Millions of people worldwide develop foodborne illnesses caused by Salmonella enterica (S. enterica) every year. The pathogenesis of S. enterica depends on flagella, which are appendages that the bacteria use to move through the environment. Interestingly, populations of genetically identical bacteria exhibit heterogeneity in the number of flagella. To understand this heterogeneity and the regulation of flagella quantity, we propose a mathematical model that connects the flagellar gene regulatory network to flagellar construction. A regulatory network involving more than 60 genes controls flagellar assembly. The most important member of the network is the master operon, flhDC, which encodes the FlhD4C2 protein. FlhD4C2 controls the construction of flagella by initiating the production of hook basal bodies (HBBs), protein structures that anchor the flagella to the bacterium. By connecting a model of FlhD4C2 regulation to a model of HBB construction, we investigate the roles of various feedback mechanisms. Analysis of our model suggests that a combination of regulatory mechanisms at the protein and transcriptional levels induce bistable FlhD4C2 levels and heterogeneous numbers of flagella. Also, the balance of regulatory mechanisms that become active following HBB construction is sufficient to provide a counting mechanism for controlling the total number of flagella produced.
Salmonella causes foodborne illnesses in millions of people worldwide each year. Flagella, which are appendages that the bacteria use to move through the environment, are a key factor in the infection process. Populations of genetically identical bacteria have been observed to contain both motile cells, generally with 6–10 flagella, and nonmotile cells, with no flagella. In this paper, we use mathematical models of the gene network that regulates flagellar construction to explore how the bacteria controls the number of flagella produced. We suggest that a bacterium must accumulate a threshold amount of a master regulator protein to initiate flagella production and failure to reach the threshold results in no flagella. Downstream mechanisms that impact the amount of master regulator protein are sufficient to determine how many flagella are produced.
Citation: Utsey K, Keener JP (2020) A mathematical model of flagellar gene regulation and construction in Salmonella enterica. PLoS Comput Biol 16(10): e1007689. https://doi.org/10.1371/journal.pcbi.1007689
Editor: Qing Nie, University of California Irvine, UNITED STATES
Received: January 24, 2020; Accepted: August 16, 2020; Published: October 22, 2020
Copyright: © 2020 Utsey, Keener. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: JPK was funded by USA National Science Foundation (https://www.nsf.gov) award DMS 1515130. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Millions of people worldwide develop foodborne illnesses caused by Salmonella enterica (S. enterica) every year . The pathogenesis of S. enterica depends on flagella, which are appendages that the bacteria use to move through the environment. Flagella are located peritrichously on the bacterium’s surface and generally number between 6 and 10 per cell . However, not all bacteria produce flagella. Partridge and Harshey describes the distributions of the number of flagella per bacterium under different growth conditions . It was observed under each growth condition that some bacteria did not produce any flagella, while the number of flagella on bacteria that produced them roughly follows a normal distribution, where the mean depended on the growth condition. This phenotypic heterogeneity enables the evasion of the host immune system during acute infection . It is not yet clear what mechanisms determine flagella heterogeneity and how the bacteria regulate the number of flagella produced.
Structurally, a flagellum is divided into three parts: basal body, hook, and filament. The basal body anchors the flagellum to the cell membrane, the hook is a flexible joint that links the rigid basal body with the filament, and the filament is a rigid helical structure which rotates to drive the cell forward. Assembly of functional flagella involves the temporally coordinated expression of more than 60 genes . In particular, these genes are divided into three classes: Class I, Class II, and Class III depending on the timing of their activation. The flhDC operon is the only member of Class I. This operon encodes the master regulator proteins FlhD and FlhC, which form a transcriptional activation complex FlhD4C2 . The FlhD4C2 complex is responsible for activation of Class II genes. Class II promoters control the expression of all the proteins required for the assembly of a functional hook-basal body (HBB) structure . The HBB includes the flagellar type III secretion system, which exports flagellar proteins from the cytoplasm through the growing structure during assembly . Once the HBB assembly is complete, FlhD4C2 activates expression of flgM, fliA, and other Class III genes. fliA encodes the flagella-specific sigma factor, σ28, which is essential for Class III promoter activation. flgM encodes an anti-sigma factor FlgM. Prior to HBB completion FlgM and σ28 form a complex that inhibits the activity of σ28. After the HBB is completed, the specificity of the secretion apparatus switches, so filament subunits and FlgM are secreted from the cell. σ28 facilitates the secretion of FlgM through the HBB, thereby acting as the FlgM type III secretion chaperone . The secretion of FlgM frees σ28 inside the cell, which leads to the activation of Class III genes. Class III genes control the assembly of the filament, the chemotaxis system, and motor proteins.
