A methodological basis for description and analysis of systems with complex switch-like interactions

 A wide range of complex systems appear to have switch-like interactions, i.e. below (or above) a certain threshold x has no or little influence on y, while above (or below) this threshold the effect of x on y saturates rapidly to a constant level. Switching functions are frequently described by sigmoid functions or combinations of these. Within the context of ordinary differential equations we present a very general methodological basis for designing and analysing models involving complicated switching functions together with any other non-linearities. A procedure to determine position and stability properties of all stationary points lying close to a threshold for one or several variables, so-called singular stationary points, is developed. Such points may represent homeostatic states in models, and are therefore of considerable interest. The analysis provides a profound insight into the generic effects of steep sigmoid interactions on the dynamics around homeostatic points. It leads to qualitative as well as quantitative predictions without using advanced mathematical methods. Thus, it may have an important heuristic function in connection with numerical simulations aimed at unfolding the predictive potential of realistic models. Received 25 January 1996; received in revised form 29 June 1997     

A general model of reaction kinetics in biological systems

Dynamic mathematical models in biotechnology require, besides the information about the stoichiometry of the biological reaction system, knowledge about the reaction kinetics. Modulation phenomena like limitation, inhibition and activation occur in different forms of competition with the key enzymes responsible for the respective metabolic reaction steps. The identification of a priori unknown reaction kinetics is often a critical task due to the non-linearity and (over-) parameterization of the model equations introduced to account for all the possible modulation phenomena. The contribution of this paper is to propose a general formulation of reaction kinetics, as an extension of the Michaelis-Menten kinetics, which allows limitation/activation and inhibition effects to be described with a reduced number of parameters. The versatility of the new model structure is demonstrated with application examples.     

A stochastic model for the sigmoidal behaviour of cooperative biological systems

A stochastic model for cooperative transitions in biological systems based on a Markov chain is proposed. This model requires only two parameters, the mean probability, p, and the coupling capacity, Deltap, which measure the probability of forming a new weak bond depending on the number of similar bonds already formed and it is also responsible for the transition. In this paper we show how the model works for a large number of identical molecules and how it can be useful for studying the noise around the centre of the transition where, increasing the degree of cooperativity, i.e. the number n in the well-known Hill equation, the width of the noise increases along with its fractal dimension. A simple relationship between the degree of cooperativity and the parameter Deltap is proposed, suggesting that the cooperativity of real biological transitions is related to the coupling capacity Deltap of the present model.     

A method for the analysis of biological transduction phenomena

Biological transduction can be defined as the triggering of a cellular response by the binding of molecules of effector substances to specific cellular sites. An example of biological transduction, analyzed in this report, is the triggering of T-cell proliferation by the binding of T-cell growth factor (TCGF) to specific TCGF-binding sites on responsive T-cells. Sigmoidal or S-shaped curves often result when measurements of biological response are plotted as a function of concentration of effector substance. Such curves suggest that effector molecules must bind a critical number of cellular sites, and this critical number of bound complexes must undergo secondary events (cross-linking, association, internalization, second messenger release, etc.) in order to initiate the biological response. The method described here estimates the critical number of cellular sites (R) and the probability of these secondary events (PS/B) as follows: (1) The total number of cellular sites (N) is estimated from binding data, and the probabilities (PB) of effector molecules binding to a site are estimated from response data. (2) The response data are assumed to follow the summed binomial distribution function, which is equated to the incomplete beta function. (3) R and PS/B are estimated by applying nonlinear regression to the incomplete beta function. The T-cell data to which the method was applied gave N = 15,000, R = 5, and PS/B = 7.22 x 10(-4). These results show that the binding of very few TCGF molecules is required for activation of T-cells and that the probability of the secondary events leading to cell proliferation is much smaller than the probability of TCGF binding to T-cells. The method described can be used to analyze any biological transduction experiments where both binding and biological response data are available.     

A logic for biological systems

This paper proposes a specification language, hybrid projection temporal logic of modelling, analyzing and verifying biological systems which can be considered, in general, to be hybrid systems consisting of a non-trivial mixture of discrete and continuous components. The syntax and semantics of the logic are presented, and some examples of hybrid systems are modelled to illustrate the formalism.     

