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Parikh vectors
3SIU_1 8SHE_1 4XAL_1 Letter Amino acid
12 33 16 A Alanine
2 11 8 H Histidine
5 23 9 T Threonine
3 10 1 Y Tyrosine
10 12 4 Q Glutamine
16 32 11 L Leucine
9 40 7 S Serine
6 18 12 R Arginine
4 19 2 C Cysteine
6 28 7 G Glycine
10 24 3 K Lycine
2 11 5 M Methionine
2 17 4 F Phenylalanine
10 29 11 V Valine
5 14 7 N Asparagine
5 27 8 D Aspartic acid
8 21 5 E Glutamic acid
9 11 2 I Isoleucine
6 7 7 P Proline
0 8 2 W Tryptophan 

3SIU_1|Chains A, D|NHP2-like protein 1|Homo sapiens (9606)
>8SHE_1|Chain A[auth N]|Guanine nucleotide-binding protein subunit beta-5|Homo sapiens (9606)
>4XAL_1|Chain A|Tegument protein VP22|Human herpesvirus 1 (strain 17) (10299)
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 \(w=f(c)\)
3SIU , Knot 66 130 0.82 38 107 125 
GSMTEADVNPKAYPLADAHLTKKLLDLVQQSCNYKQLRKGANEATKTLNRGISEFIVMAADAEPLEIILHLPLLCEDKNVPYVFVRSKQALGRACGVSRPVIACSVTIKEGSQLKQQIQSIQQSIERLLV
8SHE , Knot 169 395 0.85 40 221 367 
MCDQTFLVNVFGSCDKCFKQRALRPVFKKSQQLSYCSTCAEIMATEGLHENETLASLKSEAESLKGKLEEERAKLHDVELHQVAERVEALGQFVMKTRRTLKGHGNKVLCMDWCKDKRRIVSSSQDGKVIVWDSFTTNKEHAVTMPCTWVMACAYAPSGCAIACGGLDNKCSVYPLTFDKNENMAAKKKSVAMHTNYLSACSFTNSDMQILTASGDGTCALWDVESGQLLQSFHGHGADVLCLDLAPSETGNTFVSGGCDKKAMVWDMRSGQCVQAFETHESDINSVRYYPSGDAFASGSDDATCRLYDLRADREVAIYSKESIIFGASSVDFSLSGRLLFAGYNDYTINVWDVLKGSRVSILFGHENRVSTLRVSPDGTAFCSGSWDHTLRVWA
4XAL , Knot 62 131 0.77 40 100 123 
MHHHHHHSSGVDLGTENLYFQSMRKLHFSTAPPNPDAPWTPRVAGFNKRVFCAAVGRLAAMHARMAAVQLWDMSRPRTDEDLNELLGITTIRVTVCEGKNLLQRANELVNPDVVQDVDAATATRGRSAASR

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 \(c\).

\(|P_{f(3SIU_1)}(2) \setminus P_{f(8SHE_1)}(2)|=28\), \(|P_{f(8SHE_1)}(2) \setminus P_{f(3SIU_1)}(2)|=142\). 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 1:1010010101010111010100011011000000001001100100010011001111110101101110111100000110111000011101011001111001010010010001001000100111
Pair \(Z_2\) Length of longest common subsequence
3SIU_1,8SHE_1 170 3
3SIU_1,4XAL_1 129 3
8SHE_1,4XAL_1 191 3

Newick tree

 
[
	8SHE_1:97.51,
	[
		3SIU_1:64.5,4XAL_1:64.5
	]:33.01
]

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 \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{130}{\log_{20}130})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 3SIU_1 8SHE_1 146 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]