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Parikh vectors
4AIX_1 7LJA_1 8SQF_1 Letter Amino acid
6 10 14 D Aspartic acid
7 16 7 P Proline
10 18 12 T Threonine
2 3 16 N Asparagine
2 4 0 C Cysteine
8 12 14 Q Glutamine
11 28 16 G Glycine
4 15 19 I Isoleucine
1 4 9 M Methionine
2 18 12 F Phenylalanine
15 20 16 S Serine
7 28 21 A Alanine
7 8 6 Y Tyrosine
3 15 15 E Glutamic acid
0 6 7 H Histidine
6 45 20 L Leucine
7 23 20 V Valine
2 16 13 R Arginine
2 6 10 W Tryptophan
5 4 18 K Lycine 

4AIX_1|Chains A, B, C, D|IG LAMBDA CHAIN V-IV REGION BAU|HOMO SAPIENS (9606)
>7LJA_1|Chains A, B|Isoform 2 of Potassium channel subfamily K member 4|Homo sapiens (9606)
>8SQF_1|Chains A, B|Beta-lactamase|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
4AIX , Knot 55 107 0.80 38 82 102 
SYELTQPPSVSVSPGQTASITCSGDKLGDKYAYWYQQKPGQSPVLVIYQDSKRPSGIPERFSGSNSGNTATLTISGTQAMDEADYYCQAWDSSTAVVFGGGTKLTVL
7LJA , Knot 125 299 0.79 40 176 279 
MTTAPQEPPARPLQAGSGAGPAPGRAMRSTTLLALLALVLLYLVSGALVFRALEQPHEQQAQRELGEVREKFLRAHPCVSDQELGLLIKEVADALGGGADPETQSTSQSSHSAWDLGSAFFFSGTIITTIGYGNVALRTDAGRLFCIFYALVGIPLFGILLAGVGDRLGSSLRHGIGHIEAIFLKWHVPPELVRVLSEMLFLLIGCLLFVLTPTFVFCYMEDWSKLEAIYFVIVTLTTVGFGDYVAGADPRQDSPAYQPLVWFWILLGLAYFASVLTTIGNWLRVVSRRTSNSLEVLFQ
8SQF , Knot 121 265 0.85 38 175 261 
MRVLALSAVFLVASIIGMPAVAKEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP

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 \(c\).

\(|P_{f(4AIX_1)}(2) \setminus P_{f(7LJA_1)}(2)|=28\), \(|P_{f(7LJA_1)}(2) \setminus P_{f(4AIX_1)}(2)|=122\). 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 1:00010011010101100101000100110001010000110011111000000101110010100010010101010011001000001100001111111001011
Pair \(Z_2\) Length of longest common subsequence
4AIX_1,7LJA_1 150 3
4AIX_1,8SQF_1 171 3
7LJA_1,8SQF_1 175 3

Newick tree

 
[
	8SQF_1:90.01,
	[
		4AIX_1:75,7LJA_1:75
	]:15.01
]

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 \((0.85)(0.8)(\frac{406 }{\log_{20} 406}-\frac{107}{\log_{20}107})=91.0\)
Status Protein1 Protein2 d d1/2
Query variables 4AIX_1 7LJA_1 112 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]