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The notion of permanence is used to deal with population dynamical systems which are too complicated to allow a detailed analysis of their asymptotic behaviour. This paper offers an exposition of some general mathematical results, illustrated by applications from population genetics, ecology, sociobiology and — somewhat more detailed — from prebiotic evolution.

Proton transfer reactions in principle belong to the simplest possible chemical processes, since only a naked proton is exchanged between the two molecules concerned.

Charles Darwin erkannte in der Vielfalt der Arten die Prinzipien der Evolution: Abwandlung, Wettbewerb und Auslese. Seit Darwins Zeit ist unser Wissen nicht nur auf dem Gebiet der Molekularbiologie, sondern auch der Geophysik und Geochemie der prabiotischen Erde in einer Weise gewachsen, die im 19. Jahrhundert unvorstellbar gewesen ware. Sind wir damit in der Lage, die Evolution in jene graue Vorzeit zuruckzuverfolgen, als es noch keine Lebewesen auf der Erde gab?

Polynucleotide replication is considered as an example of primitive biological evolution which can be tested experimentally. We distinguish deterministic selection which is based on differences in the rate constants of replication and degradation and neutral selection, a stochastic phenomenon in finite populations which is a result of the nature of the replication process only. Depending on the structural details of the fitness surface and on the distance from the temporal evolutionary optimum the changes in the population are predominantly due to adaptive or neutral mutations: far from the optimum almost every advantageous mutation will be fixed by adaptive selection whereas fixation of neutral mutation dominates close to the optimum.

UV-visible and 13C NMR measurements described in the literature and our 31P NMR measurements support the following mechanism of proton transfer reactions in aqueous solutions of pyridoxamine phosphate: Only the tautomeric equilibrium between neutral form, A N, and zwitterion, A Z, which is analogous to the tautomeric equilibrium of 3-hydroxypyridine in aqueous solution, is important, and that equilibrium does not change upon the dissociation of the second phosphate proton. With these simplifying assumption, we have simulated the relaxation spectrum of the proton transfer reactions of pyridoxamine phosphate in water using parameters from analogous reactions and compared it with our ultrasound and temperature jump measurements. We have found that the relaxation process measured by the temperature jump experiment is mainly caused by the overall reaction A N=A Z (or A N - =A Z - ) and the ultrasound absorption at the isoelectric point between pK2 and pK3 is mainly caused by the overall reaction $$A^{\text{ + }} + {\text{x}} A_N^ - + \left( {1 - {\text{x}}} \right)A_Z^ - = {\text{y}}A_N + \left( {2 - {\text{y}}} \right)A_Z , 0 \leqq {\text{x}} \leqq {\text{1,}} {\text{0}} \leqq {\text{y}} \leqq 2$$ .

Mise au point. Cas ou des calculs ab initio sont possibles. Complexes a liaisons hydrogenes binaires (HF, H 2 O) et ternaires

The notion of hypercycles has been introduced (Eigen, 1971) in order to characterize a functional entity which integrates information stored in individual self-replicating elements. Commonly, these self-replicating elements compete, like polynucleotides do in test tube experiments (Spiegelman, 1971; Biebricher et al., 1981) or, like species do in nature. A single species is selected eventually together with its most frequent mutants and the less efficient competitors disappear. In order to suppress competition, specific coupling terms of second order - these are coupling terms which are proportional to the product of two population numbers or concentration - have to be introduced into the kinetic equations such a link has to extend to all members, requiring the formation of a closed catalytic cycle (Eigen and Schuster, 1979). The copying of polynucleotide sequences, like every transfer of information, can only occur with finite accuracy (Eigen et al., 1981). This physical restriction imposes a limit upon the content of information that can be transmitted. A higher content of information requires more precise replication. A more involved replication machinery, however, presupposes more information to build it. Thus, a single autocatalyst will not be able to increase its content of information over the given threshold. How does one escape from this vicious circle? Hypercycles present a solution to the problem: many information carriers coexist and may cooperate to build a more sophisticated and more precisely copying replication machinery. Optimization of growth rates has never been the goal for which hypercycles were conceived. Any first order process can proceed at a higher rate than a second order process if low enough concentrations are chosen. On the other hand, any realistic hypercycle involves both first and second order terms. At low concentrations first order terms provide for accumulation of material while at higher concentration second order terms stabilize cooperation. At which concentration level such a prevalence changes is a matter of quantitative values of rate parameters rather than of basic principle. An autocatalyst may outgrow a hypercycle under certain conditions but it can never solve the problem of integrating information.

