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Reaction of cross-inversion in a chiral chemical system
CHEMICAL
10 March 1995
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 234 (1995) 309-312
Reaction of cross-inversion in a chiral chemical system Leonid V. II'ichov Institute of Automation and Electrometry, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russian Federation Received 11 March 1994; in final form 30 December 1994
Abstract We use the master equation approach to the evolution of a model chemical system containing chiral reagents. The case when the master equation can be solved is considered. Considerable progress in understanding the origin of chiral purity of the biosphere has been achieved [1]. Our planet is inhabited by organisms containing exclusively left (L) modifications of amino acids and right (D) modifications of sugars. There are reasons for considering the chiral purity of the environment as being indispensable to the existence of even the simplest self-replicating biological structures. Thus the breaking of mirror symmetry had to take place in the pre-biological (i.e. chemical) period of evolution, Under conditions of ordinary chemical synthesis the outcome of the L-enantiomer is equal to that of the D one. A so-called racemic mixture is obtained, Investigation of hypothetical reactions causing spontaneous breaking of mirror symmetry is a useful starting point for solving the chiral purity problem. In the majority of publications on this problem rate equations for concentrations (i.e. for averaged quantities) are used. The role of inevitable chemical fluctuations in the process of mirror symmetry breaking is the subject of the present work. Consider a closed system of chiral molecules involved in the following reactions: ordinary racemization k~ L. " D, (1) kI
Elsevier Science B.V. SSDI 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 0 5 4 - 2
cross-inversion reaction k2 L+D ,L+L, L+D and the reverse reaction L + L k3 , L + D , D+D
k2
,D+D
(2)
k3 ~ D + L .
(3)
The total number of molecules N = n L + n D is constant. The following notation is used: N = 2j, n L = j + m, n D = j - m. The state of the system is specified by the quantity m ( - j ~< m ~
p(m, t), (4)
310
L. V. ll'ichov / Chemical Physics Letters 234 (1995) 309-312
where W+ (m) = vl( j - m) + v2(j2 _ m 2) + v3( j _ m)2,
The summation in (5) goes from m + m 0 to j + rain(m, m0). Expression (6) describes the relaxation of p(m, t) to a Bernoullian distribution
W _ ( m ) = v l ( j + m ) + v2(j 2 - m e ) + v 3 ( j + m) 2
(2j)!
p ( m , t) ~ 2 -2j are respectively the transition probabilities per unit time for the creation and annihilation of the L-enantiomer. The frequency v 1 does not depend on the geometry of the system nor its volume V in particular (this is not correct if racemization takes place on the walls). We can say nothing in general about the V-dependence of v 2 and v 3 if some set of elementary reactions stands behind (2) and (3). However, this dependence is definitely not the same as that of v I and so we can vary the ratios v l / v 2 . 3 by varying V. Unfortunately, the author has not succeeded in solving Eq. (4) with arbitrary l.,1, 1.,2 and v 3. One can prove that no choice of characteristic function transforms Eq. (4) to that of hypergeometric type. This circumstance reduces considerably the hope of finding a general solution to Eq. (4). Solvable cases are considered below, For completeness we begin with the case v I >> v2j, v3j, when cross_inversion can be neglected l. it is convenient to use the following characteristic
In the limit j >> 1 this distribution transforms to a Gauss±an one with width jl/2. Relaxation in the case v~ + 2 j r 3 = v 2 - v 3 =- v is of interest. We have W± (m) = v( j qEm) ( j + m + 1). (8) Introducing an effective quantum mechanical operatot of angular momentum with components J ± = f t + i J2, J0 ---J3 and corresponding set of state vectots I j m ) (m = - j . . . . . j ) we write expression (8) in the form
W ± ( m ) = v ( j m + l l J ± [jm) 2. Let us consider p(m, t) as the diagonal element ( j m [ 13[ j m ) of the density matrix of some hypothetical quantum system with spin j. Eq. (4) in our case coincides with the diagonal part of the corresponding kinetic equation for t3,
d
(5)
~-= kit.
Solving this equation with initial condition p(m, O) = 8=,~0 and inverting definition (5) we arrive at the following expression for the transition probability:
p(m, r[ too, O) (j + m0)!( j - too)! = ~ n!(j + m - n)!(j + m o - n)!(n - m - too)!
1
This is the well-knownEhrenfestproblem,
]
[
]
J - P , J+ + J - , 13J^+
+[f+13, J ] + [ f + , 1 3 J ' _ ] } .
