ISSN 10637745, Crystallography Reports, 2013, Vol. 58, No. 5, pp. 697–709. © Pleiades Publishing, Inc., 2013. Original Russian Text © K.K. Konstantinov, A.F. Konstantinova, 2013, published in Kristallografiya, 2013, Vol. 58, No. 5, pp. 682–695.
STRUCTURE OF MACROMOLECULAR COMPOUNDS
New Concept of the Origin of Life on Earth K. K. Konstantinov and A. F. Konstantinova Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 119333 Russia email:
[email protected] Received April 4, 2013
Abstract—A model of coupled autocatalytic reactions with allowance for the crystallization of diastereomer is considered. It is shown that the differences in the physical properties of diastereomers can be a 100% enan tioselective factor, which makes it possible to obtain a significant chiral polarization even at low autocatalysis enantioselectivity. In terms of a complicated model, more complex molecules should have a higher chiral polarization than simpler ones. The calculation of the dynamics of the model under consideration shows that the presence of binomial coefficients in the reaction of pair formation from two different enantiomers (pro vided that diastereomers have identical properties) leads to the occurrence of an additional 100% enantiose lective factor. Estimation shows that the theoretical difference between the right and lefthanded molecules (which is due to weak interaction), described in the literature, is sufficient to explain the directed symmetry breaking and construction of biological molecules from Lamino acids and Dsugars at the origin of life on earth. DOI: 10.1134/S1063774513050040
INTRODUCTION Many crystals are optically active (or gyrotropic): they rotate the plane of polarization when linearly polarized light propagates through them [1]. It is known that only 11 out of 18 symmetry classes of crys tals possess enantiomorphism (mirror symmetry), i.e., have left and righthanded modifications [2]. The properties of these crystals are identical, except for the rotation of the plane of polarization and somewhat different refractive indices [1]. When dissolved, these crystals become inactive [3]. The situation with biological materials is quite dif ferent: practically all are optically active in the liquid state, in solutions, or in crystals. Almost 150 years ago, Louis Pasteur [4] showed that optical activity is due to the properties of molecules and molecular structures with respect to mirror reflection; enantiomorphic molecules are formed in only living organisms; and asymmetry of a particular kind is a fundamental sign of living matter, including all plants, animals, and humans. All proteins are known to be constructed from only lefthanded amino acids in the form of L isomers (left handed) and righthanded sugars in the form of D iso mers (righthanded). This property is referred to as chiral purity or homochirality [5]. There is always only one optical isomer in living systems; i.e., only one optically active compound [5]. In this context, there are many hypotheses explaining the origin of optical activity in living nature [5–13]. Let us dwell on the studies considering different theories of the origin of life on earth. First and fore most, we should mention the work by Frank [13], who
developed a model for an autocatalytic mechanism that may lead to mirror symmetry breaking in the evo lution of homochirality. Chemical reactions may result in the formation of unequal numbers of L and D amino acids due to the phenomenon referred to as spontaneous symmetry breaking and may occur only under certain physical conditions: it is impossible in closed systems, where energy and matter supply is absent. Miller [10] synthesized amino acids from gases in his studies by providing conditions that presumably existed in the atmosphere of primeval earth. These experiments yield equal numbers of left and right handed isomers. Many theoretical models were pro posed by Soai and students, who considered autoca talysis in detail [14, 15]. In this field the most interesting theoretical studies are [5, 6, 16–18]. It was shown in them that, in terms of coupled autocatalytic reactions, chiral polarization close to 100% can be achieved only in cases where the enantioselectivity of direct (γ+) or reverse (γ–) autoca talysis is close to unity and the total enantioselectivity of autocatalysis is γ = γ+ + γ– > 1. The most important studies are those devoted to the physical properties of enantiomers. Examples include works by Blackmond [11] and Breslow [19], who showed that different physical properties of diastereomers may lead to high chiral polarization in the absence of autocatalysis. It is known in particular that mixing two solutions with approximately equal amounts of right and left handed serine yields a solution with approximately 99% chiral polarization [11]. The main stages of the origin of life on earth can be described as follows [6, 10]: abiogenic formation and
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accumulation of organic molecules in the form of racemic mixtures; mirror symmetry breaking in the racemic “soup” and the formation of only one type of asymmetric molecule (Lamino acids and Dsugars); and the formation of short molecular chains (building blocks of future DNA, RNA, and protein molecules), i.e., the occurrence of specific functions. The main difficulty in explaining the origin of life on earth can be formulated as follows: enantioselectiv ity of autocatalysis at the level of simple chemical reactions (especially with allowance for racemization) is insufficient to yield chiral polarization close to 100%, which is necessary to form specific functions [5, 6]. In other words, there is a discontinuity in chiral polarization, which cannot be explained in terms of modern theories. In this study we discuss a model that makes it possible to avoid this discontinuity. The sec ond important question is the dynamics of the origin of life on the earth: life may arise in only a thermody namically nonequilibrium system; therefore, it is nec essary to consider the dynamics of the system. The third question is the character of symmetry breaking (spontaneous or directed) in the origin of life on earth. GENERAL MODEL OF CHEMICAL REACTIONS Let us consider the general isotropic and uniform model of chemical reactions. Let there be N different chemical materials (S1, …, SN) and M possible reac tions occurring between them (R1, …, RM). The state of the system is determined by the number of molecules of each material: X(t) = (X1(t), …, XN(t) or, equiva lently, their concentrations: ρ(t) = (ρ1(t), …, ρN(t), where ρα(t) = 1X α ( t ) (Ω is the volume of the system); Ω Greek and Latin (beginning with i) subscripts indi cate, respectively, materials and reactions. Then the evolution of the system (for the uniform isotropic case, except for the dependence on spatial coordinates and other variables, e.g., temperature) can be written as [20–24] dρ α ( t ) = dt
M
∑ν
αi a i ( ρ ( t ) ),
(1)
i=1
where ναi is a stoichiometric matrix, which indicates the number of molecules of a particular material dis appearing or arising as a result of reaction, and ai is the ith reaction rate, which may depend in a complex way on ρ(t). If the number of molecules involved in reac tions is not very large (for example, when intracellular processes are simulated), one must consider the basic chemical equation [20], which allows one to take into account the probabilities of transitions and estimate the standard deviation from the mean [25]. This equa tion is not considered in our study, but the applicability limits for Eq. (1) are made. This equation also makes
