one of the tantalizing questions raised by molecular biology is whether the basic structures of life as we know them arose through a darwinian evolutionary process and , if so , what were the evolutionary pressures acting on them ?
the genetic code was initially believed to be universal throughout all living things , even though some variations in both nuclear and mitochondrial systems have recently been found ( see   for a review ) .
these variations are , however , limited and correspond essentially to the reassignment of one or a few codons to another amino acid .
the idea that the genetic code could have evolved to its present form has been repeatedly suggested .
for instance , it has been proposed that early codes were simpler , in that they coded for only a few amino acids , and that the number of amino acids coded in the genetic code increased as the code evolved .
several hypotheses have been put forward to explain the evolution of the genetic code to its present form , and to find out what the genetic code is optimized for .
one possible scenario is that the genetic code evolved so as to minimize the consequence of errors during transcription and translation . to test this hypothesis ,
some researchers have tried to estimate the percentage of optimal achievement of the natural code by quantifying the cost of single - base changes .
more recently , haig and hurst   and freeland and hurst   improved that approach by comparing the natural code with random codes . to this end , they defined a fitness function ,  , that measures the efficiency of the code in limiting the consequences of transcription and translation errors .
this function  supposedly evolved towards a minimum through evolution . to measure how close the natural code is to the actual minimum of  , they generated random genetic codes , and computed the fraction of those that are better - that is , have a smaller value of  - than the natural code .
they found that only a very small fraction of the random codes are better than the natural code , and concluded that the natural code is therefore optimal in that it minimizes the effect of translation and transcription errors .
haig and hurst   tested several fitness functions ,  , based on different physicochemical parameters , and found that single - base changes in the natural code had the smallest average effect when using , as a cost measure , the change in polarity or hydropathy between the corresponding amino acids .
these parameters , although not unique , are clearly biologically relevant , as they are related to hydrophobicity , a property known to be important in protein conformation ( see , for example ,   for reviews ) .
changing , through a transcription or a translation error , a nonpolar amino acid into a polar one at some strategic position in the sequence of a protein can have dramatic consequences on its conformation . using these parameters , and assuming that all point mutations occur with the same frequency ,
haig and hurst   found that the fraction of random codes that beat the natural code is of the order of 10 .
it has been shown experimentally that individual translation errors occur more frequently at the first and third codon positions than at the second , and that there are transition / transversion biases . taking this into account , freeland and hurst   proposed a modified fitness function  , which models more accurately the probability of translation errors .
they found that with this improved modeling , the fraction of random genetic codes that are better than the natural one decreases from 10 to 10 .
they retrieved from their calculations a well known property of the genetic code : single - base substitutions in the first and third codon position are strongly conservative with respect to changes in polarity . here , we highlight the importance of another parameter in the optimization of the genetic code , namely the frequency at which different amino acids occur in proteins .
this frequency differs from protein to protein , and even from species to species , but there is a general pattern that prevails ( table 1 ) . in figure 1
, we have plotted the number of codons coding for the same amino acid ( synonyms ) versus the amino - acid frequency .
the correlation between these two quantities , first noted by king and jukes , led us to suspect that the amino - acid frequency is an important parameter in the optimization of the genetic code , which should also be taken into account in the fitness function . our calculations indeed confirm that the genetic code is even more optimal with respect to translation errors if the amino - acid frequencies of table 1 are properly incorporated in . in addition , we bring further improvements to  by using quantities other than polarity to measure the roles of the different amino acids in protein conformation and stability .
it should be stressed that the biological relevance of the parameters used in  is crucial in the estimation of the relative robustness of the natural code .
indeed , one can always construct an artificial fitness function  such that the natural biological structure apparently lies at its minimum .
clearly , the hydrophobicity parameters used by haig and hurst   are biologically motivated , but we would like to do better by refining our cost measure . in particular , we have devised a mutation matrix describing the average cost of single amino - acid substitutions in protein stability , obtained by computer experiments .
for instance , it takes into account that mutating cysteine into any other amino acid may be very costly as it may break a disulfide bond .
such an effect would not be apparent if only a single property , say hydrophobicity , was taken into account .
we show that , with a fitness function  depending on this mutation matrix and the amino - acid frequencies , only about two out of 10 randomly generated codes are better than the natural code .
this suggests that the genetic code is even better optimized to limit translation errors than was previously thought .
it is built out of 64 codons , each consisting of three consecutive dna bases ( a , g , c , t ) or rna bases ( a , g , c , u ) .
these 64 codons are divided into 21 sets of synonyms , which each code for one of the 20 natural amino acids or correspond to a stop signal ; hence , to each codon , c , an amino acid ( or stop signal ) a is assigned through a function a(c ) .
consider now an error during transcription from dna to rna or during translation from rna to protein , in which codon c is mistaken for codon c ' .
this error thus results in amino acid a(c ) being replaced by amino acid a ' = a(c ' ) .
the associated cost is estimated by a function g(a , a ' ) , which measures the difference between the amino acids a and a ' with respect to their physicochemical properties or their role in ( de)stabilizing protein structures ; when a or a ' corresponds to a stop codon , we set g(a , a ' ) = 0 . different cost functions g will be discussed in the next section .
following freeland and hurst , the fitness  of a code is measured by the average of the cost g over all codons c and all single - base errors c - > c ' where p(c ' | c ) is the probability of misreading codon c as codon c ' .
if one focuses on transcription errors only , as do haig and hurst in , then all p(c'| c ) values must be taken as equal .
but here we consider translation errors , as do free - land and hurst in , and hence p(c ' | c ) changes according to whether c and c ' differ in the first , second or third base , and lead to a transition or a transversion .
a transition is the substitution of a purine ( a , g ) into another purine , or a pyrimidine ( c , u / t ) into another pyrimidine , whereas a transversion interchanges purines and pyrimidines .
on the basis of experimental data indicating that transitions are more common than transversions , and that errors on the third base are more frequent than errors on the first base , which are themselves more frequent than errors on the second base , freeland and hurst   have chosen the following values of p(c ' |c ) , which we also use here : p(c ' | c ) = 1/n if c and c ' differ in the 3rd base only , p(c ' | c ) = 1/n if c and c ' differ in the 1st base only and cause a transition , p(c ' | c ) = 0.5/n if c and c ' differ in the 1st base only and cause a transversion , p(c ' | c ) = 0.5/n if c and c ' differ in the 2nd base only and cause a transition , p(c ' | c ) = 0.1/n if c and c ' differ in the 2nd base only and cause a transversion , p(c ' | c ) = 0 if c and c ' differ by more than 1 base . where n is a normalization factor ensuring that c'p(c ' | c ) = 1 .
obviously , these probabilities only roughly approximate the true transition / transversion and base position biases .
however , the computed fitness of the genetic code has been shown to be relatively insensitive to their precise values .
let us now return to the correlations between the number of codons coding for an amino acid and the frequency of this amino acid ( see figure 1 ) .
king and jukes , who first noted this correlation , suggested that most of the amino acids in genomes have arisen by random mutations that do not affect the properties and function of the proteins .
as a consequence , the number of synonymous codons determines the frequency of amino acids .
an alternative interpretation , assuming a very different chain of causality , is that the amino - acid frequencies are fixed by their physicochemical properties .
for instance , tryptophan would be a rare amino acid because its specific properties are seldom needed in proteins or because it is difficult to synthesize . the correlation between the amino - acid frequencies and number of synonymous codons ( figure 1 ) would then be interpreted as being due to an adjustment of the natural genetic code to the frequency of the amino acids .
