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
- Dorota Herman^{1}Email author,
- Christopher M Thomas^{2} and
- Dov J Stekel^{3}
https://doi.org/10.1186/1752-0509-5-119
© Herman et al; licensee BioMed Central Ltd. 2011
Received: 16 May 2011
Accepted: 29 July 2011
Published: 29 July 2011
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.
Background
IncP-1 Plasmids
Plasmids are autonomous, extra-chromosomal, self-replicating DNA elements typically associated with bacteria [1]. They are important as they can maintain and transfer genes for antibiotic resistance [2] and other important phenotypes, often as part of transposable elements [3]. They are also widely used as tools for genetic manipulation [4, 5]. Finally, they serve as tractable models for a replicating chromosome and for evolution and co-evolution studies [6, 7].
The low-copy number RK2 plasmid belongs to the plasmid incompatibility group P (IncP) of Escherichia coli (IncP-1 of Pseudomonas species). It can persist in most Gram-negative bacteria [8, 9], and is thus referred to as having a broad host range. Moreover, RK2 plasmids can transfer conjugatively to Gram-positive bacteria [10] as well as to some eukaryotic cells [11] and yeasts [12]. RK2 and identical plasmids were first isolated from carbenicillin resistant Pseudomonas aeruginosa and Klebsiella aerogenes in 1969 [13]. Their complete sequence was first compiled in 1994 [13], and improved genome sequence published in 2007 [14].
The RK2 plasmid genome encodes five promoters that are known or are predicted to be cooperatively regulated by KorA and KorB dimers (korA p, kfrA p, trfA p, kleA p, klaA p). Experiments have shown at least 3.4-fold cooperativity at korA p[24]. Therefore, cooperative binding of KorA and KorB to the DNA strand enhances the repression of genes from a particular operon. The most salient example of an operon regulated by KorA and KorB dimers on a single promoter, korA p, is the korABF operon itself, which encodes the two global regulators KorA and KorB, and is known as the central control operon (cco) [13]. Thus the cco is co-operatively and negatively autoregulated by KorA and KorB dimers. KorA dimers bind in the -10 region of korA p [18], overlapping the region where RNA polymerase (RNAP) binds to initiate transcription. KorB dimers bind 39/40 bp upstream of the transcription starting site, immediately upstream of the -35 region of the RNAP binding site; it is plausible that KorB allows RNAP to bind to the DNA strand but prevents it from opening the helix [26]. Nuclear Magnetic Resonance spectroscopy (NMR) and mutational analysis experiments have identified that tyrosine 84 on KorA is crucial for cooperativity between KorA and KorB [27]. Moreover, this residue is directly involved in the protein-protein interaction and has no impact on repression activity. Although the KorB dimer has two contact sites for KorA, it is currently thought that cooperating KorA and KorB act on the DNA strand as a complex in proportion 1:1 [27].
Quantitative Experimental Measurements of the central control operon
Gene expression from the cco has been broadly examined experimentally. The abundances of KorA and KorB in exponential growth in E.coli have been measured in numbers of total monomers as 4000 (1600 nM) and 1000 (400 nM) respectively [18, 28], and the abundance of KorB in a mutant, where co-operativity between KorA and KorB was knocked out by modification of tyrosine 84, was measured as 2300 monomers (920 nM) (unpublished data from Thomas et al.). Affinities of KorA and KorB to their binding sites at the promoter korA p have also been measured [18, 22] and are 12.9 nM and 9.3 nM, respectively. Although affinities of KorA to the KorB-DNA complex and KorB to the KorA-DNA complex have not been measured, the affinity of KorB to the comparable IncC1-DNA complex has been measured as 3.1 nM [22]. Interestingly, the abundances of KorA and KorB proteins are more than two orders of magnitude higher than the affinities of these proteins to the auto-regulating binding sites.
There are varied data on the repression level of the korA p promoter. When KorA and KorB have been over-expressed relative to the wild-type, near total repression (> 800-fold) of the cco has been reported [24]. This indicates complete repression when both KorA and KorB dimers are bound to the DNA. A stronger signal of reporter protein expression has been demonstrated when either KorA or KorB dimers are bound alone. We refer to this phemonenon as partial repression. Recent quantitative work, where KorA and KorB expression was controlled on an inducible promoter at a similar copy number as in the wild type plasmid, suggests that the reduction in promoter activity by KorA and KorB at physiological levels is about 91.8-fold when compared to the case when repressors are absent [28].