J cos2j(l~o) P ( ~ , r ) = ~_, eim*p(m, z ) . m=-j For P(q~, r ) Eq. (4) takes the form
×(e -T sinh r)2J(tanh z) m+m°-2n
l v{[ A ^
d--]13= 3
function P(~p, r):
O~P( ~o, ~-) = - 2 sin(~p) O~P( q~, r ) ,
(7) ( j + m) !( j - m) ! "
(6)
(9)
This is the equation from Ref. [41 which describes the relaxation of a quantum spin placed in a magnetic field in the limit of high temperature. As a result, a uniform distribution over 'Zeeman sublevels' is reached, p( m, t) = ( jm ] 13( t ) ] j m ) ~ (2 j + 1) - 1, when t ~ o% i.e. in the steady state regime when v I + 2 j r 3 = v 2 - /J3 any possible number of L-enantiomers can be found with equal probability. It is evident that the character of evolution of the diagonal elements of the density matrix will not be disturbed if we add the expression v ( 2 J o 13-/0_./213 - 13f02) to the right-hand part of Eq. (9). The new equation can be written in the form
d
d-t 13=
2
v[J13J - j ( j + ^ ^ 1) 13],
(10)
L.V. II'ichov/ ChemicalPhysicsLetters234 (1995) 309-312
311
explicitly revealing its symmetry with respect to rotations. Solutions of Eq. (10) have the simplest form when expressed through linear combinations of ( jm [ ~( t) Jj ~ ) which realize irreducible represen-
Here n stands for nL, N-1 q( n, t) = ~ p ( n', t [ n, 0) n'= 1
tations of the SU(2)group, J p~q(t)= E (-1)J-m'(jmj-~lKq) m,m'= -j
is the probability for the system not to reach the boundaries up to the moment t under the condition
n(t=O)=n. Themeantimeis TN(n ) = f f q(n, t) dt.
×(jm[ ~(t) ljm'),
1 1
(jml ~(t) ljm') = ~,(--1)J-m'(jmj-m'lKq)pKq(t),
(11)
K,q
where ( jmj - rr[ I Kq) are Clebsch-Gordon coefficients; K = 0, 1 , . . . , 2j; q - - K , - K + 1 . . . . . x. For the quantities pKq(t) Eq. (10) takes the form d -~tpKq(t ) = - - V K ( K q - 1 ) p K q ( t ).
(12)
Combining (11) and (12) and taking into account the initial condition (jmJ~(O)[jrn')=~r, moSr,,,~o we arrive at
Using Eq. (14) we obtain the following equation for To(n):
To(n + 1)
- 2To(n ) +
try. The solution of Eq. (15) is as follows: [ /,/ N - 1 1 n-1 n-k
To(n)
"~-
p211- \
E --~ -- E
N k=l
k=l
k(-N---'-k) 1"
If n >> 1 and N - n >> 1 the following good approximation is useful: (13)
This expression seems to be useful not only in the point v I + 2 j r 3 = v 2 - 1,3 but also in some of its vicinity, when the solution of Eq. (4) terminates in an almost homogeneous distribution ( I (2j + 1)p(m, oo) - 11 << 1 for all m). Let us imagine for a moment that v I = v 3 = 0. In this case Eq. (4) describes a random walk with absorbing boundaries at m = +__j. When the system gets into these states no further evolution takes place and the system is chirally polarized. The authors of Ref. [2] called this phenomenon a 'kinetic trap'. The mean time of polarization is of interest. To find the time, the inverse master equation is needed, d
-~q(n, t) = 1 , 2 ( N - n)
× [q(n + 1, t) + q(n - 1, t) - 2q(n,
(15)
To(O)=To(N)=O in accordance with definition, To(n) = TN(N- n) due to L - D substitution symme-
~=0
×(jmj-mlrO)(jmoj-molrO).
