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J. Phys. Chem. B 2008, 112, 5098-5104
Re-Examination of Reversibility in Reaction Models for the Spontaneous Emergence of Homochirality Donna G. Blackmond*,†,‡ and Omar K. Matar‡ Department of Chemistry and Department of Chemical Engineering and Chemical Technology, Imperial College London SW72AZ, United Kingdom ReceiVed: December 18, 2007; In Final Form: January 31, 2008
The concept of reversibility in complex chemical reaction networks has recently been introduced in discussions concerning the origin of biological homochirality. In computational studies drawing on an analogy to recent experimental studies involving reversible crystallization processes, recycling of reaction educts has been suggested to provide a driving force for the spontaneous emergence of homochirality. We demonstrate here that reversible reaction networks closed to mass flow lead inexorably to a racemic state for thermally driven reactions, which must adhere to the principle of microscopic reversibility. This conclusion was reached for analogous “triangle reaction” networks studied by Onsager in 1931. Special cases such as photochemical reactions offer an exception that may have prebiotic relevance. Fundamental differences between physical and chemical systems are discussed in order to clarify the role of reversibility in each case.
Introduction The evolution of homochirality in amino acids and sugars from a presumably racemic prebiotic world is a topic that has long intrigued scientists. The emergence of homochirality requires both a mechanism for spontaneous symmetry breaking and a means of propagating the resulting imbalance of enantiomers. The importance of stochastic models for the interpretation of symmetry breaking in autocatalytic reactions has been discussed.1,2 Models for the propagation of an imbalance have for the most part relied on a deterministic approach to the question of symmetry breaking by assuming the existence of an initial imbalance, for example, as outlined in the Frank model.3 Experimental proof of these concepts came in the 1990s with two separate discoveries of “far-from-equilibrium” processes leading to single chirality. The first report documented a physical process of enantiomorph crystallization,4 and the second reported an irreversible autocatalytic chemical reaction.5 More recently and at the other end of the spectrum, a model involving thermodynamic control of the evolution of homochirality in solution has also been proposed and experimentally verified.6 Perhaps most compelling, however, is Viedma’s “chiral amnesia” model based on interplay between kinetics and thermodynamics in crystallizations close to equilibrium.7,8 A critical feature of this model is the reversibility of the crystallization process that, paradoxically, represents the driving force for achieving solid-phase homochirality.9 The concept of reversibility as a driving force for homochirality has also been discussed in the context of chemical reaction systems,10-12 an idea that is appealing because it provides a means of correcting “mistakes” that produce the wrong enantiomer. Gridnev12 has suggested that reversibility in chemical reaction systems may be analogous to the crystallization/dissolution processes in Viedma’s model for solid-phase homochirality, although this appears to be counterintuitive given the wealth of examples in solution-phase asymmetric synthesis where reverse reactions cause erosion rather than enhancement of product enantiomeric excess. * Corresponding author. E-mail:
[email protected]. † Department of Chemistry. ‡ Department of Chemical Engineering and Chemical Technology.
In this paper, we probe in general the concept of reversibility in chemical reactions by examining in closer detail the kinetic models proposed in two recent reports. In the first case, a simple autocatalytic reaction network was treated with simulations incorporating what was termed a “back reaction”.10 In a second example that treats a more sophisticated noncatalytic network, epimerization reactions were proposed as the central driving force to destabilize the racemic state.11 By theoretical arguments and reaction simulations, we provide a general assessment of the role of reversibility in the approach to a single chiral state via chemical reaction systems.13 A comparison between chemical and physical processes reveals different constraints on the role of energy input in promoting the evolution of a single homochiral state. Results Case 1: Autocatalytic Reaction. Soai’s discovery that the autocatalytic alkylation of pyrimidyl aldehydes using diisopropylzinc5 (Scheme 1) leads to increasing product alkanol ee over time represents the first and most prominent experimental proof of concept for the Frank model3 for spontaneous asymmetric synthesis. Further studies have delineated features of both the stochastic symmetry breaking and the inhibition and propagation mechanisms of this reaction, as required for asymmetric amplification.14 Blackmond, Brown, and co-workers showed that the reaction network produces a stochastic distribution of homochiral and heterochiral dimers, with homochiral dimers active as catalysts.14 The autocatalyst ee is effectively increased after in each catalytic turnover because the inactive heterochiral dimer serves to sequester a proportionately larger fraction of the minor enantiomer. The model describing the Soai reaction is a “far-fromequilibrium” irreversible reaction scenario in which production of the minor enantiomer is not forbidden; ee increases over time because selective propagation of the major enantiomer eventually overwhelms the minor enantiomer produced as a consequence of the near-racemic conditions at the outset of the reaction. Thus, a closed system containing a finite concentration of reagents will always contain a finite concentration of the minor enantiomer; a single chiral state can be achieved in this model only in an open system with an infinite source of reactants.
