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Parikh vectors
1PHT_1 5ZGO_1 7SJD_1 Letter Amino acid
6 6 11 D Aspartic acid
0 0 1 C Cysteine
10 14 6 E Glutamic acid
11 14 9 G Glycine
5 8 2 I Isoleucine
2 2 10 F Phenylalanine
4 18 12 A Alanine
3 2 6 N Asparagine
7 3 4 Y Tyrosine
7 22 13 L Leucine
4 15 2 P Proline
1 0 1 W Tryptophan
5 11 6 R Arginine
1 2 2 H Histidine
4 12 7 T Threonine
2 4 7 Q Glutamine
4 6 12 S Serine
3 28 8 V Valine
5 9 12 K Lycine
1 2 6 M Methionine 

1PHT_1|Chain A|PHOSPHATIDYLINOSITOL 3-KINASE P85-ALPHA SUBUNIT|Homo sapiens (9606)
>5ZGO_1|Chains A, B, C, D, E, F|Adenine phosphoribosyltransferase|Thermus thermophilus (strain HB8 / ATCC 27634 / DSM 579) (300852)
>7SJD_1|Chains A, B|Dehaloperoxidase B|Amphitrite ornata (129555)
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 \(w=f(c)\)
1PHT , Knot 46 85 0.80 38 75 83 
MSAEGYQYRALYDYKKEREEDIDLHLGDILTVNKGSLVALGFSDGQEARPEEIGWLNGYNETTGERGDFPGTYVEYIGRKKISPP
5ZGO , Knot 79 178 0.76 36 112 166 
MRTYPVEIAGVRRELPIVQVGPGVAVALLNLLGDTELTEAAAEALAKRLPPEVEVLVTPEVKAVPLAHALSRITGKPYVVARKTEKPYMINPVSRQVLSITTGKPQLLVLDGADIPRVRGKKVAIVDDVVSTGSTLAGLRELIESVGGEVVAVLAVFTEGTPRQDVVALGHLPLFKPE
7SJD , Knot 68 137 0.81 40 105 134 
GFKQDIATLRGDLRTYAQDIFLAFLNKYPDEKRNFKNYVGKSDQELKSMAKFGDHTEKVFNLMMEVADRATDCVPLASDASTLVQMKQHSGLTTGNFEKLFVALVEYMRASGQSFDSQSWDRFGKNLVSALSSAGMK

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 \(c\).

\(|P_{f(1PHT_1)}(2) \setminus P_{f(5ZGO_1)}(2)|=45\), \(|P_{f(5ZGO_1)}(2) \setminus P_{f(1PHT_1)}(2)|=82\). 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 1:1010100001100000000001010110110100101111110010010100111101000001001011100100110001011
Pair \(Z_2\) Length of longest common subsequence
1PHT_1,5ZGO_1 127 4
1PHT_1,7SJD_1 136 3
5ZGO_1,7SJD_1 129 4

Newick tree

 
[
	7SJD_1:67.17,
	[
		1PHT_1:63.5,5ZGO_1:63.5
	]:3.67
]

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 \((0.85)(0.8)(\frac{263 }{\log_{20} 263}-\frac{85}{\log_{20}85})=57.1\)
Status Protein1 Protein2 d d1/2
Query variables 1PHT_1 5ZGO_1 68 50.5
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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