Analysis of interaction for mixtures of agents using the linear isobole

The linear isobole that is commonly used as a reference for the study of interaction is derived from the interaction of an agent with itself. It is shown that the general use of the linear isobole in the study of the combined effects of mixtures of agents implies interaction between the agents whether the dose-effect curves of the agents are the same or not. It is difficult to generalize the interaction between two doses of the same agent to the interaction between two doses of different agents with different action mechanisms without the use of a mechanistic model. Predictions using non-interaction defined as independent action are generally different from those using linear isobole. A simple mechanistic framework based on the concept of common intermediate lesions is introduced in this paper to relate these two methods used for the analysis of synergism and antagonism. In this framework of lesion development, two agents that have no common intermediate lesion in their action will be non-interactive (referred to as independent action). When the two agents share a common intermediate, it is shown that the combined effect will follow the linear isobole (referred to as common action). This simple framework of analysis is applicable to the general study of interaction between two agents with different types of dose-effect curves.     

Contribution to the theory of random and biased nets

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multiple-axon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.     

A reinterpretation of the mathematical biophysics of the central nervous system in the light of neurophysiological findings

The fundamental equations for the interaction between neurons used in mathematical biophysics seem at first incompatible with the actual neurophysiological findings on the synaptic transmission. It is shown, however, that those equations may be readily interpreted in terms of accepted neurophysiological views. What has been termed “synapse” in mathematical biophysics must be regarded as a complicated network of internuncial neurons. It is shown that, under rather conditions, the number of those interneurons willstatistically vary with time according to the differential equation postulated for the excitatory and inhibitory factors. The latter are thus interpreted as the number of excitatory and inhibitory interneurons.     

A mathematical model for cell sorting,migration and shape in the slug stage ofDictyostelium discoideum

A mathematical model for cell sorting and migration in the slug stage of cellular slime moldsDictyostelium discoideum is proposed. Assuming that a slug is a “mixed fluid” of prespore and prestalk cells, a set of equations which describe the dynamics of cell distribution, internal pressure and velocity of hte slug are derived from the balance formula of individual cell movement. These equations are analyzed to obtain the spatial patterns of the two types of cells at dynamical equilibrium and the relationship between the migration velocity and the slug size. The body shape of the elongated slug at the migrating stage is also investigated, taking account of the law of surface tension. The stable shapes of slugs with different volumes are explicity obtaained and the existence of critical size of a slug is suggested.     

Comparative substrate-inhibitor analysis of mink liver monoamine oxidases

Comparative substrate-inhibitor analysis of catalytic properties of liver monoamine oxidases (MAO) was performed in the mature males of the American mink Mustela vison and the European mink Mustela lutreola. The action on the MAO activity was studied of alkaloids of the benzo[c]phenanthridine group: sanguinarine and chelidonine, diisoquinoline alkaloid berberine, medicinal agents “Ukrain” and “Sanguirythrin” as well as derivatives of 2-propylamine: deprenyl and chlorgylin. The latter turned out to be irreversible inhibitor of the MAO A form, whereas deprenyl-irreversible inhibitor of the MAO B form in both studied mink species. The selectivity of action of each inhibitor on the corresponding liver MAO form for the species M. vison was one order of magnitude stronger than for the species M. lutreola. All studied alkaloids as well medicinal agents on their basis have been shown to be specific irreversible inhibitors of the intermediate strength of the liver MAO A form of both mink species. They inhibit the enzymatic deamination of serotonin, tyramine, and tryptamine without affecting the deamination reaction of benzylamine and β-phenylethylamine (at concentrations of 10 mM and lower). Out of five studied isoquinoline agents, the medication “Ukrain” and alkaloid chelidonine have the highest inhibitory action; the agent “Sanguirythrin” and alkaloids berberine and sanguinarine produce the weaker monoamine oxidase effect. The revealed specificity of action of the studied inhibitors is an indirect evidence for the presence in the liver enzymes of both mink species, like in the rat liver enzyme, of several molecular forms.     

