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Parikh vectors
3RZY_1 1XYS_1 3OGT_1 Letter Amino acid
2 3 6 C Cysteine
4 18 14 Q Glutamine
12 14 10 T Threonine
2 15 8 Y Tyrosine
17 25 15 V Valine
7 20 13 R Arginine
9 11 17 E Glutamic acid
6 15 8 F Phenylalanine
11 25 16 D Aspartic acid
12 25 11 G Glycine
8 16 18 I Isoleucine
14 9 14 K Lycine
1 17 14 P Proline
10 24 24 S Serine
2 7 1 W Tryptophan
6 38 12 A Alanine
4 29 7 N Asparagine
1 7 9 H Histidine
5 23 34 L Leucine
6 6 8 M Methionine 

3RZY_1|Chain A|Fatty acid-binding protein, adipocyte|Homo sapiens (9606)
>1XYS_1|Chains A, B|XYLANASE A|Cellvibrio japonicus (155077)
>3OGT_1|Chain A|Vitamin D3 receptor|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 \(w=f(c)\)
3RZY , Knot 70 139 0.82 40 108 133 
QQMGRGSMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA
1XYS , Knot 149 347 0.83 40 205 335 
GLASLADFPIGVAVAASGGNADIFTSSARQNIVRAEFNQITAENIMKMSYMYSGSNFSFTNSDRLVSWAAQNGQTVHGHALVWHPSYQLPNWASDSNANFRQDFARHIDTVAAHFAGQVKSWDVVNEALFDSADDPDGRGSANGYRQSVFYRQFGGPEYIDEAFRRARAADPTAELYYNDFNTEENGAKTTALVNLVQRLLNNGVPIDGVGFQMHVMNDYPSIANIRQAMQKIVALSPTLKIKITCLDVRLNNPYDGNSSNDYTNRNDCAVSCAGLDRQKARYKEIVQAYLEVVPPGRRGGITVWGIADPDSWLYTHQNLPDWPLLFNDNLQPKPAYQGVVEALSGR
3OGT , Knot 114 259 0.81 40 173 247 
DSLRPKLSEEQQRIIAILLDAHHKTYDPTYSDFCQFRPPVRVNDGGGSVTLELSQLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSEDQIVLLKSSAIEVIMLRSNESFTMDDMSWTCGNQDYKYRVSDVTKAGHSLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAALIEAIQDRLSNTLQTYIRCRHPPPGSHLLYAKMIQKLADLRSLNEEHSKQYRCLSFQPECSMKLTPLVLEVFGNEIS

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 \(c\).

\(|P_{f(3RZY_1)}(2) \setminus P_{f(1XYS_1)}(2)|=49\), \(|P_{f(1XYS_1)}(2) \setminus P_{f(3RZY_1)}(2)|=146\). 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 1:0011010100111101011000010001001111110001111101011101010110100000100001011110010010100001000101011111010010100001000000001110011011000010001
Pair \(Z_2\) Length of longest common subsequence
3RZY_1,1XYS_1 195 4
3RZY_1,3OGT_1 181 3
1XYS_1,3OGT_1 168 3

Newick tree

 
[
	3RZY_1:97.18,
	[
		3OGT_1:84,1XYS_1:84
	]:13.18
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{139}{\log_{20}139})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 3RZY_1 1XYS_1 134 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]