10.1021/jp7118586 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/28/2008
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SCHEME 1: Soai Autocatalytic Reaction3 (R ) CH3, (CCSi(CH3)3)5
SCHEME 2: Case 1: Autocatalytic Reaction for Amplification of Product eea
Saito and Hyuga10 presented a simple example of an irreversible autocatalytic reaction that amplifies enantiomeric excess with the network shown in Scheme 2a, where a prochiral molecule A reacts catalytically with two molecules of a chiral product R or S to give a further molecule of that product. A second reagent B is maintained at high concentration (pseudozero-order conditions). While the Saito-Hyuga model represents a valid example of a nonlinearly autocatalytic reaction, it should be noted that it does not describe the experimental observations of the Soai reaction discussed above.15 However, this simplified model provides examples of the features important for general discussion of far-from-equilibrium closed autocatalytic systems, in that minor enantiomer production is neither forbidden nor destroyed but is gradually overwhelmed. Equation 1 gives a rationalization of the asymmetric amplification observed in the irreversible reaction shown in Scheme 2a, starting from a small imbalance between autocatalysts R and S. Because two molecules of R or S are present in the elementary reaction step, the ratio of reaction rates forming R and S is not linearly related to their concentration ratio.
(
)
rate(R) rate(S)
Scheme
2a
)
( ) [R] [S]
2
(1)
Figure 1a shows a reaction simulation confirming that under irreversible reaction conditions asymmetric amplification occurs in the reactions of Scheme 2a. This also confirms that the degree of amplification that may be achieved in this closed system is limited by the finite magnitude of the initial concentration of substrate A and by the presence of the minor product enantiomer, because the latter continues to produce itself at low conversion of A before the major enantiomer reaction pathway begins to dominate. Saito and Hyuga proposed that inclusion of the “back reactions” shown in Scheme 2b would allow unlimited amplification of ee in this network. They suggested that the reconversion of A allows further amplification of enantiomeric excess in a closed system with a finite concentration of A. According to this model, the rate of consumption of the minor enantiomer via the back reaction overwhelms its production over time. The ratio of rates for production of R and S for the network in Scheme 2b is given by eq 2. At high conversions of A, the overall production of the minor enantiomer becomes negative, meaning that it is converted back to A and then on to the major enantiomer, thus enhancing product ee. The simulation shown in Figure 1b confirms that inclusion of these reactions indeed allows the system to achieve homochirality.
(
)
rate(R) rate(S)
( )
[R] 2 ‚ 2b ) [S] Scheme
(
a The “back” reactions of Scheme 2b are the true reverse reactions of the uncatalyzed reaction in Scheme 2d.
kforward[A] -
kforward[A] -
)
kback [R] kback [S]
(2)
However, it is important to clarify that the “back reactions” given in Scheme 2b do not represent the reverse of the
Figure 1. Simulation of the evolution of enantiomeric excess over time for the nonlinearly autocatalytic system shown in Scheme 2 with [A]0 ) 1 M, [R]0 ) 0.00505 M; [S]0 ) 0.00495 M (ee0 ) 1%). (a) Solid blue line: reaction network of Scheme 2a with kautocat ) 5 M-2 s-1. (b) Dashed black line: reaction network of Scheme 2a plus “back reactions” of Scheme 2b with kback ) 0.01 s-1. (c) Open pink circles: reaction network of Scheme 2a plus the true reverse reactions of Scheme 2c with kreverse ) 0.2 M-2 s-1. (d) Solid black line with open diamonds: network combining true reversible autocatalytic reactions (Schemes 2a + 2c) and true reversible uncatalyzed reactions (Schemes 2b + 2d). The value for kuncat ) 0.25 s-1 is dictated by the values of kautocat, kreverse, and kback as required by thermodynamics.
