Showing papers in "Bellman Prize in Mathematical Biosciences in 1982"
TL;DR: Two practical branch and bound algorithms for determining minimal and near-minimal phylogenetic trees from protein sequence data are presented.
Abstract: Two practical branch and bound algorithms for determining minimal and near-minimal phylogenetic trees from protein sequence data are presented. A mathematical description and analysis of phylogenetic trees introduces these algorithms. A comment of efficiency and fine tuning completes the paper. An example is cited where computer time was reduced from an estimated 55 days for a total search, to just under 5 minutes.
539 citations
IBM1
TL;DR: In this article, the degree of agreement between dendrograms (which correspond uniquely to hierarchic nested classifications) is discussed and a new consensus index is proposed that does not have this defect.
Abstract: Methods for measuring the degree of agreement between dendrograms (which correspond uniquely to hierarchic nested classifications) are discussed. While some methods take the relative levels of the subsets into account, most presently available methods do not. A previously proposed consensus index is shown to have a maximum possible value that is (for a given number of objects) a function of the balance (the degree to which the subsets of each set are equal in size) of the consensus of the two classifications being compared. A new consensus index is proposed that does not have this defect. In addition, a new index is proposed that is the proportion of all possible classifications that contain the sets that are in the consensus of the two dendrograms being compared. A table giving the numbers of possible bifurcating dendrograms (up to t =100 objects) is furnished to assist in the computation of this latter index.
300 citations
TL;DR: In this paper, the authors analyzed the problem of obtaining unique estimates of the parameters μ m and K s of the Michaelis-Menten equation for a microbial batch growth process.
Abstract: The reason for difficulties in obtaining unique estimates of the parameters μ m and K s of the Michaelis-Menten equation are analysed for a microbial batch growth process. With the aid of simulation studies in which the influences of different types of noise on the parameter estimates are compared, it is shown that, although theoretically identifiable in the deterministic case with ideal measurements, the parameters cannot in general be correctly determined from noisy measurements. The difficulties are further illuminated by estimation examples using real data. It certain situations, in which the value of the ratio K s / s o is high or in which only few and noisy measurements are available, the linear approximation of the Michaelis-Menten equation gives a better fit. The practical difficulties in obtaining correct values of the model parameters do not limit the applicability of the Michaelis-Menten model, which in most cases explains the bacterial growth behavior excellently. Rather, they underline the fact that care must be taken when utilizing parameter estimates for biological interpretations.
290 citations
TL;DR: E eggs and sperm of the sea urchin Paracentrotus lividus of the Mediterranean are used for an in vitro study of fertilization kinetics and the role of the jelly is discussed.
Abstract: Eggs and sperm of the sea urchin Paracentrotus lividus of the Mediterranean are used for an in vitro study of fertilization kinetics. The results are analyzed in terms of two models. One of these models assumes that all sperm-egg encounters lead to permanent attachment; the other (less realistically) assumes that sperm continue their random search after an unsuccessful encounter. More than 100 spermatozoa per egg are needed to achieve a fertilization ratio of more than 95%. There are two explanations for this: only 1% of the egg surface is subject to fertilization, or only 1% of spermatozoa are intrinsically able to fertilize. In the same context, chemotactic attraction and the role of the jelly are discussed. Comparison with earlier work of Rothschild and Swann and of Hultin and Hagstrom clarifies some discrepancies between and within these papers.
189 citations
TL;DR: The integrated whole-organism model is of the comprehensive type, is nonlinear, and evidences the major crucial processes of glucose, insulin, and glucagon dynamics per se and their interrelationships.
Abstract: An integrated whole-organism model of the short-term blood glucose regulation system is presented. The model is of the comprehensive type, is nonlinear, and evidences the major crucial processes of glucose, insulin, and glucagon dynamics per se and their interrelationships. Validation of the model has been performed by dealing simultaneously with different kinds of test inputs in a variety of normal and pathological states and by looking not only at plasma accessible variables, but also at the behavior of unit processes. Current practical uses of the model in the area of carbohydrate metabolism regulation are briefly outlined.
183 citations
TL;DR: It is shown that the problem of determining a phylogeny of maximum parsimony from a given set of protein sequences is NP-complete, which implies that it is more fruitful to attempt to develop heuristic techniques than to try to develop exact algorithms for phylogeny construction.
