#
Natural Abundance Isotopic Chirality in the Reagents of the Soai Reaction^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

**1**), di(iso-propyl)zinc (

**2**) and the autocatalyst, which is the alkylation product of

**1**by

**2**, leading to the secondary alcohol (

**3**), as shown in Figure 1 [41,42]. These molecules are formed from H, C, N, O and Zn, which are present in nature as mixtures of their stable isotopes; the H, C, N, and O with dominance of the lighter nuclei, while Zn as a more equable mixture of five isotopes [12] (Table S1). As can be seen from the structures in Figure 1, the “isotopically prochiral” groups in compounds

**1**–

**3**are the tert.-Bu group in

**1**and

**3**, as well as the iso-Pr group in compounds

**2**and

**3**. These groups consist of H and C, thus the isotopic substitution by these elements will be analyzed here. N is not included in our analysis, even if a very recent study has shown that chirality caused (only) by N-isotopes can control the enantiomeric outcome of the Soai reaction [43]. Some fragments of hypothetic intermediates [44,45,46] of the Soai autocatalysis show chiral arrangement of the Zn atoms. One of these fragments, the structure of which has been essentially supported by X-ray crystallography [47], will also be included in the present study.

#### 2.1. The Tert.-Butyl Group

**1**and

**3**is a highly symmetric species. Therefore, it needs at least two different substituents on at least two of its methyl groups for the generation of a center of chirality on the central C atom. Because of the presence of three identic (methyl) groups in this functionality, the number of possible isotopically substituted derivatives amounts to 112 (Supporting Materials 1) if the isotopic substitution in the central carbon atom is neglected, and 224 if this is taken into account. Because of the necessity of (at least) double isotopic substitution for obtaining a chiral isotopomer, the probabilities of the formation of such structures with natural abundance

^{2}H and/or

^{13}C substituents is very low. Only four combinations (from 112) have probability >10

^{−8}. The sum of the probabilities is 1.027 × 10

^{−5}, deriving mostly from species with one

^{13}C substitution.

_{50%}, %) with the Pars–Mills equation [40,48] (with 50% confidence, see also Supporting Materials 2) for sample sizes that could appear in usual micropreparative work (ranging from millimol to femtomol) (Table 1).

Sample Size | e.e._{50%} |
---|---|

millimol | 8.60 × 10^{−7} |

micromole | 2.72 × 10^{−5} |

nanomol | 8.60 × 10^{−4} |

picomol | 2.72 × 10^{−2} |

femtomol | 8.60 × 10^{−1} |

**1**and/or

**3**could influence the outcome of the most sensitive variant of the Soai reaction, however, taking into regard that it is separated from the pyrimidyl unit by the rigid C

_{2}moiety, and thus from the decisive molecular events around the new stereocenter, this option appears as scarcely probable, but cannot be excluded.

#### 2.2. The Iso-Propyl Group(s)

**2**and

**3**. There is, however, both quantitative and qualitative differences between these structures. Compound

**2**contains two i-Pr groups, in organometallic bond, while autocatalyst

**3**contains only one i-Pr moiety, linked by covalent C(sp

^{3})–C(sp

^{3}) bond to the newly formed center of chirality. This is a fundamental difference, especially from the viewpoint that in

**2**the i-Pr groups are present before the C–C bond making it an alkylation step, while the i-Pr group in

**3**represent a final stage; that is, the stage when the fate of the asymmetry of the critical carbon atom has already been decided. On the other hand, compound

**3**appears as autocatalyst in the subsequent cycle(s) of the reaction in intermediate(s), the structure of which is actually known only from theoretical studies [44,45,46,47,49]

**methyl**group.

^{12}C) = p = 0.98889, P(

^{13}C) = q = 0.01111, P(

^{1}H) = u = 0.999844 and P(

^{2}D) = v = 0.000156. Thus

p_{1} = P(^{12}C^{1}H_{3}) = pu^{3} | p_{5} = P(^{13}C^{1}H_{3}) = qu^{3} |

p_{2} = P(^{12}C^{1}H_{2}^{2}D) = 3pu^{2}v | p_{6} = P(^{13}C^{1}H_{2}^{2}D) = 3qu^{2}v |

p_{3} = P(^{12}C^{1}H^{2}D_{2}) = 3puv^{2} | p_{7} = P(^{13}C^{1}H^{2}D_{2}) = 3quv^{2} |

p_{4} = P(^{12}C^{2}D_{3}) = pv^{3} | p_{8} = P(^{13}C^{2}D_{3}) = qv^{3} |

_{i}= 1 (i = 1–8). These cases, obviously, provide no chirality.

