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Parikh vectors
1YXD_1 1KLU_1 4BVP_1 Letter Amino acid
31 9 19 A Alanine
13 8 14 K Lycine
23 12 16 T Threonine
9 6 18 H Histidine
30 15 35 L Leucine
13 13 19 P Proline
15 6 28 S Serine
5 4 22 Y Tyrosine
25 17 19 V Valine
13 9 28 N Asparagine
17 12 21 D Aspartic acid
5 2 6 C Cysteine
14 18 18 E Glutamic acid
13 8 4 R Arginine
8 3 29 Q Glutamine
23 7 21 G Glycine
18 8 25 I Isoleucine
9 3 9 M Methionine
7 15 10 F Phenylalanine
1 4 7 W Tryptophan 

1YXD_1|Chains A, B|dihydrodipicolinate synthase|Escherichia coli (562)
>1KLU_1|Chain A|HLA CLASS II HISTOCOMPATIBILITY ANTIGEN, DR ALPHA CHAIN|Homo sapiens (9606)
>4BVP_1|Chains A, B|ECTONUCLEOSIDE TRIPHOSPHATE DIPHOSPHOHYDROLASE I|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 \(w=f(c)\)
1YXD , Knot 132 292 0.85 40 177 282 
MFTGSIVAIVTPMDEKGNVCRASLKKLIDYHVASGTSAIVSVGTTGESATLNHDEHADVVMMTLDLADGRIPVIAGTGANATAEAISLTQRFNDSGIVGCLTVTPYYNRPSQEGLYQHFKAIAEHTDLPQILYNVPSRTGCDLLPETVGRLAKVKNIIGIKEATGNLTRVNQIKELVSDDFVLLSGDDASALDFMQLGGHGVISVTANVAARDMAQMCKLAAEGHFAEARVINQRLMPLHNKLFVEPNPIPVKWACKELGLVATDTLRLPMTPITDSGRETVRAALKHAGLL
1KLU , Knot 85 179 0.82 40 140 175 
EHVIIQAEFYLNPDQSGEFMFDFDGDEIFHVDMAKKETVWRLEEFGRFASFEAQGALANIAVDKANLEIMTKRSNYTPITNVPPEVTVLTNSPVELREPNVLICFIDKFTPPVVNVTWLRNGKPVTTGVSETVFLPREDHLFRKFHYLPFLPSTEDVYDCRVEHWGLDEPLLKHWEFDA
4BVP , Knot 160 368 0.85 40 222 353 
MDTNPCEKHSCIAVIDAGSTGSRLHIYSYDTDDTNTPIHIEEIWNKKIKPGFASIQPNSVTIDAYLTMLLADAPIHNIPVYFYATAGMRLLPQSQQKKYYDELDYWFRQQSQWQLVEAKTITGNDEALFDWLAVNYKLDTLKSVQNKSVGVMDMGGASVQIVFPMPKNAEISKHNQVELNIYGQNINLYVHSFLGLGQTEMSHQFLNSPSCFANDYPLPDGESGQGNAPSCKEEVTSLMNSVHKVNQQIQPLLALNPVNEWYSIGGISNLASSQLFHFENSELTNQSLLQQGDNQICHQQWDILNGQYPDDEYLYQYCLLSSYYYALMVDGYGINPNQTIHYIPPEQNLDWTIGVVLHRALEHHHHHH

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 \(c\).

\(|P_{f(1YXD_1)}(2) \setminus P_{f(1KLU_1)}(2)|=101\), \(|P_{f(1KLU_1)}(2) \setminus P_{f(1YXD_1)}(2)|=64\). 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 1:1101011111011000101001010011000110100111011001001010000010111101011010111111011010101101000100011110101010000100011000101110000110110011000100111001101101001111001010100100100110001111010010110110111011101010111001101001110101101011000111100011101011110110001111100010111011000100010111001111
Pair \(Z_2\) Length of longest common subsequence
1YXD_1,1KLU_1 165 4
1YXD_1,4BVP_1 185 3
1KLU_1,4BVP_1 188 3

Newick tree

 
[
	4BVP_1:96.57,
	[
		1YXD_1:82.5,1KLU_1:82.5
	]:14.07
]

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 \((0.85)(0.8)(\frac{471 }{\log_{20} 471}-\frac{179}{\log_{20}179})=85.5\)
Status Protein1 Protein2 d d1/2
Query variables 1YXD_1 1KLU_1 110 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]