it possible to describe nonelementary kinetics, when ai are not just products of some coefficients by concen trations in certain powers. Note that, since molecules consist of atoms and the number of atoms of each type does not change in a closed system, all ναi ≠ 0 and Eq. (1) has a certain number of integrals of motion, which can be written as N
∑b
kα ρ α ( t )
= const,
(2)
α=1
where bkα (k = 1, …, NA) is the number of atoms of type k in a molecule of type α and NA is the minimum num ber of “atoms”1 necessary to describe the system. In the simple case considered here, one “atom” is suffi cient; i.e., one integral of motion arises. Chirality of Materials Let us assume that some N different materials are chiral and, as a consequence, exist in the form of two enantiomers L and D in the model under consider ation2, where vector L denotes all lefthanded enanti omers and vector D stands for all corresponding right handed ones. Enantiomers may insignificantly differ in the number of the socalled advantage factors, which manifest themselves in the dependence of αi (for the corresponding reactions) on these advantage factors. Since the differences between enantiomers (only two for each pair) are described by the same symmetry group (reflection group in space R1) as the simulta neous change in sign for all advantage factors, the ini tial system of equations has symmetry (L, D, g)
(D, L, (–g)),
(3)
where g is the vector of advantage factors. For further analysis it is convenient to replace variables and make the following replacements for each pair of enanti ρL + ρD ρL – ρD ⎞ ⎛ Y = omers:(ρL, ρD) , Y– = . For ⎝ + 2 2 ⎠ nonchiral materials, variables are not replaced. Chiral Y polarizations η = – are also often used in practice Y+ instead of Y–. Note that, to analyze stability, it is expe dient to extend the system of equations and convert advantage factors into additional (timeindependent) variables. This description makes it possible to con sider advantage factors as some initial conditions, i.e., in the same way as the initial values of all concentra tions. 1
The term “atom” is in quotes to show the importance of specif ically the integrals of motion rather than specific atomic compo sition of particular molecules. 2 Diastereomers are considered individual materials, because they often have different physical and/or chemical properties.
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State Vector of the System The initial system of equations can be written in the general form as dY ( t) = F ( Y ), dt
(4)
where Y = (Y1(t), …, YK(t)) is the state vector of the sys tem in new variables (the total number of new variables will be denoted as K ≥ N) and F(Y) is a complex vector function of Y, which is independent of t within our model. Stationarity (i.e., independence of Y on time) implies that F ( Y ) = 0.
(5)
Symmetric solution means that all Y– = 0 and all g = 0. In this case, Fa(Y) ≡ 0 for all these variables (due to symmetry). For the remaining variables Ya, we obtain a closed system of equations, which can be solved; a symmetric stationary solution always exists (because of the symmetry of the system). Let us consider the problem of stability of symmetric solution. This is a wellknown problem, which is solved by expanding F(Y) in a Taylor series in the vicinity of the stationary solution [26] and analyzing the eigenvalues of the thus formed matrix Wab at the point Y = YS, where YS is the stationary point: ∂F W ab = a . ∂Y b
(6)
In this stage, one condition is sufficient: if there are eigenvalues with a real part larger than zero, there are solutions that are exponentially unstable. In particular cases there may be situations where the real part of one or more eigenvalues is identically zero (and negative for all other eigenvalues). Then, stability is determined by the number of linear independent eigenvectors with respect to degeneracy in eigenvalues. In these (unsta ble) cases, the solutions are polynomially unstable. Splitting of Single Stable and Symmetric Solution Let us consider the stationary point YS, where a sin gle stable and symmetric solution is split into asym metric stable solutions. We will fix the number of all atoms in the system, except for the atoms of one type (there is always at least one for any real system), and find out if it is possible to determine the total concen tration of atoms of this type at which splitting occurs. Mathematically, this problem is trivial: it is reduced to a solution of the bifurcation problem (the problem of determining the value of the parameter at which the real part of at least one of the eigenvalues of matrix Wab passes through zero). However, even an insignificant complication of the system gives rise to polynomial equations with a power larger than four; i.e., there is no analytical solution in radicals. CRYSTALLOGRAPHY REPORTS
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Crystallization/Dissolution Along with simple chemical reactions, the rates of which are proportional to the products of concentra tions in particular powers, the system is affected by various physical processes. The simplest of them (along with diffusion, which is disregarded here) are liquid–vapor and liquid–solid phase transitions. If a gas or an insoluble material is formed as a result of reaction, these products precipitate. Mathematically, this means that liquid (solutions), vapor, and solid phases of particular materials should be distinguished in (1). It is convenient to model these phases by intro ducing additional “materials” corresponding to these phases, and describe the corresponding “reactions” (phase transitions). In this study we consider only the reversible crystallization/dissolution process. Its description in the simplest form calls for three param eters: limiting concentration, crystallization rate (when the concentration of material exceeds the lim iting value), and dissolution rate (when the concentra tion is below limiting). Note that this model signifi cantly simplifies the real process, because, despite the fact that crystallization (precipitation) occurs in the bulk, dissolution occurs on the surface of dissolved material and, as a consequence, depends strongly on many parameters (degree of grinding, diffusion, mix ing, flows, etc.). We neglect all these factors. Thus, the crystallization/dissolution of material from ρi (mate rial in solution) to ρ Ci (precipitate) can be described as C ρ→ ← i
ρ C ( ρ i, i
ρ Ci )
max ⎧ and ρ Ci = 0 ⎪ 0, ρ i ≤ ρ i ⎪ = ⎨ k sol ( ρ i – ρ imax ), ρ i ≤ ρ imax and ⎪ ⎪ k ( ρ – ρ max ), ρ > ρ max i i i ⎩ cryst i
(7) ρ Ci > 0,
Hence, dissolution is described in terms of the simpli fied version of the Noyes–Whitney equation [27]. max Among these coefficients, we are interested in ρ i . Let us consider possible diastereomers that can be formed from two enantiomers; they will be referred to as L and D. The simplest compounds that may exist are LL, DD, and L pairs D. Since the physical and chemi cal properties of diastereomers often differ, the values max of the coefficient ρ i for the crystallization/dissolu tion processes LL → ← CLL and LD → ← CLD may signifi 3
cantly differ (we assume that all coefficients for the LL → ← CLL and DD → ← CDD processes are pairwise equal). 3 Properties
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of enantiomers may differ by the value (dimension less) of the advantage factor g Ⰶ 1, and properties of diastere omers may differ by the dimensionless value gd ~ 1.
