ALBERT

Notes: Abstract A general mechanism underlying bursting is proposed and described. It consists of two coupled nonlinear oscillators with different frequencies, where the slower oscillator alternatively switches the faster one on and off. This mechanism is shown to work in an extended Bonhoefer-van der Pol oscillator as well as in a modified version of the Hodgkin-Huxley equations.

Notes: Abstract Pointwise upper and lower bounds for the solution of a class of nonlinear diffusion problems with Michaelis-Menten kinetics are presented. Simple analytical bounding curves are obtained and for an illustrative case the calculated values bound the recent numerical solution of P. Hiltmann and P. Lory, 1983.Bull. math. Biol. 45, 661–664.

Notes: Abstract The cellular response in terms of steady-state variance of cell mass concentration to fluctuations in incoming nutrient concentration to a chemostat has been examined. A white noise process is assumed to describe incoming nutrient concentration fluctuations and the variance of cell mass concentration has been found to depend on cell yield (a lumped measure of nutrient concentration fluctuation magnitude and lifetime) and two system time constants.

Notes: Abstract In a well-known collection of his essays in cognitive psychology Miller (The Psychology of Communication. Penguin, 1974) describes in detail a number of experiments aiming at a determination of the limits (if any) of the human brain in processing information. He concludes that the ‘channel capacity’ of human subjects does not exceed a few bits or that the number of categories of (one-dimensional) stimuli from which unambiguous judgment can be made are of the order of ‘seven plus or minus two’. This ‘magic number’ holds also, Miller found, for the number of random digits a person can correctly recall on a row and also the number of sentences that can be inserted inside a sentence in a natural language and still be read through without confusion. In this paper we propose a dynamical model of information processing by a self-organizing system which is based on the possible use of strange attractors as cognitive devices. It comes as an amusing surprise to find that such a model can, among other things, reproduce the ‘magic number seven plus-minus two’ and also its variance in a number of cases and provide a theoretical justification for them. This justification is based on the optimum length of a code which maximizes the dynamic storing capacity for the strings of digits constituting the set of external stimuli. This provides a mechanism for the fact that the ‘human channel’, which is so narrow and so noisy (of the order of just a few bits per second or a few bits per category) possesses the ability of squeezing or ‘compressing’ practically an unlimited number of bits per symbol—thereby giving rise to a phenomenal memory.

Notes: Abstract Electrical polarization of an artery or an arteriole may be modeled by the use of equations developed for two-dimensional cable theory. Two special cases have previously been solved: those corresponding to the case in which the radius is either zero (one-dimensional cable theory) or infinite. This paper presents the general solution.

Notes: Abstract The distance geometry approach for computing the tertiary structure of globular proteins emphasized in this series of papers (Goelet al., J. theor. Biol. 99, 705–757, 1982) is developed further. This development includes incorporation of some secondary structure information—the location of alpha helices in the primary sequence—in the algorithm to compute the tertiary structure of alpha helical globular proteins. An algorithm is developed which estimates the interresidue distances between chain-proximate helices. These distances, in conjunction with the global statistical average distances obtainable from a database of real proteins and determined by the primary sequence of the protein under study, are used to determine the tertiary structure. Five proteins, parvalbumin, hemerythrin, human hemoglobin, lamprey hemoglobin, and sperm whale myoglobin, are investigated. The root mean square (RMS) errors between the calculated structures and those determined by X-ray diffraction range from 4.78 to 7.56 Å. These RMSs are 0.21–2.76 Å lower than those estimated without the secondary structure information. Contact maps and three-dimensional backbone representations also show considerable improvements with the introduction of secondary structure information.

Notes: Abstract If a plane membrane consists of patches, each with a given area and a given diffusion coefficient, then the transient of the total unidirectional flux of a diffusing substance (as defined experimentally by Ussing) is predictable. Here the inverse problem is studied: given only the observed transient of the total unidirectional diffusion flux, the unknown membrane heterogeneity transverse to the flux is to be quantified. The ratio of the arithmetic and of the harmonic means (both area-weighted) of the diffusion coefficients, evaluated over the membrane, is expressed in terms of the observed transient alone and is used to characterize the heterogeneity. A unique exact solution of the inverse problem for two kinds of patches is obtained in closed form. A singular limit of this solution pertains to currently postulated models of endothelial membranes, for which a characteristically shaped transient is predicted.