autocatalytic mechanism in Scheme 2a. For true reversibility, a reaction must proceed from products to reactants through the same elementary steps as in the forward direction. This is a statement of the principle of microscopic reversibility. First articulated by Tolman in 1925,16 microscopic reversibility dictates that in a reaction system converting reactant A to products R and S such as that shown in Scheme 2a, under equilibrium conditions, the forward and reverse reactions not only have equal rates but also take identical pathways. Thus, the least-energy pathway from reactants to products also
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represents the least energy pathway back from products to reactants. Systems operating far from equilibrium, including irreversible catalytic cycles, or open systems with removal of products, are not bound by this principle. However, as shown below, the equilibrium condition provides constraints on the values of the reverse rate constants at a given temperature. The truly reversible version of the autocatalytic reactions in Scheme 2a is given by Scheme 2c, showing that the re-formation of A is not first-order, but third order, in [R] or [S]. Figure 1c shows that product ee for the truly reversible autocatalytic reaction in the absence of the uncatalyzed reaction initially follows the same course as shown for the irreversible case in Scheme 2a and Figure 1a, but at high conversion of A the reverse reaction begins to erode rather than enhance the product ee until the system approaches the racemic state. Equation 3 gives the ratio of R to S formation for this truly reversible autocatalytic case, demonstrating that it is the overall rate of production of the major enantiomer that will become negative at high conversions of A; that is, the major enantiomer will begin to be consumed, and ee will fall, as high conversion of A is approached. The reaction eventually equilibrates with a racemic mixture of R and S at a value of [A] dictated by the values of kforward and kreverse.
( ) rate(R) rate(S)
2c
)
Scheme
( )(
)
[R] 2 kforward[A] - kreverse[R] ‚ [S] kforward[A] - kreverse[S]
SCHEME 3: Network of CO-driven Peptide Formation/ degradation Pathways Following Ref 17
SCHEME 4: Case 2: Noncatalytic Dipeptide Formation and Degradation Reaction Network
(3)
While the back reactions in Scheme 2b proposed by Saito and Hyuga do not represent the reverse of the mechanistic pathway shown in Scheme 2a, they do represent the reverse of the uncatalyzed conversion of A to R or S. Thermodynamics dictates that the catalyzed and uncatalyzed pathways must exhibit identical ratios of forward and reverse rate constants, regardless of whether the system is near or far from equilibrium. Thus, the values of kautocat and kback for the reactions in Scheme 2a,b, respectively, are related to the values for the true reverse pathways for the autocatalyzed and uncatalyzed pathways shown in Scheme 2c,d, respectively, in the manner illustrated in eq 4.
