A mathematical model is developed for the folate cycle based on standard biochemical kinetics. We use the model to provide new insights into several different mechanisms of folate homeostasis. The model reproduces the known pool sizes of folate substrates and the fluxes through each of the loops of the folate cycle and has the qualitative behavior observed in a variety of experimental studies. Vitamin B 12 deficiency, modeled as a reduction in the V max of the methionine synthase reaction, results in a secondary folate deficiency via the accumulation of folate as 5-methyltetrahydrofolate (the "methyl trap"). One form of homeostasis is revealed by the fact that a 100-fold up-regulation of thymidylate synthase and dihydrofolate reductase (known to occur at the G 1 /S transition) dramatically increases pyrimidine production without affecting the other reactions of the folate cycle. The model also predicts that an almost total inhibition of dihydrofolate reductase is required to significantly inhibit the thymidylate synthase reaction, consistent with experimental and clinical studies on the effects of methotrexate. Sensitivity to variation in enzymatic parameters tends to be local in the cycle and inversely proportional to the number of reactions that interconvert two folate substrates. Another form of homeostasis is a consequence of the nonenzymatic binding of folate substrates to folate enzymes. Without folate binding, the velocities of the reactions decrease approximately linearly as total folate is decreased. In the presence of folate binding and allosteric inhibition, the velocities show a remarkable constancy as total folate is decreased.The folate cycle plays a central role in cell metabolism. Among its important functions are the delivery of one-carbon units to the methionine cycle, for use in methylation reactions, and the synthesis of pyrimidines and purines. Dietary folate deficiency or genetic polymorphisms in folate-metabolizing enzymes are associated with megaloblastic anemia, developmental abnormalities including neural tube defects and Down's syndrome, and various types of cancer, especially those of the gastrointestinal tract and leukemias (1-8). Elevated homocysteine concentrations, a biomarker of a low folate status, have been implicated in cardiovascular diseases and Alzheimer's disease (6, 9, 10). Furthermore, several enzymes in the cycle are the targets of cancer chemotherapeutic agents (11, 12). Because of its importance in human health, folate metabolism has long been the focus of clinical, nutritional, and biochemical investigations. Biochemical studies have provided extensive and detailed information about each of the enzymes and metabolites of the folate cycle. As a consequence, the components and the reaction diagram of the folate cycle are well understood (13).The structure of the folate cycle is relatively complex (Fig. 1) and consists of several interacting loops. Most of the reactions depend in a nonlinear way on the concentrations of their substrates. Therefore, the behavior of the f...

SUMMARYThe thermal convection in a layer of a third grade uid is investigated, with viscosity being a general function of temperature. We develop a non-linear stability analysis and prove that unconditional nonlinear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a uid of third grade are preferable to those for a uid of second grade or a dipolar uid.

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