Model transformations to evaluate transient thermal responses at a tissue surface

For a spatially distributed model describing the transient temperature response of a thermistor-tissue system, Wei et al. [J. Biomech. Eng., 117:74-85, 1995] obtained an approximate transformation for fast analysis of the temperature response at the tissue surface. This approximate transformation reduces the model to a single ordinary differential equation. Here, we present an exact transformation that yields a single differential-integral equation. Numerical solutions from the approximate and exact transformations were compared to evaluate the differences with several sets of parameter values. The maximum difference between the exact and approximate solutions did not exceed 15 percent and occurred for only a short time interval. The root-mean-square error of the approximate solution was no more than 5 percent and within the level of experimental noise. Under the experimental conditions used by Wei et al., the approximate transformation is justified for estimating model parameters from transient thermal responses.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

Effects of low-chloride solutions on action potentials of sheep cadiac purkinje fibers

The rapid repolarization during phase 1 of the action potential of sheep cardiac purkinje fibers has been attributed to a time- and voltage-dependent chloride current. In part, this conclusion was based on experiments that showed a substantial slowing of phase 1 when larger, presumably impermeant, anions were substituted for chloride in tyrode’s solution. We have re- examined the electrical effects of low-chloride solutions. We recorded action potentials of sheep cardiac purkinje fibers in normal tyrode’s solution and in low-chloride solutions made by substituting sodium propionate, acetylglycinate, methylsulfate, or methanesulfonate for the NaCl of Tyrode’s solution. Total calcium was adjusted to keep calcium ion activity of test solutions equal to that of control solutions. Propionate gave qualitatively variable results in preliminary experiments; it was not tested further. Low-chloride solutions made with the other anions gave much more consistent results: phase 1 and the notch that often occurs between phases 1 and 2 were usually unaffected, and the action potential duration usually increased. The only apparent change in the resting potential was a transient 3-6 mV depolarization when low-chloride solution was first admitted to the chamber, and a symmetrical transient hyperpolarization when chloride was returned to normal. If a time- and voltage-dependent chloride current exists in sheep cardiac purkinje fibers, our results suggest that it plays little role in generating phase 1 of the action potential.     

An analysis of a dendritic neuron model with an active membrane site

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.This work was supported in part by NSF Grants MCS 83-00562 and MDS 85-01535     

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

A nonlinear master equation (NLME) is proposed basedon general information measures.Classical and cut-off solutions of the NLME are considered.In the former case, the NLME exhibits uniquely defined stationary distributions. In the latter case, there are multiple stationary distributions.In particular, for classical solutions, it is shown that transient solutions converge to stationary distributions that maximize information measures (H-theorem). Cut-off distributions arestudied numerically for the Haken-Kelso-Bunz model. The Haken-Kelso-Bunz modelis known to describe multistable human motor control systems. It is shownthat a stochastic Haken-Kelso-Bunz model based on a NLME can exhibit multiplestationary cut-off distributions.In doing so, we illustrate that multistability in stochastic biological systems can beestablished by means of cut-off distributions.     

Osmosis in leaky pores: the role of pressure

Osmosis in leaky pores is often considered to involve gradients of pressure that originate from concentration gradients at the pore wall. These postulated surface pressures are incompatible with the Boltzmann equation and result from an incorrect application of the Gibbs-Duhem equation to binary solutions near surfaces. Such pressures would give rise to tensions in the surface phase that are not observed experimentally. Application of the argument to the swelling of porous systems and gels shows that this phenomenon cannot involve internal pressures within the fluid phase.     

Boundary element method analysis for the bioheat transfer equation.

In this paper, the boundary element method (BEM) approach is applied to solve the Pennes (1948) bioheat equation. The objective is to develop the BEM formulation and demonstrate its feasibility. The basic BEM formulations for the transient and steady-state cases are first presented. To demonstrate the usefulness of the BEM approach, numerical solutions for 2-D steady-state problems are obtained and compared to analytical solutions. Further, the BEM formulation is applied to model a conjugate problem for an artery imbedded in a perfused heated tissue. Analytical solution is possible when the conduction in the x-direction is negligible. The BEM and analytical results have very good agreement.     

