9196
J. Phys. Chem. B 2008, 112, 9196–9200
Oscillatory Symmetry Breaking in the Soai Reaction Ka´roly Micskei,*,† Gyula Ra´bai,† Emese Ga´l,† Luciano Caglioti,‡ and Gyula Pa´lyi§ Department of Chemistry, UniVersity of Debrecen, H-4010 Debrecen, Hungary, Department of Chemistry and Technology of Biologically ActiVe Compounds, UniVersity La Sapienza Roma, Ple. Moro 5, I-00185 Roma, Italy, and Department of Chemistry, UniVersity of Modena and Reggio Emilia, Via Campi, 183, I-41100 Modena, Italy ReceiVed: April 17, 2008
A kinetic model of spontaneous amplification of enantiomeric excess in the autocatalytic addition of diisopropylzinc to prochiral pyrimidine carbaldehydes is extended by a negative feedback process. Simulations based on the extended model result in large-amplitude oscillations both in a continuous-flow stirred tank reactor (CSTR) and in a semibatch configuration under optimized initial conditions. When sustained oscillations are maintained in a CSTR, no enantiomeric product distribution could be observed in the calculated series; the system keeps its initial enantiomeric ratio endlessly. During damped oscillations, or steady-state conditions, however, chiral amplification from a very small initial enantiomeric excess to more than 99% occurs in a semibatch configuration. Calculations indicated spontaneous enantiomeric product enrichment (i.e., accumulation of one of the enantiomers at the cost of the other one) from strictly achiral starting conditions in a semibatch configuration due to the inherent numerical error of the integrator method, which can be regarded as a model of the statistical fluctuation in the numbers of enantiomeric molecules. SCHEME 1: Asymmetric Autocatalysisa
I. Introduction Asymmetric autocatalysis observed experimentally in the alkylation of N-heterocyclic aldehydes by zinc dialkyls appears to be one of the most important chemical discoveries of the 1990s.1 The reaction is found to be autocatalytic both in the sense of the chemical reaction and enantioselectivity (Scheme 1). Later, a variant of this reaction was used2 to realize the first experimentally documented example of an “absolute asymmetric synthesis” meeting the most rigorous theoretical requirements assigned for this phenomenon.3 Recently this reaction (called now Soai reaction) attracts much interest of researchers from various aspects. An important intention is to understand its mechanism. Such an understanding may give hope for finding more chiral autocatalytic reactions or designing such reaction systems in laboratory experiments. The kinetics of the Soai reaction has throughout been studied with this aim.4 Several approaches, ranging from mono- to tetrameric key intermediates were proposed. Buhse5 has suggested a simplified kinetic model for the description of the Soai reaction. His model consists of the addition of di(isopropyl)zinc (Zn) to a pyrimidyne-5-carbaldehyde (CHO), giving pyrimidyl5-alkanol (COH), which acts also as catalyst. Enantiomeric excess of the product (or other enantiomerically pure chiral additives) can act as chiral auxiliaries (inductors). Formation of a chiral isopropoxyzinc intermediate (COZn), usually with a significantly higher enantiomeric excess than that of the chiral auxiliary at the beginning of the reaction, is observed experimentally. It is thought that the understanding of the mechanistic details of such a chemical system may be of importance in throwing more light also on the origin of biological chirality.6 * To whom correspondence should be addressed. E-mail: kmicskei@ delfin.unideb.hu. † University of Debrecen. ‡ University La Sapienza Roma. § University of Modena and Reggio Emilia.
a In the Soai reaction2 with the other enantiomer, R, the reaction proceeds analogously; R ) H, Me, t-BusCtCs).