independent of the assumed chain of causality , it is natural to expect that a codon error replacing a frequent amino - acid type with another leads to more absolute errors and thus has more consequences , at least on average , than an error affecting a rare amino acid . the frequencies with which the different amino acids occur in proteins , which are similar in different organisms ( table 1 ) , are only imperfectly taken into account in the fitness function  given by equation ( 1 ) , because of the imperfect correlation between amino - acid frequency and number of synonymous codons ( figure 1 ) .
to account properly for the amino - acid frequencies , we propose a modified fitness function  : where p(a ) is the relative frequency of amino acid a , and n(c ) is the number of codons in the block to which c belongs . in other words ,
n(c ) is the number of synonyms coding for the amino acid a(c ) that c codes for .
note that equation ( 2 ) supposes that there is no codon bias , that is , the different synonyms of a given amino acid appear with the same frequency .
to measure the effect of the amino - acid frequency on the value of the fitness function  , we define , for the sake of comparison , another fitness function  where all the amino acids are supposed equally frequent , that is , p(a ) = 1/20 : the function g(a , a ' ) in equations ( 1 ) and ( 2 ) measures the cost - as far as protein stability and structure is concerned - of substituting amino acid a by a ' .
hence , a natural choice for g consists of taking the squared difference in hydrophobicity h of the amino acids a and a ' : g ( a , a ' ) = ( h(a ) - h(a ' ) )   ( 4 ) there exist various hydrophobicity scales for amino acids .
, which is the one used by haig and hurst   and freeland and hurst . in the second scale , h(a ) is the average solvent accessibility of amino acid a derived from a set of 141 well resolved and refined protein structures with low sequence identity ( see methods ) ; solvent accessibilities are computed using survol .
although hydrophobic forces dominate in proteins , other types of interactions also contribute to protein stability ( see   for reviews ) .
we therefore also attempted to devise a better cost function g(a , a ' ) , measuring more accurately the difference between amino acids a and a ' .
this new function is inspired by recent computations of the change in free energy of a protein when a single amino acid is mutated .
it is obtained by mutating in silico , in all proteins of the aforementioned set of 141 protein structures and at all positions , the wild - type amino acids into the 19 other possible ones , and evaluating the resulting changes in folding free energy with mean force potentials derived from the same structure dataset . the matrix elements
m(a , a ' ) are obtained as the average of all the computed folding free - energy changes , which correspond to a substitution a  a ' .
details on the procedure and the value of the matrix elements m(a , a ' ) are given in the methods section .
this matrix is taken as a cost function : g ( a , a ' ) = m(a , a ' )   ( 5 ) for the purpose of comparing g with a reference matrix that exclusively reflects the structure of the genetic code , we define the cost function : g ( a , a ' ) = 3 - (a , a ' )   ( 6 ) where  ( a , a ' ) is zero when a and a ' coincide , and otherwise equal to the minimum number of bases that must be changed to transform a codon coding for a into a codon coding for a ' . as a last cost function ,
used only for comparison with earlier work , we consider the point accepted mutations 74 - 100 ( pam74 - 100 ) substitution matrix , one of the most commonly used matrices in the context of protein - sequence alignment : g ( a , a ' ) = pam74 - 100 ( a , a ' )   ( 7 ) this matrix is derived from the pattern of amino - acid substitution frequencies observed within naturally occurring pairs of highly diverged homologous protein sequences .
however , the use of this matrix for measuring the genetic code 's fitness   has been criticized , as matrices derived from substitution patterns observed in homologous proteins reflect not only the similarity between amino acids with respect to their physicochemical and energetic properties , but also the facility with which one amino acid is mutated into another and thus their proximity in the genetic code . however , as the pam74 - 100 matrix is derived from highly diverged protein sequences , this effect can be expected to be relatively limited .
this is indeed the case , as the linear correlation coefficient between the nondiagonal entries of g and g is only 0.43 .
nevertheless , the correlation coefficient between g and g is still much lower , namely 0.19 , so that g may in no way be suspected to include information on proximity in the genetic code .
finally , note that the correlation coefficient between g and g is equal to 0.60 ; these matrices thus share common features but contain also different information .
this is not surprising considering that their derivations use very different starting points ( protein three - dimensional structures in one case , sequence similarity in the other ) . to evaluate the robustness of the natural genetic code with respect to translation errors ,
we computed the fitness functions  ,  and  using equations ( 1 ) to ( 3 ) for the natural genetic code , and compared it to the corresponding fitnesses of random codes . the random codes are obtained by maintaining the codon block structure of the natural genetic code , where each block corresponds to synonyms coding for the same amino acid ( or stop signal ) . when generating a random code , the stop signal is kept assigned to the same block as in the natural genetic code , whereas the different amino acids are randomly interchanged among the 20 remaining blocks
. thus , each random code is simply specified by a different function a(c ) in equations ( 1 ) to ( 3 ) .
this is the procedure previously used by haig and hurst   and freeland and hurst .
thus , in a first stage , we computed the fitness functions  and  for the natural genetic code and for 10 randomly generated codes , using the three cost functions g , g and g. we then calculated the fraction f of random codes whose value of  is lower than that of the natural code .
this fraction is supposedly a good estimate of the relative merit of the natural genetic code comparative to other codes .
it appears that this fraction f is always smaller for  than for . this is especially true for the cost function g where f is 300 times smaller .
this result indicates that the natural code appears to be better optimized with respect to translation errors if the amino - acid frequencies are taken into account . to investigate this further ,
we have analyzed which of the cost functions g , g or g genetic code appears to be best optimized for .
we compared the fraction f of better codes for each of the cost functions using the fitness function . for the hydrophobicity functions g and g , the result is roughly the same : f is about 0.5 - 1.0 in 10 .
the relative statistical error on this value is of the order of n , where n is the number of random codes better than the natural one that were found in our sample of 10 random codes ; thus , n is about 650 - 1,200 , and the error is insignificant . for the mutational cost function g
this result shows that the natural genetic code appears even more optimal if the cost function g is used than if hydrophobicity - based cost functions are considered . as g has been computed from protein stability changes effected by point mutations
, we may conclude that the genetic code is optimized in such a way as to limit the effect of translation errors on the three - dimensional structure and stability of the coded proteins .
note that the improvement brought by the choice of g results from the fact that it probably better accounts for the cost of a mutation than a mere difference of hydrophobicity ; for example , glycine , proline and cysteine have close neighbors in hydrophobicity , whereas the cost of their mutation as accounted for by g is high .
this is due to their special role in determining protein structure : glycine and proline can adopt backbone torsion angles essentially inaccessible to other amino acids , and cysteine can form disulfide bonds . to check the significance of this result ,
we have computed the fraction f of random codes that beat the natural one for random choices of the amino - acid frequencies , distinct from the natural frequencies p(a ) .
we have generated 10 sets of random p(a ) values , and , for each of them , estimated the fraction f ( out of a sample of 10 random codes ) .
the percentage of random amino - acid frequency sets that result in a lower fraction f than the natural frequencies is shown in table 3 .
we find that a random assignment of the amino - acid frequencies does not decrease / in most ( at least 97% ) of the cases , and this tendency persists for all cost functions g. thus , the probability that the decrease of f , observed in table 2 , when passing from  to  , was due to chance is quite limited .
we may therefore conclude that the genetic code is optimized so as to take into account the natural amino - acid frequencies .
we also investigated whether the result that the natural code is better optimized if amino - acid frequencies are taken into account does not depend crucially on the amino - acid frequencies used . for this purpose , we calculated the fraction f of random codes with lower  value than the genetic code for the four sets of amino - acid frequencies listed in table 1 , which are computed from genomes of eukaryotes , archaea , bacteria and from all these genomes together .
