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.     

Inhibition, spike threshold, and stimulus selectivity in primary visual cortex

Ever since Hubel and Wiesel described orientation selectivity in the visual cortex, the question of how precise selectivity emerges has been marked by considerable debate. There are essentially two views of how selectivity arises. Feed-forward models rely entirely on the organization of thalamocortical inputs. Feedback models rely on lateral inhibition to refine selectivity relative to a weak bias provided by thalamocortical inputs. The debate is driven by two divergent lines of evidence. On the one hand, many response properties appear to require lateral inhibition, including precise orientation and direction selectivity and crossorientation suppression. On the other hand, intracellular recordings have failed to find consistent evidence for lateral inhibition. Here we demonstrate a resolution to this paradox. Feed-forward models incorporating the intrinsic nonlinear properties of cortical neurons and feed-forward circuits (i.e., spike threshold, contrast saturation, and spike-rate rectification) can account for properties that have previously appeared to require lateral inhibition.     

Carcinogenicity of the aromatic amines: from structure-activity relationships to mechanisms of action and risk assessment

Aromatic amines represent one of the most important classes of industrial and environmental chemicals: many of them have been reported to be powerful carcinogens and mutagens, and/or hemotoxicants. Their toxicity has been studied also with quantitative structure-activity relationship (QSAR) methods: these studies are potentially suitable for investigating mechanisms of action and for estimating the toxicity of compounds lacking experimental determinations. In this paper, we first summarized the QSAR models for the rodent carcinogenicity of the aromatic amines. The gradation of potency of the carcinogenic amines depended firstly on their hydrophobicity, and secondly on electronic (reactivity, propensity to be metabolically transformed) and steric properties. On the contrary, the difference between carcinogenic and non-carcinogenic aromatic amines depended mainly on electronic and steric properties. These QSARs can be used directly for estimating the carcinogenicity of aromatic amines. A two-step prediction is possible: (1) estimation of yes/no activity; (2) if the answer from step 1 is yes, then prediction of the degree of potency. The QSARs for rodent carcinogenicity were put in a wider context by comparing them with those for: (a) Salmonella mutagenicity; (b) general toxicity; (c) enzymatic reactions; (d) physical-chemical reactions. This comparative QSAR exercise generated a coherent global picture of the action mechanisms of the aromatic amines. The QSARs for carcinogenicity were similar to those for Salmonella mutagenicity, thus pointing to a similar mechanism of action. On the contrary, the general toxicity QSARs (both in vitro and in vivo systems) were mostly based on hydrophobicity, pointing to an aspecific mechanism of action much simpler than that for carcinogenicity and mutagenicity. The oxidation of the amines (first step in the main metabolic pathway leading to carcinogenic and mutagenic species) had identical QSARs in both enzymatic and physical-chemical systems, thus providing evidence for the link between simple chemical reactions and those in biological systems. The results show that it is possible to generate mechanistically and statistically sound QSAR models for rodent carcinogenicity, and indirectly that the rodent bioassay is a reliable source of good quality data.     

2D-QSAR in hydroxamic acid derivatives as peptide deformylase inhibitors and antibacterial agents

Peptide deformylase catalyzes the removal of N-formyl group from the N-formylmethionine of ribosome synthesized polypeptide in eubacteria. Quantitative structure–activity relationship (QSAR) studies have been carried out in a series of β-sulfonyl and β-sulfinyl hydroxamic acid derivatives for their PDF enzyme inhibitory and antibacterial activities against Escherichia coli DC2 and Moraxella catarrhalis RA21 which demonstrate that the PDF inhibitory activity in cell free and whole cell system increases with increase in molar refractivity and hydrophobicity. The comparison of the QSARs between the cell free and whole cell system indicate that the active binding sites in PDF isolated from E. coli and in M. catarrhalis RA21 are similar and the whole cell antibacterial activity is mainly due to the inhibition of PDF. Apart from this the QSARs on some matrixmetelloproteins (COL-1, COL-3, MAT and HME) and natural endopeptidase (NEP) indicate the possibilities of introducing selectivity in these hydroxamic acid derivatives for their PDF inhibitory activity.     

Correlation between simulated physicochemical properties and hemolycity of protegrin-like antimicrobial peptides: predicting experimental toxicity

The therapeutic, antibiotic potential of antimicrobial peptides can be prohibitively diminished because of the cytotoxicity and hemolytic profiles they exhibit. Quantifying and predicting antimicrobial peptide toxicity against host cells is thus an important goal of AMP related research. In this work, we present quantitative structure activity relationships for toxicity of protegrin-like antimicrobial peptides against human cells (epithelial and red blood cells) based on physicochemical properties, such as interaction energies and radius of gyration, calculated from molecular dynamics simulations of the peptides in aqueous solvent. The hypothesis is that physicochemical properties of peptides, as manifest by their structure and interactions in a solvent and as captured by atomistic simulations, are responsible for their toxicity against human cells. Protegrins are beta-hairpin peptides with high activity against a wide variety of microbial species, but in their native state are toxic to human cells. Sixty peptides with experimentally determined toxicities were used to develop the models. We test the resulting relationships to determine their ability to predict the toxicity of several protegrin-like peptides. The developed QSARs provide insight into the mechanism of cytotoxic action of antimicrobial peptides. In a subsequent blind test, the QSAR correctly ranked four of five protegrin analogues newly synthesized and tested for toxicity.     