) Kautocat ) Kuncat eq eq
[R]eq [A]eq
)
kuncat kautocat (Scheme 2b,d) ) kback kreverse (Scheme 2a,c) (4)
Microscopic reversibility thus dictates that the uncatalyzed “back” reaction of Scheme 2b cannot be included without inclusion of both the uncatalyzed and the autocatalytic reaction pathways as reversible reactions. This brings together all four networks in Scheme 2. The value of kuncat in Scheme 2d is dictated by our previous choice of values of kautocat, kback, and kreverse in the same example, as required by eq 4. Figure 1d shows that when the reversible uncatalyzed reaction is properly included in the reversible autocatalytic network; no amplification in ee is observed, and the trend toward a racemic mixture is inexorable. Case 2: Noncatalytic Dipeptide Formation and Degradation Reaction Network. The second example that we treat is that given by Plasson et al.,11a who developed a reaction model based on Wa¨chtersha¨user and co-workers’17 cycle of peptide formation and degradation from R-amino acids catalyzed by (Fe,Ni)S, depicted in Scheme 3. The CO-driven thermal activation of L and D amino acids produces L* and D*, suggested to be N-carboxyanhydrides, followed by reaction with a further L or D to give peptide dimers LL and DD. Degradation
occurs via further CO-driven formation of hydantoin (E) and hydrolysis to urea (F), ultimately returning back to unactivated monomers L and D. Plasson et al.11a used this example to provide a model for a system closed to mass flow but open to energy input, couched in terms of the elementary rate steps shown in Scheme 4a. Although no mass enters or leaves the system, an input of energy is required to activate of L and D enantiomers to L* and D*. These activated monomers then add to unactivated monomers to form homochiral (LL and DD) and heterochiral (DL and LD)
Spontaneous Emergence of Homochirality
Figure 2. Simulations of the unidirectional reaction system shown in Scheme 4a starting from ee values of 100, 80, 40, 20, and 1% ee. [L]0 + [D]0 ) 1 M;. a ) 10-8 s-1; b ) 5 × 10-4 ) 5 s-1; h ) 10-7 s-1; e ) 10-7 s-1; p ) 2 × 10-2 M-1 s-1; R ) 0.35; β ) 0.2; γ ) 0.3. Values of the rate constants taken from ref 11a.
dimers. Epimerization between homochiral and heterochiral dimers with forward and reverse rate constants e and γe, respectively, presumably occurs via the hydantoin intermediate E. The authors of ref 11a maintain that this reversible epimerization provides a driving force for asymmetric amplification of enantiomeric excess from a small initial imbalance in L and D monomers. Plasson and co-workers describe the mechanism shown in Scheme 4a as a system of network catalytic cycles containing what they term “dissipative” loops. These are illustrated in Scheme 4b. Decay of homochiral dimers LL and DD or heterochiral dimers DL and LD produces two unactivated monomers. However, in their model, the reverse reactions are forbidden, and dimer formation is allowed only via reaction between one unactivated and one activated monomer. Scheme 4b shows that these reactions are unidirectionally cyclic for the net reaction of the L and D monomers. The authors state that the dimer epimerization reactions link the left and right sides of this network in an “antagonistic” manner that results in a steady state where one side partially depopulates the other. They contend that while the system is closed with regard to mass flow, an input of energy allows continual reactivation of L and D to L* and D*, respectively, which maintains the system away from equilibrium. These authors suggest that this energy input serves a role analogous to that of mass flow in an open system with unidirectional cycles. Such a system conserves a steady state in cyclic intermediate species concentrations by converting an input of energy rich matter to an output of energy depleted matter. By considering only the elementary reaction steps shown in Scheme 4a,b, however, it is clear that this closed mass system does not obey microscopic reversibility. Using estimated values for the rate constants, these authors simulated the reaction of a 1% ee initial mixture of L and D at a total concentration of 1 M according to the network in Scheme 4a, showing that the system ultimately reaches a stable nonracemic value of approximately 70% ee.18 This simulation is reproduced in Figure 2 (orange line). Indeed, Figure 2 shows that the unidirectional reaction network of Scheme 4a proposed by Plasson and co-workers leads to the same nonracemic steady state, but not to homochirality, from any initial ee. The ultimate ee attained is dictated by the unidirectional flow of species coupled with the relative magnitudes of the various rate constants. Thus, both amplification and erosion of ee, but not
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Figure 3. Values of the rate constants kdissoc, k′assoc, and k′dissoc for chosen values of kassoc from Scheme 4c, as given by eqs 5-8 and the kinetic parameter values given in Figure 2.
homochirality, is predicted by the network shown in Scheme 4a under the conditions presented by Plasson et al. 11a Although the network in Scheme 4a is stated to represent a nonequilibrium steady-state condition, it is important to discuss the unidirectional elementary reactions given in the scheme in the context of the theoretical equilibrium condition characterized by forward and reverse rate constants for each elementary step in the network. Scheme 4c shows the completely reversible network for the system in Scheme 4a. Four elementary rate steps and rate constants for each side of the network, which were not included in the network of Scheme 4a, are shown with dashed reaction arrows in Scheme 4c. These include two steps for association (kassoc and k′assoc) and two for dissociation (kdissoc and k′dissoc) of homochiral and heterochiral dimers, respectively. Consideration of this system at equilibrium allows us to determine the relationships between these additional rate constants and the rate constants given for the unidirectional reaction networks in ref 11a and in Scheme 4a, given in eqs 5-8.