the results turn out to be essentially insensitive to the chosen frequency set : the fractions f differ at most by a factor of two , which leaves all conclusions unchanged .
we have also included in table 2 the results based on the fitness function . it can be argued that this function partly takes , but imperfectly , the amino - acid frequencies into account .
indeed , for this fitness function each codon is assigned the same weight , which corresponds to each amino acid being assigned a frequency proportional to the number of synonyms n(a ) coding for it . in the case of the natural genetic code , this frequency corresponds approximately to the amino - acid frequency as there is a correlation between n(a ) and p(a ) , as shown in figure 1 .
but for random codes , where the amino acids are randomly interchanged between the codon blocks , this correspondence breaks down .
thus , the way in which  takes amino - acid frequencies into account depends on the code considered .
this explains why the fraction f of random codes better than the natural one is roughly of the same order using  and . note that f is always larger for than for  , indicating again the importance of the amino acid frequencies in the optimality of the genetic code . for sake of comparison
, we have added in table 2 the values of the fraction f of random codes with a lower  value than the natural one , using the cost functions g and g , which include information about the structure of the genetic code . with g
, the fraction f is the same as with g , whereas with g , we did not find any random code better than the natural one among the 10 random codes tested . to estimate f without having to generate a larger ensemble
we computed , from the values of  for the 10 random codes , the probability function  (  ) to have a given value of . we fitted log( (  ) ) to a polynomial of fourth degree , and extrapolated this curve down to the value of  for the natural code .
this provides an estimate of the fraction f of random codes that have a lower  value .
we found with this procedure that / is of the order of 10 with g , and thus about 10 times smaller than with g andg .
it is not surprising that the fraction f of random codes that does better than the natural code is extremely small for g , as this matrix exclusively reflects the proximity of amino acids in the genetic code and renders the issue of the code 's fitness tautologous .
it can indeed be argued that g contains information on the proximity of amino acids in the genetic code , superimposed on the desired measure of their similarity in preserving protein structure , because it is computed from amino - acid substitutions in families of evolutionarily related proteins , which are more frequent between amino acids that are closer in the genetic code .
the fact that g and g yield similar f values ( see table 2 ) can be taken to indicate that this is not the case , and thus that both these cost functions can reliably be used to estimate the code 's fitness against translation errors .
it could , however , also be argued that g describes protein structure less well than g and includes somewhat more information about the genetic code ( as monitored by correlation coefficients of 0.43 and 0.19 of gand g with respect to g ) .
these two effects tend to compensate each other , and could be expected to yield similar f values for g and g. it therefore seems safer to use g as cost function , because , owing to its very definition , it seems to capture important structural information and to be independent of the code 's structure .
we also investigated how optimal the genetic code is with respect to amino - acid interchanges that do not affect codon degeneracy .
we exhaustively generated all alternative codes that preserve the amino - acid degeneracy and computed the fraction f of these codes that do better than the natural code with respect to mistranslation .
we found that f is of the order of 10 for the three fitness functions  ,  and  ( table 4 ) .
it is thus similar to the f value computed on the unrestricted set of alternative codes for  and  , and much larger for . this result simply reflects the correlation between the codon degeneracy and the amino - acid frequency ( figure 1 ) .
indeed , this correlation implies that a much larger proportion of the better codes maintain the degeneracy , if the frequency of the amino acids is taken into account in the fitness function , as in . in contrast , in  and  the amino - acid frequency is not considered and f is of the same order with the restricted and unrestricted sets .
it has been proposed that the genetic code has evolved from a simpler ancestral code , encoding only a few amino acids present at early times , and that new amino acids appearing as biosynthetic derivatives of the original ones were incorporated by subdivision and reassignment of their synonymous codons .
this so - called coevolution hypothesis is supported by the observation that biosynthetically related amino acids are close within the genetic code . to investigate the optimality of the genetic code in the coevolution framework
, we computed the fraction f of alternative codes that perform better than the natural code against translation errors and that differ from the natural code by shuffling amino acids belonging to the same biosynthetic pathway .
we found that f is equal to 2.9  10 , whereas it is equal to 2  10 for the unrestricted set allowing all shufflings ( table 4 ) .
this means that the fraction f of better codes is somewhat larger in the biosynthesis - restricted set than in the complete set , and thus that the optimality rate is slightly lower .
this result can also be viewed differently . when considering the total number of codes in the two sets ( which are of the order of 10 and 10 ) , it means that there are only six codes that beat the natural code in the restricted set , whereas there are 10 such codes in the unrestricted set .
this is particularly striking given that our definitions of cost functions and sets of biosynthetically related amino acids only constitute approximations .
we can thus conclude that the genetic code is quite robust against mistranslation in the space of all alternative codes , and is close to being fully optimal if historical biosynthetic constraints are taken into account .
a complementary measure of the optimality of the genetic code is its percentage of optimization , defined as 100%(code - mean)/(best - mean ) , where code is the fitness of the genetic code , best fitness of the best of all possible codes and mean the average fitness over all codes .
this measure indicates how close the fitness value of the genetic code is to the fitness value of the optimal code .
note however that this optimality measure has no absolute meaning and may not be compared among fitness functions  defined on the basis of different cost functions g ; to illustrate this , consider the following example : if g is defined by |h(a)-h(a')| instead of ( h(a)-h(a ' ) ) ( see equation ( 4 ) ) , the fraction f of better codes remains unchanged but the percentage of optimality does change .
it is , however , meaningful to compare the percentage of optimization of the genetic code in the unrestricted and biosynthesis restricted sets with a same g function . for  , we find that this percentage is equal to 97% and 98% in the two sets , respectively .
this indicates that the fitness value of the genetic code is not very far from that of the best possible code , whether focusing on the subset of alternative codes preserving biosynthetic proximities , or considering all possible codes .
it is built out of 64 codons , each consisting of three consecutive dna bases ( a , g , c , t ) or rna bases ( a , g , c , u ) .
these 64 codons are divided into 21 sets of synonyms , which each code for one of the 20 natural amino acids or correspond to a stop signal ; hence , to each codon , c , an amino acid ( or stop signal ) a is assigned through a function a(c ) .
consider now an error during transcription from dna to rna or during translation from rna to protein , in which codon c is mistaken for codon c ' .
this error thus results in amino acid a(c ) being replaced by amino acid a ' = a(c ' ) .
the associated cost is estimated by a function g(a , a ' ) , which measures the difference between the amino acids a and a ' with respect to their physicochemical properties or their role in ( de)stabilizing protein structures ; when a or a ' corresponds to a stop codon , we set g(a , a ' ) = 0 . different cost functions g will be discussed in the next section .
following freeland and hurst , the fitness  of a code is measured by the average of the cost g over all codons c and all single - base errors c - > c ' where p(c ' | c ) is the probability of misreading codon c as codon c ' .
if one focuses on transcription errors only , as do haig and hurst in , then all p(c'| c ) values must be taken as equal .
but here we consider translation errors , as do free - land and hurst in , and hence p(c ' | c ) changes according to whether c and c ' differ in the first , second or third base , and lead to a transition or a transversion .
a transition is the substitution of a purine ( a , g ) into another purine , or a pyrimidine ( c , u / t ) into another pyrimidine , whereas a transversion interchanges purines and pyrimidines .
