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Parikh vectors
3WOY_1 8VSV_1 4KJV_1 Letter Amino acid
15 10 3 D Aspartic acid
4 5 3 C Cysteine
9 8 3 Q Glutamine
10 6 5 F Phenylalanine
2 4 3 W Tryptophan
16 24 2 V Valine
17 8 7 R Arginine
8 8 6 N Asparagine
16 5 5 E Glutamic acid
15 4 2 I Isoleucine
6 14 5 Y Tyrosine
11 22 8 G Glycine
2 2 1 M Methionine
8 12 6 P Proline
19 34 4 S Serine
14 17 3 T Threonine
25 14 7 A Alanine
10 3 3 H Histidine
29 17 7 L Leucine
15 11 3 K Lycine 

3WOY_1|Chain A|CLIP-associating protein 2|Homo sapiens (9606)
>8VSV_1|Chains A[auth 1], C[auth 3]|IgG EEEV-373 Heavy chain|Homo sapiens (9606)
>4KJV_1|Chains A, B[auth C]|E3 ubiquitin-protein ligase XIAP|Homo sapiens (9606)
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 \(w=f(c)\)
3WOY , Knot 115 251 0.84 40 169 241 
GGAGAVDEDDFIKAFTDVPSIQIYSSRELEETLNKIREILSDDKHDWDQRANALKKIRSLLVAGAAQYDCFFQHLRLLDGALKLSAKDLRSQVVREACITVAHLSTVLGNKFDHGAEAIVPTLFNLVPNSAKVMATSGCAAIRFIIRHTHVPRLIPLITSNCTSKSVPVRRRSFEFLDLLLQEWQTHSLERHAAVLVETIKKGIHDADAEARVEARKTYMGLRNHFPGEAETLYNSLEPSYQKSLQTYLKS
8VSV , Knot 100 228 0.79 40 144 218 
QVQLVESGGGVVQPGRSLRLSCAASGFTFSSHVMYWVRQAPGKGLEWVAVITYDGGNKYYADSVRGRLTISRDNSKNTLYLQMNSLRAEDTAVYYCASPRGDSGSYYDIDYFDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC
4KJV , Knot 48 86 0.82 40 78 84 
GTIYPRNPAMYSEEARLKSFQNWPDYAHLTPRELASAGLYYTGIGDQVQCFACGGKLKNWEPGDRAWSEHRRHFPNCFFVLGRNLN

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 \(c\).

\(|P_{f(3WOY_1)}(2) \setminus P_{f(8VSV_1)}(2)|=96\), \(|P_{f(8VSV_1)}(2) \setminus P_{f(3WOY_1)}(2)|=71\). 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 1:11111100001101100110101000001000100100110000001000101100100111111100001100101101110101001000110010101101001110010011011110110111001011100101110111000011011111000000001110000101101110010000100011111001001100101010101000011100011101001000101000001000100
Pair \(Z_2\) Length of longest common subsequence
3WOY_1,8VSV_1 167 3
3WOY_1,4KJV_1 163 3
8VSV_1,4KJV_1 162 3

Newick tree

 
[
	3WOY_1:83.00,
	[
		4KJV_1:81,8VSV_1:81
	]:2.00
]

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 \((0.85)(0.8)(\frac{479 }{\log_{20} 479}-\frac{228}{\log_{20}228})=72.5\)
Status Protein1 Protein2 d d1/2
Query variables 3WOY_1 8VSV_1 91 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]