On the Forkker-Planck equation in the stochastic theory of mortality: I

This paper, consisting of two parts, gives all the mathematical details that were omitted in a previous work by G. A. Sacher and E. Trucco (“The Stochastic Theory of Mortality.”Ann. N. Y. Acad. Sci.,96, 985–1007, cited here as ST). We assume that the reader is familiar with ST, where the stochastic theory of mortality, originally proposed by Sacher, is discussed at length. We recall that the basic model presented there refers to an ensemble of particles performing Brownian motion in one dimension, with the added constraint of two absorbing barriers. These two points, collectively, are designated as the “lethal bound.” Part I (section 1 to 4) deals with the special case in which the two absorbing barriers are symmetrically located at a finite distance from the origin. The solution of the Fokker-Planck equation is obtained from the theory of eigenvalue problems. Quite generally, the eigenfunctions functions belong to the family of Kummer's confluent hypergeometric functions, but the symmetry condition imposed here results in considerable simplification and makes it possible to estimate the first few eigenvalues by a graphical procedure. In section 3 we show how perturbation theory can be applied in the limiting case of “weak homeostasis,” and section 4 deals with the opposite extreme of “strong homeostasis.” A rigorous proof is given for the result corresponding to equation (28) of ST (asymptotic or quasi-static approximation for the “force of mortality”). This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

On the spread of information with time and distance

The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.     

Life and death near a windy oasis

We propose a simple experiment to study delocalization and extinction in inhomogeneous biological systems. The nonlinear steady state for, say, a bacteria colony living on and near a patch of nutrient or favorable illumination (“oasis”) in the presence of a drift term (“wind”) is computed. The bacteria, described by a simple generalization of the Fisher equation, diffuse, divide A→A + A, die A→ 0, and annihilate A + A→ 0. At high wind velocities all bacteria are blown into an unfavorable region (“desert”), and the colony dies out. At low velocity a steady state concentration survives near the oasis. In between these two regimes there is a critical velocity at which bacteria first survive. If the “desert” supports a small nonzero population, this extinction transition is replaced by a delocalization transition with increasing velocity. Predictions for the behavior as a function of wind velocity are made for one and two dimensions. Received: 3 August 1998 / Revised version: 17 July 1999 / Published online: 4 July 2000     

A generalized Fokker-Planck equation in the case of the Volterra model

Zwanzig and Mori's projection-operator method is used in order to derive a generalized nonlinear Fokker-Planck equation for one “relevant” species in the many species conservative Volterra model. The deterministic, autonomous, Markovian equations of motion, when averaged over a suitable ensemble of initial conditions in general give rise to a non-autonomous, non-Markovian stochastic process for the evolution of this relevant species. Moreover, this relevant species may show irreversible damping, although self-interaction terms are absent in the many species model.     

Science through the internet: Researching, evaluating and citing websites

This article attempts to convey the joys and frustrations of skimming the Internet trying to find relevant information concerning an academic’s work as a scientist, a student or an instructor. A brief overview of the Internet and the “do’s and don’ts” for the neophyte as well for the more seasoned “navigator” are given. Some guidelines of “what works and what does not” and “what is out there” are provided for the scientist with specific emphasis for biologists, as well as for all others having an interest in science but with little interest in spending countless hours “surfing the net”. An extensive but not exhaustive list of related websites is provided.     

Two novel gene mutations (Glu174→Lys,Phe383→Tyr) causing the “hepatic” form of carnitine palmitoyltransferase II deficiency

Carnitine palmitoyltransferase II (CPT II) deficiency has two different clinical forms, one with “hepatic” and the other with “muscular” symptoms. We studied the molecular basis of the “hepatic” form in two Japanese siblings. Their CPT II activity in lymphoblasts was reduced to 3% of the level observed in normal controls. cDNA analysis showed that the proband was a compound heterozygote. One allele carried a new mutation, G621→A (Glu174→Lys). The other carried three single-base substitutions; a new mutation, T1249→A (Phe383→Tyr), and two previously reported polymorphisms. The brother had the same four substitutions. Neither of the two new mutations in this study was detected in the 60 alleles of 30 Japanese control subjects. Secondary structure prediction analysis of the mutated CPT II protein was different from that of the normal protein. We concluded that these mutations caused the “hepatic” form of CPT II deficiency in the probands. Received: 16 October 1995     

Binocular fusion in Panum’s limiting case of stereopsis obeys the uniqueness constraint

In the information processing procedure of stereo vision, the uniqueness constraint has been used as one of the constraints to solve the “correspondence problem”. While the uniqueness constraint is valid in most cases, whether it is still valid in some particular stimulus configuration (such as Panum’s limiting case) has been a problem of widespread debate for a long time. To investigate the problem, we adopted the Panum’s limiting case as its basic stimulus configuration, and delved into the phenomenon of binocular fusion from two distinct aspects: visual direction and orientation disparity. The results show that in Panum’s limiting case binocular fusion does not comply with the rules governing regular binocular fusion as far as visual direction and orientation disparity are concerned. This indicates that double fusion does not happen in Panum’s limiting case and that the uniqueness constraint is still valid.     

