The identification of nonlinear biological systems: LNL cascade models

Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.     

Wiener analysis of nonlinear systems using Poisson-Charlier crosscorrelation

Nonlinear systems with event-sequence input, such as are often encountered in neurophysiology, may be experimentally tested with all possible input sequences by stimulation with a Poisson process eventsequence. A complete predictive model of the system's response may be constructed from this data with the Wiener expansion based on the Poisson-Charlier polynomials. Here it is shown how this formulation leads to an efficient method for the evaluation of unknown systems by crosscorrelation, generalizing previous methods. The basic statistical properties of the procedure are demonstrated and the length of experiment required for accurate estimation of the model is computed. The procedure is translated into digital algorithms and the analogous procedures for white noise analysis are presented.     

A new algorithm for the identification of multiple input Wiener systems

Multiple-input Wiener systems consist of two or more linear dynamic elements, whose outputs are transformed by a multiple-input static non-linearity. Korenberg (1985) demonstrated that the linear elements of these systems can be estimated using either a first order input-ouput cross-covariance or a slice of the second, or higher, order input-output cross-covariance function. Korenberg's work used a multiple input LNL structure, in which the output of the static nonlinearity was then filtered by a linear dynamic system. In this paper we show that by restricting our study to the slightly simpler Wiener structure, it is possible to improve the linear subsystem estimates obtained from the measured cross-covariance functions. Three algorithms, which taken together can identify any multiple-input Wiener system, have been developed. We present the theory underlying these algorithms and detail their implementation. Simulation results are then presented which demonstrate that the algorithms are robust in the presence of output noise, and provide good estimates of the system dynamics under a wide set of conditions.     

Multiple membrane systems as biological models

Summary The current-voltage equations for double, triple, and quadruple membrane systems are derived in closed form from the flow equations of irreversible thermodynamics. Numerical examples show that the behavior of these systems is very similar to that of nerve and muscle membranes. Multiple membrane systems exhibit resting potentials which do not have a characteristic Nernst concentration dependence; nonpermeant ions play a significant role in this nonlogarithmic behavior. Furthermore, multiple membrane systems have rectification properties similar to those of biological membranes. The direction of rectification is determined by the polarity of the membrane systems, not by the ionic concentrations in the bathing solutions.     

X-ray diffraction by amorphous polymers: models for biological systems

The x-ray scattering method has been used to investigate the structure in two amorphous crosslinked polymers which are regarded as test systems to establish the power of the method as applied to amorphous biologically significant polymer associations. It is shown that structural information can be determined about the rigid regions within the polymer systems, i.e., those regions held in particular configuration by stereochemical effects. Models of such regions extending over distances with dimensions of ca 18 Å are proposed for an Araldite polymer and for crosslinked poly (methyl/butyl methacrylate). The results allow some general statements about the usefulness and limitations of the amorphous x-ray method.     

Anionic regulation of biological systems: the special role of chloride in the coagulation cascade

The discovery that previously unidentified allosteric properties of several proteins, such as fibrinogen and myoglobin, can be triggered by anions binding, has suggested the possibility to design a new "active" role of chloride in the modulation of a broad range of biological systems. The molecular bases of the anions binding to proteins depends by their charge density in turn regulating the ability to bind water molecules and interact with basic groups on proteins. This review reports the role of the physiologically relevant chloride, and of other anions, in the regulation of several proteins, with special attention to the coagulation cascade. Moreover, possible mechanisms of modification of plasma, intra- or extracellular chloride concentration are listed.     

