5.5 Materials and Methods
5.5.4 Covariance statistic calculations
Covariance statistics were calculated according to Martin et al. [19], and Dunn et al. [3].Zpxis similar to the Zres statistic of Little and Chen [15], but simpler to calculate (Supplementary Methods S1). Positions that contain gaps are not analyzed because gaps violate the assumption of orthology connecting covariation with coadaptation and coevolution (Figure S2).
Mutual Information (MI) measures the reduction in uncertainty of one variable given in- formation about another variable and was calculated as previously [19]. As shown in equation 1, in the context of protein sequence families it measures the difference between the expected entropy (H) of residues in two columnsiand jif they were independent against the observed joint entropy,Hi,j.
MIi,j = Hi+Hj−Hi,j (5.1)
true in the context of protein families since, to a first approximation, every position in a protein family shares common ancestry with every other position, and the positions in a gene for a given protein are rarely split by recombination. MIp, the product correctedMI, estimates the backgroundMIsignal caused by sequence non-independence [3] as shown in equation 2
MI pi,j = MIi,j−(MIi,x× MIj,x)/MI (5.2)
where MIi,x is the meanMI of position iwith all other positions and MI is the overall mean
MI.
TheMIp values were converted toZ scores since absoluteMIpvalues vary somewhat be- tween alignments and because the underlying distribution ofMIpvalues approximates a Gaus- sian distribution in the absence of structural and functional covariation [3]:
Zpi,j = (MI pi,j−MI p)/σ(MI p) (5.3)
where againMI pis the meanMIpandσ(MI p) is its standard deviation.
Zpx is a modification of Little and Chen’sZresstatistic [15] that is based on the residual
MI of a linear regression between MIi,j and the mean MI of positions i and j, MIi × MIj.
The plot shown in Figure 1A shows that the residual is nearly identical to MIp with a slope of 1, an intercept of 0 and anr2 value of 0.9995. Zresis the product of the Z scores derived from the residuals for each individual position [15]. Since the residual and MIpare virtually indistinguishable (Supplementary Methods S1), we substituted the more efficiently calculated
MIpfor the residual in the formula forZresas shown in equation 5 to calculateZi×j.
Zi×j = MI pi,j−MI pi σ(MI pi) × MI pi j−MI pj σ(MI pj) (5.4)
As expected from the similarity of the underlying statistics, Zres and Zi×j are extremely
ZresandZi×jvalues are the product of the twoZscores, and so these values scale geometrically
when compared to Z p. In this study we use Zpx, which is the square root of Zi×j to allow
comparison of theZ pandZi×jvalues on similar scales.
∆Z pis a measure of the difference score between successiveZpscores at positioni. ∆Z pwas calculated by placing theZpvalues of positioniwith all other positions, x, in an ordered list, with the largestZpvalue first, i.e.Li = (Z pi,x,Z pi,x+1,Z pi,x+2...Z pi,x+(N−1)), whereNis the num- ber of Zpvalues for positioni. ∆Z p is the sequential difference between list elements scaled by interquartile units as follows:
∆Z pi,x =
Z pi,x −Z pi,x+1
R (5.5)
whereRis the interquartile range calculated as 12 the difference between the 75th and the 25th
percentile for the data in Li. The interquartile range scaling factor and the median provide
robust estimates of dispersion and central tendency that compensate for variation in the distri- bution of Zp at each position while making the minimum number of assumptions about the underlying distribution. Thus∆Z pmeasures how extreme the difference inZpscores is for all pairs of positions,i,x.
Since each Zpscore is a measure of the covariation between two positions, i and j, there are two∆Z pscores for each pair of positions: one is the difference betweenZpi,j and the next
highest score for positioni, and the other is the difference between Zpj,i and the next highest
scoring position for position j. We used the greater of the∆Z pscores.
MI, and the derived statistics, MIp, ∆Z p andZpx were calculated only forungapped po- sitions in the multiple sequence alignments. Covariation analysis attempts identify those po- sitions that are coevolving for structural or functional reasons, and as shown in this report, depends upon the precise placement of homologous positions in the alignment.