1)
1 = - v2n ( N - n ) "
p(m, t l mo, O) 2j = E e-~'K(K+I)t(- 1) m-too
To(n -
t)]. (14)
TN(n)=v2
,In
[N~
[Nlnln-)+
N - n l n ( _ _ _ _ ~ N I] N ~N-n]]" (16)
We can find the system being L-polarized with probability PL = n/N and D-polarized with probability PD = 1 --PL" In reality reactions (1) and (3) never vanish. They will inevitably blur the steady state distribution p(m) =pLSmd +pD~m,_j and make it symmetric about the central point. Thus the act of cross-inversion smooths the narrow peak of distribution (7). When 1,1 + 2jr3 --- 1,2 - 1,3 fluctuations are strong enough to establish a homogeneous distribution. If cross-inversion prevails, the probability distribution demonstrates an effective attraction to the boundary points m = ±j. The master equation (4) provides a localized model of the reactions (1)-(3). Diffusion, which controls any chemical reaction in nature, is not taken into account. Because of this the model we used is hardly applicable to macroscopic systems (with N NA), where the controlling role of diffusion may not
312
L. V. ll'ichov / Chemical Physics Letters 234 (1995) 309-312
be neglected. In contrast, one may suppose conditions which provide mixing in small ensembles of chiral molecules ( N ~< 10). For such systems Eq. (4) seems suitable. All this concerns also expression (16) and reveals its limited applicability, There are noteworthy distinctions between scenarios of chiral polarization in the popular model of Frank [5] and in the case of cross-inversion. The Frank model and its numerous generalizations have some peculiarities: (i) continuous production and annihilation of chiral reagents; (ii) inevitable reaction which increases the modulus of chiral parameter of order ( n L ) -- ( n D )
; 1"/----- ( nL ) + (nD ) (iii) existence of two mean trajectories (left and right), along which the system moves losing its mirror symmetry. None of these features can be attributed to the system considered above.
It is a pleasure to acknowledge helpful discussions with S.G. Rautian and A.M. Shalagin. This work was supported by the International Science Foundation (grant RCM000), the Netherlands Organization for Scientific Research (NWO) and by the Russian Foundation for Fundamental Research (grant No. 93-02-03567).
References [1] V.I. Gol'dansky and V.V. Kuz'min, Usp. Fiz. Nauk. 157 (1989) 3. [2] I.V. Alexandrov and A.V. Pajitnov, Khim. Fiz. 6 (1987) 1243. [3] C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1985). [4] A.A. Belavin, B.Ya. Zel'dovich and A.M. Perelomov, Zh. Eksp. Teor. Fiz. 56 (1969) 264. [5] F.C. Frank, Biochim Biophys. Acta. II (1953) 459.
10 March 1995
PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 234 (1995) 309-312
Reaction of cross-inversion in a chiral chemical system Leonid V. II'ichov Institute of Automation and Electrometry, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russian Federation Received 11 March 1994; in final form 30 December 1994
Abstract We use the master equation approach to the evolution of a model chemical system containing chiral reagents. The case when the master equation can be solved is considered. Considerable progress in understanding the origin of chiral purity of the biosphere has been achieved [1]. Our planet is inhabited by organisms containing exclusively left (L) modifications of amino acids and right (D) modifications of sugars. There are reasons for considering the chiral purity of the environment as being indispensable to the existence of even the simplest self-replicating biological structures. Thus the breaking of mirror symmetry had to take place in the pre-biological (i.e. chemical) period of evolution, Under conditions of ordinary chemical synthesis the outcome of the L-enantiomer is equal to that of the D one. A so-called racemic mixture is obtained, Investigation of hypothetical reactions causing spontaneous breaking of mirror symmetry is a useful starting point for solving the chiral purity problem. In the majority of publications on this problem rate equations for concentrations (i.e. for averaged quantities) are used. The role of inevitable chemical fluctuations in the process of mirror symmetry breaking is the subject of the present work. Consider a closed system of chiral molecules involved in the following reactions: ordinary racemization k~ L. " D, (1) kI
Elsevier Science B.V. SSDI 0 0 0 9 - 2 6 1 4 ( 9 5 ) 0 0 0 5 4 - 2
cross-inversion reaction k2 L+D ,L+L, L+D and the reverse reaction L + L k3 , L + D , D+D
k2
,D+D
(2)
k3 ~ D + L .
