Queueing up for enzymatic processing: correlated signaling through coupled degradation

Abstract

High-throughput technologies have led to the generation of complex wiring diagrams as a post-sequencing paradigm for depicting the interactions between vast and diverse cellular species. While these diagrams are useful for analyzing biological systems on a large scale, a detailed understanding of the molecular mechanisms that underlie the observed network connections is critical for the further development of systems and synthetic biology. Here, we use queueing theory to investigate how 'waiting lines' can lead to correlations between protein 'customers' that are coupled solely through a downstream set of enzymatic 'servers'. Using the E. coli ClpXP degradation machine as a model processing system, we observe significant cross-talk between two networks that are indirectly coupled through a common set of processors. We further illustrate the implications of enzymatic queueing using a synthetic biology application, in which two independent synthetic networks demonstrate synchronized behavior when common ClpXP machinery is overburdened. Our results demonstrate that such post-translational processes can lead to dynamic connections in cellular networks and may provide a mechanistic understanding of existing but currently inexplicable links.

Figure 1

Coupling via common enzymatic machinery: connection to queueing theory. (A) Rate-limited processing can couple the numbers of different job types in a queue. Different jobs (yellow and cyan squares) are forming a queue to be processed and removed from the queue by a fixed number of servers with finite processing capacity. If the arrival rate exceeds the maximum processing rate, the servers become overloaded, the queue lengthens dramatically, and the numbers of jobs competing for the attention of the servers become tightly correlated. (B) Individual stochastic trajectories for a queueing system in three different conditions demonstrate correlation resonance. We consider the system with L=100 servers, each with a processing rate of μ=10 min⁻¹. Jobs renege, that is, abandon the system, at the first-order rate γ=ln2/20.0 min⁻¹. Yellow corresponds to jobs of type 1 and count x₁, while blue corresponds to jobs of type 2 and count x₂. Trajectories are normalized by their arrival rates. (Top) Underloaded condition, with λ₁=0.25 μL, the arrival rate of job 1, and λ₂=0.5 μL, the arrival rate of job 2. The total rate of arrival Λ=0.75 μL is less than the total processing rate μL. (Middle) Balanced condition, with λ₁=0.5 μL, λ₂=0.5 μL, and Λ=μL. (Bottom) Overloaded condition, with λ₁=0.75 μL, λ₁=0.5 μL, and Λ=1.25 μL.

Figure 2

The E. coli stress response network employs queueing as a signaling mechanism to ensure the most rapid response possible to adverse conditions. (A) Certain types of stress cause the accumulation of a large amount of aberrant proteins, which are targeted for rapid degradation and compete with the master stress regulator, σ^s, for a limited amount of ClpXP machinery. This leads to a decrease in the effective degradation rate of σ^s, allowing it to build up rapidly and initiate an immediate response. See the Supplementary Information for a precise definition of the model and for simulation details. (B) For a stochastic queueing model with 100 ClpXP molecules that each have processing rate μ=10 min⁻¹, with cells dividing every 20 min, a scan of the mean steady-state level of σ^s with respect to the stress level (mistranslated protein production rate λ_m) demonstrates a very sensitive response of the system once the system has crossed the balance point (respectively colored dashed lines). Different σ^s basal production rates λ_σs are indicated in the figure. (C) The dynamic response of σ^s to a 10-min pulse of stress demonstrates that queueing provides for a very fast and dynamic response. Importantly, the response is highly specific temporally, as σ^s only remains at a high level during the time that excess aberrant proteins are around. We assume the basal rate λ_σs=500 min⁻¹, while λ_m=1000 min⁻¹ during the pulse of stress but λ_m=0 min⁻¹ otherwise. (Inset) The mean response amplitude of σ^s to a periodic stress is strongest, especially at fast frequencies, when the system is on average near balance or slightly overloaded. The rate λ_m is taken to be a constant plus a sinusoid with amplitude 100 min⁻¹ and given frequency. (D) The adaptive response leads to positive correlations between σ^s and mistranslated protein levels for a broad range of parameters, peaking near the balance point (respectively colored dashed lines). Parameters are the same as those used in (B).

