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.

L is the needle length in terms of number of O proteins. n s is the number of I proteins needed for a inner rod to assemble.
The probability that a given needle lives at time t is given by e −λ B t e −β I It (β I I) ns t ns−1 (n s − 1)! + λ B e −λ B t e −β I It The probability that needle has length L given that it lives at time t is given by Using Equations (1) and (2) we can calculate the probability the a needle has lengh L: and n s and L are integers. Note that the normalization condition of P (L) is good, where and for the summation over L we have used For 1, therefore, L increases linearly with (β O O/β I I) and the slope can be used to determine n experimentally.
In this case, L is independent of n and β I I.

Calculation of the variance
To calculate the standard deviation of L, we notice that From Eq. (7), Therefore and Substituting Eq. (8) into Eq. (17) and keeping only the zeroth-order term of yields Substituting Eq. (8) into Eq. (17) and keeping only the lowest order term of 1/ yields 1.4 Distribution of the Needle Length at 1 Let From Eqs. (9) and (18), where 2 Details of the stochastic simulation We treated the bases as individual, discrete "agents" in our simulations. There are two integers associated with each base; the first represents the number of inner rod proteins, and can take values from 0 to n s . The second represents the number of needle proteins associated with that particular base, and can take any positive integer value. We maintained two separate populations of bases; the set of "immature" bases, with less than n s inner rod proteins (call this set B), and "mature" bases, with exactly n s inner rod proteins (call this Only immature bases can participate in binding reactions with inner rod or needle proteins, and when any given base binds to its "last" inner-rod protein (i.e. binds to an inner-rod protein and undergoes the transition from n s −1 to n s inner rod proteins), that base becomes mature and is moved from the immature to mature pool.
A binding event in our model always results in an increase in the needle (or inner-rod) protein number associated with the base. As a result, we ignore a number of possible scenarios that might occur during needle assembly. For instance, a needle protein might dissociate from the needle; since the needle itself is a helix, the only protein that is likely to unbind is the one at the very tip. Alternatively, a needle protein might be exported by the base but simply never attach to the growing needle. Both of these scenarios represent "unsuccessful export," in that the export event would reduce the number have no impact on the overall behavior of the system. As such, we neglect unsuccessful export events without a loss of generality.
Unsuccessful export could also lead to the accumulation of needle protein monomers in the extracellular space, which could bind to needles and extend them through a mechanism other than export. In our model, however, we assume that the extracellular volume is much larger than the intracellular volume: as a result, re-binding events are likely to be very rare and are also neglected. In any case, re-binding would simply change the relative values of β O and β O , so consideration of this effect would also not impact our results.
We implemented this simulation using the standard "Gillespie-Doob" approach for exact simulation of stochastic chemical kinetics (1). In this case, the "propensity" or "activity" of any given reaction was calculated according to the functions over the arrows in Fig. ? Table 1.