(3)
The total number of molecules N = n L + n D is constant. The following notation is used: N = 2j, n L = j + m, n D = j - m. The state of the system is specified by the quantity m ( - j ~< m ~
p(m, t), (4)
310
L. V. ll'ichov / Chemical Physics Letters 234 (1995) 309-312
where W+ (m) = vl( j - m) + v2(j2 _ m 2) + v3( j _ m)2,
The summation in (5) goes from m + m 0 to j + rain(m, m0). Expression (6) describes the relaxation of p(m, t) to a Bernoullian distribution
W _ ( m ) = v l ( j + m ) + v2(j 2 - m e ) + v 3 ( j + m) 2
(2j)!
p ( m , t) ~ 2 -2j are respectively the transition probabilities per unit time for the creation and annihilation of the L-enantiomer. The frequency v 1 does not depend on the geometry of the system nor its volume V in particular (this is not correct if racemization takes place on the walls). We can say nothing in general about the V-dependence of v 2 and v 3 if some set of elementary reactions stands behind (2) and (3). However, this dependence is definitely not the same as that of v I and so we can vary the ratios v l / v 2 . 3 by varying V. Unfortunately, the author has not succeeded in solving Eq. (4) with arbitrary l.,1, 1.,2 and v 3. One can prove that no choice of characteristic function transforms Eq. (4) to that of hypergeometric type. This circumstance reduces considerably the hope of finding a general solution to Eq. (4). Solvable cases are considered below, For completeness we begin with the case v I >> v2j, v3j, when cross_inversion can be neglected l. it is convenient to use the following characteristic
In the limit j >> 1 this distribution transforms to a Gauss±an one with width jl/2. Relaxation in the case v~ + 2 j r 3 = v 2 - v 3 =- v is of interest. We have W± (m) = v( j qEm) ( j + m + 1). (8) Introducing an effective quantum mechanical operatot of angular momentum with components J ± = f t + i J2, J0 ---J3 and corresponding set of state vectots I j m ) (m = - j . . . . . j ) we write expression (8) in the form
W ± ( m ) = v ( j m + l l J ± [jm) 2. Let us consider p(m, t) as the diagonal element ( j m [ 13[ j m ) of the density matrix of some hypothetical quantum system with spin j. Eq. (4) in our case coincides with the diagonal part of the corresponding kinetic equation for t3,
d
(5)
~-= kit.
Solving this equation with initial condition p(m, O) = 8=,~0 and inverting definition (5) we arrive at the following expression for the transition probability:
p(m, r[ too, O) (j + m0)!( j - too)! = ~ n!(j + m - n)!(j + m o - n)!(n - m - too)!
1
This is the well-knownEhrenfestproblem,
]
[
]
J - P , J+ + J - , 13J^+
+[f+13, J ] + [ f + , 1 3 J ' _ ] } .
J cos2j(l~o) P ( ~ , r ) = ~_, eim*p(m, z ) . m=-j For P(q~, r ) Eq. (4) takes the form
×(e -T sinh r)2J(tanh z) m+m°-2n
l v{[ A ^
d--]13= 3
function P(~p, r):
O~P( ~o, ~-) = - 2 sin(~p) O~P( q~, r ) ,
(7) ( j + m) !( j - m) ! "
(6)
(9)
This is the equation from Ref. [41 which describes the relaxation of a quantum spin placed in a magnetic field in the limit of high temperature. As a result, a uniform distribution over 'Zeeman sublevels' is reached, p( m, t) = ( jm ] 13( t ) ] j m ) ~ (2 j + 1) - 1, when t ~ o% i.e. in the steady state regime when v I + 2 j r 3 = v 2 - /J3 any possible number of L-enantiomers can be found with equal probability. It is evident that the character of evolution of the diagonal elements of the density matrix will not be disturbed if we add the expression v ( 2 J o 13-/0_./213 - 13f02) to the right-hand part of Eq. (9). The new equation can be written in the form
d
d-t 13=
2
v[J13J - j ( j + ^ ^ 1) 13],
(10)
L.V. II'ichov/ ChemicalPhysicsLetters234 (1995) 309-312
311
explicitly revealing its symmetry with respect to rotations. Solutions of Eq. (10) have the simplest form when expressed through linear combinations of ( jm [ ~( t) Jj ~ ) which realize irreducible represen-
Here n stands for nL, N-1 q( n, t) = ~ p ( n', t [ n, 0) n'= 1
tations of the SU(2)group, J p~q(t)= E (-1)J-m'(jmj-~lKq) m,m'= -j
is the probability for the system not to reach the boundaries up to the moment t under the condition
n(t=O)=n. Themeantimeis TN(n ) = f f q(n, t) dt.
×(jm[ ~(t) ljm'),
1 1
(jml ~(t) ljm') = ~,(--1)J-m'(jmj-m'lKq)pKq(t),
(11)
K,q
where ( jmj - rr[ I Kq) are Clebsch-Gordon coefficients; K = 0, 1 , . . . , 2j; q - - K , - K + 1 . . . . . x. For the quantities pKq(t) Eq. (10) takes the form d -~tpKq(t ) = - - V K ( K q - 1 ) p K q ( t ).
