Global transcription regulation of RK2 plasmids: a case study in the combined use of dynamical mathematical models and statistical inference for integration of experimental data and hypothesis exploration

Abstract

Background: IncP-1 plasmids are broad host range plasmids that have been found in clinical and environmental bacteria. They often carry genes for antibiotic resistance or catabolic pathways. The archetypal IncP-1 plasmid RK2 is a well-characterized biological system, with a fully sequenced and annotated genome and wide range of experimental measurements. Its central control operon, encoding two global regulators KorA and KorB, is a natural example of a negatively self-regulated operon. To increase our understanding of the regulation of this operon, we have constructed a dynamical mathematical model using Ordinary Differential Equations, and employed a Bayesian inference scheme, Markov Chain Monte Carlo (MCMC) using the Metropolis-Hastings algorithm, as a way of integrating experimental measurements and a priori knowledge. We also compared MCMC and Metabolic Control Analysis (MCA) as approaches for determining the sensitivity of model parameters.

Results: We identified two distinct sets of parameter values, with different biological interpretations, that fit and explain the experimental data. This allowed us to highlight the proportion of repressor protein as dimers as a key experimental measurement defining the dynamics of the system. Analysis of joint posterior distributions led to the identification of correlations between parameters for protein synthesis and partial repression by KorA or KorB dimers, indicating the necessary use of joint posteriors for correct parameter estimation. Using MCA, we demonstrated that the system is highly sensitive to the growth rate but insensitive to repressor monomerization rates in their selected value regions; the latter outcome was also confirmed by MCMC. Finally, by examining a series of different model refinements for partial repression by KorA or KorB dimers alone, we showed that a model including partial repression by KorA and KorB was most compatible with existing experimental data.

Conclusions: We have demonstrated that the combination of dynamical mathematical models with Bayesian inference is valuable in integrating diverse experimental data and identifying key determinants and parameters for the IncP-1 central control operon. Moreover, we have shown that Bayesian inference and MCA are complementary methods for identification of sensitive parameters. We propose that this demonstrates generic value in applying this combination of approaches to systems biology dynamical modelling.

Figure 1

Genome map of RK2 plasmid. a) RK2 whole genome map, 60,099 bp; KorA, KorB, TrbA, KorC are global regulators. Arrows to the promoters and signs indicate the binding sites of the regulators and a type of regulation (- is repression), oriV and oriT are origins of vegetative replication and transfer, respectively, blac - Tn1 (transportable element), grey - Tra1 and Tra2 (transfer genes), blue - partitioning function, red - the central control operon. b) The central control operon, consisting of korA, incC2/C1, korB, korG/F genes, is regulated by a single promoter (p) and ends at a terminator (t). The operon is negatively and co-operatively auto-regulated by KorA and KorB. c) Binding sites of KorA and KorB on the korAp promoter: KorB binds 40 bp upstream of the transcription start point (tsp), and KorA binds 33 bp downstream from KorB binding site (O_B1). KorA binds in -10 region, where RNAP binds.

Figure 2

Two distinct parameter value sets. (a) Posterior distributions for the protein synthesis rates for KorB and KorA (see Figure. 3) are bimodal. (b) KorB concentrations from model simulations using typical parameter values from the left peak for: dimers in wild type (purple); monomers in wild type (red); total monomers in wild type (black); and total monomers in the plasmid mutant (cyan). The latter three curves are indistinguishable and have been superimposed with dashed lines. For these parameter values, KorB is mainly present as monomers; the same is true for KorA (data not shown). Although the experimentally measured concentration of KorB increased in the mutant strain, the steady state concentrations from these simulations are within the 50% experimental error associated with Western blot analysis. (c) KorB concentrations from model simulations using typical parameter values from the right peak. KorB is mainly present as dimers; the same is true for KorA (data not shown). (d) Proportions of DNA occupation by KorA or KorB dimers in steady state, with left peak parameter values, for empty DNA (D), KorA-DNA complex (X), KorB-DNA complex (Y) and KorA-KorB-DNA complex (Z). The DNA is mostly unoccupied, allowing full transcription to take place, although the presence of a low concentration of KorA dimers indicates some partial repression by KorA. The promoter is very rarely occupied by KorB dimers. (e) Proportions of DNA occupation for right peak parameter values. The promoter is generally repressed, being occupied by both KorA and KorB dimers, and transcription from the unoccupied state is rare.

Figure 3

Posterior distributions of estimated parameters. (a) Joint posterior distributions of protein synthesis rates (k_A and k_B) in the logarithmic scale. Both parameters are bimodal and the two parameters are positively correlated. (b) Heat map of the joint posterior for the KorB monomerization rate (σ_B) and the KorB synthesis rate (k_B) in the logarithmic scale. The right peak in Figure 2a (high synthesis rate) is associated with low monomerization rate and vice versa.

Figure 4

Posterior distribution of the monomerization rates. The posterior distribution of the monomerization rates for KorA (blue) and KorB (red) proteins on a logarithmic scale shows the irrelevance of the monomerization rates for the model. The apparently multimodal features of the distributions result from granularity of the sampling over such a wide range; these are artefacts and are not statistically reproducible (data not shown).

Figure 5

Posterior distribution of the scaling parameters. (a) Marginal posterior distributions of the scaling parameters (π_X-blue, π_Y -red) of the protein synthesis rates for KorA-DNA and KorB-DNA complexes, respectively. The distributions do not specify clearly any specific region in the parameter space. (b) Marginal posterior distribution of KorA synthesis rate (k_A). (c) Correlated posterior distributions and selected parameter sets pointed by horizontal and vertical lines for KorA synthesis rate (k_A) and the scaling parameter of the synthesis rate from KorA-DNA complex (π_X) (d) The equivalent plot for KorA synthesis rate (k_A) and the scaling parameter of the synthesis rate from KorB-DNA complex (π_Y).

Figure 6

Concentration Control Coefficients. Concentration control coefficients for a) KorA and b) KorB shows the sensitivity of the model to each parameter: k_A - KorA synthesis rate, k_B - KorB synthesis rate, k₁-affinity of O_A1, k₂ - affinity of O_B1, k₃ - affinity of KorA to KorB-DNA complex, k₄ - affinity of KorB to KorA-DNA complex, σ_A, σ_B - monomerization rates for KorA and KorB, respectively, λ_A, λ_B - dimerization rate for KorA and KorB, respectively, γ_P - protein turnover. The synthesis and turn-over rates are particularly important while the dimerization and monomerization rates are unimportant.

Figure 7

Dependence of the unrepressed to repressed ratio on the copy number of plasmids. Models: brown - 0u, red - u0, blue - uu, purple - 11. The reported 91.8-fold repression ratio is shown as a horizontal line. The ratios for the u0 and uu models cross this line at a realistic plasmid copy number. The 0u model crosses the line at an unrealistically low plasmid copy number, and the 00 model has much higher ratios (data not shown). The 11 model crosses the line at an unrealistically high plasmid copy number; models: the first and second signs stand for expression from complexes when KorA or KorB are bound to the DNA, respectively, 1 - no repression, u - partial repression, 0 - total repression.