Mass Spectrometric and Spectroscopic Studies of the Nutritional Supplement Chromium(III) Nicotinate

Despite chromium nicotinate’s popular use as a chromium nutritional supplement, the structure and composition of chromium nicotinate have only been poorly described. As solid chromium nicotinate is intractable, being insoluble or unstable in common solvents, studies on the solid have been limited, and studies of the solution from which the “compound” precipitates have additionally provided little additional data. The results of mass spectrometric and spectroscopic investigations designed to further elucidate the structure and composition of chromium nicotinate are described. The results demonstrated that the three common methods for producing “chromium nicotinate” all yield different compounds, all of which are polymers of Cr(III), oxygen-bound nicotinate, hydroxide, and water. Implications for interpreting results of nutritional studies of “chromium nicotinate” are discussed.     

Consistency of optimal sequence alignments

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. Dedicated to Prof. Akiyoshi Wada on the occasion of his 60^th birthday.     

Law of the Minimum Paradoxes

The “Law of the Minimum” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth (Justus von Liebig, 1840, Salisbury, Plant physiology, 4th edn., Wadsworth, Belmont, 1992) and quantitatively supported by many experiments. Some generalizations based on more complicated “dose-response” curves were proposed. Violations of this law in natural and experimental ecosystems were also reported. We study models of adaptation in ensembles of similar organisms under load of environmental factors and prove that violation of Liebig’s law follows from adaptation effects. If the fitness of an organism in a fixed environment satisfies the Law of the Minimum then adaptation equalizes the pressure of essential factors and, therefore, acts against the Liebig’s law. This is the the Law of the Minimum paradox: if for a randomly chosen pair “organism–environment” the Law of the Minimum typically holds, then in a well-adapted system, we have to expect violations of this law.     

Analysis of the exponential decay model of the neuron showing frequency threshold effects

The exponential decay model of a neuron has been analyzed using the “random walk” approach of stochastic processes and an “absorbing barrier” solution is obtained forg ^T (s)—the Laplace transform of the output pulse interval density function. An expression for the mean output frequency is derived from this and a variety of input-output curves plotted which show frequency threshold effects in single neurons. Our results are compared with those of other authors obtained by computer simulation techniques, and the significance of these results discussed with reference to the possible behavior of networks constructed of such neuron units.     

Properties of the Proximate Parameter Tuning Regularization Algorithm

An important aspect of systems biology research is the so-called “reverse engineering” of cellular metabolic dynamics from measured input-output data. This allows researchers to estimate and validate both the pathway’s structure as well as the kinetic constants. In this paper, the recently published ‘Proximate Parameter Tuning’ (PPT) method for the identification of biochemical networks is analysed. In particular, it is shown that the described PPT algorithm is essentially equivalent to a sequential linear programming implementation of a constrained optimization problem. The corresponding objective function consists of two parts, the first emphasises the data fitting where a residual 1-norm is used, and the second emphasises the proximity of the calculated parameters to the specified nominal values, using an ∞-norm. The optimality properties of PPT algorithm solution as well as its geometric interpretation are analyzed. The concept of optimal parameter locus is applied for the exploration of the entire family of optimal solutions. An efficient implementation of the parameter locus is also developed. Parallels are drawn with 1-norm parameter deviation regularization which attempt to fit the data with a minimal number of parameters. Finally, a small example is used to illustrate all of these properties.     

Binocular fusion in Panum’s limiting case of stereopsis obeys the uniqueness constraint

In the information processing procedure of stereo vision, the uniqueness constraint has been used as one of the constraints to solve the “correspondence problem”. While the uniqueness constraint is valid in most cases, whether it is still valid in some particular stimulus configuration (such as Panum’s limiting case) has been a problem of widespread debate for a long time. To investigate the problem, we adopted the Panum’s limiting case as its basic stimulus configuration, and delved into the phenomenon of binocular fusion from two distinct aspects: visual direction and orientation disparity. The results show that in Panum’s limiting case binocular fusion does not comply with the rules governing regular binocular fusion as far as visual direction and orientation disparity are concerned. This indicates that double fusion does not happen in Panum’s limiting case and that the uniqueness constraint is still valid.     

