Stochastic Kinetic Analysis of the Frank Model. Stochastic Approach to

The continuous time discrete state stochastic kinetic approach and its extension to flow-through reactors was used to study a straightforward modifica...
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J. Phys. Chem. B 2009, 113, 7237–7242

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Stochastic Kinetic Analysis of the Frank Model. Stochastic Approach to Flow-Through Reactors Ga´bor Lente* and Tama´s Ditro´i Department of Inorganic and Analytical Chemistry, UniVersity of Debrecen, POB 21, Debrecen, H-4010 Hungary ReceiVed: January 12, 2009; ReVised Manuscript ReceiVed: March 24, 2009

The continuous time discrete state stochastic kinetic approach and its extension to flow-through reactors was used to study a straightforward modification of the Frank model to interpret absolute asymmetric synthesis, which is impossible using deterministic approaches. Computational methods for calculating multidimensional probability distributions and expectations for enantiomeric excess were developed. The results showed that narrow focus on the conventionally defined enantiomeric excess could lead to misleading conclusions and the yield-adjusted enantiomeric excess is often more useful. Closed systems proved to be more favorable for the formation of high enantiomeric excesses than flow-through reactors and the importance of mutual antagonism can also be questioned in the original Frank model. It was also shown that a flow-through reactor with a relatively small number of molecules predicts the behavior of much larger systems well. Introduction The overwhelming dominance of L-amino acids and D-sugars over their mirror-image counterparts in nature has attracted a lot of scientific attention ever since the discovery of chirality.1 The first fundamental question in this respect is how any enantiomeric excess could be formed from a nonchiral starting state during evolution despite the apparently symmetric forces of nature. Symmetry-breaking must have occurred abiotically, although it could later be enhanced and maintained by biological processes. The second, and notably separate, fundamental question is why exactly L-amino acids and D-sugars dominate on Earth. This might be a consequence of the laws of nature but at present seems more likely to be just a coincidence and independently formed forms of life could have the opposite dominating chirality. Theoretical attempts to answer the first question led to three concepts: enantioselective autocatalysis, chiral amplification, and absolute asymmetric reaction.2,3 In an absolute asymmetric reaction, a nonracemic mixture of enantiomers forms from nonchiral reactants in the absence of any external chiral effects. There are two experimentally known chemical examples of this phenomenon.4,5 In addition, the crystallization of NaClO3 provides a closely related, but not strictly chemical observation.6 All of these known experimental examples are inherently stochastic in nature and attempts to model them should be based on stochastic considerations. Chiral amplification means the formation of large enantiomeric excesses by the end of a process as a consequence of a much smaller initial imbalance and can be understood in deterministic terms as well. The most wellknown and widely studied experimental example of this phenomenon is the Soai reaction.2,5,7-11 Finally, enantioselective autocatalysis is a catalytic process in which an enantiomer catalyzes its own formation with greater selectivity than that for its mirror image. It should be emphasized that these three concepts are closely related but distinct. Enantioselective autocatalysis does not necessarily lead to chiral amplification * To whom correspondence should be addressed. Fax: + 36 52 489667. Tel: + 36 52 512900/22373. E-mail: [email protected].

(e.g., first-order autocatalysis alone does not amplify initial differences12,13) and chiral amplification might even have thermodynamic background, which has nothing to do with kinetic phenomena such as catalysis.14 The concepts of enantioselective autocatalysis and chiral amplification were first proposed in 1953 by F. C. Frank.2 Several extensions of the Frank model have been analyzed since then.15-17 However, all these attempts2,15-17 used the deterministic approach, which is unfortunate because any enantiomeric excess can only arise as a consequence of imposed initial fluctuations. Although this is a reasonable qualitative interpretation, it still relies on uncharacterized, a posteriori external effects to reach symmetry breaking. Strictly speaking, the deterministic approach can only hope to interpret chiral amplification but never absolute asymmetric synthesis. If a deterministic model is used to interpret the origin of homochirality, it is tantamount to stating that there has always been some imbalance of enantiomers in nature. This is not satisfactory, especially given the fact that stochastic models, the basic mathematics of which is well developed,18,19 provide a much better explanation. Theory shows that autocatalytic reactions can only be described by stochastic methods in certain cases,20 this fact was experimentally confirmed in autocatalytic processes not involving any chiral material.21,22 There are notable stochastic aspects in enzyme catalysis23 and the importance of stochastic single molecule events in biological systems was demonstrated in a recent paper.24 In the past few years, the stochastic kinetic approach was used in systems involving enantioselective autocatalysis and theoretical work showed that certain combinations of kinetic parameters lead inevitably to absolute asymmetric synthesis.25-32 These initial attempts obviously treated the simplest schemes possible. With the results of these earlier studies in mind, the present paper will attempt to give a stochastic description of the Frank model. As Frank’s original proposal postulated an open system,2 this paper will also introduce a stochastic description for flow reactors. Only results are given in the main text of this paper; lengthy mathematical proofs and calculation algorithms are deposited in the Supporting Information.

