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Parikh vectors
4UNO_1 8TRI_1 3OPH_1 Letter Amino acid
8 30 24 R Arginine
6 45 16 D Aspartic acid
1 13 4 H Histidine
6 19 4 F Phenylalanine
2 25 17 T Threonine
5 45 17 V Valine
4 20 14 Q Glutamine
4 33 16 I Isoleucine
6 11 7 K Lycine
5 7 10 M Methionine
9 47 36 L Leucine
5 21 18 S Serine
6 16 4 Y Tyrosine
6 27 21 G Glycine
5 26 12 P Proline
3 2 5 W Tryptophan
8 33 36 A Alanine
4 27 7 N Asparagine
1 0 3 C Cysteine
6 27 15 E Glutamic acid 

4UNO_1|Chain A|ETS TRANSLOCATION VARIANT 5|HOMO SAPIENS (9606)
>8TRI_1|Chains A, B, C|Cadherin-23|Mus musculus (10090)
>3OPH_1|Chain A|Beta-lactamase SHV-1|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
4UNO , Knot 54 100 0.83 40 87 98 
SMRGSLQLWQFLVTLLDDPANAHFIAWTGRGMEFKLIEPEEVARRWGIQKNRPAMNYDKLSRSLRYYYEKGIMQKVAGERYVYKFVCDPDALFSMAFPDN
8TRI , Knot 193 474 0.83 38 223 440 
MASDVNDNRPVFVRPPNGTILHIKEEIPLRSNVYEVYATDNDEGLNGAVRYSFLKTTGNRDWEYFTIDPISGLIQTAQRLDREKQAVYSLILVASDLGQPVPYETMQPLQVALEDIDDNEPLFVRPPKGSPQYQLLTVPEHSPRGTLVGNVTGAVDADEGPNAIVYYFIAAGDEDKNFHLQPDGRLLVLRDLDRETEATFSFIVKASSNRSWTPPRGPSPALDLLTDLTLQEVRVVLEDINDQPPRFTKAEYTAGVATDAKVGSELIQVLALDADIGNNSLVFYGILAIHYFRALANDSEDVGQVFTMGSVDGILRTADLFMAYSPGYFVVDIVARDLAGHNDTAIIGIYILRDDQRVKIVINEIPDRVRGFEEEFIRLLSNITGAIVNTDDVQFHVDMKGRVNFAQTELLIHVVNRDTNRILDVDRVIQMIDENKEQLRNLFRNYNVLDVQPAISVQLPDDMSALLEHHHHHH
3OPH , Knot 123 286 0.81 40 167 270 
MRYIRLCIISLLATLPLAVHASPQPLEQIKLSESQLSGRVGMIEMDLASGRTLTAWRADERFPMMSTFKVVLCGAVLARVDAGDEQLERKIHYRQQDLVDYSPVSEKHLADGMTVGELCAAAITMSDNSAANLLLATVGGPAGLTAFLRQIGDNVTRLDSWETELNEALPGDARDTTTPASMAATLRKLLTSQRLSARSQRQLLQWMVDDRVAGPLIRSVLPAGWFIADKTGAGERGARGIVALLGPNNKAERIVVIYLRDTPASMAERNQQIAGIGAALIEHWQR

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 \(c\).

\(|P_{f(4UNO_1)}(2) \setminus P_{f(8TRI_1)}(2)|=30\), \(|P_{f(8TRI_1)}(2) \setminus P_{f(4UNO_1)}(2)|=166\). 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 1:0101010110111011001101011110101101011010011001110000111000010001000000111001110001001100101110111100
Pair \(Z_2\) Length of longest common subsequence
4UNO_1,8TRI_1 196 3
4UNO_1,3OPH_1 164 4
8TRI_1,3OPH_1 150 4

Newick tree

 
[
	4UNO_1:94.92,
	[
		3OPH_1:75,8TRI_1:75
	]:19.92
]

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 \((0.85)(0.8)(\frac{574 }{\log_{20} 574}-\frac{100}{\log_{20}100})=139.\)
Status Protein1 Protein2 d d1/2
Query variables 4UNO_1 8TRI_1 171 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]