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Let us consider an arbitrary reversible chemical reaction A → ← B, where the concentrations of materials A and B are of interest. The following condition is sat isfied in the dynamic equilibrium state: ν A, A → B a A → B ( ρ ) + ν A, B → A a B → A ( ρ ) = 0.
(8)
We consider a small variation in the concentrations A and B: ρ → ρ + δAB, where δAB ≡ ( δ ρA , δ ρB ) is the variation in the concentrations of only materials A and B. Then, having expanded aA → B(ρ) and aB → A(ρ) at point ρ in a Taylor series and leaving only the terms linear in δ ρA and δ ρB , we obtain ∂a B → A ( ρ )⎞ A → B(ρ) ⎛ ∂a – δρ ⎝ ∂ρ A ∂ρ A ⎠ A ∂a A → B ( ρ ) ∂a B → A ( ρ )⎞ + ⎛ – δ ρB = 0, ⎝ ∂ρ B ∂ρ B ⎠
(9)
Here it is taken into account that νA, A → B + νA, B → A = 0 (both direct and reverse reactions are considered). The same expression is obtained for material B. Since simple chemical reactions polynomially depend on concentrations, all reaction rates are differentiable and the coefficients before δ ρA and δ ρB are definite and nonzero; i.e., in the presence of dynamic equilib rium, the variations δ ρA and δ ρB are dependent, which manifests itself in the dynamic equilibrium equation. The situation with crystallization/dissolution is quite different. Mathematically, this is due to the following: if we separate crystallization/dissolution into direct (crystallization) and reverse (dissolution) processes, the rates of both are identically zero in a certain range of concentrations and, as a consequence, their deriva tives are identically zero. As a result, situations arise where the variation in one of the concentrations must be zero, whereas the other can be arbitrary. In contrast to a simple chemical reaction, crystallization/dissolu tion process cannot be continuously distorted by changing simultaneously ρA and ρB from zero to the point with concentrations A and B. This means that, as applied to (1), crystallization/dissolution processes (at a certain relation between the properties of diastere omers) are 100% enantioselective: one process (for example, LD crystallization) starts with an increase in concentrations, whereas the other (LL and DD crys tallization) does not occur at all. Specifically this is the fundamental difference between the crystalliza tion/dissolution process and conventional chemical reactions.
MODEL OF COUPLED AUTOCATALYTIC REACTIONS WITH ALLOWANCE FOR CRYSTALLIZATION OF DIASTEREOMERS Description of the Reactions Considered in the Model As a starting point, we will take a simple and well known model of two coupled autocatalytic reactions [5, 6]. Let us assume that there is a neutral (nonchiral) material, based on which right and lefthanded and more complex materials can be constructed; we will refer to them as A, L, and D, respectively. We assume also that these materials are in a homogeneous solu tion (for example, water); therefore, only one param eter is sufficient for their description, specifically, their concentration, which will be denoted as ρA, ρL, and ρD, respectively. The temporal evolution of concentra tions involves a number of processes. Since chemical reactions in solution are reversible, a reverse process should also be considered for each direct process. First, there is synthesis of L and D from A and the reverse process of decomposition, which can be described as A → ← L and A → ← D. The corresponding synthesis and decomposition rates are determined as aA → L = kA → L ρA ,
aL → A = kL → A ρL
aA → D = kA → D ρA ,
aD → A = kD → A ρD .
(10)
Second, we take into in consideration autocatalysis and the reverse process of decomposition with allow ance for “anomalous” autocatalysis A+L A+L
→ ← L + L, A + D → ← D + L, A + D
→ ← D+D → ← L + D.
(11)
The corresponding reaction rates can be written as [6] aA + L → L + L = kA + L → L + L ρA ρL , 2
aL + L → A + L = aL + L → A + L ρL , aA + D → D + D = kA + D → D + D ρA ρD , 2
aD + D → A + D = aD + D → A + D ρD ,
(12)
aA + L → D + L = kA + L → D + L ρA ρL , aD + L → A + L = aD + L → A + L ρD ρL , aA + D → L + D = kA + D → L + D ρA ρD , aL + D → A + D = aL + D → A + D ρL ρD . Since we are interested in the ability of the system to withstand racemization, we will take this process (in the form L → ← D) into consideration; the correspond ing rates can be written as aL → D = kL → D ρL ,
aD → L = kD → L ρD .
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Situations with all (or almost all) coefficients pair wise equal are generally analyzed. In these cases the subscript S is used to denote either L or D and the sub script N is used to denote the other enantiomer in the coefficients containing both subscripts, for example kA + L → L + L = kA + D → D + D ≡ kA + S → S + S kA + L → D + L = kA + D → L + D ≡ kA + S → N + S .
