Binding of R2 to A

The ScbR homo-dimer (R₂) forms a complex with ScbA (A).

Chemical equation

$A + R_{2} \rightleftharpoons A-R_{2}$

Rate equation

$r= \frac{k^{-}_{3}}{K_{d3}}\cdot [A]\cdot [R_{2}] - k^{-}_{3}\cdot [A-R_{2}]$

Parameters

The parameters of this reaction are the dissociation constant for binding of ScbR to ScbA ($K_{d3}$) and the dissociation rate for binding of ScbR to ScbA ($k^{-}_{3}$). Since there is no concrete evidence of the existence of the ScbA-ScbR complex so far, it is possible that the interaction between the two proteins is unstable/ transient and therefore the parameter values reflect this belief. The values of such complexes according to the literature , lie in the millimolar or micromolar scale.

• 

  Ozbabacan et al. 2011
• 

  Perkins et al. 2010

Name	Value	Units	Value in previous GBL model	Remarks-Reference
$K_{d3}$	$10^{3}-10^{6}$	$nM$	$0.083 nM^{-1} s^{-1}$ (Range tested: $10^{-7}-10^{-1} nM^{-1} s^{-1}$) (Bistability range: $0.083-0.12 nM^{-1} s^{-1}$)	According to the Ozbabacan et al. association constants for transient protein protein interactions lie in the millimolar or micromolar range. Ozbabacan et al. 2011
$k^{-}_{3}$	$145-1.85 \cdot 10^{5}$	$min^{-1}$	$630 s^{-1}$ (Range tested: $0-10^{3} s^{-1}$) (Bistability range: $460-630 s^{-1}$)	According to Northrup et al. the $K_{on3}$ of protein-protein bond formations occur in the order of $10^{6} M^{-1}s^{-1}$. Northrup et al. 1992 These values are also supported by Ozbabacan et al. who claim that association rates are usually in the range $10^5-10^6 M^{-1}s^{-1}$ ($0.006-0.06 nM^{-1}min^{-1}$). Ozbabacan et al. 2011 Therefore, these values are used to generate the probability distribution for $k_{on3}$ as described in the following section. Afterwards, the log-normal distribution for the dissociation rate of the ScbR-ScbA binding $k^{-}_{3}$ is derived from the distributions of $k^{-}_{3}$ and $K_{d3}$.

Parameters with uncertainty

Since the values we are using for this parameter correspond to generic association constant values of a wide range of protein-protein interactions and not specifically to GBL or related systems, we wish to explore the whole range of values and investigate the conditions under which the ScbR-ScbA complex formation would be feasible. Therefore, we set the mode of the log-normal distribution for $K_{d3}$ to $7 \cdot 10^{4} nM$ and the confidence interval factor to $13$. Thus, the range where 95.45% of the values are found is between $5.385 \cdot 10^{3} nM$ and $9.1 \cdot 10^{5} nM$.

Similarly, by following the same reasoning, the mode of the log-normal distribution for $k_{on3}$ is set to $0.019 nM^{-1} min^{-1}$ and the confidence interval factor to $3.2$. This means that the range where 95.45% of the values are found is between $0.0059 nM^{-1} min^{-1}$ and $0.0608 nM^{-1} min^{-1}$.

Since the two parameters are interdependent, thermodynamic consistency also needs to be taken into account. This is achieved by creating a bivariate system as described here. Since no information was retrieved for $k^{-}_{3}$ and therefore is the parameter with the largest geometric coefficient of variation, this is set as the dependent parameter as per: $k^{-}_{3}=k_{on3} \cdot k_{d3}$. The location and scale parameters of $k^{-}_{3}$ (μ=8.3942 and σ=1.0961) were calculated from those of $K_{d3}$ and $k_{on3}$.

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 ($k_{d3}$,$k^{-}_{3}$) of the bivariate system is derived by employing random values generated by the two distributions.

The parameters of the distributions of the multivariate system are:

Parameter	μ	σ	Correlation matrix
$k_{on3}$	$-3.6938$	$0.51917$	N/A
$K_{d3}$	$12.088$	$0.96535$	$\begin{pmatrix} 1 & 0.871 \\ 0.871 & 1 \end{pmatrix}$
$k^{-}_{3}$	$8.3942$	$1.0961$

The multivariate system of the normal distributions ($ln(k_{d3})$ and $ln(k^{-}_{3})$) 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.