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Parikh vectors
4JYI_1 8RAL_1 4HWO_1 Letter Amino acid
6 15 28 V Valine
9 7 17 Q Glutamine
15 12 28 G Glycine
12 5 16 H Histidine
9 12 20 F Phenylalanine
17 13 22 S Serine
22 12 16 T Threonine
4 8 14 Y Tyrosine
5 9 15 N Asparagine
35 16 33 L Leucine
12 2 16 M Methionine
14 15 14 P Proline
14 3 31 R Arginine
16 16 23 D Aspartic acid
6 3 8 C Cysteine
17 9 23 K Lycine
1 4 7 W Tryptophan
14 8 18 A Alanine
20 15 40 E Glutamic acid
19 11 22 I Isoleucine 

4JYI_1|Chains A, B|Retinoic acid receptor beta|Homo sapiens (9606)
>8RAL_1|Chain A|H-2 class II histocompatibility antigen, A-B alpha chain|Mus musculus (10090)
>4HWO_1|Chains A, B|Threonine--tRNA ligase|Escherichia coli (83333)
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 \(w=f(c)\)
4JYI , Knot 119 267 0.83 40 172 257 
MGSSHHHHHHSSGLVPRGSHMESYEMTAELDDLTEKIRKAHQETFPSLCQLGKYTTNSSADHRVRLDLGLWDKFSELATKCIIKIVEFAKRLPGFTGLTIADQITLLKAACLDILILRICTRYTPEQDTMTFSDGLTLNRTQMHNAGFGPLTDLVFTFANQLLPLEMDDTETGLLSAICLICGDRQDLEEPTKVDKLQEPLLEALKIYIRKRRPSKPHMFPKILMKITDLRSISAKGAERVITLKMEIPGSMPPLIQEMMENSEGHE
8RAL , Knot 95 195 0.85 40 150 191 
EDDIEADHVGTYGISVYQSPGDIGQYTFEFDGDELFYVDLDKKETVWMLPEFGQLASFDPQGGLQNIAVVKHNLGCLTKRSNSTPATNEAPQATVFPKSPVLLGQPNTLICFVDNIFPPVINITWLRNSKSVADGVYETSFFVNRDYSFHKLSYLTFIPSDDDIYDCKVEHWGLEEPVLKHWEPEIPAPMSELTE
4HWO , Knot 175 411 0.85 40 243 391 
MARDHRKIGKQLDLYHMQEEAPGMVFWHNDGWTIFRELEVFVRSKLKEYQYQEVKGPFMMDRVLWEKTGHWDNYKDAMFTTSSENREYCIKPMNCPGHVQIFNQGLKSYRDLPLRMAEFGSCHRNEPSGSLHGLMRVRGFTQDDAHIFCTEEQIRDEVNGCIRLVYDMYSTFGFEKIVVKLSTRPEKRIGSDEMWDRAEADLAVALEENNIPFEYQLGEGAFYGPKIEFTLYDCLDRAWQCGTVQLDFSLPSRLSASYVGEDNERKVPVMIHRAILGSMERFIGILTEEFAGFFPTWLAPVQVVIMNITDSQSEYVNELTQKLSNAGIRVKADLRNEKIGFKIREHTLRRVPYMLVCGDKEVESGKVAVRTRRGKDLGSMDVNEVIEKLQQEIRSRSLKQLEELEHHHHHH

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 \(c\).

\(|P_{f(4JYI_1)}(2) \setminus P_{f(8RAL_1)}(2)|=105\), \(|P_{f(8RAL_1)}(2) \setminus P_{f(4JYI_1)}(2)|=83\). 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 1:110000000000111101001000010101001000100100001101001100000001000101011110010011000110110110011110110110010110110101111010000010000101001101000010011111100111011001111010000011101101101000010010010010011101101010000100101110111010010010101100110101011101111100110000100
Pair \(Z_2\) Length of longest common subsequence
4JYI_1,8RAL_1 188 4
4JYI_1,4HWO_1 171 6
8RAL_1,4HWO_1 189 3

Newick tree

 
[
	8RAL_1:96.99,
	[
		4JYI_1:85.5,4HWO_1:85.5
	]:11.49
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{195}{\log_{20}195})=78.0\)
Status Protein1 Protein2 d d1/2
Query variables 4JYI_1 8RAL_1 99 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]