Noise regulation by quorum sensing in low mRNA copy number systems

Abstract

Background: Cells must face the ubiquitous presence of noise at the level of signaling molecules. The latter constitutes a major challenge for the regulation of cellular functions including communication processes. In the context of prokaryotic communication, the so-called quorum sensing (QS) mechanism relies on small diffusive molecules that are produced and detected by cells. This poses the intriguing question of how bacteria cope with the fluctuations for setting up a reliable information exchange.

Results: We present a stochastic model of gene expression that accounts for the main biochemical processes that describe the QS mechanism close to its activation threshold. Within that framework we study, both numerically and analytically, the role that diffusion plays in the regulation of the dynamics and the fluctuations of signaling molecules. In addition, we unveil the contribution of different sources of noise, intrinsic and transcriptional, in the QS mechanism.

Conclusions: The interplay between noisy sources and the communication process produces a repertoire of dynamics that depends on the diffusion rate. Importantly, the total noise shows a non-monotonic behavior as a function of the diffusion rate. QS systems seems to avoid values of the diffusion that maximize the total noise. These results point towards the direction that bacteria have adapted their communication mechanisms in order to improve the signal-to-noise ratio.

Figure 1

Scheme of a simplified biochemical network of QS systems near the activation threshold. Schematic representation of the biochemical processes considered in our approach for describing the dynamics of the signaling molecule, A, in cell i. The mRNA dynamics satisfies a dichotomous process characterized by the states M_0,1 corresponding to zero and one molecules respectively. Once the autoinducer has been produced, it can diffuse in and out the cell leading to cell communication (see text).

Figure 2

Probability densities of the signaling molecule and parameter space. Panel A: Schematic representation of the different probability densities of the autoinducer concentration depending on the value of $\tilde{\alpha }$ and $\tilde{\beta }$ with respect to that of $\tilde{D}$. Given a set of values ($\tilde{\alpha }$, $\tilde{\beta }$) the dynamics of the autoinducer shows different behaviors depending on the value of the diffusion parameter since the transitions lines are located at $\tilde{\alpha }$, $\tilde{\beta }$ = 1 + $\tilde{D}$. The constraints of our modelling in terms of the parameters values make the region on the top-left corner non-accessible (see text). Panel B: Parameter space diagram ($\tilde{\alpha }$, $\tilde{\beta }$) indicating the sets of parameters used in the simulations (solid squares): γ₁ = (8, 2), γ₂ = (15, 5), γ₃ = (8, 0.5), γ₄ = (15, 0.5). The experimental values reported for the degradation rate of the mRNA leads to a biological meaningful range for $\tilde{\alpha }$ (blue region). The low constitutive expression assumption is prescribed by the constraint $\tilde{\alpha }>2\tilde{\beta }$ (red colored region).

Figure 3

Distributions and dynamics of the signaling molecule in a diffusionless system. Panel A: Distributions of c_A at steady-state for different sets of parameters ($\tilde{\alpha }$, $\tilde{\beta }$) as indicated in Figure 2B. In all cases $\tilde{D}$ = 0. The histogram obtained in simulations (blue bars) compares well with the distribution from the analytical calculations (blue line). Yet, deviations are observed due to intrinsic noise (see text). Panel B: The dynamics of the autoinducer show different behaviors depending on the region of the parameters phase space (see Figure 2). Two typical trajectories are shown with a grey-shaded background indicating the presence of a mRNA molecule in the cell.

Figure 4

Distributions and dynamics of the signaling molecule in a system with diffusion. Distributions (left column) and dynamics (center column) of c_A at steady-state for different values of $\tilde{D}$. The right most column stands for a density plot of the distribution of $c_{A_1}$ as a function of $c_{A_2}$ for discerning a putative increase in the molecular noise (see text). In all cases the parameters set ($\tilde{\alpha }$, $\tilde{\beta }$) is γ₂ (see Figure 2B). The production rate ${\tilde{k}}_{+}$ is modulated as a function of ($\tilde{\alpha }$, $\tilde{\beta }$, $\tilde{D}$) in order to maintain constant the average 〈c_A〉 = 25 nM. The histograms obtained in the stochastic simulations (blue bars, left column) are in qualitative agreement with the probability densities from the analytical calculations (blue line, left column). When increasing the diffusion coefficient, the system explores different dynamics as revealed by the trajectories shown in the center column. The grey-shaded background shown in the trajectories of c_A indicates the presence of a mRNA molecule in the cell. The density plots (right column) reveals that the diffusion does not contribute to an increase of the intrinsic noise since the spreading of the distributions in a direction perpendicular to the diagonal does not grow.

Figure 5

Noise of the signaling molecule as a function of the diffusion coecient. Noise ${\eta }_{c_A}^2$ as a function of diffusion coefficient $\tilde{D}$ for the set of parameters γ₂ (see Figure 2B): stochastic simulations (circles) and analytical expression. Eq. (18), (solid line). By using the decomposition ${\eta }_{c_A}^2={\eta }_{c_A,\mathrm{int}}^2+{\eta }_{c_A,\text{tran}}^2$ the differences between the computational and the theoretical distributions quantifies the amount of intrinsic noise (squares). As evidenced by the linear regression (blue short-dashed line) the later remains constant and is the main contribution to the total noise only for large diffusion values, $\tilde{D}$ > 10⁴ (see text).