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Parikh vectors
1SXU_1 5QXJ_1 1BZD_1 Letter Amino acid
22 6 12 A Alanine
3 6 8 P Proline
13 5 5 Y Tyrosine
3 11 5 I Isoleucine
7 5 5 F Phenylalanine
4 3 1 C Cysteine
3 3 0 Q Glutamine
6 13 12 E Glutamic acid
1 2 1 M Methionine
14 6 12 T Threonine
2 14 4 R Arginine
14 3 3 N Asparagine
3 3 4 H Histidine
16 13 7 L Leucine
21 7 8 K Lycine
11 6 12 S Serine
1 0 2 W Tryptophan
13 7 12 V Valine
17 16 5 D Aspartic acid
10 1 9 G Glycine 

1SXU_1|Chain A|Nitrophorin 4|Rhodnius prolixus (13249)
>5QXJ_1|Chain A|ATPase family AAA domain-containing protein 2|Homo sapiens (9606)
>1BZD_1|Chains A, B|PROTEIN (TRANSTHYRETIN)|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 \(w=f(c)\)
1SXU , Knot 87 184 0.82 40 126 179 
ACTKNAIAQTGFNKDKYFNGDVWYVTDYLNLEPDDVPKRYCAALAAGTASGKLKEALYHYDPKTQDTFYDVSELQVESLGKYTANFKKVDKNGNVKVAVTAGNYYTFTVMYADDSSALIHTCLHKGNKDLGDLYAVLNRNKDAAAGDKVKSAVSAATLEFSKFISTKENNCAYDNDSLKSLLTK
5QXJ , Knot 63 130 0.78 38 97 125 
SMQEEDTFRELRIFLRNVTHRLAIDKRFRVFTKPVDPDEVPDYRTVIKEPMDLSSVISKIDLHKYLTVKDYLRDIDLICSNALEYNPDRDPGDRLIRHRACALRDTAYAIIKEELDEDFEQLCEEIQESR
1BZD , Knot 64 127 0.81 38 101 123 
GPTGTSESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE

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 \(c\).

\(|P_{f(1SXU_1)}(2) \setminus P_{f(5QXJ_1)}(2)|=84\), \(|P_{f(5QXJ_1)}(2) \setminus P_{f(1SXU_1)}(2)|=55\). 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 1:1000011100110000010101101000101010011000011111101010100110000100000100100101001100010100100010101110110000101101000011100010010001101011100000111100100110110101001100000001000001001100
Pair \(Z_2\) Length of longest common subsequence
1SXU_1,5QXJ_1 139 3
1SXU_1,1BZD_1 137 4
5QXJ_1,1BZD_1 146 3

Newick tree

 
[
	5QXJ_1:72.17,
	[
		1SXU_1:68.5,1BZD_1:68.5
	]: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 \((0.85)(0.8)(\frac{314 }{\log_{20} 314}-\frac{130}{\log_{20}130})=56.8\)
Status Protein1 Protein2 d d1/2
Query variables 1SXU_1 5QXJ_1 70 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]