A comparative investigation of certain difference equations and related differential equations: Implications for model-building

Many mathematical models for physical and biological problems have been and will be built in the form of differential equations or systems of such equations. With the advent of digital computers one has been able to find (approximate) solutions for equations that used to be intractable. Many of the mathematical techniques used in this area amount to replacing the given differential equations by appropriate difference equations, so that extensive research has been done into how to choose appropriate difference equations whose solutions are “good” approximations to the solutions of the given differential equations. The present paper investigates a different, although related problem. For many physical and biological phenomena the “continuum” type of thinking, that is at the basis of any differential equation, is not natural to the phenomenon, but rather constitutes an approximation to a basically discrete situation: in much work of this type the “infinitesimal step lengths” handled in the reasoning which leads up to the differential equation, are not really thought of as infinitesimally small, but as finite; yet, in the last stage of such reasoning, where the differential equation rises from the differentials, these “infinitesimal” step lengths are allowed to go to zero: that is where the above-mentioned approximation comes in. Under this kind of circumstances, it seems more natural tobuild themodel as adiscrete difference equation (recurrence relation) from the start, without going through the painful, doubly approximative process of first, during the modeling stage, finding a differential equation to approximate a basically discrete situation, and then, for numerical computing purposes, approximating that differential equation by a difference scheme. The paper pursues this idea for some simple examples, where the old differential equation, though approximative in principle, had been at least qualitatively successful in describing certain phenomena, and shows that this idea, though plausible and sound in itself, does encounter some difficulties. The reason is that each differential equation, as it is set up in the way familiar to theoretical physicists and biologists, does correspond to a plethora of discrete difference equations, all of which in the limit (as step length→0) yield the same differential equation, but whose solutions, for not too small step length, are often widely different, some of them being quite irregular. The disturbing thing is that all these difference equations seem to adequately represent the same (physical or biological) reasoning as the differential equation in question. So, in order to choose the “right” difference equation, one may need to draw upon more detailed (physical or) biological considerations. All this does not say that one should not prefer discrete models for phenomena that seem to call for them; but only that their pursuit may require additional (physical or) biological refinement and insight. The paper also investigates some mathematical problems related to the fact of many difference equations being associated with one differential equation.     

Outline of a probabilistic approach to animal sociology: II

Under certain assumptions concerning the probabilities of “mutations,” i.e. changes of structure of bird societies, it is shown that the probability distribution for all possible structures of a society ofN individuals approaches a limit independent of the initial probability distribution. A formula for the limiting distribution is derived.     

Dynamic analysis,with feedback characterization,of hybrid human musculoskeletal frameworks by the linegraph-flowgraph procedure

The dynamic response of human musculo-skeletal framework is treated by (i) idealization of the musculo-skeletal framework as hybrid structural networks possessing feedback characteristics and then (ii) employing linegraph-flowgraph procedures for the feedback characterization of the hybrid structural networks. Topological procedures are used in which a “tree” of a network furnishes the skeleton upon which the “linkage” (muscle representing) members provide interaction. Feedback characterization (representing the sensitivity of the skeletal members to the tensile forces) is defined, between the internal “linkage” and “tree” members, by means of the flowgraph. Mikusinski operational calculus is used to facilitate representation of inertia effects by dynamic feedback characterization, with inclusion of initial conditions.     

Some remarks on the boolean algebra of nervous nets in mathematical biophysics

Recent demonstration by the author has shown that the fundamental equations of the mathematical biophysics of the central nervous system can be considered as describing the behavior of very large numbers of neurons, of which each one follows discontinuous laws, such as discussed by W. S. McCulloch and W. Pitts. In that light some of the old problems are discussed. The comparative merits of the “microscopic” and “macroscopic” approaches are discussed for the problem of the point to point correspondence between the retina and the cortex, with the number of connecting fibers much less than the number of cells. Some aspects of discrimination of intensities are also discussed. Finally, a few generalizations of the McCulloch-Pitts treatment are suggested, and a nervous network is constructed which illustrates some aspects of the perception of numbers.     

A model for unequal ventilation of the lungs assuming a common dead space and two separate dead spaces

This is a model for the time-variation of helium concentrations in lung wash-out curves. The helium (or other inert gas) is in a spirometer, which is connected by a common dead space to two separate dead spaces, each of which leads into a chamber. The chambers expand and contract, thus taking in some helium at each “breath.” Equations for the changes in helium concentration in each part of the system are set up; in this way difference equations are derived for the amount of helium in the spirometer after each breath, in and out, and complete solutions when the initial concentration is zero in all parts of the system except the spirometer. A simple solution when the chambers do not essentially differ (“equal ventilation”) is compared with the general case. The concept of “unequal lung ventilation” is discussed critically in relation to the model; some physiological interpretations are also mentioned. Numerical examples are given to show the effect of changes in various constants, in particular tidal volumes, end volumes, and the common dead space.     