Identification of nonlinear models of biological membranes by the method of fixed voltage

A mathematical substantiation of the method suggested by Hodkin and Huxley in 1952 for the identification of nonlinear systems is presented. A procedure for the application of this method was developed, which involves creating the structure of a mathematical model, carrying out a series of tests with specially chosen signals, and finding the unknown parameters. The basic requirements to admissible sets of entrance and target signals and the operator of the system were determined. It was shown that it should be quite continuous, the minimal number of unknown parameters and the minimal complexity of structure of the operator should provide the required quality of approximation. The merits and demerits of the mathematical models of Hodkin-Huxley and Noble, and the procedures used for their creation are discussed. The structure of the operator for the identification of mathematical models of excitable membranes when a large number of membrane currents is considered is offered. It was found that nonlinear electric properties of biological membranes can be identified using tests with other kinds of "fixed" parameters, for example, the method of "fixed" current, the fixed linearly increasing voltage, and others.     

A viscoelastic model for use in predicting arterial pulse waves

In nonlinear mathematical models of the arterial circulation, the viscoelasticity of the vessel walls has generally been neglected or only taken into account in a highly approximate manner. A new method is proposed to simulate the nonlinear viscoelastic properties of the wall material with the aid of a convolution integral of the creep function and the pressure history. With this simulation it is possible to properly describe the measured characteristics of arterial viscoelasticity. Moreover, it is utilized in a mathematical model of arterial pulse propagation to study the influence of the internal wall friction on the shape, amplitude and mean value of pressure and flow pulses. The corresponding predictions are in much better agreement with in-vivo measurements, especially for the distal part of the circulation, than those obtained without viscoelasticity.     

Structural and biophysical properties of a synthetic channel-forming peptide: Designing a clinically relevant anion selective pore

The design, synthesis, modeling and in vitro testing of channel-forming peptides derived from the cys-loop superfamily of ligand-gated ion channels are part of an ongoing research focus. Over 300 different sequences have been prepared based on the M2 transmembrane segment of the spinal cord glycine receptor α-subunit. A number of these sequences are water-soluble monomers that readily insert into biological membranes where they undergo supramolecular assembly, yielding channels with a range of selectivities and conductances. Selection of a sequence for further modifications to yield an optimal lead compound came down to a few key biophysical properties: low solution concentrations that yield channel activity, greater ensemble conductance, and enhanced ion selectivity. The sequence NK(4)-M2GlyR T19R, S22W (KKKKPARVGLGITTVLTMRTQW) addressed these criteria. The structure of this peptide has been analyzed by solution NMR as a monomer in detergent micelles, simulated as five-helix bundles in a membrane environment, modified by cysteine-scanning and studied for insertion efficiency in liposomes of selected lipid compositions. Taken together, these results define the structural and key biophysical properties of this sequence in a membrane. This model provides an initial scaffold from which rational substitutions can be proposed and tested to modulate anion selectivity. This article is part of a Special Issue entitled: Protein Folding in Membranes.     

Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments

Growth competition assays have been developed to quantify the relative fitness of HIV-1 mutants. In this article, we develop mathematical models to describe viral/cellular dynamic interactions in the assay system from which the competitive fitness indices or parameters are defined. In our previous HIV-viral fitness experiments, the concentration of uninfected target cells was assumed to be constant (Wu et al. 2006). But this may not be true in some experiments. In addition, dual infection may frequently occur in viral fitness experiments and may not be ignorable. Here, we relax these two assumptions and extend our earlier viral fitness model (Wu et al. 2006). The resulting models then become nonlinear ODE systems for which closed-form solutions are not achievable. In the new model, the viral relative fitness is a function of time since it depends on the target cell concentration. First, we studied the structure identifiability of the nonlinear ODE models. The identifiability analysis showed that all parameters in the proposed models are identifiable from the flow-cytometry-based experimental data that we collected. We then employed a global optimization approach (the differential evolution algorithm) to directly estimate the kinetic parameters as well as the relative fitness index in the nonlinear ODE models using nonlinear least square regression based on the experimental data. Practical identifiability was investigated via Monte Carlo simulations.     

