Role of Autoregulation and Relative Synthesis of Operon Partners in Alternative Sigma Factor Networks

Despite the central role of alternative sigma factors in bacterial stress response and virulence their regulation remains incompletely understood. Here we investigate one of the best-studied examples of alternative sigma factors: the σB network that controls the general stress response of Bacillus subtilis to uncover widely relevant general design principles that describe the structure-function relationship of alternative sigma factor regulatory networks. We show that the relative stoichiometry of the synthesis rates of σB, its anti-sigma factor RsbW and the anti-anti-sigma factor RsbV plays a critical role in shaping the network behavior by forcing the σB network to function as an ultrasensitive negative feedback loop. We further demonstrate how this negative feedback regulation insulates alternative sigma factor activity from competition with the housekeeping sigma factor for RNA polymerase and allows multiple stress sigma factors to function simultaneously with little competitive interference.


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Despite the central role of alternative sigma factors in bacterial stress response and 22 virulence their regulation remains incompletely understood. Here we investigate one of 23 the best-studied examples of alternative sigma factors: the σ B network that controls the 24 general stress response of Bacillus subtilis to uncover widely relevant general design 25 principles that describe the structure-function relationship of alternative sigma factor 26 regulatory networks. We show that the relative stoichiometry of the synthesis rates of 27 σ B , its anti-sigma factor RsbW and the anti-anti-sigma factor RsbV plays a critical role in 28

Introduction
To address these issues we develop a detailed mathematical model of the σ B 74 network and examine its dynamics to understand the mechanistic principles underlying 75 the pulsatile response. By decoupling the post-translational and transcriptional 76 components of the network we show that an ultrasensitive negative feedback between 77 the two is the basis for σ B pulsing. Moreover we find that the relative synthesis rates of 78 σ B and its operon partners RsbW and RsbV, plays a critical role in determining the 79 nature of the σ B response. We also use our model, together with previously published 80 experimental data from [13,14], to explain how the σ B network is able to encode the rate  We further develop this model to investigate how the network functions in the 95 context of other σ-factors. As in many other bacteria, σ B is one of the many σ-factors 96 that complex with RNA-polymerase core that is present in limited amounts [3,16]. 97 Therefore, when induced these alternative σ-factors compete with one another and the 98 housekeeping σ-factor σ A for RNA polymerase. We use our model to investigate how 99 the design of this network enables it to function even in the presence of competition 100 from σ A which has a significantly higher affinity for RNA polymerase [17]. Lastly, we 101 investigate how multiple alternative σ-factors compete when cells are exposed to 102 multiple stresses simultaneously. Using our model we identify design features that are 103 ubiquitous in stress σ-factor regulation and critical to bacterial survival under diverse 104 types of stresses.

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Biochemically accurate model of σ B pulsing 107 In a recent study, Locke et. al. [13] demonstrated that a step-increase in energy stress 108 results in pulsatile activation of σ B . The study also proposed a minimal mathematical  To understand the σ B network response we built on our earlier study [15] to 121 develop a detailed mathematical model that explicitly includes all known molecular 122 interactions in the network. Note that we made one significant change to the model 123 discussed in [15]. The model in [15] assumed that the synthesis rates for σ B and its 124 operon partners (RsbW and RsbV) follow the stoichiometry of their binding ratios (i.e. RsbW and RsbV concentrations respectively). However experimental measurements 127 have shown that σ B , RsbW and RsbV are produced in non-stoichiometric ratios [18]. 128 Accordingly, in contrast to our earlier study, we assumed σ B , RsbW and RsbV can be 129 produced in non-stoichiometric ratios and studied how changes in relative synthesis 130 rates of σ B operon partners affect the response of the σ B network to step-increases in 131 energy stress phosphatase levels. We note that RsbX, a negative regulator of RsbTU 132 phosphatase [20], is not included in our model. RsbX was excluded for simplicity since it 133 is not essential for the pulsatile response of the σ B network [14]. 134 Simulations of this detailed model showed that different combinations of thereby completing the σ B pulse.