An interesting feature of the model is that it contains an autocatalytic feedback. Such a feedback is known to be necessary precondition of chemical oscillations. On the basis of this fact a question arises whether oscillatory changes in species concentrations can and how it would be calculated with this model and how oscillations affect the accumulation of enantiomeric excess during the course of the reaction. Sufficient numerical details are given in this paper for proceeding with an analysis of eventual oscillatory behavior of the Soai reaction and that of oscillatory enantiomeric product enrichment. Concentration oscillations in some chemical reactions7 have been known for a century. There are periodic, aperiodic, and chaotic changes in the concentrations of some intermediates during the course of such a reaction. Chemical oscillations may serve also as simple model systems for understanding biological timing mechanisms. We show here that a slightly modified version of the core model is capable of describing large
10.1021/jp803334b CCC: $40.75 2008 American Chemical Society Published on Web 07/02/2008
Oscillatory Symmetry Breaking in the Soai Reaction
J. Phys. Chem. B, Vol. 112, No. 30, 2008 9197
TABLE 1: Reactions and Corresponding Rate Laws in the Kinetic Core Model of Soai Reaction Proposed by Buhse5 (Reaction 7 Is Added to the Model in This Work) composite reactions2
reactions
rate laws
COH + Zn f COZn CHO + Zn f COZn COZn + COZn µ (COZn)2
1 2 3
R1 ) k1[COH][Zn] R2 ) k2[CHO][Zn] R3 ) k3[COZn]2
(COZn)2 + CHO µ (COZn)2-CHO
4
R-3) k-3[(COZn)2] R4 ) k4[(COZn)2][CHO]
(COZn)2-CHO + Zn f (COZn)2 + COZn CHO + COH f COH COZn f COH
5
R-4 ) k-4[(COZn)2-CHO] R5 ) k5[(COZn)2-CHO][Zn]
6 7
R6 ) k6[COH][CHO] R7 ) k7[COZn]
amplitude oscillations in species concentrations in time both under continuous-flow conditions (CSTR) and in a semibatch configuration. Our calculations based on the extended model including chiral processing also show that any chiral asymmetry introduced into the CSTR in the form of slight enantiomeric excess of the COH reagent does not induce any amplification of enantiomeric excess. However, a slight initial enantiomeric excess introduced into a semibatch reactor results in a high excess of one of the product enantiomers. The most striking finding is that even the numerical error of the integrator method can induce accumulation of one of the enantiomers in a semibatch reactor configuration without any initial chiral asymmetry. We report here the results of our simulations. II. Methods and Models A. Methods. In our calculations we used a differential equation system which was created on the basis of the composite reactions of Buhse’s model5 taking into account the corresponding (hypothetical) rate laws of these reactions. Simulations were performed on a personal computer. The algorithm used for the numerical integration of the differential equation system was based on a semi-implicit Runge-Kutta8 method with an error parameter 10-5. In some cases, GEAR algorithm was also used in order to control the calculations. There was no observable difference in the results obtained by the two different numerical methods. B. The Kinetic Model. The kinetic core model proposed by Bushe5 along with the corresponding rate laws is summarized in Table 1. It is assumed that the formation of COZn takes place via two formally distinct pathways: a direct and uncatalysed route (reactions 1 and 2) as well as an autocatalytic pathway. The dimerization of COZn (reaction 3) and the association of CHO to the dimer species (reaction 4) followed by reaction 5 constitute the autocatalytic cycle. Reactions 3 and 4 are considered to be reversible without specifying that the rate of equilibrium was fast or slow. We added reaction 7, a new composite reaction (which is not involved in the original Buhse model), to the model. This reaction represents a simple first order removal of COZn complex providing in such a way the negative feedback necessary for the oscillation to occur. CSTR Conditions. Sustained oscillations in the species concentrations as a function of time can only be expected under flow conditions when continuous input feeds of COH, CHO, and Zn are maintained at constant flow rate. Simultaneously continuous output feed for all the species should also be maintained and taken into account in the simulations. Accordingly, differential equations describing the time evaluation of the species concentrations must contain the flow terms.
Figure 1. Calculated oscillations under flow conditions using the model, rate laws (Table 1), and rate constant values as given by Buhse:5 k1 ) 1.0 × 104 M-1 s-1, k2 ) 1.0 × 10-5 M-1 s-1, k3 ) 8.0 × 102 M-1 s-1, k-3 ) 1.1 × 102 s-1, k4 ) 1.0 × 102 M-1 s-1, k-4 ) 1.0 × 102 s-1, k5 ) 8.0 × 103 M-1 s-1, k6 ) 7.8 × 103 M-1 s-1, k7 ) 4.0 × 10-4 s-1. Input concentrations: [COH]0 ) 2.08 × 10-4 M, [CHO]0 ) 2.08 × 10-2 M, [Zn]0 ) 3.13 × 10-2 M. Flow rate (reciprocal residence time in the CSTR): k0 ) 2.0 × 10-5 s-1.
Figure 2. Calculated concentration oscillations under flow conditions at k0 ) 1.8 × 10-5 s-1. Other parameters are shown in Figure 1.
