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Parikh vectors
5YXD_1 3IXC_1 2CHC_1 Letter Amino acid
10 4 3 P Proline
2 2 4 W Tryptophan
5 5 6 Y Tyrosine
11 8 15 D Aspartic acid
12 4 3 F Phenylalanine
24 11 8 E Glutamic acid
6 23 17 G Glycine
31 14 15 L Leucine
15 7 2 K Lycine
8 10 2 M Methionine
9 16 15 V Valine
3 3 3 C Cysteine
14 4 7 Q Glutamine
10 10 3 H Histidine
12 13 6 S Serine
13 12 10 T Threonine
9 8 17 R Arginine
9 7 3 N Asparagine
11 16 23 A Alanine
15 14 8 I Isoleucine 

5YXD_1|Chain A|Bile acid receptor|Homo sapiens (9606)
>3IXC_1|Chain A|Hexapeptide transferase family protein|Anaplasma phagocytophilum (212042)
>2CHC_1|Chains A, B, C|PROTEIN RV3472|MYCOBACTERIUM TUBERCULOSIS (83332)
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 \(w=f(c)\)
5YXD , Knot 104 229 0.82 40 154 219 
ELTPDQQTLLHFIMDSYNKQRMPQEITNKILKEEFSAEENFLILTEMATNHVQVLVEFTKKLPGFQTLDHEDQIALLKGSAVEAMFLRSAEIFNKKLPSGHSDLLEERIRNSGISDEYITPMFSFYKSIGELKMTQEEYALLTAIVILSPDRQYIKDREAVEKLQEPLLDVLQKLCKIHQPENPQHFACLLGRLTELRTFNHHHAEMLMSWRVNDHKFTPLLCEIWDVQ
3IXC , Knot 88 191 0.80 40 137 183 
MAHHHHHHMGTLEAQTQGPGSMREVLVPYAGVSPSVDSTAFIAGNARIIGDVCIGKNASIWYGTVLRGDVDKIEVGEGTNIQDNTVVHTDSMHGDTVIGKFVTIGHSCILHACTLGNNAFVGMGSIVMDRAVMEEGSMLAAGSLLTRGKIVKSGELWAGRPAKFLRMMTEEEILYLQKSAENYIALSRGYL
2CHC , Knot 81 170 0.81 40 119 162 
GAMGPVDEQWIEILRIQALCARYCLTINTQDGEGWAGCFTEDGAFEFDGWVIRGRPALREYADAHARVVRGRHLTTDLLYEVDGDVATGRSASVVTLATAAGYKILGSGEYQDRLIKQDGQWRIAYRRLRNDRLVSDPSVAVNVADADVAAVVGHLLAAARRLGTQMSDT

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 \(c\).

\(|P_{f(5YXD_1)}(2) \setminus P_{f(3IXC_1)}(2)|=92\), \(|P_{f(3IXC_1)}(2) \setminus P_{f(5YXD_1)}(2)|=75\). 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 1:0101000011011100000001100100011000101000111100110001011101000111100100000111101011011110010110001101000110001000110000101110100011010100000111011111010000100001100100111011001001001001001101110100100100001011101010000101110011010
Pair \(Z_2\) Length of longest common subsequence
5YXD_1,3IXC_1 167 3
5YXD_1,2CHC_1 167 3
3IXC_1,2CHC_1 134 4

Newick tree

 
[
	5YXD_1:88.31,
	[
		3IXC_1:67,2CHC_1:67
	]:21.31
]

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 \((0.85)(0.8)(\frac{420 }{\log_{20} 420}-\frac{191}{\log_{20}191})=67.5\)
Status Protein1 Protein2 d d1/2
Query variables 5YXD_1 3IXC_1 85 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]