Geometric singular perturbation theory in biological practice

Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples.     

Noninferior periodic operation of the biological reactor

The optimal periodic operation of the biological reactor was studied from the standpoint of the two-objective programming problem. The noninferior set with respect to the cell productivity and the conversion of the substrate into the biomass was determined by use of the optimization technique due to Miele. It was shown that the noninferior set was composed in general of the repeated batch branch and the repeated fed-batch branch, which occupy the high-productivity portion and the high-conversion portion of the noninferior set, respectively. However, the latter branch disappears in the case of growth kinetics with no substrate inhibition. In addition, the extreme points of the noninferior set yielding the maximal productivity and the maximal conversion represent such operations that are equivalent to the steady-state operation (chemostat culture) and the batch operation, respectively.     

The application of infinite-order perturbation theory to biological polymers

Quantum mechanical perturbation theory of infinite order is applied to an infinitely long biological polymeric chain. The resulting ensemble-averaged density of electronic states is approximately calculated by truncation according to a method introduced by S. Y. Wu [(1974) J. Math. Phys. 15 , 947–949]. The results are found to require reasonable computational facilities and the method may be applied to a study of mutagenicity.     

Approximate expressions of the bifurcating periodic solutions in a neuron model with delay-dependent parameters by perturbation approach

This paper is interested in gaining insights of approximate expressions of the bifurcating periodic solutions in a neuron model. This model shares the property of involving delay-dependent parameters. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work so harder. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of delay models. Although Xu et al. (Phys Lett A 354:126–136, 2006) studied stability switches, Hopf bifurcation and chaos of the neuron model with delay-dependent parameters, the dynamics of this model are still largely undetermined. In this paper, a detailed analysis on approximation to the bifurcating periodic solutions is given by means of the perturbation approach. Moreover, some examples are provided for comparing approximations with numerical solutions of the bifurcating periodic solutions. It shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that of systems with delay-independent parameters only.     

Noninferior periodic operation of the metabolite producing biological reactor

The optimal periodic operation of the biological reactor in which the metabolites belonging to Type I or II in Gaden's classification are produced was investigated from the viewpoint of the multiobjective programming problem. In addition to the productivity of the desired product, its concentration and the conversion of the substrate which have a large influence on the performance of the separation process following the fermentation process were adopted as the objective functions. The growth of cells was assumed to be inhibited both by the substrate and the product, and the Luedeking-Piret model was employed for the specific production rate. The optimal periodic operation was determined by use of the optimization method due to Miele. It was clarified that the noninferior set for the periodic operation was generally composed of the repeated batch portion and the repeated fed-batch one.     

Robust detection of periodic time series measured from biological systems

Background  

Periodic phenomena are widespread in biology. The problem of finding periodicity in biological time series can be viewed as a multiple hypothesis testing of the spectral content of a given time series. The exact noise characteristics are unknown in many bioinformatics applications. Furthermore, the observed time series can exhibit other non-idealities, such as outliers, short length and distortion from the original wave form. Hence, the computational methods should preferably be robust against such anomalies in the data.     

Modelling periodic oscillation of biological systems with multiple timescale networks

In this paper, we aim to develop a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple timescale networks (MTN). Fast reactions constitute a positive feedback-loop network (PFN), while slow reactions consist of a cyclic feedback-loop network (CFN), in MTN. Multiple timescales are exploited to simplify models according to singular perturbation theory. We show that a MTN has no stable equilibrium but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of MTNs with only one PFN, and then generalise the result to MTNs with multiple PFNs. Finally, we design a biologically plausible gene regulatory network by the cI and Lac genes, to demonstrate the theoretical results. Since there is less restriction on the network structure of a MTN, it can be expected to apply to a wide variety of areas on the modelling, analysing and designing of biological systems.     

Identification of potential targets in biological signalling systems through network perturbation analysis

The network-based representation and analysis of biological systems contributes to a greater understanding of their structures and functions at different levels of complexity. These techniques can also be used to identify potential novel therapeutic targets based on the characterisation of vulnerable or highly influential network components. There is a need to investigate methods for estimating the impact of molecular perturbations. The prediction of high-impact or critical targets can aid in the identification of novel strategies for controlling the level of activation of specific, therapeutically relevant genes or proteins. Here, we report a new computational strategy for the analysis of the vulnerability of cellular signalling networks based on the quantitative assessment of the impact of large-scale, dynamic perturbations. To show the usefulness of this methodology, two complex signalling networks were analysed: the caspase-3 and the adenosine-regulated calcium signalling systems. This allowed us to estimate and rank the perturbation impact of the components defining these networks. Testable hypotheses about how these targets could modify the dynamic operation of the systems are provided. In the case of the caspase-3 system, the predictions and rankings were in line with results obtained from previous experimental validations of computational predictions generated by a relatively more computationally complex technique. In the case of the adenosine-regulated calcium system, we offer new testable predictions on the potential effect of different targets on the control of calcium flux. Unlike previous methods, the proposed approach provides perturbation-specific scores for each network component. The proposed perturbation assessment methodology may be applied to other systems to gain a deeper understanding of their dynamic operation and to assist the discovery of new therapeutic targets and strategies.     

Dissociation of thyroglobulin by tetraphenylborate ion

Purified bovine and ovine thyroglobulins (19 S) are partially dissociated into 12-S subunits after treatment with sodium tetraphenyl borate. The extent of dissociation obtained by sodium tetraphenyl borate or sodium dodecyl sulfate treatment is the same. The electrophoretic mobilities on acrylamide gels of sodium tetraphenyl borate-resistant molecules and of native thyroglobulin are identical. Sodium dodecyl sulfate-resistant molecules move more slowly than the native protein.     

