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Parikh vectors
8QYO_1 6TTN_1 7ZCL_1 Letter Amino acid
12 24 9 I Isoleucine
8 12 16 F Phenylalanine
14 18 8 Y Tyrosine
2 3 3 W Tryptophan
2 6 2 C Cysteine
20 35 20 L Leucine
5 5 5 M Methionine
14 13 15 S Serine
10 24 15 P Proline
11 8 13 R Arginine
11 16 8 Q Glutamine
18 28 17 E Glutamic acid
14 23 15 K Lycine
4 7 6 H Histidine
13 13 16 T Threonine
19 19 13 V Valine
23 15 17 A Alanine
7 13 7 N Asparagine
7 17 11 D Aspartic acid
20 18 10 G Glycine 

8QYO_1|Chains A, O|Proteasome subunit alpha type-2|Homo sapiens (9606)
>6TTN_1|Chain A|Hyoscyamine 6 beta-hydroxylase|Datura metel (35625)
>7ZCL_1|Chain A|Collariella virescens UPO|Collariella virescens (1934374)
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 \(w=f(c)\)
8QYO , Knot 106 234 0.82 40 160 226 
MAERGYSFSLTTFSPSGKLVQIEYALAAVAGGAPSVGIKAANGVVLATEKKQKSILYDERSVHKVEPITKHIGLVYSGMGPDYRVLVHRARKLAQQYYLVYQEPIPTAQLVQRVASVMQEYTQSGGVRPFGVSLLICGWNEGRPYLFQSDPSGAYFAWKATAMGKNYVNGKTFLEKRYNEDLELEDAIHTAILTLKESFEGQMTEDNIEVGICNEAGFRRLTPTEVKDYLAAIA
6TTN , Knot 141 317 0.85 40 200 305 
SNADVPIIDLQQDHLLIVQQITKACQDFGLFQVINHGVPEKLMVEAMEVYKEFFALPAEEKEKFQPKGEPAKFELPLEQKAKLYVEGERRCNEEFLYWKDTLAHGCYPLHEELLNSWPEKPPTYRDVIAKYSVEVRKLTMRILDYICEGLGLKLGYFDNELTQIQMLLANYYPSCPDPSTTIGSGGHYDGNLITLLQQDLVGLQQLIVKDDKWIAVEPIPTAFVVNLGLTLKVMSNEKFEGSIHRVVTHPIRNRISIGTLIGPDYSCTIEPIKELISQENPPLYKPYPYAEFAEIYLSDKSDYDAGVKPYKINQFPN
7ZCL , Knot 109 226 0.87 40 164 220 
LDFSKWKTRQPGEFRAPCPAMNSLANHGFIPRDGRNITVAMLVPVLQEVFHLSPELAQTISTLGLFTAQDPSKGVFTLDDLNRHNLFEHDASLSREDYYFHKDASTFRPEVFKKFMSHFKGKEYVTLEDAASARYAMVQESRKKNPTFTYTVQQRITSYGETIKYFRTIVEPATGKCPVAWIKILFEQERLPYNEGWRPPKAELSGFSMASDVLELALVTPEKLID

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 \(c\).

\(|P_{f(8QYO_1)}(2) \setminus P_{f(6TTN_1)}(2)|=63\), \(|P_{f(6TTN_1)}(2) \setminus P_{f(8QYO_1)}(2)|=103\). 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 1:110010010100101010110100111111111101110110111110000000110000010010110001111001111000111001001100001100011101011001101100000011101111011101100101011000101101110101110001010011000000010100110011101000101010000101110001110010100100011111
Pair \(Z_2\) Length of longest common subsequence
8QYO_1,6TTN_1 166 6
8QYO_1,7ZCL_1 158 4
6TTN_1,7ZCL_1 174 4

Newick tree

 
[
	6TTN_1:86.93,
	[
		8QYO_1:79,7ZCL_1:79
	]:7.93
]

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 \((0.85)(0.8)(\frac{551 }{\log_{20} 551}-\frac{234}{\log_{20}234})=90.4\)
Status Protein1 Protein2 d d1/2
Query variables 8QYO_1 6TTN_1 117 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]