While the components of the gene network have been identified, understanding the additional details and dynamics of the system requires further study. In particular, genetically identical bacteria grown in the same environment exhibit bistable expression of multiple flagellar genes . Expression of Class II and Class III genes, flhB and fliC, respectively, is bimodal, and this bimodality is likely due to bistability upstream in the network [10–13]. Other factors also impact the flagella gene network, including nutrient availability, which enhances the expression of flagellar and motility genes . Nutrients repress RflP, formerly YdiV, expression through the protein CsrA . Low nutrient levels correspond to high RflP expression, and high nutrient levels correspond to low RflP expression. The tuning of RflP expression influences the flagellar gene network because RflP promotes the degradation of FlhD4C2 through the protease ClpXP .
A series of mathematical models have been proposed to explore the dynamics of the flagellar gene network [17–19]. While these models investigated the roles of some of the feedback mechanisms in the network, they fail to describe the heterogeneity observed in bacterial populations, since all bacteria are predicted to grow flagella. Further, the number of HBBs was modeled as a continuous variable, although HBBs are discrete entities.
Meanwhile, Koirala et al. presented a model that attempted to explain how nutrients tune flagellar gene expression dynamics in Salmonella enterica . Using experimental observations and a deterministic model, the authors demonstrated that environmental nutrient levels can regulate RflP expression and that this expression can lead to bistable expression of Class II and Class III genes in the bacteria . The model does not include HBB construction, or, consequently, the additional feedback mechanisms that are activated following HBB completion.
To investigate the heterogeneous flagella quantities within populations of clonal bacteria, we propose two mathematical models of the flagellar gene regulatory network. We first consider a sub-system of the network to focus on FlhD4C2 production and degradation. We use bifurcation analysis of an ordinary differential equation model to explore the role of flhDC and rflP expression rates on overall FlhD4C2 concentration. We find bistability in FlhD4C2 concentration as a function of both expression rates. We then extend the model to consider HBB construction and additional regulatory mechanisms. We retain the bistability observed in the sub-system model and investigate factors that impact the bistable regime. We also propose a mechanism for how S. enterica controls the number of flagella produced via FliT:FliD dynamics and Class III regulation of FliZ.
We first consider a model of the key components that regulate flhDC expression and FlhD4C2 protein concentrations. The model includes the FliZ-RflP-FlhD4C2 feedback loop and RflM activity. Expression of fliZ is up-regulated by FlhD4C2 and FliZ represses expression of rflP . RflP inhibits the activity of FlhD4C2 by binding to the FlhD subunit and targeting FlhD4C2 for proteolysis via ClpXP . Thus, FliZ is involved in a negative-negative feedback loop through RflP on the FlhD4C2 protein. FlhD4C2 up-regulates expression of rflM, and RflM represses the transcription of FlhD4C2 [21–23]. Fig 1 summarizes the mechanisms included in this model.
FlhD4C2 upregulates FliZ and RflM. The formation of the RflP:FlhD4C2 complex leads to FlhD4C2 proteolysis by ClpXP and RflP recycling. The production of RflM leads to repression of flhDC expression.
We assume that on the time scale of flagellar gene regulation and construction, the bacterial cytoplasm is well-mixed. The diffusion coefficients for proteins in the bacterial cytoplasm range from 1–10 μm2s−1, and since S. enterica are 2–5 microns long, proteins can diffuse across the bacterium on the order of seconds [24, 25]. Since HBBs are completed 30 minutes after induction of flhDC expression and nascent flagellar filaments become visible at 45 minutes, spatial non-uniformity of the proteins will not be significant .