on the basis of experimental data indicating that transitions are more common than transversions , and that errors on the third base are more frequent than errors on the first base , which are themselves more frequent than errors on the second base , freeland and hurst   have chosen the following values of p(c ' |c ) , which we also use here : p(c ' | c ) = 1/n if c and c ' differ in the 3rd base only , p(c ' | c ) = 1/n if c and c ' differ in the 1st base only and cause a transition , p(c ' | c ) = 0.5/n if c and c ' differ in the 1st base only and cause a transversion , p(c ' | c ) = 0.5/n if c and c ' differ in the 2nd base only and cause a transition , p(c ' | c ) = 0.1/n if c and c ' differ in the 2nd base only and cause a transversion , p(c ' | c ) = 0 if c and c ' differ by more than 1 base . where n is a normalization factor ensuring that c'p(c ' | c ) = 1 .
obviously , these probabilities only roughly approximate the true transition / transversion and base position biases .
however , the computed fitness of the genetic code has been shown to be relatively insensitive to their precise values .
let us now return to the correlations between the number of codons coding for an amino acid and the frequency of this amino acid ( see figure 1 ) .
king and jukes , who first noted this correlation , suggested that most of the amino acids in genomes have arisen by random mutations that do not affect the properties and function of the proteins . as a consequence ,
an alternative interpretation , assuming a very different chain of causality , is that the amino - acid frequencies are fixed by their physicochemical properties .
for instance , tryptophan would be a rare amino acid because its specific properties are seldom needed in proteins or because it is difficult to synthesize .
the correlation between the amino - acid frequencies and number of synonymous codons ( figure 1 ) would then be interpreted as being due to an adjustment of the natural genetic code to the frequency of the amino acids .
independent of the assumed chain of causality , it is natural to expect that a codon error replacing a frequent amino - acid type with another leads to more absolute errors and thus has more consequences , at least on average , than an error affecting a rare amino acid . the frequencies with which the different amino acids occur in proteins , which are similar in different organisms ( table 1 ) , are only imperfectly taken into account in the fitness function  given by equation ( 1 ) , because of the imperfect correlation between amino - acid frequency and number of synonymous codons ( figure 1 ) .
to account properly for the amino - acid frequencies , we propose a modified fitness function  : where p(a ) is the relative frequency of amino acid a , and n(c ) is the number of codons in the block to which c belongs .
in other words , n(c ) is the number of synonyms coding for the amino acid a(c ) that c codes for .
note that equation ( 2 ) supposes that there is no codon bias , that is , the different synonyms of a given amino acid appear with the same frequency . to measure the effect of the amino - acid frequency on the value of the fitness function 
, we define , for the sake of comparison , another fitness function  where all the amino acids are supposed equally frequent , that is , p(a ) = 1/20 :
the function g(a , a ' ) in equations ( 1 ) and ( 2 ) measures the cost - as far as protein stability and structure is concerned - of substituting amino acid a by a ' .
hence , a natural choice for g consists of taking the squared difference in hydrophobicity h of the amino acids a and a ' : g ( a , a ' ) = ( h(a ) - h(a ' ) )   ( 4 ) there exist various hydrophobicity scales for amino acids .
, which is the one used by haig and hurst   and freeland and hurst . in the second scale , h(a ) is the average solvent accessibility of amino acid a derived from a set of 141 well resolved and refined protein structures with low sequence identity ( see methods ) ; solvent accessibilities are computed using survol .
although hydrophobic forces dominate in proteins , other types of interactions also contribute to protein stability ( see   for reviews ) .
we therefore also attempted to devise a better cost function g(a , a ' ) , measuring more accurately the difference between amino acids a and a ' .
this new function is inspired by recent computations of the change in free energy of a protein when a single amino acid is mutated .
it is obtained by mutating in silico , in all proteins of the aforementioned set of 141 protein structures and at all positions , the wild - type amino acids into the 19 other possible ones , and evaluating the resulting changes in folding free energy with mean force potentials derived from the same structure dataset . the matrix elements
m(a , a ' ) are obtained as the average of all the computed folding free - energy changes , which correspond to a substitution a  a ' .
details on the procedure and the value of the matrix elements m(a , a ' ) are given in the methods section .
this matrix is taken as a cost function : g ( a , a ' ) = m(a , a ' )   ( 5 ) for the purpose of comparing g with a reference matrix that exclusively reflects the structure of the genetic code , we define the cost function : g ( a , a ' ) = 3 - (a , a ' )   ( 6 ) where  ( a , a ' ) is zero when a and a ' coincide , and otherwise equal to the minimum number of bases that must be changed to transform a codon coding for a into a codon coding for a ' . as a last cost function ,
used only for comparison with earlier work , we consider the point accepted mutations 74 - 100 ( pam74 - 100 ) substitution matrix , one of the most commonly used matrices in the context of protein - sequence alignment : g ( a , a ' ) = pam74 - 100 ( a , a ' )   ( 7 ) this matrix is derived from the pattern of amino - acid substitution frequencies observed within naturally occurring pairs of highly diverged homologous protein sequences .
however , the use of this matrix for measuring the genetic code 's fitness   has been criticized , as matrices derived from substitution patterns observed in homologous proteins reflect not only the similarity between amino acids with respect to their physicochemical and energetic properties , but also the facility with which one amino acid is mutated into another and thus their proximity in the genetic code . however , as the pam74 - 100 matrix is derived from highly diverged protein sequences , this effect can be expected to be relatively limited .
this is indeed the case , as the linear correlation coefficient between the nondiagonal entries of g and g is only 0.43 .
nevertheless , the correlation coefficient between g and g is still much lower , namely 0.19 , so that g may in no way be suspected to include information on proximity in the genetic code .
finally , note that the correlation coefficient between g and g is equal to 0.60 ; these matrices thus share common features but contain also different information .
this is not surprising considering that their derivations use very different starting points ( protein three - dimensional structures in one case , sequence similarity in the other ) .
to evaluate the robustness of the natural genetic code with respect to translation errors , we computed the fitness functions  ,  and  using equations ( 1 ) to ( 3 ) for the natural genetic code , and compared it to the corresponding fitnesses of random codes .
the random codes are obtained by maintaining the codon block structure of the natural genetic code , where each block corresponds to synonyms coding for the same amino acid ( or stop signal ) . when generating a random code , the stop signal is kept assigned to the same block as in the natural genetic code , whereas the different amino acids are randomly interchanged among the 20 remaining blocks .
thus , each random code is simply specified by a different function a(c ) in equations ( 1 ) to ( 3 ) .
this is the procedure previously used by haig and hurst   and freeland and hurst .
thus , in a first stage , we computed the fitness functions  and  for the natural genetic code and for 10 randomly generated codes , using the three cost functions g , g and g. we then calculated the fraction f of random codes whose value of  is lower than that of the natural code .
this fraction is supposedly a good estimate of the relative merit of the natural genetic code comparative to other codes .
it appears that this fraction f is always smaller for  than for . this is especially true for the cost function g where f is 300 times smaller .
this result indicates that the natural code appears to be better optimized with respect to translation errors if the amino - acid frequencies are taken into account . to investigate this further ,
we have analyzed which of the cost functions g , g or g genetic code appears to be best optimized for .
we compared the fraction f of better codes for each of the cost functions using the fitness function . for the hydrophobicity functions g and g , the result is roughly the same : f is about 0.5 - 1.0 in 10 .
the relative statistical error on this value is of the order of n , where n is the number of random codes better than the natural one that were found in our sample of 10 random codes ; thus , n is about 650 - 1,200 , and the error is insignificant . for the mutational cost function g , f is several orders of magnitude lower , namely 2 in 10 .
