Induced mirror symmetry breaking via template-controlled copolymerization: theoretical insights
aa r X i v : . [ q - b i o . Q M ] O c t Induced mirror symmetry breaking via template-controlled copolymerization:theoretical insights
Celia Blanco ∗ and David Hochberg † Centro de Astrobiolog´ıa (CSIC-INTA), Carretera Ajalvir Kil´ometro 4, 28850 Torrej´on de Ardoz, Madrid, Spain
A chemical equilibrium model of template-controlled copolymerization is presented for describingthe outcome of the experimental induced desymmetrization scenarios recently proposed by Lahavand coworkers.
It is an empirical fact that mirror symmetry is broken in all known biological systems, where processes crucial for lifesuch as replication, imply chiral supramolecular structures, sharing the same chiral sign (homochirality). These chiralstructures are proteins, composed of aminoacids almost exclusively found as the left-handed enantiomers (S), alsoDNA, and RNA polymers and sugars with chiral building blocks composed by right-handed (R) monocarbohydrates.One scenario for the transition from prebiotic racemic chemistry to chiral biology suggests that homochiral peptidesmust have appeared before the onset of the primeval enzymes [1–5]. However, the polymerization of racemic mixtures(1:1 proportions) of monomers in ideal solutions typically yields chains composed of random sequences of both theleft and right handed repeat units following a binomial distribution [6]. This statistical problem has been overcomerecently by the experimental demonstration of the generation of amphiphilic peptides of homochiral sequence, thatis, of a single chirality, from racemic compositions. This route consists of two steps: (1) the formation of racemicparallel or anti-parallel β -sheets either in aqueous solution or in 3-D crystals [7] during the polymerization of racemichydrophobic α -amino acids followed by (2) an enantioselective controlled polymerization reaction [8–14] (Fig. 1). Thisprocess leads to racemic or mirror-symmetric mixtures of isotactic oligopeptides where the chains are composed fromamino acid residues of a single handedness. Furthermore, when racemic mixtures of different amino acid species werepolymerized, isotactic co-peptides of homochiral sequence were generated. Here a host or majority species ( R , S ),together with a given number m of minority amino acid species ( R , S ) , ( R , S ) , ... ( R m , S m ) (supplied with lesserabundance) were employed. The guest (S) and (R) molecules are enantioselectively incorporated into the chains ofthe (S) and (R) peptides, respectively, however the former are stochastically distributed within the homochiral chains.As a combined result of these two effects, the sequence of the co-peptide S and R chains will differ from each other,resulting in non-racemic mixtures of co-peptide polymer chains: non-enantiomeric pairs of chains are thus formed.By considering the sequences of these peptide chains, a statistical departure from the racemic composition of thelibrary of the peptide chains is created which varies with chain length N and with the relative concentrations of thehost/guest monomers used in the polymerization [9, 10]. The mechanism has some features in common with the FIG. 1: The scheme proposed in Ref. [7] leading to regio-enantioselection within racemic β -sheet templates. scenarios proposed by Green[15], Eschenmoser[16] and Siegel[17] in which a limited supply of material results in astochastic mirror symmetry breaking process.To address the general scenario for the generation of libraries of diastereoisomeric mixtures of peptides in accordwith that proposed in Ref.[9], consider a model with a host amino acid species and m guest amino acids. We assumeas given the prior formation of the initial templates or β -sheets, and are concerned exclusively with the subsequentrandom polymerization reactions (step (2)). The underlying nonlinear template control is implicit throughout thediscussion.We consider stepwise additions and dissociations of single monomers from one end of the (co)polymer chain,considered as a strand within the β -sheet. It is reasonable to regard the β -sheet in equilibrium with the free monomerpool[18]. ∗ Electronic address: [email protected] † Electronic address: [email protected][18] Reports a stochastic simulation of two concurrent processes: 1) an irreversible condensation of activated amino acids and 2) reversibleformation of racemic β -sheets of alternating homochiral strands, treated as a one-dimensional problem. These architectures lead to the From detailed balance, each individual monomer attachment or dissociation reaction is in equilibrium. This holdsfor closed equilibrium systems in which the free monomers are depleted/replenished