procedures we examined axonal trees from the goldfish tectum. A notion of complete partition randomness is presented as an alternative to our growth hypotheses.

Notes: Abstract Compartmental models of biological or physical systems are often described by a system of “stiff” differential equations. In this paper an algorithm for solving a system with linear coefficients is presented that employs numerical inversion of the Laplace transform of the model equations. The inversion algorithms and Gear's backward differentiation method are compared for two stiff test problems and a differential system governing a 27-compartment model of bile acid transport and metabolism. The inversion algorithm is reliable, requires modest computation time on a desktop computer and provides better accuracy than Gear's method, especially for the extremely stiff example.

Notes: Abstract Proposed model geometries are described for three types of axisymmetric and four types of flattened sickled erythrocytes. The axisymmetric types are designated as concave-convex, convex-convex and parabolic, while the flattened types carry designations of concave-convex, convex-convex, parabolic and S-shaped. Formulae are provided for the volume and surface area of each type and a figure showing the profile of each is included. Measurements of actual sickled cellsin vitro could be used to find values for the volume and surface area of each type of cell by using parameter values for the appropriate model geometry. These would give close approximations that could be useful in clinical therapies and laboratory investigations for sickle cell anemia. Surface area to volume ratios can also be found to a close approximation for each cell type.

Notes: Abstract In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature.

Notes: Abstract Currently applied three-copartment models for analyzing kinetic data derived fromin vivo positron emission tomographic (PET) studies of radioligand-neuroreceptor interactions require assumptions which may not be strictly valid. Such assumptions include very rapid kinetics for nonspecific binding and the absence of multiple specific receptors or subtypes. Computer simulations, based on an exact analytical solution of the relevant differential equations, indicate the numerical errors that can arise when the assumptions are invalid. We propose a fourcompartment model which requires fewer assumptions. A simple relationship is derived for expressing the microscopic rate constants of either the three- or four-compartment model as explicit functions of the experimentally-observed macroscopic rate constants. This could eliminate the need for time-consuming, iterative, non-linear, curve-fitting approaches and numerical integration. The usefulness of the four-compartment model is limited, however, by the sensitivity and temporal resolution of current PET imaging devices.

Notes: Abstract Various diffusion processes employed for modelling logistic growth are briefly summarized. A discrete-time, discrete-state space stochastic process for population growth is proposed and analyzed with either Bose-Einstein or Maxwell-Boltzmann statistics for the distribution of offspring in available sites in a restricted region. A diffusion approximation is constructed, which differs from those previously employed. The logistic law is a natural deterministic analog of the diffusion process.

Notes: Abstract In many biophysical and biochemical experiments one observes the decay of some ligand population by an appropriate system of traps. We analyse this decay for a one-dimensional system of randomly distributed traps, and show that one can distinguish three different regimes. The decay starts with a fractional exponential of the form exp[−(t/t 0)1/2], which changes into a fractional exponential of the form exp[−(t/t 1)1/3] for long times, which in its turn changes into a pure exponential time dependence, i.e. exp[−t/t 2] for very long times. With these three regimes, we associate three time scales, related to the average trap density and the diffusion constant characterizing the motion of the ligands.

Notes: Abstract Some key experiments of artificial production ofsitus inversus viscerum are briefly reviewed and a two-step mechanism for the explanation of the systematic asymmetric visceral arrangement in vertebrates is proposed. A two-variable reaction-diffusion system displaying a symmetry-breaking bifurcation is considered, and it is demonstrated that a slight asymmetry of the boundary conditions can give rise to a marked asymmetry in the resulting dissipative structure in both one-and three-dimensional systems. A criterion is formulated allowing classification of reaction-diffusion systems operating in a three-dimensional space with regard to their ability to incorporate slight asymmetries at the boundaries in the form of a chiral dissipative structure.