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The same analysis can be applied for different values of relative synthesis rates,

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The role of negative feedback in producing a pulsatile response also explains 252 why pulsing does not occur in strains where σ B operon is transcribed constitutively [13].

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In this case, the σ B network lacks the negative feedback necessary to produce a insensitive to BT (Fig. S3BC). Thus, the full system lacks the negative feedback and as a 266 result σ B does not pulse. Using our analytical approximation we found that this 267 phosphatase threshold is proportional to the basal level of RsbW kinase synthesis rate 268 and the ratio of the kinase and phosphatase catalytic rate constants (Fig. S3DE).

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Increase in the basal σ B operon expression rate increases the phosphatase threshold. proportionally changed the operon transcription rate to ensure that the total 408 concentrations of σ B , RsbW and RsbV are kept fixed. We found that indeed pulse 409 amplitude decreases with increase in degradation/dilution rate (Fig. 4CD). Our 410 simulations showed that Kramp, the half-maximal constant for the dependence of pulse 411 amplitude on ramp duration, was indeed sensitive to the degradation rate (Fig. 4EF).

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This suggests that the timescale separation between the post-translational and 413 transcriptional responses is the basis of ramp rate encoding into pulse amplitude.

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The design of the σ B network enables it to compete with σ A for RNA polymerase 416 The results thus far indicate that σ B network functions in the effectively negative 417 feedback regime where increase in the operon expression decreases σ B activity.

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Negative feedback loops have been shown to increase the robustness of the system to 419 perturbations. We therefore decided to investigate how the σ B network design affects its 420 performance when it faces competition for RNA polymerase from other σ-factors, e.g. To further illustrate the importance of the negative feedback in insulating the 462 network, we compared the response of the wildtype network to an "in silico" mutant 463 network wherein the σ B operon is constitutive rather than σ B dependent (Fig. 5A). 464 Consequently this network lacks any feedback between free σ B and total σ B . Our 465 simulations (Fig. 5B, right panel) show that the free σ B concentration of the no-466 feedback-network does not show adaptive pulsing and therefore σ B concentration 467 fluctuates along with the phosphatase levels. Increase in σ A did not affect this response.

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This is expected since in the absence of feedback σ A only affects the expression of σ B 469 targets in this network (Fig. 5A, right panel). Without an increase in free σ B (Fig. 5D), the 470 increased competition for RNApol at higher σ A reduced the σ B target promoter activity 471 (Fig. 5CE). Similarly a positive feedback network design is also incapable of increasing 472 free σ B in response to an increase in σ A (Fig. S5CDE). Thus fluctuations in σ A can 473 interfere with the σ B stress-response of these alternative network designs. In contrast, 474 the wildtype σ B network with its ultrasensitive negative feedback design can 475 compensate for competition effects (Fig. 5DE).