by the templated polymerization.Then we can compute the equilibrium concentrations of all the (co)-polymers in terms of equilibrium constants K i and the free monomer concentrations. The equilibrium concentration of an S -type copolymer chain of length n + n + n + ... + n m = N made up of n j molecules S j is given by p Sn ,n ,...,n m = ( K s ) n ( K s ) n ... ( K m s m ) n m /K ,where s j = [ S j ] [19]. Similarly for the concentration of an R -type copolymer chain of length n ′ + n ′ + n ′ + ... + n ′ m = N made up of n ′ j molecules R j : p Rn ′ ,n ′ ,...,n ′ m = ( K r ) n ′ ( K r ) n ′ ... ( K m r m ) n ′ m /K , where r j = [ R j ].The number of different S -type copolymers of length l with n j molecules of type S j is given by the multinomialcoefficient. Hence the total concentration of the S -type copolymers of length l is given by p Sl = X n + n + ... + n m = l (cid:18) ln , n , ..., n m (cid:19) p Sn ,n ,...,n m = 1 K ( K s + K s + ... + K m s m ) l , (1)which follows from the multinomial theorem [20]. We calculate the number of each type S j of S -monomer present inthe S -copolymer of length equal to l , for any 0 ≤ j ≤ m : s j ( p Sl ) = X n + n + ... + n m = l (cid:18) ln , n , ..., n m (cid:19) n j p Sn ,n ,...,n m = s j ∂∂s j p Sl = K j K s j l ( K s + K s + ... + K m s m ) l − . (2)Then we need to know the total amount of the S -type monomers bound within the S -type copolymers, from thedimer on up to a maximum chain length N . Using Eq.2 for the jth type of amino acid, this is given by s j ( p ST ot ) = N X l =2 s j ( p Sl ) → K j K s j a (2 − a )(1 − a ) , (3)the final expression holds in the limit N → ∞ provided that a = ( K s + K s + ... + K m s m ) <
1. This must bethe case, otherwise the system would contain an infinite number of molecules [19]. Similar considerations hold forthe R -sector, and the total amount of R monomers inside R type copolymers for the jth amino acid, is given by r j ( p RT ot ) = K j K r j b (2 − b )(1 − b ) where b = ( K r + K r + ... + K m r m ) <
1. From this we obtain the mass balance equationswhich hold for both enantiomers of the host and guest amino acids, and is our key result: s j + K j K s j a (2 − a )(1 − a ) = s jtot , r j + K j K r j b (2 − b )(1 − b ) = r jtot . (4)These equations express the fact that each type of enantiomer is either free, or is else bound inside a (co)polymerstrand within the template.The problem then consists in the following: given the total concentrations of all the m + 1 enantiomers { s jtot , r j tot } mj =0 , and the K i we calculate the free monomer concentrations { s j , r j } mj =0 from solving Eqs. (5). Denoteby s tot + ... + s m tot + r tot + ... + r m tot = c tot the total system concentration. From the solutions we can calcu-late e.g., the equilibrium concentrations of homochiral copolymers of any specific sequence or composition as wellas the resultant enantiomeric excess for homochiral chains of length l composed of the host (majority) amino acid: ee l = ( r ) l − ( s ) l ( r ) l +( s ) l . When there are no guest aminoacids, i.e., for m = 0, and when the majority species is suppliedin racemic proportions s tot : r tot = 1 : 1, then ee l must be zero: there will be no mirror symmetry breaking. Sowe turn to the scenario of Ref[9] and consider the influence of a single guest species, m = 1 being sufficient for ourpurposes.We first use our mass balance equations to calculate ee l for the same initial compositions of the monomers asreported in [9]. This is shown in top of Fig. 2. We consider a single equilibrium constant K = K = K = 1 M − forsake of simplicity, and the total system concentration, c tot = 1 M . The enantiomeric excess increases when increasingthe amount of guest species s ′ tot , obtaining a maximal symmetry breaking for the case shown with equal amountsof majority and minority S-molecules: s tot = s ′ tot . In the limit as s ′ tot → ee l for fixed l (top to bottom sequence of curves). The ee l increases monotonicallywith the chain length l in all cases. The behavior of the ee l demonstrates quite well the induced symmetry breakingmechanism proposed in Ref.[9]. formation of chiral peptides whose isotacticity increases with length. ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææàààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò rtot : stot : s'tot æ à ò ee % æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææàààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò rtot : r'tot : stot : s'tot æ à ò Oligopeptide length, l ee % FIG. 2: Calculated ee values versus chain length l from solving Eqs. (5). Top: non-racemic host r tot > s tot and one guestaminoacid s ′ tot ( m = 1) and three monomer starting compositions (in moles) r tot : s tot : s ′ tot = 0 . .
