In an earlier paper (Cohen and Bergman, Am. J. Physiol. 268 (1995) E397), we explored the relationship between the exponents in the exponential curve fit to isotopic enrichment versus time and the fractional turnover rate of the largest metabolic pool in the pathway. Here we present the analysis on a more rigorous footing and apply it to questions of cerebral and cardiac metabolism. Our emphasis in this paper is to describe and justify mathematically an approach for analysis of metabolic dynamics, not with the intention of replacing the use of numerical software for estimation of flux rates but for giving the scientist the opportunity to examine the system in an approximate manner, and thereby to check not only that the results of the numerical solution are the correct solutions to the equations but also that the equations portray the correct simplification of the metabolic pathway. We introduce the "dominant rate constant" as a tool for deriving algebraic formulas relating rates of metabolic flux, sizes of metabolic pools, and the dynamics of isotopic enrichment. Illustrations of such algebraic formulas are provided for the rates of the citric acid cycle (CAC), glycolysis and glutamine synthesis in brain, as well as the rate of the CAC in heart. In addition, we prove that formulas for estimation of rates of glycolysis and of the CAC depend critically on the fractional turnover rates of lactate and glutamate, respectively. The justification for analysis of simulated data is that we are studying the effects of simplifications of metabolic models on the accuracy of estimation of metabolic pathways. Our use of the dominant rate constant is an analytical convenience that allows us to assess proposed simplifications of metabolic pathways.

中文翻译：

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从体内主要速率常数估算代谢通量：应用于脑和心脏。

在较早的论文中（Cohen和Bergman，Am。J. Physiol。268（1995）E397），我们探索了适合于同位素富集的指数曲线中的指数与时间之间的关系，以及最大代谢池中的总代谢率途径。在这里，我们提出了一个更为严格的基础分析，并将其应用于脑和心脏代谢的问题。我们在本文中的重点是描述和数学证明一种用于分析代谢动力学的方法，其目的不是要取代使用数字软件来估算通量率，而是要让科学家有机会以近似方式检查系统，因此，不仅要检查数值解的结果是否是方程的正确解，还要检​​查方程是否描绘了代谢途径的正确简化。我们引入“主导速率常数”作为导出与代谢通量的速率，代谢池大小和同位素富集动力学有关的代数公式的工具。提供了此类代数公式的示例，说明了大脑中柠檬酸循环（CAC），糖酵解和谷氨酰胺合成的速率，以及心脏中CAC的速率。此外，我们证明了用于估计糖酵解速率和CAC的公式分别主要取决于乳酸和谷氨酸的分数周转率。分析模拟数据的理由是，我们正在研究简化代谢模型对代谢途径估计准确性的影响。我们使用显性速率常数是一种分析上的便利，使我们能够评估代谢途径的拟议简化。