In all complex organisms, the precise levels and timing of gene expression controls vital biological processes. In higher eukaryotes, including the fruit fly Drosophila melanogaster, the complex molecular control of transcription (the synthesis of RNA from DNA) and translation (the synthesis of proteins from RNA) events driving this gene expression are not fully understood. In particular, for Drosophila melanogaster, there is a plethora of experimental data, including quantitative measurements of both RNA and protein concentrations, but the precise mechanisms that control the dynamics of gene expression during early development and the processes which lead to steady-state levels of certain proteins remain elusive. This study analyzes a current mathematical modeling approach in an attempt to better understand the long-term behavior of gene regulation. The model is a modified reaction-diffusion equation which has been previously employed in predicting gene expression levels and studying the relative contributions of transcription and translation events to protein abundance [10,11,24]. Here, we use Matrix Algebra and Analysis techniques to study the stability of the gene expression system and analyze equilibria, using very general assumptions regarding the parameter values incorporated into the model. We prove that, given realistic biological parameter values, the system will result in a unique, stable equilibrium solution. Additionally, we give an example of this long-term behavior using the model alongside actual experimental data obtained from Drosophila embryos.

中文翻译：

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使用简单的反应扩散模型在果蝇早期胚胎中分析基因表达的稳定性。

在所有复杂的生物中，基因表达的精确水平和时间控制着至关重要的生物学过程。在包括果蝇果蝇（Drosophila melanogaster）在内的高级真核生物中，尚未完全理解驱动该基因表达的复杂的分子控制转录（由DNA合成RNA）和翻译（由RNA合成蛋白质）事件。特别是对于果蝇（Drosophila melanogaster），有大量的实验数据，包括对RNA和蛋白质浓度的定量测量，但是在早期发育过程中控制基因表达动态的精确机制以及导致稳态水平的过程。某些蛋白质仍然难以捉摸。这项研究分析了当前的数学建模方法，以期更好地了解基因调控的长期行为。该模型是修改后的反应扩散方程，以前已用于预测基因表达水平和研究转录和翻译事件对蛋白质丰度的相对贡献[10,11,24]。在这里，我们使用矩阵代数和分析技术来研究基因表达系统的稳定性，并使用关于纳入模型的参数值的非常一般的假设来分析平衡。我们证明，给定现实的生物学参数值，系统将产生一个独特的，稳定的平衡解决方案。另外，