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Parikh vectors
5OIT_1 2KNI_1 4NXW_1 Letter Amino acid
1 6 7 C Cysteine
10 0 14 Q Glutamine
27 1 54 L Leucine
12 2 15 T Threonine
17 3 22 V Valine
15 2 21 D Aspartic acid
11 4 20 E Glutamic acid
20 1 15 I Isoleucine
13 3 21 P Proline
4 2 5 W Tryptophan
37 0 32 A Alanine
8 1 9 N Asparagine
10 0 5 M Methionine
7 1 8 F Phenylalanine
16 2 23 S Serine
13 4 23 R Arginine
28 3 17 G Glycine
5 1 5 H Histidine
9 5 14 K Lycine
6 0 12 Y Tyrosine 

5OIT_1|Chains A[auth B], B[auth D], C[auth F], D[auth H]|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (83332)
>2KNI_1|Chain A|Psalmotoxin-1|Psalmopoeus cambridgei (179874)
>4NXW_1|Chain A|Mitochondrial dynamic protein MID51|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 \(w=f(c)\)
5OIT , Knot 114 269 0.79 40 167 253 
MAGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL
2KNI , Knot 25 41 0.75 32 33 38 
SEDCIPKWKGCVNRHGDCCEGLECWKRRRSFEVCVPKTPKT
4NXW , Knot 147 342 0.83 40 191 329 
GSARKGQVDLKKSRLRMSLQEKLLTYYRNRAAIPAGEQARAKQAAVDICAELRSFLRAKLPDMPLRDMYLSGSLYDDLQVVTADHIQLIVPLVLEQNLWSCIPGEDTIMNVPGFFLVRRENRGSSYWDRCVVGGYLSPKTVADTFEKVVAGSINWPAIGSLLDYVIRPAPPPEALTLEVQYERDKHLFIDFLPSVTLGDTVLVAKPHRLAQYDNLWRLSLRPAETARLRALDQADSGCRSLCLKILKAICKSTPALGHLTASQLTNVILHLAQEEADWSPDMLADRFLQALRGLISYLEAGVLPSALNPKVNLFAELTPEEIDELGYTLYCSLSEPEVLLQT

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 \(c\).

\(|P_{f(5OIT_1)}(2) \setminus P_{f(2KNI_1)}(2)|=155\), \(|P_{f(2KNI_1)}(2) \setminus P_{f(5OIT_1)}(2)|=21\). 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 1:11111010011101110000111011011000110111011001011001000111011110101000001101110100111110010111001111100111101110110101001101010001011011111101110111101010011110011011001100100111001100110001111111001110111111110011101011001100011111010010111001011100111100101101011100011
Pair \(Z_2\) Length of longest common subsequence
5OIT_1,2KNI_1 176 3
5OIT_1,4NXW_1 168 3
2KNI_1,4NXW_1 182 3

Newick tree

 
[
	2KNI_1:91.27,
	[
		5OIT_1:84,4NXW_1:84
	]:7.27
]

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 \((0.85)(0.8)(\frac{310 }{\log_{20} 310}-\frac{41}{\log_{20}41})=87.5\)
Status Protein1 Protein2 d d1/2
Query variables 5OIT_1 2KNI_1 108 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]