The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT.

Hybrid methods have been known for a long time as very efficient algorithms for attenuation correction in single-photon emission computed tomography, but only recently have efforts been made to formulate them with more rigorous mathematics. This has allowed us to explain their efficiency in terms of approximate inversion, and to establish a convergence condition. The present study focuses on the convergence problem and emphasizes the question of symmetry. Hybrid method operators are not symmetrical; therefore the convergence condition is not easily verified. New schemes based on a modified conjugate gradient method are presented. Convergence is proved and performances are shown to be at least as good as the standard hybrid schemes on perfect and noisy simulated data.

Pattern formation in an immobilized bienzyme system. A morphogenetic model.

Experimental and theoretical studies of a reaction-diffusion model of two immobilized enzymes participating in the cellular acid-base metabolism, namely glutaminase and urease, are presented. The system shows an unstable steady state at pH 6.0, where any perturbation will drive the system towards a more alkaline or more acidic pH, owing to the autocatalytic behaviour with respect to pH exhibited by both enzymes. When diffusion is coupled to reaction by means of immobilization, different patterns of the internal pH profile appear across the membrane. If the bienzymic membrane is subjected to a perturbation at its boundaries, of the same amplitude but in opposite directions, the internal pH evolves through an asymmetric pattern to attain a nearly symmetric distribution of pH. The pH value at the final steady state is more acidic or more alkaline than the initial state according to the initial and boundary conditions. The final nearly symmetric state is attained more rapidly when less enzyme is immobilized (1.8 x 10(-4) M.s-1 as against 3.3 x 10(-4) M.s-1 of total enzyme activity in the membrane volume). The experimental results agree rather well qualitatively with numerical predictions of the model equations.

Electrical excitability of artificial enzyme membranes. IV. Theoretical approach of the membrane potential of synthetic proteinic films.

This paper deals with the theoretical approach of the membrane potential of artificial proteinic film. Programming techniques using finite difference simulations for the steady state and transient solutions of the Nernst-Planck and Poisson equations were used and solved by the collocation and corrector methods. This approach allows one to calculate the membrane potential without any discontinuity between the Donnan and the diffusion potentials, the thickness of the boundary layers being automatically determined by the intrinsic properties of the solution and of the membrane. The theoretical results are compared with experimental potentials measured on proteinic artificial films.

Mathematical modeling of immobilized enzyme systems.

We model the interaction of reaction and diffusion in immobilized enzyme systems by P.D.E.s or simpler equations. We present models with only one steady state, others with multiple steady states (in particular one related to morphogenesis) and some with periodic solutions. The usefulness of continuation methods to resolve this complexity of behaviors is pointed out.

Multiple steady states and oscillatory behavior of a compartmentalized phosphofructokinase system.

This paper discusses interacting diffusion and reaction in an open enzyme system. The enzyme, rabbit muscle phosphofructokinase (PFK; ATP:D-fructose-6-phosphate 1-phosphotransferase, EC 2.7.1.11), is inhibited strongly by excess of the substrate ATP. Metabolites diffuse through an inert membrane separating the enzyme from the bulk reacting medium. We demonstrate that such a simple system is able to account for the existence of both oscillatory behavior (limit cycle) and multiple steady states (hysteresis) as well as for the sudden transitions between stable and periodic behaviors. Experimental evidence for time oscillations is given.

Determination of substrate concentrations by a computerized enzyme electrode.

A numerical treatment of the signal produced by an electrode onto which an artificial enzyme membrane is mounted can give the concentration of the substrate (glucose, saccharose, lactose, amino acids, etc.) in solution. In the example of a glucose analyzer, in which glucose oxidase catalyzes the oxidation of glucose, the computer receives pO(2) level data from the electrode and calculates the glucose concentration. The transient electrode signal, measured as the enzyme membrane is exposed to a solution of glucose, is least-square approximated by a third-degree polynomial whose slope at inflection point is characteristic of the external glucose concentration. A calibration procedure provides a cubic spline approximation of glucose concentration as a function of slope, thus enabling automatic measurement of samples. The computer performs the calculations, and actuates valves for air rinsing, introduction of the sample, and water rinsing.

Spatiotemporal behaviors in immobilized enzyme systems.

The immobilization of enzymes within an artificial membrane, with a homogeneous distribution of the active sites, allows a simple modelling in a well defined context. The systems are described by non-linear PDE'S, taking into account enzyme reaction and metabolite diffusion. These equations can exhibit several types of behaviors, qualitatively different from those observed in solution, such as hysteresis, oscillations and pattern formations. Preliminary experimental results have shown the existence of sustained oscillations and instabilities with immobilized acetylcholinesterase and phosphofructokinase.

Hysteresis, oscillations, and pattern formation in realistic immobilized enzyme systems.

Hysteresis, oscillations, and pattern formation in realistic biochemical systems governed by P.D.E.s are considered from both numerical and mathematical points of view. Analysis of multiple steady states in the case of hysteresis, and bifurcation theory in the cases of oscillations and pattern formation, account for the observed numerical results. The possibility to realize these systems experimentally is their main interest, thus bringing further arguments in favor of theories explaining basic biological phenomena by diffusion and reaction.

The kinetic behavior of an artificial bienzyme membrane.

Artificial membranes bearing immobilized enzymes can be used to study some effects of membrane structure on enzyme kinetic behavior. The bienzyme system described is a mixture of beta-glucosidase and glucose oxidase. Gluconolactone, the product of thesecond enzyme, is an inhibitor of the first one. The resulting feedback effect has been compared using a mixed two-enzyme membrane, two separated one-enzyme membranes, and astirred bienzyme solution. The feedback effect is quicker and more efficient in the two-enzyme membrane than in solution; it is slower and less efficient in the case of the separated one-enzyme membranes. Effects of enzyme proximity in the structure are discussed. Conclusions are drawn concerning the efficiency of feedback mechanisms when enzymes are embedded within a single structure.