# A Bacterial Form I' Rubisco Has a Smaller Carbon Isotope Fractionation than Its Form I Counterpart
## 2 Materials and Methods
### Calculation of KIE  
We first pre-processed the data by assessing which data points to fit. We expected the \(\delta^{13}\)C of CO\({}_{2}\) to increase following an exponential curve that eventually reaches an asymptote, but the last few data points start to decrease in \(\delta^{13}\)C. This may be due to a variety of reasons, including: (1) Ambient CO\({}_{2}\) contaminating the exetaer containers as they are left out after the reaction; (2) re-equilibration of the aqueous and gaseous inorganic carbon pools; or (3) instrument error upon needle sampling of exetaer vial. Because exponential curves are linear in a log-log space, we therefore log-transformed the data points then systematically fit a linear regression through varying sets of data and calculated the resulting error (adjusted R\({}^{2}\) value). The adjusted R\({}^{2}\) value consistently decreased after data point 9 for L\({}_{8}\), and after data point 8 for L\({}_{8}\)S\({}_{8}\) (Figure S1B,C). Therefore, we proceeded to use data points 1-9 for L\({}_{8}\) and 1-8 for L\({}_{8}\)S\({}_{8}\).  
We then converted time to \(f\), the fraction of the inorganic C pool remaining. Since RuBP was the limiting substrate, we could calculate the moles of CO\({}_{2}\) consumed if we assume: i) A 1:1 ratio of RuBP to CO\({}_{2}\) was utilized by Rubisco, and ii) full consumption of the RuBP pool. For each rubisco assay, 125 \(\upmu\)mol of RuBP and 9.84 \(\upmu\)mol of NaHCO\({}_{3}\) were added. Therefore, 7.87% of the initial CO\({}_{2}\) pool was consumed, or \(F\) = 0.9213. We then assume that \(f\) = 1 at \(t\) = 0, and \(f\) = 0.9213 at the upper bound of the fit. A general model of y = a*EXP(-b*x) + c was applied to the data, though with carbon isotope data in the \({}^{13}\)R format instead of the \(\delta^{13}\)C format because \({}^{13}\)R values can be manipulated arithmetically while \(\delta^{13}\)C values cannot [40]. The model was then fitted using the non-linear least squares function (call: \(nls\)); R Statistical Software (v4.1.0; R Core Team 2021, [41])). Model outputs are shown in Table S4 and Figure S2.  
Time was then converted to \(f\) using the equation:  
\[f=1-\left(\frac{R_{i}-R_{0}}{R_{upper}-R_{0}}\times(1-F)\right)\]  
where \(R_{0}\) is the first measured \({}^{13}R\) value in each set of data, \(R_{upper}\) is the fitted parameter \(c\) from the model and \(F\) = 0.9213, which is calculated from the amount of RuBP added to the assay.  
Next, a correction was done to account for the C isotope fractionation between CO\({}_{2}\) and HCO\({}_{3}\)\({}^{-}\) at equilibrium, where CO\({}_{2}\) is \(\sim\)8% lighter (more negative \(\delta^{13}\)C value) than HCO\({}_{3}\)\({}^{-}\)[42]. We followed the correction outlined in [25] where the adjustment is applied before linear regression in a Rayleigh plot:  
\[R/R_{0}~{}adj.=\frac{(fR/R_{0})^{C}}{f}\]  
where \(C\) = \((1.009+10^{\prime}(\mathrm{pK}-\mathrm{pH}))/(1+10^{\prime}(\mathrm{pK}- \mathrm{pH}))\). The pK is that of carbonic acid, for which we used a value of 6.4 [43]. The pH of the L\({}_{8}\)S\({}_{8}\) assay was 8.49, and the pH of the L\({}_{8}\) assay was 8.52.  
Finally, a Rayleigh plot was made for each rubisco plotting \(\mathrm{ln}(^{13}R/^{13}R_{0})_{adj.}\); \({}^{10}\)00 vs. \(-\mathrm{ln}(f)\) (Figure S3). The best fit slope, \(D\), was calculated using a linear regression (call: \(lm\)); R Statistical Software (v4.1.0; R Core Team 2021, [41])). \(D\) was then converted to \(\Delta\), the KIE, using the equation \(\Delta\) = \(D/(1-D/1000)\)[25]. Doing so, we found the KIE of the L\({}_{8}\)S\({}_{8}\) rubisco to be 22.42 \(\pm\) 2.37, and 16.25 \(\pm\) 1.36 for the L\({}_{8}\)s rubisco. Results are shown in Table 1.