(12)
Combining (11) and (12) and taking into account the initial condition (jmJ~(O)[jrn')=~r, moSr,,,~o we arrive at
Using Eq. (14) we obtain the following equation for To(n):
To(n + 1)
- 2To(n ) +
try. The solution of Eq. (15) is as follows: [ /,/ N - 1 1 n-1 n-k
To(n)
"~-
p211- \
E --~ -- E
N k=l
k=l
k(-N---'-k) 1"
If n >> 1 and N - n >> 1 the following good approximation is useful: (13)
This expression seems to be useful not only in the point v I + 2 j r 3 = v 2 - 1,3 but also in some of its vicinity, when the solution of Eq. (4) terminates in an almost homogeneous distribution ( I (2j + 1)p(m, oo) - 11 << 1 for all m). Let us imagine for a moment that v I = v 3 = 0. In this case Eq. (4) describes a random walk with absorbing boundaries at m = +__j. When the system gets into these states no further evolution takes place and the system is chirally polarized. The authors of Ref. [2] called this phenomenon a 'kinetic trap'. The mean time of polarization is of interest. To find the time, the inverse master equation is needed, d
-~q(n, t) = 1 , 2 ( N - n)
× [q(n + 1, t) + q(n - 1, t) - 2q(n,
(15)
To(O)=To(N)=O in accordance with definition, To(n) = TN(N- n) due to L - D substitution symme-
~=0
×(jmj-mlrO)(jmoj-molrO).
1)
1 = - v2n ( N - n ) "
p(m, t l mo, O) 2j = E e-~'K(K+I)t(- 1) m-too
To(n -
t)]. (14)
TN(n)=v2
,In
[N~
[Nlnln-)+
N - n l n ( _ _ _ _ ~ N I] N ~N-n]]" (16)
We can find the system being L-polarized with probability PL = n/N and D-polarized with probability PD = 1 --PL" In reality reactions (1) and (3) never vanish. They will inevitably blur the steady state distribution p(m) =pLSmd +pD~m,_j and make it symmetric about the central point. Thus the act of cross-inversion smooths the narrow peak of distribution (7). When 1,1 + 2jr3 --- 1,2 - 1,3 fluctuations are strong enough to establish a homogeneous distribution. If cross-inversion prevails, the probability distribution demonstrates an effective attraction to the boundary points m = ±j. The master equation (4) provides a localized model of the reactions (1)-(3). Diffusion, which controls any chemical reaction in nature, is not taken into account. Because of this the model we used is hardly applicable to macroscopic systems (with N NA), where the controlling role of diffusion may not
312
L. V. ll'ichov / Chemical Physics Letters 234 (1995) 309-312
be neglected. In contrast, one may suppose conditions which provide mixing in small ensembles of chiral molecules ( N ~< 10). For such systems Eq. (4) seems suitable. All this concerns also expression (16) and reveals its limited applicability, There are noteworthy distinctions between scenarios of chiral polarization in the popular model of Frank [5] and in the case of cross-inversion. The Frank model and its numerous generalizations have some peculiarities: (i) continuous production and annihilation of chiral reagents; (ii) inevitable reaction which increases the modulus of chiral parameter of order ( n L ) -- ( n D )
; 1"/----- ( nL ) + (nD ) (iii) existence of two mean trajectories (left and right), along which the system moves losing its mirror symmetry. None of these features can be attributed to the system considered above.
It is a pleasure to acknowledge helpful discussions with S.G. Rautian and A.M. Shalagin. This work was supported by the International Science Foundation (grant RCM000), the Netherlands Organization for Scientific Research (NWO) and by the Russian Foundation for Fundamental Research (grant No. 93-02-03567).
References [1] V.I. Gol'dansky and V.V. Kuz'min, Usp. Fiz. Nauk. 157 (1989) 3. [2] I.V. Alexandrov and A.V. Pajitnov, Khim. Fiz. 6 (1987) 1243. [3] C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1985). [4] A.A. Belavin, B.Ya. Zel'dovich and A.M. Perelomov, Zh. Eksp. Teor. Fiz. 56 (1969) 264. [5] F.C. Frank, Biochim Biophys. Acta. II (1953) 459.