10.1021/jp900276h CCC: $40.75  2009 American Chemical Society Published on Web 04/24/2009

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SCHEME 1:

A f BR

ν1 ) ku[A]

(R1)

A f BS

ν2 ) ku[A]

(R2)

A + BR f 2BR

ν3 ) kc[A][BR]

(R3)

A + BS f 2BS

ν1 ) kc[A][BS]

(R4)

BR + BS f 2C

ν1 ) kd[BR][BS]

(R5)

Results and Discussion The Frank Model. Scheme 1 shows a slightly modified version of the Frank model,2 which was studied in the present work. Steps R1,R2 and R3,R4 are symmetric and therefore have the same rate constants. Steps R1 and R2 involve the formation of chiral B from nonchiral precursor A. Steps R3 and R4 are enantioselective autocatalytic pathways of the same overall process. Step R5 is the reaction of two different enantiomers of B producing nonchiral product C, originally termed mutual antagonism by Frank.2 It should be noted that only steps R3-R5 were present in Frank’s original proposal.2 However, a model based on these three reactions can only interpret chemical change if the chiral products BR and BS are already present in the beginning, which is in conflict with the objective of modeling absolute asymmetric synthesis. To avoid this conceptual problem, reactions R1 and R2 (formation of chiral molecules without autocatalysis) were included in Scheme 1, very similarly to a recently published deterministic study on the effect of noise in the Frank model.17 One additional minor modification of the original model is that the mutual antagonism step R5 shows two molecules of C as product rather than one. This modification does not change the essence of the scheme and was introduced for an entirely technical reason. In this way, the overall number of molecules does not change in any of the reaction steps, which makes the enumeration of possible states (see later) in the stochastic approach easier and reduces the computation time by orders of magnitude. Stochastic Approach in Flow Reactors. The usual deterministic approach to chemical kinetics is based on using continuous concentration-time functions. Despite being in obvious contrast with the particle-based, noncontinuous view of matter, this approach is satisfactory for the overwhelming majority of problems because the number of particles is usually quite large. Under certain conditions, especially when reactions at low particle numbers are important, this deterministic approach fails.20 Several different mathematical methods for such cases have been developed under the name of stochastic kinetics.19 The continuous time discrete state (CDS) stochastic approach is probably the closest to the accepted particle-based view of matter.19 The mathematics of this approach has been developed in detail and only a very brief summary is given here.19 The CDS approach identifies a state of the system at a given time by listing the exact number of each particle present. A differential equation for the probability of every state is written based on the kinetic scheme. These differential equations are linear and can in principle be solved analytically, but this is

