Antisense interaction between r and a

The full length mRNA of scbR (r) binds to the full length mRNA of scbA (a) and form a complex which prevents further translation of both mRNAs.

Chemical equation

$a + r \rightleftharpoons r-a$

Rate equation

$r= \frac{ k^{-}_{ar}}{K_{ar}}\cdot [r]\cdot [a]-k^{-}_{ar} \cdot [r-a]$

Parameters

The parameters of this reaction is the binding and unbinding rate constant for the interaction of scbR mRNA with scbA mRNA ($K_{ar}$) and ($k^{-}_{ar}$). The parameter values are derived from various RNA/RNA interaction studies in bacteria.

Name	Value	Units	Value in previous GBL model	Remarks-Reference
$K_{ar}$	Failed to parse (Cannot store math image on filesystem.): 0.40-89	$nM$	$0.001 nM^{-1} s^{-1}$ (Bistability range: $6.5 \cdot 10^{-4}-0.16 nM^{-1} s^{-1}$)	In a 1991 review, Y. Eguchi reported association rates for double stranded RNA binding in the range of $1 \cdot 10^{5}-3 \cdot 10^{6} M^{-1} s^{-1}$ (Failed to parse (Cannot store math image on filesystem.): 0.006-0.18 nM^{-1} min^{-1} ). A similar range (~$10^{6} M^{-1} s^{-1}$) is published by S. Brantl in a 2007 review on regulatory mechanisms employed by antisense RNAs. Y. Eguchi 1991 S. Brantl 2007 These values are also in agreement with an in vitro study by Franch et al. on the effect of a U-turn loop structure in RNA/RNA interactions, which employed E. coli hok-RNAs and Sok-RNAs. The study reported binding rate constants $0.15 \cdot 10^{6}-5 \cdot 10^{6} M^{-1} s^{-1}$ ($0.009-0.3 nM^{-1} min^{-1}$) and suggested that ~$10^{6} M^{-1} s^{-1}$ is the upper limit for this interaction. Franch et al. 1999 Franch et al. 1999 Finally, Nordgren et al. published a study on the kinetics of antisense RNA interactions by using Surface Plasmon Resonance (in wild-type and mutant RNA pairs), where the values reported for the association rate constants are in the range of $4.5 \cdot 10^{4}-9.2 \cdot 10^{5} M^{-1} s^{-1}$ (Failed to parse (Cannot store math image on filesystem.): 0.0027-0.0552 nM^{-1} min^{-1} ) and the resulting equilibrium dissociation constant (calculated from the other parameters and measured by gel shift assay) are in the range of $4 \cdot 10^{-10}-8.9 \cdot 10^{-8} M$ (Failed to parse (Cannot store math image on filesystem.): 0.4-46 nM ). The experiments measured in vitro the rate of complex formation and in vivo the efficiency of control. Nordgren et al. 2001 Finally, Lima et al. (see table below) in a study measuring kinetics on synthetic oligonucleotides, reported a range of association constants between Failed to parse (Cannot store math image on filesystem.): 6 \cdot 10^{6}-1 \cdot 10^{7} M^{-1} s^{-1} (Failed to parse (Cannot store math image on filesystem.): 0.36-0.6 nM^{-1} min^{-1} ). Therefore, the overall range of values for the association rate $k_{on_{ar}}$ is Failed to parse (Cannot store math image on filesystem.): 0.0027-0.6 nM^{-1} min^{-1} .
$k^{-}_{ar}$	Failed to parse (Cannot store math image on filesystem.): 0.006-6	$min^{-1}$	$0.01 s^{-1}$ (Bistability range: $0-0.2 s^{-1}$)	In the study by Nordgren et al., dissociation rate constants in the range Failed to parse (Cannot store math image on filesystem.): 4 \cdot 10^{-4}-7.7 \cdot 10^{-3} s^{-1} (0.024-0.0462 min^{-1}) were reported (see figure above). These values are also consistent with the ones published by Lima et al. in a study on antisense oligonucleotide hybridization kinetics, who reported dissociation rate constant values in the range $1 \cdot 10^{-4}-2 \cdot 10^{-2} s^{-1} (0.006-1.2 min^{-1})$ Lima et al. 2001 Although the values retrieved by literature are between $0.006$ and $1.2 min^{-1}$, we wish to explore a wider range of values ($0.006-6 min^{-1}$) in order to take into account the possibility that high GC contents may affect the kinetic rate constants.

