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Parikh vectors
2HKN_1 7NBK_1 4EGS_1 Letter Amino acid
8 1 15 A Alanine
4 0 7 I Isoleucine
7 0 18 S Serine
2 0 5 Y Tyrosine
1 0 8 M Methionine
6 0 9 D Aspartic acid
1 2 2 C Cysteine
4 0 3 Q Glutamine
14 2 18 G Glycine
3 0 9 H Histidine
3 0 15 L Leucine
5 0 15 K Lycine
1 0 1 W Tryptophan
6 0 7 R Arginine
1 0 2 N Asparagine
4 0 7 F Phenylalanine
8 1 5 T Threonine
10 0 11 V Valine
6 0 16 E Glutamic acid
3 0 7 P Proline 

2HKN_1|Chains A, B|Dynactin-1|Homo sapiens (9606)
>7NBK_1|Chains A, B|DNA (5'-D(*CP*GP*TP*AP*(FFC)P*G)-3')|synthetic construct (32630)
>4EGS_1|Chains A, B|Ribose 5-phosphate isomerase RpiB|Thermoanaerobacter tengcongensis (273068)
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 \(w=f(c)\)
2HKN , Knot 50 97 0.78 40 79 94 
GSHMSAEASARPLRVGSRVEVIGKGHRGTVAYVGATLFATGKWVGVILDEAKGKNDGTVQGRKYFTCDEGHGIFVRQSQIQVFEDGADTTSPETPDS
7NBK , Knot 5 6 0.49 8 4 4 
CGTACG
4EGS , Knot 84 180 0.80 40 125 171 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMRVLFVCTGNTCRSPMAEGIFNAKSKALGKDWEAKSAGVFAPEGFPASSEAVEVLKKEYGIDISDHRAKSLREEDLKGADLVLAMAFSHKRSLVSQYPEYADKIFTIKEFVGLEGDVEDPYGMPLEVYKKTAEELSGLIDKLIEKL

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 \(c\).

\(|P_{f(2HKN_1)}(2) \setminus P_{f(7NBK_1)}(2)|=78\), \(|P_{f(7NBK_1)}(2) \setminus P_{f(2HKN_1)}(2)|=3\). 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 1:1001010101011011001011101001011011101110101111110010100010101000100001011110000101100110000100100
Pair \(Z_2\) Length of longest common subsequence
2HKN_1,7NBK_1 81 2
2HKN_1,4EGS_1 132 4
7NBK_1,4EGS_1 127 2

Newick tree

 
[
	4EGS_1:71.03,
	[
		2HKN_1:40.5,7NBK_1:40.5
	]:30.53
]

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 \((0.85)(0.8)(\frac{103 }{\log_{20} 103}-\frac{6}{\log_{20}6})=38.4\)
Status Protein1 Protein2 d d1/2
Query variables 2HKN_1 7NBK_1 48 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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