this result shows that the natural genetic code appears even more optimal if the cost function g is used than if hydrophobicity - based cost functions are considered . as g has been computed from protein stability changes effected by point mutations
, we may conclude that the genetic code is optimized in such a way as to limit the effect of translation errors on the three - dimensional structure and stability of the coded proteins .
note that the improvement brought by the choice of g results from the fact that it probably better accounts for the cost of a mutation than a mere difference of hydrophobicity ; for example , glycine , proline and cysteine have close neighbors in hydrophobicity , whereas the cost of their mutation as accounted for by g is high .
this is due to their special role in determining protein structure : glycine and proline can adopt backbone torsion angles essentially inaccessible to other amino acids , and cysteine can form disulfide bonds . to check the significance of this result ,
we have computed the fraction f of random codes that beat the natural one for random choices of the amino - acid frequencies , distinct from the natural frequencies p(a ) .
we have generated 10 sets of random p(a ) values , and , for each of them , estimated the fraction f ( out of a sample of 10 random codes ) .
the percentage of random amino - acid frequency sets that result in a lower fraction f than the natural frequencies is shown in table 3 .
we find that a random assignment of the amino - acid frequencies does not decrease / in most ( at least 97% ) of the cases , and this tendency persists for all cost functions g. thus , the probability that the decrease of f , observed in table 2 , when passing from  to  , was due to chance is quite limited .
we may therefore conclude that the genetic code is optimized so as to take into account the natural amino - acid frequencies .
we also investigated whether the result that the natural code is better optimized if amino - acid frequencies are taken into account does not depend crucially on the amino - acid frequencies used . for this purpose , we calculated the fraction f of random codes with lower  value than the genetic code for the four sets of amino - acid frequencies listed in table 1 , which are computed from genomes of eukaryotes , archaea , bacteria and from all these genomes together .
the results turn out to be essentially insensitive to the chosen frequency set : the fractions f differ at most by a factor of two , which leaves all conclusions unchanged .
we have also included in table 2 the results based on the fitness function . it can be argued that this function partly takes , but imperfectly , the amino - acid frequencies into account .
indeed , for this fitness function each codon is assigned the same weight , which corresponds to each amino acid being assigned a frequency proportional to the number of synonyms n(a ) coding for it . in the case of the natural genetic code
, this frequency corresponds approximately to the amino - acid frequency as there is a correlation between n(a ) and p(a ) , as shown in figure 1 .
but for random codes , where the amino acids are randomly interchanged between the codon blocks , this correspondence breaks down .
thus , the way in which  takes amino - acid frequencies into account depends on the code considered .
this explains why the fraction f of random codes better than the natural one is roughly of the same order using  and . note that f is always larger for than for  , indicating again the importance of the amino acid frequencies in the optimality of the genetic code . for sake of comparison
, we have added in table 2 the values of the fraction f of random codes with a lower  value than the natural one , using the cost functions g and g , which include information about the structure of the genetic code . with g
, the fraction f is the same as with g , whereas with g , we did not find any random code better than the natural one among the 10 random codes tested . to estimate f without having to generate a larger ensemble
we computed , from the values of  for the 10 random codes , the probability function  (  ) to have a given value of . we fitted log( (  ) ) to a polynomial of fourth degree , and extrapolated this curve down to the value of  for the natural code .
this provides an estimate of the fraction f of random codes that have a lower  value .
we found with this procedure that / is of the order of 10 with g , and thus about 10 times smaller than with g andg .
it is not surprising that the fraction f of random codes that does better than the natural code is extremely small for g , as this matrix exclusively reflects the proximity of amino acids in the genetic code and renders the issue of the code 's fitness tautologous .
it can indeed be argued that g contains information on the proximity of amino acids in the genetic code , superimposed on the desired measure of their similarity in preserving protein structure , because it is computed from amino - acid substitutions in families of evolutionarily related proteins , which are more frequent between amino acids that are closer in the genetic code .
the fact that g and g yield similar f values ( see table 2 ) can be taken to indicate that this is not the case , and thus that both these cost functions can reliably be used to estimate the code 's fitness against translation errors .
it could , however , also be argued that g describes protein structure less well than g and includes somewhat more information about the genetic code ( as monitored by correlation coefficients of 0.43 and 0.19 of gand g with respect to g ) .
these two effects tend to compensate each other , and could be expected to yield similar f values for g and g. it therefore seems safer to use g as cost function , because , owing to its very definition , it seems to capture important structural information and to be independent of the code 's structure .
we also investigated how optimal the genetic code is with respect to amino - acid interchanges that do not affect codon degeneracy .
we exhaustively generated all alternative codes that preserve the amino - acid degeneracy and computed the fraction f of these codes that do better than the natural code with respect to mistranslation .
we found that f is of the order of 10 for the three fitness functions  ,  and  ( table 4 ) .
it is thus similar to the f value computed on the unrestricted set of alternative codes for  and  , and much larger for . this result simply reflects the correlation between the codon degeneracy and the amino - acid frequency ( figure 1 ) .
indeed , this correlation implies that a much larger proportion of the better codes maintain the degeneracy , if the frequency of the amino acids is taken into account in the fitness function , as in . in contrast , in  and  the amino - acid frequency is not considered and f is of the same order with the restricted and unrestricted sets . it has been proposed that the genetic code has evolved from a simpler ancestral code , encoding only a few amino acids present at early times , and that new amino acids appearing as biosynthetic derivatives of the original ones were incorporated by subdivision and reassignment of their synonymous codons . this so - called coevolution hypothesis is supported by the observation that biosynthetically related amino acids are close within the genetic code . to investigate the optimality of the genetic code in the coevolution framework
, we computed the fraction f of alternative codes that perform better than the natural code against translation errors and that differ from the natural code by shuffling amino acids belonging to the same biosynthetic pathway .
we found that f is equal to 2.9  10 , whereas it is equal to 2  10 for the unrestricted set allowing all shufflings ( table 4 ) .
this means that the fraction f of better codes is somewhat larger in the biosynthesis - restricted set than in the complete set , and thus that the optimality rate is slightly lower .
this result can also be viewed differently . when considering the total number of codes in the two sets ( which are of the order of 10 and 10 ) , it means that there are only six codes that beat the natural code in the restricted set , whereas there are 10 such codes in the unrestricted set .
this is particularly striking given that our definitions of cost functions and sets of biosynthetically related amino acids only constitute approximations .
we can thus conclude that the genetic code is quite robust against mistranslation in the space of all alternative codes , and is close to being fully optimal if historical biosynthetic constraints are taken into account .
a complementary measure of the optimality of the genetic code is its percentage of optimization , defined as 100%(code - mean)/(best - mean ) , where code is the fitness of the genetic code , best fitness of the best of all possible codes and mean the average fitness over all codes .
this measure indicates how close the fitness value of the genetic code is to the fitness value of the optimal code .
note however that this optimality measure has no absolute meaning and may not be compared among fitness functions  defined on the basis of different cost functions g ; to illustrate this , consider the following example : if g is defined by |h(a)-h(a')| instead of ( h(a)-h(a ' ) ) ( see equation ( 4 ) ) , the fraction f of better codes remains unchanged but the percentage of optimality does change .
it is , however , meaningful to compare the percentage of optimization of the genetic code in the unrestricted and biosynthesis restricted sets with a same g function . for 
, we find that this percentage is equal to 97% and 98% in the two sets , respectively .
this indicates that the fitness value of the genetic code is not very far from that of the best possible code , whether focusing on the subset of alternative codes preserving biosynthetic proximities , or considering all possible codes .
our results confirm and specify those of freeland and hurst : the genetic code seems structured so as to minimize the consequences of translation errors on the three - dimensional structure and stability of the coded proteins .
we have shown that , using the cost function g which best reflects the roles of various amino acids in protein structures , and taking amino - acid frequencies into account , about 2 out of 10 random codes do better than the natural code .