Semibatch Reactor Conditions. Continuous input flow of the reagents is maintained, but no output flow is taken into account in the simulations. Consequently, unlimited accumulation of the product takes place in the reactor. III. Results and Discussion Calculated Sustained Oscillations in a CSTR. We fixed the values of the rate constants as given by Buhse (see Figure 1) and carried out calculations using different flow rate values. The flow rate is characterized by k0 (flow rate/reactor volume). The input flow of a species was given as k0[species]0. The output flow was given as k0[species], where [species]0 ) the input concentration, [species] ) the actual concentration in the reaction mixture. Results of systematic calculations show that concentration of all species involved in the model exhibit period oscillations in time in a given range of “optimum” flow rates and input concentrations. The shape, period time, and the amplitude are strongly affected by the flow rate. A typical oscillatory trace, which was calculated with k0 ) 2.0 × 10-5 s-1, is shown in Figure 1. When the k0 value is lower just a little, both the period time and the amplitude are much larger (Figure 2). Such a sensitivity of the behavior of an oscillatory reaction on the flow rate is very rare.7 Step by Step Accumulation of Chirality. Oscillatory periods in the concentration of the key intermediate COZn may be regarded as “separate” catalytic cycles. These, however, are accompanied by the amplification of chirality in the Soai
9198 J. Phys. Chem. B, Vol. 112, No. 30, 2008
Micskei et al. TABLE 2: Reaction Steps and Corresponding Rate Constants Used for the Kinetic Model Including Chiral Processing for Soai Reaction Proposed by Buhse5 (Reaction 7 Is Added to the Model in This Work) composite reactions2
Figure 3. Results of step-by-step calculations for the evolution of enantiomeric excess in consequent autocatalytic cycles with one molecule as starting enantiomeric excess (eestart ) 1.66 × 10-22 %) in a moderate Soai system9a (B ) 1). Points belong to the left-hand vertical axis and indicate the accumulation of enantiomeric excess. Oscillatory curve belongs to the right-hand vertical axis.
reaction, even without chiral additive. The first intermediate where this amplification becomes detectable is just the COZn alkoxyzinc intermediate (by its R C atom). One may recognize a striking analogy with that chiral amplification, where Soai and co-workers6 have repeatedly reacted newer and newer portions of the reactants in the same reaction mixture, realizing thus a chain of chiral autocatalytic reactions in the same system. The evolution of chirality in such systems could be described by a simple empirical equation found recently by our groups.9 Equation 1 relates the initial (start), maximum (max), and actual product (prod.) enantiomeric excesses (ee %) of a given chiral autocatalytic system and enables to follow (eq 2) the evolution of chirality (ee) from cycle-to-cycle in consecutive catalytic cycles (in the same reaction mixture). These considerations prompted us to try to analyze the oscillating model of the Soai reaction by the same empirical treatment (eqs 1 and 2)
eeprod ) eemax
eestart B + eestart
(1)
where eeprod is the enantiomeric excess of the product in the individual reaction cycle (%), eemax is the calculated maximum enantiomeric excess achieved in the given system (%), eestart is the starting enantiomeric excess of the product at the beginning of the reaction (%) (for the first reaction cycles, we define this parameter as the percentage of added enantiopure product with respect to the starting substrate), ee (as usual) ) R-S/(R+S) × 100 or S-R/(R+S) × 100, where R and S are the molar quantities of the R and S enantiomers formed in the reaction. In this work these are representing the corresponding (R-COH) and (S-COH) alkanols. B is a constant, the ee value when the reaction reaches one-half of eemax. This formula enables to calculate the evolution of the enantiomeric excess following a chain of steps of chiral autocatalysis
eeprod(i) ) eemax
eeprod(i-1) B + eeprod(i-1)
(2)
