Interfacial models of nerve fiber cytoskeleton.

A new approach, basing on a resemblance between cytoskeleton structures associated with plasma membranes and interfacial layers of coexisting phases, is proposed. In particular, a lattice model, similar to those of the theory of surface properties of pure liquids and nonelectrolyte solutions (Ono, S., and S. Kondo. 1960. Handbuch der Physik.), has been developed to describe nerve fiber cytoskeleton. The preliminary consideration of the model shows the existence of submembrane cytoskeleton having increased peripheral densities of microtubules (compared with the bulk density) which is in qualitative agreement with the data in literature. Some additional possibilities of the approach proposed are briefly discussed.     

The extracellular potential of a myelinated nerve fiber in an unbounded medium and in nerve cuff models

A model is presented for the calculation of single myelinated fiber action potentials in an unbounded homogeneous medium and in nerve cuff electrodes. The model consists of a fiber model, used to calculate the action currents at the nodes of Ranvier, and a cylindrically symmetrical volume conductor model in which the fiber's nodes are represented as point current sources. The extracellular action potentials were shown to remain unchanged if the fiber diameter and the volume conductor geometry are scaled by the same factor (principle of corresponding states), both in an unbounded homogeneous medium and in an inhomogeneous volume conductor. The influence of several cuff electrode parameters, among others, cuff length and cuff diameter, were studied, and the results were compared, where possible, with theoretical and experimental results as reported in the literature.     

Realization of biophysical models of cardiac electrical activity

Considered are the principles of realization of biophysical models of heart ventricle electrical activity in the form of a double electric layer on the surface of the electrically active myocardium (epicardium and endocardium) and the boundary surfaces dividing the model compartments with different electrophysiological characteristics. The model parameters are the electrophysiological and anatomical characteristics of the heart such as the geometry of the ventricles and the specialized His-Purkinje conduction system, the velocity of depolarization spread over myocardium, the ratio of the velocities of excitation transmission through the Myocardium / His / Purkinje elements of the model, the shape of transmembrane action potentials on the boundary surfaces, the orientation of the intrinsic anatomical axes of the heart relative to the initial set of coordinates, and some other biophysical characteristics of the myocardium. This model is the main unit of a computer simulation system, which includes databases of real and simulated electrocardiosignals.     

The effect of active substances on nerve fiber regeneration in nerve tissue cultures

Explants of the ganglion trigeminale from chick embryos and hippocampi from embryonal rats were incubated in maximowchambers with semisynthetic media. The different parts of nervous tissue were influenced experimentally by addition of biological extracts and by substances with known composition. The regeneration of nerve fibers was investigated by the index of growth. The growth index was calculated from the ratio of nerve fibre index of the test cultures to that of the influenced cultures. Under these conditions biological extracts enhanced the growth of nerve fibres. In the same way the growth of nerve fibres was statistically significantly stimulated by substances with known composition of aminoacids, orotic acid, sodium orotate and cyclic monophosphates. A stimulating effect of cyclic guanosinmonophosphate seems to exist only in CNS explants and only in young fetal rats and consists in an increased migration and proliferation of cells as well as in the formation of fibres from neuroblasts. The investigations gave strong evidence for the in vitro testing to be very useful in studies of nerve fibre regeneration. However the choice of suitable reference systems, suitable quantitative parameters, optimal concentration and periods of application of effective substances and the age of the animals are of fundamental importance for the evaluation of the results. Experiments regarding the stimulation of the differentiation of neurons and the growth of nerve fibres are of practical clinical and therapeutical interest.     

Water fluxes in nerve fiber

Summary Van der Waals energies of interaction between model cell surfaces are calculated for various distances of separation, layer thicknesses and compositions of cell surfaces and intercellular media. In these calculations the cell peripheries are considered to consist of two layers: (1) A phospholipid-cholesterol-protein plasma membrane and (2) a surface coat, which consists of protein, sugar and water. The required Van der Waals parameters of sugars, phospholipids and cholesterol are derived from refractive indices of their solutions in the visible and ultraviolet regions. Polarizabilities and Van der Waals parameters of these substances are determined and shown to be almost independent of concentration of solutions. Resulting isotropic polarizabilities differ by less than five percent from values obtained by the addition of bond polarizabilities.The magnitude of Van der Waals interactions between cell surfaces has been found to vary with composition according to the following sequence: water–15 ergs and 6×10^–14 ergs at 50 Å distance of separation, which corresponds to free energies per unit area of 210-1600kT/ ²     

Osmotic relations of nerve fiber

Summary The volumetric elastic modulus of the sheath and the osmotic swelling pressure of the axoplasmic polymer network of the giant axon ofLoligo vulgaris were measured. Evidence was obtained that (1) the elastic modulus of the sheath, (2) the swelling pressure of axoplasm, and (3) theeffective osmotic pressure difference due to mobile solutes determine axonal volume. The contributions of the sheath and the axoplasm were significant because theeffective osmotic pressure due to mobile solutes was a small fraction of thetheoretical bulk osmotic pressure due to these solutes. The giant axon was converted from an imperfect to a near perfect osmometer by minimizing the contribution of the sheath and the axoplasmic gel.The early experiments were undertaken in the Stazione Zoologica, Naples, and the Democritus Nuclear Research Center, Athens, Greece; the more recent experiments were undertaken in facilities provided by Prof. N. Nikolaou, Aegina, Greece.     

