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the scheme of the experimental approach . a violin plot of the measured levels of fluorescence for genotypes carrying different number of missense mutations .
log - fluorescence and d evolutionary conservation expressed as shannon entropy and e fraction of mutant amino acid states found in avgfp orthologues ( y - error bars in e show binomial proportion confidence interval level , 68% , and se elsewhere ) .
a , pairs of amino acid sites for which we assayed at least one combination of mutations ( in blue , upper left ) .
the distribution of the maximum level of epistasis observed between sites ( blue scale , lower right ) and unknown values ( white ) .
b , pairs of sites under exceptionally strong epistatic interaction ( e < 2 ) connected by a blue line on the gfp structure .
c , the distribution of distances in the gfp structure between sites with at least one pair of epistatically interacting mutations ( red ) and all pairs of sites in the structure ( grey ) .
d , epistasis between pairs of mutations as a function of their individual fluorescence . the contribution of internally and externally oriented amino acid residues in the avgfp structure relative to pairs of missense mutations showing no epistasis ( |e| < 0.3 ) , weak ( 0.3 < |e| < 0.7 ) and strong ( |e| > 0.7 ) epistasis .
a. a multiple linear regression where fluorescence is linear combination of effects of individual single mutations .
b , a multiple regression where mutations contribute linearly to a fitness potential and fluorescence is a sigmoidal function of p where f ~ e/(1+e ) .
c , d , the predicted fluorescence by a neural network approach . predicted fitness function by a neural network with one hidden neuron and two neurons in the outer layer .
the genotype data was passed to the input layer of the neural network as an array of 0 s and 1 s corresponding to the absence or presence of amino acid mutations in the genotype , respectively .
the first hidden layer consisted of a single neuron which calculated the weighted sum of inputs using weights obtained during training .
output of the first hidden layer was passed through an output subnetwork that transformed this value with a non - linear function to make the final prediction of fluorescence .
the output subnetwork consisted of several neurons with a sigmoidal transfer function , allowing the subnetwork to approximate a broad range of non - linear functions .
the final mapping of the hidden value to fluorescence was determined by the weights of connections between neurons inside the output subnetwork . during training all weights
f , correlation between hidden value of the neural network and rosetta - predicted g for single mutants .
fitness landscapes1,2 , depictions of how genotypes manifest at the phenotypic level , form the basis for our understanding of many areas of biology27 yet their properties remain elusive . 
 studies addressing this issue often consider specific genes and their function as proxy for fitness2,4 , experimentally assessing the impact on function of single mutations and their combinations in a specific sequence2,5,815 or in different sequences2,3,5,1618 . 
 however , systematic high - throughput studies of the local fitness landscape of an entire protein have not yet been reported . here , 
 we chart an extensive region of the local fitness landscape of the green fluorescent protein from aequorea victoria ( avgfp ) by measuring the native function , fluorescence , of tens of thousands of derivative genotypes of avgfp . 
 we find that its fitness landscape is narrow , with half of genotypes with two mutations showing reduced fluorescence and half of genotypes with five mutations being completely non - fluorescent . 
 the narrowness is enhanced by epistasis , which was detected in up to 30% of genotypes with multiple mutations arising mostly through the cumulative impact of slightly deleterious mutations causing a threshold - like decrease of protein stability and concomitant loss of fluorescence . 
 a model of orthologous sequence divergence spanning hundreds of millions of years predicted the extent of epistasis in our data , indicating congruence between the fitness landscape properties at the local and global scales . 
 the characterization of the local fitness landscape of avgfp has important implications for a number of fields including molecular evolution , population genetics and protein design .