Mathematical Modeling of Length Control in the Type III Secretion System

Abstract

Type III secretion secretion (T3SS) system is a protein export pathway that helps bacterial cells construct many structures, like the flagellar hook and the injectisome, that aid in crucial physiological processes such as locomotion and pathogenesis. Both, the flagellar hook and the injectisome, involve long extracellular channels and the length of these channels is highly regulated to allow these structures to perform their intended functions. Numerous experiments have been performed to understand the structural details of this nanomachine during the past decade. Despite the concerted efforts of molecular and structural biologists, several crucial aspects of the assembly of these structure, such as the regulation of the length of the needle and the flagellar hook, remain unclear. There are two leading models for how length control is achieved in the flagellar hook and T3SS needle: the substrate switching model, where the length is controlled by assembly of an inner rod, and the ruler model, in which a molecular ruler controls the length. While there is qualitative experimental evidence to support both models, there is a lack of detailed quantitative characterization of these models that could be used to unambiguously test these mechanisms experimentally. In this work, we used a combination of mathematical and computational techniques to better understand these length control mechanisms. Based on a set of straightforward assumptions, we constructed a mathematical model for length control based on the timing of substrate switching. Our model made predictions about commonly measured quantities such as the average needle lengths and the variance in lengths. In particular our model predicted for the substrate switching mechanism that the variance scales quadratically with the average length. Our model also predicted the form of the needle length distribution based on this mechanism, and found excellent agreement with available experimental data from Salmonella typhimurium with only a single free parameter. We also 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 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 lead to the evolution of different length control mechanisms in different bacterial species.