**iso-propyl**group containing two methyl groups, gives 8 × 8 = 64 cases if the possible carbon or hydrogen isotope substitution in the central CH group is disregarded; if only the

^{12}C/

^{13}C or

^{1}H/

^{2}D exchange is taken into account, the number of cases is 2 × 64 = 128, while both of these exchanges yield 2 × 128 = 256 cases. We shall calculate below on the basis of 128 cases.

_{i}× p

_{j}and q × p

_{i}× p

_{j}, where i = 1,2,…,8 and j = 1,2,…,8. If i = j, the i-Pr group becomes symmetric, which are 16 cases from 128, leaving 112 as asymmetric cases. From these, obviously, 56 are of R and 56 of S configuration. The sum of the probabilities of the symmetric (achiral) cases is ∑(p × p

_{i}

^{2}+ q × p

_{i}

^{2}) = 0.977131 (since i = j in the symmetric cases).

_{chiral}= 1 – 0.977131 = 0.022869. The abundance of the symmetric species with central

^{12}C is thus 0.98889 × 0.977131 = 0.96627566, while with central

^{13}C it is 0.01111 × 0.977131 = 0.01085593 and the sum of these (as it should be) equals 0.977131.

**0.022869 ≈ 2.3%**.

**2,**of the Soai reaction, however, contains

**two iso-propyl**groups. This generates 128 × 128 = 16,384 cases if only the

^{12}C/

^{13}C exchange possibility on the central (CH) carbon is considered; if the

^{1}H/

^{2}D exchange is also taken into account, this increases the number of cases to 4 × 128 × 128 = 65,536. We shall use the former number (128) in the following calculations.

- (i)
- Each (of the two) i-Pr groups can be symmetric “internally”. In this case, each i-Pr group provides 16 (8 + 8) symmetric cases. Altogether, these are 16
^{2}= 256 cases. In the calculations with only one i-Pr group, we have seen that this leads to ∑(p × p_{i}^{2}+ q × p_{i}^{2}) = 0.977131 probability. The probability of that both i-Pr groups are of “internal” symmetry is the square of this value, 0.977131^{2}= 0.954785. This is then the probability of the achiral cases. Consequently, the probability of the chiral cases, which are 16,384 – 256 = 16,128 structures, is P_{chiral}= 1 – 0.954785 =**0.045215****≈****4.5%**. - (ii)
- If the symmetry between the two i-Pr groups is considered, this requires a “corresponding pair” to each of the configurations in one of the groups: 64 cases to each of the 64 combinations in one i-Pr, that is 642 = 4096, with “unique” CH group, or 128
^{2}= 16,384 with one isotope exchange in the methylene group, or 256^{2}= 65,536, if both C and H exchange is considered. In more explicit terms, P_{achiral}= ∑[(p × p_{i}× p_{j})^{2}+ (q × p_{i}× p_{j})^{2}] = p^{2}∑(p_{i}× p_{j})^{2}+ q^{2}∑(p_{i}× p_{j})^{2}= (p^{2}+ q^{2}) ∑(p_{i}× p_{j})^{2}, which is numerically for the above mentioned (one isotope exchange in the CH group) case: 0.978 × 0.9771312 =**0.93380****≈****93.4%**.