A general kinetic model for biological nutrient removal activated sludge systems: model evaluation

Hu et al. (2007) presented a general kinetic model for biological nutrient removal (BNR) activated sludge (AS) systems in general, but for external nitrification (EN) BNRAS (ENBNRAS) systems in particular. In this article, this model is evaluated against a large number of experimental data sets. In this evaluation, the model is first used to simulate a wide variety of conventional internal nitrification (IN) BNRAS systems to evaluate its predictions and also evaluate the model parameters suggested by Hu et al. (2007), and to calibrate those constants for which values are not available in the literature. Simulation results indicate that the model, with appropriately calibrated parameters, is capable of predicting COD removal, nitrification and denitrification and two types of biological excess phosphorus removal (BEPR), namely aerobic and anoxic/aerobic P uptake BEPR. The model is then used to simulate the ENBNRAS systems to evaluate its capacity of simulating the behaviour of this system. Simulation results show that the model is capable of simulating the behaviour of the ENBNRAS systems, including COD, nitrification, denitrification and BEPR, particularly anoxic P uptake BEPR, with the values of kinetic and stoichiometric parameters obtained in modelling conventional BNRAS systems, except for micro(NIT), K(MP), eta(PAO) and eta(H) which required calibration.     

A canonical model of multistability and scale-invariance in biological systems

Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena.     

A general kinetic model for biological nutrient removal activated sludge systems: model development

In this article, a kinetic model is developed and presented for biological nutrient removal (BNR) activated sludge (BNRAS) systems in general, but for external nitrification (EN) BNRAS (ENBNRAS) systems in particular. The model is based on the UCTPHO model, but includes some significant modifications, such as anoxic P uptake and associated denitrification by phosphorus accumulating organisms (PAOs). Some key features of the model are described and discussed before the model is presented. Model evaluation will be addressed in another article (Hu et al., 2007).     

A neural net model for masking phenomena

Some aspects of masking phenomena are considered in terms of the simplest possible model of two-factor neural elements. The effect of a number of variables can be accounted for, but the introduction of an internuncial element results in a masking function which need not be symmetric about zero delay interval. As an illustration, the results for a special case are compared with available data. In general, such a model results in a masking function which depends on the intensity, area, and duration of the stimuli, as well as on the temporal and spatial separation between them.     

A quantitative test for superoxide radicals produced in biological systems.

The preparation and properties of a partially succinoylated cytochrome c, suited for the detection of superoxide anion radicals in liver microsomes, is reported. By succinoylation of 45% of the primary amino groups of horse heart cytochrome c the activity towards solubilized NADPH--cytochrome P-450 reductase was diminished by 99% compared with native cytochrome c. The capacities of cytochrome b5 and cytochrome c oxidase to reduce the succinoylated ferricytochrome c and oxidize succinoylated ferrocytochrome c respectively were decreased to a similar extent. However, the bimolecular rate constant for the reduction of the partially succinoylated ferricytochrome c by O2-. was estimated to be one-tenth of the value for the reaction of O2-. with native ferricytochrome c a pH 7.7. On this basis the quantification of O2-. generated by NADPH-supplemented liver microsomes became possible. The initial rates of succinoylated ferricytochrome c reduction determined at various finite concentrations of the cytochrome c derivative can be extrapolated to obtain true rates of O2-. generation in a homogeneous system. The problems encountered in the quantitative determination of O2-. produced in biological membranes, e.g. microsomes, are discussed.     

A switch-like dynamic mechanism for the initiation of replicative senescence

Telomeres are specialized structures protecting chromosomes against genome instability. Telomeres shorten with cell division, and replicative senescence is induced when telomeres are badly eroded. Whereas TRF2 (telomeric-repeat binding factor 2), ATM (ataxia telangiectasia mutated) and p53 have been identified involved in senescence induction, how it is triggered remains unclear. Here, we propose an integrated model associating telomere loss with senescence trigger. We characterize the dynamics of telomere shorting and the p53-centered regulatory network. We show that senescence is initiated in a switch-like manner when both the shortest telomere becomes uncapped and the TRF2-ATM-p53-Siah1 positive feedback loop is switched on. This work provides a coherent picture of senescence induction in terms of telomere shortening and p53 activation.     

