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Parikh vectors
7GYK_1 5JVF_1 3USY_1 Letter Amino acid
41 31 22 A Alanine
17 8 10 N Asparagine
29 18 12 D Aspartic acid
26 12 11 T Threonine
1 1 0 W Tryptophan
23 17 8 R Arginine
3 2 0 C Cysteine
6 7 2 H Histidine
27 9 21 I Isoleucine
28 12 22 K Lycine
13 14 6 P Proline
15 12 12 Q Glutamine
30 24 11 G Glycine
10 4 4 Y Tyrosine
21 9 23 E Glutamic acid
30 35 23 L Leucine
4 8 8 M Methionine
19 9 9 F Phenylalanine
22 17 14 S Serine
29 36 16 V Valine 

7GYK_1|Chain A|Heat shock 70 kDa protein 1A|Homo sapiens (9606)
>5JVF_1|Chain A|Site-determining protein|Pseudomonas aeruginosa PAO1 (208964)
>3USY_1|Chains A, B|Flagellar motor switch protein|Helicobacter pylori (210)
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 \(w=f(c)\)
7GYK , Knot 164 394 0.83 40 210 374 
GPLGSMAKAAAIGIDLGTTYSCVGVFQHGKVEIIANDQGNRTTPSYVAFTDTERLIGDAAKNQVALNPQNTVFDAKRLIGRKFGDPVVQSDMKHWPFQVINDGDKPKVQVSYKGETKAFYPEEISSMVLTKMKEIAEAYLGYPVTNAVITVPAYFNDSQRQATKDAGVIAGLNVLRIINEPTAAAIAYGLDRTGKGERNVLIFDLGGGTFDVSILTIDDGIFEVKATAGDTHLGGEDFDNRLVNHFVEEFKRKHKKDISQNKRAVRRLRTACERAKRTLSSSTQASLEIDSLFEGIDFYTSITRARFEELCSDLFRSTLEPVEKALRDAKLDKAQIHDLVLVGGSTRIPKVQKLLQDFFNGRDLNKSINPDEAVAYGAAVQAAILIKSTRAAAS
5JVF , Knot 122 285 0.80 40 176 265 
GPLGSMKQMGSMHPVQVIAVTGGKGGVGKTNVSVNLALALADLGRRVMLLDADLGLANVDVLLGLTPKRTLADVIEGRCELRDVLLLGPGGVRIVPAASGTQSMVHLSPMQHAGLIQAFSDISDNLDVLVVDTAAGIGDSVVSFVRAAQEVLLVVCDEPTSITDAYALIKLLNRDHGMTRFRVLANMAHSPQEGRNLFAKLTKVTDRFLDVALQYVGVIPYDESVRKAVQKQRAVYEAFPRSKASLAFKAVAQKVDSWPLPANPRGHLEFFVERLVQHPATGSAV
3USY , Knot 108 234 0.84 36 156 224 
GPLGSMQKNFAYLGKIKPQQLADFIINEHPQTIALILAHMEAPNAAETLSYFPDEMKAEISIRMANLGEISPQVVKRVSTVLENKLESLTSYKIEVGGLRAVAEIFNRLGQKSAKTTLARIESVDNKLAGAIKEMMFTFEDIVKLDNFAIREILKVADKKDLSLALKTSTKDLTDKFLNNMSSRAAEQFVEEMQYLGAVKIKDVDVAQRKIIEIVQSLQEKGVIQTGEEEDVIE

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 \(c\).

\(|P_{f(7GYK_1)}(2) \setminus P_{f(5JVF_1)}(2)|=87\), \(|P_{f(5JVF_1)}(2) \setminus P_{f(7GYK_1)}(2)|=53\). 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 1:1111011011111101100000111100101011100010000100111000001110110001110100011010011100110111000100111011001001010100010001101001001110010011010110110011101110100000010001111111011011001011111011000101000111101111010101101001110101011000111001000110011001000000010000011001001000100010000010101001101101000100101001000110001011001100101001010011111100011010011001101001000101001110111101111100001110
Pair \(Z_2\) Length of longest common subsequence
7GYK_1,5JVF_1 140 6
7GYK_1,3USY_1 148 6
5JVF_1,3USY_1 152 6

Newick tree

 
[
	3USY_1:76.60,
	[
		7GYK_1:70,5JVF_1:70
	]:6.60
]

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 \((0.85)(0.8)(\frac{679 }{\log_{20} 679}-\frac{285}{\log_{20}285})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 7GYK_1 5JVF_1 137 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]