The velocity of conduction in nerve fiber and its electric characteristics

The theory developed in this paper shows that the propagation of spike potential along a nerve fiber and the conduction of an electric wave along an inert inorganic conductor follow a common quantitative relationship. This result gives further support to the belief that propagation of excitation is an electrical process. The basic idea of the theory is derived from the consideration that velocity has, by its mathematical definition, a local meaning; conduction in a nerve is completely determined by the local characteristics of the latter, as well as those of the wave. The final formula derived does not make use of any other field of science beyond the fundamental principles of electricity. It gives the conduction velocity in terms of the electric characteristics of the fiber and of the duration of the spike potential. The formula is in agreement with the known dependence of the conduction velocity on various parameters characterizing the axon. The computed velocity agrees with the measured ones on the squid giant axon, crab nerve axon, frog muscle fiber and Nitella cell. The membrane inductance appears as a velocity controling agent which prevents also a possible distortion of the spike potential during conduction. The structural meaning of the electric characteristics of the axon membrane is discussed from the viewpoint of the diffusion theory. A formula for the velocity of spread of the electrotonus is also derived.     

Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach

A major goal of biophysics is to understand the physical mechanisms of biological molecules and systems. Mechanistic models are evaluated based on their ability to explain carefully controlled experiments. By fitting models to data, biophysical parameters that cannot be measured directly can be estimated from experimentation. However, it might be the case that many different combinations of model parameters can explain the observations equally well. In these cases, the model parameters are not identifiable: the experimentation has not provided sufficient constraining power to enable unique estimation of their true values. We demonstrate that this pitfall is present even in simple biophysical models. We investigate the underlying causes of parameter non-identifiability and discuss straightforward methods for determining when parameters of simple models can be inferred accurately. However, for models of even modest complexity, more general tools are required to diagnose parameter non-identifiability. We present a method based in Bayesian inference that can be used to establish the reliability of parameter estimates, as well as yield accurate quantification of parameter confidence.     

The impact of exclusion processes on angiogenesis models

Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.

     

Electrocardiographic image of myocardial ischemia: Real measurements and biophysical models

The peculiarities of the genesis of electrocardiosignal under myocardial ischemia in connection with an increase of its electrical inhomogeneity, as well as the electrophysiological mechanisms of morphological changes of the T wave of the electrocardiogram, including its “symmetrization” have been considered. A systemic approach to the problem has been used, which combines the mathematical, computer, and physiological modeling of the cardiac electrical activity with studies of the electric field of the human heart in terms of biophysical models. A database for the repolarization parameters of experimental electrocardiosignals (“Norm” and “Ischemia” samples) has been formed. The parameters of the ST-T interval and T wave, which could characterize the symmetry of the latter, and some additional properties of the repolarization process have been obtained. The methods of mathematical modeling were used also. Computer experiments were carried out on a system for 3D modeling of the cardiac electrical activity at different structural levels of the object. By the results of preliminary analysis, the β _T index, which is calculated as the ratio of two maximum absolute values of the derivative of the cardiosignal at the left and right of the T wave apex, has been chosen as one of the main diagnostic markers of ischemia. There is reason to believe that the use of the β _T index allows one to recognize those deviations from the norm that are usually hidden from a physician in traditional ECG analysis. The ratios of repolarization intervals inside of the generalized QT interval have also appeared potentially informative. With the purpose to test and correct the hypotheses for conducting further investigations, some preliminary experiments on a low-resolution model for several alternatives of the degree, localization, and extensiveness of ischemia have been carried out. The results obtained at the first stage of the team-work are essential to the understanding of mechanisms of the genesis of an electrocardiographic image for myocardial ischemia and of interest for the biophysically sound development of new algorithms for computer cardiodiagnostics.     

Colchicine inhibition of nerve fiber formation in vitro

Inhibition of nerve fiber (neurite) formation by colchicine and Colcemid was studied in monolayer cultures of dissociated spinal ganglia of the chick. Replica cultures were fixed after appropriate incubation and alkaloid treatment. Quantitative estimates of the mean total neurite length per neuron (MNL) were made by use of camera lucida tracing. MNL values plotted against time of incubation gave control curves with an initial lag period, a phase of rapid increase, and a final phase in which MNL increase was retarded. Colchicine at 0.01–0.05 µg/ml (2.4 x 10^-8-1.2 x 10^-7 M) caused reversible, concentration dependent, inhibition of the increase in MNL when applied during the lag period or phase of rapid increase. At the highest concentration there was a net decrease in MNL. The effect of Colcemid at 0.05 µg/ml was similar to that of colchicine, but more rapidly reversible. In most experiments there was no loss of neurons during the period of inhibition of MNL increase by colchicine or Colcemid. Therefore selective destruction of neurons was not involved in the inhibition of neurite growth. Prolonged incubation after treatment with the highest concentration used resulted in a 50% loss of neurons, in part through detachment of viable cells. Quantitative radioautography of the alkaloid-treated neurons with leucine-¹⁴C indicated little or no inhibition of incorporation into protein during inhibition of MNL increase. The results strongly suggest that inhibition of neurite growth involves a specific effect of colchicine, presumably the disruption of microtubules. They are thus consistent with the hypothesis that the polymerization of microtubules is essential to the formation of nerve fibers.     

Dimensional analysis of nerve models

General equations for (i) a uniform patch of nerve membrane, (ii) a continuous (unmyelinated) axon and (iii) a noded (myelinated) axon are analyzed using dimensional analysis. The original dimensioned equations are transformed to dimensionless equations. These equations contain dimensionless constants called similarity parameters which are functions of the physical constants or parameters of the system. (The similarity parameters are analogous to such quantities as the Reynolds number and Mach number used in fluid dynamics.) There is one similarity parameter for each of cases (i) and (ii), and four for case (iii). All dimensioned systems having the same values of all the similarity parameters form a similarity class.Once a quantity such as threshold stimulus or conduction velocity is computed for one member of any similarity class, the same quantity can be easily computed for any other member of the same class, by a simple formula containing the physical constants of the system, called a generating equation, or by an even simpler expression of proportionality, called a scaling relation.