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Parikh vectors
7BEL_1 3TLP_1 1KBO_1 Letter Amino acid
12 8 25 A Alanine
7 3 7 H Histidine
13 13 25 L Leucine
15 7 10 Y Tyrosine
16 4 14 V Valine
17 8 9 N Asparagine
8 1 1 C Cysteine
10 7 13 P Proline
12 2 9 T Threonine
10 17 24 K Lycine
0 9 6 M Methionine
14 4 17 F Phenylalanine
15 10 17 S Serine
9 10 9 R Arginine
9 13 12 D Aspartic acid
6 3 12 Q Glutamine
7 13 17 E Glutamic acid
16 8 21 G Glycine
7 10 19 I Isoleucine
2 0 6 W Tryptophan 

7BEL_1|Chains A[auth R], F[auth X]|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3TLP_1|Chains A, B|Protein polybromo-1|Homo sapiens (9606)
>1KBO_1|Chains A, B, C, D|NAD(P)H dehydrogenase [quinone] 1|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 \(w=f(c)\)
7BEL , Knot 94 205 0.81 38 142 195 
ETGHHHHHHTNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGKK
3TLP , Knot 72 150 0.80 38 111 141 
MGDSMISSATSDTGSAKRKSKKNIRKQRMKILFNVVLEAREPGSGRRLCDLFMVKPSKKDYPDYYKIILEPMDLKIIEHNIRNDKYAGEEGMIEDMKLMFRNARHYNEEGSQVYNDAHILEKLLKEKRKELGPLPDDDDMASPAENLYFQ
1KBO , Knot 122 273 0.83 40 177 264 
VGRRALIVLAHSERTSFNYAMKEAAAAALKKKGWEVVESDLYAMNFNPIISRKDITGKLKDPANFQYPAESVLAYKEGHLSPDIVAEQKKLEAADLVIFQFPLQWFGVPAILKGWFERVFIGEFAYTYAAMYDKGPFRSKKAVLSITTGGSGSMYSLQGIHGDMNVILWPIQSGILHFCGFQVLEPQLTYSIGHTPADARIQILEGWKKRLENIWDETPLYFAPSSLFDLNFQAGFLMKKEVQDEEKNKKFGLSVGHHLGKSIPTDNQIKARK

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 \(c\).

\(|P_{f(7BEL_1)}(2) \setminus P_{f(3TLP_1)}(2)|=92\), \(|P_{f(3TLP_1)}(2) \setminus P_{f(7BEL_1)}(2)|=61\). 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 1:0010000000010111011010011010110000100011000110001010010001101001001010010100111010010011110010110000011001010111100001000111000010011000010110001000100110010011011000111000110100111001001111010110111010100
Pair \(Z_2\) Length of longest common subsequence
7BEL_1,3TLP_1 153 4
7BEL_1,1KBO_1 177 4
3TLP_1,1KBO_1 162 3

Newick tree

 
[
	1KBO_1:87.43,
	[
		7BEL_1:76.5,3TLP_1:76.5
	]:10.93
]

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 \((0.85)(0.8)(\frac{355 }{\log_{20} 355}-\frac{150}{\log_{20}150})=62.1\)
Status Protein1 Protein2 d d1/2
Query variables 7BEL_1 3TLP_1 81 69.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]