Temperature Control of Fimbriation Circuit Switch in Uropathogenic Escherichia coli: Quantitative Analysis via Automated Model Abstraction

Uropathogenic Escherichia coli (UPEC) represent the predominant cause of urinary tract infections (UTIs). A key UPEC molecular virulence mechanism is type 1 fimbriae, whose expression is controlled by the orientation of an invertible chromosomal DNA element—the fim switch. Temperature has been shown to act as a major regulator of fim switching behavior and is overall an important indicator as well as functional feature of many urologic diseases, including UPEC host-pathogen interaction dynamics. Given this panoptic physiological role of temperature during UTI progression and notable empirical challenges to its direct in vivo studies, in silico modeling of corresponding biochemical and biophysical mechanisms essential to UPEC pathogenicity may significantly aid our understanding of the underlying disease processes. However, rigorous computational analysis of biological systems, such as fim switch temperature control circuit, has hereto presented a notoriously demanding problem due to both the substantial complexity of the gene regulatory networks involved as well as their often characteristically discrete and stochastic dynamics. To address these issues, we have developed an approach that enables automated multiscale abstraction of biological system descriptions based on reaction kinetics. Implemented as a computational tool, this method has allowed us to efficiently analyze the modular organization and behavior of the E. coli fimbriation switch circuit at different temperature settings, thus facilitating new insights into this mode of UPEC molecular virulence regulation. In particular, our results suggest that, with respect to its role in shutting down fimbriae expression, the primary function of FimB recombinase may be to effect a controlled down-regulation (rather than increase) of the ON-to-OFF fim switching rate via temperature-dependent suppression of competing dynamics mediated by recombinase FimE. Our computational analysis further implies that this down-regulation mechanism could be particularly significant inside the host environment, thus potentially contributing further understanding toward the development of novel therapeutic approaches to UPEC-caused UTIs.


Modeling of Temperature-dependent FimB and FimE Regulation
The basic reaction-level subnetwork of FimB and FimE regulation is given in Figure 3. The main mode of temperature control in this process is enabled via a small protein H-NS, which represses the expression of both fimB and fimE by occupying DNA regions containing fimB and fimE promoters and preventing RNA polymerase (RNAP) from binding [2]. It has been reported that the hns gene is auto-regulated with [H-NS] generally remaining constant, except during certain specific conditions, such as cold shock [3]. Importantly, however, H-NS DNA binding affinity is controlled by the ambient temperature and, consequently, so is the production of FimB and FimE [3,4]. Furthermore, this temperature-dependent transcriptional regulation of fimB and fimE by H-NS is effected differentially. That is, it has been observed that the expression of fimB increases nearly two-fold (119 vs. 195 Miller units) as temperature increases from 30 • C to 37 • C, while the expression of fimE decreases about four-fold (226 vs. 61 Miller units) under the same conditions [2].
H-NS rate constants for binding and unbinding to P B (i.e., k 2 and k −2 ) were inferred on the bases of previously reported experimental results as follows. First, we noted that in the case of chromosomal DNA binding: [P * ] = 1. That is, for a given level of bound promoter [P * -H-NS] (which, in turn, sets the level of downstream switching), K D ∼ [H-NS]. So, we next used the empirical temperature-dependent, but non-specific K ns D values for binding of H-NS to DNA at 28 • C, 37 • C, and 42 • C [4] to estimate the amount of [H-NS] ns that would be required in this case to generate switching probabilities consistent with those measured experimentally [5]. Finally, we deduced the correct (specific) K D values by multiplying K ns D by the ratio [H-NS]/[H-NS] ns , where the true physiological [H-NS] levels were identified as: 20, 000 molecules at 37 • C (as reported in [6] for E. coli in exponentially growing phase); 30, 000 at 28 • C (based on the observations in [7] that levels of [H-NS] held steady between 23 • C and 30 • C at about 1.5 of those observed at 37 • C); and 18, 000 at 45 • C (by interpolating results in [8] on the relative protein expression during heat-shock induction to 50 • C). The H-NS rate constants at P E (i.e., k 4 and k −4 ) are inferred analogously, though-unlike the fimB case-the negative modulation by H-NS is increased at higher temperature. These K D values of H-NS binding to P B and P E are shown in Table S1. Note that the so obtained values of disassociation constants K D for specific H-NS binding at P * are indeed significantly lower than those reported for non-specific DNA interactions [2] as might otherwise be expected from general considerations. The binding rate constants are derived from these K D values by assuming a rapid unbinding rate and by setting the unbinding rate constant to 10s −1 . This rapid unbinding rate constant is set by the fastest timescale in the system in order to capture the fast adaptation of recombinases to temperature perturbation, which may be crucial to rapidly adjusting the fimbriation levels in response to ambient change [5,9,10].
In order to estimate the rate constants for the H-NS bindings to promoter regions of the two recombinases at other temperature points, we have performed an exponential curve fitting based on K D values at the three temperature points shown in Table S1. The results are shown in Figure S1. These K D values are utilized in the same way as the first three temperature points to infer the values of k 2 and k 4 at the other 7 temperature points. At 28 • C, fimE is estimated to produce 200 proteins in one cell generation, while at 37 • C, it produces 61 proteins. These numbers are chosen to be comparable with the ratio of the fimE expression data at 30 • C and 37 • C from [2]. To reduce the FimE production even further at 42 • C, fimE is assumed to produce 25 proteins in one cell generation. A complete summary of reaction rate constants involved in the FimE-FimB regulation network is provided in Tables 4 and 5.
In order to further evaluate the robustness and fidelity of described estimates for H-NS binding at the two promoter sites, P B and P E , we have used the abstracted model to perform a sensitivity analysis of the ON-to-OFF switching frequency (the main objective variable in the problem) with respect to variations of these parameters and across a range of temperature points ( Figure S2). The results show that combinations of up to ±20% perturbations in both P E -H-NS binding and P B -H-NS binding constants have limited effects on the predicted levels of total ON-to-OFF switching probability. As our results are also consistent with those observed empirically (Table 2), this supports a conclusion that our estimates of binding rate constants are indeed robust against small perturbations as well as faithful to the underlying process dynamics.
The initial concentration of RNAP is taken to be 30nM, which has been previously established as the physiologically available amount of these holoenzymes in E. coli grown in minimal medium, and is the same as the level successfully used in previous work analyzing phage λ developmental decision pathway model [11,12] The initial concentration of H-NS as well as the RNAP binding and unbinding rate constants for both promoter sites (i.e., k 1 , k −1 , k 3 , and k −3 ) are derived by assuming 50 percent occupancy of H-NS and 25 percent occupancy of RNAP at P B at 37 • C. This configuration is found to be effective for modeling the thermoregulation of fimB and fimE expression by H-NS.
As reported in [2], the ratio of FimB levels in hns + versus hns − mutants at 37 • C is approximately 2. The value of k 5 was derived by matching this ratio, given that FimB is produced around 200 times in one cell generation at 37 • C. The production rate constant of FimE (i.e., k 6 ) is chosen to be the same as that of FimB.
The value of the degradation rate constant of FimB (i.e., k d1 ) is chosen so that its production and degradation reactions equilibrate when the concentration of FimB is 100nM at 37 • C. This number is chosen as the best fit from the range of 1 -100nM thought to be a reasonable value for [FimB] and [FimE] [13]. The degradation rate constant of FimE (i.e., k d2 ) is then taken to be the same as that of FimB.
Finally, the average initial concentrations of the two recombinases in the ON state (i.e. before we begin monitoring the fimS switch shutdown rate) are determined by first starting in a state without any recombinase activity and running an ODE simulation of the [FimB] and [FimE] regulation model for two cell generations. The concentrations of the two recombinases are then retrieved at the end of Table S1. Temperature-dependent K D for H-NS binding to P B and P E derived from reported data.

