Edwards R, Machina A, McGregor G, van den Driessche P
Department of Mathematics and Statistics, University of Victoria, STN CSC, PO Box 1700, Victoria, BC, V8W 2Y2, Canada,
Bull Math Biol. 2015 Jun;77(6):953-83. doi: 10.1007/s11538-015-0073-9. Epub 2015 Mar 11.
Qualitative models of gene regulatory networks have generally considered transcription factors to regulate directly the expression of other transcription factors, without any intermediate variables. In fact, gene expression always involves transcription, which produces mRNA molecules, followed by translation, which produces protein molecules, which can then act as transcription factors for other genes (in some cases after post-transcriptional modifications). Suppressing these multiple steps implicitly assumes that the qualitative behaviour does not depend on them. Here we explore a class of expanded models that explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. We mainly deal with regulation functions that are steep sigmoids or step functions, as is often done in protein-only models. We find that flow cannot be constrained to switching domains, though there can still be asymptotic approach to singular stationary points (fixed points in the vicinity of switching thresholds). This avoids the thorny issue of singular flow, but leads to somewhat more complicated possibilities for flow between threshold crossings. In the infinitely fast limit of either mRNA or protein rates, we find that solutions converge uniformly to solutions of the corresponding protein-only model on arbitrary finite time intervals. This leaves open the possibility that the limit system (with one type of variable infinitely fast) may have different asymptotic behaviour, and indeed, we find an example in which stability of a fixed point in the protein-only model is lost in the expanded model. Our results thus show that including mRNA as a variable may change the behaviour of solutions.
基因调控网络的定性模型通常认为转录因子直接调控其他转录因子的表达，而不涉及任何中间变量。事实上，基因表达总是涉及转录过程，转录产生信使核糖核酸（mRNA）分子，随后是翻译过程，翻译产生蛋白质分子，这些蛋白质分子随后可作为其他基因的转录因子（在某些情况下是在转录后修饰之后）。忽略这些多步骤过程隐含地假设了定性行为不依赖于它们。在此，我们探索一类扩展模型，该模型明确纳入了转录和翻译过程，并跟踪mRNA和蛋白质的浓度。我们主要处理像在仅蛋白质模型中常做的那样，为陡峭的S型函数或阶跃函数的调控函数。我们发现流不能被限制在切换域，尽管仍然可以存在向奇异驻点（切换阈值附近的不动点）的渐近趋近。这避免了奇异流这个棘手问题，但导致了阈值穿越之间的流有更复杂的可能性。在mRNA或蛋白质速率的无限快极限情况下，我们发现在任意有限时间间隔内，解一致收敛到相应仅蛋白质模型的解。这留下了极限系统（一种变量无限快）可能有不同渐近行为的可能性；实际上，我们发现了一个例子，其中仅蛋白质模型中一个不动点的稳定性在扩展模型中丧失。因此，我们的结果表明将mRNA作为一个变量纳入可能会改变解的行为。