Tungsten vs. Molybdenum in models for biological systems.

Biological systems show a marked preference for molybdenum over tungsten. Studies with methyliminodiacetic acid and L-cysteine have shown that the formation constants of the complexes with Mo(Vi) and W(VI) are very similar. These results imply that these elements would be bound with roughly equal strengths to an apoenzyme or a carrier whether or not these proteins contain a ligating sulfhydryl group. Similarly, transport across a membrane would not be expected to distinguish compounds of these metals providing they are carried in the same oxidation states. However, molybdenum could be distinguished from tungsten through the greater ease of reduction of the compounds of molybdenum.     

A quantitative model for partition in aqueous multiphase systems.

A model for the partition of charged molecules in aqueous multiphase systems has been developed. The partition coefficient of one component, or the overall partition coefficient of a number of components, between two arbitrary phases is expressed in terms of the difference in electrical potential between the phases (due to electrolytes present in the system), the net charges of the partitioned components and their partition coefficients in a (sometimes hypothetical) uncharged state. The fraction of material in one phase has also been described as a function of the net charges of the partitioned components. The model fits well to experimental data for partition of chromate, pyridine, ribonuclease A, two types of CO-hemoglobin and an enzyme mixture (yeast lysate) in three-phase systems consisting of poly(ethylene glycol), dextran, Ficoll and water. Minor deviations from the model are construed to be a pH-dependent uptake of ions. The data have also been used to detect differences in solvation of similar proteins, as well as the presence of several forms of some glycolytic enzymes present in yeast lysate.     

Millisecond fading and recovery phenomena in fluorescent biological objects.

Cytofluorometric signals derived from some frequently used fluorophores were studied during illumination times in the millisecond range. These rapid signals were recorded on a storage oscilloscope. The objects studied included (1) Berberine sulphate stained mast cell heparin, (2) Acriflavine-Feulgen stained DNA, (3) Acridine orange stained mast cell heparin, (4) Acridine orange stained DNA and (5) Fluorescein isothiocyanate-conjugated IgG in an antinuclear factor test. A new rapid fading phenomenon, appearing as an initial peak upon the familiar slowly declining fluorescence signal, is reported. This fading, which had a duration of about 10 ms, also showed a very rapid recovery. The influence of this phenomenon on fluorometric measurement techniques is discussed. The millisecond fading phenomenon occurred in all the fluorophores studied except Fluorescein isothiocyanate-conjugated IgG. In the case of acridine orange the phenomenon was present when the dye was bound to nuclear DNA but absent when the dye was bound to mast cell heparin. This suggests that the millisecond fading and recovery phenomenon may be used in fluorescent microprobe studies.     

Three-dimensional multiple-wavelength fluorescence microscopy for the structural analysis of biological phenomena.

Cellular events are accomplished by the coordinated interactions of cellular components within the three-dimensional context of a cell. Simultaneous observation of multiple components in three dimensions can be essential for understanding such interactions. Toward this end, we have developed a computerized microscope workstation capable of recording three-dimensional images of multiple cellular components in fixed and living cells. All aspects of microscope control, data collection, image processing and analysis can be performed on the one workstation. In this report, we describe the components and capabilities of this integrated system. In addition, we discuss some general problems of multiple-wavelength, three-dimensional imaging and our application of this technology to the analysis of chromosome organization in Drosophila melanogaster. Three-dimensional imaging of fixed embryos stained by indirect immunofluorescence has revealed the structural organization of chromosomes, microtubules, and the nuclear lamins. Imaging of living embryos injected with fluorescently labelled proteins has confirmed and extended these results by allowing the study of these structures throughout the cell cycle. The combination of the molecular specificity of fluorescence microscopy and the three-dimensional structural information obtained by our workstation has provided novel insights into the dynamic aspects of chromosome behavior during the cell cycle. We believe this system has many important applications in the study of the molecular basis of cellular events.