often extremely difficult to handle because the number of possible states and differential equations is very large. It should be noted that the CDS approach is superior to the usual deterministic approach in a sense that it incorporates the particulate nature of matter without making any assumptions not present in the deterministic approach.19 In theory, every conclusion drawn with the deterministic approach can be reached as a limiting case of the CDS approach for very high particle numbers.19,33 However, only the stochastic approach provides meaningful interpretations for low molecule numbers.19 Although some attempts have been reported for using stochastic kinetics in open systems,34 we have found no literature precedent for using the CDS approach in a flow-through system. However, the extension of the stochastic principles for CSTR reactors is quite straightforward. The effect of flow is mathematically identical to a series of first-order chemical reactions which replace any molecule in the reactor by the molecule(s) present in the feed. The only problem that needs further attention is ensuring that the volume of the reactor does not change. In deterministic cases, the rates of inflow and outflow (measured in volume/time) are set equal. In the CDS approach, the stoichiometry of the “flow reaction” can serve the same purpose. Generally, a linear combination of the numbers of molecules present, defined by the stoichiometries of the chemical reactions in the reactor, should be constant. In Scheme 1, however, none of the chemical reactions change the overall number of molecules present and only molecule A is present in the feed. Therefore, the effect of flow can be thought of as first-order reactions B f A, C f A, and D f A, all having the same rate constant κf (the possible “flow reaction” A f A can be excluded as it leads to no change). Stochastic Description of the Frank Model. In our CDS description, P(a,r,s,t) denotes the possibility that the reactor contains exactly a molecules of A, r molecules of BR and s molecules of BS at time instant t. Initially, only n molecules of A are present and consequently P(n,0,0,0) ) 1. It is not necessary to specify the number of C molecules in identifying a state as conservation of mass ensures c ) n - a - r - s. In this system, the overall number of possible states is given as

M)

( )

n+3 (n + 3)(n + 2)(n + 1) ) 3 6

(1)

The full differential master equation19 of the stochastic model can be written as dP(a, r, s, t) ) -{2aκu + arκc + asκc + rsκd + (n - a)κf} × dt P(a, r, s, t) + {(a + 1)κu + (a + 1)(r - 1)κc} ×

P(a + 1, r - 1, s, t) + {(a + 1)κu + (a + 1)(s - 1)κc} × P(a + 1, r, s - 1, t) + {(r + 1)(s + 1)κd}P(a, r + 1, s + 1, t) + (r + 1)κfP(a - 1, r + 1, s, t) + (s + 1)κfP(a - 1, r, s + 1, t) + (n - a - r - s + 1)κf P(a - 1, r, s, t) (2)

It should be noted that eq 2 also describes a closed system without flow if κf ) 0 is set. The relationship between stochastic and deterministic rate constants is straightforward:

κu ) ku

κc ) kc ⁄ (NAV)

κd ) kd ⁄ (NAV) κf ) rflow ⁄ V (3)

where NA is the Avogadro constant, V is the volume of the reactor, and rflow is the flow rate. It can be shown that the solution of eq 2 is symmetric for r and s:

Stochastic Kinetic Analysis of the Frank Model

P(a, r, s, t) ) P(a, s, r, t)

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(4)

This symmetry is not as obvious as it might seem at first sight (see notes in the Supporting Information) and is not guaranteed if the initial conditions are different. A symmetry-reduced version of eq 2 can be used to accelerate numerical calculations. Most of the considerations in this paper will focus on stationary (time-independent) conditions in flow systems and the final states for closed systems. Selected descriptors of the reactor, rather than the probability of all possible states, will be used in discussing the results. The most important descriptor is based on the enantiomeric excess, which is defined for a particular state as

ee(a, r, s) )

|r - s| r+s

(5)

The expectation for enantiomeric excess in the reactor at any given time is

ee(t) )

∑

ee(a, r, s)P(a, r, s, t)

(6)

all M states

This paper will show that the exclusive use of this traditional enantiomeric excess can be grossly misleading, a point also made qualitatively for closed systems in a recent paper.35 Another descriptor, the yield-adjusted enantiomeric excess (E) will also be discussed, which is formally the product of ee and the overall yield of the chiral product:

E(a, r, s) ) E(t) )

∑

|r - s| n

E(a, r, s)P(a, r, s, t)

(7) (8)

all M states

Finally, the stationary (or, in closed systems, final) values of these descriptors are defined as

ee ) lim ee(t)

(9)

E ) lim E(t)