Abstract: The problem of determining a phylogeny of maximum parsimony from a given set of protein sequences is defined. It is shown that this problem is what is called, in computer science, NP-complete. The implication of this result is that it is equivalent in difficulty to a host of other problems in combinatorial optimization which are notorious for their intractability. This implies that it is more fruitful to attempt to develop heuristic techniques (which do not guarantee maximum parsimony but which do run in reasonable computer time) than to try to develop exact algorithms for phylogeny construction
157 citations
TL;DR: A population dynamics model for a heterogeneous population is used to compare the effectiveness of six prevention methods for gonorrhea involving population screening and contact tracing of selected groups.
Abstract: A population dynamics model for a heterogeneous population is used to compare the effectiveness of six prevention methods for gonorrhea involving population screening and contact tracing of selected groups. The population is subdivided according to sex, sexual activity, and symptomatic or asymptomatic infection. For this model contact tracing of certain groups is more effective than general population screening.
103 citations
TL;DR: In this paper, it is shown that the divergence angle occurring in nature (and leading to the appearance of the Fibonacci numbers) gives the theoretically most efficient packing of the seeds.
Abstract: In the mathematical description given by Vogel [1], the seeds on a sunflower head are centered at points on a “cyclotron spiral,” with constant divergence angle between any two successive seeds. It is shown that the divergence angle occurring in nature (and leading to the appearance of the Fibonacci numbers) gives the theoretically most efficient packing of the seeds. The inadequacy of Vogel's explanation for the occurrence of the Fibonacci angle is also discussed.
81 citations
TL;DR: In this paper, the authors proposed a crown shape that maximizes the light absorption when as even as possible light absorption is preconditioned for each moment throughout a single day or the whole growing period.
Abstract: Maximization of light absorption throughout the whole growing period is used as the criterion in a theoretical study of the optimum crown shape. A tall, narrow conelike crown shape of constant volume maximizes the light absorption when as even as possible light absorption is preconditioned for each moment throughout a single day or the whole growing period.
58 citations
TL;DR: In this paper, the authors present some population models, containing both age structure and diffusion, for which there are stable age distributions, which are distributions which are products of functions of age only with functions of space and time only.
Abstract: The paper presents some population models, containing both age structure and diffusion, for which there are stable age distributions. This means distributions which are products of functions of age only with functions of space and time only. For these models a formal calculation is presented showing that the stable distributions will always be the large-time limits. Two kinds of diffusion are studied: random, and directed (which means diffusion to avoid crowding). The principal specialization of the age structure is that the death modulus can be decomposed into the sum of two terms. The first is a function of age only and represents deaths by natural causes, while the second is a function of total population only and represents environmental effects.
57 citations
TL;DR: It is shown that the velocity of label release is v = v max T T /( K 1 2 + T T ), where both V max and K 1 1 2 are linear functions of the effector- cell population and T T is the initial target-cell population.
Abstract: A mathematical model of the 51Cr-release microcytotoxicity assay is utilized to find conditions under which the kinetics of this assay resemble the kinetics of a classical enzyme-substrate reaction. Assuming a steady-state approximation, that “bystander” effector cells do not bind markedly better than the cytotoxic effector cells, and that the programming of the target cells for lysis is irreversible, it is shown that the velocity of label release is v = vmaxTT/(K 1 2 +TT), where both Vmax and K 1 2 are linear functions of the effector-cell population and TT is the initial target-cell population. Moreover, the expressions for K 1 2 and Vmax are expressed in terms of natural kinetic parameters of the process and attributes of the noncytotoxic bystanders.
TL;DR: The development and application of three powerful algorithms for the analysis and simulation of mathematical models consisting of ordinary differential equations and an unconstrained global optimization algorithm are reported, showing that the algorithms are not only practicable for high-order models, but also useful as conceptual tools.
Abstract: This paper reports the development and application of three powerful algorithms for the analysis and simulation of mathematical models consisting of ordinary differential equations. First, we describe an extended parameter sensitivity analysis: we measure the relative sensitivities of many dynamical behaviors of the model to perturbations of each parameter. We check sensitivities to parameter variation over both small and large ranges. These two extensions of a common technique have applications in parameter estimation and in experimental design. Second, we compute sensitivity functions, using an efficient algorithm requiring just one model simulation to obtain all sensitivities of state variables to all parameters as functions of time. We extend the analysis to a behavior which is not a state variable. Third, we present an unconstrained global optimization algorithm, and apply it in a novel way: we determine the input to the model, given an optimality criterion and typical outputs. The algorithm itself is an efficient one for high-order problems, and does not get stuck at local extrema. We apply the sensitivity analysis, sensitivity functions, and optimization algorithm to a sixth-order nonlinear ordinary differential equation model for human eye movements. This application shows that the algorithms are not only practicable for high-order models, but also useful as conceptual tools.
