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Parikh vectors
4QQB_1 8OQH_1 6JMZ_1 Letter Amino acid
0 7 22 I Isoleucine
0 0 19 F Phenylalanine
0 7 17 S Serine
0 5 5 R Arginine
0 5 16 N Asparagine
0 3 9 Q Glutamine
0 2 14 E Glutamic acid
0 1 11 D Aspartic acid
0 2 11 Y Tyrosine
4 5 21 A Alanine
0 7 32 K Lycine
0 3 10 T Threonine
0 3 1 M Methionine
0 10 12 P Proline
0 0 0 W Tryptophan
0 8 11 V Valine
2 8 0 C Cysteine
4 7 19 G Glycine
0 4 11 H Histidine
0 4 21 L Leucine 

4QQB_1|Chains A[auth C], C[auth P]|msl2 mRNA|null
>8OQH_1|Chains A, B|Type 1 phosphatidylinositol 4,5-bisphosphate 4-phosphatase|Homo sapiens (9606)
>6JMZ_1|Chain A|Peptidase M23|Campylobacter jejuni (197)
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 \(w=f(c)\)
4QQB , Knot 7 18 0.37 8 10 11 
UUUUUUUGAGCACGUGAA
8OQH , Knot 49 91 0.81 36 78 89 
GSHMSPDSGSAPMITCRVCQSLINVEGKMHQHVVKCGVCNEATPIKNAPPGKKYVRCPCNSLLIAKVTSQRIACPRPYCKRIINLGPVHPG
6JMZ , Knot 117 262 0.83 36 156 247 
MELIKGQALFLELDKKDFLSLKNNDKNIPTFAHPKNQEKILAIFSLPYKNPPQNTKLIAFYKDKKEEIFIKTLEGNYKSEKLQVENKKIFPPKTIQERIAKELKEANAIYSSYTPKALFNGAFNIPLNSFITSDFGKARTFNEKVASYHSGTDFRAATGTPIYAANSGVVKIAKDRYFAGNSVVIDHGFGIYSQYYHLSKIDVKVGQKIKKGELIGLSGASGRVSGPALHFGILAGGKQVDPLDFVSKFNAIFQLEHHHHHH

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 \(c\).

\(|P_{f(4QQB_1)}(2) \setminus P_{f(8OQH_1)}(2)|=8\), \(|P_{f(8OQH_1)}(2) \setminus P_{f(4QQB_1)}(2)|=76\). 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 1:000000011101010111
Pair \(Z_2\) Length of longest common subsequence
4QQB_1,8OQH_1 84 2
4QQB_1,6JMZ_1 160 2
8OQH_1,6JMZ_1 172 3

Newick tree

 
[
	6JMZ_1:92.78,
	[
		4QQB_1:42,8OQH_1:42
	]:50.78
]

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 \((0.85)(0.8)(\frac{109 }{\log_{20} 109}-\frac{18}{\log_{20}18})=34.6\)
Status Protein1 Protein2 d d1/2
Query variables 4QQB_1 8OQH_1 47 26
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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