A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

Analysis of a mathematical model for transport of I-albumin

The hypotheses and results given are motivated by the study of the distribution of albumin in man which represents a class of delay-differential systems. The approach used is to study the behavior of the solutions of nonlinear delay-differential systems with variable coefficients under the assumptions of continuity and boundedness of coefficients. The criterions are conditions on the roots of a certain “quasi-polynomial”, i.e., a polynomial in a variable and exponential of that variable. These criterions bear a resemblance to the ones in the case of constant coefficients and retardations and are applicable to this case also. The method is based on Lyapunov type functional with appropriate properties.     

Approximation for limit cycles and their isochrons

Local analysis of trajectories of dynamical systems near an attractive periodic orbit displays the notion of asymptotic phase and isochrons. These notions are quite useful in applications to biosciences. In this note, we give an expression for the first approximation of equations of isochrons in the setting of perturbations of polynomial Hamiltonian systems. This method can be generalized to perturbations of systems that have a polynomial integral factor (like the Lotka-Volterra equation).     

Determination of the quaternary phase diagram of the water-ethylene glycol-sucrose-NaCl system and a comparison between two theoretical methods for synthetic phase diagrams

Characterization of the thermodynamic properties of multi-solute aqueous solutions is of critical importance for biological and biochemical research. For example, the phase diagrams of aqueous systems, containing salts, saccharides, and plasma membrane permeating solutes, are indispensible in the field of cryobiology and pharmacology. However, only a few ternary phase diagrams are currently available for these systems. In this study, an auto-sampler differential scanning calorimeter (DSC) was used to determine the quaternary phase diagram of the water-ethylene glycol-sucrose-NaCl system. To improve the accuracy of melting point measurement, a “mass-redemption” method was also applied for the DSC technique. Base on the analyses of these experimental data, a comparison was made between the two practical approaches to generate phase diagrams of multi-solute solutions from those of single-solute solutions: the summation of cubic polynomial melting point equations versus the use of osmotic virial equations with cross coefficients. The calculated values of the model standard deviations suggested that both methods are satisfactory for characterizing this quaternary system.     

Predicting Physical Time Series Using Dynamic Ridge Polynomial Neural Networks

Forecasting naturally occurring phenomena is a common problem in many domains of science, and this has been addressed and investigated by many scientists. The importance of time series prediction stems from the fact that it has wide range of applications, including control systems, engineering processes, environmental systems and economics. From the knowledge of some aspects of the previous behaviour of the system, the aim of the prediction process is to determine or predict its future behaviour. In this paper, we consider a novel application of a higher order polynomial neural network architecture called Dynamic Ridge Polynomial Neural Network that combines the properties of higher order and recurrent neural networks for the prediction of physical time series. In this study, four types of signals have been used, which are; The Lorenz attractor, mean value of the AE index, sunspot number, and heat wave temperature. The simulation results showed good improvements in terms of the signal to noise ratio in comparison to a number of higher order and feedforward neural networks in comparison to the benchmarked techniques.     

Translated Chemical Reaction Networks

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analyzed via a variety of techniques, including stoichiometric network analysis, deficiency theory, and algebraic techniques (e.g., Gröbner bases). In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network’s capacity to permit a particular class of steady states, called toric steady states, to topological properties of a generalized network called a translated chemical reaction network. These networks share their reaction vectors with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.     

Monitoring in a predator–prey systems via a class of high order observer design

The goal of this work is the monitoring of the corresponding species in a class of predator–prey systems, this issue is important from the ecology point of view to analyze the population dynamics. The above is done via a nonlinear observer design which contains on its structure a high order polynomial form of the estimation error. A theoretical frame is provided in order to show the convergence characteristics of the proposed observer, where it can be concluded that the performance of the observer is improved as the order of the polynomial is high. The proposed methodology is applied to a class of Lotka–Volterra systems with two and three species. Finally, numerical simulations present the performance of the proposed observer.     