(14)
Since this model was investigated (with racemiza tion disregarded) in [6], we report here one of the main results obtained. The system described by reactions (10)–(12) (i.e., with racemization disregarded) becomes unstable when the following condition is sat isfied (in the first approximation in Kδ+ and Kδ–): ( 1 – Kδ + )γ + + ( 1 – Kδ – )γ– ≥ 1,
kA + S → S + S – kA + S → N + S γ + = , kA + S → S + S + kA + S → N + S kS + N → A + N – kS + S → A + S , γ – = kS + S → A + S + kS + N → A + N
(16)
are the enantioselectivities of the direct and reverse reac kA → S tions, respectively; δ+ = ; ( k A + S → S + S + k A + S → N + S )ρ A kS → A . and δ– = ( k S + S → A + S + k S + N → A + N )ρ A We will supplement the reaction model with pro cesses that take into account a possible difference between the physical properties of LL (DD) and LD. There are reversible chemical reactions of pair forma tion: L+L → ← LL, D + D → ← DD, L + D → ← LD. (17) The corresponding reaction rates can be written as 2
a L + L → LL = k L + L → LL ρ L , a LL → L + L = a LL → L + L ρ LL , 2
a D + D → DD = k D + D → DD ρ D ,
In addition, there are processes of precipitation and reverse dissolution, which can be written in the form LL → ← C LL ,
(18)
a DD → D + D = a DD → D + D ρ DD , a L + D → LD = k L + D → LD ρ L ρ D ,
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LD → ← C LD ,
(19)
a LL → ← C LL = C LL → ← C LL ( ρ LL, ρ C LL ), a DD → ← C DD = C DD → ← C DD ( ρ DD, ρ C DD ),
(20)
a LD → ← C LD = C LD → ← C LD ( ρ LD, ρ C LD ), where ρ CLL , ρ CDD , and ρ CLD are analogs of concentra tions for precipitate (the total amount of a particular material in the solid phase divided by the total volume of liquid) and the functions C LL → ← C LL , C DD → ← C DD , and C LD → ← C LD determine the dynamics of precipitation and reverse dissolution with the corresponding crystalliza tion/dissolution rates and maximum concentrations. Since we are interested in the ability of the system to withstand racemization, it is necessary to consider the corresponding racemization reactions: LL → ← LD, C LL → ← C LD ,
DD → ← LD, C → ←C , DD
(21)
LD
which take into account the fact that one of the mole cules L or D from a pair may be transformed into the other in both solution and precipitate. Direct reac tions LL → ← DD and CLL → ← CDD may also occur. How ever, they are less likely than reaction (21), because they require the simultaneous transformation of both molecules into their enantiomers. These reactions are disregarded in the model discussed here. Note that the racemization rate of L and D in compounds may differ from that of L and D in solution. This possibility is also neglected, and the racemization rate of molecules L and D is assumed to be identical, regardless of the material they enter. This limitation gives rise to a factor of 2 because of the binomial coefficients for racemiza tion of a number of pairs. It is also important (for model consistency) to directly include the decomposition of LL; LD; DD; and, especially, CLL, CDD, and CLD in A:
a LD → L + D = a LD → L + D ρ LD . In a more general case we should consider reac tions of the L1 + L2 → ← L3 type and similarly for other pairs. Therefore, the use of this simplified model sug gests that similar behavior of the system should be expected in a more complex case.
DD → ← C DD ,
where CLL, CDD, and CLD stand for materials LL, DD, and LD, respectively, in the solid phase. Processes of precipitation and reverse dissolution are physical and have radically different dynamics. Their rates can for mally be written as
(15)
kS + S → A + S + kS + N → A + N where K = is the reversibility kA + S → S + S + kA + S → N + S coefficient for the autocatalytic stage; the quantities γ+ and γ–, defined as
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LL → A, DD → A, LD → A C LL → A, C DD → A, C LD → A.
(22)
Formally, these chemical reactions should be reversible; nevertheless, we assume that the rate of reverse reactions is much lower than the rate of reac
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tions (17), and, thus, the channels of additional “dou ble” synthesis can be neglected. It is important to take into consideration the reac tion of formation of material A from some other mate rial that is not spent (it will be referred to as Z): Z → A.
(23)
This reaction, like the irreversible decomposition reaction (22), does not affect the analysis of the stabil ity of the system but shifts the total equivalent concen tration of material A (the sum of the concentrations of all materials, multiplied by the number of “molecules” of material A that is necessary to form a particular material). However, this reaction is important for understanding the dynamics of directed symmetry breaking and it should be taken into consideration for the following reasons. First, the chemical reactions of formation of chiral organic molecules call for the pres ence of simpler (nonchiral) organic molecules (mole cule A in our model). However, the conditions at the beginning of the formation of the earth were too unfa vorable for organic compounds to arise. As a conse quence, it is unlikely that simple organic molecules were initially in excess. They were synthesized during the earth’s formation from very simple molecules and atoms. Accordingly, more correct dynamics is observed when material A, being initially absent, is gradually accumulated for very simple reactions, which are modeled by the Z → A process. Second, when the concentration of material A reaches a critical value (either due to its formation from material Z or due to the change in parameters, for example, temper ature), symmetry breaking begins in the system. Under these conditions, it is the total amount of mate rial A that remains constant, tot
ρA = ( ρA + ρL + ρD + 2 ( ρ LL + ρ DD + ρ LD + ρ CLL + ρ CDD + ρ CLD ) ),
(24)
rather than ρA. To describe this process, the reaction model must take into account the change in the amount of material A and the fact that the system was closed4; i.e., there were no materials that could not be transformed in any way (for example, CLL in the absence of the reaction CLL → A). We assume also that simple precipitation in the form of L or D is impossi ble: this process must be preceded by the formation of pairs LL (or DD), and only then precipitate can be formed. In reality, more complex versions may occur. As was shown in [6], the total enantioselectivity must exceed unity to ensure the formation of an unsta ble system; therefore, the existence of one mechanism (crystallization/dissolution) that can singly provide 100% enantioselectivity leads to the following: even small additional enantioselectivity in the catalytic 4
A system can be closed only with respect to the materials exist ing in it. However, its evolution requires constant energy influx.