We model transcription and translation processes as Michaelis-Menten functions and protein binding, unbinding, and degradation processes linearly. FlhD4C2 is produced at a basal rate of ρ1, and the production rate is divided by the concentration of RflM, scaled by η1. FlhD4C2 binds to RflP at rate κ1. FliZ is produced as a function of FlhD4C2 concentration. RflP is produced at a basal rate of ρ2, and the production is repressed by FliZ. External nutrient availability also represses rflP expression, so ρ2 is a proxy for nutrient availability . RflP forms a complex with FlhD4C2 and is recycled at rate δ1 where FlhD4C2 is degraded by ClpXP. We assume that the complex does not dissociate spontaneously. RflM is produced as a function of FlhD4C2 concentration. All components, including the RflP:FlhD4C2 complex, are assumed to degrade at a small, basal rate, γi, where i = 1, 2, …, 5. Table 1 lists the model parameters with descriptions and values.
Bifurcation analysis of sub-system model.
Since some bacteria produce flagella and others do not, we suspect that FlhD4C2 concentration is bistable. We use resultant analysis of the model to determine what parameter regimes support bistability . To apply resultant analysis, we take the model to steady state, reduce the system to a single polynomial in one variable, and perform a rescaling to reduce the number of parameters. We then solve the equation for a rescaled parameter of interest. To find double roots of the first derivative, we then take the resultant of the first and second derivatives of the right side of the equation. The curve in parameter space on which the first and second derivatives have a common root corresponds to the separation between monotonic and triphasic behavior. We first take the model to steady state and find Substituting into , we find
We reduce the number of parameters by taking the rescaling x = k1w and letting , , , , , and . We find We solve for P2 to find We define f(w) as We next use resultant analysis of f′(w) and f″(w) to determine when f(w) is monotone and when it is triphasic. The triphasic regions correspond to bistability, and the monotone regions correspond to monostability.
We first consider the case in which η1 = 0 (A2 = 0), which means that RlfM does not impact flhDC expression. f(w) simplifies to We next calculate the resultant of the numerators of the first and second derivatives of fA2=0(w) using Maple (2019; Maplesoft, Waterloo, ON), and we find where The function Rf,A2=0 = 0 separates parameter space into regions for which fA2=0(w) is monotone increasing (Rf,A2=0 < 0) and triphasic (Rf,A2=0 > 0). In the triphasic region, there is bistability, which will be verified later. The solid curve in Fig 2 shows the zero level contour for Rf,A2=0.
The solid curve indicates the A2 = 0 case, and the dashed curve indicates the A2 = 0.2 case. Rf < 0 below and to the left of the curve and Rf > 0 above and to the right of the curve.
Rf,A2=0 is negative to the left of the curve and is positive to the right of the curve. f(w) is monotone for values of P1 and A1 that fall to the left of the curve, whereas for values that fall to the right of the curve, f(w) is triphasic, and there is bistability. Thus, bistability is possible under constitutive expression of flhDC (i.e. when A2 = 0).
We note that in the case that D = 0, the resultant reduces to For positive values of A1 and P1, Rf,A2=0,D=0 is always negative, and either A1 = 0 or P1 = 0 satisfy Rf = 0. This indicates that we must have D > 0 to have bistability. In terms of the original parameters, this means that γ4 > 0 is required to have bistability.
We next consider the case in which η1 ≠ 0 (A2 ≠ 0), meaning RflM represses the expression of flhDC. We calculate the resultant of the numerators of the first and second derivatives of f(w), but it is too long to show here. The dashed curve in Fig 2 is the zero level contour for Rf when RflM activity is included. We see that the Rf = 0 curve moves up and to the right as A2 increases, which decreases the size of the parameter space in which there is bistability. In particular, the repressive activity of RflM requires an increase in the baseline production rates of FlhD4C2 (P1) and FliZ (A1) to have bistability.
We next use the parameters in Table 1 and XPPAUT (8.0) to visualize the bistable regime identified by the resultant analysis. Fig 3 shows the steady state concentration of FlhD4C2 plotted as a function of the parameters ρ1 and ρ2, demonstrating bistability for certain regions of ρ1, ρ2 parameters. Fig 3 also shows that the value of the FlhD4C2 production term, , is multivalued for the same regions of ρ1 and ρ2.
(A) Bistability is exhibited as ρ1 varies, where ρ2 = 30. (B) Bistability is also found as ρ2 varies and ρ1 = 45 is fixed.
