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Parikh vectors
2BAM_1 6URL_1 8XVP_1 Letter Amino acid
4 3 72 T Threonine
0 1 29 Y Tyrosine
0 1 86 N Asparagine
0 3 32 Q Glutamine
0 8 31 E Glutamic acid
0 1 7 H Histidine
0 6 15 F Phenylalanine
0 3 65 S Serine
0 6 47 V Valine
2 9 46 G Glycine
0 1 34 I Isoleucine
0 7 63 D Aspartic acid
0 10 48 L Leucine
0 9 50 K Lycine
0 2 18 M Methionine
4 6 81 A Alanine
0 5 18 R Arginine
2 4 0 C Cysteine
0 2 41 P Proline
0 2 15 W Tryptophan 

2BAM_1|Chains A[auth C], B[auth D]|DNA (5'-D(*TP*AP*TP*GP*GP*AP*TP*CP*CP*AP*TP*A)-3')|
>6URL_1|Chains A, B|Barrier-to-autointegration factor|Homo sapiens (9606)
>8XVP_1|Chains A, B|Inulosucrase|Limosilactobacillus reuteri (1598)
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 \(w=f(c)\)
2BAM , Knot 7 12 0.48 8 8 10 
TATGGATCCATA
6URL , Knot 49 89 0.82 40 77 85 
MTTSQKHRDFVAEPMGEKPVGSLAGIGEVLGKKLEERGFDKAYVVLGQFLVLKKDEDLFREWLKDTCGANAKQSRDCFGCLREWCDAFL
8XVP , Knot 290 798 0.81 38 272 689 
MLERKEHKKMYKSGKNWAVVTLSTAALVFGATTVNASADTNIENNDSSTVQVTTGDNDIAVKSVTLGSGQVSAASDTTIRTSANANSASSAANTQNSNSQVASSAAITSSTSSAASSNNTDSKAAQENTNTAKNDDTQKAAPANESSEAKNEPAVNVNDSSAAKNDDQQSSKKNTTAKLNKDAENVVKKAGIDPNSLTDDQIKALNKMNFSKAAKSGTQMTYNDFQKIADTLIKQDGRYTVPFFKASEIKNMPAATTKDAQTNTIEPLDVWDSWPVQDVRTGQVANWNGYQLVIAMMGIPNQNDNHIYLLYNKYGDNELSHWKNVGPIFGYNSTAVSQEWSGSAVLNSDNSIQLFYTRVDTSDNNTNHQKIASATLYLTDNNGNVSLAQVANDHIVFEGDGYYYQTYDQWKATNKGADNIAMRDAHVIEDDNGDRYLVFEASTGLENYQGEDQIYNWLNYGGDDAFNIKSLFRILSNDDIKSRATWANAAIGILKLNKDEKNPKVAELYSPLISAPMVSDEIERPNVVKLGNKYYLFAATRLNRGSNDDAWMNANYAVGDNVAMVGYVADSLTGSYKPLNDSGVVLTASVPANWRTATYSYYAVPVAGKDDQVLVTSYMTNRNGVAGKGMDSTWAPSFLLQINPDNTTTVLAKMTNQGDWIWDDSSENLDMIGDLDSAALPGERDKPVDWDLIGYGLKPHDPATPNDPETPTTPETPETPNTPKTPKTPENPGTPQTPNTPNTPEIPLTPETPKQPETQTNNRLPQTGNNANKAMIGLGMGTLLSMFGLAEINKRRFN

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 \(c\).

\(|P_{f(2BAM_1)}(2) \setminus P_{f(6URL_1)}(2)|=6\), \(|P_{f(6URL_1)}(2) \setminus P_{f(2BAM_1)}(2)|=75\). 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 1:010111000101
Pair \(Z_2\) Length of longest common subsequence
2BAM_1,6URL_1 81 2
2BAM_1,8XVP_1 270 3
6URL_1,8XVP_1 233 3

Newick tree

 
[
	8XVP_1:14.70,
	[
		2BAM_1:40.5,6URL_1:40.5
	]:10.20
]

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 \((0.85)(0.8)(\frac{101 }{\log_{20} 101}-\frac{12}{\log_{20}12})=34.7\)
Status Protein1 Protein2 d d1/2
Query variables 2BAM_1 6URL_1 46 25.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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