Mathematical analysis and modeling of motion direction selectivity in the retina

Motion detection is one of the most important and primitive computations performed by our visual system. Specifically in the retina, ganglion cells producing motion direction-selective responses have been addressed by different disciplines, such as mathematics, neurophysiology and computational modeling, since the beginnings of vision science. Although a number of studies have analyzed theoretical and mathematical considerations for such responses, a clear picture of the underlying cellular mechanisms is only recently emerging. In general, motion direction selectivity is based on a non-linear asymmetric computation inside a receptive field differentiating cell responses between preferred and null direction stimuli. To what extent can biological findings match these considerations? In this review, we outline theoretical and mathematical studies of motion direction selectivity, aiming to map the properties of the models onto the neural circuitry and synaptic connectivity found in the retina. Additionally, we review several compartmental models that have tried to fill this gap. Finally, we discuss the remaining challenges that computational models will have to tackle in order to fully understand the retinal motion direction-selective circuitry.     

QSAR of the testosterone binding globulin affinity by means of correlation weighting of local invariants of the graph of atomic orbitals

Numerical values of the testosterone binding globulin affinity have been modeled as a mathematical function of molecular structure in two versions of molecular structure elucidation: first, by hydrogen-filled molecular graphs (HFG); second, by the so-called graphs of atomic orbitals (GAO). Increased orders of Morgan extended connectivity in the HFG and GAO have been examined as local invariants. Using optimisation of the correlation weights of the above-mentioned invariants, quantitative structure–activity relationships (QSAR) have been obtained. Best statistical characteristics of these QSARs are derived in the case of the Morgan extended connectivity of first order in the GAO. They are as follows: n = 11, r² = 0.6540, s = 0.824, F = 17 (training set); n = 9, r² = 0.8791, s = 0.388, F = 51 (test set).     

Optimization of batch fermentation processes by graphical method

A graphical method is proposed for batch fermentation process optimization. Its objective function is the maximum output of the product when the time of process is fixed or free. The technique of optimization is based on Bellman's dynamic programming concepts and the assumption that the process can be described by mathematical models of generalized structure. An example of the proposed technique's application is presented.     

Methods for comparing Salmonella mutagenicity data sets using nonlinear models

A variety of linear and nonlinear mathematical models have been proposed to characterize Salmonella mutagenicity data sets, but no systematic procedure has been suggested for comparing two or more data sets across experiments, laboratories, occasions, mutagens or treatment conditions. In this paper, a general method for data-set comparison is provided. Nonlinear regression techniques are applied to real data sets. Data-set and parameter equivalence are described in depth. Confidence-band construction for nonlinear models and other graphical techniques are presented as auxiliary tools. Key Statistical Analysis System (SAS) code programs are provided.     

Evidence for a highly elastic shell-core organization of cochlear outer hair cells by local membrane indentation

Cochlear outer hair cells (OHCs) are thought to play an essential role in the high sensitivity and sharp frequency selectivity of the hearing organ by generating forces that amplify the vibrations of this organ at frequencies up to several tens of kHz. This tuning process depends on the mechanical properties of the cochlear partition, which OHC activity has been proposed to modulate on a cycle-by-cycle basis. OHCs have a specialized shell-core ultrastructure believed to be important for the mechanics of these cells and for their unique electromotility properties. Here we use atomic force microscopy to investigate the mechanical properties of isolated living OHCs and to show that indentation mechanics of their membrane is consistent with a shell-core organization. Indentations of OHCs are also found to be highly nonhysteretic at deformation rates of more than 40 microm/s, which suggests the OHC lateral wall is a highly elastic structure, with little viscous dissipation, as would appear to be required in view of the very rapid changes in shape and mechanics OHCs are believed to undergo in vivo.     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Novel quantitative tools for engineering analysis of hepatocyte cultures in bioartificial liver systems

Extracorporeal bioartificial liver devices (BAL) are perhaps among the most promising technologies for the treatment of liver failure, but significant technical challenges remain in order to develop systems with sufficient processing capacity and of manageable size. One key limitation is that during BAL operation, when the device is exposed to plasma from the patient, hepatocytes are prone to accumulate intracellular lipids and exhibit poor liver-specific functions. Based on hepatic intermediary metabolism, we have utilized mathematical programming techniques to optimize the biochemical environment of hepatocyte cultures towards the desired effect of increased albumin and urea synthesis. To investigate the feasible range of optimal hepatic function, we have obtained a Pareto optimal set of solutions corresponding to liver-specific functions of urea and albumin secretion in the metabolic framework using multiobjective optimization. The importance of amino acids in the supplementation and the criticality of the metabolic pathways have been investigated using logic-based programming techniques. Since the metabolite measurements are bound to be patient specific, and hence subject to variability, uncertainty has to be integrated with system analysis to improve the prediction of hepatic function. We have used the concept of two stage stochastic programming to obtain robust solutions by considering extracellular variability. The proposed analysis represents a new systematic approach to analyze behavior of hepatocyte cultures and optimize different operating parameters for an extracorporeal device based on real-time conditions.