25 : 0 .
25 (filled circles),0 . .
45 : 0 .
05 (squares) and 0 . .
475 : 0 .
025 (triangles) for the equilibrium constant K = 1 M − and the total monomerconcentration c tot = 1 M . Compare to Fig. 13 of Ref.[9]. Bottom: racemic host r tot = s tot and m = 1 guest r ′ tot , s ′ tot . Startingcompositions r tot : r ′ tot : s tot : s ′ tot = 0 . . . . . .
15 : 0 . .
25 (squares) and 0 . .
18 : 0 . . The solutions of the mass balance equations (5) can be used to evaluate the average chain lengths as functions ofinitial monomer compositions and the equilibrium constants. The average chain lengths of the S -type copolymers < l S > , composed of random sequences of the S j type monomers, and that of the R -type copolymers < l R > composedof random sequences of the R j type monomers, are derived in the Supplementary Information. Results for the m = 1three monomer cases are shown there in Table I. There is a marked increase in the average chain length whenincreasing K , we moreover observe how the average chain length corresponding to each monomer species increaseswhen increasing its own starting proportion. In the case of additives of only one handedness (three monomer case)and for the different compositions considered ( r tot : s tot : s ′ tot = 0 . .
25 : 0 .
25, 0 . .
45 : 0 .
05, 0 . .
475 : 0 . S -type copolymers and the R -type polymers will be the same. This follows since K is the same for both monomer types and the amount of S -type and R -type molecules in the starting compositions isthe same, r tot = s tot + s ′ tot , so the average chain length must be the same: < l S > = < l R > .By a further example, we carry out an analysis for the case of one guest m = 1 and all four enantiomers, treat-ing a majority species R, S in strictly racemic proportions and a single guest amino acid R ′ , S ′ in various relativeproportions. We solve Eq. (5) and then calculate ee l for the different chain lengths l for three different startingmonomer compositions. In Fig. 2 (bottom) we show the results obtained from calculating ee l for K = 1 M − and c tot = 1 M . The behavior is qualitatively similar to that previously commented, the greater the relative dis-proportion of the minority species r ′ tot , s ′ tot , the greater is the enantiomeric excess. Values for the average chainlengths are calculated for four molecules, with the abundances r tot : r ′ tot : s tot : s ′ tot = 0 . . . . r tot : r ′ tot : s tot : s ′ tot = 0 . , . , . .