The physical mechanisms for partial repression are unclear. Weaker reporter gene signals might result from a mixture of DNA molecules that are either repressor free and therefore "on" or occupied with repressor and therefore "off". Alternatively it may be that occupancy of the promoter by just one or other of the repressors bound to its operator site might just reduce the chance of promoter activity rather than complete inhibition. The difficulty of experimental investigation of these mechanisms arises because KorA or KorB alone would only be bound to the DNA for a small proportion of time.
Mathematical Modelling Approaches
In this work we use mathematical models to better articulate our understanding of regulation of the cco by KorA and KorB. The models facilitate integration of the experimental data in the context of a formal description of the molecular processes of cco regulation. For the model itself, the approach we take is to derive a set of ordinary differential equations (ODEs) describing the dynamical processes that give rise to the measured protein concentrations, and thus to obtain a mathematical description of a biological mechanisms of interest - namely transcription regulation and protein synthesis.
Importantly, some of the available experimental data, such as affinities of binding sites, refer to dynamical processes of gene regulation, which can be thought of as parameters of a dynamical model. Other data, such as measurements of protein expression, are phenotypic, and can be thought of as outputs of a dynamical model. Some information, e.g. the rates of protein synthesis, is experimentally unavailable, and need to be estimated from the data that is available. This situation motivates the use of Bayesian inference [29] to integrate experimental measurements with the model and infer unknown parameters. Bayesian inference provides a robust probabilistic framework for capturing different sources of uncertainty in a modelling environment. It provides outputs in the form of posterior distributions for the parameters, that describe how likely it is that each of the parameters takes particular values, given the experimental measurements and model structure. The Bayesian framework has been recently used in systems biology [30] for different models including dynamical models described by ODEs [31] and stochastic models [32]. These uses include parameter inference for dynamical models of metabolic pathways using steady state measurements [33], for a dynamical model of gene regulation where time series measurements were used [34], or stochastic models of gene expression in single cells [32]. In this work, we employ a Bayesian Monte Carlo Markov Chain (MCMC) method known as the Metropolis-Hastings algorithm [35].
The posterior distributions of parameters obtained from Bayesian inference can be used to determine how sensitive a model is to that parameter: where the distribution is broad, then the model is less sensitive to that parameter, and where it is narrow it is more sensitive to that parameter. If when a prior distribution is supplied, for example for a measured parameter, the posterior distribution is similar to the prior distribution then no further information has been gained. However, if the posterior distribution is different, then additional information has been obtained.
A conceptually different approach to determining the sensitivity of a biological system model to its parameters is provided by Metabolic Control Analysis (MCA; [36, 37]). This measures the extent to which an outcome of the model is sensitive to changes in parameter values. It gives sensitivity read-outs, referred to as control coefficients, based on analytic properties of a set of differential equations. These two approaches have been developed by distinct research communities and have rarely been applied for analysis of the same system.
Aims of this study
In this study, we combine dynamical modelling, Bayesian inference and MCA to integrate the experimental data with our mechanistic knowledge of cco regulation, with the aim of addressing five specific research questions. First, are the measured data compatible with our understanding of the key mechanisms underlying the system operation? Specifically, can the cooperative model explain the high abundance of KorA and KorB proteins relative to their binding strengths? Second, is there sufficient information to robustly estimate unknown and unmeasured parameters, in particular the rates of protein synthesis and the rates at which KorA and KorB dimers separate into monomers, from the available data and molecular mechanisms. If not, what additional information is required? Third, to which parameters is the system sensitive, and to what degree? This knowledge will act as a guide for future experiments to determine what parameters to measure. Fourth, can we use the information about de-repression in the absence of KorA and KorB dimers to learn more about partial repression by these proteins, and thus refine the model? Fifth and finally, to what extent do Bayesian inference and MCA provide comparable or complementary information about parameter sensitivities of the same system of equations fitted to experimental data?