where eeprod(i) is the enantiomeric excess of the product in the (ith) step (%) and eeprod(i-1) is the starting enantiomeric excess in the Ith step, which is the product at the (i-1) step of the reaction (%). Figure 3 shows step-by-step accumulation of enantiomeric excess in consequent autocatalytic cycles with one molecule as starting enantiomeric excess along with the oscillations in CSTR mode. First, we investigated whether accumulation of chirality takes place during oscillations in a CSTR configuration. Reaction steps
(R)-COH + Zn f (R)-COZn (S)-COH + Zn f (S)-COZn CHO + Zn f (R)-COZn CHO + Zn f (S)-COZn (R)-COZn + (R)-COZnµ (R)(R)-(COZn)2 (S)-COZn + (S)-COZnµ (S)(S)-(COZn)2 (R)-COZn + (S)-COZnµ (R)(S)-(COZn)2 (R)(R)-(COZn)2 + CHOµ (R)(R)-(COZn)2-CHO (S)(S)-(COZn)2 + CHOµ (S)(S)-(COZn)2-CHO (R)(S)-(COZn)2 + CHOµ (R)(S)-(COZn)2-CHO (R)(R)-(COZn)2-CHO + Zn f (R)(R)-(COZn)2 + (R)-COZn (S)(S)-(COZn)2-CHO + Zn f (S)(S)-(COZn)2 + (S)-COZn (R)(S)-(COZn)2-CHO + Zn f (R)(S)-(COZn)2 + (R)-COZn (R)-CHO + COH f COH (S)-CHO + COH f COH (S)-COZn f COH (R)-COZn f COH
rate parameter
reactions
k1 k1 k2 k2 k3k-3
1 1 2 2 3
k3k-3
3
k3k-3
3
k4k-4
4
k4k-4
4
k4k-4
4
k5
5
k5
5
k5
5
k6 k6 k7 k7
6 6 7 7
and corresponding rate constants used for the kinetic model including chiral processing for Soai reaction summarized in Table 2. On the basis of this model, no accumulation of chirality can be simulated in the CSTR mode. When enantiomeric excess is applied in the input flow, it does not induce any ee enrichment, rather the initial ee remains unchanged during oscillations The reason is that oscillatory periods in a CSTR do not mimic that process when newer and newer portions of the reactants are repeatedly reacted in the same reaction mixture because of the continuous outflow of the products. However, in a semibatch configuration, which is obviously more similar to the step by step catalytic cycles of chiral amplifications because no outflow of the products is maintained, results show quick oscillatory separation of the two enantiomers (R-COZn and S-COZn) even in the case of a slight initial excess of R-COH over S-COH (Figure 4). Oscillatory separation of the enantiomers was calculated on a longer run (see Figure 5) when the initial enantiomeric excess is smaller than that shown in Figure 4. The most striking result is that the calculation indicates ee accumulation even if no initial ee is added. Thus, Figure 6 shows that, after a very long quasistationary state, sudden accumulation of enantiomeric excess takes place in semi batch configuration with no initial ee. Our control calculations indicated that this accumulation is found with other numerical methods as well, but the unpredictability of the chirality sign was noticed as a general phenomenon in the calculations (Figure 7). Each run was found to be reproducible with the same calculation parameters. However, a slight variation of the internal calculation parameters, such as the integration step size or error parameter was sufficient to trigger the system into an opposite final ee. Similar simulation results were reported in a nonoscillatory model4d in a batch
Oscillatory Symmetry Breaking in the Soai Reaction
Figure 4. Calculated damped-oscillatory separation of R-COZn (top curve) and S-COZn (bottom curve) in a semibatch reactor in the case of a slight initial excess of R-COH over S-COH, k0 ) 2.6 × 10-5 s-1. Input concentrations: [R-COH]0 ) 1.10 × 10-4, [S-COH]0 ) 1.04 × 10-4 M, [Zn]0 ) 2.1 × 10-2, [CHO]0 ) 2.08 × 10-2 M. Calculations were carried out with a model shown in Table 2 using rate laws and rate constant values as given by Buhse:5 k1 ) 1.0 × 104 M-1 s-1, k2 ) 1.0 × 10-5 M-1 s-1, k3 ) 8.0 × 102 M-1 s-1, k3 ) 1.1 × 102 s-1, k4 ) 1.0 × 102 M-1 s-1, k-4 ) 1.0 × 102 s-1, k5 ) 8.0 × 103 M-1 s-1, k6 ) 7.8 × 103 M-1 s-1, k7 ) 4.0 × 10-4 s-1.
Figure 5. Calculated damped-oscillatory separation of R-COZn (top curve) and S-COZn (bottom curve) in a semibatch reactor in the case of a slight initial excess of R-COH over S-COH. Input concentrations: [R-COH]0 ) 1.06 × 10-4, [S-COH]0 ) 1.04 × 10-4 M, [Zn]0 ) 2.1 × 10-2, [CHO]0 ) 2.08 × 10-2 M. Other parameters are the same as shown in Figure 4.