(10)

tf∞

tf∞

Notably, the deterministic approach would predict ee ) E ) 0 under all circumstances for Scheme 1 with identical initial conditions. Solution for a Closed System. A solution for the closed system model without mutual antagonism (κd ) 0, κf ) 0) was published earlier.26,27,30 The closed system proceeds to a limited number of final states, from which no further change can arise. No state can occur more than once in the course of the process. The distribution of BR and BS molecules in the final states is the most important feature this system and can be calculated without determining every P(a,r,s,t) function. Similarly to previous lines of thought,26,27 let Q(a,r,s) denote the probability that the system goes through state (a,r,s) at any time during the process. It can be shown that

Q(a, r, s) ) lim P(a, r, s, t) + tf∞

n

E)

∑

1 2iQ(0, i, 0) n i)1

(12)

From eqs 2 and 11, a recursive formula can be derived for Q(a,r,s) values:

Q(a, r, s) ) {(a + 1)κu + (a + 1)(r - 1)κc}Q(a + 1, r - 1, s, t) + 2(a + 1)κu + (a + 1)(r + s - 1)κc + (r - 1)sκd {(a + 1)κu + (a + 1)(s - 1)κc}Q(a + 1, r, s - 1) + 2(a + 1)κu + (a + 1)(r + s - 1)κc + r(s - 1)κd (r + 1)(s + 1)κdQ(a, r + 1, s + 1) (13) 2aκu + a(r + s + 2)κc + (r + 1)(s + 1)κd With an appropriate calculation sequence, all Q(a,r,s) values can be computed in reasonable time for n values up to 1000. It is seen that eq 2 is homogeneous in the stochastic rate constants. Therefore, the parameters can be scaled with one of the κ values (suitably κu in this case) without any loss of information. Figure 1 shows the probability distributions of final states calculated for n ) 200 at several different κd/κu values with κc/κu ) 0.01 fixed. Predictably, the distribution at low κd/κu values (10-2 and lower) resembles very much the one calculated for the model without mutual antagonism.26 Under these conditions, practically all A is consumed before any C is formed and the only role of mutual antagonism is to consume BR or BS fully (they are present together for an extended time) without changing the r - s difference. At higher κd/κu, the maximum at r ) 0 splits into a symmetric pair, and this bimodal distribution is unchanged when κd/κu is increased above 101. At these parameter values, BR or BS can only coexist for very short time intervals compared to the time scale of their formation. More examples of the probability distribution at different parameter values are given in the Supporting Information. Figure 2 shows E as a function of both κd/κu and κc/κu in a three-dimensional graph. It is clear that E is dominantly determined by κc/κu and κd/κu plays only a marginal role: in a rather limited range of κc/κu, an increase in κd/κu results in a small increase in E. Figure 2 also emphasizes that the exclusive use of ee to assess an enantioselective autocatalytic model can be misleading. The value of ee is practically 1 for all points because final states can only contain BR or BS molecules, but not the two together. Still, E values for a huge region of the parameter space are quite low. In these cases, hardly any chiral material forms in the reaction. Figure 3 shows how E values change with increasing molecule numbers. There are two different ways of increasing

∫0∞ (2aκu + arκc + asκc + rsκd)P(a, r, s, t) dt (11)

Obviously, Q(n,0,0) ) 1 holds because (n,0,0) the certain initial state. There can be no A molecules in final states, and BR and BS molecules cannot coexist either. Therefore, ee ) 1 - Q(0,0,0) = 1 is valid in a closed system.36 The value of E can be calculated as follows:

Figure 1. Probability distribution in a closed system. κc/κu ) 0.01.

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S(a, r, s) ) lim P(a, r, s, t) tf∞

(14)

The stationary probabilities, which are conceptually very different from the stationary concentrations used in the deterministic approach to flow-through reactors, can be calculated for all states by writing 0 on the left side of eq 2:

0 ) -{2aκu + arκc + asκc + rsκd + (n - a)κf}S(a, r, s) + {(a + 1)κu + (a + 1)(r - 1)κc}S(a + 1, r - 1, s) + {(a + 1)κu + (a + 1)(s - 1)κc}S(a + 1, r, s - 1) + {(r + 1)(s + 1)κd}S(a, r + 1, s + 1) + (r + 1)κf × S(a - 1, r + 1, s) + (s + 1)κf S(a - 1, r, s + 1) + (n - a - r - s + 1)κf S(a - 1, r, s) (15) Figure 2. Expectation for the yield-adjusted enantiomeric excess in a closed system. n ) 200.