Modeling of the fimS Configuration Dynamics
In order to analyze the fimbriation shutdown model, we examine the ON-to-OFF switching dynamics, Figure 4, through 6 possible transition states (out of the 18 available fimS configurations), Table 6. Their equilibrium thermodynamics characteristics are based on those given in Wolf & Arkin [13]. In our fim switch configuration model, the fimS binding and unbinding rate constants are estimated from the standard free energy relationship, ∆G = −RT ln (k f /k r ) using a rapid unbinding rate constant of 1.0s −1 . This unbinding rate constant is chosen so that it is an order of magnitude smaller than that in the FimB and FimE regulation for relatively fast adaptation of the two recombinases to temperature perturbation. Since only states 3-8-where IHF and either recombinase species are bound to the switch DNA regionare configured to invert the fim switch from ON to OFF, the values of k p are set to 0 for states 1-2, and 9-18, while the values of k p for states 3-8 are derived using our qualitative knowledge on the switching regulation, and chosen so that results from our detailed model fit those observed empirically. For example, since the switching rates are faster when Lrp occupies Lrp-I and/or Lrp-II, but not Lrp-III, the values of k p5 is chosen to be much greater than those of k p3 and k p6 . Also, in our model, the initial concentration of IHF is set to 10nM to match the ON-to-OFF frequency from the experimental observation at 37 • C [5]. The initial concentration of Lrp is modeled as an increasing function of temperature to indirectly capture the upregulation of lrp expression at higher temperatures owing to the reduction in H-NS-based thermorepression [4,13,14]. In our model, the Lrp DNA-binding configurations are simplified to be in one of three states: (1) no Lrp is bound; (2) 2 molecules of Lrp are bound; and (3) 3 molecules of Lrp are bound. The concentration of Lrp is quantified for each temperature setting based on the tuning mechanism of Lrp as illustrated in Figure S3(a). The concentration of Lrp at 37 • C is chosen to be 5nM as this value is determined to be the physiologic concentration of free Lrp in the cell at 37 • C in [13]. The concentration of Lrp at the other two temperature settings is set so that it qualitatively agrees with observations of the temperature tuning mechanism in [13]. At 28 • C, [Lrp] 0 is set to 2nM, which also serves as the lower bound on [Lrp] at lower temperatures, so that Lrp molecules are unlikely to occupy Lrp-I and Lrp-II, and moreover to prevent Lrp molecules from binding to Lrp-III, while at 42 • C [Lrp] 0 is set to 20nM, so that Lrp molecules are likely to occupy all three Lrp binding sites. Using the concentration of Lrp at these three temperature points, the concentration of Lrp at temperature points higher than 28 • C is obtained via exponential curve fitting as shown in Figure S3(b). A summary of the resulting [Lrp] values at various temperature points is given in Table 7.    Figure S3. Temperature tuning mechanism of Lrp. (a) At very low concentration Lrp is unlikely to occupy the Lrp binding sites and the fim switching rate is low [13]. When the concentration is around 5nM, Lrp tends to occupy Lrp-1 and/or Lrp-2, but not Lrp-3-the configuration which activates fimS switching. As the concentration of Lrp increases even further, Lrp is likely to occupy Lrp-3 as well as Lrp-1 and Lrp-2, which inhibits switching [15]. (b) Estimation of [Lrp] 0 at temperature points higher than 28 • C using exponential curve-fitting (with the three temperature data points at 28 • C, 37 • C, and 42 • C as input).