## 9 Rubisco Kinetics
The postulated \"trade-off\" between CO\({}_{2}\)-specificity and carboxylation reaction rate of Rubisco was criticized by the capstone study published by [PERSON] and [PERSON] in 2018 [49]. Giving a complete description of the reaction elementary steps and related microscopic kinetic constants, the study aimed at evaluating whether the Rubisco enzyme is tightly constrained by its active-site chemistry or whether the implementation of powerful tools as directed evolution techniques may have the opportunity to break the above cited \"trade-off\" and improve simultaneously both \(k_{cat,C}\) and \(\mathrm{S_{C/O}}\) parameters. Results of the analysis point out that simultaneous improvement of CO\({}_{2}\)-specificity and turnover rate is possible considered that, more likely, the stronger molecular constraint relay on the \"activity-molecular stability\" trade off. Moreover, the authors gave a complete derivation of the Michaelis-Menthen type rate, law for the Rubisco catalytic activity. Very recently, the increased availability of measured kinetic parameters for the RuBP carboxylation and oxygenation reactions from a variety of C3- and C4-plants, algae (both green and red algae), bacteria and cyanobacteria allowed the analysis of a greater collection of data among different species. [PERSON] et al., have investigated the possible relationship between kinetic parameters measured for approximately 325 species at 25 \({}^{\circ}\)C and pH = 8 [51]. The analysis was developed by using the Rubisco reaction mechanism and nomenclature described by [PERSON] and [PERSON] in 2018 [49] (see Section S6). The qualitative energetic diagram shown in Figure 17a evidences that the carboxylation (line green profile) and oxygenation (line red profile) reactions proceed through two \"effective\" energetic barriers. Considering the carboxylation reaction after RuBP enolization, the first energetic barrier is related to CO\({}_{2}\)-addition to the enediolate, while the second \"rate limiting\" energetic barrier is related to the C2-carboxylic acid hydration and C2-C3-gemdiolate bond cleavage (Figure 17a). Given standard assumptions, the authorsderived the catalytic efficiency (defined as \(k_{cat}/\text{K}_{\text{M}}\)) is related to the height of the first effective barrier, while \(k_{cat}\) is related to the second (Figure 17b). Thanks to the increased amount of kinetic data analyzed, [PERSON] observed that the previously assessed strong \(k_{cat,C}\)-\(\text{S}_{\text{C/O}}\) negative correlation was significantly weakened. The previously proposed correlation between \(k_{cat,C}\) and \(\text{K}_{\text{C}}\) or \(k_{cat,C}\) and \(\text{S}_{\text{C/O}}\) appeared inconsistent if analyzed among a larger amount of data. An important outcome of the study was that a short range of variations clearly emerged for most kinetic parameters if analysed among organisms with similar physiology. For example, \(k_{cat,C}\) showed variation by only 50% among Form I Rubiscos and \(\text{S}_{\text{C/O}}\) showed variation of less than 30% among Form I Rubiscos (Figure 18a). Considering the \(\text{S}_{\text{C/O}}\) definition (see Table S3) the following power-law relationship could be derived: \(\log(k_{cat,C}/\text{K}_{\text{C}})=\log(k_{cat,O}/\text{K}_{\text{O}})+\log( \text{S}_{\text{C/O}})\). When considering a set of data grouped for similar cellular physiology such as C3- and C4-plants and cyanobacteria (Figure 18b), having \(\text{S}_{\text{C/O}}\) values roughly constant, experimentally a strong positive correlation was found between \(k_{cat,C}/\text{K}_{\text{C}}\) and \(k_{cat,O}/\text{K}_{\text{O}}\) (in a log-log diagram). As matter of fact, Form I Rubiscos display a high-confidence power-law relationship between \(k_{cat,C}/\text{K}_{\text{C}}\) and \(k_{cat,O}/\text{K}_{\text{O}}\) (R = 0.94, \(p<10^{-10}\)).