Analysis of Transient Changes in Fluid Exudation from Isolated Maize Roots

It was found that the fluid exudation rate from isolated maizeroots rapidly decreases when the external concentration of varioussolutes is increased; thereafter a relatively slow increaseto a new value occurs. An analysis of this transient phenomenonhas been achieved on the basis that there is an active transportof salt (KCl) into a compartment within the root. Further, ithas been assumed that fluid exudation is created by a net osmoticwater flow into this compartment. Our analysis of the experimentaldata indicates that the volume of the compartment has a similarmagnitude to the estimated total volume of the xylem vesselsin the root. The transient curve obtained with some solutes (methanol, magnesiumsulphate, mannitol, and raffinose) were significantly differentfrom the usual response. An attempt to apply an appropriate diffusion equation to theinitial phase of the transient curves showed that the time courseof this phase may be interpreted as being controlled by therate of solute diffusion towards an osmotic barrier within theroot.     

A stochastic theory of phase transitions in human hand movement

The order parameter equation for the relative phase of correlated hand movements, derived in a previous paper by Haken et al. (1985), is extended to a time-dependent stochastic differential equation. Its solutions are determined close to stationary points and for the transition region. Remarkably good agreement between this theory and recent experiments done by Kelso and Scholz (1985) is found, and new predictions are offered.     

Neurohumoral control of ionic pumping in molluscan muscle—II. evidence for an electrogenic sodium pump in radular protractor muscle of Busycon canaliculatum

1. A sucrose gap technique was used to study the effects of brief periods of superfusion with solutions in which the potassium content of artificial sea water was reduced or omitted.2. Stepwise reduction in bath potassium had a complex effect, culminating in the response to potassium-free solution. This was composed of a rapid initial hyperpolarizing phase, overtaken by a slower depolarizing phase, which was accompanied by force.3. Readmission of bath potassium induced a transient after-hyperpolarization.4. There was a high degree of individual variability in RPM preparations from different animals. This was particularly evident in cases in which either the hyperpolarizing phase or the depolarizing phase predominated, in the response to zero-potassium, but the muscles from any one animal showed reproducible responses.5. The RPM behaved as predicted on Nernst equation grounds, to the extent that initial hyperpolarization showed stepwise increases with stepwise reduction in [K⁺]₀, but as the steps approached zero-potassium there was a stepwise increase in the slower depolarizing response, suggesting reduction in electrogenic Na-K exchange.6. In Na-free solution the depolarizing phase of the response to zero-K was abolished, leaving only an enhanced hyperpolarizing phase.7. Abrupt chilling had a depolarizing effect.8. There was only a slight increase in resistance during the action of zero K.     

Prediction of ice content in biological model solutions when frozen under high pressure

High pressure is, at least, as effective as cryoprotective agents (CPAs) and are used for decreasing both homogenous nucleation and freezing temperatures. This fact gives rise to a great variety of possible cryopreservation processes under high pressure. They have not been optimized yet, since they are relatively recent and are mainly based on the pressure–temperature phase diagram of pure water. Very few phase diagrams of biological material are available under pressure. This is owing to the lack of suitable equipment and to the difficulties encountered in carrying out the measurements. Different aqueous solutions of salt and CPAs as biological models are studied in the range of 0°C down to ‐35°C, 0.1 up to 250 MPa, and 0–20% w/w total solute concentration. The phase transition curves of glycerol and of sodium chloride with either glycerol or sucrose in aqueous solutions are determined in a high hydrostatic pressure vessel. The experimental phase diagrams of binary solutions were well described by a third‐degree polynomial equation. It was also shown that Robinson and Stokes' equation at high pressure succeeds in predicting the phase diagrams of both binary and ternary solutions. The solute cryoconcentration and the ice content were calculated as a function of temperature and pressure conditions during the freezing of a binary solution. This information should provide a basis upon which high‐pressure cryopreservation processes may be performed and the damages derived from ice formation evaluated. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Sonochemical free radical formation in aqueous solutions

