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Parikh vectors
6AKH_1 3QWF_1 6SBC_1 Letter Amino acid
7 1 5 C Cysteine
6 12 14 I Isoleucine
19 14 23 T Threonine
22 33 18 V Valine
13 15 15 D Aspartic acid
15 12 10 H Histidine
21 8 17 P Proline
4 3 3 M Methionine
3 11 17 N Asparagine
30 7 20 Q Glutamine
45 16 37 L Leucine
27 27 24 G Glycine
12 16 9 K Lycine
18 12 13 F Phenylalanine
22 22 22 S Serine
5 2 15 W Tryptophan
28 27 53 A Alanine
27 12 18 R Arginine
27 14 21 E Glutamic acid
5 6 13 Y Tyrosine 

6AKH_1|Chain A|DNA-directed DNA/RNA polymerase mu|Homo sapiens (9606)
>3QWF_1|Chains A, B, C, D, E, F, G, H|17beta-hydroxysteroid dehydrogenase|Cochliobolus lunatus (5503)
>6SBC_1|Chain A|MstE|Scytonema sp. PCC 10023 (1680591)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
6AKH , Knot 147 356 0.80 40 198 344 
GSAAAPLSPAWMPAYACQRPTPLTHHNTGLSEALEILAEAAGFEGSEGRLLTFCRAASVLKALPSPVTTLSQLQGLPHFGEHSSRVVQELLEHGVCEEVERVRRSERYQTMKLFTQIFGVGVKTADRWYREGLRTLDDLREQPQKLTQQQKAGLQHHQDLSTPVLRSDVDALQQVVEEAVGQALPGATVTLTGGFRRGKLQGHDVDFLITHPKEGQEAGLLPRVMCRLQDQGLILYHQHQHSCCESPTRLAQQSHMDAFERSFCIFRLPQPGSWKAVRVDLVVAPVSQFPFALLGWTGSKLFQRELRRFSRKEKGLWLNSHGLFDPEQKTFFQAASEEDIFRHLGLEYLPPEQRNA
3QWF , Knot 120 270 0.83 40 167 254 
MPHVENASETYIPGRLDGKVALVTGSGRGIGAAVAVHLGRLGAKVVVNYANSTKDAEKVVSEIKALGSDAIAIKADIRQVPEIVKLFDQAVAHFGHLDIAVSNSGVVSFGHLKDVTEEEFDRVFSLNTRGQFFVAREAYRHLTEGGRIVLTSSNTSKDFSVPKHSLYSGSKGAVDSFVRIFSKDCGDKKITVNAVAPGGTVTDMFHEVSHHYIPNGTSYTAEQRQQMAAHASPLHRNGWPQDVANVVGFLVSKEGEWVNGKVLTLDGGAA
6SBC , Knot 153 367 0.82 40 210 339 
GSTLQPLENSTRQEKLLYPKLNQLSNSINAAVAFLLEARNLEGWWQDFNFPQAASIGDEWVTAYVGTMLATLPYAHVHEALMQAWELLKIRDHRPTGEWGYNYILCGDADTTGWALQLAAAVGASDSERAQQARAALATHLQPNGGIATFAEESIRAYIKVPDLANVSFQGWCGAHTCVSAAVAALPEFRSRLHDYLRVTQTSQGNWEGYWWSDHEYTTALTAEALAAGGQAADQPSIEQAVAWGLKRLCPQGFVATSKHPNGSTFATAWCLRLLLLNTVDAEVKAARAAAIGWLLEQQRPNGSWVSSAYLRIPYPFDRNPNQFPHWRYYDEIEGDKRFEGSIIFDHNSIFTTATVVNSLVKAAPML

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(6AKH_1)}(2) \setminus P_{f(3QWF_1)}(2)|=95\), \(|P_{f(3QWF_1)}(2) \setminus P_{f(6AKH_1)}(2)|=64\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:10111110111111010001011000001100110111011110100101101001101101110110010010111011000001100110011000100100000000101100111111001001000110010010001001000001110000010011100010110011001110111110101011100101010010111001001001111101100100011110000000000010011000010110001011011011010110101111110011111111010011000100100000111100011101000011011000011001110011100001
Pair \(Z_2\) Length of longest common subsequence
6AKH_1,3QWF_1 159 3
6AKH_1,6SBC_1 160 4
3QWF_1,6SBC_1 151 4

Newick tree

 
[
	6AKH_1:81.11,
	[
		3QWF_1:75.5,6SBC_1:75.5
	]:5.61
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{626 }{\log_{20} 626}-\frac{270}{\log_{20}270})=99.7\)
Status Protein1 Protein2 d d1/2
Query variables 6AKH_1 3QWF_1 127 110
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]