_{chiral}= 1 – 0.93380 =

**0.06620**

**≈**

**6.6%**.

Case (i) | Case (ii) | ||
---|---|---|---|

p = 0.045215 | p = 0.06662 | ||

Sample size | e.e._{50%} | Sample size | e.e._{50%} |

millimol | 1.29595 × 10^{−8} | millimol | 1.07 × 10^{−8} |

micromole | 4.09814 × 10^{−7} | micromole | 3.39 × 10^{−7} |

nanomol | 1.29595 × 10^{−5} | nanomol | 1.07 × 10^{−5} |

picomol | 4.09814 × 10^{−4} | picomol | 3.39 × 10^{−4} |

femtomol | 1.29595 × 10^{−2} | femtomol | 1.07 × 10^{−2} |

**S**-

**E4**), shown in Figure 2. We considered that variant of the i-Pr groups, where only

^{1}H/

^{2}D or

^{12}C/

^{13}C exchange occurs in the central CH groups. In this case, the number of all possibilities is 128

^{4}= 268,435,456. Several of these structures are “degenerated” (equivalent to each other).

**S-E4**can be deduced as follows:

- (a)
- Chirality caused by the Zn-bound i-Pr groups, and only by these. The number of the Zn-bound cases is 128
^{2}, subtracting the number of the symmetric cases and multiplying the rest with the number of the symmetric cases of the C-bound i-Pr groups we obtain: (128^{2}– 16^{2}) × 128 = 2,064,384. The probability of the symmetric (achiral) cases is the same, which was calculated above for case (i) of the instance where only the internal symmetry of two i-Pr groups was considered: 0.9771312 =**0.954785****≈****95.5%**, while for the chiral cases (1 − 0.954785) =**0.045215****≈****4.5%**. - (b)
- Chirality is caused by C-bound i-Pr groups, and only by these. In this case, each of the C-bound 128 i-Pr groups can be paired with 127 different configurations, choosing achiral Zn-bond partners we obtain 128 × 127 × 16
^{2}= 4,161,536 chiral cases. The probability of the C-bound achiral cases according to this approach is:**0.93380****≈****93.4%**, and of the chiral cases (1 − 0.93380) =**0.06620****≈****6.6%**. - (c)
- If the combination of both the chirality of the Zn-bound and the C-bound structures (i-Pr groups) is considered, the number of the chiral cases amounts: (128
^{2}− 16^{2}) × 128 × 127 =**262,176,768**. The probability of this case equals**0.00300**because the probability of chirality in case (a) is 0.045215 and in case (b) is 0.06620, thus the probability of joint event equals 0.045215 × 0.06620 =**0.00300**.

_{∑}= 04222 + 0.06320 + 0.00300 =

**0.10842**

**≈**

**10.8%**.

**0.891572 ≈ 89.2%**, while for the probability of the chiral structures we obtain P

_{∑}= (1 − 0.891572) =

**0.108428**

**≈**

**10.8%**.

- (d)
- The number of the achiral cases is 128 for the C-bound structures and 16 × 16 = 256 for the Zn-bound ones. Thus the total number of the achiral cases is 128 × 256 =
**32,768**.

**268,402,688**. This number plus the number of the achiral structures from case (d) gives exactly

**268,435,456**, as calculated from pure combinatorial considerations.

**Figure 2.**Schematic structure of the dimeric intermediate of the Soai reaction, according to Schiaffino and Ercolani [44] (

**S-E4**).

**S-E4,**with the Pars–Mills equation (with 50% confidence, see also Supporting Materials 2) the expectable e.e.

_{50%}values from sample sizes ranging from millimol to femtomol, for the chiral fraction of the molecules (Table 3):

Sample Size | e.e._{50%} (%) |
---|---|

millimol | 8.36900 × 10^{−9} |

micromole | 2.64651 × 10^{−7} |

nanomol | 8.36900 × 10^{−6} |

picomol | 2.64651 × 10^{−4} |

femtomol | 8.36900 × 10^{−3} |

**S-E4**are valid only if the isotopic composition of the central O

_{2}Zn

_{2}ring is disregarded. At the present level of our calculations we did so.

#### 2.3. Zinc Isotopic Chirality?

**S-E9**), as shown in Figure 3A. In this substructure, four Zn atoms and four O atoms form a virtual cube-like polyhedron, as shown in Figure 3B. Interestingly this hypothetical structural element also appears in the X-ray structure of the “enantiomeric” model compound of the Soai autocatalysis, reported very recently by Matsumoto et al. [47].

