Thermodynamic mechanism for solution phase chiral SPECIAL FEATURE amplification via a lattice model a b a,1 Thomas G. Lombardo , Frank H. Stillinger , and Pablo G. Debenedetti Departments of aChemical Engineering and bChemistry, Princeton University, Princeton, NJ 08544b; Edited by Benjamin Widom, Cornell University, Ithaca, NY, and approved February 24, 2009 (received for review December 18, 2008) We formulate a two-dimensional lattice model to study the equilib- reaction product by the catalyzing role of amino acid enantiomor- rium phase behavior of a ternary mixture composed of two enan- phic excess present in the solution. A basic conclusion of that study tiomeric forms of a chiral molecule and a nonchiral liquid solvent. was that the presence of a symmetrical pair of eutectic points† in Numerical solution of the model invoking the mean-field approxi- the ternary phase diagram for the enantiomorphic pair of amino mation of statistical mechanics allows the calculation of a ternary acids and their non-chiral solvent played a key role in the amplifica- phase diagram. A prominent feature of the phase diagram is the tion process. In accord with that conclusion, this article introduces appearance of a mirror-image pair of triple points involving coexis- and analyzes a simple lattice model that, as shown below, is capable tence of a liquid phase enriched in one of the enantiomers with two of reproducing, at least qualitatively, such ternary phase diagrams. solid phases: a racemic and an enantiopure crystal. Thus, over broad Although schematic phase diagrams consistent with the type of ranges of initial composition, including liquid mixtures containing solid–liquid equilibrium discussed in this work can be found in almost equal amounts of the two enantiomers, the equilibrium the stereochemistry literature (9, 10), our focus is on the micro- state of the system produces liquid-phase chiral amplification. The scopic origin of such behavior, which has not received comparable calculations predict that chiral amplification is favored at low tem- attention. peratures, and by strengthening those molecular interactions that The following section defines the lattice model chosen for study. stabilize the racemic crystal. The phase behavior that we obtain Although it is conceptually simple, its large-system-limit thermo- is qualitatively identical to that reported in a recent experimental dynamic behavior is currently not amenable to a mathematically study of solutions of amino acids, aimed at probing the liquid-phase exact solution. Consequently, for its analysis we have invoked a CHEMISTRY control of asymmetric catalysis. Those results, and the present cal- standard mean-field approximation, described in the third section. culations that provide molecular-level insight into their underlying We believe that this approximation preserves the overall essential causes, suggest a possible thermodynamic scenario for the liquid- behavior of the model’s thermodynamics, which can be extracted phase emergence of chiral imbalance in a prebiotic and presumably from straightforward numerical calculations. The results of those nearly racemic world. calculations are then presented. Finally, a summary of our conclu- sions and remarks about extensions of our model that we believe chirality | asymmetric | behavior are feasible are discussed. The technical details are included in supporting information (SI) Appendix. hiral asymmetry choices exhibited by molecules that are C present in living organisms constitute a vivid and scientifi- cally challenging set of observations. Such geometric preferences Lattice Model favoring one enantiomer over its mirror image are obvious in the Todemonstrate that an enantiomorph-solution ternary phase dia- observed structures of amino acids, sugars, and the biopolymers gram of the type required for liquid-phase chiral amplification can that they form. The DNA double helix constitutes another meso- arise from simple molecular interactions, we have constructed and scopic example of unvarying chirality. These facts automatically analyzed an elementary two-dimensional lattice model. In partic- generate fundamental questions about how those chiral asymme- ular, this involves a simple square lattice on the sites of which tries in the terrestrial biosphere spontaneously arose, and to what three species can exist: right- and left-handed enantiomorphs (to extent liquid phases were involved in that scenario. It is also impor- be denoted “D” and “L” for simplicity), and a nonchiral sol- tant to know whether the exclusion of the alternate mirror-image vent species “S”. Fig. 1A illustrates the elements of this model, enantiomorphs amounts to a basic prerequisite for the origina- showing in minimalist fashion how the three species