stage should be sufficient to induce significant chiral polarization. Calculation Method and the Values of Parameters Used in the Calculations The obtained system of equations is rather difficult for analytical consideration. Below we report numeri cal calculations for a number of parameters. Note that, if the model complexity increases, an analytical solution to the problem of stability rapidly becomes impossible, because polynomial equations of an order larger than four arise, and the numerical solutions to these equations become unstable (large numerical errors arise with an increase in the polynomial power). An alternative way is to solve the dynamic problem with a small deviation from the symmetric one in either parameters (advantage factors) or initial condi tions. Within this approach it is also inconvenient to use chiral variables, because one must apply different approaches to the analysis of numerical errors and convergence for chiral polarizations and concentra tions. The initial equations for concentrations (with out some additional transformations) are most conve nient for a numerical solution to the dynamic prob lem: they make it possible to describe uniformly any problem and solve it by standard numerical methods. We used this approach. The Wolfram Mathematica software was applied to solve the system of differential equations and perform symbolic transformations. As was mentioned above, the ability of the system to withstand racemization and the dependence of autocatalytic reactions on enantioselectivity are of interest. As a parameter determining enantioselectiv γ ity, we will use γ, provided that γ+ ≡ γ– ≡ . In this 2 case, we will fix the values kA + S → S + X ≡ kA + S → S + S + kA + S → N + S and kS + X → A + X ≡ 2 kS + S → A + S + kS + N → A + N : average rates of direct and 2 reverse autocatalytic reactions. All calculations are performed with the same values of coefficients, except for those varied in terms of a particular modification of the model or a particular plot. The values of the dimensionless parameters used in the calculations are listed in the table. Calculation Results The number of possible parameters is fairly large; therefore, the versions are discussed as they become more complicated. Let us consider the situation where LL, DD, and LD pairs are not formed and crystalliza tion is absent. This case corresponds to the models considered in [5, 6]; however, we additionally take into account racemization. We will mainly trace the depen
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Values of parameters used in calculations Parameter
Value
Comments
γ
from 0 to 2
Values from 1 to 2 correspond to the mode in which coupled autocatalytic reactions lead to symmetry breaking. At γ < 1, coupled autocatalytic reactions (without crystallization) are stable in the symmetric state.
m
from 0 to 200
Racemization rate. We use the same value for the racemization of L and D, regardless of the composition of the material they exist in.
kA → S
1
Direct synthesis rate.
kS → A
1
Direct decomposition rate.
ΔkS → A
–5
10
Relative difference in the direct decomposition rates.
kA + S → S + X
0.5
Mean direct autocatalysis rate.
kS + X → A + S
0.5
Mean reverse autocatalysis rate.
kL + D → LD
20
Formation rate of LD pairs. Within the model under consideration, this rate is twice as high er as the corresponding formation rates of LL and D pairs due to binomial coefficients.
kS + S → SS
10
Formation rate of LL and DD pairs.
kLD → L + D
100
Decomposition rate of LD pairs.
kSS → S + S
100
Decomposition rate of LL and DD pairs.
max ρ LD
10
cLD
1000
dLD
10
Maximum LD concentration at which precipitation begins. LD crystallization (precipitation) rate. LD dissolution rate.
max ρ SS
1000
Maximum concentration of LL and DD at which precipitation begins. Within this study we assume that LL and DD are readily dissolved, while LD dissolves poorly.
cSS
1000
Crystallization (precipitation) rate of LL and DD.
dSS
10
Dissolution rate of LL and DD.
k cLD → A + A
1
Decomposition rate of LD precipitate.
k cSS → A + A
1
Decomposition rate of LL and DD precipitate.
tot ρA
1000
Total amount of material A in the system.
ρL – ρD dences for the chiral polarizations η = and ρL + ρD ρ LL – ρ DD . ηB = ρ LL + ρ DD Figure 1 shows the dependence of chiral polariza tion η on the coefficient of autocatalysis enantioselec tivity γ and the racemization coefficient m; ηB ≡ 0 because the formation of pairs is disregarded. Note that within this model one can fairly easily obtain an analytical dependence of the critical total concentra
η 1.0
tot ρA ,
tion at which bifurcation occurs, on the model parameters. The expression is fairly cumbersome, and we omitted it for this reason. With the coefficient val ues used and γ = 2 taken as an example, we arrive at the tot
expression ρ A = 3(1 + 2m), which corresponds to the maximally possible (dimensionless) racemization CRYSTALLOGRAPHY REPORTS
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2.0
0.5 0 0
1.5 50
γ
100 m
150 200
1.0
Fig. 1. Dependences of chiral polarization η on the auto catalysis enantioselectivity γ and the racemization coeffi cient m.
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KONSTANTINOV, KONSTANTINOVA η 1.0
(b)
(a)
0.8 ηB
0.6
ηB
η
η
0.4 0.2 0 0
0.5
1.0
1.5
2.0 γ
0
0.5
1.0
1.5
2.0 γ
Fig. 2. Dependences of chiral polarizations η and ηB on the autocatalysis enantioselectivity γ at the racemization coefficient m = 0 for (a) the model with pair formation and (b) the model with pair formation and crystallization.
coefficient (m ≈ 166), at which bifurcation begins for tot the value ρ A = 1000 used by us. Then the system was supplemented with the pair formation reaction (without crystallization). As a result, we obtained similar plots (omitted here) for the chiral polarization η and the “secondgeneration” chiral polarization ηB.5 The main differences from the previous plot and conclusions are as follows. The for mation of pairs (i.e., complication of the system with the same amount of available material (total concen tot tration ρ A ) significantly reduces the ability of the sys tem to withstand racemization, because the formation of pairs compensates for the autocatalysis nonlinearity. This conclusion is of great importance. In the beginning of the formation of the earth, conditions were highly unfavorable for the existence of organic molecules. Racemization rate is known to increase with an increase in temperature [5]. In the prebiotic stage, the racemization rate was much higher than under current conditions. However, this means the following: if complex organic molecules did not require an almost 100% chirally polarized medium for its formation, the ability of a simpler system to with stand racemization would be much lower, because simple molecules would serve as building blocks for more complex molecules. Thus, only the simple mol ecules whose “descendants” (more complex mole cules) could not be formed in a racemic medium had a significant advantage during symmetry breaking. The simple addition of pair formation (increase in the non linearity of the model) does not increase the ability of the system to withstand insufficient enantioselectivity. At γ = 1, asymmetric solutions disappear, as in the first case. Chiral polarization of the second generation exceeds that of the first generation in the entire range of existence of asymmetric solutions. This means that, 5 The term second generation will be applied to LL and DD.
