Fig 1.
The σB regulatory network consists of a core circuit, which is activated by two distinct upstream pathways.
Under non-stress conditions, σB is bound, and held inactive by, its anti-sigma factor (RsbW, W in the figure). An anti-anti-sigma factor (RsbV, V in the figure) is inactivated by a phosphate group. The upstream pathways are triggered by two distinct types of stress (environmental and energy stress), each activating their respective phosphatase (RsbTU and RsbP, respectively). These will dephosphorylate, and thus activate, RsbV. Once activated, RsbV binds RsbW, which simultaneously releases σB in a partner switching mechanism. This permits σB to activate the general stress response of B. subtilis. RsbW is a kinase that re-phosphorylates RsbV, allowing RsbW to re-bind σB and shut the activation off. In addition, σB activates the production of itself, RsbW, and RsbV, creating a mixed positive/negative feedback loop. The environmental stress response phosphatase complex, RsbTU, consists of the co-factor RsbT and the phosphatase RsbU (T and U, respectively, in the figure). The availability of RsbT is controlled by a multi-protein complex called the stressosome and the phosphatase RsbX (both not depicted in the figure for simplicity). The energy stress response phosphatase, RsbP (P in the figure), also depends on a second protein, RsbQ (not depicted in this figure).
Fig 2.
The core σB circuit is capable of generating both response behaviours.
(A,B) For various noise amplitudes (η), the CLE adaptation of the Narula model is exposed to a stress step (at t = 0, red dashed line). (A) Mean [σB] in response to the input (average over n = 150 simulations). The amplitude of the single response pulse increases with noise. (B) For each value of η, four different simulations are shown. There is little variation between the individual trajectories. (C,D) Illustration of our measures for the degree with which the system exhibits the single response pulse (C) and stochastic pulsing (D) behaviours. A simulation is divided into a transient phase (t ∈ [0.0, 5.0]) and an asymptotic phase (t ∈ [5.0, 200.0], but we note that in this figure the x-axis is cut to t = 50 to better display both phases). Next, we find the maximum activity in the transient phase, the maximum activity in the asymptotic phase, and the mean activity in the asymptotic phase. The single response pulse measure (C) is defined as the maximum transient activity (orange line) divided by the maximum asymptotic activity (magenta line). The stochastic pulsing measure (D) is defined as the maximum asymptotic activity (orange line) divided by the mean asymptotic activity (magenta line). In practice, a mean measure over several (n > 50) simulations is always used. (E,F) The parameters η (noise amplitude) and kK2 (a proxy for the system’s proneness to oscillations) are varied. For each parameter combination, the maximum magnitude of the single response pulse (E) and stochastic pulsing (F) behaviours that can be achieved by varying the parameter pstress (20.0 μM< pstress < 200.0 μM) is found and plotted. (G) The single response pulse behaviour is maximised at (kK2, η) = (15.0hr−1, 0.04). For these values, four simulations are shown. (H) The stochastic pulsing behaviour is maximised at (kK2, η) = (9.0hr−1, 0.06). For these values, four simulations are shown. (G,H) These simulations demonstrate that the CRN of the Narula model can generate both behaviours while exposed to intrinsic noise only. (I,J) It is possible to recreate both response behaviours, single pulse response (I) and stochastic pulsing (J), using the Gillespie algorithm. This demonstrates that the responses are not dependent on the modelling approach used. Parameter values and other details on simulation conditions for this figure are described in S1–S3 Tables.
Fig 3.
The effective dephosphorylation rate is the main determinant of which behaviour is produced.