are accom- tion, survival, propagation, and evolution of living organisms, or modated on the lattice. The two distinct enantiomorphs, D and whether it is an incidental by-product of the appearance of life. L, appear as bent arrows that conform to the square-lattice Not surprisingly, these issues have generated an historically bond directions, whereas the nonchiral solvent is represented lengthy and diverse literature (e.g., (1, 2)). Several mechanisms by small open circles at the lattice sites. One and only one of have been proposed to explain the spontaneous appearance of chi- these species resides at each lattice site. Although thismodel ral bias from a presumably unbiased prebiotic terrestrial environ- postulates that molecular occupancy patterns can reconfigure ment. One intriguing scenario involves chemical reactions subject freely on the lattice, the individual enantiomorphic molecules are to asymmetric autocatalysis, a phenomenon realized experimen- not capable of transforming to their mirror-image forms. Thus, tally by Soai and collaborators (3). Another invokes chiral crystal- an initially established molecular composition remains invariant lization and Ostwald ripening (4). A third possibility focuses on thereafter. the equilibrium phase behavior of enantiomorphs with a nonchi- ral third component that acts as a relatively poor solvent (5, 6). In addition to experimental studies, a number of computational Author contributions: F.H.S. and P.G.D. designed research; T.G.L. performed research; T.G.L. examinations of chirality have been published (e.g., (7, 8)). In this analyzed data; and T.G.L., F.H.S., and P.G.D. wrote the paper. study we focus on liquid-phase phenomena and adopt a theoretical The authors declare no conflict of interest. approach to calculate the phase behavior. This article is a PNAS Direct Submission. The theoretical investigation reported in this article was stim- 1To whom correspondence should be addressed. E-mail: [email protected]. ulated by the experimental study reported in ref. 5 concern- † In this article the term eutectic point refers to the composition of a solution phase that ing asymmetric amplification in liquid solutions by amino acid is in equilibrium with two distinct solid phases. This terminology is adopted from ref. 9. catalysis. Specifically, that investigation quantitatively probed the This article contains supporting information online at www.pnas.org/cgi/content/full/ liquid-phase control of the enantiomorphic excess of a chiral aldol 0812867106/DCSupplemental. www.pnas.org / cgi / doi / 10.1073 / pnas.0812867106 Early Edition 1of5 Downloaded by guest on October 1, 2021 The postulated form, Eq. 1, vanishes for pure solvent, as well as for any solution configurations in which all D and L molecules present are fully solvated (i.e., surrounded by 4 nearest-neighbor solvents). Choosing the three coupling parameters v0, v1, and v2 to be negative and sufficiently large in magnitude will ensure that at any temperature the solvent will only be able thermodynamically to support dilute solutions of the enantiomorphs. If the lattice hosts only solvent molecules and just one of the enantiomorphs, the ground state (i.e., the global minimum) will exhibit coexist- ing phases, respectively, that are pure solvent and a perfect crystal of identically oriented D or L molecules. When equal numbers of D and L molecules are present, with or without solvent, we require that the global minimum exhibit a stable crystal whose structure involves DL alternation around every nearest-neighbor square it occupies, as illustrated in Fig. 1C. Because of the 4-site nature of the DL interaction, this requirement translates into the following inequality: v2 < 2v1 < 0. [2] It should be mentioned in passing that invoking a 4-site interac- tion to stabilize a 1:1 racemic crystal, instead of a 2-site interaction Fig. 1. Schematic of two-dimensional lattice model. (A) The 9 possible occu- favoring contact of chiral opposites, is necessary to suppress high pancy states for each lattice site. Each of the enantiomers can take on 4 concentrations of chiral-impurity point defects in the otherwise different orientations, which are designated by the direction of the back- pure enantiomorph crystals. bone: “southeast” (SE), “southwest” (SW), “northwest” (NW), and “north- The model’s thermodynamic properties, including ternary east” (NE). Rotations are only permitted in the two-dimensional plane, thus phase diagrams, can be extracted from the Helmholtz free energy preventing a switch of chirality. B The interaction between molecules of the ( ) F. This in turn follows from the classical canonical partition func- same chirality and orientation (v1). (C) The interaction between molecules of tion Q(β)(β = 1/kBT, where