Neural network models for biological waste-gas treatment systems

This paper outlines the procedure for developing artificial neural network (ANN) based models for three bioreactor configurations used for waste-gas treatment. The three bioreactor configurations chosen for this modelling work were: biofilter (BF), continuous stirred tank bioreactor (CSTB) and monolith bioreactor (MB). Using styrene as the model pollutant, this paper also serves as a general database of information pertaining to the bioreactor operation and important factors affecting gas-phase styrene removal in these biological systems. Biological waste-gas treatment systems are considered to be both advantageous and economically effective in treating a stream of polluted air containing low to moderate concentrations of the target contaminant, over a rather wide range of gas-flow rates. The bioreactors were inoculated with the fungus Sporothrix variecibatus, and their performances were evaluated at different empty bed residence times (EBRT), and at different inlet styrene concentrations (C(i)). The experimental data from these bioreactors were modelled to predict the bioreactors performance in terms of their removal efficiency (RE, %), by adequate training and testing of a three-layered back propagation neural network (input layer-hidden layer-output layer). Two models (BIOF1 and BIOF2) were developed for the BF with different combinations of easily measurable BF parameters as the inputs, that is concentration (gm(-3)), unit flow (h(-1)) and pressure drop (cm of H(2)O). The model developed for the CSTB used two inputs (concentration and unit flow), while the model for the MB had three inputs (concentration, G/L (gas/liquid) ratio, and pressure drop). Sensitivity analysis in the form of absolute average sensitivity (AAS) was performed for all the developed ANN models to ascertain the importance of the different input parameters, and to assess their direct effect on the bioreactors performance. The performance of the models was estimated by the regression coefficient values (R(2)) for the test data set. The results obtained from this modelling work can be useful for obtaining important relationships between different bioreactor parameters and for estimating their safe operating regimes.     

Identification of nonlinear models of biological membranes using the voltage-clamp method

The method for identification of nonlinear systems proposed in 1952 by Hodgkin and Huxley is mathematically justified. A procedure for the application of this method is developed, including the development of the structure of a mathematical model, carrying out a series of tests with special chosen signals, and determination of unknown parameters. Basic requirements for the admissible sets of input and output signals and to the system operator have been determined. It is shown that this operator should be totally continuous and that the minimum number of unknown parameters and the minimum complexity of the operator structure should give an approximation of the necessary quality. The pros and cons of the Hodgkin-Huxley and Noble mathematical models and the methods used for their development are discussed. A structure for the operator for the identification of mathematical models of excitable membranes with a large number of membrane currents is proposed. It is found that the nonlinear electrical properties of biological membranes can be identified using tests with other types of “clamped” parameters, such as the current, ramp voltage, etc.     

A tunable multivariable nonlinear robust observer for biological systems

This paper presents a robust nonlinear asymptotic observer with adjustable convergence rate with a great potential of applicability for biological systems in which the main state variables are difficult and expensive to measure or such measurements do not exist. This observer scheme is based on the classical asymptotic observer, which is modified to allow the tuning of the convergence rate. It is shown that the proposed observer provides fast and satisfactory estimates when facing load disturbances, system failures and parameter uncertainty while maintaining the excellent robustness and stability properties of the classical asymptotic observer. The implementation of the tunable observer is carried out by numerical simulations of a mathematical model of an anaerobic digestion process used for wastewater treatment. The key results are examined and further developed.     

Calibration of dynamic models of biological systems with KInfer

Methods for parameter estimation that are robust to experimental uncertainties and to stochastic and biological noise and that require a minimum of a priori input knowledge are of key importance in computational systems biology. The new method presented in this paper aims to ensure an inference model that deduces the rate constants of a system of biochemical reactions from experimentally measured time courses of reactants. This new method was applied to some challenging parameter estimation problems of nonlinear dynamic biological systems and was tested both on synthetic and real data. The synthetic case studies are the 12-state model of the SERCA pump and a model of a genetic network containing feedback loops of interaction between regulator and effector genes. The real case studies consist of a model of the reaction between the inhibitor κB kinase enzyme and its substrate in the signal transduction pathway of NF-κB, and a stiff model of the fermentation pathway of Lactococcus lactis.     

