Gene regulatory circuits must contend with intrinsic noise that arises due to finite numbers of proteins. While some circuits act to reduce this noise, others appear to exploit it. A striking example is the competence circuit in Bacillus subtilis, which exhibits much larger noise in the duration of its competence events than a synthetically constructed analog that performs the same function. Here, using stochastic modeling and fluorescence microscopy, we show that this larger noise allows cells to exit terminal phenotypic states, which expands the range of stress levels to which cells are responsive and leads to phenotypic heterogeneity at the population level. This is an important example of how noise confers a functional benefit in a genetic decision-making circuit.

Fluctuations, or “noise”, in the response of a system is usually thought to be harmful. However, it is becoming increasingly clear that in single-celled organisms, noise can sometimes help cells survive. This is because noise can enhance the diversity of responses within a cell population. In this study, we identify a novel benefit of noise in the competence response of a population of Bacillus subtilis bacteria, where competence is the ability of bacteria to take in DNA from their environment when under stress. We use computational modeling and experiments to show that noise increases the range of stress levels for which these bacteria exhibit a highly dynamic response, meaning that they are neither unresponsive, nor permanently in the competent state. Since a dynamic response is thought to be optimal for survival, this study suggests that noise is exploited to increase the fitness of the bacterial population.

Funding: MK was supported by NIH biophysics training grant T32GM008297. LH was supported by NSF REU grant PHY-1460899. GMS was supported by the National Institute of General Medical Sciences Grant R01 GM088428 and the National Science Foundation Grant MCB-1450867. Work in Paris was supported by MCCIG grant no. 303561. This work was also supported by the San Diego Center for Systems Biology (NIH Grant P50 GM085764). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Here, using stochastic modeling and quantitative fluorescence microscopy, we study the effects of intrinsic noise on the competence dynamics and the ensuing population heterogeneity of B. subtilis. We uncover a novel effect of noise that goes beyond architecture-dependent stochastic effects in a single cell. Specifically, we find that at both low and high stress levels, noise prevents cells from becoming unresponsive or indefinitely responsive to the stress, and instead allows cells to respond dynamically. These effects expand the range of stress levels over which the population of cells maintains a heterogeneous response distribution, which is critical to the population viability (see Fig 1B ). The use of efficient numerical methods and stochastic simulation at several levels of model complexity allows us to elucidate the mechanisms behind these effects. A central prediction from our modeling is that these effects are rooted in noise arising from low numbers of molecules. We verify this prediction using quantitative fluorescence microscopy by comparing the population response of native B. subtilis with that of synthetic mutants harboring the less-noisy SynExSlow circuit. Taken together, these results constitute a fundamental example of how noise can increase the functionality of a phenotypic response.

The competence circuit includes a negative feedback loop, which is known to play a critical role in controlling the exit from competence [ 15 ], the duration of competence [ 13 ], the variability of the duration of competence [ 16 ], and the integration of signals into competence that permit the existence of a transient competent state [ 17 ]. However, the effects of noise on the dynamics of the competence response are only partially understood. Previous work has shown that noise can trigger excitations into the competent state when the circuit is tuned to the excitable regime [ 15 ]. This transition into the competent state was shown to be governed by intrinsic noise rather than extrinsic noise [ 18 ]. Later work showed further that these excitations have a high variability in their duration, and that this variability is directly linked to the architecture of the competence circuit [ 16 ]. In particular, this work employed an analogous synthetic excitable circuit, termed SynEx, to provide evidence that the duration variability is due to intrinsic noise from low molecule numbers in the native circuit. Additionally, a similar strain, SynExSlow, was created that had a mean competence duration comparable to the native strain at the cost of additional complexity in order to reduce the efficiency of a proteolytic negative feedback loop [ 16 ]. However, the ability of this intrinsic noise to trigger sustained or repeatable excitations has not yet been quantified. Moreover, the generic effects of intrinsic noise on the three dynamic regimes, and how these effects translate to the physiological function of B. subtilis at the population level, are unknown.

(A) When all cells in a population exhibit either no response (left) or a high response (right), then the population is homogenous. In contrast, if individual cells exhibit a dynamic response (middle), this leads to a heterogenous population, with a fraction f of cells in the responsive state at any given time. (B) Intrinsic noise affects the dynamics of the response. Without intrinsic noise, the viable stress level response range is narrow as indicated by the black dashed lines since it is limited to deterministic dynamics. However, for the B. subtilis competence response, we find in this study that noise expands the viable response range: the range of stress levels over which f remains neither 0 nor 1. This expansion is illustrated by the blue dashed lines indicating the extent where f, solid blue line, remains between 0 and 1. f permits a heterogeneous population when between 0 and 1.