TL;DR: Results and techniques from the theory of Volterra integral equations are used to investigate the existence, uniqueness, positivity, and boundedness of solutions, the existence of equilibrium solutions, and the stability of equilibria of a nonlinear age-dependent model for bacterial growth in a continuous fermentation process.
Abstract: Using results and techniques from the theory of Volterra integral equations, we investigate the existence, uniqueness, positivity, and boundedness of solutions, the existence of equilibrium solutions, and the stability of equilibria of a nonlinear age-dependent model for bacterial growth in a continuous fermentation process. The demographic parameters of the model, such as the growth, death, and fission rates of the cells, depend (in a nonlinear way) on the substrate concentration in the reactor tank.
TL;DR: In this paper, it was shown that no one strategy need be eliminated in the limit of an evolutionary game and that the geometric mean of the frequencies of certain strategies approaching zero approaches zero.
Abstract: Domination among strategies in an evolutionary game implies that the geometric mean of the frequencies of certain strategies—the unfit—approaches zero. However, as we show by example no one strategy need be eliminated in the limit.
TL;DR: In this paper, a case-by-case analysis of single-input experiments on all possible three-compartment models is presented, and the structural causes of nonuniqueness are discussed by reference to examples from the latter catalogue.
Abstract: While a choice of techniques exists for checking the deterministic (structural) identifiability of a specific linear, time-invariant model from a specific experiment, and some progress has been made towards topological criteria for identifiability, no method at present available allows quick and reliable checking of a range of models for globally unique identifiability from a range of experiments. Even individual cases are sometimes difficult and tedious to check. The reasons are examined by exhaustive case-by-case analysis of single-input experiments on all possible three-compartment models. All patterns of loss to the environment are covered, and all combinations of observed compartments. Catalogues of minimal observation sets for globally unique identifiability, and of nonuniquely identifiable cases, are presented. The structural causes of nonuniqueness are discussed by reference to examples from the latter catalogue. Methods are given for shortening the derivation of the structural equations giving rise to nonunique parameters. From the diversity of behavior found, it is concluded that the prospects of obtaining a comprehensive set of necessary and sufficient structural conditions for globally unique identifiability are poor.
TL;DR: In the proposed table, parity of women is the main criterion used to summarize the reproductive experience of a study population and can be used to test the temporal difference and difference between countries in fertility and to evaluate the effectiveness of family planning programs.
Abstract: Birth order instead of age of women is considered the basic variable in the study of human reproduction. In the proposed table, parity of women is the main criterion used to summarize the reproductive experience of a study population. The elements of the table serve to describe the fertility pattern of a female population; to determine the population distribution by completed family size; and to estimate the parity progression ratios, the parity specific fertility rates, the mean waiting time between the ith birth and the completion of family, and others. The table can be used to test the temporal difference and difference between countries in fertility and to evaluate the effectiveness of family planning programs. A fertility table based on current U.S. data has been constructed for illustration.
TL;DR: In this article, a new family of stock-recruitment curves is proposed for the growth term in the differential equation for biomass change with time, which is based on a new class of stock recruitment curves and the resultant yield curves of a general production model are investigated.
Abstract: A form for the growth term in the differential equation for biomass change with time is proposed, which is based on a versatile new family of stock-recruitment curves. The resultant yield curves of a general production model are investigated. The form of the curves is more variable than is usual in such models, but depends systematically on the values adopted for biologically well-defined parameters such as natural mortality and natural rate of increase. The consequences for management are explored by examining the magnitude of the maximum sustainable yield (MSY) and suitable values for the yield⧸biomass ratio. It is concluded that the MSY is of the order of but usually somewhat less than 0.5 MB 0 , although variation by a factor of three either way is possible. MSY is usually attained at a yield⧸biomass ratio no more than a few times M (the natural mortality), and at biomasses less than half the pristine biomass ( B 0 ). However, exploitation of stocks having domed stock-recruitment curves at MSY is dangerous, because the ability of the population to recover from perturbations is impaired. This is associated with an increase in the return time for the population.