Multi-model recursive identification for nonlinear systems with non-uniformly sampling

A recursive least squares based on Multi-model is proposed for non-uniformly sampled-data nonlinear (NUSDN) systems. The corresponding state space model of an NUSDN system is derived using lifting technique. Taking advantage of the Fuzzy c-Mean Clustering algorithm, NUSDN is divided into several local models. The basic idea is that the NUSDN system is viewed as a model switching system under a given rule. Once the local models are identified, the global model is determined. A pH neutralization process validate the performance of the proposed algorithm.     

mlegp: statistical analysis for computer models of biological systems using R

Gaussian processes (GPs) are flexible statistical models commonly used for predicting output from complex computer codes. As such, GPs are well suited for the analysis of computer models of biological systems, which have been traditionally difficult to analyze due to their high-dimensional, non-linear and resource-intensive nature. We describe an R package, mlegp, that fits GPs to computer model outputs and performs sensitivity analysis to identify and characterize the effects of important model inputs. AVAILABILITY: http://www.biomath.org/mlegp     

Template-based intervention in Boolean network models of biological systems

Motivation

A grand challenge in the modeling of biological systems is the identification of key variables which can act as targets for intervention. Boolean networks are among the simplest of models, yet they have been shown to adequately model many of the complex dynamics of biological systems. In our recent work, we utilized a logic minimization approach to identify quality single variable targets for intervention from the state space of a Boolean network. However, as the number of variables in a network increases, the more likely it is that a successful intervention strategy will require multiple variables. Thus, for larger networks, such an approach is required in order to identify more complex intervention strategies while working within the limited view of the network’s state space. Specifically, we address three primary challenges for the large network arena: the first challenge is how to consider many subsets of variables, the second is to design clear methods and measures to identify the best targets for intervention in a systematic way, and the third is to work with an intractable state space through sampling.

Results

We introduce a multiple variable intervention target called a template and show through simulation studies of random networks that these templates are able to identify top intervention targets in increasingly large Boolean networks. We first show that, when other methods show drastic loss in performance, template methods show no significant performance loss between fully explored and partially sampled Boolean state spaces. We also show that, when other methods show a complete inability to produce viable intervention targets in sampled Boolean state spaces, template methods maintain significantly consistent success rates even as state space sizes increase exponentially with larger networks. Finally, we show the utility of the template approach on a real-world Boolean network modeling T-LGL leukemia.

Conclusions

Overall, these results demonstrate how template-based approaches now effectively take over for our previous single variable approaches and produce quality intervention targets in larger networks requiring sampled state spaces.

     

Stable oscillations in mathematical models of biological control systems

Summary Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.     

Converting differential-equation models of biological systems to membrane computing

This paper presents a method to convert the deterministic, continuous representation of a biological system by ordinary differential equations into a non-deterministic, discrete membrane computation. The dynamics of the membrane computation is governed by rewrite rules operating at certain rates. That has the advantage of applying accurately to small systems, and to expressing rates of change that are determined locally, by region, but not necessary globally. Such spatial information augments the standard differentiable approach to provide a more realistic model. A biological case study of the ligand–receptor network of protein TGF-β is used to validate the effectiveness of the conversion method. It demonstrates the sense in which the behaviours and properties of the system are better preserved in the membrane computing model, suggesting that the proposed conversion method may prove useful for biological systems in particular.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

A method for constructing data-based models of spiking neurons using a dynamic linear-static nonlinear cascade

This paper describes a general nonlinear dynamical model for neural system identification. It describes an algorithm for fitting a simple form of the model to spike train data, and reports on this algorithm's performance in identifying the structure and parameters of simulated neurons. The central element of the model is a Wiener-Bose dynamic nonlinearity that ensures that the model is able to approximate the behaviour of an arbitrary nonlinear dynamical system. Nonlinearities associated with spike generation and transmission are treated by placing the Wiener-Bose system in cascade with pulse frequency modulators and demodulators, and the static nonlinearity at the output of the Wiener-Bose system is decomposed into a rectifier and a multinomial. This simplifies the model without reducing its generality for neuronal system identification. Model elements can be characterised using standard methods of dynamical systems analysis, and the model has a simple form that can be implemented and simulated efficiently. This model bears a structural resemblance to real neurons; it may be regarded as a connectionist neuron that has been generalized in a realistic way to enable it to mimic the behaviour of an arbitrary nonlinear system, or conversely as a general nonlinear model that has been constrained to make it easy to fit to spike train data. Tests with simulated data show that the identification algorithm can accurately estimate the structure and parameters of neuron-like nonlinear dynamical systems using data sets containing only a few hundred spikes.