Monotone and near-monotone biochemical networks

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion. Supported in part by NSF Grants DMS-0504557 and DMS-0614371.     

Chain processes and their biophysical applications: Part I. General theory

A mathematical theory applicable to the biological effects of radiations as chain processes is developed. The theory may be interpreted substantially as a “hit theory” involving the concepts of “sensitive volume” or “target area”. The variability of the sensitivity of the organism to the radiation and its capacity of recovery between single hits is taken into account. It is shown that in a continuous irradiation of a biological aggregate in which the effect of each single hit cannot be observed, recovery and variation of sensitivity are formally equivalent to each other so that a discrimination between these two phenomena is possible only by discontinuous irradiation or by using different radiation intensities. Methods for the calculation of the “number of hits” and for the determination of the kinetics of the processes from “survival curves” or similar experimental data are given. The relation between the recovery and the Bunsen-Roscoe law is discussed. The case in which the injury of the organism is dependent on the destruction of more than one “sensitive volume” is also considered.     

The representation of biological systems from the standpoint of the theory of categories

A mathematical framework for a rigorous theory of general systems is constructed, using the notions of the theory of Categories and Functors introduced by Eilenberg and MacLane (1945,Trans. Am. Math. Soc.,58, 231–94). A short discussion of the basic ideas is given, and their possible application to the theory of biological systems is discussed. On the basis of these considerations, a number of results are proved, including the possibility of selecting a unique representative (a “canonical form”) from a family of mathematical objects, all of which represent the same system. As an example, the representation of the neural net and the finite automaton is constructed in terms of our general theory.     

Molecular set theory: II. An aspect of the biomathematical theory of sets

In an earlier paper (Molecular Set Theory: I.Bull. Math. Biophysics,22, 285–307, 1960) the author proposed a “Molecular Set Theory” as a formal mathematical meta-theoretic system for representing complex reactions not only of biological interest, but also of general chemical interest. The present paper is a refinement and extension of the earlier work along more formal algebraic lines. For example the beginnings of an algebra of molecular transformations is presented. It also emphasizes that this development, together with the genetical set theory of Woodger's and Rashevsky's set-theoretic contributions to Relational Biology, points to the existence of a biomathematical theory of sets which is not deducible from the general mathematical, abstract theory of sets.     

A mathematical model of HIV dynamics in the presence of a rescuing virus with replication deficiency

Recently, an enzyme (Cre recombinase) has been developed by directed evolution that successfully removes the HIV genome from the nuclear DNA of infected cells. To explore this idea further, we hypothesized that a replication deficient virus (called “police virus”), added externally, can deliver such a recombinase which excises the integrated HIV DNA from the genome of infected cells. Such a “police virus” could attack and remove the integrated provirus which is not possible using contemporary strategies. The hypothesis was tested by developing a mathematical model that describes the dynamics of virus-host cell interaction and the consequences of introducing the “police virus”. The simulations show that such a therapeutic vector may eradicate all HIV viruses from the system in the long term. All components of the HIV infection (free virus, latently, and actively infected cells) can be cleared and the system ends up only with susceptible CD4+ cells. The proposed model may provide new insights in the dynamical behavior and future alternative treatments of HIV.     

Chaos in abstract kinetics: Two prototypes

“Spiral-type chaos” and “screw-type chaos” constitute two simple types of nonperiodic oscillatory flow in 3-variable continuous systems. The former type is exhibited, for example, by auniversal system in the switching mode, when the regimens of flow on the two stable branches of the slow manifold in state space are made to differ in an appropriate manner. Screw-type chaos occurs in ahysteresis oscillator between two stable limit cycles, if the rotation gain is positive. For either case, an analogous 2-dimensional “branched papersheet flow” exists. Both flows are determined by a single-variable discrete dynamical system of the Lorenz-Li-Yorke type (as a cross-section), as well as by an equivalent new map. Numerical simulations of two abstract reaction systems giving rise to non-idealized (that is, truly 3-dimensional) flows of either type are presented. The corresponding discrete dynamical systems (Poincaré maps) are 2-dimensional now, having the form of a flattened hairpin (“horseshoe”) in the simplest case. Thus, two actual examples for 3-dimensional flows suspended by a horseshoe diffeomorphism seem to have been found. One contains just a single functionally effective nonlinearity. Real systems of either type may be found in physics, chemistry, biochemistry, biophysics and economy.     