Depending on the values of ρ1 and ρ2, the steady state value of FlhD4C2 concentration has one solution or three solutions. As shown in Fig 3A, the FlhD4C2 concentration is low for small ρ1 values. This means that the bacteria do not produce enough FlhD4C2 to initiate flagella production. With ρ1 increased sufficiently, a saddle node bifurcation occurs, and bistability arises. The lower solution represents the case in which the bacterium produces no flagella, and the upper solution represents the case in which the bacterium produces flagella. Increasing ρ1 further results in another saddle node bifurcation, and a return to a single steady state value for FlhD4C2 concentration. The expectation is that for high flhDC production rates, the bacterium will have a high FlhD4C2 concentration, and, consequently, produce flagella. Saddle node bifurcations are also observed for ρ2, as shown in Fig 3B. In this case, small ρ2 values result in a high steady state FlhD4C2 concentration, intermediate ρ2 values result in bistability, and large ρ2 values result in a low FlhD4C2 concentration.
Many factors control flhDC expression, so it is reasonable to expect that clonal bacteria may have different values for ρ1. In this case, the variation in ρ1 values could contribute to heterogeneity in flagella number, assuming the bacteria on the lower branch have no flagella and bacteria on the upper branch have enough FlhD4C2 to produce flagella. Also, since nutrient availability represses rflP expression, small variations in nutrient availability could result in differing values for ρ2. Variability in ρ2 would result in different concentrations of FlhD4C2, and, consequently, the presence or absence of flagella. We note that, as with the case A1 = 0, γ4 > 0 is required for bistability.
We also consider the η1 = 0 case, which corresponds to a knockout of RflM or constitutive expression of flhDC via an inducible promoter. Under this condition, the FlhD4C2 production term reduces to ρ1. Fig 4 shows the steady state concentration of FlhD4C2 is bistable for certain regions of ρ1 and ρ2, but the FlhD4C2 production term is not multivalued.
Bistability is exhibited for FlhD4C2 concentration, but not FlhD4C2 production, as ρ1 varies, where ρ2 = 30 (A), and as ρ2 varies and ρ1 = 49.5 (B).
In this case, constitutive expression of flhDC can lead to bistable FlhD4C2 levels. In particular, the η1 = 0 case shows bistability in FlhD4C2 even when the production of flhDC is not multivalued. Therefore, we suggest that RflM knockout cells should still exhibit bistability in the protein level of FlhD4C2.
In anticipation of a need to understand the role of FlhD4C2 degradation, we next investigate the impact of the basal degradation rate of FlhD4C2 on FlhD4C2 concentration. Fig 5A shows bistability in FlhD4C2 concentration as a function of the parameter γ1.
(A) Steady state solution for concentration of FlhD4C2, where ρ2 = 30. Bistability is seen as γ1 varies. (B) Two parameter bifurcation diagram. The region of parameters within the curve support bistable FlhD4C2 steady state concentrations, whereas parameter values above and below the curve result in a single steady state for FlhD4C2 concentration.
A saddle node bifurcation is observed for FlhD4C2 concentration as a function of γ1. Small γ1 values result in bistable steady state FlhD4C2 concentration, and increasing γ1 sufficiently results in a low FlhD4C2 concentration. We next investigate the impact of γ1 on the bistability observed in FlhD4C2 as a function of ρ2. Fig 5B shows a cusp bifurcation in the γ1-ρ2 plane. This demonstrates that the bistability previously observed in FlhD4C2 as a function of ρ2 is sensitive to γ1.
To explore how the bacteria regulate the total number of flagella constructed, we extend the model to consider the construction of HBBs and additional regulatory mechanisms that become active following HBB completion. In particular, FlhD4C2 up-regulates the fliDST operon, which encodes the FliT and FliD proteins. Before HBB completion, FliT and FliD are bound together, which inhibits the activity of FliT. Following HBB completion, FliD is secreted through the HBB and is assembled as the cap of the flagellum. The secretion of FliD frees FliT in the cell to bind to FlhD4C2 and target FlhD4C2 for proteolysis via ClpXP [27–29]. FliT is recycled during FlhD4C2 degradation . FlhD4C2 also up-regulates production of FlgM and σ28, which form a complex, and complete HBBs secrete FlgM, resulting in free σ28 in the bacterium. σ28 up-regulates the expression of Class III genes, including fliC. FliC is also secreted through the HBB to become the flagellar filament. σ28 also up-regulates fliZ, which means fliZ is under both Class II and Class III regulation. The gene network for the full model is shown in Fig 6.
