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Parikh vectors
8UMZ_1 6ELR_1 4RXR_1 Letter Amino acid
15 25 35 E Glutamic acid
12 16 42 I Isoleucine
4 29 41 K Lycine
9 12 24 F Phenylalanine
0 3 5 W Tryptophan
14 11 26 A Alanine
16 13 23 N Asparagine
8 11 26 Q Glutamine
22 30 42 L Leucine
22 18 33 V Valine
10 13 30 R Arginine
18 18 38 D Aspartic acid
10 21 32 G Glycine
0 8 9 C Cysteine
20 14 19 T Threonine
12 16 27 S Serine
11 13 25 Y Tyrosine
11 8 24 H Histidine
3 7 12 M Methionine
13 15 26 P Proline 

8UMZ_1|Chains A, B|Protocadherin-15 EC4-EC7|Mus musculus (10090)
>6ELR_1|Chains A, B|Tyrosine-Protein Kinase JAK1|Homo sapiens (9606)
>4RXR_1|Chains A, B|Deoxynucleoside triphosphate triphosphohydrolase SAMHD1|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 \(w=f(c)\)
8UMZ , Knot 105 230 0.82 36 153 220 
MDENNQSPYFTMPSYQGYILESAPVGATISESLNLTTPLRIVALDKDIEDTKDPELHLFLNDYTSVFTVTPTGITRYLTLLQPVDREEQQTYTFLITAFDGVQESEPVVVNIRVMDANDNAPVFDPYLPRNLSVVEEEANAFVGQVRATDPDAGINGQVHYSLGNFNNLFRITSNGSIYTAVKLNREARDHYELVVVATDGAVHPRHSTLTLYIKVLDIDDNLEHHHHHH
6ELR , Knot 133 301 0.84 40 203 291 
DIVSEKKPATEVDPTHFEKRFLKRIRDLGEGHFGKVELCRYDPEGDNTGEQVAVKSLKPESGGNHIADLKKEIEILRNLYHENIVKYKGICTEDGGNGIKLIMEFLPSGSLKEYLPKNKNKINLKQQLKYAVQICKGMDYLGSRQYVHRDLAARNVLVESEHQVKIGDFGLTKAIETDKEYYTVKDDRDSPVFWYAPECLMQSKFYIASDVWSFGVTLHELLTYCDSDSSPMALFLKMIGPTHGQMTVTRLVNTLKEGKRLPCPPNCPDEVYQLMRKCWEFQPSNRTSFQNLIEGFEALLK
4RXR , Knot 215 539 0.83 40 265 510 
MGSSHHHHHHSSGENLYFQGSQIHVDTMKVINDPIHGHIELHPLLVRIIDTPQFQRLRYIKQLGGGYYVFPGASHNRFEHSLGVGYLAGCLVHALGEKQPELQISERDVLCVQIAGLCHDLGHGPFSHMFDGRFIPLARPEVKWTHEQGSVMMFEHLINSNGIKPVMEQYGLIPEEDIYFIKEQIVGPLESPVEDSLWPYKGRPENKSFLYEIVSNKRNGIDVDKWDYFARDCHHLGIQNNFDYKRFIKFARVCEVDNELRICARDKEVGNLYDMFHTRNSLHRRAYQHKVGNIIDTMITDAFLKADDYIEITGAGGKKYRISTAIDDMEAYTKLTDNIFLEILYSTDPKLKDAREILKQIEYRNLFKYVGETQPTGQIKIKREDYESLPKEVASAKPKVLLDVKLKAEDFIVDVINMDYGMQEKNPIDHVSFYCKTAPNRAIRITKNQVSQLLPEKFAEQLIRVYCKKVDRKSLYAARQYFVQWCADRNFTKPQDGDVIAPLITPQKKEWNDSTSVQNPTRLREASKSRVQLFKDDPM

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 \(c\).

\(|P_{f(8UMZ_1)}(2) \setminus P_{f(6ELR_1)}(2)|=63\), \(|P_{f(6ELR_1)}(2) \setminus P_{f(8UMZ_1)}(2)|=113\). 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 1:10000001010110001011001111101000101001101111000100000101011100000110101011000101101100000000011101101100001111010110100011110101100101100010111101010010111010100011010011010001010011010001000001111100111010000101010110100010000000
Pair \(Z_2\) Length of longest common subsequence
8UMZ_1,6ELR_1 176 3
8UMZ_1,4RXR_1 172 6
6ELR_1,4RXR_1 160 4

Newick tree

 
[
	8UMZ_1:89.21,
	[
		4RXR_1:80,6ELR_1:80
	]:9.21
]

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 \((0.85)(0.8)(\frac{531 }{\log_{20} 531}-\frac{230}{\log_{20}230})=86.2\)
Status Protein1 Protein2 d d1/2
Query variables 8UMZ_1 6ELR_1 112 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]