Notes: Abstract Major types of sequence similarity searching (often, and incorrectly, called ‘homology’ searching) are reviewed and examples of each are presented. The features and limitations of each type of program, and individual implementations of each type are discussed. Two pairs of sequences are used as examples to show how implementations of each type differ in their results and their presentation. Both local and global alignment programs are examined, and the programs reviewed run on many different types of computer architectures, from laboratory computers such as the IBM PC, minicomputers such as the VAX, to large mainframe computers such as DEC-10/20 series.

Notes: Abstract Criteria for the existence of globally stable equilibria in classical Volterra predator-prey systems represented by loop graphs are provided by comparing the community matrix with a matrix belonging to matrix classS W .

Notes: Abstract Ordinary reaction-diffusion mechanisms do not account for size invariance properties of morphogenetic fields. We show that such a failure results from ignoring cell individuality. By considering purely topological factors, such as the number of intercellular contacts or the extent of the cell surface in contact with neighbouring cells, size invariance exists in reaction-diffusion systems. Our results are general, model independent and may be applied to any multi-unit ensemble exhibiting coherent behaviour.

Notes: Abstract The artery is treated as a thick-walled cylindrical shell. Using the large deformation theory, an analytical expression for the pulse wave speed is obtained and the effect of twist on the wave speed is discussed. Numerical results indicate that although phase velocity increases with pressure, it decreases with increasing twist angle.

Notes: Abstract The effect of the shape of stenosis on the resistance to blood flow through an artery with mild local narrowing has been studied. It is shown that the resistance to flow decreases as the shape of stenosis changes and the maximum resistance is attained in the case of symmetric stenosis.

Notes: Abstract A set of 2n−2 relations (edges) and a set ofn−1 hypothetical taxonomic units (HTUs) derive from the estimation of a binary phylogeny of a set ofn operational taxonomic units (OTUs). We propose an easy way for numbering thesen−1 hypothetical taxonomic units, as well as for then−2 interior points of an unrooted binary phylogeny. We also present an alternative method to the one proposed by Rohlf (Bull. math. Biol. 45, 33–40, 1983) for numbering the π i=1 n (2i−3) possible rooted binary phylogenies and the π i=1 n−1 (2i−3) possible unrooted binary phylogenies conerning a set ofn operational taxonomic units. An illustrative example of the method is presented. It is hoped that some studies in phylogenetics will become more accessible, from the viewpoint of computational economy, by the use of this method.

Notes: Abstract The heterogeneity of rat brain opiate receptors was examined by analyzing competition data. The binding of three prototypical tritiated opioid agonists, [3H]-dihydromorphine ([3H]-DHM), [3H]-D-ala2-D-leu5-enkephalin ([3H]-DADLE), and [3H]-ethylketocyclazocine ([3H]-EKC) was determined in the presence of varying concentrations of each of these unlabeled ligands, generating nine displacement curves. A computer program was then used to find the best fit of a model system to these data, assuming two, three or four independent binding sites. The best fit was a four-site model. One of these sites is specific for DHM; two are relatively selective for DHM and DADLE respectively, but also bind EKC. The remaining site binds only EKC with high affinity. These results, together with displacement data using naloxone, FK33824, and D-ala2-met5-enkephalinamide, are discussed in terms of current opiate receptor models.

Notes: Abstract This work continues with an examination of capillary exchange models as operators, namely the operatorsO k andK αk relating extravascular and intravascular concentration to input for the Krogh cylinder model of a single capillary, a model basic to many organ models. Fundamental algebraic and analytic properties are presented: the operators belong to a commutative Banach algebra; an addition theorem holdsK αk +K βk =K α+β,k ; the operatorK αk has an inverse;K αk -1 , (as an operator on LebesgueL p space or on the locally integrable functions); partial derivatives are given forK αk [f](t) andO k [f](t) (sensitivity functions); and inequalities are established for the derivatives. Dominance relations between model curves are inferred. Error bound formulas are presented forK andO as bounds on ‖K αk f-K βl f‖ p and ‖O k f-O l f‖ p for allL p . Consequent limitations on relative errors are shown. The implications for operators on a finite time interval are deduced.

