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Parikh vectors
4NMP_1 2OTE_1 8PYN_1 Letter Amino acid
2 11 14 N Asparagine
3 8 21 S Serine
3 10 11 T Threonine
4 7 18 R Arginine
4 15 15 D Aspartic acid
3 2 9 Q Glutamine
8 18 33 E Glutamic acid
9 13 26 L Leucine
0 5 20 M Methionine
1 11 15 F Phenylalanine
1 13 14 Y Tyrosine
9 15 12 I Isoleucine
5 23 16 K Lycine
4 12 16 P Proline
4 9 20 A Alanine
1 0 4 C Cysteine
12 18 21 G Glycine
5 8 3 H Histidine
0 2 5 W Tryptophan
9 16 28 V Valine 

4NMP_1|Chains A, B|Golgi-associated PDZ and coiled-coil motif-containing protein|Homo sapiens (9606)
>2OTE_1|Chains A, B|GFP-like fluorescent chromoprotein cFP484|Clavularia sp. (86521)
>8PYN_1|Chain A[auth AAA]|Insulin-like growth factor 1 receptor beta chain|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 \(w=f(c)\)
4NMP , Knot 46 87 0.78 36 73 85 
GPIRKVLLLKEDHEGLGISITGGKEHGVPILISEIHPGQPADRCGGLHVGDAILAVNGVNLRDTKHKEAVTILSQQRGEIEFEVVYV
2OTE , Knot 101 216 0.83 38 152 211 
GVIKPDMKIKLKMEGNVNGHAFVIEGEGEGKPYDGTNTINLEVKEGAPLPFSYDILTNAFAYGNRAFTKYPDDIPNYFKQSFPEGYSWERTMTFEDKGIVKVKSDISMEEDSFIYEIHLKGENFPPNGPVMQKKTLKWEPSTEILYVRDGVLVGDIKHKLLLEGGGHHRVDFKTIYRAKKAVKLPDYHFVDHRIEILNHDKDYNKVTVYESAVARY
8PYN , Knot 140 321 0.84 40 195 308 
MASVNPEYFSAADVYVPDEWEVAREKITMSRELGQGSFGMVYEGVAKGVVKDEPETRVAIKTVNEAASMRERIEFLNEASVMKEFNCHHVVRLLGVVSQGQPTLVIMELMTRGDLKSYLRSLRPEMENNPVLAPPSLSKMIQMAGEIADGMAYLNANKFVHRDLAARNCMVAEDFTVKIGDFGMTRDIYETDYYRKGGKGLLPVRWMSPESLKDGVFTTYSDVWSFGVVLWEIATLAEQPYQGLSNEQVLRFVMEGGLLDKPDNCPDMLFELMRMCWQYNPKMRPSFLEIISSIKEEMEPGFREVSFYYSEENKAENLYFQ

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 \(c\).

\(|P_{f(4NMP_1)}(2) \setminus P_{f(2OTE_1)}(2)|=38\), \(|P_{f(2OTE_1)}(2) \setminus P_{f(4NMP_1)}(2)|=117\). 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 1:111001111000001111010110001111110010110110001110110111110110100000001101100001010101101
Pair \(Z_2\) Length of longest common subsequence
4NMP_1,2OTE_1 155 3
4NMP_1,8PYN_1 184 3
2OTE_1,8PYN_1 171 3

Newick tree

 
[
	8PYN_1:92.27,
	[
		4NMP_1:77.5,2OTE_1:77.5
	]:14.77
]

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 \((0.85)(0.8)(\frac{303 }{\log_{20} 303}-\frac{87}{\log_{20}87})=68.3\)
Status Protein1 Protein2 d d1/2
Query variables 4NMP_1 2OTE_1 90 61.5
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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