in the case of a more complicated system, more com plex molecules possess higher chiral polarization. Let us consider the effect of supplementing the model with the crystallization/dissolution process. Note that, in the racemization model under consider ation6, the formation of, for example, precipitate CLD should lead (due to the racemization) to the occur rence of CLL and CDD. Thus, one must consider all three crystallization/dissolution processes, even if the critical concentration for LL and DD is much higher than that for LD (LL and DD are readily dissolved, while LD dissolves poorly). The results for the first generation chiral polarization and the secondgenera tion chiral polarization ηB are also similar in appear ance to the previous ones. The fundamental difference of this version is not in the fact that the system became more stable to racemization (the maximum value m corresponding to γ = 2 increased), but in the fact that far in the range γ < 1 (under conditions of obviously insufficient autocatalysis enantioselectivity) the sys tem involved in the crystallization/dissolution process can reach a significant chiral polarization. Here, like in the previous case (with the formation of pairs but without crystallization), the secondgeneration chiral polarization exceeds the firstgeneration one. For convenience in comparing these results, we report twodimensional dependences of the chiral polariza tions η and ηB on the autocatalysis enantioselectivity γ for the racemization coefficient m = 0 (Figs. 2a, 2b). Let us consider the dependences of the chiral polarizations on the enantioselectivity coefficient γ max
and the maximum solubility of LL (and DD), ρ SS , for the racemization coefficients m = 0 (Fig. 3a) and m = 1 (Fig. 3b)7. In the absence of racemization, the 6 Identical
racemization of L or D, regardless of the composition of the material they enter. 7 The plots are rotated for clearness.
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(a)
(b)
ηB 1.0
ηB 0.8 2.0
2.0
0.5
0.4 1.5
0 50
1.0 40
γ
0
ρmax ss
0
γ
0.5
30
20 10
1.0 40
0.5
30 ρmax ss
1.5 0 50 20 10 0
0
Fig. 3. Dependence of chiral polarization ηB on the autocatalysis enantioselectivity γ and the maximum solubility of LL DD) – max
ρ SS , for the model with pair formation and crystallization at racemization coefficients m = (a) 0 and (b) 1.
max
situation is fairly simple. Until ρ SS reaches some crit ical concentration (which is determined by the dynamic equilibrium in the system for certain values of other parameters), the precipitation of CLD starts after that of CLL (and CDD) and makes a negative contribu tion to enantioselectivity; i.e., the system remains in the symmetric state. After the critical concentration is exceeded, CLD begins to precipitate before CLL (and CDD). Since this is a 100% enantioselective process, the symmetric state on the system is split into two asym max metric states. However, if the excess of ρ SS above the critical concentration is insufficiently large, splitting stops, because the concentration of LL (or DD) begins max to rise after reaching the ρ SS value. Due to the pres ence of racemization, the range of existence of asym max metric solutions shifts to larger ρ SS and γ values, leading simultaneously to a decrease in the chiral polarization of both generations, in agreement with the previous plots. ORIGIN OF LIFE IN A NONEQUILIBRIUM SYSTEM As was noted in [28], living matter is able to store, read, and reproduce information. From the thermo dynamic point of view, this ability, on the one hand, requires a constant influx of “structured” energy, the spectrum of which differs to a great extent from the thermodynamic spectrum of the system. At the same time, a constant outflux of “thermodynamic” energy (thermal energy of the system) is also required. Thus, when considering the origin of life on earth, the fact that the system is not closed is of critical importance. CRYSTALLOGRAPHY REPORTS
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Concerning a dynamic system, two likely scenarios (and, more likely, their combination) can be imple mented in it. The first is the case where material A is initially absent. This situation corresponds to the fact that there were not even any simple organic materials during the formation of the earth because of highly unfavorable conditions. In the course of time, condi tions for synthesizing organic materials arose, in par ticular, due to the energy influx in the form of high energy UV light and γ radiation, simultaneously with a decrease in the general temperature to the level at which organic molecules stop decomposing. This sys tem is out of thermodynamic equilibrium, but is char acterized by certain (different!) absorbed and emitted energy spectra. In the model under consideration, this scenario corresponds to reaction (23) and zero initial concentrations of all materials, except for ρZ. In this scenario, influx of energy with a certain spectrum (UV and γradiation) is a more important factor. The second scenario is related to the dependence of race mization on the external conditions. There are differ ent opinions on the thermal state of the earth (hot or cold) in the beginning of the formation of life; never theless, it is known that the UV flux was much more intense in that stage and that, fairly likely, volcanic activity was also much higher [10]. In other words, decomposition and racemization processes should occur more rapidly at that time. CALCULATION RESULTS FOR THE DYNAMIC MODEL Different Solubilities of Diastereomers A common feature of the aforementioned scenar ios is that they both imply transition through the sta
2013
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KONSTANTINOV, KONSTANTINOVA ρAtot 3000
ρA 80
(a)
(b)
2500 60
2000 1500
40
1000 20
500
0
0 0
500
1000 1500 2000 2500
0
3000 t
500
1000
1500
2000
2500
3000 t
tot
Fig. 4. Time dependences of (a) ρ A and (b) ρA.
ρ
ρ 100
30
(b)
(a) ρL
25
ρLL
80
20
60
15
40
10 ρLD
20
ρD
5
0
0 0
500
1000
1500
2000
2500
0
3000 t
500
1000
1500
ρDD
2000
2500
3000 t
2000
2500
3000 t
Fig. 5. Time dependences of (a) ρL and ρD and (b) ρLL, ρDD, and ρLD.
ρ
η 1.0
1200
(a)
(b)
1000
0.8
800 600
ηB
0.6
ρCLD
η
0.4 400
ρCDD
0.2
ρCLL
200
0
0 0
500
1000
1500
2000
2500
Fig. 6. Time dependences of (a) ρ C , ρ C LL
0
3000 t DD
, and ρ C
bility boundary from the side of one symmetric solu tion to the side of two asymmetric solutions. Both sce narios were implemented (and, likely, simultaneously) when life arose on earth. In this study we report an example for only the first scenario, when simple organic molecules are gradually accumulated. The results are presented in Figs. 4–6. The calculations were performed with the following values of parame
LD
500
1000
1500
and (b) chiral polarizations η and ηB.
ters (differing from the values in the table): γ = 0.58, m = 0, and kZ → A = 1. Figure 4 shows the time dependences of (a) the tot total concentration ρ A in the system and (b) the con γ value is much smaller than the critical value (γ = 1), at which chiral splitting occurs for simple autocatalysis.
8 This
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ρL
(a)
10 8
707
ρLD
(b) ρLL
ρD
ρDD
6 4 2 0 0
500
1000
1500
2000
2500
3000 t
0
500
1000
1500
2000
2500
Fig. 7. Time dependences of (a) ρL and ρD and (b) ρLL, ρDD, and ρLD.
