Correction for Laurent et al., Emergence of homochirality in large molecular systems

PHYSICS Correction for “Emergence of homochirality in large molecular systems,” by Gabin Laurent, David Lacoste, and Pierre Gaspard, which published January 11, 2021; 10.1073/pnas.2012741118 (Proc. Natl. Acad. Sci. U.S.A. 118, e2012741118).

The authors note that, due to a printer’s error, parts of the legends for Figs. 3 and 4 appeared incorrectly. In the legend for Fig. 3, line 1, “autocatalytic network [22]-[22]-[23]” should instead appear as “autocatalytic network [21]-[22]-[23].” In the legend for Fig. 4, lines 1–2, “the generalized Frank model network [22]-[22]-[23]” should instead appear as “the generalized Frank model network [21]-[22]-[23].” The figures and their corrected legends appear below. The online version has been corrected.

Fig. 3.

Dynamical simulations of the autocatalytic network [21]-[22]-[23]. Typical time evolution of two species contained in the autocatalytic network as a function of time above the threshold concentration A₀ (A) and below it (B) is shown. The solid lines represent one of the two enantiomers for a given species and the dashed line the other enantiomer. Both simulations were carried out with an initial enantiomeric excess ε = 10⁻², and concentrations of all chiral species were initialized at D₀ = 2 + ε and L₀ = 2 – ε. The unactivated achiral species was initialized at Ã₀ = 0 and the activated one at A₀ = 80 in A and A₀ = 45 in B. All of the constants $k_{+ijk}$ and ${\overline{k}}_{-ij}$ follow a log-normal distribution of parameters µ = –10.02 and σ = 1.27 (i.e., corresponding to a log-normal distribution with $\langle k_{+}\rangle =\langle {\stackrel{\sim }{k}}_{-}\rangle ={10}^{-4}$ and ${\sigma }_{k_{+}}={\sigma }_{{\stackrel{\sim }{k}}_{-}}=2\times {10}^{-4}$), with ${\stackrel{\sim }{k}}_{ij}={\stackrel{\sim }{k}}_{ji}$ to satisfy the mirror symmetry described in SI Appendix, Eq. S16. The number of chiral species was set up to N_C = 20.

Fig. 4.

Probability of instability of the racemic state for the generalized Frank model [21]-[22]-[23]. Probability of the initial racemic state to be unstable by mechanism (i) as a function of the normalized value of the control parameter A₀ for the expanded Frank’s model in the irreversible regime and for different values of number of chiral species N_C: N_C = 10 (magenta), N_C = 20 (red), N_C = 30 (green), N_C = 40 (yellow), and N_C = 100 (blue). The control parameter A₀ has been normalized by the theoretical threshold at the transition, defined by the equality in the inequality of Eq. 6. An average over 1,000 realizations of the rate constants following a log-normal distribution such that $\langle k_{+}\rangle =\langle {\stackrel{\sim }{k}}_{-}\rangle ={10}^{-4}$ and ${\sigma }_{k_{+}}={\sigma }_{{\stackrel{\sim }{k}}_{-}}=2\times {10}^{-4}$ has been performed. (Inset) Comparison between the observed control parameter value A₀ at the transition (red circles) with the theoretical prediction given by Eq. 6 (blue solid line) after averaging over 100 realizations of the rate constants.