Notes: Abstract The nonlinear nature of the hydraulic permeability, as well as the corresponding pressure and displacement fields, in a soft tissue are studied for steady-state permeation. From a two-phase continuum model analytical expressions are derived that can be used with the results from a permeation experiment to determine the dependence of the permeability on the strain. In the process it is found that, because of the compaction of the tissue arising from fluid flow, it is necessary to distinguish between the apparent and intrinsic permeability. The former, which is an averaged quantity, is the permeability usually obtained in permeation studies. However, as shown from the analysis, it can differ substantially from the latter, which is the local permeability in the tissue.

Notes: Abstract The analytic, eccentric spheres model of the torso was used to examine the validity of approximating the ‘infinite medium’ potential by integrating ‘finite medium potentials’ over the torso surface. Although idealized, the analytic model is sophisticated enough for all important torso conductivity and geometry parameters to be preserved in the formulation. The model generates both ‘finite medium’ potentials (for which the torso is surrounded by air) and also ‘infinite medium’ potentials (for which the outermost layer of the torso extends outward to infinity). The finite medium torso potentials were integrated over the torso surface in accordance with the approximation used by many investigators in an effort to make the surface distribution more representative of the primary cardiac sources. The resulting potential distribution was compared with the true infinite medium potential, in which the effects of internal inhomogeneities (secondary sources) were taken into account. The difference between the two representations was found to be significant, and caution should be used when interpreting such data.

Notes: Abstract Environ analysis, an input-output analysis for models of ecological systems, has been previously formulated for linear systems. This note has a twofold purpose: first, we indicate that a variation of parameters technique can be applied, at least in principle, to computeboth input and output environs; and second, we show that this technique may be used for computation of environs in nonautonomous, nonlinear compartment models. This nonlinear theory, obtained as a direct extension of dynamical system developments, allows the traditional environ partitioning of compartmental storages and flows. An example of a nonlinear nutrient-producer-consumer system whose output environs can be computed asymptotically is presented to illustrate these concepts.

Notes: Abstract Prolonged exposure to cyanide leads to a delayed but reversible disappearance of tetanic hyperpolarization in theXenopus node (G. M. Schoepfle,Am. J. Physiol. 231, 1033–1038, 1976). This effect is attributed to a pronounced decline in the absolute values of the ATP and ADP concentrations, such that the ATP-driven ion translocation is no longer possible, regardless of the existing values for (Na)i, (K)i and the (ATP)/(ADP) ratio. Mathematically, this would imply a vanishing of a constant pump conductance gp in the exression for electrogenic pump current densitityJ p, whereJ p=g p (V m −E p) in whichV m is membrane potential andE p is an ATP-and sodium-dependent e.m.f.

Notes: Abstract For the two-parameter (A, α) exponentially-stiffening constituitive relation, typical of many biological materials, it is shown that the uniaxial stress-strain behavior of an initially curved strip is significantly changed by the residual bending stresses. Closedform theoretical results depend on the thickness to radius ratio (h/R) and the relative strain level ε(h/R). The bending stresses tend to obscure accurate measurement ofA and α unless care is taken. However, it is shown that by changing co-ordinates to (dℝ/d∈, ℝ)-space, bothA and α can be recovered from the high stress data, and α alone can be recovered from the low stress data. This has practical application to the mechanics of cornea, sclera, and heart muscle.

Notes: Abstract An axisymmetric flow of a power law fluid through circular tubes under constant pressure gradient with the flow parameters varying radially is analyzed theoretically. The main finding is that for the Fahraeus-Lindqvist (F-L) effect to occur, it is necessary to have at least one of the parametersK (consistency) andn (index) as a discontinuous function ofr in the absence of wall slip; and with slip condition the parameters could be continuous functions ofr under specific conditions. In both the cases the existence of more than one discontinuity cannot be ruled out. The results obtained are consistent with experimental findings of blood flow through narrow tubes.

Notes: Abstract This is a study of the properties of a zygotic algebra of two linked autosomal loci with different recombination rates in males and females, without selection or mutation and with random mating. The above-mentioned zygotic algebra contains a genetic subalgebra. A canonical basis of this subalgebra is constructed and the train roots are calculated.

Notes: Abstract Among the conformations which the DNA molecule can adopt, the transition beween the A and B families, controlled by water content (relative humidity), seems to be implicated in the transcription process. Focusing on the main structural difference involved (tilting of base normals with respect to the helix axis), a model is constructed, solitary wave solutions of the resulting equation of motion are demonstrated and possible experimental implications indicated.