26, and are displayed in Table II in Supplementary Information, where otherchoices for the K i and c tot are employed (see Tables III-VI).In summary, we consider a multinomial sample space for the distribution of equilibrium concentrations of homochiralcopolymers formed via template control. We deduce mass balance equations for the enantiomers of the individualamino acid species, and their solutions are used to evaluate the sequence-dependent copolymer concentrations, in termsof the total species concentrations. Measurable quantities signalling the degree of mirror symmetry breaking such asthe ee and average chain lengths are evaluated. This approach provides a quantitative basis for the template-controlledinduced desymmetrization mechanisms advocated by Lahav and coworkers [8–14].We are indebted to Meir Lahav for suggesting a mathematical approach to this problem. CB has a Calvo Rod´esscholarship from INTA. DH acknowledges a grant AYA2009-13920-C02-01 from the MICINN and forms part of theCOST Action CM0703 “Systems Chemistry”. [1] J. Bada and S. Miller, Biosystems , 21 (1987).[2] V. Avetisov, V. Goldanskii, and V. Kuzmin, Dokl. Akad. Nauk USSR , 282 (1985).[3] V. Goldanskii, V. Avetisov, and V. Kuzmin, FEBS Lett. , 181 (1986).[4] V. Avetisov and V. Goldanskii, Proc. Natl. Acad. Sci. USA , 11435 (1996).[5] L. Orgel, Nature , 203 (1992).[6] A. Guijarro and M. Yus, The Origin of Chirality in the Molecules of Life (RSC Publishing, Cambridge, 2009), 1st ed.[7] I. Weissbuch, R. Illos, G. Bolbach, and M. Lahav, Acc. Chem. Res. , 1128 (2009).[8] H. Zepik, E. Shavit, M. Tang, T. Jensen, K. Kjaer, G. Bolbach, L. Leiserowitz, I. Weissbuch, and M. Lahav, Science ,1266 (2002).[9] J. Nery, G. Bolbach, I. Weissbuch, and M. Lahav, Chem. Eur. J. , 3039 (2005).[10] J. Nery, R. Eliash, G. Bolbach, I. Weissbuch, and M. Lahav, Chirality , 612 (2007).[11] I. Rubinstein, R. Eliash, G. Bolbach, I. Weissbuch, and M. Lahav, Angew. Chem. Int. Ed. , 3710 (2007).[12] I. Rubinstein, G. Clodic, G. Bolbach, I. Weissbuch, and M. Lahav, Chem. Eur. J. , 10999 (2008).[13] R. Illos, F. Bisogno, G. Clodic, G. Bolbach, I. Weissbuch, and M. Lahav, J. Am. Chem. Soc. , 8651 (2008).[14] R. Illos, G. Clodic, G. Bolbach, I. Weissbuch, and M. Lahav, Orig. Life Evol. Biosph. , 51 (2010).[15] M. M. Green and B. A. Garetz, Tetrahedron Letters , 2831 (1984).[16] M. Bolli, R.Micura, and A. Eschenmoser, Chem. Biol. , 309 (1997).[17] J. Siegel, Chirality , 24 (1998).[18] N. Wagner, B. Rubinov, and G. Ashkenasy, ChemPhysChem , 2771 (2011).[19] A. Markvoort, H. ten Eikelder, P. Hilbers, T. de Greef, and E. Meijer, Nat. Commun. (2011).[20] Tech. Rep., National Institute of Standards and Technology, http://dlmf.nist.gov/26 (technical report). Supplementary Information I. β -SHEET CONTROLLED COPOLYMERIZATION FIG. 3: Regio-enantioselection within racemic β -sheet templates. The proposed regio-enantioselection within racemic beta sheets is graphically illustrated by Fig 3. For sake of simplicity, we consider ahost majority species ( L , R ) and a minority guest species ( L , R ) of amino acids both provided in ideally racemic proportions. Theamino acids of a given handedness attach to sites of the same chirality within the growing beta sheet leading to the polymerization ofoligomer strands of a single chirality, in the alternating fashion as depicted. The vertical line segments denote hydrogen bonds betweenadjacent strands. Since the polymerization in any given strand is random and the guest molecules are less abundant than the hosts, theformer will attach in a random fashion, leading to independent uncorrelated random sequences in each strand. The overall effect leads tonon-enantiomeric pairs of chiral copolymers, so mirror symmetry is broken in a stochastic manner. II. AVERAGE CHAIN LENGTHS
We can calculate the average copolymer chain lengths as functions of initial monomer compositions s jtot , r jtot , for the jth species,0 ≤ j ≤ m , and the equilibrium constants K j , using the solutions of our mass balance equations: s j + K j K s j a (2 − a )(1 − a ) = s jtot , r j + K j K r j b (2 − b )(1 − b ) = r jtot , (5)where a = K s + K s + ... + K m s m < b = K r + K r + ... + K m r m < S -type copolymers, composed of random sequences of the S j type monomers, andthat of the R -type copolymers composed of random sequences of the R j type monomers are given by: < l S > = P Nl =2 ( s ( p Sl ) + s ( p Sl ) + ... + s m ( p Sl )) P Nl =2 p Sl → ( s + K K s + ... + K m K s m ) a (2 − a )(1 − a ) a (1 − a ) K = 2 − a − a , (6) < l R > = P Nl =2 ( r ( p Rl ) + r ( p Rl ) + ... + r m ( p Rl )) P Nl =2 p Rl → ( r + K K r + ... + K m K r m ) b (2 − b )(1 − b ) b (1 − b ) K = 2 − b − b , (7)respectively. We also obtain an expression for the average length of the polymer chains composed exclusively by the S j or R j monomersfor a given fixed amino acid type j : < l s j S > = P Nl =2 s j ( p S ( s j ) l ) P Nl =2 p S ( s j ) l = P Nl =2 K j K s j l ( K j s j ) l − P Nl =2 ( K j s j ) l K → ( s j K j ) (2 − K j s j )(1 − K j s j ) ( K j s j ) (1 − K j s j ) = 2 − K j s j − K j s j , (8) TABLE I: Average chain lengths for the three different starting compositions as a function of K for c tot = 1 Mr tot : s tot : s ′ tot = 0 . .
