Adaptation for protein synthesis efficiency in a naturally occurring self-regulating operon

Abstract

The korAB operon in RK2 plasmids is a beautiful natural example of a negatively and cooperatively self-regulating operon. It has been particularly well characterized both experimentally and with mathematical models. We have carried out a detailed investigation of the role of the regulatory mechanism using a biologically grounded mechanistic multi-scale stochastic model that includes plasmid gene regulation and replication in the context of host growth and cell division. We use the model to compare four hypotheses for the action of the regulatory mechanism: increased robustness to extrinsic factors, decreased protein fluctuations, faster response-time of the operon and reduced host burden through improved efficiency of protein production. We find that the strongest impact of all elements of the regulatory architecture is on improving the efficiency of protein synthesis by reduction in the number of mRNA molecules needed to be produced, leading to a greater than ten-fold reduction in host energy required to express these plasmid proteins. A smaller but still significant role is seen for speeding response times, but this is not materially improved by the cooperativity. The self-regulating mechanisms have the least impact on protein fluctuations and robustness. While reduction of host burden is evident in a plasmid context, negative self-regulation is a widely seen motif for chromosomal genes. We propose that an important evolutionary driver for negatively self-regulated genes is to improve the efficiency of protein synthesis.

Figure 1. Model representation.

(a) Multi-scale stochastic model of gene regulation of the RK2 central control operon. The population dynamic layer contains plasmid replication as an independent Poisson process and continuous host cell growth at an exponential rate with cell division when the cell doubles its initial size. After each division, only one daughter cell is tracked. The plasmid molecular dynamics layer contains protein synthesis, dimerization, cooperative binding of KorA and KorBdimers to the promoter, and repression of the operon. (b) The four models for comparison: CCO is the wild type system; CCOnoC is a system with two regulators but with no cooperativity between then regulators; CCOregB is a system with just a single dimeric regulator; CCOnoR is a system with no regulation. In order to ensure comparability, the rates of protein synthesis are tuned so that protein abundance is the same in all models (see Methods ). (c) Stochastic model formulation for the CCO model: A₁, A₂, B₁, B₂– KorA and KorB monomer and dimer, respectively; D, X, Y, Z – states of the DNA strand: empty DNA, KorA-DNA, KorB-DNA, KorA-KorB-DNA complexes respectively; k_p – plasmid replication rate, k_A, k_B – maximal KorA and KorB synthesis rates, π_X, π_Y - scaling parameters for the protein synthesys, k_onP – protein dimerization rate, k_onD – protein association rate to the DNA; k_off1, k_off2, k_off3, k_off4– KorA, KorB dissociation rates, KorA for KorA-DNA complex, KorB from KorB-DNA complex, KorA from KorA-KorB-DNA complex and KorB from KorA-KorB-DNA complex, respectively. Each of the other models is derived by appropriate simplification of this model. For CCOnoC, k_off1 = k_off3 and k_off2 = k_off4; for CCOregB, the reactions between KorAdimers and DNA are removed from the model; for CCOnoR, the reactions between KorBdimers and DNA are also removed from the model.

Figure 2. Results on the various optimizations by the central control operon regulation.

Comparison of the ability of the four models to optimize different desirable properties. With the exception of (c), bar heights are means and error bars are standard errors across 20 replicates. (a) Fluctuations in KorB regulator concentration in their steady state for models that include plasmid replication and (b) exclude plasmid replication. There is little improvement in protein fluctuations between the models, with a minor improvement observed when a second regulator is introduced when plasmid copy number fluctuations are also present. (c) Dynamics of KorA total monomer concentrations after a new host transfection; means of arising concentrations over time are indicated by solid lines, the shadows show standard deviations. The horizontal solid line indicates a mean KorA concentration and dashed line a half of the mean concentration. (d) Times of reaching a half of a mean KorA concentration with standard errors; the systems with strong regulation reach a half of mean KorA concentration quicker than with weak or without regulation. (e) Number of mRNA produced per generation after the model has reached steady state. CCOnoR (grey) has no regulation; CCOregB (yellow) has a single regulatory protein; CCOnoC (cyan) has two regulatory proteins; CCO (red) has two regulatory proteins with cooperativity. The translation rate is tuned so that the total protein abundance is held constant. There is a small reduction in mRNA usage after the introduction of a single regulator; a second regulator brings a 20-fold improvement in mRNA usage; the introduction of cooperativity brings a further 3-fold improvement. (f) The transcription rate is tuned so that protein abundance is held constant. All four architectures are equivalent.

Figure 3. Robustness of central control operon regulation results.

Comparisons between the different models are robust to uncertainty in parameter values. Bar heights are means; error bars are standard deviations from 1000 resamples of the parameter values. (a) Differences between the models in control of protein fluctuations show a similar pattern to Figure 2(a), demonstrating that the result is robust to uncertainty in parameter values. (b) A similar pattern is plasmid replication is included, as per Figure 2(b). (c) Difference in response times between the four models is comparable to that shown in Figure 2(d). Note, however, that the difference in mean response time between the models with and without cooperativity is small relative the variability due to uncertainty in parameter values. (d) Difference in mRNA usage between the four models when transcription rate is held constant shows the same overall result as Figure 2(e). The difference between the CCOnoC and CCO model results are more than 3-fold, comparable with Figure 2(e), and which is difficult to see in the bar plot. A comparison with translation rate held constant, as per Figure 2(f), is shown in Figure S2.

Figure 4. Optimization of the mRNA production by the central control operon regulation.

mRNA production per one generation for a single negative regulation for different repressor-DNA affinities and different levels of reduction in expression while a repressor is bound to the DNA. The red arrows represent statistically significant differences of mRNA production. The top-left hand corner approximates an unregulated system while the bottom-right corner approximates the wild-type system of RK2. The implied evolutionary trajectory is that first increased efficacy of repression would evolve following which the protein-DNA interaction would become stronger.