Bacterial Secretion and the Role of Diffusive and Subdiffusive First Passage Processes

<div><p>By funneling protein effectors through needle complexes located on the cellular membrane, bacteria are able to infect host cells during type III secretion events. The spatio-temporal mechanisms through which these events occur are however not fully understood, due in part to the inherent challenges in tracking single molecules moving within an intracellular medium. As a result, theoretical predictions of secretion times are still lacking. Here we provide a model that quantifies, depending on the transport characteristics within bacterial cytoplasm, the amount of time for a protein effector to reach either of the available needle complexes. Using parameters from <em>Shigella flexneri</em> we are able to test the role that translocators might have to activate the needle complexes and offer semi-quantitative explanations of recent experimental observations.</p> </div

Dynamics of 100 sub-diffusive effectors escaping from a sphere with radius = 0.5 m and targets on the boundary.

<p>Left panel: contour plot of the time (in seconds) until the first 50 random walkers have escaped from the sphere (see legend) as a function of the target radius and sub-diffusion coefficient . Right panel: average escape time (in seconds) of all 100 effectors (see legend) as a function of the target radius and sub-diffusion coefficient .</p

Isobars of the spatially averaged mean first passage time of a Brownian particle in a sphere with radius and targets on the boundary, as a function of target radius and diffusion coefficient .

<p>Its respective values (in seconds) along each isobar are shown by text labels. The parameter is set to in all plots; hence we assume that the particle can start its motion from anywhere in the domain. Left to right: , 42 and 92. Row (A): is fixed to 0.5 m. Row (B):  = 1.1 m.</p

A random trajectory, representing an effector protein's movement, confined to a disk

<p>(<b>A</b>) <b>or sphere</b> (<b>B</b>) <b>with radius .</b> The trajectory, assumed to be Brownian here, is shown by the blue trace. The 10 equidistant red circles in panel (A) and 12 red spheres in panel (B) are the target sites, representing the needle complex bases, whose centroids are placed on the boundary of the confining domain. We label their radius by the parameter .</p

Dependence of the GMFPT , expressed in seconds, as function of the number equidistant circular targets on the boundary with two different radius of the initial localization area : the extreme case when and when , where is the disk radius.

<p>Panels A and C represent the case, corresponding to an effector that starts at the origin, whereas panels B and D correspond to an initial particle localization being anywhere inside the bacterium. Each of the panels shows the GMFPT as function of for four choices of target radius  = 10, 50, 100 and 150 Å (line colour, see legend in panel A). The black curves represent a limiting case in which the entire boundary of the domain is a target; i.e. becomes the average time required to arrive at the boundary. In the top row we have considered the ‘fast’ model with m/s, whereas the bottom row displays results for the ‘slow’ model with m/s. In all four panels m.</p

Placement of targets

<p>(<b>red objects</b>) <b>on the boundary of a sphere.</b> In the top left corner of panel (A), a single target is displayed. In the top right and bottom left corner of panel (A) 42 and 92 targets are placed according to a geodesic grid. In the bottom right corner of panel (A), the geodesic grid for 25 targets is used but only to half of the sphere. The fourth panel shows 25 targets on the upper hemisphere – to study localized activation of needle complexes. (B) If we have no information about the starting location of a random walker in a disk, we assume that it is uniformly random within a smaller disk of radius . (C) The same idea applied to a random walker in a sphere; we assume that it can start anywhere within a smaller spherical volume of radius .</p

The MFPT function of a Brownian particle in a sphere with small spherical targets on its boundary

<p>(<b>not shown in the various plots</b>)<b>.</b> A visualization of the symmetric arrangements of the targets on the surface of the sphere has been sketched in the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0041421#s4" target="_blank">Materials and Methods</a> section. The MPFT values are plotted in colour code for each subfigure as function of the particle's starting position within the sphere. Starting positions are chosen in a smaller spherical region with radius for clarity. The top row contains values of of the ‘fast’ model in which  = 0.5 m, m/s and  = 150 Å with , 42 and 92 targets corresponding, respectively, to panel A, B and C. The bottom row shows the results for the ‘slow’ model in which  = 1.1 m, m/s and  = 15 Å with , 20 and 50, corresponding, respectively, to panel D, E and F.</p

Parameter ranges in our (sub)diffusive models.

<p>Parameter ranges in our (sub)diffusive models.</p

SK-channel blockade enhances CaM activation during LTP-inducing stimuli.

<p><i>(A)</i> SK-channel blockade enhances spine membrane depolarization, in particular during bAPs, which substantially increases activation of HVA L-type VGCCs (EPSP-2bAPs, third panel down). The additional Ca²⁺ -influx results in an order of magnitude increase in fully-activated CaCaM both globally and in channel nanodomains. The effect is not observed in the bAP-EPSP protocol since there is no SK-priming presynaptic stimulus before the postsynaptic spike. <i>(B)</i> Bar chart summarizing SK-blockade effect on peak global [CaCaM] levels across different stimuli. [CaCaM] enhancement by SK-blockade is greatest for compound stimuli with the form of STDP protocols which classically induce LTP.</p

Number of effectors left in the sphere as function of time from a bacterium of spherical shape with radius = 0.5 m.

<p>Panel A represents the case of 92 needle complexes distributed uniformly on the sphere. Panel B represents the case of 25 needle complexes distributed uniformly only in the upper hemisphere. In both cases the needle complexes have radius  = 50 Å. The solid black curves represent the secretion of diffusive effectors (I), the dash and dash-dotted lines represent the secretion of diffusive effectors with sequentially activated needle complexes (II), and the blue line represents the secretion of subdiffusive effectors (III).</p