Notes: Abstract Balls are removed one-at-a-time at equal time intervals from an urn initially containingw 0 white balls and a large number b of black balls and each black or white ball is immediately replaced by a black ball. The distribution of the number of white balls remaining aftert iterations (under certain limiting operations) is taken from the literature. The problem is to use this result to find the time required to remove a fixed number of white ballsw 1 from the urn. We then find the mean and variance of this distribution and also look at the special case whenw 1 =w 0.

Notes: Abstract Approximate equations for epithelial solute and water transport have been combined with the relations of mass conservation to yield a single differential equation representing volume flow along the proximal tubule. This flow equation is first order, quasilinear and may be integrated directly. For the steady state, the result is an implicit relation between volume flow and distance along the tubule. For two time-dependent problems (step change of tubule inlet velocity or osmolality) the trajectories (distance as a function of transit time) of a fluid element starting at the inlet are obtained. Differentiation of the steady-state relation with respect to the inlet velocity yields a first-order differential equation relating inlet and outlet velocity. This equation is considered in detail, particularly with regard to the influence of solute-linked water reabsorption. Model calculations with parameters representing rat proximal tubule indicate that it will be difficult to discern coupled water flux in this epithelium from only outlet and inlet flows. Calculations using lower transport rates and lower permeabilities suggest that this equation may be useful in quantifying coupled water flow in proximal tubules from other species.

Notes: Abstract The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.

Notes: Abstract Galerkin's finite element-Laplace transform technique (GAFELTTE) has been used to study transient temperature distribution in human skin and subcutaneous tissues. This study incorporates heat conduction, heat carried by perfusion of blood in the capillary beds and metabolic heat generation in the tissues. Different values of various quantities have been considered in all three parts, namely epidermis, dermis and subcutaneous tissues, depending on physiological considerations. The GAFELTTE provides interface temperatures for a wide range of the values of skin surface temperatures. These values have been used to obtain temperature profiles in the region considered. Steady-state temperature distribution has been deduced from the solution obtained by GAFELTTE and has been compared with the results obtained by using different methods.

Notes: Abstract Necessary and sufficient conditions are given for three equilibria to occur in a predatorprey model and conditions are given for two of these to be stable. The existence of two stable equilibria requires predator intraspecific competition for either space or food, and the lower the prey growth rate the stronger this predator self-regulation must be. A prey growth rate that is skewed to the right, the ability of a few predators to survive at low prey densities, and predators with high searching effectiveness, long handling times, and large maximum per capita rate of increase all make two stable equilibria more likely.

Notes: Abstract Drawing evidence from a variety of cardiovascular studies on the heart rate in homeothermic animals, the author establishes the following thesis. The servocontrol (i.e. the autonomic and reflex control) by the medulla oblongata of the heart (rate) is a negative feedback dynamic which is isomorphic (i.e. ‘diffeomorphic’) to the dyamic underlying the heat rate control in those animals (cf. Kuyk,Bull. math. Biol. 46, 81–102, 1984). In fact, unlike in the heat rate case, the qualitative evidence supporting this thesis can not be fully complemented by quantitative data stemming from experiments, because of a lack of pertinent experiments—which, indeed, should measuresimultaneously the heart rate state parameter and thefour control parameters at the input side of the medulla. The results of some of the existing experiments on animal preparations can nevertheless be adduced to recognize that this dynamic can be graphed by the five-dimensional butterfly catastrophe type. The theory leads to new ways of looking at experiments in the field and/or setting up such experiments in the future.

Notes: Abstract A model based upon minimization of surface energy is proposed as an explanation for compaction and internalization of cells during mammalian embryo development. The model is used to simulate and graphically display these phenomena on a computer.

Notes: Abstract This paper describes a growth model for binary topological trees. The model defines the branching probability of all segments in the tree. The branching probability of a segment is formulated as a function of two variables, one indicating its type (intermediate or terminal), the other representing its order, i.e. the topological distance to the root segment. The function is determined by two parameters, namely the ratio of branching probabilities of intermediate and terminal segments and the strength of the order dependency, implemented in an exponential form. Expressions are derived for the calculation of symmetry properties of the partitions and it is indicated which part of the parameter domain results in predominantly symmetrical trees.