TL;DR: A more realistic model is developed which has the advantages of a stable steady state point and an unstable extinction point for predator-prey interactions in which the predator population feeds on the eggs of the prey population.
Abstract: This manuscript is intended to provide further insight into the dynamics of predator-prey interactions in which the predator population feeds on the eggs of the prey population. Reasons are provided for the results shown in a previous article. It is further shown that, by making a slight change in the predator harvesting function of the previous model, a more realistic model is developed which has the advantages of a stable steady state point and an unstable extinction point. Comments are also made regarding the special case of this model for which the predator population is constant with respect to the harvest of the prey's eggs. Lastly, a proposition is set forth by which the results of this work could be tested.
TL;DR: It is shown that ligamentious structures contribute to the stability of the system and help maintain the integrity of the joint.
Abstract: A nine-link planar biped model is studied. Its nonlinear differential equations are derived. Constraints due to the connections of the links and the contact between the model and the ground are analyzed, and forces of constraint are specified as functions of the state and inputs. With large external forces acting on the model, connection constraints are maintained by the ligaments and other soft tissue structures. It is shown that ligamentious structures contribute to the stability of the system and help maintain the integrity of the joint. By using linear feedback control, the nine-link model is stabilized around the vertical stance. The stable motion of the system in the vicinity of the vertical is studied by computer simulation of walking and tiptoe gaits.
TL;DR: In this paper, a model of coupled fluid flow in the capillaries and tissue of the microcirculation is presented, and the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary is investigated.
Abstract: Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.
TL;DR: In this article, a stochastic model for tumor growth is derived as a diffusion approximation of a continuous-time, density dependent branching process with a Gompertz growth law as the deterministic part.
Abstract: A stochastic model for tumor growth is derived as a diffusion approximation of a continuous-time, density dependent branching process with a Gompertz growth law as the deterministic part. For the diffusion process, the conditional probabilities of extinction, reaching a size c , and doubling are computed along with the expected time of these events. The results are given in terms of integrals, which are evaluated by numerical methods that account for the logarithmic singularity introduced by the Gompertz growth law. When the variance of the branching process is small compared to the deterministic term, simplified asymptotic expressions are given using methods that are modified for the Gompertzian logarithm. The results are used to find the probability of implant take in limiting-dilution assay experiments and the probability of a tumor becoming detectable in carcinogenesis.
TL;DR: An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here, intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities ofcapillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature.
Abstract: Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.
TL;DR: In the present study, biologically interpretable models, the SAR framework, and relevant statistical techniques are integrated in a form which suits the practical research worker.
Abstract: Considerable attention has been paid in the literature to the development of biologically interpretable mathematical models of spontaneous stationary neuronal spike-train activity. However, these models work poorly in applied spike-train analysis. In order to improve matters, we need (1) to select appropriate statistical techniques, and (2) to fit the models into the semialternating renewal (SAR) framework. In the present study, biologically interpretable models, the SAR framework, and relevant statistical techniques are integrated in a form which suits the practical research worker. The theory is illustrated with reference to the analysis of a number of spike trains.
TL;DR: In this article, a model of a microcirculatory unit was developed to study oxygen exchange processes within the upper part of the skin, including the loop-shaped capillary structure of stratum papillare, the nonlinear binding of oxygen by hemoglobin, and the shift of the oxygen hemoglobin dissociation curve due to temperature variations.
Abstract: A model of a microcirculatory unit has been developed to study oxygen exchange processes within the upper part of the skin. The model includes the loop-shaped capillary structure of stratum papillare, the nonlinear binding of oxygen by hemoglobin, and, in particular, the shift of the oxygen hemoglobin dissociation curve due to temperature variations. The corresponding nonlinear elliptic boundary value problem is defined and the existence of at least one solution assured. After describing the numerical procedure to calculate an approximation to the solution, results of several calculations representing different supply situations of the upper skin are presented and discussed.
TL;DR: In this article, the dynamics of the discrete, scalar population model xk + 1 = ax2k(1 − xk) are investigated and some similarities to and differences from previous models are observed.