Dissociation of proteinase-inhibitor complexes by trichloroacetate

It was demonstrated that the addition of high concentrations of the chaotrope, sodium trichloroacetate, to proteinase assays provided for a dissociation of proteinase-inhibitor complexes. The complexes evaluated contained a heat-stable, polypeptide inhibitor of cysteine proteinases isolated from the cellular slime mold, Dictyostelium discoideum. The proteinases that were present in separate complexes included either D. discoideum proteinases or the plant proteinase papain. The general assay procedures described may be useful in detection of endogenous proteinase-inhibitor complexes in many systems.     

Enzymic Dissociation of Zea Shoot Cell-Wall Polysaccharides V. Dissociation of Xyloglucan by Urea

A procedure has been devised to extract and identify structuralcomponents of the xyloglucan of Zea mays L. (hybrid B73 ? Mo17) shoot cell-walls. A water-insoluble fraction of Zea shootcell-walls, after pretreatment with purified Bacillus subtilis(1 3),(1 4)-ß-D-glucan 4-glucanohydrolase, purifiedB. subtilis endo-(l 4)-ß-xylanase and an enzyme preparationfrom B. subtilis enriched in glucuronoxylanase (Kato and Nevins1984a, Nishitani and Nevins 1991), was subsequently treatedwith 7 M urea. The carbohydrates (0.8% of the water-insolublefraction of Zea shoot cell-walls) liberated by the urea treatment,were comprised of xyloglucan polymers with molecular weightswhich varied from 1.0 ? 10⁴ to 4.0 7times; 10⁴ Da. Other wallfragments associated with the isolated polymer suggest covalentbonding of xyloglucan to other polysaccharides. Structural analysesof the xyloglucan polymers reveal a cellulose-like backbonewith about 35% of the C-6 positions substituted with xyloseand other sugars. About 80% of xyloglucan present in the enzyme-pretreatedwater-insoluble fraction of Zea shoot cell-walls was liberatedby the urea treatment. The procedure avoids the use of alkaliin the solubilization of xyloglucan. ¹Supported in part by National Science Foundation research grantsPCM 7818588 and DMB 8505901. (Received September 10, 1990; Accepted May 15, 1991)     

Dissociation of yeast hexokinase by hydrostatic pressure

The pressure-induced dissociation of the isozymes P1 and P2 of hexokinase was investigated by studies of the spectral shift of the intrinsic protein fluorescence and by the fluorescence polarization of dansyl conjugates. The free energy of association of the monomers at atmospheric pressure, Katm, was -14.2 kcal mol-1 at 20 degrees C and -11.4 kcal mol-1 at 0 degrees C. The positive enthalpy indicates that the association of the monomers is entropy-driven, overcoming the negative enthalpy of hydration of the subunit interfaces. At 0 degrees C and 1 bar, glucose stabilizes the association by -1.1 kcal mol-1 and the binding of both adenosine 5'-(beta, gamma-methylenetriphosphate) (AMPPCP) and glucose by an even larger amount, -1.34 kcal mol-1. Paradoxically, adenosine 5'-triphosphate (ATP), or AMPPCP, in the absence of glucose destabilizes the association by +0.34 kcal mol-1, while adenosine 5'-diphosphate (ADP) stabilizes it by -0.6 kcal mol-1. Comparison of dV0, the apparent standard volume of association, at different pHs and temperatures indicates that its value (115-160 mL mol-1) is strongly dependent upon the ionization of a group at the subunit interface with a pK near neutrality. Under dissociating pressures, trypsin action results in permanent dissociation of the dimer, confirming earlier observations of Colowick by less direct methods. The P1 and P2 enzymes differ in Katm and dV0 and markedly so in the effects of salt upon the stability of the dimer.(ABSTRACT TRUNCATED AT 250 WORDS)     

Dissociation of polysome aggregates by protease k

Apparent large size-classes of zein-synthesizing polysomes from developing kernels of Zea mays L. were converted to smaller polysomes after treatment with Protease K. The reduction in polysome size was not a result of ribonuclease activity, inasmuch as the enzyme did not affect the free polysomes or the size of the mRNA from the membrane-bound polysomes. High concentrations of MgCl(2) in polysome buffer inhibited ribonuclease activity and appeared to cause protein interaction between nascent zein polypeptides. Although Protease K inhibited the polysome's capacity for protein synthesis, it was a useful reagent for determining if polysomes were aggregated by protein.     

Dissociation of human hemoglobin by different organomercurials

The reaction of free and masked SH groups of human hemoglobin with a variety of organomercurials promotes dissociation into α and β chains which can be followed by starch-gel electrophoresis.     

Use of cluster and discriminant analyses to compare rhizosphere bacterial communities following biological perturbation

We present an approach to comparing the diversity and composition of bacterial communities from different habitats and for identifying which members of a community are most affected by an introduced bacterium. We use this method to explore both previously published and new data from field and growth chamber experiments in which we isolated heterotrophic bacteria from samples of root-free soil, roots of nontreated soybean seedlings, and from the roots of soybean seedlings grown from Bacillus cereus UW85nl—treated seeds. We characterize bacterial isolates for 40 physiological attributes, and grouped the isolates hierarchically using two-stage density-linkage cluster analysis. Multivariate analysis of variance and discriminant analysis of the relative frequencies of the clusters in the soil and rhizosphere habitats were then used to determine whether there were differences among the bacterial communities from the various habitats, and which of the clusters were most useful in discriminating among the communities. We used rarefied estimates of richness as a measure of community diversity in the various habitats. Introduction of UW85n 1 affected the composition and/or diversity of rhizosphere communities in three of four experiments. Present address: Department of Environmental Science, Policy and Management, 108 Hilgard Hall, University of California, Berkeley, CA 94720 Correspondence to: G.S. Gilbert.