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Parikh vectors
3BAS_1 6HTS_1 9FRG_1 Letter Amino acid
3 33 19 A Alanine
4 18 35 R Arginine
16 44 22 E Glutamic acid
2 10 11 H Histidine
0 0 12 W Tryptophan
2 29 34 T Threonine
4 21 30 D Aspartic acid
3 36 18 G Glycine
13 41 21 K Lycine
5 13 22 M Methionine
1 17 25 P Proline
1 8 27 F Phenylalanine
4 24 43 S Serine
1 12 20 Y Tyrosine
4 13 15 N Asparagine
0 6 3 C Cysteine
6 18 15 Q Glutamine
2 40 26 I Isoleucine
14 42 46 L Leucine
4 31 35 V Valine 

3BAS_1|Chains A, B|Myosin heavy chain, striated muscle/General control protein GCN4 chimera|Argopecten irradians (31199)
>6HTS_1|Chains A, C, E|RuvB-like 1|Homo sapiens (9606)
>9FRG_1|Chains A, B, C, D, E|Gamma-aminobutyric acid receptor subunit rho-1|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 \(w=f(c)\)
3BAS , Knot 46 89 0.77 36 61 85 
GSHMPLLSIARQEEEMKEQLKQMDKMKEDLAKTERIKKELEEQNVTLLEQKNDLFGSMKQLEDKVEELLSKNYHLENEVARLKKLVGER
6HTS , Knot 185 456 0.82 38 231 434 
MKIEEVKSTTKTQRIASHSHVKGLGLDESGLAKQAASGLVGQENAREACGVIVELIKSKKMAGRAVLLAGPPGTGKTALALAIAQELGSKVPFCPMVGSEVYSTEIKKTEVLMENFRRAIGLRIKETKEVYEGEVTELTPCETENPMGGYGKTISHVIIGLKTAKGTKQLKLDPSIFESLQKERVEAGDVIYIEANSGAVKRQGRCDTYATEFDLEAEEYVPLPKGDVHKKKEIIQDVTLHDLDVANARPQGGQDILSMMGQLMKPKKTEITDKLRGEINKVVNKYIDQGIAELVPGVLFVDEVHMLDIECFTYLHRALESSIAPIVIFASNRGNCVIRGTEDITSPHGIPLDLLDRVMIIRTMLYTPQEMKQIIKIRAQTEGINISEEALNHLGEIGTKTTLRYSVQLLTPANLLAKINGKDSIEKEHVEEISELFYDAKSSAKILADQQDKYMK
9FRG , Knot 197 479 0.84 40 237 452 
MLAVPNMRFGIFLLWWGWVLATESRMHWPGREVHEMSKKGRPQRQRREVHEDAHKQVSPILRRSPDITKSPLTKSEQLLRIDDHDFSMRPGFGGPAIPVGVDVQVESLDSISEVDMDFTMTLYLRHYWKDERLSFPSTNNLSMTFDGRLVKKIWVPDMFFVHSKRSFIHDTTTDNVMLRVQPDGKVLYSLRVTVTAMCNMDFSRFPLDTQTCSLEIESYAYTEDDLMLYWKKGNDSLKTDERISLSQFLIQEFHTTTKLAFYSSTGWYNRLYINFTLRRHIFFFLLQTYFPATLMVMLSWVSFWIDRRAVPARVPLGITTVLTMSTIITGVNASMPRVSYIKAVDIYLWVSFVFVFLSVLEYAAVNYLTTVQERKEQKLREKLPCTSGLPPPRTAMLDGNYSDGEVNDLDNYMPENGEKPDRMMVQLTLASERSSPQRKSQRSSYVSMRIDTHAIDKYSRIIFPAAYILFNLIYWSIFS

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 \(c\).

\(|P_{f(3BAS_1)}(2) \setminus P_{f(6HTS_1)}(2)|=13\), \(|P_{f(6HTS_1)}(2) \setminus P_{f(3BAS_1)}(2)|=183\). 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 1:10011110110000010001001001000110000100010000101100000111010010001001100000100011010011100
Pair \(Z_2\) Length of longest common subsequence
3BAS_1,6HTS_1 196 3
3BAS_1,9FRG_1 208 3
6HTS_1,9FRG_1 166 3

Newick tree

 
[
	3BAS_1:10.38,
	[
		6HTS_1:83,9FRG_1:83
	]:23.38
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{89}{\log_{20}89})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 3BAS_1 6HTS_1 171 100.5
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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