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Parikh vectors
1DTM_1 1YPK_1 8UQW_1 Letter Amino acid
12 1 29 G Glycine
9 1 27 I Isoleucine
18 5 31 L Leucine
6 2 25 S Serine
5 1 23 T Threonine
3 1 5 Y Tyrosine
8 0 25 V Valine
1 0 6 N Asparagine
7 2 17 D Aspartic acid
0 1 2 C Cysteine
5 0 10 Q Glutamine
17 1 35 A Alanine
4 2 28 R Arginine
14 5 18 E Glutamic acid
19 3 6 K Lycine
2 0 7 M Methionine
6 1 15 F Phenylalanine
2 0 4 W Tryptophan
11 0 6 H Histidine
4 1 14 P Proline 

1DTM_1|Chain A|RECOMBINANT SPERM WHALE MYOGLOBIN VARIANT H93G|Physeter catodon (9755)
>1YPK_1|Chain A[auth L]|Prothrombin light chain|Homo sapiens (9606)
>8UQW_1|Chains A, B|Phosphotriesterase variant PTE-R18|Brevundimonas diminuta (293)
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 \(w=f(c)\)
1DTM , Knot 72 153 0.79 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSGATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
1YPK , Knot 19 27 0.77 28 25 25 
ADCGLRPLFEKKSLEDKTERELLESYI
8UQW , Knot 141 333 0.82 40 187 312 
MGDRINTVRGPITISEVGFTLTHEHICGSSAGFLRAWPEFFGSREALVEKAVRGLRRARAAGVRTIVDVSTFDLGRDVRLLAEVSRAADVHIVAATGVWLDPPLSIRMRSVEELTQFFLREIQYGIEDTGIRAGIIKVAITGKVTPFQELVLRAAARASLATGVPVITHTAGSQRGGEQQAAIFESEGLSPSRVCIGHSDETDDLSYLTALAARGYLIGLDRIPHSAIGLEDNASATAFMGSRSWQTRALLIKALIDQGYMKQILVSNDWLFGISSYVTNFMDVMDSVNPDGMAFIPLRVIPFLREKGIPQETLAGITVTNPARFLSPTLRAS

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 \(c\).

\(|P_{f(1DTM_1)}(2) \setminus P_{f(1YPK_1)}(2)|=100\), \(|P_{f(1YPK_1)}(2) \setminus P_{f(1DTM_1)}(2)|=14\). 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 1:110010101110111010101110100111011000100100100100100010101000100011011011111100010001010111001100001110010110011101100001101110101110011011000111000011001
Pair \(Z_2\) Length of longest common subsequence
1DTM_1,1YPK_1 114 3
1DTM_1,8UQW_1 170 4
1YPK_1,8UQW_1 180 3

Newick tree

 
[
	8UQW_1:95.57,
	[
		1DTM_1:57,1YPK_1:57
	]:38.57
]

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 \((0.85)(0.8)(\frac{180 }{\log_{20} 180}-\frac{27}{\log_{20}27})=53.9\)
Status Protein1 Protein2 d d1/2
Query variables 1DTM_1 1YPK_1 66 38
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]