Arrows indicate positive regulation or protein binding/unbinding processes. Blunt arrows indicate repression of gene expression.
For simplicity, we assume that FliT and FliD are produced as a complex (FliT:FliD), rather than modeling the production of each protein and formation of the FliT:FliD complex. Similarly, σ28 and FlgM are assumed to be produced as the complex σ28:FlgM.
We use the variables defined in the sub-system model and set v = [FliT], c2 = [FliT:FlhD4C2], c3 = [FliT:FliD], c4 = [σ28:FlgM], a = [σ28], and f = [FliC]. Referring to Fig 6, we construct the following model (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) As before, we assume that the system is well-mixed, and we model transcription and translation processes as Michaelis-Menten functions and protein binding, unbinding, and degradation processes linearly. J is the number of complete HBBs that are capable of secreting FlgM, FliD, and FliC, and we assume that the secretion of the three components, FlgM, FliD, and FliC, through the HBBs is competitive and saturating. We assume that these three components have the same secretion affinity, σ. As in the sub-system model, we assume that every compound, including the complexes, degrade at some basal rate.
The model involves both protein level dynamics, including binding and degradation, and transcription and translation. We assume that the protein level dynamics occur on a faster time scale than transcription and translation, so we take a quasi-steady-state approximation for the protein level dynamics. We assume that c1 = [RflP:FlhD4C2], y = [RflP], and c2 = [FliT:FlhD4C2] are fast equilibrating, so that they are at equilibrium. We therefore assume that meaning that We also assume that meaning that We now let u = v + c2, the sum of [FliT] and [FliT:FlhD4C2], so we have We then assume that meaning that Then we have so that Using the quasi-steady-state approximation, the model no longer includes equations for c1 = [RflP:FlhD4C2], y = [RflP], and c2 = [FliT:FlhD4C2]. The reduced model with competitive, saturating secretion, is (17) (18) (19) (20) (21) (22) (23) (24)
Since little is understood about HBB initiation and construction, we propose a simple model that assumes the environment within the bacteria is well-mixed. We assume that a general HBB protein is produced at a rate that depends on the concentration of FlhD4C2 in the bacterium. Mouslim and Hughes showed that flagella construction requires a threshold of flhDC expression, and, consequently, Class II expression . This finding suggests that the concentration of HBB proteins must reach some threshold before initiation of HBB construction. We assume that once the concentration of the HBB protein reaches a threshold, nucleation of a HBB occurs, and the free HBB protein concentration drops to zero. The newly nucleated HBB grows by recruiting free HBB proteins until it reaches a threshold level, at which point the HBB is considered complete [30, 31]. Complete HBBs do not recruit free HBB proteins but secrete FlgM, FliD, and FliC, which results in free σ28 and FliT within the bacterium. Depending on the HBB protein production rate and recruitment rate of incomplete HBBs, the bacteria are capable of growing multiple HBBs simultaneously. Let h be the concentration of free HBB proteins in the bacterium, and bi be the number of HBB proteins in the ith incomplete HBB. We then have (25) (26) where α5 is the production rate of free HBB proteins, k7 is the Michaelis constant for HBB protein expression, r is the recruitment rate of free HBB proteins to incomplete HBBs, kh is the Michaelis constant for HBB protein recruitment, γ12 is the degradation rate of free HBB proteins, and gi is zero or one, depending on whether the ith HBB is growing. K is the number of incomplete HBBs, K = ∑i gi. The nucleation and completion thresholds are hnuc and hmax, respectively.
We use all zero initial conditions, unless specified otherwise, and set K = 0 and J = 0. We then numerically integrate Eqs 17–26 until h reaches the nucleation threshold, hnuc, which indicates the first nucleation event. Using the state of the system just prior to nucleation and setting h → 0, b1 → hnuc, g1 → 1, K → 1, we numerically integrate the system again until h = hnuc or bi = hmax. If h = hnuc, another HBB is nucleated, meaning h → 0, bj → hnuc, gj → 1, K → K + 1, where the jth HBB is newly nucleated. If bi = hmax, we complete construction of the ith HBB, meaning bi → 0, gi → 0, K → K − 1, J → J + 1. The process continues until the system reaches steady-state.