Results
Mathematical model for the regulation of KorA and KorB
We derived the model by first writing a chemical reaction scheme for the biological mechanisms; as this is not used further, the scheme is supplied as an additional file (see Additional file 1: Chemical reaction scheme). The model consists of equations for KorA monomers (A_{1}), KorA dimers (A_{2}), KorB monomers (B_{1}) and KorB dimers (B_{2}). The processes and parameters included in the model are: (i) associations and dissociations of KorA and KorB proteins to/from the DNA strand; (ii) their binding cooperativity; (iii) expression of KorA (k_{A}) and KorB (k_{B}) from the empty DNA strand (D); (iv) from the DNA strand bound by either KorA (X) or KorB (Y) dimers; (v) dimerizations (λ_{A}, λ_{B}) and monomerizations ( σ_{A}, σ_{B}) of the KorA and KorB proteins; (vi) degradations/dilutions of all species (γ_{P}). The equations are:
Equation 1 describes the rate of change of KorA monomers as a function of time. Protein synthesis is considered as a consolidated process that includes transcription, translation and mRNA turnover. This choice of abstraction is appropriate because we have experimental data on protein abundance and dilution, but no available data on mRNA abundance or turnover. The first term (Dk_{A}) represents protein synthesis from DNA to which neither KorA nor KorB is bound. The second term (Xπ_{X}k_{A}) represents protein synthesis from DNA to which KorA but not KorB is bound; the protein synthesis rate k_{A} is scaled by the partial repression parameter π_{X}. The third term (Yπ_{Y}k_{A}) is similar to the second term, but for protein synthesis from DNA to which KorB is bound. No synthesis is possible when both KorA and KorB are bound so there is no term in the equation for this. The fourth term (2σ_{A}A_{2}) represents monomerization, in which a single dimer (A_{2}) dissociates into two monomers. The fifth term (-λ_{A}A_{1}^{2}) represents dimerization, in which two molecules of KorA bind to form a KorA dimer. The final term (-γ_{P}A_{1}) represents protein turnover. Only dilution is taken into account: KorA and KorB are relatively stable proteins and the data modelled come from exponential phase cells, which are growing rapidly. This approximation would not be appropriate if we were modelling stationary phase cells, where the effects of dilution are smaller and so the rate of protein degradation could be more important. Equation 2 describes the rate of change of KorA dimers as a function of time. The terms represent the processes of dimerization, monomerization and protein loss, respectively, and correspond to the equivalent terms in Equation 1. Equations 3 and 4 are similar to Equations 1 and 2; they describe the dynamics for KorB monomers and dimers, respectively.
Parameters and data in simulations
Parameter/Data | Symbol | Distribution | Value(mode) | cv | Unit |
---|---|---|---|---|---|
RK2 abundance | D _{0} | lognormal | 4.5 | 0.5 | nM |
KorA abundance | A_{tot} | lognormal | 1600 [18] | 0. 5 | nM |
KorB abundance | B_{tot} | lognormal | 400 [28] | 0. 5 | nM |
KorB abundance (mutant) | BM_{tot} | lognormal | 920 | 0. 5 | nM |
Scaling parameter for KorA synthesis | π _{X} | uniform | [0,1] | - | - |
Scaling parameter for KorB synthesis | π _{Y} | uniform | [0,1] | - | - |
KorA synthesis | k _{A} | - | - | - | s^{-1} |
KorB synthesis | k _{B} | - | - | - | s^{-1} |
KorA affinity to DNA | k _{1} | lognormal | 12.9 [18] | 0. 5 | nM |
KorB affinity to DNA | k _{2} | lognormal | 9.3 [22] | 0. 5 | nM |
KorA affinity to KorB-DNA | k _{3} | lognormal | 3.1 | 0. 5 | nM |
KorB affinity to KorA-DNA | k _{4} | lognormal | 3.1 [22] | 0. 5 | nM |
KorA monomerization | σ _{A} | - | - | - | s^{-1} |
KorB monomerization | σ _{B} | - | - | - | s^{-1} |
KorA dimerization | λ _{A} | lognormal | 0.001 | 0.05 | nM^{-1}s^{-1} |
KorB dimerization | λ _{B} | lognormal | 0.001 | 0.05 | nM^{-1}s^{-1} |
Protein degradation | γ _{P} | lognormal | 0.0003875 | 0.05 | s^{-1} |
Two distinct sets of parameter values fit and explain the experimental data
Monomerization rates are not important and can be excluded
KorA protein abundance for varying monomerization rates
σ_{A}= σ_{B}[s^{-1}] | A_{1}[nM] | A_{2}[nM] | A_{tot}[nM] |
---|---|---|---|
0 | 10 | 838 | 1686 |
0.001 | 19 | 838 | 1695 |
0.01 | 54 | 835 | 1724 |
Analyses of joint posteriors is essential for estimation of partial repression parameters
Metabolic Control Analysis reveals complementary important and unimportant parameters
The protein synthesis rates show positive control coefficients for the abundance of the associated protein, and a negatively control coefficient for the abundance of the alternate protein. There is also moderate control of the protein abundances by the cooperative affinity of KorA and KorB to the DNA (k_{3}, k_{4}). This results from the occupation of this state for the greatest proportion of time (Figure 2e). The other affinities are less important because the DNA is only in unoccupied or partly occupied states for a small proportion of time. MCA also revealed the low impact of the monomerization rates on the model (Figure 6). This is in agreement with the results obtained with the MCMC-based approach and thus provides further justification for the exclusion of these parameters in future work.
