Resilience of parity-violation-induced chiral selectivity to nonequilibrium temperature fluctuations in open systems

We consider the sensitivity of chemical reactions, able to undergo spontaneous mirror symmetry breaking, to chiral bias in the presence of nonequilibrium temperature fluctuations and derive a selectivity criterion. For this, we estimate the magnitude of fluctuations $\delta k$ in chemical reaction rate constants $k$ arising from the nonequilibrium temperature fluctuations $\delta T$ about a mean value $T$. To leading order, the relative rate constant fluctuations $\delta k_i/k_i$ for each reaction $i$ are given by the product of the activation enthalpy $\mathrm{\Delta }{H}_i^{‡}/RT$ for the $i\mathrm{th}$ reaction multiplied by the relative rms temperature fluctuations $\delta T_{\mathrm{rms}}/T$. The latter are determined by the system's specific heat at constant volume: $C_V$. We test this criterion with simulations carried out for an open-flow fully reversible Frank model, and for a range of parity-violating energy differences (PVED) within the theoretically estimated upper and lower bounds. Depending on the relative magnitudes of the deterministic PVED bias and the temperature fluctuations, the PVED bias can either (i) select the final stable chiral outcome deterministically or (ii) select one of two possible stable chiral outcomes with an asymmetric statistical weighting. For larger temperature fluctuations, the PVED bias loses its selectivity, and the final stable chiral outcomes are stochastic and equally probable. This paper points towards the possible design of small volume chemical flow reactors capable of detecting the elusive PVED bias in bulk systems, provided other sources of fluctuations can be sufficiently controlled and attenuated.

Figure 1

Reversible version of the Frank model in a continuous open-flow reactor of volume $V$ at mean temperature $T$, assuming instant and perfect diffusion of all the species in solution. Achiral resource $A$ flows in at fixed concentration ${\left[A\right]}_{\mathrm{in}}$; all species $A$, $L$, $D$, and $P$ flow out with their instantaneous concentrations. The matter flow maintains the system out of chemical equilibrium. The fluid volumes entering and exiting the reactor per unit time are the same.

Figure 2

Reaction coordinate diagrams corresponding to the noncatalytic or spontaneous (top) and the autocatalytic (bottom) reactions, respectively. Left: Reactants, transition states $(‡),$ and products in the absence of PVED. Right: The effect of the PVED on the energies of the individual enantiomers $L$ and $D$ and the transition states. These (minuscule) energy differences $\mathrm{\Delta }\mathrm{\Delta }{G}^{‡}>0$ in the enantiomers appear only in those processes involving the enantiomers.

Figure 3

Characteristic dynamic outcome of deterministic mirror symmetry breaking triggered by a minuscule PVED bias $g={10}^{-19}$ [Eq. (55)], in the absence of all fluctuations, for the open flow Frank model (see Sec. 4). (a) The evolution of all four chemical species. (b) The percent of enantiomeric excess $\text{ee}(\%)=\left(\right[L]-[D]/[L]+[D\left]\right)\times 100$. The outcome is homochiral in favor of the $D$ enantiomer. Changing the sign of the PVED bias to $g=-{10}^{-19}$ leads to the opposite homochiral outcome, now in favor of the $L$ (not shown). See text for related comments, the parameter values employed, and also Fig. 7 in Appendix pp4 for a characteristic imperfect (biased) bifurcation.

Figure 4

Distributions of the outcomes from multiple simulations $n=200$ of the reaction model. (a) Subject exclusively to chiral bias $g={10}^{-19}$ and zero fluctuations $\xi =0$. (b) Subject to pure thermal fluctuations $\xi ={10}^{-22}$ and in the absence of any chiral bias $g=0$. The red (blue) column corresponds to total number of final homochiral $D$ ($L$) states. See text for related comments and Fig. 7 in Appendix pp4 for a structure of imperfect and perfect bifurcations.

Figure 5

Distributions of the outcome from multiple simulations $n$ of the competition between the PVED [Eq. (55)] and the product of the rms temperature fluctuation times the activation enthalpy factor ${\xi }_j$ [Eq. (34)]. The level of chiral bias corresponds to $g={10}^{-12}$. (a) $\xi ={10}^{-13}$ and $n=60$. (b) $\xi ={10}^{-12}$ and $n=60$. (c) $\xi ={10}^{-11}$ and $n=120$. (d) $\xi ={10}^{-10}$ and $n=30$. The red (blue) bar centered at $-1$ ($+1$) represents the total number of stationary homochiral outcomes for which the final majority enantiomer is $D$ or $L$. See text for related comments.

Figure 6

Distributions of the outcome from multiple simulations $n=200$ of the competition between the PVED [Eq. (55)] and the product of the rms temperature fluctuation times activation enthalpy factor ${\xi }_j$ [Eq. (34)]. Chiral bias corresponds to $g={10}^{-17}$. (a) $\xi ={10}^{-18}$. (b) $\xi ={10}^{-17}$. (c) $\xi ={10}^{-16}$. (d) $\xi ={10}^{-15}$. The red (blue) bar centered at $-1$ ($+1$) represents the total number of stationary homochiral outcomes for which the final majority enantiomer is $D$ or $L$. See text for further remarks.

Figure 7

Solution $\alpha \left(\lambda \right)$ of the bifurcation equation (D1). Red dashed curves: Perfect bifurcation in the absence of any chiral influence $g=0$. Blue solid curves: The imperfect bifurcation diagram due to the chiral symmetry breaking interaction $g>0$. Note that the racemic composition, or racemate, (represented by the abscissa $\alpha =0$) is not a stationary solution of Eq. (D1) for $g>0$, and hence does not form part of the imperfect bifurcation diagram. Instead, the racemate $\alpha =0$ lies inside the basin of attraction of the upper stable scalemic branch (blue).

Figure 8

Dynamic outcome of spontaneous mirror symmetry breaking triggered by pure temperature fluctuations in the rate constants, for the simplified open flow Frank model [Eqs. (E1) and (E2)]. (a) The evolution of both enantiomer concentrations $\left[L\right]$ and $\left[D\right]$ starting from an initial (unstable) racemic configuration for 12 independent runs. (b) The enantiomeric excess $\text{ee}=\left(\right[L]-[D]/[L]+[D\left]\right)$ calculated at time $t=600\text{s}$. In these simulations, $D$ is the majority enantiomer for five runs ($\text{ee}=-1$) and $L$ is the majority for seven runs ($\text{ee}=+1$). $k_a\left(A\right)=0.18$, $k_{-a}=0.09$, $k_1=0.14$, initial ${\left[L\right]}_0={\left[D\right]}_0=0.5$, and noise amplitude $\xi ={10}^{-6}$.