Use of the β‐function to estimate the skewness of species responses

Abstract. Response of a species to an environmental variable may be modeled and predicted using a wide spectrum of different functions. Contrary to other functions (Gaussian, polynomial etc), all parameters of the β‐function are interpretable in ecological terms. However, computational difficulties in the determination of the β‐function parameters initiated controversial debates on the applicability and usefulness of this function in vegetation modelling and gradient analysis. We propose a simple algorithm for fitting the β‐function to observed data. Analytic properties of the algorithm (its ability to recover the known species responses along gradients) are tested using a series of simulated data. In most cases the algorithm correctly estimated parameters of the simulated responses.     

Fixed-point bifurcation analysis in biological models using interval polynomials theory

The paper proposes a systematic method for fixed-point bifurcation analysis in circadian cells and similar biological models using interval polynomials theory. The stages for performing fixed-point bifurcation analysis in such biological systems comprise (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system’s nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To obtain a clear view about the values of the roots of the characteristic polynomial and about the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov’s stability theorem is proposed. In this approach, the study of the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the study of the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed approach for the analysis of fixed-point bifurcations in nonlinear models of biological neurons is tested through numerical and simulation experiments.     

A probabilistic model for fitting MWC polynomials in protein-ligand binding

Given a binding polynomial in Adair form, A(x) = 1 + beta 1 x + ... + beta n x n, beta i greater than or equal to 0, a basic problem is to determine a method of fitting a model polynomial to A(x) and a quantitative measure of the goodness of fit. This paper presents such a method for fitting Monod-Wyman-Changeux (MWC) model polynomials when A(x) is of degree three or four. The method of fitting is based on the property that the zeros of an MWC polynomial of any degree lie on a circle in the complex plane. The parameters in the MWC model are determined so that if possible this circle coincides with the circle on which lie the zeros of A(x). The measure of goodness of fit is provided by a probabilistic model which gives the probability that a binding polynomial has its zeros on a circle on which lie the zeros of an MWC polynomial and if so, the probability that the juxtaposition of the two sets of zeros can occur by chance alone.     

Simulation of tagasaste pulping using soda-anthraquinone

In this work, published experimental result data of the pulping of tagasaste (Chamaecytisus proliferus L.F.) with soda and anthraquinone (AQ) have been used to develop a model using a neural network. The paper presents the development of a model with a neural network to predict the effects that the operational variables of the pulping reactor (temperature, soda concentration, AQ concentration, time and liquid/solid ratio) have on the properties of the paper sheets of the obtained pulp (brightness, traction index, burst index and tear index). Using a factorial experimental design, the results obtained with the neural network model are compared with those obtained from a polynomial model. The neural network model shows a higher prediction precision that the polynomial model.     

Optical Characteristics of Thiamine in Model Systems and in Holoenzyme

The optical properties of thiamine diphosphate-dependent enzymes change significantly on their interaction with cofactors (thiamine, bivalent metal ions) and substrates. These changes are connected with structural alterations of the active site and the mechanism of its functioning, and in some cases they reflect changes in the optical properties of the coenzyme itself within the protein. The use of optical characteristics, especially together with model systems, appeared to be a rather promising approach for investigation of the active site of thiamine diphosphate-dependent enzymes and the mechanism of its functioning. So, it seemed to be useful to summarize the literature data concerning the optical characteristics of thiamine (thiamine diphosphate) in model systems and the efficiency of their application for study of thiamine diphosphate-dependent enzymes.     

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).     

Structural identifiability of polynomial and rational systems

Since analysis and simulation of biological phenomena require the availability of their fully specified models, one needs to be able to estimate unknown parameter values of the models. In this paper we deal with identifiability of parametrizations which is the property of one-to-one correspondence of parameter values and the corresponding outputs of the models. Verification of identifiability of a parametrization precedes estimation of numerical values of parameters, and thus determination of a fully specified model of a considered phenomenon. We derive necessary and sufficient conditions for the parametrizations of polynomial and rational systems to be structurally or globally identifiable. The results are applied to investigate the identifiability properties of the system modeling a chain of two enzyme-catalyzed irreversible reactions. The other examples deal with the phenomena modeled by using Michaelis–Menten kinetics and the model of a peptide chain elongation.     

Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.     

An atoms-in-molecules study of the genetically-encoded amino acids: I. Effects of conformation and of tautomerization on geometric, atomic, and bond properties

The theory of Atoms-In-Molecules (AIM) is a partitioning of the real space of a molecule into disjoint atomic constituents as determined by the topology of the electron density, rho(r). This theory identifies an atom in a molecule with a quantum mechanical open system and, consequently, all of the atom's properties are unambiguously defined. AIM recovers the basic empirical cornerstone of chemistry: that atoms and functional groups possess characteristic and additive properties that in many cases exhibit a remarkable transferability between different molecules. As a result, the theory enables the theoretical synthesis of a large molecule and the prediction of its properties by joining fragments that are predetermined as open systems. The present article is the first of a series (in preparation) that explore this possibility for polypeptides by determining the transferability of the building blocks: the amino acid residues. Transferability of group properties requires transferability of the electron density rho(r), which in turn requires the transferability of the geometric parameters. This article demonstrates that these parameters are conformation-insensitive for a representative amino acid, leucine, and that the atomic and bond properties exhibit a corresponding transferability. The effects of hydrogen bonding are determined and a set of geometrical conditions for the occurrence of such bonding is identified. The effects of transforming neutral leucine into its zwitter-ionic form on its atomic and bond properties are shown to be localized primarily to the sites of ionization.     

Statistical model building and model criticism for human circadian data

Mathematical models have played an important role in the analysis of circadian systems. The models include simulation of differential equation systems to assess the dynamic properties of a circadian system and the use of statistical models, primarily harmonic regression methods, to assess the static properties of the system. The dynamical behaviors characterized by the simulation studies are the response of the circadian pacemaker to light, its rate of decay to its limit cycle, and its response to the rest-activity cycle. The static properties are phase, amplitude, and period of the intrinsic oscillator. Formal statistical methods are not routinely employed in simulation studies, and therefore the uncertainty in inferences based on the differential equation models and their sensitivity to model specification and parameter estimation error cannot be evaluated. The harmonic regression models allow formal statistical analysis of static but not dynamical features of the circadian pacemaker. The authors present a paradigm for analyzing circadian data based on the Box iterative scheme for statistical model building. The paradigm unifies the differential equation-based simulations (direct problem) and the model fitting approach using harmonic regression techniques (inverse problem) under a single schema. The framework is illustrated with the analysis of a core-temperature data series collected under a forced desynchrony protocol. The Box iterative paradigm provides a framework for systematically constructing and analyzing models of circadian data.     

Evaluation of the EMG-force relationship of trunk muscles during whole body tilt

The study was aimed at the identification of the electromyography (EMG)-force relationship of five different trunk muscles. EMG-force relationships differ depending on changes in firing rate and the concurrent recruitment of motor units, which are linear and S-shaped, respectively. Trunk muscles are viewed as belonging to either the local or global muscle systems. Based on such assumptions, it would be expected that these functionally assigned muscles use different activation strategies. Thirty-one healthy volunteers (16 women, 15 men) were investigated. Forces on the trunk were applied with the use of a device that gradually tilts the body to horizontal position. Rotation capability enabled investigation of forward and backward as well as right and left sideward tilt directions. Surface EMG (SEMG) of five trunk muscles was taken. Root mean square (rms) values were computed and relative amplitudes, according to the measured maximum amplitudes, were calculated individually. Back muscles were characterized by a linear SEMG-force relationship during forward tilt. Abdominal muscles showed an S-shaped polynomial SEMG-force relationship for backward tilt direction. Sideward tilt directions evoked lesser SEMG levels with polynomial curve characteristics for all investigated muscles. Therefore, the SEMG-force relationship possibly is also subject to force vector in relation to fiber direction.