In this way, the original set of M simultaneous differential equations is transformed into M homogeneous linear algebraic equations, any one of which can be obtained by summing the remaining M - 1 equations. One more independent equation can be written based on the fact that all the states have been listed: the sum of all probabilities is one.

∑

S(a, r, s) ) 1

(16)

all M states

Figure 3. Expectation for the yield adjusted enantiomeric excess as a function of the number of molecules in a closed system. κc/κu ) 1 (a), 100/n (b); κd/κu ) 1 (a), 100/n (b).

molecule numbers in stochastic kinetic models. The first is to keep the concentrations constant and increase the volume (curve a in Figure 3). In this case, the second-order stochastic rate constants also change with the increase in molecule numbers (see eq 3). The yield-adjusted enantiomeric excess E decreases as n is increased at constant concentration. This is not surprising as Kurtz’s theorem33 states that the deterministic description of a chemical reaction system, which would predict the formation of 0 enantiomeric excess in Scheme 1, is the infinite volume limit of the CDS approach. The curve also confirms the general inadequacy of the deterministic approach to study these phenomena. The other way of increasing molecule numbers in stochastic models is to keep the volume constant and increase the concentrations (curve b in Figure 3). The slight increase in E with growing n is rationalized by noting that second-order reactions, most importantly the enantioselective autocatalysis in steps R3 and R4, become more dominating over first-order processes when the initial concentrations increase. Solution for a Flow-Through System. There is a substantial difference between the CDS descriptions of Scheme 1 in closed and flow-through systems. In the latter, there are no final states at all and each state can arise from or give rise to another state(s). Stationary conditions, in which none of the probabilities change any more, play a similar role in flow-through reactors as final states do in closed systems. The S(a,r,s) stationary probabilities are defined as

Simultaneous eqs 15 and 16 have a unique solution; therefore, the stationary probabilities are independent of the initial conditions. Although these equations are linear, finding their numerical solution can be rather challenging computationally. First of all, an efficient method for the enumeration of all possible states is needed to list the variables and equations in a systematic way. The set of equations obtained thus needs to be solved with a numerical method. On a normal personal computer, an elimination method can be used for this purpose up to about n ) 20 (M ) 1771). For larger systems, a special iteration method was developed, which could be used up to n ) 100 (M ) 176 851). It is unlikely that using these solution methods on more powerful computers would improve the stated limits significantly as the fundamental problem is not computation time but memory requirements, which are proportional to n6 for the elimination method and n3 for the iteration process. In the present work, the precision of the solutions found by the elimination method was only limited by the precision of computational number representation (roughly (10-15%). The precision of the iteration method, however, was limited by computation time and was not better than (1%. Figure 4 shows a multidimensional probability distribution calculated for n ) 20. The front right triangle has 0 values in this graph because r + s > 20 is impossible. In Figure 4, r ) s is a plane of symmetry in agreement with eq 4. This distribution has a peak around racemic states meaning that the expected value of enantiomeric excess is low for the particular combination of parameters. Different values of parameters yield completely different distributions (Supporting Information). Figure 5 shows E and ee as a function of both κu/κf and κc/κf at fixed κd/κf and n ) 100. Similar surfaces are given in Figure 6, but κd/κf and κc/κf are the independent variables and κu/κf is fixed. The front left portions of both graphs provide additional examples when considering ee only would result in questionable conclusions. As expected, relatively high values of κc (efficient enantioselective autocatalysis) are the key to getting high E or ee values. The role of mutual antagonism is more controversial. For low values of κd, unlike in a closed system, reaction R5 has no influence on the course of the overall process at all, and could even be excluded from the model. The most surprising

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Figure 4. Probability distribution in an open system. κu/κf ) 10; κc/κf ) 0.1; κd/κf ) 0.1; n ) 20.

Figure 6. Expectation for the yield-adjusted and conventional enantiomeric excess in an open system. κu/κf ) 1; n ) 100.