Partial repression models give very different protein synthesis rates and suggest model elimination
Estimated protein synthesis rates and their adequate scaling parameters for five different models
Model | k_{A}[s^{-1}] | π _{X} | k_{B}[s^{-1}] | π _{Y} | ||||
---|---|---|---|---|---|---|---|---|
m | cv | m | cv | m | cv | m | cv | |
11 | 7.9 | 0.2 | - | - | 2.3 | 0.2 | - | - |
uu | 11.5 | 1.4 | 0.72 | 0.5 | 3.2 | 1.3 | 0.72 | 0.4 |
u0 | 14.0 | 2.4 | 0.75 | 0.45 | 4.0 | 2.5 | - | - |
0u | 43.0 | 1.5 | - | - | 11.6 | 1.6 | 0.8 | 0.4 |
00 | 735.0 | 0.3 | - | - | 231.0 | 0.2 | - | - |
Ratio of protein abundance in the model without repression to the model under consideration
Model | 11 plasmids | |||
---|---|---|---|---|
k_{A}[s^{-1}] | A_{tot} [nM] | A_{tot}(no repression) [nM] | Ratio | |
11 | 7.9 | 1576 | 91741 | 58 |
uu | 11.5 | 1649 | 133548 | 81 |
u0 | 14 | 1649 | 162582 | 99 |
0u | 43 | 1627 | 499355 | 307 |
00 | 735 | 1649 | 133548 | 5308 |
Discussion
In this work, we have developed a mathematical model for the regulation of the global regulators KorA and KorB of RK2 plasmids, a natural example of an autogenous, negatively self-regulated transcriptional system. The combination of Bayesian inference using MCMC with dynamical models has allowed us to integrate mechanisms and data in a systems biology context, and enabled us to explore hypotheses about unknown parameters and mechanisms.
One great advantage of the Metropolis-Hastings algorithm is that, if used carefully, it can reveal multimodal distributions of parameter values providing good fit to data, which can be thought of as 'local optima' within classical parameter estimation such as least squares. In our case, these had different biological interpretations. Although we could rule out one set of parameter values based on the knowledge that KorA and KorB are mostly present as dimers, it is easy to envisage applications where such relevant information would not be available. Had that been the case, the methodology would have highlighted a crucial experiment to carry out.
The MCMC approach can also assess the sensitivity of the model to parameter estimates using the spread of the posterior distributions. We have found that our model is sensitive to the rates of protein synthesis, and insensitive to the rates of monomerization. However, there are two limitations. First, MCMC does not indicate the direction of parameter changes. Second, where tight prior information has been provided, such as in our case for the protein turnover rates, we were unable to glean further information.
These limitations have been overcome by the complementary use of MCA for parameter sensitivity analysis, which has provided us with quantitative and qualitative information about parameters. MCA was able to confirm the importance of the protein synthesis rates and the irrelevance of the monomerization rates. In addition, MCA identified negative correlation between the protein synthesis rates for KorA/KorB and the concentration of KorB/KorA, respectively. Finally, MCA revealed the sensitivity of the system to the protein turnover rates, highlighting the need for careful measurement of these parameters. In this model, the protein turnover parameters represent the process of dilution due to cell growth. Since the carriage of plasmids has a potential impact on host cell growth rate, this would suggest that the plasmid system itself is highly sensitive to that impact, and could imply that there would be strong evolutionary pressure to reduce the impact.