The competence response of the gram-positive bacterium Bacillus subtilis provides a striking example of dynamically maintained phenotypic heterogeneity. Under stress, B. subtilis undergoes a natural and transient differentiation event, termed competence, that allows the organism to incorporate exogenous genes into its genome. Previous studies have shown that entry into the competent state is controlled by a genetic circuit that that can be tuned to one of three dynamical regimes [ 13 ]: an excitable regime at low stress levels, where cells rarely and transiently enter the competent state; an oscillatory regime at intermediate stress, where cells oscillate in and out of the competent state; and a mono-stable regime at high stress, where cells remain in the competent state. Importantly, oscillatory (and repeatably excitable) dynamics lead to phenotypic heterogeneity, since cells are dynamically transitioning in and out of the competent state (see Fig 1A ). This heterogeneity is especially important to the survival of B. subtilis: if no cells respond, competence is not exploited, and the population may succumb to the stress. On the other hand, if all cells are permanently in the competent state, this can also be fatal to the population, since competence has been shown to reduce the cell growth rate and prevent cell division due to the inhibition of FtsZ [ 14 , 15 ]. Therefore, maintaining a dynamic competence response, and therefore a heterogenous population, is thought to be crucial to survival under stress.

The ability of molecular noise to cause stochastic phenotype changes has been demonstrated in a number of biological systems. In the context of enzymes, several studies have explored how intrinsic noise due to low numbers of molecules, or even a single molecule, can have dramatic effects through the amplified actions of a few enzymes [ 7 , 8 ]. Moreover, studies of bacterial operons, including in the context of bacterial persistence, have suggested that stochasticity could be encoded in the interactions between genes in a genetic regulatory network by ensuring that certain operon states are exposed to low numbers of molecules [ 9 – 11 ]. Recently, a theoretical study has demonstrated the conditions for when deterministic approaches to modeling genetic circuit dynamics break down, due to amplified effects of rare events caused by a small number of regulators [ 12 ]. Together, these works suggest that phenotypic heterogeneity could be rooted in low-molecule-number noise, and that this noise could in turn be encoded in the architecture of genetic regulatory networks.

Snapshots of bacterial populations often reveal large phenotypic heterogeneity in the gene expression states of its composite individuals. Such phenotypic heterogeneity in a clonal population of bacterial cells in a single environment has significant consequences for how well the organisms can adapt and survive. On the one hand, a population with little or no heterogeneity may allow for all cells to take advantage of certain optimal conditions to which the population is exposed. In this case, heterogeneity is suboptimal and therefore detrimental to fitness. On the other hand, numerous recent studies have shown that heterogeneous populations allow for cells to account for uncertainty in future environmental conditions [ 1 – 6 ]. In this case, heterogeneity is beneficial to fitness. A straightforward way to maintain high phenotypic heterogeneity is for each cell to exhibit a dynamic response. This allows each cell in its turn to transition among the various states of the population, e.g. via switching, pulsing, or oscillatory dynamics. The heterogeneity is intrinsically encoded in each cell, and is often enhanced by, or even entirely due to, stochasticity, or “noise”, at the molecular level [ 2 – 4 , 6 ].

Results

Entry of B. subtilis cells into the competent state occurs at high expression levels of the ComK protein. This protein activates a set of downstream genes allowing for the uptake of DNA [15]. ComK is typically expressed at a basal level, and stress in the environment alters the level of expression. In our genetic circuit design, as described below, increasing the stress level is mimicked by inducing comK expression using an increasing amount of a lactose analogue, Isopropyl β-D-1-thiogalactopyranoside (IPTG), in the environment.

In the native competence circuit, ComK activates its own expression, and represses the expression of another protein, ComS. ComS and ComK compete to be rapidly degraded by the MecA protein complex [15] (see Fig 2A, bottom). Therefore, high concentrations of ComS hinder the degradation of ComK, effectively providing positive feedback to ComK by allowing ComK levels to build up. These interactions are summarized in Fig 2A (top). Intrinsic fluctuations in comK mRNA have been shown to be another possible source of variations in ComK proteins [18].

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larger image TIFF original image Download: Fig 2. Architectures and model parameters of the native and SynEx circuits. The top row summarizes the regulatory interactions, while the bottom row depicts the model details. (A) In the native circuit, ComK is produced with the induction rate α k and activates its own expression with Hill function parameters β k , k k , and h. ComS is expressed at the basal rate α s and is repressed by ComK with Hill function parameters β s , k s , and p. ComK and ComS are degraded at rates λ k and λ s , respectively, and, additionally, both compete for binding to the degradation enzyme MecA. MecA degrades ComK and ComS with maximal rates δ k and δ s , respectively, and with Michaelis-Menten constants Γ k and Γ s , respectively. (B) In the SynEx circuit, ComK is produced with the induction rate α k and activates its own expression with Hill function parameters β k , k k , and h. MecA is expressed at the basal rate α m and is activated by ComK with Hill function parameters β m , k m , and p. ComK and MecA are degraded at rates λ k and λ m , respectively, and MecA enzymatically degrades ComK with rate δ. https://doi.org/10.1371/journal.pcbi.1004793.g002

In the SynEx circuit, as described in [16], the repression of ComS by ComK is removed by gene knockout. Then, the expression of MecA is placed under the control of ComK. This causes ComK to activate MecA, which in turn represses ComK via active protein degradation (see Fig 2B, bottom). These interactions are summarized in Fig 2B (top). Note that in the native circuit, ComK represses its own activator (ComS), while in the SynEx circuit, ComK activates its own repressor (MecA).

