A NEW DEFINITION OF LIFE 255 hrin a word by the same hm vmbol of e the infom rest can be ignore tion is e infor 2 Fig.1.Tetrahedral and Square Planar Substitutions Spatial information would be im ble to quantify (as H=-∑log2p be subo tent of a fre 3 g:固+g:()】-1 If we place the ongin at the e ch on are assumed to be equal (eg uivalen t to requir and says nothing about the knowledge co ained in the in ads win aton of each bst n if a coin flip is olved at all.To se eg.1in a c Sullivan et6)who calculates the information content of and the switching of the the last tw group occurs ecaus ontent of chiral ca CHIRALITY AND INFORMATION THEORY H=-2g-22og2 (5 to the calculation of the For carbon molecules with more than one chiral center ie.The information con- ncreas multipl of 2 (e es F H=-∑2og2=名 -2 sets of c oordina or rds the the numbe eg.4 (as for a die with 1.1.2.3 on its faces): of chiral centers times 2 all si atation of substituents is able and H=-1og257-log2-1og21=15 (④ 0s and s,a 1 uch as3 responding to tho wtAB,A and AI nt information as.Note that whila ncemning one dimension only equal to that given by eq.6: his is becau three of he grou H=-23g5=158 uent information content of the peptide resides solely Chirality DOI 10.1002/chirH ¼
Xn i¼1 pi log 2 pi ð1Þ where the pi are the probabilities of each event (or symbol) that occurs. In the case of the coin flip, H is given by eq. 2:
1 2 log 2 1 2 8 >: 9 >; þ 1 2 log 2 1 2 8 >: 9 >;
¼ 1 ð2Þ Thus, 1 bit, either a 0 or a 1, is sufficient to convey the in￾formation. Note that H is only a quantity of information, and says nothing about the knowledge contained in the in￾formation, for example, if heads wins the last bottle of beer, or whether a 0 should represent heads or tails, or even if a coin flip is involved at all. To use eq. 1 in a chemi￾cal setting, we could, for example, follow the method of Sullivan (Ref. 6) who calculates the information content of molecular conformations by assigning a probability to each possible conformation and summing as in eq. 1. CHIRALITY AND INFORMATION THEORY A chiral carbon molecule has four different substituents attached to it, as in Figure 1. Upon first consideration, a method of calculating the amount of information contained in 1 might seem to be analogous to the calculation of the information given by a four-sided die. The information con￾tent of such a die is given by eq. 3: H ¼
X 4 1 1 4 log 2 1 4 ¼ 2 ð3Þ And likewise the information content of a carbon molecule 2 with Cs symmetry (such as glycine) would be given by eq. 4 (as for a die with 1,1,2,3 on its faces): H ¼
1 2 log 2 1 2
1 4 log 2 1 4
1 4 log 2 1 4 ¼ 1:5 ð4Þ However, a hypothetical square planar molecule such as 3 also has 4 different substituents and contains the same amount of substituent information as 1. Note that while an individual amino acid has the information content given by eq. 3, in the context of a peptide, three of the groups con￾tribute no information. This is because three of the groups of each amino acid are the same, and therefore the substit￾uent information content of the peptide resides solely in the side chain of each amino acid. An analogy could be made to our own language if we were to precede each let￾ter in a word by the same three symbols, for example, rather than ‘‘cat’’, we wrote ‘‘@#$c@#$a@#$t’’; the informa￾tion resides solely in the last symbol of each four and the rest can be ignored. If the frequencies of each amino acid were equal, the information content would be log2(1/20) for each residue. This substituent information is the infor￾mation discussed by Yockey in Ref. 3. What I am consider￾ing and what we need to be concerned with in analyzing the existence of chiral building blocks in biology, is the spatial information contained in structures 1–3. Spatial information would be impossible to quantify (as any other quantity other than infinity) because in any coor￾dinate system the coordinates may be subdivided infin￾itely, and thus the information content of a free object would be infinite. However, molecules such as 1–3 have constraints of bond lengths and geometries such that a fi- nite number of spatial coordinates specifies the location of the substituents relative to an origin placed at the center of the point group. If we place the origin at the chiral cen￾ter of 1, for example, and if the bond lengths from the cen￾tral carbon are assumed to be equal (equivalent to requir￾ing r to be constant in a spherical polar coordinate sys￾tem), then two angles, y and f, suffice to locate the first atom of each substituent. Two pairs of such angles exist, corresponding to each enantiomer, which we may label as yAyByCyD, yAyByDyC and fAfBfCfD, fAfBfDfC, where, for example, yA represents the angle y that the group labeled A occupies in Figure 1, and the switching of the coordinates of the last two groups occurs because exchanging any two groups of structure 1 creates its enan￾tiomer. Thus, the spatial information content of chiral car￾bon molecules with a single chiral center is given by eq. 5: H ¼
X 2 1 1 2 log 2 1 2
X 2 1 1 2 log 2 1 2 ¼ 2 ð5Þ For carbon molecules with more than one chiral center, the number of unique sets of coordinates increases as powers of 2, and the information content therefore increases as multiples of 2 (excluding meso forms). For example, a carbon molecule with 4 chiral centers has 24 or 16 possible unique sets of coordinates, and therefore log2(24 ) or 4 bits per coordinate or 8 total bits of spatial information. In other words, the total number of bits is the number of chiral centers times 2. Structure 2 contains no spatial information at all, since every permutation of substituents is superimposable and thus there is only one set of the ys and fs, and 1*log(1) 5 0. The square planar molecule 3 consists of three distinct isomers, corresponding to those with AB, AC, and AD on the diagonal, respectively, and therefore there is spatial in￾formation concerning one dimension only equal to that given by eq. 6: H ¼
X 3 1 1 3 log 2 1 3 ¼ 1:58 ð6Þ For planar molecules such as 3, even if there were many more substituents in the plane, the number of bits of infor￾Fig. 1. Tetrahedral and Square Planar Substitutions. A NEW DEFINITION OF LIFE 355 Chirality DOI 10.1002/chir