kassoc‚kdissoc )
a‚h‚p b
k′assoc‚k′dissoc ) R‚β
a‚h‚p b
(5)
(6)
k′dissoc R ) kdissoc γ
(7)
k′assoc ) γ‚β kassoc
(8)
Lack of inclusion of these steps in the reaction network presented in Scheme 4a implies that all four rate constants exhibit values that may be regarded as negligible under the reaction conditions. However, only three of these four equations are independent, meaning that choosing a value of one of the four rate constants (kassoc, k′assoc, kdissoc, k′dissoc) determines the other three values. Figure 3 reveals an inverse relationship
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Blackmond and Matar SCHEME 5: Onsager’s Triangle Reaction22 and the Reciprocal Relation Derived for the Rate Constants
Figure 4. Simulation of the reaction network in Scheme 4c for different values of the association constant kassoc, with kdissoc, k′assoc, and k′dissoc given from Figure 3 and eqs 5-8; all other constants as in Figure 2.
between the magnitudes of association and the dissociation rate constants; therefore, assigning a very small value to an association rate constant in the network necessarily provides large values for the rate constants for the two dissociation steps. Thus, all reverse steps may not simultaneously be neglected in simulations of the reaction network regardless of whether the system operates near or far from equilibrium. Figure 4 demonstrates that when the full network with all elementary step rate constants is properly taken into account, a stable nonracemic steady state will not result. The system inexorably approaches the racemic state for all initial ee values and all values of the rate constants, tested for a range of 12 orders of magnitude. The magnitude of the value of kassoc simply dictates whether the system is driven toward monomers or toward dimers in its final racemic state. Plasson et al.11 based their model of Scheme 4a on the experimental reports of Wa¨chtersha¨user and co-workers17 of peptide formation/degradation shown in Scheme 3. Those authors found experimental evidence of both reversibility and racemization in this thermally activated network. Thus, both theoretical considerations and experimental evidence suggest that recycling in the experimental reaction network of Scheme 3 will not lead to emergence of a stable nonracemic state. Discussion The examples treated here rely on the concept of activated recycling and network autocatalysis proposing that a nonequilibrium, nonracemic steady state may be maintained in a closed mass network by a continuous input of energy. Maintaining a system away from equilibrium is crucial to the concept of “hypercycles”, which is the term introduced by Eigen and Schuster19 to describe linked catalytic cycles representing a minimum requirement for a macromolecular organization that may accumulate, preserve, and process information. Their treatment of hypercycles is based on the net flow of energyrich matter toward energy-poor matter across the open system. As long as mass flow pulls products away from one cycle and into the next, a unidirectional steady-state may be maintained. This is the case for any “irreversible” catalytic cycle: the catalyst is regenerated in the final step, which is different from the first step in which the catalyst was consumed by binding a substrate. Such an irreversible cycle clearly operates outside equilibrium conditions and does not by itself obey microscopic reversibility. Metabolic cycles such as the Krebs cycle offer typical examples (although the Krebs cycle operates in reverse under some
conditions). In this context, however, it is important to note that the maximum efficiency of a catalytic cycle will be reached when the system remains close to equilibrium, as shown theoretically as well as experimentally by Knowles and Albery20 for the triosephosphate isomerase system, an enzyme that has evolved to near perfect efficiency. In order to assess these concepts for a closed mass system driven by energy flow, the nature of the energy-activating process must be considered. In their recent review of models for homochirality, Plasson et al.11b discussed both of the examples we treat here. For Saito and Hyuga’s autocatalytic reaction of Case 1, they conceded that Scheme 2b does not represent the true reverse of Scheme 2a but suggested that such “back” reactions could result from an application of an energy source