Notes: Abstract The problem of cellular differentiation and consequent pattern generation during embryonic development has been mathematically investigated with the help of a reaction-diffusion model. It is by now a well-recognized fact that diffusion of micromolecules (through intercellular gap junctions), which is dependent on the spatial parameter (r), serve the purpose of ‘positional information’ for differentiation. Based on this principle the present model has been constructed by coupling the Goodwin-type equations for RNA and protein synthesis with the diffusion process. The homogeneous Goodwin system can exhibit stable periodic solution if the value of the cooperativity as measured by the Hill coefficient (ρ) is greater than 8, which is not biologically realistic. In the present work it has been observed that inclusion of a negative cross-diffusion can drive the system into local instability for any value of ρ and thus a time-periodic spatial solution is possible around the unstable local equilibrium, eventually leading to a definite pattern formation. Inclusion of a negative cross-diffusion thus makes the system biologically realistic. The cross-diffusion can also give rise to a stationary wave-like dissipative structure.

Notes: Abstract A nes software system is described for building simulation programs on micro- and minicomputers. Model equations are written as C subroutines, compiled and linked to the SCoP package to produce a menu-driven, interactive program. The system maintains a database of names, values, and units for all model parameters and variables. Run-time options include several methods for interactive parameter modification and both graphic and tabular outputs, with output values presented as they are calculated. Simulation output values can be compared with experimental data graphically and a companion program SCoPFit is provided for formal optimization of parameter values.

Notes: Abstract We consider the existence and global stability of aq-member equilibrium (1≤q≤n) in partially closed food-chains of lengthn having an abiotic component as resource. We observe that such existence demands bounds of resource supply rate and these bounds are weighted sums of interaction coefficients. Particular results of global sector-stability of partially feasible equilibria of simple food-chains obeying Lotka-Volterra dynamics are shown. Lastly the elasticity of such food-chains when a new species is introduced at the highest trophic level is investigated.

Notes: Abstract Criteria are established for three classes of models of single-species dynamics with a single discrete delay to have a globally asymptotically stable positive equilibrium independent of the length of delay.

Notes: Abstract The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically).

Notes: Abstract In this paper a general class of semi-Markov compartmental systems is studied. Two models for different input processes are analysed. Attention has been paid to the recurrence times associated with each compartment and to the distribution of the number of particles in each compartment. As an example, a three-compartment system is discussed to study the movement between three health states of patients with chronic diseases.

Notes: Abstract A strategy is presented for searching the gene and protein sequence data banks which combines the use of two previously described algorthms. The implementation of this strategy is thoroughly evaluated with respect to sensitivity, specificity and speed. The establishment of standard benchmarks for comparing programs that rearch the sequence data banks for homology is proposed.

Notes: Abstract During functional linkage, ligand receptors are coupled to other receptors and to the cell's metabolic-transport apparatus. The linkage guides the cellular processing of matter, energy and information. Previous conceptions of functional linkage have used the ideas of classical physics appropriate to macroscopic objects. This study presents an initial quantum mechanical model of functional linkage in the case of ligands moving through lipid bilayers and hydrophilic transmembrane channels (‘pores’) of molecular dimensions. On the basis of permeability data, energy surfaces consisting of piecewise-constant potential regions are used to model the lipid bilayers and transmembrane channels. The centre-of-mass wavefunction for a ligand on such energy surfaces is analysed and the permeability coefficients calculated from the wavefunction's transmission characteristics. It is found that quasi-bound states in the several ligand-binding regions of a bilayer or pore system can functionally link to facilitate the passage of the molecule across the permeability barrier. Appearance of the linkage is a sensitive function of the ligand's energy. If the centre-of-mass energies are distributed as in a thermalized fluid, the flux via the quantum functional linkage can equal or exceed that of a classical flux for proton transport through rigid pores in which the intrasite barriers are relatively high (0.25–1 eV) and narrow (0.1–1 Å). The functional linkage plays a less important role in bilayer (rather than pore) energy surfaces and at higher molecular weights. If the ligand-receptor interaction is accompanied by energy transfer to or from ligands, the flux via the quantum functional linkage can equal or exceed the classically expected flux at all relevant ligand molecular weights. These findings are discussed in relation to earlier work and the limitations of the model emphasized.