The phenomena of stable and transient acoustic cavitation in liquids exposed to ultrasound are briefly explained. The role of micronuclei, resonant bubble size, and rectified diffusion in the initiation of transient cavitation is reviewed. In aqueous solutions transient cavitation initially generates hydrogen atoms and hydroxyl radicals that may recombine to form hydrogen and H2O2 or may react with solutes in the gas phase, at the gas-liquid boundary, or in the bulk of the solution. The analogies and differences between sonochemistry and ionizing radiation chemistry are explored. The use of spin trapping and electron spin resonance to conclusively identify hydrogen atoms and hydroxyl radicals and to detect cavitation produced by continuous wave and by pulsed ultrasound is described in detail.     

Steady-state stability of substrate inhibited enzyme catalysis in open reaction systems

Substrate inhibited enzyme reactions occurring in systems open to mass transfer may display multiple steady-state behavior. For a simple one-substrate case it is shown theoretically that the unstable steady-state region always lies within the conversion range of 50–100%. A criterion for the stability of a steady-state point is given. Numerical solutions of the appropriate transient equation show how the system approaches stable steady state in instances where there are three possible steady-state points. The consequences of the existence of an unstable region on the systems response to changes in its parameters is discussed.     

Three-Dimensional Structures of the Spatiotemporal Nonlinear Schr?dinger Equation with Power-Law Nonlinearity in PT-Symmetric Potentials

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell.     

Measurement of membrane potential and estimation of effective fixed-charge density in membranes.

Electrical potentials Em arising across cross-linked phenolsulfonate membrane separating NaCl solutions of molality M1 and M2 have been measured at 25 degrees C. These values of Em have been used in the Nernst equation to calculate values for the apparent transport number ti(app) for the counterion or the co-ion in the membrane. Values of ti(app) together with the limiting value for the cation transport number in the aqueous phase have been used in the equation developed by Kobatake and co-workers to evaluate the membrane permselectivity Ps as a function of external electrolyte concentration. With the help of the equation relating Ps to phiX, the effective fixed-charge density in the membrane (where phi is a constant, 0 less than phi less than 1, and X is the membrane stochiometric charge density and can be evaluated by chemical analysis of the membrane phase), values for phiX and phi have been determined. Values of phi were low in dilute solutions and increased with increase in the concentration of the external solution. Similar behavior was noted in the case of another membrane system, cross-linked polymethacrylic acid in contact with KOH solutions. On the other hand, the membrane system, "untreated" collodion in contact with KCl solutions, exhibited a behavior in which the values of phi, low in dilute solutions, increased and then decreased following a gradual increase in the external concentration. This slight divergence in its behavior was attributed to the heterogeneity of the collodion membrane structure. The reliability of this potentiometric method to estimate effective fixed-charge density in membranes has been discussed in relation to a similar but old method due to Teorell, Meyer and Sievers. Also the significance of the values derived for phi has been pointed out.     

Transient kinetic studies of heme reduction in Escherichia coli nitrate reductase A (NarGHI) by menaquinol