J. Phys. Chem. B, Vol. 112, No. 30, 2008 9199
Figure 7. Steady-state separation of enantiomers in a semibatch reactor with no initial enantiomeric excess. Calculations were carried out with a model shown in Table 2 using rate laws and rate constant values as given by Buhse:5 k1 ) 1.0 × 104 M-1 s-1, k2 ) 1.0 × 10-5 M-1 s-1, k3 ) 8.0 × 102 M-1 s-1, k-3 ) 1.1 × 102 s-1, k4 ) 1.0 × 102 M-1 s-1, k-4 ) 1.0 × 102 s-1, k5 ) 8.0 × 103 M-1 s-1, k6 ) 7.8 × 103 M-1 s-1, k7 ) 4.0 × 10-4 s-1, [(R)-COH]0 ) 1.04 × 10-4 M, [(S)-COH]0 )1.04 × 10-4, [CHO]0 ) 2.08 × 10-2 M, [Zn]0 ) 2.1 × 10-2 M, k0 ) 2.9 × 10-5 s-1. [R-COZn] top, [S-COZn] bottom.
or achiral substrate concentration can drive the system into an optically active state from achiral initial state. A very recent publication10 reports on an analogous effect of “added noise” which gives an additional support to our interpretation (and vice versa). The results of the present study provide an interesting connection to prebiotic or early biotic chemistry. It is one of the fundamental signatures of life that spatially and temporally well-organized cyclic reactions operate in a concerted manner.11 A decisive fraction of these reactions is characterized by a high degree of enantioselectivity.12 The origin of this selectivity is not yet fully cleared up,6,12 one of the most likely possibilities is considering statistical fluctuations of enantiomers in achiralto-chiral reactions.13 The combination of the period behavior of life-related reactions with these fluctuations could very efficiently lead to high enantioselectivities in a relatively short time, as demonstrated by the present study. The combination of chiral oscillatory dynamics with stochastic in vivo phenomena opens very actual perspectives for biochemical14 and biological15 research. Acknowledgment. Financial support is acknowledged to the Italian National Research Council (CNR, Rome) and to the Italian Ministry of Education and University (MIUR, contract No. RBPR05NWWC). This research project was also supported by the [Hungarian] Scientific Research Foundation (Grants OTKA, No. T046325). References and Notes
Figure 6. Calculated oscillatory separation of the enantiomers with no initial enantiomeric excess in a semibatch reactor. Calculations were carried out with a model shown in Table 2 using rate laws and rate constant values as given by Buhse:5 k1 ) 1.0 × 104 M-1 s-1, k2 ) 1.0 × 10-5 M-1 s-1, k3 ) 8.0 × 102 M-1 s-1, k3 ) 1.1 × 102 s-1, k4 ) 1.0 × 102 M-1 s-1, k-4 ) 1.0 × 102 s-1, k5 ) 8.0 × 103 M-1 s-1, k6 ) 7.8 × 103 M-1 s-1, k7 ) 4.0 × 10-4 s-1. Input concentrations: [(R)-COH]0 ) 1.04 × 10-4 M, [(S)-COH]0 ) 1.04 × 10-4, [CHO]0 ) 2.08 × 10-2 M, [Zn]0 ) 2.1 × 10-2 M. Flow rate/reactor volume: k0 ) 2.6 × 10-5 s-1. [R-COZn] top, [S-COZn] bottom.
reactor. Such an effect has been explained as a consequence of dynamic instabilities in the model. A bifurcation can be recognized in the model behavior in which any kinetic parameter