The bistability observed in the sub-system model is maintained in the extended model. Depending on the initial conditions, HBBs are either produced or not. Since we do not explicitly model the construction of the filament, the number of HBBs is a proxy for the number of flagella. We first assume an initial condition of zero for all components of the gene network. Due to basal production of FlhD4C2, there is some expression of Class II genes (Fig 7A), but an insufficient concentration of FlhD4C2 is accumulated to initiate HBB production (Fig 7B).
(A) The protein concentration increases and reaches steady state. The protein concentrations are not large enough to initiate HBB production. (B) The tally of incomplete and complete HBBs remains at zero for all time.
In contrast, when we assume a small initial concentration of FliZ and FlhD4C2, FlhD4C2 accumulates sufficiently to initiate HBB production (Fig 8A), resulting in the production of seven HBBs (Fig 8B). This value falls within the experimentally observed range of 6–10 per bacterium .
(A) The protein concentrations increase until completion of the first HBB. The concentrations then decrease and reach a steady state that does not support further HBB nucleation. (B) Seven HBBs are produced.
The differing outcomes confirm that the FlhD4C2-FliZ-RflP feedback loop, as predicted by the sub-system model, determines if HBBs are produced. We then compute the separatrix in [FlhD4C2]-[FliZ] initial condition space to determine what initial conditions are necessary to stimulate HBB production (solid curve in Fig 9).
The solid curve indicates the ρ2 = 37.5 case and the dashed curve indicates the ρ2 = 45 case. Initial conditions to the right of the curve lead to HBB production, while initial conditions to the left of the curve do not.
We next consider the impact of nutrient availability on HBB production by changing the RflP production rate, ρ2. We use ρ2 as a proxy for environmental nutrient conditions in which increased nutrient availability decreases ρ2 and decreased nutrient availability increases ρ2. A 20% decrease in ρ2 results in the production of eight HBBs (Fig 10A), which is one more in comparison to the baseline case, whereas increasing ρ2 by 20% results in no HBB production (Fig 10B).
(A) Eight HBBs are completed when ρ2 = 30. (B) HBBs are not produced when ρ2 = 45.
Since changing ρ2 results in a different number of HBBs produced, we know that ρ2 impacts the separatrix between no HBB production and HBB production. Increasing ρ2 by 20% shifts the curve to require larger initial concentrations of FlhD4C2 and FliZ to produce HBBs (dashed curve in Fig 9). A 20% decrease in ρ2 lowers the curve out of the positive parameter quadrant, so that HBBs are produced under any biologically reasonable initial conditions.
We next investigate the impact of the FliT:FliD production rate, α3, on HBB production. Decreasing α3 results in more HBBs (Fig 11A), while increasing α3 results in fewer HBBs (Fig 11B), in comparison to the case with baseline α3.
(A) Eight HBBs are completed when α3 = 24. (B) Six HBBs are completed when α3 = 36.
The change in HBB production in response to changes in α3 suggests that the FliT:FliD concentration, and, consequently, the free FliT concentration, is a key factor in controlling the termination of HBB production. This is exemplified by setting α3 to zero (Fig 12). In this case, HBB construction continues indefinitely, which means the repressive activity of FliT on FlhD4C2 is required to stop HBB production.
(A) The protein concentrations increase indefinitely and FliT concentration remains zero. (B) The tally of incomplete and complete HBBs increases indefinitely.
To further consider the impact of FliT on FlhD4C2 concentration, we recall the sub-system model (Eqs 1–5). Even though FliT is not explicitly included in the model, increasing the basal rate of degradation of FlhD4C2, γ1, can be interpreted as a proxy for the repressive activity of FliT on FlhD4C2. As described earlier, Fig 5A shows that FlhD4C2 concentration is bistable as a function of γ1. This means that as HBBs are completed and free FliT levels increase, γ1 increases, which results in a saddle-node bifurcation and collapse of the system to a single, low steady state value. Similarly, Fig 5B shows that the bistability of FlhD4C2 concentration as a function of ρ2 is sensitive to γ1. As γ1 increases, the system moves outside the region in parameter space that supports bistability. These results suggest that there is a threshold level of free FliT required to shut-down HBB production.