(A,B) The magnitude of the single pulse response (A) and stochastic pulsing (B) behaviours across kP-pstress-space. The regions where either behaviour occurs are similar to that of the kP ⋅ pstress = C curve (show for various values, green dashed lines). (C,D) Parameter substitutions generate a parameter pprod = pstress ⋅ kP, to which both behaviours are sensitive, and a parameter pfrac = pstress/kP, to which both behaviours are insensitive. While the regions corresponding to either behaviour are adjacent, they do not overlap. The stochastic pulsing behaviour exists for slightly larger values of pprod, as compared to the single response pulse behaviour. (E,F) For 4096 different parameter sets, we characterise both behaviours’ sensitivity to change (, E, and
, F) in the parameters pprod, pfrac, kB5, and kD5 (Section 4.5.4). We do this by evaluating
and
for the four different parameters across all 4096 parameter sets. We then put each set of 4096 evaluations in ascending order and plot them in E and F. For a few parameter sets, the behaviours show some sensitivity to kB5. However, pprod has the far largest effect on either behaviour. In both cases, changes to pfrac and kD5 have little effect on the system. Hence, these lines coincide (both following the x-axis closely). Parameter values and other details on simulation conditions for this figure are described in S2 and S5 Tables.
Fig 4.
The system transitions through a range of behaviours as the effective dephosphorylation rate is varied.
(A) The magnitude of the two behaviours for our selected parameter set (kK2, η, pfrac, kB5, kD5) = (7.0 h-1, 0.025, 100.0 μM h-1, 3600.0 μM-1h-1, 18.0 h-1). 12 different selected values of pprod (used in B-M) are marked with grey lines. (B-M) For 12 different values of pprod a single simulation is displayed (stress added at t = 0, red dashed line). (B) For pprod small, the system does not respond. (C,D) As pprod is increased, the system exhibits a single response pulse. The amplitude increase with pprod. (E,F) For larger values of pprod, stochastic pulsing is exhibited. The frequency of the pulses increases with pprod. (G-I) As the stress is increased further, the system enters a limit cycle. (J-L) For large pprod, the system exhibits a single response pulse, and then enters a persistent state of elevated σB activity. The activity in this state increases with pprod. (M) For large enough stresses, the system saturates at some maximum activity. An expanded version of this figure, including bifurcation analysis of system steady state properties, can be found in S16 Fig. A similar transition, but generated through Gillespie algorithm simulations, can be found in S17 Fig. Parameter values and other details on simulation conditions for this figure are described in S4 and S5 Tables.
Fig 5.
The properties of the noise in the upstream pathways may bias the system towards either response behaviour.
(A,B) The amplitude (ηamp) and frequency (ηfreq) of the upstream pathway’s noise is varied. Plots show, for each parameter combination, the distinctness of the single response pulse (A) and stochastic pulsing (B) behaviours. The distinctness (of either behaviour) designates the system’s ability to uniquely generate that behaviour (and not the other behaviour) as a parameter is varied (here pprod). This measure is described in detail in Section 4.5.3 (there designated Dsrp or Dsp). Light green dots mark parameter sets optimising either behaviour’s distinctness. (C,D) For the parameter sets that maximise the distinctness of the single response pulse (C) and stochastic pulsing (D) behaviours, the magnitudes of the two behaviours are shown as functions of pprod. For each parameter set, 7 selected values of pprod are marked with grey lines. (E,F) For the 7 parameter sets marked in C (E), and D (F), a single simulation is displayed (stress added at t = 0, red dashed line). As pprod is varied, the two behaviours are generated much more robustly than what they were for the parameter set in Fig 4. Parameter values and other details on simulation conditions for this figure are described in S6 and S7 Tables.
Fig 6.
Core circuit noise compared to upstream noise.
(A,B) For each combination of core circuit noise amplitude (η) and upstream pathway noise amplitude (ηamp) we vary the parameter pprod (over the interval 20 < pprod < 200) and calculate each behaviour’s distinctness (Section 4.5.3). The total amount of noise in the system, rather than how it is distributed between the core and upstream pathway, is an important determinant for both behaviours’ occurrence. (A) The single pulse response behaviour is distinct when both types of noise are low. (B) The stochastic pulsing behaviours require some amount of noise (in either pathway), but do diminish if the total amount of noise becomes too large. It is especially prominent when core circuit noise amplitude (η) is small and upstream pathway noise amplitude (ηamp) is intermediately valued. Parameter values and other details on simulation conditions for this figure are described in S7 Table.
Table 1.
The reactions of the CLE adaptation of Narula model of the σB circuit.
Table 2.
The parameters of the CLE adaptation of the Narula model of the σB circuit.
Table 3.
The reactions added in our modified Narula model.
Table 4.
The parameters added in our modified Narula model.