Some views on the role of noise in “self”-organizing systems

This paper deals with information transfer from the environment and “self”-organization in open, nonlinear systems far from thermodynamic equilibrium — in the presence of either non-stationary phase jitter noise, or amplitude stationary noise. By “self”-organization we mean here the progressive formation within the system of sequential, ordered (coherent) relationships between appropriate dynamical variables-like for example, the phase differences between the oscillating components of the system. We take up (in Section II) the classical Laser as a specific example and examine in detail the influence of phase jitter noise in the mode (phase) locking process. We find—as expected—that phase fluctuations in the cavity cause degradation of the coherent behaviour (i.e. increase the entropy) of the system — which, however, levels off, or saturates with time. Further (in Section III) we examine systems where the number of self-sustained oscillating components may vary with time in such a way that the maximum entropy of the system increases faster than the overall instantaneous entropy. We put forth the hypothesis that in such cases — because of the increase of the redundancy — the system gets organized not just in spite of, but merely because of the presence of Noise. Possible applications in biological systems (especially concerning a model of cerebral organization) are briefly discussed. It is understood here, that the system has to display some preliminary dynamical structure before the organizing procedure takes over. What happens afterwards is the subject of this paper.     

On the Semio-Mathematical Nature of Codes

The relational structure of RNA, DNA, and protein bears an interesting similarity to the determination problem in category theory. In this paper, we present this deep-structure similarity and use it as a springboard for discussing some abstract properties of coding in various systems. These abstract properties, in turn, may shed light on the evolution of the DNA world from a semiotic perspective. According to the perspective adopted in this paper, living systems are not information processing systems but “meaning-making” systems. Therefore, what flows in the genetic system is not “information” but “value.” We define meaning, meaning-making, and value and then use these terms to explain the abstract dynamics of coding, which can illuminate many forms of sign-mediated activities in biosystems.     

Concept of energy in biological systems and the effects of irradiations of low energies on enzyme-substrate systems

The recent mathematical formalization of the concepts of matter and extrinsical energy, which are used for the relational representation of biological systems, is employed in the analysis of the important experimental discoveries of Comorosanet al. related to low energy electromagnetic irradiations on enzyme substrates. By means of the present analysis one of the properties inherent to the experimental phenomena is more precisely exposed, and theoretical developments corresponding to “energetical evolutions” in a biological system (Leguizamón, 1976) may now have an experimental basis. Important limitations are introduced for the validity of the commutativity and associativity of cartesian product of sets, when they represent matter and its linked extrinsical energy. In connection with this last aspect, new important knowledge is obtained for the relational mathematical representation of biological systems.     

Ecosystem-like behavior of a random-interaction model I

The behavior of large systems of randomly-interacting variables is examined using an intentionally simplified model. The stable positive solutions are found to exhibit to a significant degree some well-known properties of ecological systems. This resemblance (including for example the predominance of “predator-prey” interactions) is all the more striking in view of the lack of biological “data” at the input end. The findings suggest it advisable to distinguish two kinds of properties in ecosystems. One kind would depend on specifically biological mechanisms; the other would characterize a wide class of persistent systems, and arise from the need for a dynamic balance between positive and negative feedback.     

Question 9: Prospects for the Construction of Artificial Cells or Protocells

The construction of artificial cells or protocells that are a simplified version of contemporary cells will have implications for both the understanding of the origins of cellular Life and the design of “cell-like” chemical factories. In this short communication, we discuss the progress and remaining issues related to the construction of protocells from metabolic products. We further outline the de novo design of a simple chemical system that mimics the functional properties of a living cell without being composed of molecules of biological origin, thereby addressing issues related to Life’s origins. Presented at: International School of Complexity—4th Course: Basic Questions on the Origins of Life; “Ettore Majorana” Foundation and Centre for Scientific Culture, Erice, Italy, 1–6 October 2006.     

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.     

Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells—I. Reduced systems

This study is related to a model describing the behavior of barium-treatedAplysia neurons generating regular burst-plateau patterns. The model is represented by an autonomous dynamical system, defined inR ⁴ and depending on a small parameter. This paper is restricted to the qualitative study of three “reduced systems” deduced from the “complete system”. Part of the study is performed with the use of the qualitative theory of singular perturbations. The predicted behaviors are compared with experimental results.     

The probabilistic approach to the effects of radiations and variability of sensitivity

The calculation of the size of the “sensitive volume” or “control center” in biological effects of radiations is discussed from the viewpoint of the probabilistic theory of these phenomena based on the concept of random “effective events”. On the bases of that theory, the resistivity of a microorganism to radiation is defined as its “mean life” under a radiation of one roentgen per minute. This mean is calculated for processes with and without recovery. The case of variable sensitivity, as it occurs for instance during mitosis, is discussed in detail. Methods are given to calculate this variability from survival curves or similar experimental data. The theory is applied to experiments of A. Zuppinger on irradiation ofAscaris eggs with X-rays.     

