School of Medicine, University of South Carolina, Columbia, SC, 29209, USA.
School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA.
Bull Math Biol. 2023 Mar 6;85(4):29. doi: 10.1007/s11538-023-01122-x.
The tryptophan (trp) operon in Escherichia coli codes for the proteins responsible for the synthesis of the amino acid tryptophan from chorismic acid, and has been one of the most well-studied gene networks since its discovery in the 1960s. The tryptophanase (tna) operon codes for proteins needed to transport and metabolize it. Both of these have been modeled individually with delay differential equations under the assumption of mass-action kinetics. Recent work has provided strong evidence for bistable behavior of the tna operon. The authors of Orozco-Gómez et al. (Sci Rep 9(1):5451, 2019) identified a medium range of tryptophan in which the system has two stable steady-states, and they reproduced these experimentally. In this paper, we will show how a Boolean model can capture this bistability. We will also develop and analyze a Boolean model of the trp operon. Finally, we will combine these two to create a single Boolean model of the transport, synthesis, and metabolism of tryptophan. In this amalgamated model, the bistability disappears, presumably reflecting the ability of the trp operon to produce tryptophan and drive the system toward homeostasis. All of these models have longer attractors that we call "artifacts of synchrony", which disappear in the asynchronous automata. This curiously matches the behavior of a recent Boolean model of the arabinose operon in E. coli, and we discuss some open-ended questions that arise along these lines.
色氨酸（trp）操纵子在大肠杆菌中编码负责从色氨酸酸合成氨基酸色氨酸的蛋白质，并且自 20 世纪 60 年代发现以来，一直是研究最深入的基因网络之一。色氨酸酶（tna）操纵子编码需要运输和代谢它的蛋白质。这两种蛋白质都已使用延迟微分方程在质量作用动力学假设下进行了单独建模。最近的工作为 tna 操纵子的双稳态行为提供了强有力的证据。Orozco-Gómez 等人的作者（Sci Rep 9(1):5451, 2019）确定了一个中等范围的色氨酸，其中系统有两个稳定的稳态，他们在实验中重现了这些。在本文中，我们将展示布尔模型如何捕获这种双稳态。我们还将开发和分析 trp 操纵子的布尔模型。最后，我们将这两个模型结合起来创建一个色氨酸运输、合成和代谢的单个布尔模型。在这个组合模型中，双稳态消失了，大概反映了 trp 操纵子产生色氨酸并使系统趋向于体内平衡的能力。所有这些模型都有更长的吸引子，我们称之为“同步的人工制品”，它们在异步自动机中消失。这与大肠杆菌中阿拉伯糖操纵子的最近布尔模型的行为非常匹配，我们讨论了一些沿着这些思路出现的未解决的问题。
