Bimodal gene expression in noncooperative regulatory systems

Abstract

Bimodality of gene expression, as a mechanism contributing to phenotypic diversity, enhances the survival of cells in a fluctuating environment. To date, the bimodal response of a gene regulatory system has been attributed to the cooperativity of transcription factor binding or to feedback loops. It has remained unclear whether noncooperative binding of transcription factors can give rise to bimodality in an open-loop system. We study a theoretical model of gene expression in a two-step cascade (a deterministically monostable system) in which the regulatory gene produces transcription factors that have a nonlinear effect on the activity of the target gene. We show that a unimodal distribution of transcription factors over the cell population can generate a bimodal steady-state output without cooperative transcription factor binding. We introduce a simple method of geometric construction that allows one to predict the onset of bimodality. The construction only involves the parameters of bursting of the regulatory gene and the dose-response curve of the target gene. Using this method, we show that the gene expression may switch between unimodal and bimodal as the concentration of inducers or corepressors is varied. These findings may explain the experimentally observed bimodal response of cascades consisting of a fluorescent protein reporter controlled by the tetracycline repressor. The geometric construction provides a useful tool for designing experiments and for interpretation of their results. Our findings may have important implications for understanding the strategies adopted by cell populations to survive in changing environments.

Fig. 1.

Nonlinear transcription rate transforms a unimodal distribution of TFs into a bimodal distribution of target proteins. Transcription is regulated by TFs (here, repressors) R, which bind to n binding sites within operator region, with or without cooperativity. Transcription rate is proportional to a nonlinear function of TF number (transfer function) h(R). The figure shows the most simple case: noncooperative binding, with h(R) given by Eq. 2 with n = 1, TF binding rate k_on, and unbinding rate k_off. mRNA M degrades with rate k_dm. The target protein P is produced with rate k_p and degrades with rate k_dp. The transfer function h(R) acts as a nonlinear noise filter that transforms the unimodal input [TF distribution p(R) generated by the regulatory gene] into a bimodal output [distribution q(h) of the rates of transcription from the target gene]. Consequently, the distributions of mRNA p_mrna(M) and the target protein p_prot(P) are bimodal.

Fig. 2.

Geometric construction of the conditions for bimodal distribution of transcription rates q(h) in systems with noncooperative (A and B) and cooperative (C and D) binding of n repressors. The extrema of q(h) are the points of intersection of the rescaled transfer function z(R) (Eq. 5) and the straight line L(R) (Eq. 6) (noncooperative case), or the transfer function h(R) (Eq. 3) and the straight line L(R) (Eq. 8) (cooperative case). A and B show the simplest case of noncooperativity, with n = 1 and z(R) = h(R). The height at which L(R) intersects the vertical axis depends on the number α of repressor production bursts per cell cycle. The slope of L(R) depends on the number β of repressor molecules produced per burst. The dashed and gray solid lines show examples without intersections, which generate monomodal distributions. (A) Noncooperative binding. For α < 1, a maximum at h = 1 exists, independent of the intersections, and another maximum emerges at the lower point of intersection. (B) Noncooperative binding. For α > 1, there is always one intersection, defining the only maximum, and thus bimodality is impossible. (C) Cooperative binding. For α < n, a maximum at h = 1 exists, independent of the intersections, and another maximum emerges at the lower point of intersection. (D) Cooperative binding. For α > n, all extrema are defined by the intersections. Two maxima can emerge only when L(R) intersects h(R) three times. Parameters used for A: n = 1, c = 0.1, α = 0.5, β₁ = 50 (solid line), and β₂ = 15 (dashed line). Parameters used for B: n = 1, c = 0.1, β = 15, and α = 1.2. Parameters used for C: n = 2, c = 0.005, α = 0.9, β₁ = 20 (solid line), and β₂ = 10 (dashed line). Parameters used for D: n = 3, c = 0.0025, α = 4, β₁ = 2 (black solid line), β₂ = 1.6 (dashed line), and β₃ = 2.7 (gray solid line).

Fig. 3.

The variance of the transcription-rate distribution q(h) as a measure of bimodality of q(h) as well as mRNA and protein distributions. (A) The variance of q(h) depending on α and βc, noncooperative case with n = 1. White line, boundaries of the bimodal region. Above the white line, q(h) is bimodal. (B) The bimodality of q(h) increases as the of q(h) increases. q(h) is shown for α and βc marked in A by the points a, b, and c (α_a = 0.101, α_b = 0.25, α_c = 0.7, β = 20). (C and D) Protein distribution recovers the bimodality lost because of intrinsic noise at the mRNA level. (C) The distribution p₁(M,h = 1) has the variance , the distribution p₂(P,h = 1) has the variance , and . (D) causes the loss of bimodality of the mRNA distribution p_mrna(M), whereas at the protein distribution p_prot(P) is bimodal. k_m = 1, k_dm = 0.1, k_p = 0.5, k_dp = 0.01, k_on = 1, k_off = 10, k_mr = 5 × 10^-5, k_dmr = 0.01, k_r = 0.5, k_dr = 10^-4, and the other parameters are the same as in Fig. 2A with β₁.

Fig. 4.

When TF degrades in a comparable time scale as the target proteins, the regulatory noise is still present, but the bimodality disappears. (A) The bimodality of the protein distribution decreases as the degradation rate k_dp of the protein becomes slower than the degradation rate of TFs (solid line, theoretical prediction, Eq. 14). As the TF degradation becomes much faster then that of the target protein, the target gene does not react to variations in R (Fig. S7B), and it only experiences the mean transcription rate k_m〈h(R)〉. The unimodal distribution is then theoretically calculated from Eq. 12 with 〈h(R)〉 (dashed line). ω = k_dp/k_dr is the ratio of the slowest timescales of protein, and TF. k_dr = 10^-4 was constant, whereas k_dp was varied. At the same time, k_p, k_m, and k_dm were rescaled in such a way that the mean numbers of mRNA and proteins were constant. (B) The presence of regulatory noise in the system, depending on the ratio of TF/protein lifetimes. When the TF lifetime is much shorter than the protein lifetime, the protein noise tends to its lower bound; i.e., the intrinsic noise η_int is only present. When the TF lifetime is much longer than the protein lifetime, the protein noise tends to its upper bound η_int + η_h, where η_h is the maximum of the regulatory noise. At intermediate time scales, where the regulatory noise is between zero and η_h, the bimodality may not be present any longer (A). In particular, the bimodality is not present when the TF lifetime is comparable to the protein lifetime.

Fig. 5.

Change in the concentration of inducer or corepressor causes transitions between unimodal and bimodal gene expression. (A) In a noncooperative system, bimodality emerges as the ratio c of TF binding to unbinding rates increases. (B) In a cooperative system, bimodality emerges and then disappears as c increases. Arrows indicate the increase of c. Parameters for A are the same as in Fig. 2A with β₁, and parameters for B are the same as in Fig. 2D with β₁, a = 500, and b = 1.