 2  10 possible codes preserving the codon block structure , which means that we can expect about 10 better codes overall . moreover , if the codon block structure is not preserved , the number of possible codes is larger by orders of magnitude , and therefore the number of codes better than the natural one will certainly be much larger .
however , if we preserved the block structure and in addition restricted the space of alternative codes by interchanging only amino acids belonging to the same biosynthetic pathway , we found that there are only six codes performing better than the natural code .
the genetic code thus seems quite robust with respect to mistranslation compared to alternative codes , and almost fully optimal if the constraint is imposed that biosynthetically related amino acids are encoded in codons that are close within the genetic code .
this does not prove , but is in agreement with , the coevolution hypothesis , which assumes that the genetic code has evolved from an simpler ancestral code of only a few amino acids , by subdivision and reassignment of synonymous codons , and that the present genetic code has kept imprints of this evolution .
so we can assert from our analysis that the genetic code has been optimized through evolution up to a certain point , even though it is probably not fully optimal , at least with respect to the parameters considered here , except perhaps if historical , biosynthesis - related , constraints are imposed .
our analysis does not , however , give us much information about the mechanism of this evolution as there is unfortunately no trace left of evolution of the code or amino - acid frequencies in early times .
for instance , we do not know whether the relative frequency of occurrence of amino acids in proteins adapted so as to increase the optimality of the genetic code with respect to translation errors , or , on the contrary , whether the genetic code evolved to take into account pre - existing amino - acid frequencies .
we can , however , argue that if the amino - acid frequencies adapted to the genetic code , as assumed by king and jukes , a discrepancy in amino - acid composition between frequently and infrequently expressed genes might be detectable today ( unless the period during which evolution took place was long enough for this discrepancy to vanish ) . if , alternatively , the genetic code adapted to the amino - acid frequencies , and thus if these frequencies acted as an evolutionary pressure , one can imagine two scenarios .
either the code optimized to take into account the prebiotic frequencies of the amino acids that became involved in it , or it optimized for the amino - acid frequencies of already formed proteins ( or of a subset of them ) that were important for life and maybe linked to the code 's control .
perhaps can we assume , more realistically , that the genetic code and amino - acid frequencies evolved together during some evolutionary period , thereby approaching an optimal code / amino - acid relation .
more generally , the parameters that acted as evolutionary pressure on the genetic code probably included all the mechanisms that encode and maintain the genetic information , and were not just restricted to the frequency of amino acids and the preservation of protein structure .
for example , the genetic code is obviously related to the translation apparatus , composed of the ribosomes and trnas , whose action we described schematically here by the probabilities p(c ' | c ) to misread codon c as c ' .
all these mechanisms probably evolved together with the genetic code during the early stages of life .
although the code still evolves today , as reflected by its departure from universality in some organisms , its evolution is very limited and concerns only the reassignment of a few codons .
as the same change sometimes recurs in different lineages , the code seems to have reached the bottom of a funnel in the evolutionary landscape that contains several roughly equivalent optimal codes .
but apart from such restricted modifications , the code no longer evolves significantly , and has not undergone important modifications since an early stage in the development of life .
this stability probably arose because even small modifications in the code would have entailed loss of functionality of genes that were already being expressed .
moreover , the advent of more sophisticated transcription / translation control mechanisms , which involve huge protein systems , could have decreased the evolutionary pressure on the genetic code . even though our present information on the genetic code is insufficient to discriminate between evolutionary scenarios , our analysis enables us to put some constraints on the situation at the time when evolution of the code was pretty much frozen . in particular
, it appears that the frequencies of the amino acids that were used in proteins synthesized at that time were similar to the present frequencies .
we do not know what determines the present amino - acid frequencies , but presumably they result , at least in part , from the amino acids ' physicochemical properties .
for instance , the ratio of hydrophobic to hydrophilic amino acids is intrinsically related to the globular structure of proteins and certainly contributes to the pressure on amino - acid frequencies .
thus , we can assert that some of the pressures that determine the present amino - acid frequencies were already present at the time the code took on its definitive form .
in addition , the increased optimality of the genetic code with respect to g implies that the three - dimensional structure of proteins probably played an equally important role in fixing the structure of the code .
as the three - dimensional structure of a protein essentially determines its function , this suggests , more generally , that the protein function acted as a main evolutionary pressure on the code structure .
consequently , at the time when the genetic code took its present form , primitive life was presumably already synthesizing complex proteins .
this provides a tentative picture of primitive life at that time : the translation apparatus was similar to the present one , and organisms were made of complex proteins whose amino - acid frequencies were comparable to the present ones .
the derivation is based on a dataset of 141 high - resolution protein structures determined by x - ray crystallography and listed in . to avoid bias ,
these 141 proteins are chosen to present either less than 20% sequence identity or less than 25% sequence identity and no structural similarity .
the protein main chains are described by their heavy atoms , and each side chain is represented by a pseudo - atom c. for a given amino - acid type , the c has a well defined position relative to the main chain , corresponding to the geometric average of all heavy side - chain atoms of this type in the dataset ; for glycine , the c pseudo - atom is positioned on the c. side - chain degrees of freedom are thus neglected . each residue , at each position of each of the 141 proteins , is mutated in turn into the 19 non - wild - type amino acids .
the mutations are made by keeping the main - chain structure unchanged , and substituting the c of the given amino acid by that of the mutant amino acid . for each of these mutations ,
the change in folding free energy is evaluated using the database - derived potentials and the procedure detailed below . for each substitution of amino acid
a into a ' , the average of all computed changes in folding free energy , at all protein positions , is computed and defined as minus the matrix element m(a , a ' ) .
we then symmetrize m by setting m(a , a ' ) = [ m(a , a')+m(a',a)]/2 and only consider the lower half of m(a < a ' ) .
this procedure does not define the diagonal elements of m. on the basis of the principle that the structural role of a given amino acid is fulfilled by no other amino acid better than by itself , we assign to all the diagonal element the same maximum value : m(a , a ) = max [ m(a',a " ) ] + 1 . then
, to simplify m without modifying its structure , we center it around its mean value : finally , we multiply all matrix elements m(a , a ' ) by 2 and replace them by the closest integer .
the potentials we use to evaluate the protein conformations are derived from observed frequencies of sequence and structure patterns in the aforementioned dataset of 141 proteins .
we consider two types of potentials , called torsion   and c -c   potentials .
they take into account the propensities of single residues and residue pairs to be associated with a (  ,  ,  ) backbone torsion angle domain .
we use two variants of the torsion potential , called torsionshort - range and torsionmiddle - range . both
are computed from propensities of a (  ,  ,  ) domain ti , at position i along the sequence , or pairs of domains ( ti , tj ) , at positions i and j , to be associated with an amino acid ak at position k. but we have k - 1  i , j  k + 1 for the torsionshort - range potential and k - 8  i , j  k + 8 for the torsionmiddle - range potential .
the folding free energy  g(s , c ) of a sequence s in the conformation c computed from these propensities is expressed as : where p are normalized frequencies , n is the number of residues in the sequence s , k is the boltzmann constant and t is a conformational temperature taken to be room temperature .
the normalization factor k ensures that the contribution to g(s , c ) of each residue in the window [ k - 1 , k + 1 ] for the torsionshort - range potential or [ k - 8 , k + 8 ] for the torsionmiddle - range potential is counted once .