25 : 0 . r tot : s tot : s ′ tot = 0 . .
45 : 0 . r tot : s tot : s ′ tot = 0 . .
475 : 0 . K ( M − ) < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > .
37 2 .
37 2 .
37 2 .
15 2 .
15 2 .
37 2 .
37 2 .
37 2 .
32 2 .
03 2 .
37 2 .
37 2 .
37 2 .
34 2 .
015 3 .
16 3 .
16 3 .
16 2 .
37 2 .
37 3 .
16 3 .
16 3 .
16 2 .
93 2 .
06 3 .
16 3 .
16 3 .
16 3 .
04 2 . .
79 3 .
79 3 .
79 2 .
47 2 .
47 3 .
79 3 .
79 3 .
79 3 .
37 2 .
07 3 .
79 3 .
79 3 .
79 3 .
56 2 . .
52 6 .
52 6 .
52 2 .
69 2 .
69 6 .
52 6 .
52 6 .
52 4 .
80 2 .
09 6 .
52 6 .
52 6 .
52 5 .
51 2 . .
59 8 .
59 8 .
59 2 .
77 2 .
77 8 .
59 8 .
59 8 .
59 5 .
57 2 .
10 8 .
59 8 .
59 8 .
59 6 .
71 2 . .
32 17 .
32 17 .
32 2 .
88 2 .
88 17 .
32 17 .
32 17 .
32 7 .
44 2 .
10 17 .
32 17 .
32 17 .
32 10 .
24 2 . .
87 23 .
87 23 .
87 2 .
92 2 .
92 23 .
87 23 .
87 23 .
87 8 .
18 2 .
11 23 .
87 23 .
87 23 .
87 11 .
92 2 . K for c tot = 1 Mr tot : r ′ tot : s tot : s ′ tot = 0 . . . . r tot : r ′ tot : s tot : s ′ tot = 0 . .
14 : 0 . . K ( M − ) < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > .
38 2 .
42 2 .
31 2 .
17 2 .
17 2 .
21 2 .
06 2 .
37 2 .
40 2 .
33 2 .
18 2 .
15 2 .
20 2 .
085 3 .
18 3 .
30 3 .
00 2 .
40 2 .
40 2 .
60 2 .
14 3 .
16 3 .
25 3 .
07 2 .
42 2 .
35 2 .
54 2 . .
82 4 .
00 3 .
56 2 .
50 2 .
50 2 .
84 2 .
18 3 .
80 3 .
92 3 .
66 2 .
54 2 .
43 2 .
74 2 . .
57 7 .
00 6 .
00 2 .
71 2 .
71 3 .
50 2 .
25 6 .
54 6 .
82 6 .
22 2 .
80 2 .
62 3 .
23 2 . .
64 9 .
26 7 .
84 2 .
78 2 .
78 3 .
78 2 .
27 8 .
61 9 .
00 8 .
15 2 .
88 2 .
68 3 .
42 2 . .
41 18 .
83 15 .
65 2 .
89 2 .
89 4 .
32 2 .
30 17 .
35 18 .
24 16 .
34 3 .
02 2 .
78 3 .
76 2 . .
00 26 .
00 21 .