Autocatalysis is a rather exceptional phenomenon in chemical kinetics and, when it appears in reaction networks, various characteristic phenomena like oscillations, spatial pattern formation or eventually chaotic behaviour are commonly observed. Biochemistry and biology, in contrary, have to deal with a class of molecules for which selfreplication became obligatory. These molecules, autocatalysts in a sense, are the polynucleotides, the nucleic acids or, later on in evolution, the genes. Self-organization, the most interesting attribute of the biosphere, appears to be essentially based on the capability of self-replication.

More than ten years ago Eigen (1971) started to develop a dynamical model of molecular self-organization which is based mainly on chemical kinetics and the presently known properties of biopolymers. Later on, this theoretical concept was extended (Eigen & Schuster, 1979,1982) and supported by experimental studies (Biebricher et al.,1981, 1982; Biebricher, 1983; for a popular review see Eigen et al.,1981). The model starts out from the simplest prerequisites: polynucleotides are present and activated monomers are available in sufficient amounts. Several principles of self-organization can be derived by straightforward analysis. We just enumerate them here.

Selection and molecular evolution are often considered as processes on potential surfaces which are characterized as “fitness landscapes”. Two classes of potentials are particularly useful: “selection potential” for the selection process within a given population and “value landscapes” for evolutionary adaption. Among the various types of selection dynamics we distinguish rare and frequent mutation scenarios as well as different mechanisms for replication. The selection potential for independently replicating entities is a linear function of their concentrations. Evolutionary optimization leads to “corner equilibria” which represent pure states in the rare mutation scenario, or “quasispecies” if mutations occur frequently. The existence of a selection potential is a direct proof for the absence of complicated dynamics and dissipative structures. Dynamical systems for which no potential functions can be found are interesting in their turn because they may lead to oscillations and chaotic dynamics. Value landscapes provide direct insight into the course of evolutionary optimization. They are, however, very hard to determine even for the most simple examples which deal with “test-tube evolution”. We can discuss here only the results of a computer model which is thought to be representative also for real systems.

Two model systems for hypercyclic organisation constructed from a series of uni- and bimolecular reaction steps were studied under the condition of unlimited growth by means of qualitative analysis and numerical integration of the corresponding differential equations. It is shown that both models lead, within wide ranges of parameter and initial conditions, to the same characteristic dynamical behaviour as the elementary hypercycles and hypercycles with translation introduced in 1978 by Eigen and Schuster.

Why do millions of species, plants and animals, exist, while there is only one basic molecular machinery of the cell: one universal genetic code and unique chiralities of the macromolecules?

The aim of this contribution is to analyse some conventional kinetic equations of reaction mechanisms which include autocatalytic steps. In case of first-order or linear autocatalysis represented by the elementary step $$A + I\underset{{{\text{f'}}}}{\overset{{\text{f}}}{\longleftrightarrow}}{\text{2}}I$$ (1) the dynamics is very simple: we find a uniquely defined asymptotically stable steady state under almost all conditions. Higher order autocatalytic processes, e.g. the trimolecular elementary step $$A + 2I\underset{{{\text{f'}}}}{\overset{{\text{f}}}{\longleftrightarrow}}3I$$ (2) which has been used in several investigations of chemical non-equilibrium phenomena (SCHLOGL [1,2]; PRIGOGINE and LEFEVER [3]) lead to much richer dynamics. In higher dimensional systems we encounter limit cycles and chaotic attractors.