Notes: Abstract A new measure of subalignment similarity is introduced. Specifically, similaritys(l,c) is defined as the logarithm to the basep of the probability of findingc or fewer mismatches in a subalignment of lengthl, wherep is the probability of a match. Previous algorithms can not use this measure to find locally optimal subalignments because, unlike Needleman-Wunsch and Sellers similarities, this measure is nonlinear. A new pattern recognition algorithm is described for finding all locally optimal subalignments of two nucleotide sequences. The DD algorithm can uses(l, c) or any other reasonable similarity function to assess the relative interest of subalignments. The DD algorithm searches only the diagonal graph, which lacks insertions and deletions. This search strategy greatly decreases the computation time and does not require an arbitrary choice of gap cost. The paths of the resulting DD graph usually draw attention to likely locations for insertions and deletions. A heuristic formula is derived for estimating significance levels fors(l, c) in the context of the lengths of the two aligned sequences. The DD algorithm has been used to find interesting subalignments between the nucleotide sequences for human and murine interleukin 2.

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 A dynamical model of the left ventricle as a thick-walled cylinder contracting radially is used to derive the P-V (pressure-volume) relation in the left ventricular cavity during contraction. It is shown how the mathematical results derived could apply to experimental results.

Notes: Abstract We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.

Notes: Abstract We consider efficient methods for computing a difference metric between two sequences of symbols, where the cost of an operation to insert or delete a block of symbols is a concave function of the block's length. Alternatively, sequences can be optimally aligned when gap penalties are a concave function of the gap length. Two algorithms based on the ‘candidate list paradigm’ first used by Waterman (1984) are presented. The first computes significantly more parsimonious candidate lists than Waterman's method. The second method refines the first to the point of guaranteeingO(N 2 lgN) worst-case time complexity, and under certain conditionsO(N 2). Experimental data show how various properties of the comparison problem affect the methods' relative performance. A number of extensions are discussed, among them a technique for constructing optimal alignments inO(N) space in expectation. This variation gives a practical method for comparing long amino sequences on a small computer.

Notes: Abstract It is well documented, in the biological literature, that many species throughout the animal kingdom exhibit Gompertzian or Weibull-like population level total survival distributions. Many researchers have long assumed, believed, or otherwise postulated that an individual organism, in such a population, survived according to an exponential survival distribution. Using well-known results from reliability theory, it is shown that if every individual in the population has an exponentially distributed lifespan, then a Gompertzian or Weibull-like group/population level dynamics (or any other dynamics with a strictly increasing mortality rate for some interval) is not possible. This implies that, for species with a population level Gompertzian or Weibull (with the mortality rate strictly increasing) survival curve, some or all of the individual organisms must have non-exponentially distributed lifespans.

Notes: Abstract In flow cytometric measurement of cell DNA distribution one of the major problems is accounting for the effect of fragmentation in the staining process. This work considers a recent probabilistic model that has been proposed for the fragmentation process and species under which conditions it is possible to uniquely identify the DNA distributions of the original population using flow cytometric data. Attention is given both to the normal and to the polyploid case.

Notes: Abstract The nonlinear behavior of a particular Kolmogorov-type exploitation differential equation system assembled by May (1973,Stability and Complexity in Model Ecosystems, Princeton University Press) from predator and prey components developed by Leslie (1948,Biometrica 35, 213–245) and Holling (1973,Mem. Entomol. Soc. Can. 45, 1–60), respectively, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature dependent mite interaction on fruit trees. The most significant result of this analysis is that, in addition to the temperature ranges over which the single community equilibrium point of the system iseither globally stableor gives rise to a globally stable limit cycle, there can also exist a range wherein multiple stable states occur. These stable states consist of a focus (spiral point) and a limit cycle, separated from each other in the phase plane by an unstable limit cycle. The ecological implications of such metastability, hysteresis and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system and the biological control of mite populations are discussed. It is further suggested that a model of this sort which possesses a single community equilibrium point may be more useful for representing outbreak phenomena, especially in the presence of oscillations, than the non-Kolmogorov predator-prey systems possessing three community equilibrium points, two of which are stable and the other a saddle point, traditionally employed for this purpose.

Notes: Abstract This note is concerned with a simple mathematical model of how a population of bacterial spores decrease with time when subjected to a uniform temperature. The model assumes that there is a Boltzman distribution of energy among water or other molecules surrounding the assumed single lethal target in a spore; it assumes that repair is not possible; and that only molecules with energies above a critical level cause inactivation. The model provides new insight concerning the ‘kill-rate’ of spores during ultra heat treatment.

