Formation of homo-dimer R2

Two ScbR (R) proteins bind together to form an ScbR homo-dimer (R₂).

Chemical equation

$2R \rightleftharpoons R_{2}$

Rate equation

$r= \frac{k^{-}_{6}}{K_{d6}}\cdot [R]^{2} - k^{-}_{6}\cdot [R_{2}]$

Parameters

The parameters of this reaction are the dissociation constant for binding of one ScbR to another ($K_{d6}$) and the dissociation rate for binding of one ScbR to another ($k^{-}_{6}$).

Name	Value	Units	Value in previous GBL models	Remarks-Reference
$K_{d6}$	$2.58-42.9$	$nM$	N/A	Majka et al. published a study on dimerization of the initiator Protein DnaA of Streptomyces and on its mutants, where they report dissociation constants in the range $35.3-114 nM$. Majka et al. 2001 These values agree with Ozbabacan et al. and Nooren et al. who state that strong protein-protein interactions such as homodimerization have equilibrium dissociation constants $< 10^{-6} M$ and mostly in the nanomolar range. As the protein homodimer is quite stable, we will focus more on the lower $K_{d}$ values (<43nM).
$k^{-}_{6}$	$0.144-51.2$	$min^{-1}$	N/A	According to Northrup et al. the $k_{a}$ of protein-protein bond formations occur in the order of $10^{6} M^{-1}s^{-1}$. Northrup et al. 1992 Additionally, according to Ozbabacan et al. the binding rates can reach $10^{9} M^{-1}s^{-1}$ in fast binding reactions. Ozbabacan et al. 2011 Therefore, the range $0.5 \cdot 10^{6} M^{-1}s^{-1}-10^{9} M^{-1}s^{-1} (0.03-60 nM^{-1} min^{-1})$ is used to generate the probability distribution of $k_{on6}$ as described in the following section. Afterwards, the log-normal distribution for the dissociation rate $k^{-}_{6}$ of the ScbR homo-dimer formation is derived from the distributions of $K_{d6}$ and $k_{on6}$.

Parameters with uncertainty

When deciding how to describe the uncertainty for this parameter we must take into consideration that the values reported in literature correspond to in vitro testing of different protein-protein interaction and dimerization reactions than ScbR, although they refer to another Streptomyces protein (DnaA). This means that there might be a 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 distributions. By assigning the appropriate weights to the parameter values and using the method described here, the appropriate probability distributions were designed.

More specifically, for $K_{d6}$ the mode of the log-normal distribution is $3.89 nM$ (value that is close to the wild type protein homodimerization). However, in order to explore sa larger part of the nanomolar scale when sampling for parameter values, the Spread is calculated to be $1.9$. In this way, the range where 68.27% of the values are found is between $2$ and $7.4nM$.

With regards to the parameter $k_{on6}$, in order to explore the full range of plausible values, the mode of the log-normal distribution is set to $0.3 nM ^{-1 } min^{-1}$ and the Spread is $15.1$. Thus the range where 68.27% of the values are found is between $0.02$ and $4.66$ $nM ^{-1 } min^{-1}$.

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 $k^{-}_{6}$ and therefore is the parameter with the largest geometric coefficient of variation, this is set as the dependent parameter as per: $k^{-}_{6}=k_{on6} \cdot K_{d6}$. The location and scale parameters of $k^{-}_{6}$ (μ=2.5303 and σ=1.5324) were calculated from those of $K_{d6}$ and $k_{on6}$.

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

The values retrieved from literature and their weights are indicated by the blue dashed lines, and the uncertainty for each value is indicated using the reported experimental error (green lines) or a default value of 10% error (orange lines). The correlation matrix which is necessary to define the relationship between the two marginal distributions ($k_{d6}$,$k^{-}_{6}$) of the bivariate system is derived by employing random values generated by the two distributions.

The parameter information of the distributions of the multivariate system is:

Parameter	Mode	Spread	μ	σ	Correlation matrix
$k_{on6}$	$0.307$	$15.1$	$0.85865$	$1.4273$	N/A
$K_{d6}$	$3.89$	$1.9$	$1.6716$	$0.55779$	$\begin{pmatrix} 1 & 0.2176 \\ 0.2176 & 1 \end{pmatrix}$
$k^{-}_{6}$	N/A	N/A	$2.5303$	$1.5324$

The multivariate system of the normal distributions ($ln(k_{d6})$ and $ln(k^{-}_{6})$) 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.