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Parikh vectors
2DEC_1 1VZH_1 9EFK_1 Letter Amino acid
14 2 27 T Threonine
17 12 52 A Alanine
12 2 30 R Arginine
7 5 40 Q Glutamine
28 13 37 E Glutamic acid
5 5 3 H Histidine
24 8 34 I Isoleucine
8 4 18 M Methionine
15 2 22 F Phenylalanine
17 5 34 P Proline
28 1 26 S Serine
16 5 20 Y Tyrosine
25 12 27 K Lycine
23 14 35 V Valine
11 6 29 N Asparagine
11 5 32 D Aspartic acid
3 6 2 C Cysteine
18 9 29 G Glycine
40 8 47 L Leucine
3 2 10 W Tryptophan 

2DEC_1|Chains A, B|325aa long hypothetical protein|Pyrococcus horikoshii (70601)
>1VZH_1|Chains A, B|DESULFOFERRODOXIN|DESULFOVIBRIO BAARSII (453230)
>9EFK_1|Chains A, AA[auth B], BA[auth C], CA[auth D], DA[auth E], EA[auth F], FA[auth G], GA[auth H], HA[auth I], IA[auth J], JA[auth K], Z[auth L]|orf12|Legionella pneumophila (446)
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 \(w=f(c)\)
2DEC , Knot 138 325 0.81 40 191 314 
MKTLIEIKQTPDGIIKADKVFNKVKDKISLPNRILYLGCGSSHFLSKLLAMVTNMHGGLGIALPCSEFLYSKETYPIGEVELAVGISRSGETTEILLALEKINVKKLGITTRESSLTRMCDYSLVVPAIEESVVMTHSFTSFYFAYLQLLRYSYGLPPLNAGEISKATEKSLEYERYIREIVESFDFQNIIFLGSGLLYPVALEASLKMKEMSIFWSEAYPTFEVRHGFKAIADEKTLVVLMVEEPFEWHEKLVKEFKNQGAKVLVISNSPQDLGQDYSIELPRLSKDANPIPYLPIVQLLSYYKAVSRGLNPDNPRFLDKVVRW
1VZH , Knot 67 126 0.85 40 101 123 
MPERLQVYKCEVCGNIVEVLNGGIGELVCCNQDMKLMSENTVDAAKAKHVPVIEKIDGGYKVKVGAVAHPMEEKHYIQWIELLADDKCYTQFLKPGQAPEAVFLIEAAKVVAREYCNIHGHWKAEN
9EFK , Knot 222 554 0.84 40 266 523 
MVDVKKLLGKRDNSKLYEEIRKETKPKLDVNRLCRMRNDAKSDLDMWRSILQTAYHYAMPDYNPFENYGLAGFLTPGQQYNADIYDLTLPIAHKRLADKMLMNMVPQGQQWVKFTPGDEFGEPGTPLYQRALDATQRMTDHFFKIIDRSNFYLAVGESLQDVLISTGIIAINEGNRKRPVRYEAVPPAQVMFQGDAEGQVDAIFRDWYQVRIENIKSMWPKAEVAKLNKKPEDKVDIWECAWIDYEAPEKERYQYVVMTSSKDVLLEQSNSSWPWVVYRMRRLTGEIRGRGPSLSAYPTAATINQALEDELVAAAFQANPMYMAASDSAFNQQTFTPRPGSIVPVQMVQGEWPIKPFEQSGNIQFNALLVNDFRQQINELLYAFPLGAVNSPTRTATEAEIRYTENLESFSAMVPRLQNEFFIPVIQRTLWVINKVLPETFANIPDDIRNKMISVDGQILGLSFDTPLMTAKGQVKTAALLGFYQAAASLLGPEAATASLDPVEVLTNLADNQGIDVRNIKTREELEQLLQAAGQIAQQEAAQQGVIIGSEQPQ

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 \(c\).

\(|P_{f(2DEC_1)}(2) \setminus P_{f(1VZH_1)}(2)|=135\), \(|P_{f(1VZH_1)}(2) \setminus P_{f(2DEC_1)}(2)|=45\). 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 1:1001101000101110100110010001011001101101000110011111001011111111000110000001110101111100010000111110010100111000000100100001111110001110001001011010110000111110110100100001000001001100101001111101110111101010100101110010101010011011100001111110011010001100100011011110001001100001011010001011101111011000011001101001011001101
Pair \(Z_2\) Length of longest common subsequence
2DEC_1,1VZH_1 180 3
2DEC_1,9EFK_1 161 4
1VZH_1,9EFK_1 225 4

Newick tree

 
[
	1VZH_1:10.06,
	[
		2DEC_1:80.5,9EFK_1:80.5
	]:27.56
]

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 \((0.85)(0.8)(\frac{451 }{\log_{20} 451}-\frac{126}{\log_{20}126})=97.2\)
Status Protein1 Protein2 d d1/2
Query variables 2DEC_1 1VZH_1 123 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]