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Parikh vectors
4WMI_1 3JBM_1 3ROI_1 Letter Amino acid
15 23 24 S Serine
12 11 18 Q Glutamine
17 28 45 G Glycine
34 29 44 L Leucine
21 11 10 F Phenylalanine
22 28 33 V Valine
12 11 34 I Isoleucine
19 22 24 P Proline
5 8 0 W Tryptophan
11 9 7 Y Tyrosine
15 26 22 R Arginine
10 16 17 N Asparagine
15 22 25 D Aspartic acid
10 32 26 T Threonine
14 12 20 K Lycine
10 4 13 M Methionine
21 27 44 A Alanine
5 4 7 C Cysteine
22 11 20 E Glutamic acid
16 4 8 H Histidine 

4WMI_1|Chain A|Xyloside xylosyltransferase 1|Mus musculus (10090)
>3JBM_1|Chains A, B, C|virus-like particle of orange-spotted grouper nervous necrosis virus|Orange-spotted grouper nervous necrosis virus (421750)
>3ROI_1|Chains A, B|3-phosphoshikimate 1-carboxyvinyltransferase|Coxiella burnetii (227377)
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 \(w=f(c)\)
4WMI , Knot 138 306 0.86 40 204 293 
SLEGGVVVPVDYHLLMMFTKAEHNAPLQAKARVALSSLLRLAKFEAHEVLNLHFVSEEASREVAKALLRELLPPAAGFKCKVIFHDVAVLTDKLFPVVEAMQKYFSAGSGTYYSDSIFFLSVAMHQIMPKEIPRIIQLDLDLKYKTNIRELFEEFDNFLPGAVIGIAREMQPVYRHTFWQFRHENPKTRVGDPPPEGLPGFNSGVMLLNLEAMRQSPLYSHLLEPSWVQQLADKYHFRGHLGDQDFFTMIGMEHPELFHVLDCTWNRQLCTWWRDHGYSDVFQAYFRCEGHVKIYHGNCNTPIPED
3JBM , Knot 146 338 0.83 40 200 324 
MVRKGEKKLAKPATTKAANPQPRRRANNRRRSNRTDAPVSKASTVTGFGRGTNDVHLSGMSRISQAVLPAGTGTDGYVVVDATIVPDLLPRLGHAARIFQRYAVETLEFEIQPMCPANTGGGYVAGFLPDPTDNDHTFDALQATRGAVVAKWWESRTVRPQYTRTLLWTSSGKEQRLTSPGRLILLCVGNNTDVVNVSVLCRWSVRLSVPSLETPEETTAPIMTQGSLYNDSLSTNDFKSILLGSTPLDIAPDGAVFQLDRPLSIDYSLGTGDVDRAVYWHLKKFAGNAGTPAGWFRWGIWDNFNKTFTDGVAYYSDEQPRQILLPVGTVCTRVDSEN
3ROI , Knot 183 441 0.84 38 220 403 
SNAMDYQTIPSQGLSGEICVPGDKSISHRAVLLAAIAEGQTQVDGFLMGADNLAMVSALQQMGASIQVIEDENILVVEGVGMTGLQAPPEALDCGNSGTAIRLLSGLLAGQPFNTVLTGDSSLQRRPMKRIIDPLTLMGAKIDSTGNVPPLKIYGNPRLTGIHYQLPMASAQVKSCLLLAGLYARGKTCITEPAPSRDHTERLLKHFHYTLQKDKQSICVSGGGKLKANDISIPGDISSAAFFIVAATITPGSAIRLCRVGVNPTRLGVINLLKMMGADIEVTHYTEKNEEPTADITVRHARLKGIDIPPDQVPLTIDEFPVLLIAAAVAQGKTVLRDAAELRVKETDRIAAMVDGLQKLGIAAESLPDGVIIQGGTLEGGEVNSYDDHRIAMAFAVAGTLAKGPVRIRNCDNVKTSFPNFVELANEVGMNVKGVRGRGGF

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 \(c\).

\(|P_{f(4WMI_1)}(2) \setminus P_{f(3JBM_1)}(2)|=89\), \(|P_{f(3JBM_1)}(2) \setminus P_{f(4WMI_1)}(2)|=85\). 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 1:010111111100011111001000111010101110011011010100110101100010001101110011111111000111001111000111110110001011010000001111011100111001101101010100000100110010011111111110010110000110100001000110111011111001111101011000110001101011001100001010110001101111001011011000100010011000100011010100010101001000011100
Pair \(Z_2\) Length of longest common subsequence
4WMI_1,3JBM_1 174 4
4WMI_1,3ROI_1 180 4
3JBM_1,3ROI_1 154 3

Newick tree

 
[
	4WMI_1:92.03,
	[
		3JBM_1:77,3ROI_1:77
	]:15.03
]

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 \((0.85)(0.8)(\frac{644 }{\log_{20} 644}-\frac{306}{\log_{20}306})=93.9\)
Status Protein1 Protein2 d d1/2
Query variables 4WMI_1 3JBM_1 118 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]