Is there a general theory of community ecology?

Community ecology entered the 1970s with the belief that niche theory would supply a general theory of community structure. The lack of wide-spread empirical support for niche theory led to a focus on models specific to classes of communities such as lakes, intertidal communities, and forests. Today, the needs of conservation biology for metrics of “ecological health” that can be applied across types of communities prompts a renewed interest in the possibility of general theory for community ecology. Disputes about the existence of general patterns in community structure trace at least to the 1920s and continue today almost unchanged in concept, although now expressed through mathematical modeling. Yet, a new framework emerged in the 1980s from findings that community composition and structure depend as much on the processes that bring species to the boundaries of a community as by processes internal to a community, such as species interactions and co-evolution. This perspective, termed “supply-side ecology”, argued that community ecology was to be viewed as an “organic earth science” more than as a biological science. The absence of a general theory of the earth would then imply a corresponding absence of any general theory for the communities on the earth, and imply that the logical structure of theoretical community ecology would consist of an atlas of models special to place and geologic time. Nonetheless, a general theory of community ecology is possible similar in form to the general theory for evolution if the processes that bring species to the boundary of a community are analogized to mutation, and the processes that act on the species that arrive at a community are analogized to selection. All communities then share some version of this common narrative, permitting general theorems to be developed pertaining to all ecological communities. Still, the desirability of a general theory of community ecology is debatable because the existence of a general theory suppresses diversity of thought even as it allows generalizations to be derived. The pros and cons of a general theory need further discussion.     

Problems in the assessment of heavy-metal levels in estuaries and the formation of a pollution index

Most estuaries receive a high heavy-metal input from industries. This is reflected in the relatively high levels found in numerous estuarine organisms and in sediments. Many indicators have been suggested for facilitating the detection of heavy-metal pollution, but the problems in using these indicators to evaluate the metal loading of estuaries are considerable. Variations in species composition, and conditions at different sites, differences in season of sampling, and age of organism, as well as different metal levels in different parts of the organism, make the interpretation of results difficult. The levels reported here, similar to those in other unpolluted estuaries, have been used to suggest a baseline concentration for heavy metals in estuaries. The concept of a baseline is fundamental to the formation of a “Biological Quality Index” and “Pollution Load Index,” and a formula for such an index is suggested and tested at a preliminary level against published data for an English and a European estuary.     

Electrical Concepts in the Surface Electromyographic Signal

There are frequently used electrical terms in the biofeedback literature. Often it is assumed that the reader has detailed knowledge of these terms. The difficulty begins when seemingly familiar terms are used as a basis for an in-depth explanation of the process of electromyography. For example, the concept of impedance is based on three building blocks of electricity: current, voltage and resistance. The term “impedance” is found in every manual for biofeedback equipment with the suggestion that the electrode site be kept “low” and the encoder input “high”. A little electrical knowledge can explain why this is so and in the process formulate a more thorough understanding of the equipment used everyday with a client.     

The Morphostatic Limit for a Model of Skeletal Pattern Formation in the Vertebrate Limb

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.     

Fire regime: history and definition of a key concept in disturbance ecology

“Fire regime” has become, in recent decades, a key concept in many scientific domains. In spite of its wide spread use, the concept still lacks a clear and wide established definition. Many believe that it was first discussed in a famous report on national park management in the United States, and that it may be simply defined as a selection of a few measurable parameters that summarize the fire occurrence patterns in an area. This view has been uncritically perpetuated in the scientific community in the last decades. In this paper we attempt a historical reconstruction of the origin, the evolution and the current meaning of “fire regime” as a concept. Its roots go back to the 19th century in France and to the first half of the 20th century in French African colonies. The “fire regime” concept took time to evolve and pass from French into English usage and thus to the whole scientific community. This coincided with a paradigm shift in the early 1960s in the United States, where a favourable cultural, social and scientific climate led to the natural role of fires as a major disturbance in ecosystem dynamics becoming fully acknowledged. Today the concept of “fire regime” refers to a collection of several fire-related parameters that may be organized, assembled and used in different ways according to the needs of the users. A structure for the most relevant categories of parameters is proposed, aiming to contribute to a unified concept of “fire regime” that can reconcile the physical nature of fire with the socio-ecological context within which it occurs.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.     

