Modeling spontaneous chiral symmetry breaking and deracemization phenomena: Discrete versus continuum approaches

We derive the class of population balance equations (PBE), recently applied to model the Viedma deracemization experiment, from an underlying microreversible kinetic reaction scheme. The continuum limit establishing the relationship between the micro- and macroscopic processes and the associated particle fluxes erases the microreversible nature of the molecular interactions in the population growth rate functions and limits the scope of such PBE models to strict kinetic control. The irreversible binary agglomeration processes modeled in those PBEs contribute an additional source of kinetic control. These limitations are crucial regarding the question of the origin of biological homochirality, where the interest in any model lies precisely in its ability for absolute asymmetric synthesis and the amplification of the tiny inherent statistical chiral fluctuations about the ideal racemic composition up to observable enantiometric excess levels.

Figure 1

Solid curve: typical profile of the Gibbs free energy of nucleation and growth by monomer addition to clusters as a function of cluster size $i$ according to the classical nucleation theory [21] and for a given value of the supersaturation $S>1$. The energy levels indicate the free-energy differences $\Delta G^0$ in going from $n$ to $n+1$ monomer units. The difference between adjacent levels $\Delta G_{n\to n+1}^0$ yields the ratio of forward and reverse rates at each step of monomer-to-cluster aggregation/deaggregation; see Eq. (5). The underbraces indicate a convenient isodesmic approximation employed in the simulations [14]: within each indicated cluster size subrange $i$, the ratio of the forward to reverse rates is a constant (compare with Fig. 4, which lists the monomer-cluster reactions for each isodesmic range).

Figure 2

Formation of spherical chiral clusters (by monomer attachment and detachment) from the blue (dark gray) and red (light gray) enantiomers in solution. At the molecular level, both cluster growth (molecule uptake) and dissolution (detachment of molecules) are operative simultaneously at generally different rates; see Eq. (3). The solution phase racemization of the enantiomers can be visualized as the periodic (and rapid) interconversion of the single molecules with rate $k_R$.

Figure 3

Fluxes: (dimensionless) crystal growth rates $G_{R,P}^{*}$ vs particle size $y$ employed in Ref. [11] for a given supersaturation $S>1.01$ and ${\alpha }^{*}={10}^{-2}$. The arrows denote the direction of the corresponding advective fluxes $J_y=G^{*}\left(y\right)f(\tau ,y)$: for $y$ greater than a critical length (here equal to unity), the crystal growth rate is positive (to the right), otherwise it is negative (to the left), indicating dissolution back to monomers. But the inverse processes or counterfluxes, i.e., (i) precritical cluster growth up to the critical size and (ii) partial dissolution for clusters greater than the critical size, are not included in $G^{*}$.

Figure 4

Scheme 1: kinetic scheme based on the free energy diagram for classical theory of primary nucleation. A racemizing monomer pair $(L_1,D_1)$ in solution (rate $k_R=k_{mR}$) and monomer-cluster aggregation yielding chiral aggregates of a maximum size $(L_{N\text{max}},D_{N\text{max}})$. The network approximates the free energy curve in Fig. 1 by four isodesmic regions (indicated by the reaction steps grouped within the vertical brackets), whose end points are (i) a precritical cluster size we fix to be four monomer units, (ii) the critical cluster size denoted $p=i_{\text{crit}}$, (iii) a postcritical cluster of $p_2>p$ units, and (iv) chiral clusters of maximum size $N_{\text{max}}$. $(2\to 4)$: rates $k_1,k_{m1}$; $(4\to p)$: rates $k_{3a},k_{m3a}$; $(p\to p_2)$: rates $k_{3b},k_{m3b}$; and $(p_2\to N_{\text{max}})$: rates $k_{3c},k_{m3c}$.

Figure 5

Kinetic control: simulation showing the effect of using irreversible rates (i.e., setting $k_1=k_{3a}=k_{m3b}=k_{m3c}=0$ in Fig. 4). That is, we shut off growth of precritical clusters up to the critical size and degradation of postcritical clusters back down to the critical size. Bottom: the enantiomeric excesses ${\text{ee}}_n(\%)$ for clusters of size $2\le n\le N_{\text{max}}$. The gradient from small to large $n$ is indicated by the arrows.

Figure 6

Thermodynamic control: simulation showing the effect of using fully reversible rates, i.e., for $k_1={10}^4$, $k_{3a}={10}^2$, $k_{m3b}=90$, $k_{m3c}=90$ in Fig. 4. Bottom: enantiomeric excesses ${\text{ee}}_n(\%)$ for clusters of size $2\le n\le N_{\text{max}}$. The gradient from small to large $n$ is indicated by the arrows. Compare with Fig. 5.

Figure 7

Full kinetic model [14] with reversible agglomeration. Initial conditions: tiny initial $\text{ee}$: $\text{ee}\%=1\times {10}^{-12}\%$, type (a) (see text). This provides a striking example of absolute asymmetric synthesis.

Figure 8

Full kinetic model [14] without agglomeration. Initial conditions are large, $\text{ee}=33\%$: type (b) (see text). Kinetic controlled temporary chiral amplification (chiral excursion).

Figure 9

Full model without grinding: Initial conditions are large, $\text{ee}=33\%$: type (b) (see text). Short-lived chiral amplification followed by the final racemic state.