to the closed-mass system, without commenting further on the nature of the energy source or on how its application could avoid the full autocatalytic and noncatalytic system of reactions of the combined Scheme 2a-d. Saito and Hyuga’s own work10a addressed neither microscopic reversibility nor the question of energy input to their system. In his citation of earlier work by Marcelin,21 Tolman likened such cyclical maintenance of a catalytic state to the emi- and immigration of people between two countries separated by a mountain range. “For migration in either direction, the top of the range must be crossed, in analogy with the equal energies of the activated molecules for the two opposing reactions.”16 Contrary to the premise of Saito and Hyuga’s model,10a autocatalytic reaction products will not preferentially return to reactants by a higher energy noncatalytic route when afforded the lower energy catalytic route. The main features of the unidirectional reaction network presented by Plasson et al. in the second example treated here have been studied previously in the context of microscopic reversibility. It may be noted that each quadrant of the reaction network shown in Scheme 4 (neglecting the connecting epimerization reactions) in fact may be treated as the classic “triangle reaction” discussed by Onsager22 in 1931 in the derivation of a general class of reciprocal relations in irreversible processes based on microscopic reversibility and the detailed balance of all flows at equilibrium (Scheme 5).23 This is a topological property of the system, constraining the relationship between the rate constants such that only five out of the six shown in Scheme 5 are independent.,24 The incorrect mathematical description that results when microscopic reversibility is not properly taken into account can predict behavior including oscillations and product amplification that will not be physically meaningful. Plasson et al. imply that any applied energy source may be used to activate L and D to produce L* and D* to allow the unidirectional system of Scheme 4 to operate in a closed mass system. However, the thermal energy sources applied in the experimental Wa¨chtersha¨user reactions on which this network is based, and those most commonly applied in chemical reactions, must obey microscopic reversibility under equilibrium conditions. Such systems closed to mass flow necessarily incorporate the reverse reactions in an approach to equilibrium
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rather than steady-state. Our simulations in Figure 1d and Figure 4 reveal that reversibility induced by the elementary reaction steps proposed in the kinetic models will lead inexorably to a racemic state in a system closed to mass flow. Activating energy sources that are not subject to microscopic reversibility include photochemical reactions, external magnetic fields, and Coriolis forces. In a photochemical reaction, electronic excitation imparts energy to reacting species via absorption of photons rather than via intermolecular collisions. Thus, if the activated L* and D* species in Case 2 could be produced by photochemical activation, the linked unidirectional network shown in Scheme 4b could be feasible in a closed mass system, as long as the dissipative part of the cycle does not create sufficient thermal energy to allow the forbidden reactions represented by dashed arrows in Scheme 4c.25 Dimerization of anthracene activated by ultraviolet radiation is a one of the oldest known photochemical examples26 and is analogous to the homochiral dimerization steps in Scheme 4a. In a photochemical reaction, however, the activation rate to produce L* and D* in Scheme 4a cannot be described by the elementary unimolecular kinetics used in the simulations by Plasson et al. The rate of a photochemical reaction is given by the number of photons involved in activation per volume per time. The rate law is obtained by combining this relation with the Beer-Lambert law as shown in eq 9. An analogous equation for activation of D may be written, with I0 as the incident radiation intensity, l as the path length, and R as a constant proportional to the extinction coefficient. Thus, a photochemical rate process exhibits a more complex dependence on substrate concentration than does a simple unimolecular elementary reaction governed by thermal energy.
