15Single nucleotide substitutions in protein-coding genes can be divided into synonymous (S), 16with little fitness effect, and non-synonymous (N) ones that alter amino acids and thus generally 17 have a greater effect. Most of the N substitutions are affected by purifying selection that 18 eliminates them from evolving populations. However, additional mutations of nearby bases can 19 modulate the deleterious effect of single substitutions and thus might be subject to positive 20 selection. To elucidate the effects of selection on double substitutions in all codons, it is critical 21 to differentiate selection from mutational biases. We approached this problem by comparing the 22 fractions of double substitutions within codons to those of the equivalent double S substitutions 23 in adjacent codons. Under the assumption that substitutions occur one at a time, all within-24 codon double substitutions can be represented as "ancestral-intermediate-final" sequences and 25 can be partitioned into 4 classes: 1) SS: S intermediate -S final, 2) SN: S intermediate -N 26 final, 3) NS: N intermediate -S final, 4) NN: N intermediate -N final. We found that the 27 selective pressure on the second substitution markedly differs among these classes of double 28 substitutions. Analogous to single S substitutions, SS evolve neutrally whereas, analogous to 29 single N substitutions, SN are subject to purifying selection. In contrast, NS show positive 30 selection on the second step because the original amino acid is recovered. The NN double 31 substitutions are heterogeneous and can be subject to either purifying or positive selection, or 32 evolve neutrally, depending on the amino acid similarity between the final or intermediate and 33 the ancestral states. The general trend is that the second mutation compensates for the 34 deleterious effect of the first one, resulting in frequent crossing of valleys on the fitness 35 landscape. 36 37 38 39 40the DF values of the SS substitutions can be explained by the frequency of multi-nucleotide 129 mutations suggesting that SS double substitutions evolve (nearly) neutrally, similar to single 130 synonymous substitutions (Fig. S1). Additionally, there was no significant difference between 131the DF values of SS double substitutions in fast and slow evolving genes (Fig. 3A) which is 132 compatible with the neutral evolutionary regime. Nevertheless, although the bulk analysis of the 133 SS substitutions yields results compatible with neutrality, most of the individual SS cases seem 134 to involve weak positive selection after the BH correction, which can be linked to codon bias 135 ( Fig. S2). 136 SN: synonymous substitution followed by a non-synonymous one 137