50 2 .
92 2 .
92 4 .
49 2 .
31 23 .
91 25 .
17 22 .
48 3 .
06 2 .
80 3 .
86 2 . < l r j R > = P Nl =2 r j ( p R ( r j ) l ) P Nl =2 p R ( r j ) l = P Nl =2 K j K r j l ( K j r j ) l − P Nl =2 ( K j r j ) l K → ( r j K j ) (2 − K j r j )(1 − K j r j ) ( K j r j ) (1 − K j r j ) = 2 − K j r j − K j r j . (9)To complete the list, we can calculate the chain length averaged over all the copolymers in the system: < l > = P Nl =2 ( s ( p Sl ) + s ( p Sl ) + ... + s m ( p Sl ) + r ( p Rl ) + r ( p Rl ) + ... + r m ( p Rl )) P Nl =2 ( p Sl + p Rl ) → ( s + K K s + ... + K m K s m ) a (2 − a )(1 − a ) + ( r + K K r + ... + K m K r m ) b (2 − b )(1 − b ) a (1 − a ) K + b (1 − b ) K = a (2 − a )(1 − b ) + b (2 − b )(1 − a ) a (1 − b ) (1 − a ) + b (1 − b )(1 − a ) . (10)The right-hand most expressions ( → ) hold in the limit of N → ∞ and for a < b < m = 1 guest and equal equilibrium constants K = K = K . In the case ofadditives of only one handedness (chiral additives, r tot : s tot : s ′ tot ), and for the three different cases considered in the Communication(0 . .
25 : 0 .
25, 0 . .
45 : 0 .
05 and 0 . .
475 : 0 . S -type copolymers and the R -type polymers willbe the same, see Table I. This follows since the equilibrium constant is the same for both monomer types and the amount of S -type and R -type molecules in the starting compositions is the same r tot = s tot + s ′ tot , so the total average chain length must be the same: < l S > = < l R > = < l > . In the particular case of r tot : s tot : s ′ tot = 0 . .
25 : 0 .
25, that is, for the same starting amounts s tot = s ′ tot , theaverage length for the chains exclusively composed of S or S ′ is the also same: < l sS > = < l s ′ S > (fifth and sixth columns in Table I). Wecan appreciate a clear increase in the average chain length when increasing K (top to bottom rows), we observe moreover that theaverage chain length corresponding to each monomer species increases when increasing its starting proportion; see Table I, from left toright in the groups.In the particular case of r tot : r ′ tot : s tot : s ′ tot = 0 . . . .
3, that is the same starting amounts of r , s and s ′ , the average chainlength for the chains exclusively composed of s or s ′ is the same, < l sS > = < l s ′ S > . Numerical results for the cases r tot : r ′ tot : s tot : s ′ tot = 0 . . . . r tot : r ′ tot : s tot : s ′ tot = 0 . , . , . .
26 are shown in Table II.We consider the effect of different equilibrium constants K = K and a much smaller total system concentration c tot = 10 − M in TableIII. The dependence on varying c tot for fixed but distinct equilibrium constants K = K is displayed in Table IV. These should becompared to the previous Table I, since they refer to the same starting monomer compositions as used in that Table. Finally Tables Vand VI have been calculated for the same starting compositions as Table II and can be compared with the latter. TABLE III: Average chain lengths for the three different starting compositions as a function of K for K = K / c tot = 10 − Mr tot : s tot : s ′ tot = 0 . .
25 : 0 . r tot : s tot : s ′ tot = 0 . .
45 : 0 . r tot : s tot : s ′ tot = 0 . .
475 : 0 . K ( M − ) < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 . .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 . .
04 2 .
04 2 .
05 2 .
02 2 .
01 2 .
05 2 .
05 2 .
05 2 .
04 2 .
00 2 .
05 2 .
05 2 .
05 2 .
04 2 . .
34 2 .
31 2 .
37 2 .
17 2 .
10 2 .
36 2 .
36 2 .
37 2 .
32 2 .
02 2 .
36 2 .
36 2 .
37 2 .
34 2 . .
76 3 .
73 3 .
79 2 .
51 2 .
42 3 .
79 3 .