With statistical modelling, the identification of correlated parameters can indicate a need to reformulate the model. In this work, the identification of correlations in the joint posterior distributions for the rate of protein synthesis and degree of partial repression raises the question as to whether the model is best specified as described, or whether an alternative formulation might be more suited for inference in which the posterior distributions for the parameters would not be correlated. Two alternative model formulations could be considered. The first formulation would be to specify different parameters for the protein synthesis of KorA and KorB in each of the partial repression scenarios, with appropriate joint priors to ensure that KorA and KorB act as repressors in the model. However, this formulation has two problems. The first is that the resulting model would not be biologically realistic: korA and korB are transcribed together on the same operon, and then translated separately. Thus partial repression has the same impact on both genes, which is captured in the formulation used, but not in this alternative. From that perspective, it perhaps not surprising that the correlation emerges, as the synthesis of the two proteins are (biologically) correlated through joint transcription. The second problem is that the model would have 6 parameters instead of 4, which would significantly reduce the potential for successful parameter estimation. The second alternative formulation would be to build a model that explicitly includes separate equations for transcription of the korABF operon, and translation of each of the proteins. However, such a model would also have many more parameters than the one used. In the absence of experimental data of mRNA abundance and turnover, it would be impossible to infer the unknown parameters. Taking these aspects of alternative model formulations into account, we conclude that our original model is likely to be optimal: it combines biological realism with the capacity to infer parameters for which there is no prior knowledge.
In this model, the interactions between KorA and KorB dimers take place only on the DNA strand. It has been suggested that KorA and KorB dimers interact in solution (David Scott, personal communication). However, since we predict that the DNA is bound by both proteins 98% of the time (Figure 2e), we expect that such potential interactions should not affect the results.
The analysis of various model scenarios representing different possible mechanisms of partial repression have revealed that partial repression by KorA dimers is more compatible with the data than complete repression by KorA. This result appears surprising as KorA binds in the -10 region of the promoter, precisely overlapping the RNAP binding site [18]. While RNAP is necessary to open the DNA strand for transcription initiation, one might expect complete repression by KorA. It is important to note, however, that the model does not explicitly include competition between KorA and RNA polymerase at their overlapping binding sites. As a consequence, the prediction of partial repression by KorA can be explained as an indication that the competition between KorA and RNA polymerase is relevant for limiting transcription.
The simulations used for the model comparison shown in Figure 7 were carried out using point estimates of the model parameters chosen from the posterior distributions for each of the models. An alternative approach is to resample from the posterior distributions to obtain a statistical ensemble of model parameters and base model comparisons on that ensemble. The results of that approach (Additional File 3) lead to the same conclusions: the models that include partial repression by KorA are most compatible with the repression ratio data.
Conclusions
We have devised a mathematical model for the transcription regulation of the central control operon of RK2 plasmids by the global regulators KorA and KorB. By using an approach that couples mechanistic dynamical models with Bayesian inference, we have answered the five specific research questions we posed. First, the experimental data available for the cco system are compatible with our knowledge about its regulatory mechanisms. Second, they enable us to estimate unknown parameters such as protein synthesis and monomerization rates. Moreover, two distinct sets of parameter values can explain the experimental data, highlighting the importance of measuring the relative abundance of dimers to monomers. Third, the monomerization rate is not particularly relevant to the model formulation and can thus be neglected; and estimation of partial repression is dependent on the estimation of the protein synthesis rates and thus these parameters cannot be estimated independently. By using MCA we have also revealed the sensitivity of the model to the protein turn-over rates. Fourth, total repression by KorB alone is incompatible with the de-repression data, and it is likely that competition between KorA and RNA polymerase is an important factor in this particular regulatory system. Fifth, and finally, we have shown that MCMC and MCA are complementary approaches for parameter sensitivity analysis for our model. In summary, we highlight the potential of combining dynamical modelling, statistical inference and sensitivity analyses for deepening our understanding of gene regulatory systems and exploring biological hypotheses about their mechanisms of action.
Methods
Distributions for protein expression data
The phenotyopic read-outs are the total protein monomer abundance of KorA (A_{tot}), KorB (B_{tot}) and KorB mutant KorB (BM_{tot}) (cooperativity between KorA and KorB knock out); these have been measured experimentally [28; unpublished data from Thomas et al.]. Distributions for the data are defined by lognormal distributions with mode equal to measured value and a coefficient of variation equal to 50%, reflecting the level of variability observed experimentally (Table 1). For the simulations used to estimate the scaling parameters, the coefficient of variation for data and the measured parameters was reduced to 10% to aid convergence. Lognormal distributions are chosen because the experimental error is expressed as a percentage of measured abundance. Protein abundances were measured as the numbers of total monomers per cell based on total monomers measured for a known number of bacteria. Molar concentrations of proteins were calculated for the estimated cell volume in the appropriate growth phase, 4.15 μm^{3} for E.coli [28].
