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Parikh vectors
8FKG_1 7KIJ_1 4OKA_1 Letter Amino acid
2 10 11 S Serine
2 10 23 A Alanine
1 8 5 R Arginine
0 1 7 Q Glutamine
1 3 3 P Proline
1 6 2 F Phenylalanine
0 3 0 C Cysteine
1 11 5 E Glutamic acid
0 2 2 H Histidine
2 2 4 I Isoleucine
0 2 6 W Tryptophan
1 8 10 N Asparagine
4 4 7 D Aspartic acid
3 9 8 L Leucine
2 6 7 K Lycine
0 6 9 V Valine
2 10 21 G Glycine
1 1 3 M Methionine
0 5 20 T Threonine
0 4 6 Y Tyrosine 

8FKG_1|Chain A[auth D]|Nuclear receptor corepressor 1|Homo sapiens (9606)
>7KIJ_1|Chains A, C|ATP-dependent helicase Rep|Muscovy duck circovirus (257468)
>4OKA_1|Chain A|Streptavidin|Streptomyces avidinii (1895)
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 \(w=f(c)\)
8FKG , Knot 17 23 0.77 26 22 21 
DPASNLGLEDIIRKALMGSFDDK
7KIJ , Knot 61 111 0.86 40 91 107 
MAKSGNYSYKRWVFTINNPTFEDYVHVLEFCTLDNCKFAIVGEEKGANGTPHLQGFLNLRSNARAAALEESLGGRAWLSRARGSDEDNEEFCAKESTYLRVGEPVSKGRSS
4OKA , Knot 74 159 0.78 38 109 154 
MASMTGGQQMGRDEAGITGTWYNQLGSTFIVTAGADGALTGTYESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNAHSATTWSGQYVGGAEARINTQWLLTKGTTEANAWKSTLVGHDTFTKVKPSAASIDAAKKAGVNNGNPLDAVQQ

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 \(c\).

\(|P_{f(8FKG_1)}(2) \setminus P_{f(7KIJ_1)}(2)|=15\), \(|P_{f(7KIJ_1)}(2) \setminus P_{f(8FKG_1)}(2)|=84\). 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 1:01100111001100111101000
Pair \(Z_2\) Length of longest common subsequence
8FKG_1,7KIJ_1 99 2
8FKG_1,4OKA_1 117 2
7KIJ_1,4OKA_1 126 2

Newick tree

 
[
	4OKA_1:64.11,
	[
		8FKG_1:49.5,7KIJ_1:49.5
	]:14.61
]

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 \((0.85)(0.8)(\frac{134 }{\log_{20} 134}-\frac{23}{\log_{20}23})=40.7\)
Status Protein1 Protein2 d d1/2
Query variables 8FKG_1 7KIJ_1 51 30.5
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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