(1) Soai, K.; Shibata, T.; Morioka, H.; Choji, K. Nature 1995, 378, 767–768. (2) (a) Soai, K.; Shibata, T.; Kowata, Y. Jpn. Kokai Tokkyo, Koho 1997, 9, 268–179. (b) Soai, K.; Sato, I.; Shibata, T.; Komiya, S.; Hayashi, M.; Matsueda, Y.; Imamura, H.; Hayase, T.; Morioka, H.; Tabira, H.; Yamamoto, J.; Kowata, Y. Tetrahedron: Asymmetry 2003, 14, 185–188. (c) Kawasaki, T.; Suzuki, M.; Ishikawa, K.; Soai, K. Chirality 2006, 18, 479–482. (3) (a) Pa´lyi, G.; Micskei, K.; Ze´ka´ny, L.; Zucchi, C.; Caglioti, L. Magyar Ke´m. Lapja 2005, 60, 17–24. (b) Mislow, K. Coll. CzechosloV. Chem. Commun. 2003, 68, 849–864. (4) (a) Sato, I.; OmiyaD.; Tsukiyama, K.; OgiY.; Soai, K. Tetrahedron: Asymmetry 2001, 12, 1965–1969. (b) Blackmond, D. G.; McMillan, C. R.; Ramdeehul, S.; Schorm, A.; Brown, J. M. J. Am. Chem. Soc. 2001, 123, 10103–10104. (c) Sato, I.; Omiya, D.; Igarashi, H.; Kato, K.; Ogi, Y.; Tsukiyama, K.; Soai, K. Tetrahedron: Asymmetry 2003, 14, 975–979. (d) Rivera-Islas, J.; Lavabre, D.; Grevy, J. M.; Hernandez-Lamoneta, R.; Royas-
9200 J. Phys. Chem. B, Vol. 112, No. 30, 2008 Cabrera, H.; Micheau, J.-C.; Buhse, T. Proc. Natl. Acad. Sci. USA 2005, 102, 13743–13748. (e) Blackmond, D. G. Tetrahedron Asymmetry 2006, 17, 587–589. (5) Buhse, T. Tetrahedron: Asymmetry 2003, 14, 1055–1061. (6) (a) Soai, K.; Shibata, T.; Sato, I. Acc. Chem. Res. 2000, 33, 382– 390. (b) Soai, K. Sato, I. ViVa Origino 2002, 30, 186–198. (c) Soai, K.; Pa´lyi, G.; Zucchi, C.; Caglioti, L. Progress in Biological Chirality; Elsevier: Oxford, 2004; pp 355-364. (d) Lente, G. J. Phys. Chem. A 2006, 110, 12711–12713. (e) Iwamoto, K. Bull. Chem. Soc. Jpn. 2007, 80, 1099–1102. (f) Blackmond, D. G. Matar, O.K. J. Phys. Chem. B 2008, 112, 5098– 5104. (7) (a) Field, R. J.; Burger, M. Oscillations and TraVeling WaVes in Chemical Systems; Wiley and Sons 1985. (b) Orba´n, M. Magyar Ke´m. Folyo´irat 2006, 112, 44–54. (c) Qian, H. J. Phys. Chem. B 2006, 110, 15063–15074. (8) Kaps, P.; Rentrop, P. Numer. Math. 1979, 33, 55. (9) (a) Micskei, K.; Po´ta, G.; Caglioti, L.; Pa´lyi, G. J. Phys. Chem. A 2006, 110, 5982–5984. (b) Micskei, K.; Maioli, M.; Zucchi, C.; Caglioti, L.; Pa´lyi, G. Tetrahedron: Asymmetry 2006, 17, 2960–2962. (c) Barabas, B.; Caglioti, L.; Zucchi, C.; Maioli, M.; Ga´l, E.; Micskei, K.; Pa´lyi, G. J. Phys. Chem. B 2007, 111, 11506–11510.
Micskei et al. (10) Cruz, J. M.; Parmananda, P.; Buhse, P. J. Phys. Chem. A 2008, 112, 1673–1676. (11) (a) Ga´nti, T. J. Theor. Biol. 1997, 17, 583–593. (b) Ga´nti, T. Chemoton Theory. In Theory of LiVing Systems; Kluwer Academic/Plenum Press: Dordrecht-New York, 2003. (12) (a) Keszthelyi, L. Q. ReV. Biophys. 1995, 28, 473–507. (b) Bada, J. L. Nature 1995, 374, 594–595. (c) Pa´lyi, G.; Zucchi, C.; Cagliotti, L. AdVances in BioChirality; Elsevier: Amsterdam, 1999. (d) Pa´lyi, G.; Zucchi, C.; Cagliotti, L. Progress in Biological Chirality; Elsevier: Oxford, 2004. (13) (a) Mills, W. H. Chem. Ind. 1932, 750–759. (b) Frank, F. C. Biochim. Biophys. Acta 1953, 11, 459–463. (c) Siegel, J. S. Chirality 1998, 10, 24–27. (d) Caglioti, L.; Hajdu, C.; Holczknecht, O.; Ze´ka´ny, L.; Zucchi, C.; Micskei, K.; Pa´lyi, G. ViVa Origino 2006, 34, 62–80. (14) Recent references:(a) Velonia, K. Angew. Chem., Int. Ed 2005, 44, 560–564. (b) Samoilov, M.; Plyasunov, S.; Arkin, A. P. Proc. Natl. Acad. Sci. USA 2005, 102, 2310–2315. (c) Chaudhury, S.; Cherayil, B. J. J. Chem. Phys. 2006, 125, 024904-1-024904-9. (15) Turner, T. E.; Schell, S.; Burrage, K. Computat. Biol. Chem. 2004, 28, 165–178.
JP803334B