The notion of self-organizing systems was created in the forties mainly through the theoretical studies by Turing (1936) and von Neumann (1951). Since that time, interest in problems of self-organization has never ceased. What has changed over the years, however, are the preferred objects of interest: the self-replicative automata, more or less abstract concepts in computational physics have been replaced by experimental systems from various fields. The exciting development of molecular biology and biophysics initiated by the ingenious postulate of the double helical structure of DNA through Watson and Crick (1953) revealed more and more data on the structure and properties of biomolecules. Eigen (1971) applied chemical kinetics to molecular self-organization and formulated a general theory of precellular evolution. Thanks to the pioneering work of Spiegelman (1971) self-replication of RNA, biomolecules closely related to DNA which appear as the genetic material of some bacteriophages, can be studied by the experimental techniques used commonly in physics and chemistry. Darwinian evolution may be investigated now at the molecular level.

Stochastic phenomena are central to biological modeling. Small numbers of molecules regulate and control genetics, epigenetics, and cellular metabolism, and small numbers of well-adapted individuals drive evolution. As far as processes are concerned, the relation between chemistry and biology is a tight one. Reproduction, the basis of all processes in evolution, is understood as autocatalysis with an exceedingly complex mechanism in the language of chemists, and replication of the genetic molecules, DNA and RNA, builds the bridge between chemistry and biology. The earliest stochastic models in biology applied branching processes in order to give answers to genealogical questions like, for example, the fate of family names in pedigrees. Branching processes, birth-and-death processes, and related stochastic models are frequently used in biology and they are defined, analyzed, and applied to typical problems. Although the master equation is not so dominant in biology as it is in chemistry, it is a very useful tool for deriving analytical solutions, and most birth-and-death processes can be analyzed successfully by means of master equations. Kimura’s neutral theory of evolution makes use of a Fokker–Planck equation and describes population dynamics in the absence of fitness differences. A section on coalescent theory demonstrates the applicability of backwards modeling to the problem of reconstruction of phylogenies. Unlike in the previous chapter we shall present and discuss numerical simulations here together with the analytical approaches. Simulations of stochastic reaction networks in systems biology are a rapidly growing field and several new monographs have come out during the last few years. Therefore only a brief account and a collection of references are given.

The P300 ERP was measured in 10 subjects each for 9 days. The selection of instructions for subjects, the recording technique, the elimination of a few single trials significantly contaminated by eye movements, and the use of a correction procedure for ocular artifacts with calculable reliability and validity resulted in a set of data, in which 94% of the single trials were suitable for further analysis. The correction procedure relies on regression analysis. To reduce coherence between eyeblink activity and ongoing EEG, VEOG and EEG are averaged on eyeblinks. This yields a high reliability and validity of regression factors, determined per day, subject, and lead. In addition, this correction procedure allows for an estimation of the maximal error that must be taken into account. The efficiency of the procedure is demonstrated for single trials and averaged potentials.

A living cell can be considered a physicochemical system. Disregarding its initial development, it exists in a dynamic “steady state” depending on conditions imposed by the environment. Life is bound to a high spatial order of molecular structures, and to a high temporal order, in the sense of a well-defined coordination of physical and chemical processes, such as transport of matter and energy, and sequences of catalyzed reactions. There is no experience suggesting other principles being active in living systems beyond the well-known basic laws of physics: the quantum mechanics of atoms and molecules, and the conservation laws.

