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Parikh vectors
5WVO_1 4OUC_1 4TPM_1 Letter Amino acid
4 12 14 R Arginine
7 21 20 I Isoleucine
7 26 19 K Lycine
1 10 12 Y Tyrosine
5 20 17 D Aspartic acid
1 9 13 M Methionine
3 12 14 P Proline
0 4 7 W Tryptophan
4 17 17 V Valine
6 15 17 Q Glutamine
6 23 25 E Glutamic acid
5 23 15 G Glycine
9 41 36 L Leucine
2 19 15 F Phenylalanine
2 13 21 A Alanine
2 14 14 N Asparagine
1 9 11 C Cysteine
1 19 20 H Histidine
3 29 22 S Serine
7 21 22 T Threonine 

5WVO_1|Chains A, B|Ubiquitin|Homo sapiens (9606)
>4OUC_1|Chain A|Serine/threonine-protein kinase haspin|Homo sapiens (9606)
>4TPM_1|Chains A, B|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|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 \(w=f(c)\)
5WVO , Knot 40 76 0.76 38 61 74 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGC
4OUC , Knot 153 357 0.84 40 215 344 
MHHHHHHSSGVDLGTENLYFQSMGECSQKGPVPFSHCLPTEKLQRCEKIGEGVFGEVFQTIADHTPVAIKIIAIEGPDLVNGSHQKTFEEILPEIIISKELSLLSGEVCNRTEGFIGLNSVHCVQGSYPPLLLKAWDHYNSTKGSANDRPDFFKDDQLFIVLEFEFGGIDLEQMRTKLSSLATAKSILHQLTASLAVAEASLRFEHRDLHWGNVLLKKTSLKKLHYTLNGKSSTIPSCGLQVSIIDYTLSRLERDGIVVFCDVSMDEDLFTGDGDYQFDIYRLMKKENNNRWGEYHPYSNVLWLHYLTDKMLKQMTFKTKCNTPAMKQIKRKIQEFHRTMLNFSSATDLLCQHSLFK
4TPM , Knot 153 351 0.85 40 221 337 
MKHHHHHHHDEVDTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED

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 \(c\).

\(|P_{f(5WVO_1)}(2) \setminus P_{f(4OUC_1)}(2)|=20\), \(|P_{f(4OUC_1)}(2) \setminus P_{f(5WVO_1)}(2)|=174\). 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 1:1011100101001010101000100101010000111100001111100100100100001000001011101010
Pair \(Z_2\) Length of longest common subsequence
5WVO_1,4OUC_1 194 3
5WVO_1,4TPM_1 212 4
4OUC_1,4TPM_1 156 6

Newick tree

 
[
	5WVO_1:10.32,
	[
		4OUC_1:78,4TPM_1:78
	]:30.32
]

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 \((0.85)(0.8)(\frac{433 }{\log_{20} 433}-\frac{76}{\log_{20}76})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 5WVO_1 4OUC_1 143 85.5
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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