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 It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

Notes: Abstract The classical metaphor of the genetic program written in the DNA nucleotidic sequences is reconsidered. Recent works on algorithmic complexity and logical properties of computer programs and data are used to question the explanatory value of that metaphor. Structural properties of strings are looked for which would be necessary to apply to DNA sequences if the metaphor is to be taken literally. The notion of sophistication is used to quantify meaningful complexity and to distinguish it from classical computational complexity. In this context, the distinction between program and data becomes relevant and an alternative metaphor of DNA as data to a parallel computing network embedded in the global geometrical and biochemical structure of the cell is discussed. An intermediate picture of an evolving network emerges as the most likely where the output of the cellular computing network can produce, at a different time scale, changes in the structure of the network itself by means of changes in the DNA activity patterns.

Notes: Abstract When two strings of symbols are aligned it is important to know whether the observed number of matches is better than that expected between two independent sequences with the same frequency of symbols. When strings are of different lengths, nulls need to be inserted in order to align the sequences. One approach is to use simple approximations of sampling for replacement. We describe an algorithm for exactly determining the frequencies of given numbers of matches, sampling without replacement. This does not lead to a simple closed form expression. However we show examples where sampling with, or without, replacement give very similar results and the simple approach may be adequate for all but the smallest cases.

Notes: Abstract Pairwise optimal alignments between three or more sequences are not necessarily consistent as a whole, but consistent and inconsistent residues are usually distributed in clusters. An efficient method has been developed for locating consistent regions when each pairwise alignment is given in the form of a “skeletal representation” (Bull. math. Biol. 52, 359–373). This method is further extended so that the combination of pairwise alignments that gives the greatest consistency is found when possibly many alignments are equally optimal for each pairwise comparison. A method for acceleration of simultaneous multiple sequence alignment is proposed in which consistent regions serve as “anchor points” limiting application of direct multi-way alignment to the rest of “inconsistent” regions.

Notes: Abstract Sufficient conditions are given for the unlimited growth or otherwise in multitype population size dependent Galton-Watson processes. These conditions are given in terms of moments of offspring distributions and extend known conditions for processes with one type.

Notes: Abstract The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.

Notes: Abstract Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S. A. Maggelakis and J. A. Adam,Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters:Q, γ, b, andδ, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radiusR 0(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics ofin vitro multicellular spheroids. This will be one objective of future studies.

Notes: Abstract Workers of the speciesLeptothorax acervorum show age-polyethism, they start their life as broodworkers and later on they become nestworkers and foragers. Nestworkers and foragers of this ant species are inactive for 72% and 15% of the total time respectively. The short bursts of activity within the nest do not occur randomly but are synchronized so that the whole nest population exhibits nonperiodic pulses of activity: the ants were seen to wake each other actively. In addition starvation experiments were done to assess whether ants react upon food availability. In appeared that during a longlasting period of starvation the proportion of active ants in the nest is at a higher approximately constant level.

Notes: Abstract The statistical correlation of nucleotides in a DNA sequence is described by a set of redundanciesD 1,D 2,D 3,... By calculation of {D n} of 2341 coding regions of nucleic acid sequences it is demonstrated that about 2/3 of sequences has correlation length ≤2, 10% of sequences—correlation with 3-periodicity and others—long range aperiodic correlations. The implications of the results from the interactions of random mutation and natural selection are discussed briefly.

Notes: Abstract A basic characteristic of biological memory is that it has a graded duration, which, even for socalled short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the “strength” of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where “learning” is produced by adecrease in an appropriate membrane conductance. This paper provides a quantitative analysis of a simple kinetic scheme where by a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only over a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more plausible the possibility that the effects described here may operate in actual biological systems.

Notes: Abstract A mathematical model (Kliemann, W. 1987.Bull. math. Biol. 49, 135–152.) that predicts the quantitative branching pattern of dendritic tree was evaluated using the apical and basal dendrites of rat hippocampal neurons. The Wald statistics for χ2-test was developed for the branching pattern of dendritic trees and for the distribution of the maximal order of the tree. Using this statistic, we obtained a reasonable, but not excellent, fit of the mathematical model for the dendritic data. The model's predictability of branching patterns was greatly enhanced by replacing one of the assumptions used for the original model “splitting of branches for all dendritic orders is stochastically independent”, with a new assumption “branches are more likely to split in areas where there is already a high density of branches”. The modified model delivered an excellent fit for basal dendrites and for the apical dendrites of hippocampal neurons from young rats (30–34 days postpartum). This indicates that for these cells the development of dendritic patterns is the result of a purely random and a systematic component, where the latter one depends on the density of dendritic branches in the brain area considered. For apical dendrites there is a trend towards decreasing pattern predictability with increasing age. This appears to reflect the late arrival of afferents and subsequent synaptogenesis proximal on the apical dendritic tree of hippocampal neurons.