they are based on propensities of pairs of amino acids ( ai , aj ) at position i and j along the sequence to be separated by a spatial distance dij , calculated between the pseudo - atoms c. we consider two variants of c - c potentials .
the first one , called c - clong_range potential , describes purely nonlocal interactions along the sequence , and only takes into account residues separated by at least 15 residues along the sequence , that is j  i + 16 .
the second one , simply called c - c potential , though dominated by nonlocal interactions , possesses a local interaction component .
the nonlocal component is obtained by considering together the frequencies of all residues separated by seven sequence positions and more , thus with j  i + 8 .
the local component is obtained by computing separately the frequencies of residues separated by one to six positions along the sequence , for i + 1 < j < i + 8 .
the folding free energies are expressed as : with j  i + 16 and the normalized frequencies p independent of j - i for the c - clong_range potential , and i + 1 < j and the normalized frequencies p independent of j - i for j  i + 8 for the c - c potential .
the discretization of the spatial distances dij is performed by dividing the distances between 3 and 8  into 25 bins of 0.2  width and merging the distances greater than 8  . to increase the reliability of the statistics ,
these bins are smoothed by combining the counts in each bin with those of the 10 flanking bins at each side .
the predominance of the central bin is preserved by weighting the counts from each flanking bin by a factor 1/n , where n is the position relative to the central bin ; n is equal to 1 for the two closest bins and to 10 for the two most distant bins .
the so - defined folding free energies are reliable for common amino acids and structure motifs , but not for less common ones . to correct for the sparse data , we substitute the sequence - specific frequencies p(c , s ) , where s denotes a sequence pattern and c a structure motif , which appear in expressions ( 9 ) and ( 10 ) immediately above defining the torsion and c - c folding free energies , by a linear combination of these frequencies and the product of the separate frequencies of s and c , denoted p(s ) and p(c ) respectively : where m is the number of occurrences of the sequence pattern s in the dataset , and  a parameter .
this expression ensures that the sequence - specific contribution dominates for rare sequence patterns and tends to zero for frequent ones .
this behavior is modulated by the parameter  , which we consider here equal to 50 . to estimate the stability changes caused by a single - site mutation , we compute the folding free energy changes as : where cm and cw are the mutant and wild - type conformations and sm and sw the mutant and wild - type sequences , respectively . with this convention ,
g is positive when the mutation is destabilizing , and negative when it is stabilizing .
the conformations cm and cw of the mutant and wild - type protein are assumed to be nearly identical .
more precisely , the backbone conformations are taken as identical and only the position of the c pseudo - atom , which is amino - acid dependent , is different in the mutant and wild - type structures .
the folding free energies of the wild - type and mutant proteins are computed with linear combinations of the torsion and c - c potentials described in the previous section .
previous analyses   have shown that the combination that gives the best evaluation of the g values depends on the solvent accessibility , a of the mutated residue ; a is defined as the solvent - accessible surface in the protein structure , computed by survol , multiplied by 100 and divided by its solvent - accessible surface in an extended tripeptide gly - x - gly .
when the mutated residue is at the surface , with a solvent accessibility a equal to or larger than 50% , the optimal folding free - energy change has been shown to be equal to : when the mutated residue is half buried , half exposed to the solvent , with a solvent accessibility between 20 and 40% , the optimal folding free energy is : finally , when the mutated residue is totally buried in the protein core , with a solvent accessibility less than or equal to 20% , the optimal folding free energy is : when the mutated residue has a solvent accessibility comprised between 40 and 50% , we do not evaluate its folding free energy .
we have indeed observed that in this case , the solvent accessibility of the mutated residue is not a good measure to guide the choice of the optimal potential .
the derivation is based on a dataset of 141 high - resolution protein structures determined by x - ray crystallography and listed in . to avoid bias ,
these 141 proteins are chosen to present either less than 20% sequence identity or less than 25% sequence identity and no structural similarity .
the protein main chains are described by their heavy atoms , and each side chain is represented by a pseudo - atom c. for a given amino - acid type , the c has a well defined position relative to the main chain , corresponding to the geometric average of all heavy side - chain atoms of this type in the dataset ; for glycine , the c pseudo - atom is positioned on the c. side - chain degrees of freedom are thus neglected . each residue , at each position of each of the 141 proteins , is mutated in turn into the 19 non - wild - type amino acids .
the mutations are made by keeping the main - chain structure unchanged , and substituting the c of the given amino acid by that of the mutant amino acid . for each of these mutations ,
the change in folding free energy is evaluated using the database - derived potentials and the procedure detailed below . for each substitution of amino acid
a into a ' , the average of all computed changes in folding free energy , at all protein positions , is computed and defined as minus the matrix element m(a , a ' ) .
we then symmetrize m by setting m(a , a ' ) = [ m(a , a')+m(a',a)]/2 and only consider the lower half of m(a < a ' ) .
this procedure does not define the diagonal elements of m. on the basis of the principle that the structural role of a given amino acid is fulfilled by no other amino acid better than by itself , we assign to all the diagonal element the same maximum value : m(a , a ) = max [ m(a',a " ) ] + 1 . then
, to simplify m without modifying its structure , we center it around its mean value : finally , we multiply all matrix elements m(a , a ' ) by 2 and replace them by the closest integer .
the potentials we use to evaluate the protein conformations are derived from observed frequencies of sequence and structure patterns in the aforementioned dataset of 141 proteins .
we consider two types of potentials , called torsion   and c -c   potentials .
they take into account the propensities of single residues and residue pairs to be associated with a (  ,  ,  ) backbone torsion angle domain .
we use two variants of the torsion potential , called torsionshort - range and torsionmiddle - range . both
are computed from propensities of a (  ,  ,  ) domain ti , at position i along the sequence , or pairs of domains ( ti , tj ) , at positions i and j , to be associated with an amino acid ak at position k. but we have k - 1  i , j  k + 1 for the torsionshort - range potential and k - 8  i , j  k + 8 for the torsionmiddle - range potential .
the folding free energy  g(s , c ) of a sequence s in the conformation c computed from these propensities is expressed as : where p are normalized frequencies , n is the number of residues in the sequence s , k is the boltzmann constant and t is a conformational temperature taken to be room temperature .
the normalization factor k ensures that the contribution to g(s , c ) of each residue in the window [ k - 1 , k + 1 ] for the torsionshort - range potential or [ k - 8 , k + 8 ] for the torsionmiddle - range potential is counted once .
they are based on propensities of pairs of amino acids ( ai , aj ) at position i and j along the sequence to be separated by a spatial distance dij , calculated between the pseudo - atoms c. we consider two variants of c - c potentials .
the first one , called c - clong_range potential , describes purely nonlocal interactions along the sequence , and only takes into account residues separated by at least 15 residues along the sequence , that is j  i + 16 .
the second one , simply called c - c potential , though dominated by nonlocal interactions , possesses a local interaction component .
the nonlocal component is obtained by considering together the frequencies of all residues separated by seven sequence positions and more , thus with j  i + 8 .
the local component is obtained by computing separately the frequencies of residues separated by one to six positions along the sequence , for i + 1 < j < i + 8 .
the folding free energies are expressed as : with j  i + 16 and the normalized frequencies p independent of j - i for the c - clong_range potential , and i + 1 < j and the normalized frequencies p independent of j - i for j  i + 8 for the c - c potential .
the discretization of the spatial distances dij is performed by dividing the distances between 3 and 8  into 25 bins of 0.2  width and merging the distances greater than 8  . to increase the reliability of the statistics ,
these bins are smoothed by combining the counts in each bin with those of the 10 flanking bins at each side .