78 3 .
80 3 .
40 2 .
06 3 .
79 3 .
79 3 .
79 3 .
58 2 . .
57 8 .
56 8 .
59 2 .
78 2 .
75 8 .
59 8 .
58 8 .
59 5 .
60 2 .
10 8 .
59 8 .
59 8 .
59 6 .
73 2 . c tot for K = 100000 and K = K / r tot : s tot : s ′ tot = 0 . .
25 : 0 . r tot : s tot : s ′ tot = 0 . .
45 : 0 . r tot : s tot : s ′ tot = 0 . .
475 : 0 . c tot ( M ) < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > < l > < l S > < l R > < l sS > < l s ′ S > − .
34 2 .
31 2 .
37 2 .
17 2 .
10 2 .
36 2 .
36 2 .
36 2 .
32 2 .
02 2 .
36 2 .
36 2 .
37 2 .
34 2 . − .
76 3 .
73 3 .
79 2 .
51 2 .
42 3 .
79 3 .
78 3 .
79 3 .
40 2 .
06 3 .
79 3 .
79 3 .
79 3 .
58 2 . − .
57 8 .
56 8 .
59 2 .
78 2 .
75 8 .
59 8 .
58 8 .
59 5 .
60 2 .
09 8 .
59 8 .
59 8 .
59 6 .
73 2 . − .
86 23 .
86 23 .
87 2 .
92 2 .
91 23 .
87 23 .
86 23 .
87 8 .
18 2 .
11 23 .
87 23 .
87 23 .
87 11 .
93 2 . − .
21 72 .
21 72 .
21 2 .
97 2 .
97 72 .
22 72 .
22 72 .
21 9 .
88 2 .
11 72 .
24 72 .
27 72 .
21 16 .
79 2 . K for K = K / c tot =10 − M r tot : r ′ tot : s tot : s ′ tot = 0 . . . . r tot : r ′ tot : s tot : s ′ tot = 0 . .
14 : 0 . . K ( M − ) < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 . .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 .
00 2 . .
04 2 .
04 2 .
03 2 .
03 2 .
01 2 .
03 2 .
00 2 .
04 2 .
04 2 .
04 2 .
03 2 .
01 2 .
03 2 . .
33 2 .
36 2 .
28 2 .
19 2 .
12 2 .
22 2 .
04 2 .
33 2 .
35 2 .
30 2 .
20 2 .
10 2 .
22 2 . .
77 3 .
94 3 .
53 2 .
53 2 .
45 2 .
88 2 .
16 3 .
75 3 .
86 3 .
61 2 .
58 2 .
40 2 .
78 2 . .
62 9 .
23 7 .
83 2 .
80 2 .
77 3 .
81 2 .
27 8 .
58 8 .
79 8 .
13 2 .
89 2 .
67 3 .
44 2 . c tot for K = 100000 and K = K / r tot : r ′ tot : s tot : s ′ tot = 0 . . . . r tot : r ′ tot : s tot : s ′ tot = 0 . .
14 : 0 . . c tot ( M ) < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > < l > < l S > < l R > < l sS > < l s ′ S > < l rR > < l r ′ R > − .
33 2 .
36 2 .
28 2 .
19 2 .
12 2 .
22 2 .
04 2 .
33 2 .
35 2 .
30 2 .
20 2 .
10 2 .
22 2 . − .
77 3 .
94 3 .
53 2 .
53 2 .
45 2 .
88 2 .
16 3 .
75 3 .
86 3 .
61 2 .
58 2 .
40 2 .
78 2 . − .
62 9 .
23 7 .
83 2 .
79 2 .
77 3 .
81 2 .
26 8 .
58 8 .
97 8 .
13 2 .
89 2 .
67 3 .
44 2 . − .
99 25 .
99 21 .
50 2 .
92 2 .
92 4 .
49 2 .
31 23 .
90 25 .
16 22 .
48 3 .
06 2 .
80 3 .
86 2 . − .
58 78 .
96 64 .
75 2 .
97 2 .
97 4 .
82 2 .
33 72 .
34 76 .
33 67 .
83 3 .
12 2 .
85 4 .
05 2 ..