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Parikh vectors
4KGX_1 4TOA_1 8DMH_1 Letter Amino acid
34 5 10 A Alanine
14 18 4 E Glutamic acid
11 10 7 H Histidine
15 12 12 I Isoleucine
38 28 20 L Leucine
20 5 25 S Serine
18 3 19 T Threonine
22 5 8 V Valine
2 2 4 W Tryptophan
15 5 12 R Arginine
21 13 8 D Aspartic acid
15 9 17 G Glycine
8 5 9 M Methionine
12 1 3 P Proline
8 5 10 Y Tyrosine
15 7 15 N Asparagine
1 2 8 C Cysteine
14 9 9 Q Glutamine
15 12 11 K Lycine
12 2 14 F Phenylalanine 

4KGX_1|Chains A, C|Aspartate carbamoyltransferase|Escherichia coli (595495)
>4TOA_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X|Bacterioferritin|Pseudomonas aeruginosa (208964)
>8DMH_1|Chains A, E[auth B], I[auth C]|Glycoprotein G1|Lymphocytic choriomeningitis virus (3052303)
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 \(w=f(c)\)
4KGX , Knot 139 310 0.85 40 190 299 
ANPLYQKHIISINDLSRDDLNLVLATAAKLKANPQPELLKHKVIASCFFEASTRTRLSFETSMHRLGASVVGFSDSANTSLGKKGETLADTISVISTYVDAIVMRHPQEGAARLATEFSGNVPVLNAGDGSNQHPTQTLLDLFTIQETQGRLDNLHVAMVGDLKYGRTVHSLTQALAKFDGNRFYFIAPDALAMPQYILDMLDEKGIAWSLHSSIEEVMAEVDILYMTRVQKERLDPSEYANVKAQFVLRASDLHNAKANMKVLHPLPRVDEIATDVDKTPHAWYFQQAGNGIFARQALLALVLNRDLVL
4TOA , Knot 79 158 0.84 40 111 155 
MKGDKKVIQHLNKILGNELIAINQYFLHSRMWNDWGLKRLGAHEYHESIDEMKHADKLIERILFLEGLPNLQDLGKLLIGENTQEMLQCDLNLELKATKDLREAIVHCEQVHDYVSRDLLKDILESEEEHIDYLETQLGLIQKVGLELYLQSHMHEDD
8DMH , Knot 103 225 0.82 40 161 217 
MKLCILLAVVAFVGLSLGMYGLKGPDIYKGVYQFKSVEFDMSHLNLTMPNACSANNSHHYISMGTSGLELTFTNDSIISHNFCNLTSAFNKKTFDHTLMSIVSSLHLSIRGNSNYKAVSCDFNNGITIQYNLTFSDAQSAQSQCRTFRGRVLDMFRTAFGGKYMRSCWGWTGSDGKTTWCSQTSYQYLIIQNRTWENHCTYAGPFGMSRILLSQEKTKFLTRRRR

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 \(c\).

\(|P_{f(4KGX_1)}(2) \setminus P_{f(4TOA_1)}(2)|=124\), \(|P_{f(4TOA_1)}(2) \setminus P_{f(4KGX_1)}(2)|=45\). 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 1:1011000011010010000101111011010101010110001110011010000010100010011101111000100011001001100101100010111100100111011001010111101101000010001101101000010100101111101001001001001110101001011110111110011011000111101000100111010110100100001010001010101110100100101010110111010011001000101101001101111001111111000111
Pair \(Z_2\) Length of longest common subsequence
4KGX_1,4TOA_1 169 4
4KGX_1,8DMH_1 185 3
4TOA_1,8DMH_1 188 4

Newick tree

 
[
	8DMH_1:95.99,
	[
		4KGX_1:84.5,4TOA_1:84.5
	]:11.49
]

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 \((0.85)(0.8)(\frac{468 }{\log_{20} 468}-\frac{158}{\log_{20}158})=91.4\)
Status Protein1 Protein2 d d1/2
Query variables 4KGX_1 4TOA_1 116 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]