Robustness and the evolution of length control strategies in the T3SS and flagellar hook.

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.

Mathematical Model for Length Control by the Timing of Substrate Switching in the Type III Secretion System.

Type III Secretion Systems (T3SS) are complex bacterial structures that provide gram-negative pathogens with a unique virulence mechanism whereby they grow a needle-like structure in order to inject bacterial effector proteins into the cytoplasm of a host cell. 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 this structure, such as the regulation of the length of the needle itself, remain unclear. In this work, we used a combination of mathematical and computational techniques to better understand length control based on the timing of substrate switching, which is a possible mechanism for how bacteria ensure that the T3SS needles are neither too short nor too long. In particular, we 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. Although our findings provide preliminary evidence in support of the substrate switching model, they also make a set of quantitative predictions that, if tested experimentally, would assist in efforts to unambiguously characterize the regulatory mechanisms that control the growth of this crucial virulence factor.

Construction of a two-parameter empirical model of left ventricle wall motion using cardiac tagged magnetic resonance imaging data.

BACKGROUND: A one-parameter model was previously proposed to characterize the short axis motion of the LV wall at the mid-ventricle level. The single parameter of this model was associated with the radial contraction of myocardium, but more comprehensive model was needed to account for the rotation at the apex and base levels. The current study developed such model and demonstrated its merits and limitations with examples. MATERIALS AND METHODS: The hearts of five healthy individuals were visualized using cardiac tagged magnetic resonance imaging (tMRI) covering the contraction and relaxation phases. Based on the characteristics of the overall dynamics of the LV wall, its motion was represented by a combination of two components - radial and rotational. Each component was represented by a transformation matrix with a time-dependent variable α or ß.Image preprocessing step and model fitting algorithm were described and applied to estimate the temporal profiles of α and ß within a cardiac cycle at the apex, mid-ventricle and base levels. During this process, the tagged lines of the acquired images served as landmark reference for comparing against the model prediction of the motion. Qualitative and quantitative analyses were performed for testing the performance of the model and thus its validation. RESULTS: The α and ß estimates exhibited similarities in values and temporal trends once they were scaled by the radius of the epicardium (r(epi))and plotted against the time scaled by the period of the cardiac cycle (T(cardiac)) of each heart measured during the data acquisition. α/repi peaked at about Δt/T(cardiac)=0.4 and with values 0.34, 0.4 and 0.3 for the apex, mid-ventricle and base level, respectively. ß/r(epi) similarly maximized in amplitude at about Δt/T(cardiac)=0.4, but read 0.2 for the apex and - 0.08 for the base level. The difference indicated that the apex twisted more than the base. CONCLUSION: It is feasible to empirically model the spatial and temporal evolution of the LV wall motion using a two-parameter formulation in conjunction with tMRI-based visualization of the LV wall in the transverse planes of the apex, mid-ventricle and base. In healthy hearts, the analytical model will potentially allow deriving biomechanical entities, such as strain, strain rate or torsion, which are typically used as diagnostic, prognostic or predictive markers of cardiovascular diseases including diabetes.