Figure 3

Coupled enzymatic degradation of yellow and cyan LAA-tagged fluorescent proteins by ClpXP machinery in E. coli. (A) Schematic network diagram of the synthetic circuit. YFP is produced by the P_LtetO−1 promoter, which is repressed by TetR in the absence of doxycycline. CFP is produced by the P_lac/ara−1 promoter, which is activated by AraC in the presence of arabinose. Both CFP and YFP molecules are tagged with identical LAA tags and are targeted for degradation by the ClpXP complexes. (B) Induction plots for a single fluorescent protein (GFP) produced by the P_lac/ara−1 promoter. IPTG is held at 1 mM to fully relieve repression by lacI, and production is tuned by addition of arabinose. Blue and red symbols indicate untagged and LAA-tagged protein, respectively. Squares are mean protein counts, while diamonds are median protein counts. Solid lines are steady-state model fits to the data (including those in C and D). The red line stochastic queueing model prediction for enzymatic protein degradation compares favorably with the data. (C) Mean steady-state expression of CFP as a function of doxycycline concentration at three different levels of arabinose in triplicate flow cytometry measurements. Strong coupling is observed between CFP and YFP. Protein counts are reported using a combination of two-color flow cytometry and western blots. (D) The means of CFP and YFP increase simultaneously as the doxycycline concentration is increased. The color of the symbols corresponds to (C). In both (C) and (D), results for the stationary state of a fitted stochastic queueing model are included as solid lines. Supposing a doubling time τ_d≈30 min, we find an enzymatic degradation rate μ=7.6 × 10³ min⁻¹ for the model provides a good fit to the plotted results. Values for the production rates of YFP at given dox concentrations and for the production rates of CFP at given arabinose concentrations were determined from a best fit to the data. The qualitative result of this fit is that CFP only measurably increases when YFP becomes comparable in magnitude, consistent with a slightly overloaded queue. See Supplementary Information (Supplementary Figures S1, S5, and S6) for fitting details and parameters. Source data is available for this figure in the Supplementary Information.

Figure 4

Dynamic behavior of a synthetic signaling network. (A) Using a microfluidic platform capable of generating a time-dependent induction signal and multi-color single-cell fluorescence measurements, a large population of E. coli expressing the synthetic network were subject to a periodic series of doxycycline pulses (red), such that the system at zero doxycycline relaxed to a dim state, but the system at high doxycycline relaxed to a bright state. The total YFP and CFP fluorescence, integrated over the entire colony, demonstrates the direct response of the P_LtetO−1 promoter, producing YFP (green) as TetR is periodically deactivated, as well as the indirect response of the CFP signal (blue) due to the time-dependent saturation of the ClpXP machinery. (B) Trajectories for several individual cells demonstrate the response at a single-cell level. Source data is available for this figure in the Supplementary Information.

Figure 5

Correlations between two independent reporter proteins. (A) A two-dimensional histogram depicts correlations between YFP and CFP levels in individual cells throughout the entire experiment duration from Figure 4. The value in each rectangular bin is log scaled (value of log₁₀(1+n), with n the number of counts in a bin). (B) The Pearson's correlation coefficient between the two reporters as a function of time. Correlation is reported for the middle 80% of cells by CFP brightness (eliminating outliers that contribute bias to the correlation coefficient), and the estimated correlation coefficient for typical cells is observed to oscillate with the drive period.

Figure 6

Coupled degradation can serve to indirectly couple gene circuits. (A) Circuit diagram for a variant of a synthetic gene oscillator discussed previously (Stricker et al, 2008), expressed alongside an AHL-inducible, LAA-tagged CFP (see Supplementary Information for details). Competition for enzymatic degradation among LAA-tagged proteins allows CFP to interact with the oscillator by slowing the degradation of the oscillator components. Since fast enzymatic decay is thought to strongly influence the period and robustness of certain gene oscillators (Wong et al, 2007; Mather et al, 2009), the character of GFP oscillations should be closely tied to CFP expression. (B) Cells containing the circuit in (A) were grown in a microfluidic device to test the influence of CFP production on GFP oscillations (see Supplementary Information for details). Two single-cell trajectories for GFP and CFP fluorescence (solid lines) show regular oscillations in GFP fluorescence in the absence of external AHL, that is, at low CFP fluorescence. However, addition of 15 nm AHL (time of induction start is indicated by a vertical dotted line) introduces CFP into the system, causing the GFP oscillations to slow and the CFP signal to oscillate as a result of indirect coupling due to queueing. Coupling is also observed in the mean fluorescence across a region of cells (mean fluorescence as dashed lines), where increasing mean CFP fluorescence is associated with an increase in mean GFP fluorescence. (C) A representative trajectory where both CFP and GFP become and remain bright simultaneously. Positive coupling between trajectories is apparent.