Bacterial cells construct many structures, such as the flagellar hook and the type III secretion system (T3SS) injectisome, that aid in crucial physiological processes such as locomotion and pathogenesis. Both of these structures involve long extracellular channels, and the length of these channels must be highly regulated in order for these structures to perform their intended functions. There are two leading models for how length control is achieved in the flagellar hook and T3SS needle: the substrate switching model, in which the length is controlled by assembly of an inner rod, and the ruler model, in which a molecular ruler controls the length. Although there is qualitative experimental evidence to support both models, comparatively little has been done to quantitatively characterize these mechanisms or make detailed predictions that could be used to unambiguously test these mechanisms experimentally. In this work, we constructed a mathematical model of length control based on the ruler mechanism and found that the predictions of this model are consistent with experimental data—not just for the scaling of the average length with the ruler protein length, but also for the variance. Interestingly, we found that the ruler mechanism allows for the evolution of needles with large average lengths without the concomitant large increase in variance that occurs in the substrate switching mechanism. In addition to making further predictions that can be tested experimentally, these findings shed new light on the trade-offs that may have led to the evolution of different length control mechanisms in different bacterial species.

细菌细胞构建了许多结构，例如鞭毛钩和 III 型分泌系统 (T3SS) 注射体，有助于运动和发病机制等关键生理过程。这两种结构都涉及长的细胞外通道，并且必须高度调节这些通道的长度才能使这些结构发挥其预期功能。在鞭毛钩和 T3SS 针中如何实现长度控制有两种主要模型：底物切换模型，其中通过组装内杆来控制长度，以及尺子模型，其中分子尺控制长度. 尽管有定性的实验证据支持这两种模型，相对而言，很少有人对这些机制进行定量表征或做出可用于通过实验明确测试这些机制的详细预测。在这项工作中，我们构建了一个基于标尺机制的长度控制数学模型，发现该模型的预测与实验数据一致——不仅是平均长度与标尺蛋白长度的标度，而且对于方差。有趣的是，我们发现标尺机制允许进化具有大平均长度的针，而不会伴随基板切换机制中发生的方差大幅增加。除了做出可以通过实验测试的进一步预测之外，