**Figure 3.**(

**A**) Schematic structure of a tetrameric intermediate of the Soai reaction according to Schiaffino and Ercolani [44] (

**S-E9**) and (

**B**) schematic structure of the O

_{4}Zn

_{4}“twisted cube” fragment in compound

**S-E9**.

_{4}Zn

_{4}“cube” of compound

**S-E9**can be deduced as follows.

**Simple**

**case**(all O atoms are regarded equal, or isotopic differences in these O atoms are disregarded). This approach enables calculating the contribution of the metal atoms to chirality.

^{64}Zn as a,

^{65}Zn as b,

^{66}Zn as c,

^{67}Zn as d, and

^{68}Zn or

^{70}Zn as e (we shall later use similar notation for oxygen isotopes:

^{16}O u,

^{17}O v, and

^{18}O w).

**P**

_{∑}**=**

**0.002530375**

**≈**

**0.25%**.

Sample Size | e.e._{50%} (%) |
---|---|

millimol | 5.47817 × 10^{−8} |

micromole | 1.73235 × 10^{−6} |

nanomol | 5.47817 × 10^{−5} |

picomol | 1.73235 × 10^{−3} |

femtomol | 5.47817 × 10^{−2} |

**Complex**

**case**(isotopes of the oxygen atoms are considered). This situation can be approached by the following partial calculations:

- (i)
- The “cube” contains four different Zn isotopes, no matter which oxygen isotopes are present. This is equal to the “simple” case. The “cube” is chiral, 10 structures can be identified, the sum of the probabilities is P
_{(i)}= 0.002530375 =**0.2530375****≈****0.25%**. - (ii)
- Three different Zn isotopes in the “cube”; that is to say, within the four Zn atoms, two are of the same isotope. The two equal isotopes are placed in any case on one of the diagonals of one of the faces. Let us turn the cube in such position that the two equal isotopes occupy positions 1 and 3 (Figure 3b). Taking into regard that Zn has five isotopes, this results in 60 cases. If only the Zn atoms are regarded, this situation results in a symmetry plane in plane [2-6-8-4]. The “cube” becomes chiral if this symmetry is destroyed by putting different O isotopes to positions 5 and 7. Let us now consider the possibilities given by the three different O isotopes. This results in 54 additional cases coupled with each of the 60 cases derived from combinations of the Zn isotopes. Thus. the total number of chiral structures is 60 × 54 = 3240. The highest probability is provided by the combination (aabc + uuuw). This taken two times gives P
_{(ii)}= 1.10662 × 10^{−5}≈ 0.0011%. - (iii)
- Two different Zn isotopes, pairwise, that is 2 + 2 equal Zn isotopes, aabb and so on. This gives 10 cases, considering the three O isotopes, one obtains 36 additional possibilities for each of the Zn-derived 10, thus the total number of combinations is 10 × 36 = 360. The most probable case is the combination (aabb + uuww) two times, which gives P
_{(iii)}= 1.53122 × 10^{−7}≈ 0.000015%. - (iv)
- Two different Zn isotopes in the manner that three Zn isotopes are equal, that is the structure is of aaab type, which gives 20 cases, which is of relatively high symmetry; this can be spoiled in 54 ways for each structure, giving a total of 20 × 54 = 1080 cases. The highest probability is at the (aaab + uuuw) combination, taken two times P
_{(iv)}= 1.31637 × 10^{−4}≈ 0.013%. - (V)
- If all Zn atoms are of the same isotope, a “fourth” O isotope would be needed to obtain chiral structure, which is impossible, providing, consequently 0 cases and 0 probability.