Notes: Abstract In order for immune cells to carry out many of their functions, including clearance of infectious agents from tissue, they must first encounter their targets in the tissue. This encounter process is often the rate-limiting step in the overall function. Most immune cells exhibit chemotactic ability, and previous continuum models for encounter rates and dynamics have shown that chemotaxis can be a great advantage to cells by greatly increasing encounter rates relative to those for randomly moving cells. This paper describes computer simulations of discrete cell-target encounter events in two dimensions, for the two cases considered by the continuum models: where only a single cell and a single target are present, and where many cells and targets are present. The results of these simulations verify our previous model predictions that a small amount of chemotactic bias dramatically decreases the encounter time, while further increases in the amount of bias have a much smaller effect. Chemotactic ability is shown to be an important determinant of the kinetics of target clearance, and its effects depend on the initial cell-target ratio and the initial distributions of cells and targets. To the best of our knowledge, this work provides the first computer simulations of particle-target encounter in which there is biased motion of particles toward their targets, and is therefore of general interest beyond specific application to immune cell function.

Notes: Abstract The concept of a one-way block, arising from a region of depressed tissue, has remained central to theories for cardiac arrhythmias. We show that both the geometry of a depressed region and spatial heterogeneities in depression are key factors for inducing such a block. By using an asymptotic approximation, known as the eikonal equation, to model qualitatively the movement of a depolarization wave-front down a Purkinje fibre bundle, we show how a one-way block in conduction may result from asymmetric constriction in the width of a depressed bundle. We demonstrate that this theory is valid for biologically relevant parameters and simulate a one-way block by numerically solving the eikonal approximation. We consider the case of non-uniform depression, where the planar travelling wave speed is spatially dependent. Here, numerical simulations indicate that such a spatial dependency may, in itself, be sufficient to produce a one-way block.

Notes: Abstract The MacArthur-Wilson equilibrium theory of island biogeography has been one of the more influential concepts in modern biogeography and ecology. In this paper, we synthesize the theory and examine effects of different immigration/extinction rate-species diversity curves on original predictions from the theory by using the System Dynamics simulation modeling approach. Moreover, we develop a comprehensive and generic System Dynamics model to incorporate a variety of recent modifications and extensions of the theory, including area effect, distance effect, competition effect, habitat diversity effect, target effect, and rescue effect. Through computer simulation with STELLA, a more profound understanding of the theory of island biogeography can be gained. The System Dynamics modeling approach is especially appropriate for such a study because it maximizes the utilization of the ecological data by incorporating qualitative information so that a complex, imprecisely-defined ecological system can be studied quantitatively, effectively, and comprehensively. Our simulation results show that different monotonic rate-species diversity curves do not affect the essence of the theory of island biogeography, while the magnitude of equilibrium species diversity may be greatly affected. Non-monotonic rate-species diversity curves may result in potential multiple equilibria of species diversity. In addition, our model suggests that a non-monotonic relationship may exist between the equilibrium turnover rate and island area and between the equilibrium turnover rate and distance.

Notes: Abstract An organism persists only if it satisfies internal and external constraints. Within the organism networks of processes meet the constraints. In such networks a principle of matching often obtains: the pattern of coupling among processes matches the correlation among constraints. That is, a module—a cluster of coupled processes—meets a constraint. Dissociable modules meet dissociàble constraints. A hierarchy of modules meets a hierarchy of constraints. We have inquired whether such matching is predicted by an optimality criterion in a simple example. We find that in an ensemble of networks with unreliable processes, the networks that meet the constraints with highest reliability obey the principle of matching. The difference in reliability between modular and nonmodular networks that meet the same constraints is a function of the probability of success per process. Our results suggest that this difference is maximal at a probability of success that increases monotonically with the number of processes in the network.

Notes: Abstract A method is described for estimating the distribution and hence testing the statistical significance of sequence similarity scores obtained during a data-bank search. Maximum-likelihood is used to fit a model to the scores, avoiding any costly simulation of random sequences. The method is applied in detail to the Smith-Waterman algorithm when gaps are allowed, and is shown to give results very similar to those obtained by simulation.