Selective heterogeneity in exoprotease production by Bacillus subtilis

Abstract

Bacteria have elaborate signalling mechanisms to ensure a behavioural response that is most likely to enhance survival in a changing environment. It is becoming increasingly apparent that as part of this response, bacteria are capable of cell differentiation and can generate multiple, mutually exclusive co-existing cell states. These cell states are often associated with multicellular processes that bring benefit to the community as a whole but which may be, paradoxically, disadvantageous to an individual subpopulation. How this process of cell differentiation is controlled is intriguing and remains a largely open question. In this paper, we consider an important aspect of cell differentiation that is known to occur in the gram-positive bacterium Bacillus subtilis: we investigate the role of two master regulators DegU and Spo0A in the control of extra-cellular protease production. Recent work in this area focussed the on role of DegU in this process and suggested that transient effects in protein production were the drivers of cell-response heterogeneity. Here, using a combination of mathematical modelling, analysis and stochastic simulations, we provide a complementary analysis of this regulatory system that investigates the roles of both DegU and Spo0A in extra-cellular protease production. In doing so, we present a mechanism for bimodality, or system heterogeneity, without the need for a bistable switch in the underlying regulatory network. Moreover, our analysis leads us to conclude that this heterogeneity is in fact a persistent, stable feature. Our results suggest that system response is divided into three zones: low and high signal levels induce a unimodal or undifferentiated response from the cell population with all cells OFF and ON, respectively for exoprotease production. However, for intermediate levels of signal, a heterogeneous response is predicted with a spread of activity levels, representing typical "bet-hedging" behaviour.

Figure 1. Schematic of the DegU - Spo0A intersecting control networks.

The interaction of the main components of the model are illustrated. Interactions enclosed in dashed boxes are not explicitly considered. See text for further explanation.

Figure 2. Dynamics of the DegU and exoprotease systems.

Temporal evolution of the DegU - exoprotease system as modelled by equations (1)–(6). The levels of (A) mRNA, (B) DegU, (C) DegUP and dimer of DegUP and (D) exoprotease. All parameters as in Table 1.

Figure 3. The steady state values of DegU and exoprotease as functions of the phosphorylation rate as predicted by the full system.

Steady state solutions for (A) (DegU) and (B) (exoprotease) of system (1)–(6) as functions of . (C) Plot of DegU versus the level of exoprotease ( versus ). Solid lines represents stable solutions, dashed lines represent unstable solutions and a region of bistability. The blue and yellow dots individually indicate system output for two different values of . They are provided for ease of comparison between the graphs (see text). All parameter values from Table 1.

Figure 4. How the steady state curves change with the dephosphorylation rate .

Steady state solutions for (A) (DegU) and (B) (exoprotease) of the full system (1)–(6) as functions of . Solid lines represent stable solutions, dashed lines represent unstable solutions and a region of bistability. All parameter values from Table 1 except (green); (red); (blue).

Figure 5. The steady state values of DegU and exoprotease as functions of the phosphorylation rate as predicted by the minimal system.

Steady state solutions derived using (9). (A) (DegU) and (B) (exoprotease - computed using (S7)–(S9)) as functions of . In each figure, all parameter values from Table 1 except (green); (red); (blue).

Figure 6. Temporal evolution of protein levels subject to intrinsic noise.

Output from system (1)–(6) computed using the Gillespie SSA. Histograms show the levels of (A) DegU and (B) exoprotease collated from 1000 simulations at 10 time points. Each simulation can be considered to be equivalent to the temporal evolution of transcription in a single cell. Note that distributions for both DegU and exoprotease display no significant further change after approximately hrs. All parameter values from Table 1 except . Initial protein values set to zero.

Figure 7. Intrinsic noise effects on steady state protein levels for different phosphorylation rates.

Output from system (1)–(6) computed using the Gillespie SSA. For comparison, the DegU levels predicted by the deterministic system with the response for the chosen values indicated by the red, blue and green dots, respectively, are shown in (A). Histograms show the levels of: (B) DegU and (C) exoprotease collated from 1000 simulations at hours. All parameter values from Table 1 except in A, B and C red indicates ; blue indicates and green indicates . Initial values set to correspond to the equivalent deterministic steady state.

Figure 8. Temporal evolution of protein levels subject to extrinsic noise.

Output from system (1)–(6) computed using the Gillespie SSA. For comparison, the DegU levels predicted by the deterministic system with the response for the chosen values indicated by the red dots is shown in (A). Histograms show the levels of (B) DegU and (C) exoprotease collated from 1000 simulations at 4 time points. All parameter values from Table 1 except for each simulation, was selected randomly from the values . Initial values of the variables set to correspond to the deterministic steady state for .

Figure 9. Extrinsic noise effects on steady state protein levels: perception to discrete signal changes.

Output from system (1)–(6) subject to discrete steps in signal level/perception computed using the Gillespie SSA. For comparison, the DegU levels predicted by the deterministic system with the response for the chosen values indicated by the red, blue, green and yellow dots, respectively, are shown in (A). Histograms show the levels of (B) DegU and (C) exoprotease collated from 1000 simulations at hours. All parameter values from Table 1 except for each simulation, was selected randomly from the values: (red); (blue); (green); (yellow). Initial levels set to correspond to the deterministic steady state associated with the midpoint values.

Figure 10. Extrinsic noise effects on steady state protein levels: perception to continuous signal changes.

Output from system (1)–(6) subject to normally distributed signal level/perception computed using the Gillespie SSA. For comparison, the DegU levels predicted by the deterministic system with the response for the chosen values indicated by the red, blue, green and yellow bands, respectively, are shown in (A). Histograms show the levels of (B) DegU and (C) exoprotease collated from 1000 simulations at hours. All parameter values from Table 1 except for each simulation, was selected from normally distributed values with mean values (red); (blue); (green); (yellow). In each case the relative standard deviation of the signal input was set at . Initial levels set to correspond to the deterministic steady state associated with the mean values.

Figure 11. Effects of AbrB and SinR on selective heterogeneity.

Effects on the steady state response of the deterministic system (1)–(6): (A) The DegU response curve is unaffected by changes in AbrB and SinR. (B) The level of exoprotease expression as predicted by the full system. (C) as computed from equation (S8) in the minimal system. Colours represent: Spo0A High (SinR  = 0 and AbrB  = 0) (red); Spo0A Mid (SinR  = 7 and AbrB  = 7) (green); Spo0A Low (SinR  = 14 and AbrB  = 70) (blue) and Spo0A Off (SinR  = 21 and AbrB  = 700) (yellow). (D) and (E) shows output from system (1)–(6) computed using the Gillespie SSA. Histograms show the levels of (D) DegU and (E) exoprotease collated from 1000 simulations of the stochastic model at hours. All parameter values from Table 1 except for each simulation, was drawn from a normally distributed set of values with mean and rsd  = 20%. As above, Spo0A Off (AbrB  = 700, SinR  = 21); Spo0A Low (AbrB  = 70, SinR  = 14); Spo0A Mid (AbrB  = 7, SinR  = 7); Spo0A High (AbrB  =  SinR  = 0). Note the change of scale for exoprotease copy number in the last figure in (E). Initial values set to correspond to the deterministic steady state associated with the midpoint value.