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Parikh vectors
7ZSN_1 8DTL_1 1DGN_1 Letter Amino acid
3 2 0 W Tryptophan
17 2 5 R Arginine
18 7 6 E Glutamic acid
12 0 9 K Lycine
15 0 3 T Threonine
22 1 5 A Alanine
11 0 3 N Asparagine
26 2 11 L Leucine
12 0 3 M Methionine
15 1 2 P Proline
8 0 3 H Histidine
12 1 6 I Isoleucine
15 2 2 F Phenylalanine
13 3 4 S Serine
13 1 9 D Aspartic acid
4 2 4 C Cysteine
13 2 3 Q Glutamine
28 2 5 G Glycine
9 2 0 Y Tyrosine
23 1 6 V Valine 

7ZSN_1|Chain A|Purine nucleoside phosphorylase|Homo sapiens (9606)
>8DTL_1|Chains A[auth C], D|Insulin mimetic peptide S597|synthetic construct (32630)
>1DGN_1|Chain A|ICEBERG (PROTEASE INHIBITOR)|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 \(w=f(c)\)
7ZSN , Knot 130 289 0.85 40 194 278 
MENGYTYEDYKNTAEWLLSHTKHRPQVAIICGSGLGGLTDKLTQAQIFDYGEIPNFPRSTVPGHAGRLVFGFLNGRACVMMQGRFHMYEGYPLWKVTFPVRVFHLLGVDTLVVTNAAGGLNPKFEVGDIMLIRDHINLPGFSGQNPLRGPNDERFGDRFPAMSDAYDRTMRQRALSTWKQMGEQRELQEGTYVMVAGPSFETVAECRVLQKLGADAVGMSTVPEVIVARHCGLRVFGFSLITNKVIMDYESLEKANHEEVLAAGKQAAQKLEQFVSILMASIPLPDKAS
8DTL , Knot 20 31 0.73 30 29 29 
SLEEEWAQIECEVYGRGCPSESFYDWFERQL
1DGN , Knot 48 89 0.80 36 81 87 
ADQLLRKKRRIFIHSVGAGTINALLDCLLEDEVISQEDMNKVRDENDTVMDKARVLIDLVTGKGPKSCCKFIKHLCEEDPQLASKMGLH

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 \(c\).

\(|P_{f(7ZSN_1)}(2) \setminus P_{f(8DTL_1)}(2)|=176\), \(|P_{f(8DTL_1)}(2) \setminus P_{f(7ZSN_1)}(2)|=11\). 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 1:1001000000000101110000001011110101111100010010110010110110001110110111111010101110101010010111010111011011110011100111110101011011110001011110100110110000110011110010000100011001001100001001001111110100110001100111011110011011110001101111011000111000010010000111110011001001101111011110010
Pair \(Z_2\) Length of longest common subsequence
7ZSN_1,8DTL_1 187 3
7ZSN_1,1DGN_1 181 3
8DTL_1,1DGN_1 100 2

Newick tree

 
[
	7ZSN_1:10.24,
	[
		1DGN_1:50,8DTL_1:50
	]:52.24
]

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 \((0.85)(0.8)(\frac{320 }{\log_{20} 320}-\frac{31}{\log_{20}31})=94.6\)
Status Protein1 Protein2 d d1/2
Query variables 7ZSN_1 8DTL_1 120 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]