Distributions for model parameters
The affinities of KorA to the DNA strand (k_{1}), KorB to the DNA strand (k_{2}), KorA to the KorB-DNA complex (k_{3}), KorB to the KorA-DNA complex (k_{4}) and also DNA abundance (D_{0}) have also been measured experimentally [18, 22], and their prior distributions are also lognormal (Table 1). The affinities of KorA and KorB to the complexes were assumed to be equal to the affinity of KorB to the IncC1-DNA complex at O_{B}1 operator since IncC1 increases the affinity binding factor to 3.2 [22] when KorA increases the binding factor to about 3.4 [16]. Other parameters such as the dimerization rates were defined by diffusion properties (λ_{A}, λ_{B}) and the protein degradation rate (γ_{P}) with the coefficient of variation equal to 5%. The protein degradation rate γ_{P}, which is based only on dilution due to the fact that KorA and KorB are relatively stable proteins, was estimated from bacteria population doubling time of around 43 min. The protein synthesis and monomerization rates were entirely unknown and foe this reason, non-informative uniform priors were used for these parameters. Priors for the scaling parameters π_{X}, π_{Y} were defined by uniform distributions between 0 and 1.
Monte Carlo Simulations
Parameter estimations were carried out using the Metropolis-Hastings algorithm [29] coupled with the steady state of the dynamical model described in the Results. For a selection of the next set of the rate parameters, lognormal proposal distributions were used based on logarithm of random values selected from a normal distribution with mean 0 and standard deviation 0.05; this can be achieved either by adding a normal deviate to the log of the parameter, or by multiplying the parameter by the exponential of the normal deviate. Lognormal proposals are chosen because: (i) they ensure parameters remain positive; (ii) we have no knowledge of the scale of the parameters with un-informative priors and lognormal proposals allow searching over all orders of magnitude in an unbiased way; and (iii) for parameters with informative priors the experimental errors are expressed as percentage of mean and therefore lognormal distributions are more natural. For the two scaling parameters with uniform bounds and priors, a normal proposal distribution with mean 0 and standard deviation 0.08 was used. The variances of these distributions were empirically chosen to ensure acceptance probabilities close to 0.234 [38]. The parameter set was divided into three blocks: parameters defined by lognormal prior distributions and two separate blocks with a parameter defined by uniform prior distribution. Every iteration parameters only from one block were updated while other parameters remained unchanged. The systems did not require any additional searching techniques since satisfactory convergence was achieved. Simulations that consisted of 4,000,000 iterations were long enough to ensure repeatability.
Likelihood calculations
The joint likelihood function was given by the product of three terms, one for each data point, consisting of the probability density for the steady state concentrations of KorA and KorB in the wild-type model, and KorB in the mutant model, under lognormal distributions centred on the measured data. Calculations of the steady states of the system used multidimensional root finding were carried out using the discrete Newton algorithm of the GSL library [39] encoded in C++. For the same set of parameters, the root finding calculations were carried separately for wild type and mutant models. The only differences between these models were the values of the parameters describing protein affinities of cooperative DNA binding - the affinity of either KorA or KorB to DNA-KorB or DNA-KorA complexes, respectively. These affinities were set to the same values as the affinities of KorA or KorB binding to the naked DNA strand (k_{3} = k_{1} and k_{4} = k_{2}).
Calculations using dynamical simulations
Additionally, for model simulations with specific parameter value sets (data presented in Table 2, Table 4, Additional file 2, Figure 2b, c, d, e, and Figure 6), ODE calculations were run in C++ using the cvode solver with Newton iterations provided by the Sundials library [40]. Moreover, for quantitative sensitivity analyses, the MCA [36, 37] were carried out by direct calculations of the concentration control coefficient from the calculations using ODE simulations.
Declarations
Acknowledgements
DH is founded by The Darwin Trust of Edinburgh. Computer simulations were carried out using the Birmingham Biosciences High Performance Compute Cluster, funded by BBSRC REI grant BB/D524624/1. We thank Theodore Kypraios for discussions and advice on using MCMC, and Pawel Herman for helpful comments on the manuscript.
Authors’ Affiliations
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