Evolution seems to occur in bursts, separated by relatively quiescent periods. From the simpler prokaryotes (organisms without a nuclear membrane around their genetic material) the fully nucleated eukaryote came—but how rapidly? Prokaryotic evolution was preceded by the chemical evolution leading to polynucleotide self-replication, although no fossils are recognized from that period. Auto-catalysis must have been required at that stage. But autocatalytic chemical processes are not simple, in vitro. Today there is enormous structural variability of biopolymers and polynucleotide synthesis can occur on several different levels of increasing efficiency (faster rates). Systems for these syntheses are complex; models of them have ten-dimensional kinetic equation systems, even in fitting data from simple experiments. Under restricted conditions, these equations show stationarity of relative polynucleotide concentrations. However, a relatively simple equation system predicts an early phase of exponential growth followed by linear growth of polynucleotide concentrations. Very general equations governing the growth of polynucleotide concentrations can be analyzed, but that analysis is much simplified by studying systems having constant total nucleotide concentration. With increasing degrees of realism in the simulations, models predict that: (1) the polynucleotide having the largest excess of synthesis over degradation becomes dominant; (2) if only a few copying errors are made, a whole “quasispecies” of polynucleotides comes to dominate; (3) if many errors are made, the system merely drifts, thus limiting the length of duplicatable polymers; (4) other results will be obtained, such as symmetric self-replication, hypercyclic cooperation, “frozen accidents,” Eigen and Schuster’s elementary hypercycle, the gamelike dynamics of Maynard Smith, or Lotka-Volterra dynamics.

Secondary structures of polynucleotides can be viewed as a class of planar vertex-labeled graphs. We compute recursion formulae for enumerating a variety sub-classes of and classes of sub-graphs (structural elements) of secondary structure graphs. First order asymptotics are derived and their dependence on the composition of the underlying nucleic acid sequences is discussed.

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f (h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.

Systems biology is a new discipline built upon the premise that an understanding of how cells and organisms carry out their functions cannot be gained by looking at cellular components in isolation. Instead, consideration of the interplay between the parts of systems is indispensable for analyzing, modeling, and predicting systems' behavior. Studying biological processes under this premise, systems biology combines experimental techniques and computational methods in order to construct predictive models. Both in building and utilizing models of biological systems, inverse problems arise at several occasions, for example, (i) when experimental time series and steady state data are used to construct biochemical reaction networks, (ii) when model parameters are identified that capture underlying mechanisms or (iii) when desired qualitative behavior such as bistability or limit cycle oscillations is engineered by proper choices of parameter combinations. In this paper we review principles of the modeling process in systems biology and illustrate the ill-posedness and regularization of parameter identification problems in that context. Furthermore, we discuss the methodology of qualitative inverse problems and demonstrate how sparsity enforcing regularization allows the determination of key reaction mechanisms underlying the qualitative behavior.

The liquid state is notoriously difficult to describe on a molecular level. This is true for pure liquids as well as for liquid mixtures and solutions. In contrast to the vapor phase and the crystal there is no easy to define idealized model system for the liquid state. In order to overcome this disadvantage combined efforts by theoretical studies and experimental in— vestigations are even more adviseable than in other fields. Therefore, one of the main ideas had been to encourage the dialogue between theorists and experimentalists when the late Prof.Carpeni initiated this series of conferences on specific solute—solvent interactions by organizing the first meeting in Marseille. Six years later these problems have not at all lost their importance. Again it was a predominant goal of the present meeting to bring scientists from different fields together and to initiate further improvement of the molecular interpre— tation of the various phenomena observed in solution chemistry. This is reflected already by the eight plenary lectures presented here which deal with quantum chemical studies on intermolecular forces (H.Lischka), statistical mechanics of liquids (F.Kohler), diffraction studies on ionic solutions (A.H.Narten and R.Triolo), static and dynamic dielectric properties of liquids (L.Sobczyk), infrared and n.m.r. spectroscopy of ionic solutions (M.C.R.Symons), a thermodynamical theory of transport coefficients in dilute ionic solution (J.C.Justice and M.C.Justice), empirical models for solvent effect on chemical reactions (U.Mayer) and specific solvent influences on the mechanisms of inorganic reactions (D.W.Watts).