**4690**. The sum of the probabilities of these structures is

**P**

_{∑(i)–(iv)}= 0.002672827**≈**

**0.27%**.

Sample Size | e.e._{50%} (%) |
---|---|

millimol | 5.33019 × 10^{−8} |

micromole | 1.68555 × 10^{−6} |

nanomol | 5.33019 × 10^{−5} |

picomol | 1.68555 × 10^{−3} |

femtomol | 5.33019 × 10^{−2} |

**S-E9**discussed above, see also Supporting Materials 5), contain such Zn atoms, which are linked to two different ligands and to two oxygen atoms of equal “chemical environment”. These Zn atoms become chiral only if the two O atoms are different isotopes. Since all four Zn atoms in the intermediate

**S-E9**are of this kind, we analyzed this aspect too.

_{achiral}=

**0.995190**

**≈**

**99.5%**, while that of the chiral ones is P

_{chiral}= 1 – P

_{achiral}=

**0.0048099**

**≈**

**0.5%**. The probability that one of the four zinc atoms in the imaginary “cube” becomes chiral is: P

_{”cube”}= 4 × P

_{chiral}=

**0.01924**

**≈**

**2%**. The probability that all four Zn atoms in the “cube” become chiral is, however, very low: 1.37 × 10

^{−4}%. Whether mutual induction effects are influencing the configuration of these Zn atoms is not yet (?) clear.

Sample Size | e.e._{50%} (%, one Zn) | e.e._{50 %} (%, four Zn) |
---|---|---|

millimol | 3.97338 × 10^{−8} | 1.98669 × 10^{−8} |

micromole | 1.25649 × 10^{−6} | 6.28247 × 10^{−7} |

nanomol | 3.97338 × 10^{−5} | 1.98669 × 10^{−5} |

picomol | 1.25649 × 10^{−3} | 6.28247 × 10^{−4} |

femtomol | 3.97338 × 10^{−2} | 1.98669 × 10^{−2} |

## 3. Conclusions

**S-E4**and

**S-E9**. The first compound (

**S-E4**) has shown an incredible richness of possible chiral structures by

^{1}H/

^{2}D and/or

^{12}C/

^{13}C exchange, leading to fairly high levels of isotopically chiral components (up to ~11%). The second structure (

**S-E9**) shows possibility for statistical chirality, generated by different Zn isotopes. If our hypothesis holds, this would be the first case where chirality due to different Zn isotopes evolved.

^{1}H/

^{2}D,

^{12}C/

^{13}C, oxygen or Zn isotope-generated chirality can be controlled experimentally, by preparative (product chirality distribution) or chiroptical techniques. Obviously, model experiments with enantiomerically enriched isotopically chiral reagents could also be involved, particularly for the cases of

^{1}H/

^{2}D or

^{12}C/

^{13}C problems.

^{1}H/

^{2}D and/or

^{12}C/

^{13}C exchange generated chirality. This feature is that, at sufficiently high number of parallel experiments (84) [20], a clear tendency of systematic preference for the S-enantiomer in the product was detected [53]. This observation has not yet been suitably explained. Excess statistical isotope chirality in (one or more of) the reagents may be one of the possible reasons how the distribution degeneracy of the product enantiomers is violated, without supposing charge/parity generated energy differences between the product (or intermediate) enantiomers [54].

_{50%}values, calculated by the Pars–Mills equation [40,48] (SM2) demonstrated an important fact, which is also evident from the mathematical structure of this formula, in samples where the percentual quantity of the chiral species is higher, the expectable enantiomeric excess is lower at a given sample size. This is in good agreement with the previously discussed idea about the exceptional role of the first [22,24,55,56,57,58] or very few [59,60,61,62] chiral molecules in achiral-to-chiral reactions, as inductors of chirality in the product.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Barabás, B.; Kurdi, R.; Pályi, G.
Natural Abundance Isotopic Chirality in the Reagents of the Soai Reaction. *Symmetry* **2016**, *8*, 2.
https://doi.org/10.3390/sym8010002

**AMA Style**

Barabás B, Kurdi R, Pályi G.
Natural Abundance Isotopic Chirality in the Reagents of the Soai Reaction. *Symmetry*. 2016; 8(1):2.
https://doi.org/10.3390/sym8010002

**Chicago/Turabian Style**

Barabás, Béla, Róbert Kurdi, and Gyula Pályi.
2016. "Natural Abundance Isotopic Chirality in the Reagents of the Soai Reaction" *Symmetry* 8, no. 1: 2.
https://doi.org/10.3390/sym8010002