ρ 120
3000 t
η (a)
100
(b)
0.4
80
0.3
60
η
0.2
40
ρCLD
20
ηB
0.1
ρCLL
0
0 0
500
1000
1500
2000
2500
Fig. 8. Time dependences of (a) ρ C
LL
and ρ C
tot
centration ρA. As was noted above, ρ A can be referred to as the total number of “atoms” of type A in the sys tem. Thus, we consider a model with gradual (linear in time) accumulation of simple organic material. Fig ure 5 shows the dependences of the concentrations of (a) L and D and (b) LL, DD, and LD. One can see sev eral distinct modes of system evolution. For the t val ues from zero to ≈100, all materials are gradually accu mulated in solution. At t ≈ 100, the LD concentration reaches the critical value and CLD starts precipitating (Fig. 6a). In this case, an additional 100% enantiose lective factor arises, which is why the total enantiose lectivity of the system becomes sufficient to offer con siderable chiral polarization. Since the advantage factor used in the calculations is sufficiently small, splitting occurs at t ≈ 500, i.e., some time after the mechanism of additional enanti oselectivity is switched off. The concentrations of L and LL start increasing: the newly formed simple organic material (A) is absorbed by the “conqueror” enantiomer (L) and its derivatives (LL). As a result, after rapid splitting (which occurs during a short time interval near t ≈ 500), chiral polarizations η and ηB gradually increase. The secondgeneration chiral polarization ηB significantly exceeds the chiral polar CRYSTALLOGRAPHY REPORTS
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LD
500
1000
1500
2000
2500
3000 t
and (b) chiral polarizations η and ηB.
ization of the first generation (η) and becomes close to unity at t ≈ 1000–1200 (Fig. 6b). At t ≈ 1800, the concentration of LL reaches its critical value. The precipitation of CLL begins (Fig. 6a) and the enantioselectivity of crystallization/dissolu tion starts decreasing. This can clearly be seen in Fig. 6b, which demonstrates how chiral polarizations begin to decrease when the value t ≈ 1800 is reached. Identical Solubilities of Diastereomers We assume the aggregation probabilities for L and D molecules upon their collisions are identical; as a result, we have a doubled factor for the formation rate of LD in comparison with the synthesis rates for LL or DD. In reality, there may be other values. The next example illustrates the situation where all parameters with similar meaning for L and D have identical values. Under the assumption that all other coefficients for L, D, and their derivatives (LL, DD, LD, and their crys talline forms) are identical pairwise or in triads (i.e., having identical solubilities for LL, DD, and LD), we obtain again (only due to the additional factor 2 in the formation rate of LD) an additional 100% enantiose lectivity. Several plots for this version are shown in Figs. 7 and 8. All parameter values are the same as in
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Figs. 4–6, except for the maximum concentration LL (DD), which is the same as the maximum LD concen tration in our calculations. Figure 7 shows the dependences of the concentra tions of (a) L and D and (b) DD, DD, and LD. Having compared them with the data in Fig. 5, we should note the following. First, by the moment of splitting (t ≈ 500), the corresponding plots coincide. However, the dynamics of the system with the values used is such that immediately after splitting the ρLL value very rap max
max
idly reaches the solubility boundary ( ρ LL = ρ LD ). In the plots, this occurs in a very narrow range near t ≈ 500. However, as soon as ρLL reaches maximum solu bility, the precipitation of CLL begins. In contrast to the situation considered in Figs. 4– 6, the precipitation of CLL begins in this case much earlier, because it is assumed that LL, DD, and LD have identical properties. However, as soon as the pre cipitation of CLL begins, the crystallization enantiose lectivity begins to decrease until the total enantiose lectivity crosses the stability boundary at t ≈ 2000 (Figs. 7, 8). Figure 8 shows the time dependence of the concentrations of (a) ρ CLL and ρ CDD and (b) chiral polarizations η and ηB. It can clearly be seen that, when the LL concentration reaches the critical value (near t ≈ 500, almost immediately after splitting), the chiral polarizations begin to decrease (almost to zero); i.e., the enantioselectivity of crystallization/dissolu tion disappears when a certain concentration is exceeded. INFLUENCE OF WEAK INTERACTION AT DIRECTED SYMMETRY BREAKING Simple organic material was accumulated gradu ally, and only the advantage factors that are global in time and space could be significant at the intersection of the instability boundary by the system. There is only one such factor: weak interaction. The difference in the energy levels between L and D (g ~ 10–17) leads to slightly differing decomposition probabilities for L and D; correspondingly, Lamino acids and Dsugars, which are building blocks for all biological com pounds, are somewhat more stable [29–32]. The small g value indicates that only globalscale systems, with numbers of molecules N Ⰷ g–2 ~ 1034, can distinguish this small difference between enantiomers caused by weak interaction [16–18]. However, if there is a system of this scale and organic material is accumulated in it, as is shown in Figs. 4–6, the system passing through the stability boundary will find itself with a statistical reliability in a preferred (with respect to energy) state. Although the influence of weak interaction on the pro cesses of life forming on the earth was analyzed a fairly long time ago, it is unclear why the results have remained hypothetical. The relative accuracy of directed symmetry breaking during life formation on
the earth can be estimated better than, for example, the accuracy of an uptodate neutrino detector or any other modern highprecision experiment. Note that the occurrence of one chiral molecule with its subsequent autocatalytic multiplication is rather often discussed in the literature [33]. This approach may become insufficiently justified on the scale of the earth, because statistical fluctuations in the number of molecules are N Ⰷ 1; therefore, the dif ference in one molecule is negligible. In addition, large jumps in the external conditions in the beginning of earth’s formation also make this suggestion dubi ous, because jumps in conditions lead to multiple transitions through the instability boundary. In this case, any amplifications of individual fluctuations will be, in essence, lost and, as a consequence, individual fluctuations will be averaged over both time and space. On the other hand, specifically multiple transitions through the stability boundary make it possible to dis tinguish the difference between L and D. As was noted in [6], complex organic molecules call for almost 100% chiral polarization. Thus, even if this “super molecule” was formed as a result of fluctuations in the stage when chiral polarization close to 100% was not reached, it could not be stably replicated. At the same time, if chiral polarization close to 100% is achieved, there is no need for this “supermolecule,” because the assembly of long chains becomes possible. CONCLUSIONS We considered a model of coupled autocatalytic reactions with allowance for crystallization (precipita tion) of a compound of L and D. This model is fairly urgent, because many organic materials form insolu ble (or poorly soluble) diastereomers. The difference in the physical properties of diaste reomers is a 100% enantioselective factor. For identical properties of diastereomers, the presence of binomial coefficients in the case of the formation of pairs from two different enantiomers leads to the occurrence of an additional 100% enantioselective factor. As a con sequence, even a simple model of two coupled auto catalytic reactions with allowance for crystallization of diastereomers may yield a significant chiral polariza tion in the range of weak enantioselectivity of autocat alytic reactions. The presence of reaction channels that make it possible to form more complex molecules significantly reduces the ability of the system to withstand racem ization. Thus, the ability of complex organic mole cules to function only at almost 100% chiral polariza tion is due to the following: only those complex mole cules that could not be formed in any medium except for a medium chirally polarized by almost 100% “allowed” simpler molecules to “conquer” under conditions of strong racemization, which should exist at the beginning of the earth’s formation.