Abstract: The dynamics of the discrete, scalar population model xk + 1 = ax2k(1 – xk) are investigated. In addition to density dependence, which has been studied previously by many, this equation models the threshold phenomenon. Some similarities to and differences from previous models are observed. In particular, for large a values this model exhibits chaos which is restricted to a nowhere dense Cantor set of measure 0. In order to explain this, a piecewise linear simplification of the model is considered. Other models exhibiting similar dynamics are also mentioned.
TL;DR: A population-balance mathematical model of microbial growth in a flow reactor is formulated which incorporates an asymmetric-division, budding-cycle model of coordinated cell and nuclear division cycles for the budding yeast Saccharomyces cerevisiae, providing new bases for evaluating budding-yeast-cell cycle models.
Abstract: A population-balance mathematical model of microbial growth in a flow reactor is formulated which incorporates an asymmetric-division, budding-cycle model of coordinated cell and nuclear division cycles for the budding yeast Saccharomyces cerevisiae. Analytical solutions are obtained for limiting nutrient and cell-number concentrations in the reactor as functions of basic cell cycle parameters. Frequency functions for cell mass and DNA content in the resident yeast population are also derived under different assumptions concerning cell mass and DNA synthesis and bud scar accumulation. These results, which correspond to experimentally observable medium and population variables, provide new bases for evaluating budding-yeast-cell cycle models and for deducing kinetics of mass and DNA synthesis in single cells growing in steady-state, asynchronous populations.
TL;DR: In this article, a general method to solve kin-number problems in stable, exponentially growing populations is developed with the help of branching processes, and an assortment of applications are made.
Abstract: If an individual is sampled randomly from a population, what is the probability that he or she is first born? What is the probability distribution of his number of sisters or cousins? These are examples of kin-number problems. A general method to solve such problems in stable, exponentially growing populations is developed with the help of branching processes. An assortment of applications is made.
TL;DR: Whether a mathematical model which represents endothelial extraction by Michaelis-Menten kinetics could explain the time variation in the instantaneous extraction ratio curves and a method for estimating the kinetic parameters of the saturable transfort process in such an organ are evaluated.
Abstract: When a bolus containing a nonpermeating indicator and an indicator which permeates the endothelial cell membrane by a saturable process is injected into the blood flowing into the lung, the instantaneous extraction ratio curves measured in the pulmonary venous outflow are asymmetric with respect to the nonpermeating indicator curve. If the bolus contains a sufficient quantity of the permeating indicator that the capillary concentration begins to saturate the transfort mechanism, the extraction ratio curves are concave upward as well. The purpose of this study was to determine whether a mathematical model which represents endothelial extraction by Michaelis-Menten kinetics could explain the time variation in the instantaneous extraction ratio curves. The venous concentration curves were assumed to be the result of the endothelial transfort and distributed capillary input and transit times. In addition, we evaluated a method for estimating the kinetic parameters ( K m and V max ) of the saturable transfort process in such an organ. The results of simulations indicate that the important features of the data can be reproduced by the model, and that useful estimates of the kinetic parameters will be obtained from linear multiple regression analysis of the venous concentration curves if the standard deviation of the capillary input time distribution is not less than that of the capillary transit time distribution.
TL;DR: In this paper, structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system are treated and results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem.
Abstract: This article treats structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system. Newly formulated conditions are presented for certain of these parameters to be zero or positive. Also results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem. Answers to some questions raised in the recent literature about coefficient dependence are discussed.
TL;DR: In this paper, a continuous bilinear model in state space is used to describe the cell kinetics of a tumor-cell population under the effects of chemotherapy, and an optimal strategy for cancer treatment is derived, based on the need to balance the effects on both cancerous and normal tissues.
Abstract: A continuous bilinear model in state space is used to describe the cell kinetics of a tumor-cell population under the effects of chemotherapy. Firstly, the time-course behavior of a Chinese-hamster-ovary (CHO) cell population is simulated to demonstrate the utility of the model. Then, an optimal strategy for cancer treatment is derived, based on the need to balance the effects on both cancerous and normal tissues. The performance index minimized is the sum of the weighted tumor population and the weighted total drug dosage. The optimization problem has resulted in a two-point boundary-value problem (TPBVP) with a bang-bang control policy, which is solved by the switching-time variation method (STVM). Computer simulation of CHO cells is also carried out as a numerical example of determining optimal cancer chemotherapy.