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Parikh vectors
3HZH_1 5IDR_1 3DMX_1 Letter Amino acid
5 15 8 E Glutamic acid
6 4 5 M Methionine
10 14 6 S Serine
0 1 3 W Tryptophan
3 5 6 Y Tyrosine
9 27 17 A Alanine
12 16 10 D Aspartic acid
18 13 10 I Isoleucine
9 26 15 L Leucine
11 12 12 T Threonine
13 13 9 V Valine
7 9 12 N Asparagine
2 2 0 C Cysteine
9 6 11 G Glycine
7 2 1 H Histidine
14 23 13 K Lycine
5 10 3 P Proline
5 7 13 R Arginine
7 5 5 F Phenylalanine
5 14 5 Q Glutamine 

3HZH_1|Chain A|Chemotaxis response regulator (CheY-3)|Borrelia burgdorferi (139)
>5IDR_1|Chains A, B, C, D, E, F|DsbA-like protein|Proteus mirabilis ATCC 29906 (525369)
>3DMX_1|Chain A|Lysozyme|Bacteriophage T4 (10665)
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 \(w=f(c)\)
3HZH , Knot 77 157 0.82 38 115 150 
MRGSHHHHHHGSIQKTTIAADSSSKPRGINYDTGIPFNVLIVDDSVFTVKQLTQIFTSEGFNIIDTAADGEEAVIKYKNHYPNIDIVTLDITMPKMDGITCLSNIMEFDKNARVIMISALGKEQLVKDCLIKGAKTFIVKPLDRAKVLQRVMSVFVK
5IDR , Knot 101 224 0.81 40 155 220 
SNAALNAAQEKEVRALVRDTLVSNPEILEEAIMALQTKKADEQQAQFRQALASEHDALYNDAASPRIGAKDAKLVLVSFTDYNCPYCKRFDPLLEKITEQYPDVAVIIKPLPFKGESSAKASQAVLSVWKEDPKAFLALHQRLMQKKTMLDNASIEDAMKSTNTSKIKLTDDSLKTLQNNLELSRKLGIQGTPATVIGDTILPGAVDYDQLEIIVKEQLAKVKK
3DMX , Knot 77 164 0.79 38 116 157 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAAAINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNL

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 \(c\).

\(|P_{f(3HZH_1)}(2) \setminus P_{f(5IDR_1)}(2)|=54\), \(|P_{f(5IDR_1)}(2) \setminus P_{f(3HZH_1)}(2)|=94\). 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 1:1010000000101000011100000101100001111011110001101001001100011011001101001110000001010110101011010110010011010001011110111000110001101100111011001011001101110
Pair \(Z_2\) Length of longest common subsequence
3HZH_1,5IDR_1 148 4
3HZH_1,3DMX_1 151 3
5IDR_1,3DMX_1 153 4

Newick tree

 
[
	3DMX_1:76.65,
	[
		3HZH_1:74,5IDR_1:74
	]:2.65
]

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 \((0.85)(0.8)(\frac{381 }{\log_{20} 381}-\frac{157}{\log_{20}157})=67.3\)
Status Protein1 Protein2 d d1/2
Query variables 3HZH_1 5IDR_1 86 71
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]