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If the system of equations describing the chemical reactions in the system is unstable, the instability is exponential in most cases. Thus, the instability evolu
10. S. L. Miller and H. J. Cleaves, Prebiotic Chemistry on the Primitive Earth, Systems Biology, Vol. I: Genomics (Series in Systems Biology) (Oxford Univ. Press, 2006).
tion time is ~ log ⎛ 1 ⎞ instead of ~ 1 , where g is a small ⎝ g⎠ g parameter distinguishing enantiomers.
11. D. G. Blackmond, Philos. Trans. R. Soc. B 2857 (2011).
More complex molecules possess higher chiral polarization. Although this conclusion can often be met in the framework of a simplified analysis, we present qualitative characteristics for this result in dif ferent modes in this study. The smaller the difference in the properties of dias tereomers is, the narrower the range of concentrations tot ρA
at which the crystallization of diastereomers acts as an enantioselective factor is. Within an estimation of the number of molecules that could be involved in reactions during the forma tion of life on earth, statistical accuracy is many orders of magnitude higher than the splitting between the right and lefthanded enantiomers due to weak inter action, which is an advantage factor global in time and space. It is concluded that specifically weak interac tion is responsible for the observed chirality of living matter (dominance of Lamino acids and Dsugars).
12. E. M. Galimov, Phenomenon of Life: Between Equilib rium and Nonlinearity. Origin and Principles of Evolution (Izdvo URSS.ru, Moscow, 2006) [in Russian]. 13. F. C. Frank, Biochim. Biophys. Acta 11 (4), 459 (1953). 14. T. Kawasaki, K. Suzuki, M. Shimizu, et al., Chirality, No. 7, 479 (2006). 15. K. Suzuki, K. Hatase, D. Nishiyama, et al., J. Syst. Chem. 1, 5 (2010). 16. L. L. Morozov, V. V. Kuzmin, and V. I. Goldanskii, Sov. Sci. Rev. D 357 (1984). 17. L. L. Morozov, V. V. Kuz’min, and V. I. Goldanskii, Origins Life 13, 119 (1983). 18. L. L. Morozov, V. V. Kuz’min, and V. I. Gol’danskii, Pis’ma Zh. Eksp. Teor. Fiz. 39, 344 (1984). 19. R. Breslow, Tetrahedron Lett. 523 (17), 2028 (2011). 20. S. S. Andrews, T. Dinh, and A. P. Arkin, Stochastic Models of Biological Processes (2009), p. 8730. 21. D. T. Gillespie, J. Chem. Phys. 113, 297 (2000). 22. D. T. Gillespie, Phys J. Chem. A 106, 5063 (2002). 23. C. Gadgil, C. H. Lee, and H. G. Othmer, Bull. Math. Biol. 67, 901 (2005).
REFERENCES
24. T. Jahnke and W. Huisinga, J. Math. Biol. 54, 1 (2007).
1. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media, 2nd ed. (Nauka, Moscow, 1982; Pergamon, New York, 1984).
25. R. Erban, S. J. Chapman, and P. K. Maini, http://arxiv.org/abs/0704.1908.
2. F. I. Fedorov, Optics of Anisotropic Media (Izdvo AN BSSP, Minsk, 1958) [in Russian]. 3. C. W. Bunn, Crystals, Their Role in Nature and Science (Academic, New York, 1964; Mir, Moscow, 1970). 4. L. Paster, Selected Works, Vol. 1 (Izdvo AN SSSR, Moscow, 1960) [in Russian]. 5. V. I. Gol’danskii and V. V. Kuz’min, Usp. Fiz. Nauk 1, 157 (1989). 6. V. A. Avetisov and V. I. Gol’danskii, Usp. Fiz. Nauk 166, 8 (1996). 7. G. Palyi, C. Zucchi, and L. Caglioti, Elsevier Sci. (2004). 8. S. Toxvaerd, Int. J. Astrobiol. 4 (1), 43 (2005).
26. R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: SteadyState and Time Dependent Problems (Classics in Applied Mathematics) (SIAM, 2007). 27. A. Dokoumetzidis and P. Macheras, Int. J. Pharmaceu tics 321 (1–2), 1 (2006). 28. V. B. Bajic and T. T. Wee, Information Processing and Living Systems (World Scientific, 2005). 29. S. F. Mason and G. F. Tranter, Mol. Phys. 53, 1091 (1984). 30. V. S. Letokhov, Phys. Lett. A 53, 275 (1975). 31. G. E. Tranter, Nature (London) 318, 172 (1985). 32. A. J. MacDermott, G. F. Tranter, and S. J. Trainor, Chem. Phys. 194, 152 (1992). 33. L. Caglioti and G. Palyi, Rend. Fis. Acc. Lincei. 24, 1 (2013).
9. U. Meierhenrich, Amino Acids and the Asymmetry of Life: Caught in the Act of Formation (Springer, 2008). CRYSTALLOGRAPHY REPORTS
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Translated by Yu. Sin’kov
