## 2 Materials and Methods
### Calculations and Statistical Analysis  
#### 2.5.1 Modeling pH Variation with DIC Change  
The composition of DIC was calculated based on the following equations [26]:  
\[\left[\text{CO}_{2}\right]=\frac{DIC}{1+\frac{K_{1}}{\left[\text{H}^{+}\right]}+ \frac{K_{1}K_{2}}{\left[\text{H}^{+}\right]^{2}}} \tag{1}\]  
\[\left[\text{HCO}_{3}^{-}\right]=\frac{DIC}{1+\frac{\left[\text{H}^{+}\right]}{ K_{1}}+\frac{K_{2}}{\left[\text{H}^{+}\right]}} \tag{2}\]  
\[\left[\text{CO}_{3}^{2-}\right]=\frac{DIC}{1+\frac{\left[\text{H}^{+}\right]}{ K_{2}}+\frac{\left[\text{H}^{+}\right]^{2}}{K_{1}K_{2}}} \tag{3}\]  
where \(K_{1}\) and \(K_{2}\) are stoichiometric constants for bicarbonate and carbonate, respectively. In this study, \(\text{p}K_{1}=6.38\)[27] and \(\text{p}K_{2}=9.8\) were assumed. The \(\text{p}K_{2}\) value was derived from NaOH titration (cf. Supplementary material Section S1).  
Based on the above equations, pH variation with CO\({}_{2}\) absorption/desorption was modeled. The buffering function of phosphate and borate was assumed to be negligible because their concentrations were about 100 times lower than the DIC concentration. In the current experiment, DIC concentration can be expressed as:  
\[DIC=\left[\text{CO}_{2}\right]+\left[\text{HCO}_{3}^{-}\right]+\left[\text{ CO}_{3}^{2-}\right]=c_{0}+\Delta DIC \tag{4}\]  
where \(c_{0}\) is the initial DIC concentration by the addition of NaHCO\({}_{3}\) (0.27 mol L\({}^{-1}\)), and \(\Delta DIC\) is the change in DIC concentration due to CO\({}_{2}\) absorption or photosynthetic assimilation (mol L\({}^{-1}\)). Charge balance in the medium can be expressed as:  
\[\left[\text{H}^{+}\right]+\left[\text{Na}^{+}\right]=\left[\text{H}^{+} \right]+c_{0}+c_{b}=\left[\text{OH}^{-}\right]+\left[\text{HCO}_{3}^{-}\right] +2\left[\text{CO}_{3}^{2-}\right] \tag{5}\]  
where \(c_{b}\) is the final concentration of NaOH in the modified SOT medium that raised pH to 10.5 (0.215 mol L\({}^{-1}\)). Using the ion product constant of water \(\left(K_{w}=\left[\text{H}^{+}\right]\left[\text{OH}^{-}\right]=1.0\times 10^{ -14}\text{ mol}^{2}\text{ L}^{-2}\right)\), Equation (12) can be expressed as:  
\[\left[\text{H}^{+}\right]+c_{0}+c_{b}=\frac{\text{K}_{w}}{\left[\text{H}^{+} \right]}+\left[\text{HCO}_{3}^{-}\right]+2\left[\text{CO}_{3}^{2-}\right] \tag{6}\]  
Finally, Equations (6), (10), (11) and (13) can be formulated into a quartic equation:  
\[\left[\text{H}^{+}\right]^{4}+\left(K_{1}+c_{0}+c_{b}\right)\left[\text{H}^{ +}\right]^{3}+\left(K_{1}K_{2}+K_{1}c_{b}-K_{w}-K_{1}\Delta DIC\right)\left[ \text{H}^{+}\right]^{2} \tag{7}\]  
In the current experiment, all variables in Equation (7) except for \(\Delta DIC\) and \(\left[\text{H}^{+}\right]\) remained constant. Therefore, pH variation with CO\({}_{2}\) absorption/desorption can be estimated based on changes in DIC concentration (\(\Delta DIC\)) with this model. The quartic equation was solved with Mathematica 8.0 (Wolfram Research, Champaign, IL, USA).  
Buffer intensity, \(\beta\), was calculated based on the following approximation for diprotic acids [28]:  
\[\beta=-\frac{d\text{C}_{\text{A}}}{d\text{pH}}\cong 2.3\left(\left[\text{H}^{+} \right]+\left[\text{OH}^{-}\right]+\frac{\left[\text{CO}_{2}\right]\left[ \text{HCO}_{3}^{-}\right]}{\left[\text{CO}_{2}\right]+\left[\text{HCO}_{3}^ {-}\right]}+\frac{\left[\text{HCO}_{3}^{-}\right]\left[\text{CO}_{3}^{2-} \right]}{\left[\text{HCO}_{3}^{-}\right]+\left[\text{CO}_{3}^{2-}\right]}\right) \tag{8}\]where \(d\)C\({}_{\rm A}\) is the infinitesimal amount of acids required to decrease \(d\)pH. The equilibrated CO\({}_{2}\) gas concentration was calculated according to [PERSON]'s law:  
\[p_{\rm CO_{2}}=[\rm CO_{2}]\times\frac{1}{H^{cp}} \tag{9}\]  
where \(p_{\rm CO_{2}}\) is the partial pressure of CO\({}_{2}\) under equilibrium condition (Pa), [CO\({}_{2}\)] is the aqueous concentration of CO\({}_{2}\) (mol L\({}^{-1}\)), and \(H^{cp}\) is [PERSON]'s law solubility constant (\(3.30\times 10^{-7}\) mol L\({}^{-1}\) Pa\({}^{-1}\)) ([PERSON] 2015).