Author Correction: An improved theory for calculating leaf gas exchange more precisely accounting for small fluxes

A Correction to this paper has been published: https://doi.org/10.1038/s41477-021-00899-w.

Correction to: Nature Plants https://doi.org/10.1038/s41477-021-00861-w, published online 1 March 2021.

In the version of this Article originally published, there were errors in equations (11), (17) and (18), and in Supplementary Note 7.

In equation (11), the minus sign in the numerator of the third term should have been a plus sign. In equation (17), both instances of ‘g_cw–ab γ’ should have been ‘g_cw–ad’, and both instances of ‘g_cw–ab’ should have been ‘g_cw–ad γ’; that is, the equation should have read:

$$\begin{array}{l}C_{{\rm{s}} - {\rm{ad}}} - \frac{{A_{{\rm{T}} - {\rm{ad}}} + C_{{\rm{s}} - {\rm{ad}}}E_{{\rm{T}} - {\rm{ad}}} - C_{{\rm{s}} - {\rm{ad}}}D_{\rm{ad}}g_{{\rm{cw}} - {\rm{ad}}}}}{{G_{{\rm{i}} - {\rm{ad}}} - \alpha _{\rm{ad}}g_{{\rm{cw}} - {\rm{ad}}}}}\\ \quad \quad \quad = \left( {C_{{\rm{s}} - {\rm{ab}}} - \frac{{A_{{\rm{T}} - {\rm{ab}}} + C_{{\rm{s}} - {\rm{ab}}}E_{{\rm{T}} - {\rm{ab}}} - C_{{\rm{s}} - {\rm{ab}}}D_{\rm{ab}}g_{{\rm{cw}} - {\rm{ad}}}\gamma }}{{G_{{\rm{i}} - {\rm{ab}}} - \alpha _{\rm{ab}}g_{{\rm{cw}} - {\rm{ad}}}\gamma }}} \right)\kappa \end{array}$$

In equation (18), ‘$g_{{\mathrm{cw}}{\mbox{-}}{\mathrm{ab}}}^2$’ should have been ‘$g_{{\mathrm{cw}}{\mbox{-}}{\mathrm{ad}}}^2$’, and ‘g_cw–ab’ should have been ‘g_cw–ad’. In addition, in the expression for b, the term γ was in the wrong location in three instances: ‘(C_s−ad D_ad G_i−ab γ + (A_T−ad + C_s−ad E_T−ad) α_ab)’ should have been ‘(C_s−ad D_ad G_i−ab + (A_T−ad + C_s−ad E_T−ad) α_ab γ)’; ‘(α_ad G_i−ab γ + G_i−ad α_ab)’ should have been ‘(α_ad G_i−ab + G_i−ad α_ab γ)’; and ‘(C_s−ab D_ab G_i−ad + (A_T−ab + C_s−ab E_T−ab) α_ad γ)’ should have been ‘(C_s−ab D_ab G_i−ad γ + (A_T−ab + C_s−ab E_T−ab) α_ad)’. The equation should have read:

$$ag_{{\rm{cw}} - {\rm{ad}}}^2 + bg_{{\rm{cw}} - {\rm{ad}}} + c = 0$$

where

$$\begin{array}{ll}a = \left( {C_{{\rm{s}} - {\rm{ad}}} - C_{{\rm{s}} - {\rm{ab}}}\kappa } \right)\alpha _{\rm{ad}}\alpha _{\rm{ab}}\gamma + \kappa C_{{\rm{s}} - {\rm{ab}}}D_{{\rm{ab}}}\alpha _{\rm{ad}}\gamma - C_{{\rm{s}} - {\rm{ad}}}D_{\rm{ad}}\alpha _{\rm{ab}}\gamma \\ b = \left( {C_{{\rm{s}} - {\rm{ad}}}D_{\rm{ad}}G_{{\rm{i}} - {\rm{ab}}} + \left( {A_{{\rm{T}} - {\rm{ad}}} + C_{{\rm{s}} - {\rm{ad}}}E_{{\rm{T}} - {\rm{ad}}}} \right)\alpha _{\rm{ab}}\gamma } \right)\\ \quad \quad - \left( {C_{{\rm{s}} - {\rm{ad}}} - C_{{\rm{s}} - {\rm{ab}}}\kappa } \right)\left( {\alpha _{\rm{ad}}G_{{\rm{i}} - {\rm{ab}}} + G_{{\rm{i}} - {\rm{ad}}}\alpha _{\rm{ab}}\gamma } \right)\\ \quad \quad {\mathrm{ }} - \kappa \left( {C_{{\rm{s}} - {\rm{ab}}}D_{\rm{ab}}G_{{\rm{i}} - {\rm{ad}}}\gamma + \left( {A_{{\rm{T}} - {\rm{ab}}} + C_{{\rm{s}} - {\rm{ab}}}E_{{\rm{T}} - {\rm{ab}}}} \right)\alpha _{\rm{ad}}} \right)\\ c = \left( {C_{{\rm{s}} - {\rm{ad}}} - C_{{\rm{s}} - {\rm{ab}}}\kappa } \right)G_{{\rm{i}} - {\rm{ad}}}G_{{\rm{i}} - {\rm{ab}}} - \left( {A_{{\rm{T}} - {\rm{ad}}} + C_{{\rm{s}} - {\rm{ad}}}E_{{\rm{T}} - {\rm{ad}}}} \right)G_{{\rm{i}} - {\rm{ab}}} + \kappa \left( {A_{{\rm{T}} - {\rm{ab}}} + C_{{\rm{s}} - {\rm{ab}}}E_{{\rm{T}} - {\rm{ab}}}} \right)G_{{\rm{i}} - {\rm{ad}}}\end{array}$$

These errors have been corrected, and Supplementary Note 7, which contains the derivation of equation (18), has been updated accordingly. All calculations presented in the Article remain correct.