A reconsideration of “Races” and their impact on the origins of the Chalcolithic in the Levant using available anthropological and archaeological data

Populations of the Chalcolithic Levant as defined by archaeological excavations has in many cases reinforced the traditional scheme that a number of “races” are present. This scheme is usually based not only on differential cultural traditions as identified by archeologists, but also on the available skeletal evidence as discussed by physical anthropologists. Recently this view has been challenged and it has been suggested that the metrical and anatomical range of variability as identified within Chalcolithic populations can be subsumed into a single population or “racial” range. This paper examines both the available biological and archaeological evidence from the Chalcolithic Levant and concludes that there is no strong archaeological or biological evidence to support a multiple “racial” origin for the Chalcolithic of the Levant.     

Abstract mathematical molecular biology: II some relational consequences of the “one gene-one enzyme” hypothesis

Following the program outlined in a previous paper (Bull. Math. Biophysics,23, 237–260, 1961), a further abstract study is made of some simple relational systems which possess some properties of living organisms. It is shown that the “one gene-one enzyme” hypothesis leads to the conclusion that either all genes are built of the same chemical building blocks, or that at least all genes have a number of building blocks in common. A consistent relational application of the “one gene-one enzyme” hypothesis leads moreover to the conclusion that replication is not an inherent property of a gene. Rather there must be a set of enzymes which “copy” the genes. The number of enzymes in this set must be less than the number of genes and therefore the activity of those “copying” enzymes cannot be absolutely specific.     

A mathematical study of non-equimolar ternary gas diffusion

In view of the increasing evidence that multicomponent diffusion effects could be significant in biological gas exchange systems, a non-equimolar film model of multicomponent diffusion was derived. “Osmotic” ternary diffusion was studied for the gas systems He−N₂−O₂, He−SF₆−O₂, and N₂−SF₆−O₂. Diffusional fluxes and concentration profiles were calculated under both the “square-root” and the “product” flux conditions. Results were also compared with those obtained using the equimolar flux condition. It was found that the greater the difference of the diffusibilities between the two active components in a system, the greater the osmotic fluxes, and also the more alinear the concentration profiles. These results support the suggestion that the “product” condition applies to molecular diffusion in free space, the “square-root” condition to molecular diffusion in pores, and the equimolar flux condition to closed diffusion systems.     

Stochastic models for chemical reactions: I. Theory of the unimolecular reaction process

A stochastic model for the basic unimolecular chemical reaction is derived. This model provides a mathematical basis, altogether missing in the current kinetic theory, for the analysis of inherent random fluctuations about the strict concentration-time course prescribed by the existing deterministic theory. Limits on the extent of the predicted inherent variability are obtained and compared with those usually expected purely on the basis of random experimental errors of extraneous origin (not associated with the mechanism of reaction). The results support the extrapolation to chemical systems of a principle of statistical inaccuracy for physical systems which has been called by E. Schroedinger “the Law of Physics.”     

Iota-carrageenan from a newly farmed, rare variety of eucheumoid seaweed—“endong”

A seaweed with the local name “endong” is a rare eucheumoid variety that has recently been farmed in Tawi-Tawi, southern Philippines. Local farmers describe it as “eel-like” because of its long, slender and smooth thalli. Being somewhat similar in appearance to the dominantly farmed seaweed “tambalang” (Kappaphycus alvarezii var. tambalangii), farmers mix their harvests of “endong” with those of “tambalang”. We analyzed the hydrocolloid extract from “endong” using spectroscopic and chemical techniques. The extract’s infrared and nuclear magnetic resonance (¹H and ¹³C) spectra, and sulfate and 3,6-anhydrogalactose contents revealed carrageenan of the iota-type, similar to extracts from another farmed species, Eucheuma denticulatum. This result implies that “endong” carrageenan exhibits physico-chemical properties different from those of kappa-carrageenan of “tambalang”. The findings of this study are of significance to the seaweed industry, as carrageenan quality problems could potentially arise from a mixed harvest. Seaweed farmers are advised to separate their harvests of “endong” and “tambalang”. Presented at the 19th International Seaweed Symposium, Kobe, Japan     

Protention and retention in biological systems

This article proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called “extended present” as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of “biological inertia”.