Figure 5. Expectation for the yield-adjusted and conventional enantiomeric excess in an open system. κd/κf ) 1; n ) 100.

feature of Figure 6 is the “valley” at about κd/κf ) 10-4 and κc/κf > 1. In this region, it is instructive to consider the effect of increasing κd/κf at a fixed value of κc/κf (e.g., 106). At low values of κd/κf, reaction R5 is entirely negligible. At intermediate values, however, it can consume some of the BR and BS molecules, and the overall effect is negative for both E and ee because the autocatalyst is removed from the system. However, κd/κf is not large enough here to make the coexistence of BR and BS molecules impossible. At even larger values, BR and BS cannot be present together, which again favors high ee values and somewhat lower, but still high E values because the reactor contains some C molecules as well. Figure 7 shows E and ee as a function of n at constant concentration and volume. In the constant volume curves, both

Figure 7. Expectation for the yield-adjusted and conventional enantiomeric excess as a function of the number of molecules in an open system. Curves: E for constant volume (a), ee for constant volume (b), E for constant concentration (c), ee for constant concentration (d), κu/κf ) 1; κc/κf ) 1 (a,b), 10/n (c,d); κd/κf ) 1 (a,b), 10/n (c,d).

E and ee approach 1 as the second-order reaction of enantioselective autocatalysis becomes more important with increasing initial concentrations. The constant concentration curves, however, seem to level out with little change above n ) 30. A comparison with Figure 3 highlights that the CDS model of an open system is not bound by Kurtz’s theorem.33 The displayed results are important for another reason: the independence of n implies that the conclusions drawn from small molecule numbers might be transferred to much larger systems. This observation can be further strengthened by a mathematical line of thought

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Lente and Ditro´i Acknowledgment. G.L. thanks the Hungarian Academy of Sciences for a Bolyai Ja´nos research fellowship. Mr. La´szlo´ Ze´ka´ny is gratefully acknowledged for his help. The Hungarian Tempus Foundation helped the participation of T.D. in this project under grant No UT60093/06-07. Supporting Information Available: Extra figures, proofs, and derivations referred to in the article. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

Figure 8. Expectation for the yield-adjusted and conventional enantiomeric excess as a function of the flow rate parameter in an open system. κu ) 1; κc ) 1; κd ) 1; n ) 100.

as well, which is presented in the Supporting Information. The minor “jump” that occurs on some of the curves between n ) 20 and n ) 21 in Figure 7 is most probably caused by the fact the elimination method was used to solve the linear equation until n ) 20 and the iteration method was used from n ) 21. As discussed earlier, the precisions of these two methods are different. Figure 8 shows the influence of changing flow rate on E or ee with all the rest of the parameters kept constant. It is clear that both descriptors are decreasing functions of κf. This effect can be easily understood by noting that chiral molecules (BR and BS) are removed and only nonchiral molecules (A) are fed in as a consequence of flow. Therefore, increasing the flow rate can only be detrimental to the formation of asymmetric states. In fact, Figure 8 clearly implies that the highest enantiomeric excesses in the Frank model can be obtained in a closed system. This observation may have even more importance in a broader concept. A flow system is mathematically analogous to recycling in enantioselective autocatalytic systems. The idea of recycling has been strongly criticized based on both the principle of microscopic reversibility35,37 and thermodynamic considerations.38 The stochastic approach presented here shows that any recycling may actually lower the expectation for enantiomeric excess. Conclusion In summary, the Frank model was studied by the continuous time discrete state (CDS) stochastic approach and its extension to flow-through reactors, which, unlike the deterministic approach, actually predict absolute asymmetric synthesis in a broad parameter range and can be used to calculate statistical distributions of enantiomers. It was shown quantitatively that narrow focus on the conventionally defined enantiomeric excess could lead to grossly misleading conclusions. This warning might apply to any model in which the chiral material is not necessarily the final product. It was also concluded that closed systems are more favorable for the formation of high enantiomeric excesses under reasonable conditions. The importance of mutual antagonism can be questioned in the original Frank model, its significance may be larger as an inspiration for including catalytically active or inactive dimers in later models.8-13,16

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