the predominance of the central bin is preserved by weighting the counts from each flanking bin by a factor 1/n , where n is the position relative to the central bin ; n is equal to 1 for the two closest bins and to 10 for the two most distant bins .
the so - defined folding free energies are reliable for common amino acids and structure motifs , but not for less common ones . to correct for the sparse data , we substitute the sequence - specific frequencies p(c , s ) , where s denotes a sequence pattern and c a structure motif , which appear in expressions ( 9 ) and ( 10 ) immediately above defining the torsion and c - c folding free energies , by a linear combination of these frequencies and the product of the separate frequencies of s and c , denoted p(s ) and p(c ) respectively : where m is the number of occurrences of the sequence pattern s in the dataset , and  a parameter .
this expression ensures that the sequence - specific contribution dominates for rare sequence patterns and tends to zero for frequent ones .
this behavior is modulated by the parameter  , which we consider here equal to 50 .
to estimate the stability changes caused by a single - site mutation , we compute the folding free energy changes as : where cm and cw are the mutant and wild - type conformations and sm and sw the mutant and wild - type sequences , respectively . with this convention ,
g is positive when the mutation is destabilizing , and negative when it is stabilizing .
the conformations cm and cw of the mutant and wild - type protein are assumed to be nearly identical .
more precisely , the backbone conformations are taken as identical and only the position of the c pseudo - atom , which is amino - acid dependent , is different in the mutant and wild - type structures .
the folding free energies of the wild - type and mutant proteins are computed with linear combinations of the torsion and c - c potentials described in the previous section .
previous analyses   have shown that the combination that gives the best evaluation of the g values depends on the solvent accessibility , a of the mutated residue ; a is defined as the solvent - accessible surface in the protein structure , computed by survol , multiplied by 100 and divided by its solvent - accessible surface in an extended tripeptide gly - x - gly .
when the mutated residue is at the surface , with a solvent accessibility a equal to or larger than 50% , the optimal folding free - energy change has been shown to be equal to : when the mutated residue is half buried , half exposed to the solvent , with a solvent accessibility between 20 and 40% , the optimal folding free energy is : finally , when the mutated residue is totally buried in the protein core , with a solvent accessibility less than or equal to 20% , the optimal folding free energy is : when the mutated residue has a solvent accessibility comprised between 40 and 50% , we do not evaluate its folding free energy .
we have indeed observed that in this case , the solvent accessibility of the mutated residue is not a good measure to guide the choice of the optimal potential .
the relative frequency p(a ) ( in % ) of amino acid a ( right - hand column of table 1 ) , as a function of the number of synonyms n(a ) that code for it .
the mean frequencies of the individual amino acids ( p(a ) ) in the genomes of living organisms the frequencies p(a ) were computed as averages over the frequencies observed in genomes of archaea ( aeropyrum pernix k1 , archaeoglobus fulgidus , halobacterium sp .
nrc-1 , methanococcus jannaschii , methanobacterium thermoautotrophicum , pyrococcus abyssi , pyrococcus horikoshi and thermoplasma acidophilum ) , bacteria ( aquifex aeolicus , bacillus halodurans , bacillus subtilis , borrelia burgdorferi , buchnera aphidicola , campylobacter jejuni , chlamydia trachomatis , deinococcus radiodurans , escherichia coli k-12 , haemophilus influenzae , mycobacterium leprae , mycobacterium tuberculosis , mycoplasma genitalium , mycoplasma pneumoniae , pasteurella multocida , pseudomonas aeruginosa , rickettsia prowazekii , thermotoga maritima , treponema pallidum , ureaplasma parvum , vibrio cholerae and xylella fastidiosa ) and eukaryotes ( arabidopsis thaliana , caenorhabditis elegans , drosophila melanogaster , homo sapiens and saccharomyces cerevisiae ) .
fraction of random codes that are fitter than the genetic code fraction f of random codes that have a lower value of the fitness function ,  or  ) than the natural code , using each of the four cost functions g , g , g , g and g. values marked with an asterisk have been obtained by extrapolation as explained in text .
the number of randomly generated codes is equal to 10 and the amino - acid frequencies used are the average ones listed in the right - hand column of table 1 .
percentage of random amino - acid frequency assignments yielding lower fractions f than the natural one percentage of the sets of random amino - acid frequency assignments for which the fraction f of random codes that beat the natural code is lower than the corresponding fraction computed with the natural frequency p(a ) values .
this percentage is estimated for the four cost functions - g , g , g and g - on the basis of 100 random frequencies and , for each of them , 10 random codes .
for all cost functions except g , we were only able to give an upper bound ( estimated to be equal to 1% ) , because our sample of random codes is too small and we did not find any random frequency set for which f is lower than that obtained with the natural frequencies .
fraction f for different sets of allowed amino - acid interchanges in the alternative codes fraction f of random codes that have a lower value of the fitness function (  ,  or  ) than the natural code , using the cost function g. for the unrestricted set , the f values were estimated from 10 randomly generated codes , where the only constraint is the preservation of the code 's block structure ( as in table 2 ) .
for the biosynthesis - restricted set , only permutations of amino acids sharing the same metabolic pathway were considered , that is , interchanges of amino acids contained in one of the four sets { f , s , y , c , w } , { l , p , h , q , r } , { i , m , t , n , k } , { v , a , d , e , g } ( single - letter amino - acid notation ) . as
the number of alternative codes is reasonable ( 207,360,000 ) , they have not been randomly chosen , but all have been tested .
the degeneracy - restricted set contains results obtained by shuffling only amino acids with the same degeneracy in the natural code , corresponding to the sets { m , w } , { c , d , e , f , h , k , n , q , y } , { i } , { a , g , p , t , v } , { l , r , s}. here also , all 522,547,200 possible codes have been systematically tested .
the percentage of optimization of the natural code compared to the optimal alternative ones , as defined in the text , is given in parentheses for . for the two restricted sets , for which all alternative codes were exhaustively generated , the  value of the optimal code was computed exactly . for the unrestricted set ,
the optimal  value was taken as the best of the unrestricted and two restricted sets .
 backgroundthe genetic code is known to be efficient in limiting the effect of mistranslation errors . 
 a misread codon often codes for the same amino acid or one with similar biochemical properties , so the structure and function of the coded protein remain relatively unaltered . 
 previous studies have attempted to address this question quantitatively , by estimating the fraction of randomly generated codes that do better than the genetic code in respect of overall robustness . 
 we extended these results by investigating the role of amino - acid frequencies in the optimality of the genetic code.resultswe found that taking the amino - acid frequency into account decreases the fraction of random codes that beat the natural code . 
 this effect is particularly pronounced when more refined measures of the amino - acid substitution cost are used than hydrophobicity . to show this 
 , we devised a new cost function by evaluating in silico the change in folding free energy caused by all possible point mutations in a set of protein structures . with this function , which measures protein stability while being unrelated to the code 's structure , we estimated that around two random codes in a billion ( 109 ) are fitter than the natural code . 
 when alternative codes are restricted to those that interchange biosynthetically related amino acids , the genetic code appears even more optimal.conclusionsthese results lead us to discuss the role of amino - acid frequencies and other parameters in the genetic code 's evolution , in an attempt to propose a tentative picture of primitive life . 
Background
Results
Fitness of the genetic code with respect to translation errors
Incorporating amino-acid frequencies in the fitness function
Cost of substituting an amino acid with another
The genetic code versus random codes
Discussion
Materials and methods
Derivation of the mutation matrix
Database derived potentials
Evaluation of folding free energy changes
Acknowledgements
Figures and Tables