Stochastic chiral symmetry breaking process besides the deterministic one

Abstract

In chiral symmetry breaking, populations with initial enantiomeric excess (EE) are probabilistically favored if statistical fluctuation is present, as in nature. Stochastic methods correctly describe chiral symmetry breaking by taking into account the quantitative enantiomeric difference (excess or deficiency) and the statistical fluctuation amplitude, which is inversely proportional to the absolute size of the populations involved. From this, we obtain a law, which indicates that such a favoring probability decreases exponentially [P(EE) = 1/(e + 1)] with an initial enantiomeric deficiency mediated by statistical fluctuation. Obviously, chiral symmetry breaking equally favors populations without enantiomeric excess [P(0) = 1/2]. However, if deterministic methods are considered, chiral symmetry breaking will strictly favor the population with an initial enantiomeric excess (EE). To study these stochastic chiral symmetry breaking processes the autocatalytic Frank model was considered. Summarizing, our results show that the initial enantiomeric excesses are not entirely responsible for the final state configuration of autocatalytic finite systems.

Chiral symmetry conservation principle

“…We can infer that this is not strictly correct based on the stochastic considerations explained above. Of course, by increasing the ( N ) system, the relative amplitude fluctuation ( N −1/2 ) decreases, and the stochastic results tend to deterministic ones for N → ∞ 18 . However, the probability is always conserved for any N , implying the CSCP.…”

“…Sometimes, chiral states can appear locally and spontaneously if nonlinear processes amplify statistical fluctuations even in the absence of EE. 8,18,19,[22][23][24] The appearance of chiral states characterizes the spatial segregation of chirality.…”

“…is obtained from stochastic simulations, 18 where P L (or P D ) is the probability of L (or D) enantiomer domains of the final homochiral subsystem with P L þ P D ¼ 1 and consequently…”

“…The α parameter increases by $ N 1/2 and depends on the specific system. 18,29 The exponential law given in Equation ( 8) 18 is assumed to be general. Returning to subsystems I, II, and III, to exponential laws, and considering Equation (6) as…”

“…4,15,16 The main point of this work is to show that any chiral property must be conserved when considering the complete natural history in both space and time. The use of a kinetic model 17,18 does not limit the conclusions because all chiral properties can be correlated or at least they can behave similarly. We omit any mechanism on the spread of chirality; we consider only simple arguments but perhaps general enough.…”