Finally, we investigate the role of Class III regulation of FliZ. In particular, we set the Class III production rate of FliZ, β1, to zero. Without Class III regulation of FliZ, the FliZ concentration drops dramatically following the first HBB completion (Fig 13A), and, as a result, only five HBBs are produced (Fig 13B). The reduction in the number of HBBs produced, in comparison to the baseline case, is due to the lack of HBB nucleation following the first HBB completion event. Following HBB completion, FliT activity decreases the concentration of FlhD4C2. This slows the production of FliZ, and, without Class III regulation, the FliZ concentration drops dramatically. The drop in FliZ concentration further contributes to the decrease in FlhD4C2 concentration. FlhD4C2 then drops to a level that does not support further HBB nucleation.
(A) The protein concentrations increase until the first HBB is completed. The concentrations then rapidly decrease to a level that does not support HBB nucleation. (B) Five HBBs are constructed. No nucleation events occur after completion of the first HBB.
Together, these results suggest that the FlhD4C2-FliZ-RflP feedback loop determines whether flagella production will be initiated, and the balance of FliT:FliD dynamics and Class III regulation of FliZ control the total number of flagella produced.
S. enterica is a common foodborne pathogen that impacts millions of people each year, and the flagellum is an essential component of S. enterica pathogenesis. Motility is critical in enabling bacteria to reach the site of infection and establish disease. Therefore, understanding how S. enterica regulate and construct flagella is of interest for disease prevention and treatment. An interesting feature of flagella regulation in S. enterica is the presence of a heterogeneous quantity of flagella per bacterium in a clonal population .
Here, we developed mathematical models of the gene network that regulates flagella construction to improve our understanding of flagella heterogeneity. Analysis of a sub-system model (Eqs 1–5) of the network identified the possibility of bistable FlhD4C2 concentrations. The bistability was determined by the production rates of FlhD4C2 and RflP, and could explain why some bacteria within a clonal population grow flagella while others do not. This finding could be tested by measuring the expression of flhDC in a population of rflM knockout bacteria.
We then extended the sub-system model to include the construction of HBBs and additional regulatory mechanisms. Simulations of this model (Eqs 17–26) suggest that FliT activity and Class III regulation of FliZ provide a counting mechanism by which the bacteria can control how many flagella are produced. Class III production of FliZ prevents underproduction of flagella, and FliT prevents overproduction. Hence, FlhD4C2 is regulated by multiple negative feedback loops to ensure that protein levels in the bacteria can be tightly controlled, and both under- and overproduction of flagella is avoided.
The flhDC operon is primarily transcribed from two promoters, P1flhDC and P5flhDC, and many additional factors are shown to positively and negatively regulate flhDC expression, including HilD, RcsDBC, RtsB, and LrhA [21, 32–34]. Our model includes the main components involved in the expression of flhDC from P1flhDC. Expression from the P5flhDC, which is regulated via a FliZ-HilD positive feedback loop , plays a role in virulence rather than motility. Therefore, we focus on expression from P1flhDC. Future models could include expression from P5flhDC and additional regulatory mechanisms.
The limited availability of data meant that parameter values were selected manually, rather than measured or estimated. This study investigated the possible behaviors of the gene network, and, thus, does not rely on accurate parameter estimates. As additional data becomes available, the parameter values could be updated. Further, due to a limited understanding of HBB construction, we have proposed a simple model of the process. As we learn more about HBB construction, the model could also be updated.
Our results explain the bimodality of flagella count data and how the bacteria regulate the total number of flagella produced. However, since the model is deterministic, it cannot capture the spread of 6–10 flagella in the count data. Future models could incorporate sources of stochasticity, such as cell division, to understand the variability in the number of flagella. Since the system is assumed to be well-mixed but HBBs are discrete entities, cell division could be modeled by equally dividing the proteins in the parent bacterium and randomly splitting the HBBs between the two daughter cells. The randomness of HBB inheritance could be sufficient to generate the observed flagella distributions.
This paper has proposed two mathematical models of the flagellar gene network in S. enterica. We predict that the FlhD4C2-FliZ-RflP feedback loop determines whether flagella are produced, and the combination of FliT activity and Class III regulation of FliZ determines the total number of flagella produced.
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