Both the native and SynEx circuits have architectures characteristic of molecular oscillators. Therefore we expect both circuits to allow for a dynamic response of each individual in a population. However, the main difference is that in the native circuit, when ComK levels are high, ComS levels are low, which leads to large amounts of intrinsic noise. In contrast, in the SynEx circuit, when ComK levels are high, MecA levels are also high, corresponding to less intrinsic noise. Previous work showed that this difference in architecture causes the native circuit to display a broad range of competence durations, whereas the SynEx circuit displays a relatively narrow range of competence durations [16]. However, the effects of noise and architecture on the ranges of dynamic response and the ensuing population heterogeneity in these systems remained unknown.

Noise expands the response range To elucidate the effects of noise in each of the native and SynEx circuits (Fig 2), we develop a stochastic model of each circuit, which includes noise, and then compare each to its deterministic analog, which does not include noise. As described in Materials and Methods, we develop the stochastic models at several levels of complexity to investigate the robustness of our findings to our modeling assumptions, and we solve each model using a combination of efficient numerical solution and stochastic simulation. We first describe the behavior of the deterministic models. As shown in Fig 3A, a standard linear stability analysis of the deterministic model for each circuit reveals three dynamical regimes, depending on the value of the control parameter, the ComK induction rate α k . At low induction, each circuit is excitable, resulting in a transient differentiation event into and out of the competent (high-ComK) state. At intermediate induction, each circuit is oscillatory, periodically entering and exiting the competent state. At high induction, each circuit is mono-stable, staying in the competent state indefinitely; relaxation to the non-competent state does not occur. These three dynamical regimes have been confirmed in experimental studies of the native competence circuit [13]. PPT PowerPoint slide

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larger image TIFF original image Download: Fig 3. Stochastic modeling of B. subtilis competence. (A) The deterministic model of each circuit (see Eqs 6 and 7) exhibits three dynamic regimes (excitable, oscillatory, and mono-stable), depending on the ComK induction rate α k , which models stress level. (B) The stochastic model (see Eqs 1–5) reveals the ensuing distribution of ComK levels in each of the three dynamic regimes (excitable, oscillatory, and mono-stable). The fraction of the distribution in the responsive state f (determined by the inflection points, see Materials and Methods) is shaded. (C) Whereas the deterministic model exhibits sharp transitions between the dynamic regimes (dashed lines), the stochastic model exhibits a continuous dependence of f on induction rate. We see that for both circuits, stochasticity extends the viable response range (0 < f < 1) beyond the transitions predicted by the deterministic model, in both directions, by the factors given above the arrows (see Materials and Methods). Parameters are as in [16] and are given in S1 Text. In A and B, from left to right, the values of the control parameter are α k = {0.072, 1.15, 36}/hour (native) and α k = {0.036, 1.8, 36}/hour (SynEx). In A, from left to right, the initial conditions are ComK molecules and ComS molecules (native), and ComK molecules and MecA molecules (SynEx); in the excitable regime (left), the initial conditions are chosen to demonstrate the single, transient excitation. https://doi.org/10.1371/journal.pcbi.1004793.g003 We find that these deterministic dynamics are reflected in the steady-state solutions to the minimal stochastic models. As shown in Fig 3B, the three types of dynamics correspond to three shapes of steady-state probability distributions of ComK levels. Excitable dynamics correspond to a distribution confined to low ComK molecule numbers, oscillatory dynamics correspond to a distribution mixed between low and high molecule numbers, and mono-stable dynamics correspond to a distribution centered at high molecule numbers. As described in Materials and Methods, we calculate the fraction f of the distribution in the high-molecule-number state (see the shaded regions in Fig 3B). Within our model, f represents the fraction of time a single cell spends in the competent state, or equivalently, the fraction of an isogenic population of cells found in the competent state at a given time. Importantly, f is the indicator of population heterogeneity, since unresponsive (f = 0) or fully competent (f = 1) populations are homogeneous, while mixed populations (0 < f < 1) are heterogeneous. We define the range of induction rate α k for which 0 < f < 1 as the viable response range, since unresponsive cells (f = 0) do not benefit from competence, while long-term competence (f = 1) is known to have a detrimental effect on growth rate and cell division [14, 15]. In Fig 3C, we compare the viable response range of the stochastic model with the boundaries between dynamical regimes predicted by the deterministic model. We see that for both the native and the SynEx circuit, the stochastic range extends beyond the deterministic range for both low and high induction rate α k . At low induction rate, the range expands by more in the SynEx circuit than in the native circuit (16 times vs. 8.4 times). In contrast, at high induction rate, the range expands by more in the native circuit than in SynEx circuit (20 times vs. 3.5 times). The latter effect is much stronger, such that the total expansion of the viable range is three times larger in the native circuit than in the SynEx circuit (8.4 × 20 = 168 times vs. 16 × 3.5 = 56 times). Taken together, these observations imply that noise expands the range of stress levels to which cells can respond in a dynamic way, and that this expansion depends on circuit architecture. In the next section, we elucidate the mechanisms behind this expansion.