“Not <Emphasis Type="Italic">Here</Emphasis>”: Making the Spaces and Subjects of “Global Health” in Botswana

This essay argues that what makes “global health” “global” has more to do with configurations of space and time, and the claims to expertise and moral stances these configurations make possible, than with the geographical distribution of medical experts or the universal, if also uneven, distribution of threats to health. Drawing on a study of public–private partnerships supporting Botswana’s HIV/AIDS treatment program, this essay demonstrates ethnographically the processes by which “global health” and its quintessential spaces, namely “resource-limited” or “resource-poor settings,” are constituted, reinforced, and contested in the context of medical education and medical practice in Botswana’s largest hospital. Using Silverstein’s work on orders of indexicality, I argue that the terms of “global health” are best understood as chronotopic, and demonstrate how actors orient themselves and others spatio-temporally, morally, and professionally by using or refuting those terms. I conclude by arguing that taking “global health” on its own terms obscures the powerful forces by which it becomes intelligible. At stake are the frames within which medical anthropologists understand their objects of study, as well as the potential for the spaces of “global health” intervention to expand ever outward as American medical personnel attempt to calibrate their experiences to their expectations.     

A note on the dispersionless growth law for single cells

A population of initially synchronized cells is considered wherein each cell grows according to a dispersionless growth law and the probability of cell division is determined by cell age. The first and second moments of the distribution of birth volumes are considered as functions of time and it is shown that it is impossible for both moments to approach finite, nonzero limits ast→∞. This implies that the volume distribution of the population will not approach a limiting distribution on any finite, nonzero volume interval and that the population will not attain balanced exponential growth. An illustrative example is worked out in detail. The distribution of birth volumes is also analyzed as a function of generation number and it is found that the logarithm of the birth volume in thejth generation is normally distributed asj→∞, with an unbounded variance. Generalizations and implications of these results are briefly discussed. Work supported by the U.S. Atomic Energy Commission.     

Biosemiosis and the Cellular Basis of Mind

The human brain is a complex organ made up of neurons and several other cell types, and whose role is processing information for use in elicitation of behaviors. To accomplish this, the brain requires large amounts of energy, and this energy is obtained by the oxidation of glucose (Glc). However, the question of how the oxidation of Glc by individual neurons in brain results in their collective ability to rapidly generate feats of cognition that allow them to recognize the nature of the universe in which they live and to communicate this information remains unclear. In this article, insights into this process are provided by first considering the brain’ s homeostatic “operating system” for supply of energy to stimulated neurons, and how this system defines the basic unit of brain “structure”. This is followed by consideration of the brain’s “two-cell” neuronal communication mechanism which defines the basic unit of brain “function”. Finally, an analysis of the nature of frequency-encoded “neuronal languages” that enable ensembles of neurons to translate energy derived from the oxidation of Glc into a collective “mind”, the aggregate of all brain processes including those involving perception, thought, insight, foresight, imagination and behavior.     

Redefining the X Axis: “Professionals,” “Amateurs” and the Making of Mid-Victorian Biology – A Progress Report

A summary of revisionist accounts of the contextual meaning of`“professional” and “amateur,” as applied to the mid-Victorian X Club, is followed by an analysis of the liberal goals and inner tensions of this coalition of gentlemen specialists and government teachers. The changing status of amateurs is appraised, as are the new sites for the emerging laboratory discipline of “biology.” Various historiographical strategies for recovering the women’s role are considered. The relationship of science journalism to professionalization, and the constructive engagement of X Club publicists with their empowering audiences, are discussed. Finally, the article assesses how far the content and boundary closure of ``biology,' forged by Thomas Henry Huxley, were related to `professional' and political goals. Purebiology’s social and medical roots are examined, and the way inter-professional and wider Darwinian conflicts resulted in anew lexicon of words for the X Clubbers around 1870, including“evolution” and “agnosticism,” as well as “biology.” Biology’srole in the forging of British national identity is discussed, as are its relationship to the social strategies of liberal, Dissenting, and industrial groups in the country, whose authority sustained the new laboratory rhetoric. This revised version was published online in July 2006 with corrections to the Cover Date.