We address the problem of pattern formation on the surface of a sphere using Turing equations. By considering a generic reaction-diffusion model, we numerically investigate the patterns formed under different conditions on the parameter values. Our results show that a closed surface with curvature, as a sphere, imposes geometrical restrictions on the shape of the pattern. This is important in some biological systems where curvature plays an important role in guiding chemical, biochemical, and embryological processes.

The contents of citric acid cycle compounds were measured in rat hind limb muscle in situ during rest, exercise, and recovery from exercise. The following changes in citric acid cycle intermediates were observed during exercise for 15 min. The contents of fumarate and malate increased fourfold over resting values. The contents of citrate, isocitrate, and succinate rose 68 'I<, 48 and 87 "' , respectively, but the increase in these intermediates was delayed relative to the increase in fumarate and malate. The content of 2-oxoglutarate did not change significantly. The content of oxaloacetate increased from less than 1 nmol/g dry weight in resting muscle to 8 nmol/g dry weight during exercise. The orthophosphate content doubled. The sum of the contents of citric acid cycle intermediates doubled. The ratios of the contents of malate/fumarate and citrate/isocitrate remained constant. The constancy of the citrate/isocitrate ratio indicates that the concentration of intracellular free Mg2+ is not affected by exercise. All metabolite levels returned to resting values during recovery. Comparison of these data with the rate of operation of the purine nucleotide cycle leads us to conclude that the increase in citric acid cycle intermediates can be accounted for by the operation of the purine nucleotide cycle.The rate of operation of the purine nucleotide cycle in skeletal muscle increases during exercise and recovery following exercise [l, 21. The cycle catalyzes the net reaction : aspartate + GTP + H20 + NH3 + fumarate + G D P + Pi. We now show that the total level of citric acid cycle intermediates in rat skeletal muscle rises by a factor of two during exercise. The changes in citric acid cycle intermediates are compared with the rate of formation of IMP in the absence and presence of hadacidin, an inhibitor of adenylosuccinate synthetase. On the basis of this comparison it is concluded that the operation of the purine nucleotide cycle is sufficient to account for the observed increases in citric acid cycle intermediates in skeletal muscle. Other possible reactions for replenishing or increasing citric acid cycle intermediates are considered in the Discussion.ions participate in the reactions of the citric acid and the purine nucleotide cycles and they exert direct and indirect effects on adenylate deaminase [3]. Determinations of the contents of these substances in skeletal muscle during rest, exercise, and recovery from exercise are also presented. Orthophosphate and Mg2 MATERIALS AND METHODSMale rats of the Sprague-Dawley strain were obtained from Charles River Breeding Laboratories (Wilmington, MA). The animals received food and water ad libitum, and weighed 175-200 g at the time of use.Rats were anaesthetized by intraperitoneal injection of sodium pentobarbital (5 mg/100 g body weight) and the skin from the right hind leg was removed. The sciatic nerve was exposed and a Dastre's electrode (Palmer Co., London, UK) was attached around the nerve in its gluteal course. The leg was fixed to a platform at the ankle with ...

In this paper we address the problem of pattern formation in confined Turing systems in two dimensions, when one assumes the enhancement of the concentration of one of the chemicals at some of the confining surfaces. This model is suitable to study biological systems, such as the skin patterns shown by some marine fish. We also study numerically the dynamical growth of the system by changing the size of the confined region while dynamical diffusion and reaction phenomena take place. This idea is tested in two different models. This allows one to estimate the robustness of stripe formation. ͓S1063-651X͑97͒06807-4͔ PACS number͑s͒: 47.54.ϩr, 82.40.Bj, 82.20.Mj In 1952, Turing established the basis to explain biological patterns using two interacting chemicals ͓1͔. The experimental observation of a ''Turing pattern'' occurred in a chemical system nearly 40 years after their prediction by Turing ͓2,3͔, but it was not until very recently that the example of a Turing pattern in a biological system was confirmed in skin patterns of the angelfish ͑Pomacanthus͒ by Kondo and Asai ͓4͔. In this work the authors propose and solve a system of two reaction-diffusion equations in a growing onedimensional domain to explain the insertion of new stripes between the older ones during the growth of Pomacanthus semicirculatus and the rearrangement of the stripe pattern of Pomacanthus imperator.Kondo and Asai's interpretation was subject to criticism by Höfer and Maini ͓5͔, who did not find enough evidence to say that reaction-diffusion systems could provide a mechanistic basis for the strip-doubling phenomenon. In particular, they claim that a two-dimensional simulation would be a more realistic representation of the fish skin than a onedimensional domain. Höfer and Maini argue that a mechanism that sets the distance between adjacent stripes and some kind of ''memory'' that conserves the location of old stripes is needed in order to explain the patterning dynamics of the Pomacanthus. Accordingly, in this work we shall show that a reaction-diffusion system is capable of describing the main features of the phenomena observed in the Pomacanthus skin. For that goal, we consider two sets of Turing equations known to form different kinds of patterns; these are solved in a two-dimensional spatial domain that simulates the fish shape, with zero flux boundary conditions. The key feature of our simulation is the enforcement of an enhanced source of the activator along some of the boundaries of the domain. This idea has close parallels with the mechanism of stripe formation in the Drosophila embryo where the pattern of the anteroposterior ͑head-tail͒ segmentation is caused by a high concentration of the Bicoid protein along the anterior ͑head͒ side ͓6͔.To see how the boundary conditions and domain shape affect the stationary patterns from an initially homogeneous state, we study a simplified version of a model for glycolisis as the specific reaction mechanism, which has been extensively studied numerically by Dillon et al. ͓7͔ in one dimension...

The rapid development in our understanding of the regulation of enzyme activity makes it a high priority to ascertain whether the behavior of purified enzymes reflects their functional characteristics in vivo. Enzyme concentration is usually the most significant difference between routine in vitro assays and in vivo conditions, as it is well known that many intracellular enzymes are present in vivo at much higher concentrations than used in vitro. Various procedures are suitable for kinetic analysis at physiological concentrations of enzyme. Those more frequently used have been cell per‐meabilization, the utilization of purified enzymes at concentrations close to the in vivo range, and the addition of polyethylene glycol to increase the local protein concentration. In this review we briefly summarize observations on enzymes reported to exhibit concentration‐dependent activity. The effect of enzyme concentration has been most thoroughly investigated in the case of phosphofructokinase. These studies may provide insight into the regulation of this important enzyme in the cell. The implications of both homologous and heterologous protein‐protein interactions for the effect of enzyme concentration and their roles in the control of enzyme activity in vivo are also discussed.—Aragon, J. J.; Sols, A. Regulation of enzyme activity in the cell: effect of enzyme concentration. FASEB J. 5: 2945‐2950; 1991.

Passive parity-time symmetry breaking transitions, where long-lived eigenmodes emerge in a locally dissipative system, have been extensively studied in recent years. Conventional wisdom says that they occur at exceptional points. Here we report the observation of multiple transitions showing the emergence of slowly decaying eigenmodes in a dissipative, Floquet electronic system with synthetic components. Remarkably, in our system, the modes emerge without exceptional points. Our setup uses an electrical oscillator inductively coupled to a dissipative oscillator, where the time-periodic inductive coupling and resistive-heating losses are independently controlled. With a Floquet dissipation, slowly-decaying eigenmodes emerge at vanishingly small dissipation strength in the weak coupling limit. With a moderate Floquet coupling, multiple instances of their emergence and disappearance are observed. With an asymmetric dimer model, we show that these transitions, driven by avoided-levelcrossing in purely dissipative systems, are generically present in static and Floquet domains.