We have studied the transient kinetics of quinol-dependent heme reduction in Escherichia coli nitrate reductase A (NarGHI) by the menaquinol analogue menadiol using the stopped-flow method. Four kinetic phases are observed in the reduction of the hemes. A transient species, likely to be associated with a semiquinone radical anion, is observed with kinetics that correlates with one of the phases. The decay of the transient species and the formation of the second reduction phase of the hemes can be fitted to a double-exponential equation giving similar rate constants, k(1) = 9.24 +/- 0.9 s(-1) and k(2) = 0.22 +/- 0.02 s(-1) for the decay of the transient species, and k(1) = 9.23 +/- 0.9 s(-1) and k(2) = 0.22 +/- 0.02 s(-1) for the formation of the reduction phase. The quinol-binding-site inhibitors 2-n-heptyl-4-hydroxyquinoline-N-oxide (HOQNO) and stigmatellin have significant and different inhibitory effects on the reduction kinetics. The kinetics of heme reduction in NarI expressed in the absence of the NarGH catalytic dimer (NarI(DeltaGH) exhibits only two kinetic phases, and the decay of the transient species also correlates kinetically with the second reduction phase of the hemes. We have also studied nitrate-dependent heme reoxidation following quinol-dependent heme reduction using a sequential stopped-flow method. HOQNO elicits a much stronger inhibitory effect than stigmatellin on the reoxidation of the hemes. On the basis of our results, we propose schemes for the mechanism of NarGHI reduction by menaquinol and reoxidation by nitrate.     

Instability and excitation propagation in a catalytic reaction model. II. Distributed system model]

The regime of excitation propagation in monomeric model of active medium is considered. Conditions exist when the phase plane falls into two regions in both of which the equations are linear and their automodel solutions are found analytically. After connecting the solutions a transcendental algebraic equation is obtained, from which the values of excitation propagation rate can be found. The trigger wave has one rate value the propagating impulse--two (physically realised--one). There is a parameter region with four rates (three physically realised ones).     

An artificial membrane-cable: decay and delay of electrical potentials along a lipid bilayer with ion channels

A model membrane is described which exhibits the properties of a neurite with respect to passive propagation of electrical potentials. A groove in a glass plate is covered by a black lipid membrane of glycerol monooleate. Gramicidin is incorporated. The stationary and transient response of the assembly is tested by two experiments: (i) One end of the groove is clamped at a constant voltage. The voltage at the other end and the total electrical current are measured. (ii) A charge pulse is injected at one end of the groove. The time-dependent voltage at the other end is measured. The results with respect to the lateral decay and delay of voltage are in quantitative agreement with the stationary and transient solutions of Kelvin's equation for a homogeneous cable. If gramicidin is incorporated unevenly along the membrane, the lateral decay of voltage is found to be asymmetric with respect to both directions. The cable is a partial one-way transmission line.     

A computer analysis of the validity of the integrated Michaelis-Menten equation

The validity of assumptions made in integrating the Michaelis-Menten equation (i.e. the steady-state assumption) was examined by simulating the model on the digital computer. Time courses thus obtained by numerical integration were compared with data generated by the three most common forms of the integrated equation. Agreement was good within specified limits, the chief exception being the early transient phase of the reaction. The observed differences were very much less than theoretical estimates of the maximum error, even when rate constants were chosen that should exaggerate that error.     

Transient self-inhibition of the growth of Lactobacillus delbrueckii subsp. bulgaricus in a pH-regulated fermentor

An industrial strain of Lactobacillus delbrueckii subsp. bulgaricus was grown in a synthetic medium on lactose as carbon substrate, in a pH-regulated fermentor. Growth proceeded in two distinct phases separated by a transient stationary phase. Various experimental approaches were used to identify the cause of this growth arrest. Growth experiments in L. bulgaricus culture supernatant fluids collected at different cultivation times in fermentor, and supplemented or not with various nutritional solutions, enabled us to discard the possibility of a nutritional limitation. Tube cultures of L. bulgaricus in medium supplemented with various lactic acid concentrations showed a potential inhibition by this metabolic end product but confirmed that this inhibition was not responsible for the cessation of growth. It was concluded that at least one inhibitory compound was produced during the growth phase of the strain, and this compound disappeared from the medium in the transient stationary phase, enabling the growth to start again later in the culture. Indeed, the stoichiometric analysis of the culture showed, firstly, that unidentified carbon compounds were produced from lactose during growth, which were probably converted in lactic acid during the transient stationary phase and, secondly, that part of the amino acids consumed gave catabolic end products. Finally, bacteriocin-like compounds were not considered to be responsible for this growth arrest.