ractivation )
(
)
I0 ‚(1 - e-R[L]l ) l‚NA‚hν
(9)
Since extinction coefficients can vary by many orders of magnitude at different wavelengths of radiation, both the overall activation rate and its sensitivity to monomer concentrations L and D can vary widely. Two limiting cases occur when the extinction coefficient is very large and very small; in these cases, the photochemical reaction rate is approximately zero-order and first-order in monomer concentration, respectively, and a stable nonracemic state may indeed be achieved in a system closed to mass flow as described by Scheme 4a. The ultimate ee under steady state will be determined by the magnitudes of the constants in Scheme 4a (with the activation pathway described by eq 9 rather than by a unimolecular rate law). Photochemical reactions are of tremendous biological importance, with the production of carbohydrates from CO2 and H2O via photosynthesis the most prominent example. This process employs a complex and highly evolved molecular catalyst, chlorophyll, to impart energy absorbed from light specifically and efficiently to reacting molecules. The question of whether more primitive versions of such processes could have aided in the evolution of homochirality has been addressed. Careri and Wyman27 noted that a simple unidirectional steadystate photochemical cycle involving a primitive enzyme (the “turning wheel”24c) could provide a feasible model for prebiotic evolution provided that the rates of energy absorption and enzyme-substrate binding are of the same order of magnitude. They estimated these rates for the one-photon excitation of acetanilide, a model for the backbone of a protein, at appropriate prebiotic concentrations. They concluded that such unidirectional photochemical processes may have been involved in symmetry breaking.
Physical vs Chemical Processes. Plasson and co-workers concede that experimental support for asymmetric enhancement due to the reversibility in closed-mass chemical reaction systems has not yet been reported. By contrast, the role of reversibility in the physical processes leading to a single chiral state by energy input into a system closed to mass flow has been demonstrated experimentally and rationalized theoretically. Viedma7 has studied the evolution of a single chiral solid state in the inorganic NaClO3/water system. NaClO3 is an intrinsically achiral compound that forms enantiomorphic crystals in the solid phase. An input of mechanical energy into a closed-mass system by attrition of the solid-liquid slurry activates the crystal dissolution process. This produces smaller crystals that, according to the Gibbs-Thomson rule, dissolve faster than larger ones. The increased local solution concentration of NaClO3 produced by this dissolution in turn provides an increased driving force for the crystal growth process via Ostwald ripening, which favors large crystals regardless of their hand. The net result is that small crystals disappear and the large crystals grow larger. A small imbalance in the handedness of large compared with small crystals, which may be achieved stochastically, allows “depopulation” of one chiral state (the one randomly favored by the small crystals) toward the other (the one randomly favored by the larger crystals). The key point is that the application of mechanical energy that causes smaller crystals to dissolve does not accelerate the symmetrically reverse process, which would be the growth of those same smaller crystals because the physical process of Ostwald ripening intervenes, much in the way coupling of cycles out of equilibrium relaxes the constraint of microscopic reversibility on an individual cycle in the sequence. Thus, the analogy between reversibility in physical and chemical processes is not general, and caution must be used in attempting to extend the Viedma model to thermally activated reaction systems. Conclusions The concept of reversibility as a driving force for the evolution of a single chiral state in closed mass reacting systems operating away from equilibrium has been examined. A system that allows energy input but is closed to mass flow can operate in a unidirectional fashion and maintain a nonracemic steady state only if the energy source driving the network is not subject to the principle of microscopic reversibility. Physical cycles of crystallization and dissolution, or unidirectional catalytic cycles activated in photochemical reactions, meet this criterion. It follows that chemical reactions driven by thermal energy cannot lead to a nonracemic steady state under conditions of closed mass even when operating far from equilibrium. Two literature examples treating reversibility in chemical reaction networks are re-examined in this context. These concepts should be considered in future discussions of reaction systems that might serve as a model for the evolution of biological homochirality.28 Acknowledgment. D.G.B. is grateful for stimulating conversations with J. M. Brown, which led to the formulation of the ideas in this work. D.G.B. also thanks J. S. Bradley, C. Viedma, and L. J. Broadbelt for stimulating discussions. Funding from AstraZeneca is gratefully acknowledged. D.G.B. is a Wolfson Research Merit Award holder. References and Notes (1) (a) Barabas, B.; Caglioti, L.; Zucchi, C.; Maioli, M.; Gal, E.; Micskei, K.; Palyi, G. J. Phys Chem. B 2007, 111, 11506; (b) Saito, Y.; Sugimori, T.; Hyuga, H. J. Phys. Soc. Jpn. 2007, 76, 044802; (c) Shao, J.; Liu, L. J. Phys. Chem. A 2007, 111, 9570; (d) Caglioti, L.; Hajdu, C.;
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