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We used this simple model to study the response when cells are simultaneously 515 exposed to multiple stresses creating competition for RNApol. For these simulations we 516 fixed σ A levels and studied how activation signals for one alternative σ-factor affects the 517 activity of another. As before (Fig. S5AB), increased availability of one stress σ-factor 518 leads to a competition for RNA polymerase and as a result reduces the activity of 519 another stress σ-factor (Fig. S6EF). However, when negative feedback loops are 520 present, surprisingly, increasing the stress signal for one σ-factor did not lead to any 521 significant change in the activity of another σ-factor. For example, increasing stress 522 signaling protein PB while keeping PW fixed leads to an increase in free σ B but also 523 results in a small increase in free σ W (Fig. 6C). This response can be explained by the  Thus the two stress σ-factors are able to function simultaneously despite the 537 scarcity of RNApol. The mechanism minimizing competition between stress σ-factors 538 becomes clearer when we track the changes in σ-RNApol complexes as a function of 539 the stress signaling protein PB. As PB increases, more free-σ B becomes available and binds to RNApol (Fig. 6G). However this RNApol must be accounted for by the RNApol 541 lost by the other operating σ W and σ A factors. Comparing the contributions of each σ-542 factor shows that despite the fact that σ A has a much higher affinity for RNApol, most of 543 the RNApol in the σ B -RNApol complex is drawn from the σ A -RNApol pool rather than 544 σ W -RNApol pool (Fig. 6G). Thus the negative feedback design allows stress σ-factors to σ-factor σ A competing with each other for RNA polymerase. σ B and σ W activities are regulated 553 by negative and positive feedbacks in (A) and (B) respectively. In both cases, signaling proteins 554 P B and P W control the stress-signal driven activation of σ B and σ W respectively. C, D. 555 Dependence of free σ B and σ W levels on P B at fixed P W (= 2µM). In the wildtype negative 556 feedback system (C), increase in σ B phosphatase leads to an increase in both free σ B (green 557 curve) and free σ W (red curve). In the positive feedback system (D), increase in σ B phosphatase 558 leads to an increase in free σ B (green curve) and a decrease in free σ W (red curve). E, F. σ B and 559 σ W target promoter activities as a function of P B at fixed P W in the wildtype negative feedback 560 system (E), and the positive feedback system (F). G, H. RNA polymerase bound σ B (Rpol-σ B ) as 561 a function of P B at fixed P W in the wildtype negative feedback system (G) and the positive 562 feedback system (H). Increase in σ B phosphatase (P B ) leads to an increase in Rpol-σ B (green 563 curve) and corresponding decreases ∆Rpol-σ W in Rpol-σ W (red area) and ∆Rpol-σ A in Rpol-σ A 564 (blue area).

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The role of the negative feedback in producing this response becomes clear 567 when we compare the response of an "in silico" mutant network with positive feedback loops between σ B and BT and σ W and WT. These positive feedback loops are expected 569 to display no homeostatic properties and as a result, in this network activation of σ B 570 should significantly decrease σ W activity. Indeed, our simulation for the positive 571 feedback network (Fig. 6D) demonstrates that with increase in stress signaling protein 572 PB and the resulting increase in free σ B , the free σ W concentration decreases. As a 573 result of the increased competition for RNApol and the decreased free σ W , σ W target 574 promoter activity in this network decreases as a function of PB (Fig. 6F). Moreover 575 comparing changes in σ-RNApol complexes as a function of stress signaling protein PB 576 we find that most of the RNApol in the σ B -RNApol complex is drawn from the σ W -577 RNApol pool rather than σ A -RNApol pool (Fig. 6H). Thus the negative feedback designs 578 are essential for stress σ-factors not only to tolerate competition from σ A , but also to 579 avoid competing with each other when the cell is simultaneously exposed to multiple 580 types of stresses.  The research was supported by NIH grants GM 096189 to OAI. The authors are grateful 653 to James Locke for sharing raw data from Ref. [13] and Chet Price for feedback on the 654 manuscript.

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Mathematical model of the σ B network 659 The details of all biochemical reactions in the model and the corresponding differential The events shown in Figure 1A can be described by the following set of biochemical Here 0 v is the basal synthesis rate, f is the fold change in protein synthesis due to 701 positive autoregulation and K is the equilibrium dissociation constant for the binding of 702 σ B to the promoter DNA.

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The stress signals were assumed to control the concentrations of stress phosphatases We assume mass-action kinetics for all the above reactions (equations 1-10) to obtain 715 the following set of equations that describe network dynamics:   All model parameters are summarized in Table 1.

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To study the effects of competition for RNA polymerase, the σ B network model was  Ornstein-Uhlenbeck process as in [13]. This gamma distributed Ornstein-Uhlenbeck

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Note that at the initial phosphatase level the σ B~0 and BT is at the basal level of σ B operon transcription.

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The   The following set of equations that describe network dynamics of this extended model:  Model equations for the model of competition between σ B , σ W and σ A

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To model the competition for RNA polymerase between σ B the housekeeping σ-factor 1149 σ A and the alkaline stress response σ-factor σ W (Figs. 6 and S6), we simplified the