Parameters with uncertainty

When deciding how to describe the uncertainty for these parameter we must take into consideration the fact that many of the reported values are derived from in vitro experiments and correspond to mRNA sequences or fragments of different bacteria species. This means that there might be some difference between actual parameter values and the ones reported in literature. These facts influence the quantification of the parameter uncertainty and therefore the shape of the corresponding distribution.

With regards to the $K_{ar}$ the values that correspond to the wild type RNAs are between $1-10 nM$, therefore we put the weight of the distribution in this range and we consider as least likely the larger values as they correspond to mutant RNAs. Therefore, the mode of the log-normal distribution is set to $5 nM$ and the confidence interval factor is $10$. Thus the range where 95.45% of the values are found is between Failed to parse (Cannot store math image on filesystem.): 0.5 and Failed to parse (Cannot store math image on filesystem.): 50 .

Similarly, the values reported for $k^{-}_{ar}$ are within the range $0.06-1.2 min^{-1}$, we put the weight of the distribution within this range but also explore the whole range of values. Therefore, the mode of the log-normal distribution is set to Failed to parse (Cannot store math image on filesystem.): 0.6 and the confidence interval factor is $10$. Thus the range where 95.45% of the values are found is between $0.06$ and Failed to parse (Cannot store math image on filesystem.): 6 nM .

Finally, the probability distribution for $k_{on_{ar}}$ is defined accordingly, in order to allow the exploration of the full range of the values retrieved from literature. Therefore, the mode is set to Failed to parse (Cannot store math image on filesystem.): 0.04 nM^{-1} min^{-1} and the confidence interval factor is $12$. In this way the range where 95.45% of the values are found is between $0.0033$ and Failed to parse (Cannot store math image on filesystem.): 0.48 nM^{-1} min^{-1} . The initial distribution parameters are μ=-2.3296 and σ=0.9431.

Since the three parameters are interdependent, thermodynamic consistency also needs to be taken into account. This is achieved by creating a bivariate system as described here. Since $k_{on_{ar}}$ is the parameter with the largest geometric coefficient of variation, this is set as the dependent parameter as per: $k_{on_{ar}}=\frac{k^{-}_{ar}}{K_{ar}}$, and an updated probability distribution is defined. The location and scale parameters of $k_{on_{ar}}$ (μ=-2.1202 and σ=1.26021) were calculated from those of $K_{ar}$ and $k^{-}_{ar}$.

The probability distributions for the three parameters, adjusted accordingly in order to reflect the above values, are the following:

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The correlation matrix which is necessary to define the relationship between the two marginal distributions ($K_{ar}$,$k^{-}_{ar}$) of the bivariate system is derived by employing random values generated by the two distributions.

The final parameters of the distributions of the multivariate system are:

Parameter	μ	σ	Correlation matrix
$K_{ar}$	Failed to parse (Cannot store math image on filesystem.): 2.4035	$0.8911$	N/A
$k_{on_{ar}}$	Failed to parse (Cannot store math image on filesystem.): -2.1202	Failed to parse (Cannot store math image on filesystem.): 1.26021	Failed to parse (Cannot store math image on filesystem.): \begin{pmatrix} 1 & 0.5937 \\ 0.5937 & 1 \end{pmatrix}
$k^{-}_{ar}$	Failed to parse (Cannot store math image on filesystem.): 0.2833	$0.8911$

The multivariate system of the normal distributions ($ln(k_{on_{ar}})$ and $ln(k^{-}_{ar})$) and the resulting samples of values are presented in the following figure:

In this way, a system of distributions is created where each distribution is described and constrained by the other two. Therefore, the parameters will be sampled by the two marginal distributions in a way consistent with our beliefs and with the relevant thermodynamic constraints. However, since the model's reaction rate requires the parameters $K_{ar}$ and $k^{-}_{ar}$, and not the $k_{on_{ar}}$, the value for $K_{ar}$ is calculated by the parameters sampled from the other two distributions in an additional step, as per $K_{ar}=\frac{k^{-}_{ar}}{k_{on_{ar}}}$.