Dynamic role of “illite-like” clay minerals in temperate soils: facts and hypotheses

Analysis of new data and reinterpretation of published information for clay minerals found in temperate climate soil profiles indicates that there is often a gradient of “illite-like” clay minerals with depth. We used the term “illite-like” because these observations are based on X-Ray Diffractogram patterns and not on layer charge measurements which allow to define properly illite. It appears that “illite-like” layers are concentrated in the upper, organic - rich portion of the soil profile both under grassland and forest vegetation. “Illite-like” layer quantity seems directly related to soil potassium status. Indeed, intensive agriculture practises without potassium fertilization reduce “illite-like” content in surface soils, whereas several years of potassic fertilization without plant growth can increase “illite-like” content. The potassic soil clay mineral, illite, is particularly important in that it can be the major source of readily available potassium for plants. Spatial and temporal dynamics of clay minerals should be related to the potassium cycle. We propose that the frequently observed general trend of increasing exchangeable potassium in the top soil can be correlated with an increase in “illite-like” in the clays and that the decrease of potassium caused by intensive agricultural practices leads to “illite-like” layer destabilization. This vision of “illite-like” layer as a potassium reservoir refueled by plants and emptied by intensive cropping renews the concept of potassium availability and indicates a need to be discussed as well in natural ecosystems as in cultivated ecosystems.     

Biomechanical analysis of movement strategies in human forward trunk bending. I. Modeling

 Two behavioral goals are achieved simultaneously during forward trunk bending in humans: the bending movement per se and equilibrium maintenance. The objective of the present study was to understand how the two goals are achieved by using a biomechanical model of this task. Since keeping the center of pressure inside the support area is a crucial condition for equilibrium maintenance during the movement, we decided to model an extreme case, called “optimal bending”, in which the movement is performed without any center of pressure displacement at all, as if standing on an extremely narrow support. The “optimal bending” is used as a reference in the analysis of experimental data in a companion paper. The study is based on a three-joint (ankle, knee, and hip) model of the human body and is performed in terms of “eigenmovements”, i.e., the movements along eigenvectors of the motion equation. They are termed “ankle”, “hip”, and “knee” eigenmovements according to the dominant joint that provides the largest contribution to the corresponding eigenmovement. The advantage of the eigenmovement approach is the presentation of the coupled system of dynamic equations in the form of three independent motion equations. Each of these equations is equivalent to the motion equation for an inverted pendulum. Optimal bending is constructed as a superposition of two (hip and ankle) eigenmovements. The hip eigenmovement contributes the most to the movement kinematics, whereas the contributions of both eigenmovements into the movement dynamics are comparable. The ankle eigenmovement moves the center of gravity forward and compensates for the backward center of gravity shift that is provoked by trunk bending as a result of dynamic interactions between body segments. An important characteristic of the optimal bending is the timing of the onset of each eigenmovement: the ankle eigenmovement onset precedes that of the hip eigenmovement. Without an earlier onset of the ankle eigenmovement, forward bending on the extremely narrow support results in falling backward. This modeling approach suggests that during trunk bending, two motion units – the hip and ankle eigenmovements – are responsible for the movement and for equilibrium maintenance, respectively. Received: 1 July 1999 / Accepted in revised form: 23 October 2000     

Response time and threshold of a random net

The response time of a random net is defined as the expected time (measured in the number of synaptic delays) required for the excitation in the net (measured by the fraction of neurons firing per unit time) to reach a certain level. The response time is calculated in terms of the net parameters as a function of the intensity of the outside stimulation. Two principal types of cases are studied, 1) an instantaneous initial stimulation, and 2) continuously applied stimulation. It is shown that for a certain type of net where the required level of excitation is small, the response time-intensity equation reduces to the one derived on the basis of the “one-factor” theory applied to a neural connection. More general assumptions, however, give different types of equations. The concept of the “net threshold” is defined, and its calculation indicated. The net threshold for instantaneous stimulation is, in general, greater than that for continuous stimulation. The results are discussed with reference to existing theories of reaction times.