Filtering input fluctuations in intensity and in time underlies stochastic transcriptional pulses without feedback

Abstract

Stochastic pulsatile dynamics have been observed in an increasing number of biological circuits with known mechanism involving feedback control and bistability. Surprisingly, recent single-cell experiments in Escherichia coli flagellar synthesis showed that flagellar genes are activated in stochastic pulses without the means of feedback. However, the mechanism for pulse generation in these feedbackless circuits has remained unclear. Here, by developing a system-level stochastic model constrained by a large set of single-cell E. coli flagellar synthesis data from different strains and mutants, we identify the general underlying design principles for generating stochastic transcriptional pulses without feedback. Our study shows that an inhibitor (YdiV) of the transcription factor (FlhDC) creates a monotonic ultrasensitive switch that serves as a digital filter to eliminate small-amplitude FlhDC fluctuations. Furthermore, we find that the high-frequency (fast) fluctuations of FlhDC are filtered out by integration over a timescale longer than the timescale of the input fluctuations. Together, our results reveal a filter-and-integrate design for generating stochastic pulses without feedback. This filter-and-integrate mechanism enables a general strategy for cells to avoid premature activation of the expensive downstream gene expression by filtering input fluctuations in both intensity and time so that the system only responds to input signals that are both strong and persistent.

Keywords: filtering; gene regulation; stochastic modeling; transcriptional pulses; ultrasensitivity.

Fig. 1.

Comparison between experimental data and model results. (A) Distributions of the normalized class 2 activity $A_2/⟨A_2⟩$ in the presence (Left) and absence (Right) of YdiV for small ($P1$, red), medium ($P4$, green), and large ($P7$, blue) class 1 promoter strengths. Data are taken from ref. . We use the black dashed lines to indicate the positions of its mean value and zero for $A_2$. (B) The corresponding distributions obtained from our model. (C) Comparison of typical time series of $C_1\left(t\right)$ and $A_2\left(t\right)$ from experiments and model for the $P4$ strain with and without YdiV. (D) Response functions obtained by fitting our model to experimental data (19) with ($f_{+}$, solid line) and without ($f_{-}$, dashed line) YdiV. The circles (+YdiV) and crosses (−YdiV) indicate the mean value from different mother cells in experiments. Note that $f_s$ does not pass through the center of the data points, due to its high nonlinearity $\left(⟨f_s\left(C_1\right)⟩\right.\ne f_s\left(⟨C_1⟩\right)$. The color code for each promoter is the same as in A and B.

Fig. 2.

The bin-averaged response curves and the ratio of two timescales ${\tau }_2/{\tau }_1$. (A) Model results for ${\tau }_2=3.5{\tau }_1$. Inset shows the same behavior from experimental data. (B) Model results for ${\tau }_2=0.1{\tau }_1$. We used −YdiV strains here; see SI Appendix, Fig. S7 for similar results for +YdiV strains. (C and D) The normalized power spectra (green lines) from our model for (C) ${\tau }_2=3.5{\tau }_1$ and (D) ${\tau }_2=0.1{\tau }_1$ for the $P4$ strain. The normalized power spectrum (black line) from experiments ($P4$) is also shown for comparison. We used ${\tau }_1=45\hspace{0.17em}\text{min}$ within the range of $C_1$ correlation time estimated from experiments.

Fig. 3.

The YdiV-mediated sequestration mechanism for switch like behavior. (A) Schematic representation of the mechanism of transcriptional regulation of class 2 promoters by FlhDC. The class 1 promoter (purple) initiates the expression of the monomers FlhC and FlhD (yellow and orange, respectively) that forms the heterohexamer ${\mathrm{F}\mathrm{l}\mathrm{h}\mathrm{D}}_4{\mathrm{C}}_2$ (FlhDC). The timescale ${\tau }_1$ may be related to the degradation rate of the monomers (FlhD and FlhC), while ${\tau }_2$ may be related to the degradation rate of FlhDC or its unbinding rate to the promoters, both labeled by red arrows. The activation of class 2 genes, which depends on binding of the FlhDC complex to the class 2 promoter (green), is inhibited when FlhDC is sequestered by binding to YdiV (blue) instead (gray box). (B) Response curves obtained from the YdiV sequestration model for different values of YdiV concentrations ($P_t$ is the total class 2 promoter concentration); see Methods for details. (Inset) The response functions, $f_{+}\left(C_1\right)$ (blue) and $f_{-}\left(C_1\right)$ (red), in the presence and absence of YdiV, respectively, from fitting experimental data (the same as in Fig. 1D). (C) Hill coefficient ($H$) and the half-maximum concentration ($K_{1/2}$) as a function of the YdiV concentration.

Fig. 4.

The filter-and-integrate mechanism for pulse generation. In the presence of YdiV, the FlhDC concentration $C_1\left(t\right)$ (blue line), which fluctuates with a fast timescale ${\tau }_1$, is filtered by the inhibitory effect of YdiV. The filtered signal $f_{+}\left(C_1\left(t\right)\right)$ (black line), which has a pulse-like pattern, is then integrated over a longer timescale ${\tau }_2$ to determine the class 2 activity $A_2\left(t\right)$ (red line). In the absence of YdiV, there is no filtering effect, and the resulting $A_2$ dynamics is noisier, with no pulse. Without integration (${\tau }_2\ll {\tau }_1$), $A_2$ would have many short spikes when the signal is filtered, and it resembles the original noisy signal ($C_1$) in the absence of filtering. Results presented here are obtained from simulations of our model using parameters fitting the $P4$ strain.

Fig. 5.

The predicted responses to step stimuli. (A) Predicted dynamics of $C_1$ and $A_2$ in response to a time-dependent (step function) stimulus in which the class 1 promoter strength $\mu$ is changed at $t=0$ from a low value ${\mu }_0=\mu \left(P1\right)=2.6$ to a higher value ${\mu }_1$. The delay time ${\tau }_r$ is the time duration from the stimulus ($t=0$) to the onset of class 2 expression, which is defined as the time when $A_2$ crosses a threshold $A_2^{*}=max\left(f_{+}\right)/4$ (qualitative results do not depend on the choice of $A_2^{*}$). (B) Distributions of the delay time ${\tau }_r$ (normalized by ${\tau }_1$) for different values of ${\mu }_1$. (C) Both the average and variance of ${\tau }_r$ decreases with ${\mu }_1$ or the average FlhDC concentration. The values of $\mu$ for strains $P4$ and $P7$ used in our study are marked for reference.

Fig. 6.

The different gene regulation strategies for E. coli and Salmonella. (A) The E. coli network has the YdiV-mediated inhibition of the class 1 master regulator (FlhDC) but not the feedback mechanism from class 2 gene. As a result, E. coli has a steep but monotonic response function $f_{+}$ with a threshold $C^{*}$; that is, class 2 gene expression is activated (green) when $C_1>C^{*}$ or inhibited (red) when $C_1<C^{*}$. For an FlhDC signal that passes the threshold $C^{*}$ briefly, the class 2 activity is only turned on for a short time, which is not enough to start the flagella synthesis, given the long integration time ${\tau }_2$ for class 2 gene activation. (B) In Salmonella, there is a feedback (blue line) from a class 2 protein (FliZ) to suppress the inhibitor YdiV. This feedback mechanism leads to a bistable (hysteretic) response of class 2 genes in the range $C_l^{*}<C_1<C_h^{*}$. As a result, a brief period of elevated FlhDC concentration over the (upper) threshold $C_h^{*}$ can cause a prolonged class 2 gene activation as long as $C_1>C_l^{*}$, which can lead to flagella synthesis.