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Parikh vectors
5ZKY_1 6ZJP_1 7AAK_1 Letter Amino acid
21 65 17 P Proline
35 58 21 S Serine
22 72 20 R Arginine
27 69 19 D Aspartic acid
21 70 23 E Glutamic acid
18 28 13 H Histidine
7 16 19 K Lycine
5 26 3 W Tryptophan
36 89 26 V Valine
7 24 3 Y Tyrosine
37 111 33 A Alanine
3 5 4 C Cysteine
45 106 30 G Glycine
27 26 20 I Isoleucine
27 90 43 L Leucine
3 14 7 M Methionine
19 35 9 F Phenylalanine
31 22 13 N Asparagine
5 37 19 Q Glutamine
24 47 21 T Threonine 

5ZKY_1|Chain A|DFA-IIIase|Arthrobacter chlorophenolicus A6 (452863)
>6ZJP_1|Chain A|Beta-galactosidase|Arthrobacter sp. 32cB (1492190)
>7AAK_1|Chains A, B|Porphobilinogen deaminase|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 \(w=f(c)\)
5ZKY , Knot 176 420 0.84 40 207 398 
MPSNNRYDVTEWPAGNPAKDIGEVINSIIADIKARQGAADVDDGGKPGAVIYLPPGDYHLRTQVLIDISFLRIEGSGHGFTSSSIRFNVPEEEWPDLHELWPGGSRVIVDLPAGGAGDSAAGAAFLVAREGSPRISSVEFSNFCIDGLHFTADGSGRHPENTYANGKTGIHVASANDSFRVTDMGFVYLENALTIHKADALSIHHNFIAECGSCIELRGWGQASKITDNLVGAGPRGHSIYAENHGGLLVTANNVFPRGASSVHFKGVTRSSVTNNRLHAFYPGMVRLEENSSENLVATNHFLRDHEPWTPFFGVDNGLDDLTGLLSISGNNNSVIGNHFSEVVDANEIRPEGATPVIIRLTAGTGNFVSTNHVVSNVDTQHVLVDPGSARNTILDSGSDTQVVADRAVNAIRATPTVGF
6ZJP , Knot 360 1010 0.82 40 292 869 
MSVETPSALADSSPHTAPGSAGRSLELGAADIQDLESFEAGRGALPARAYLQSDAPRLSLNGEWQFRLSPGSRVAPDDGWQLGEALNGFESLPVPSSWPMHGHGAPAYTNVQFPFAVEPPHVPEANPIGDHLVVFEAGPEFFPHALLRFDGIESAGTVWLNGVELGTTRGSRLAHEFDVSGILEQGENTLAVRVAQFSAASYVEDQDMWWLPGIFRDVTLQARPAAGIDDVFVHAGYDHITGEGILKVEASRGGQAIDAVVRVPELALELAAGTEVRVPAVEPWSAEVPKLYEAAVSAAGESVALQIGFRSIAIEDAQFKVNGRRILLRGVNRHEHHPRLGRVVPRDVVEAELRLMKQHNINAIRTSHYPPHPQFLALADQLGFYVVLECDLETHGFESAGWAQNPSDDPQWEDALVDRMRRTVERDKNHASVVMWSLGNEAGTGRNLAAMSRWTKDRDPSRPIHYEGDWSSEHVDVYSRMYASQAETALIGQGIEPALNDAALDARRRAMPFVLCQYVHAMGNGPGGMSEYQALFEKYPRLMGGFVWEWLEHGITVSTADGVDHYGYGGDFGEEVHDGNFVTDGLVDADRRPRPGLLDFKKVIEPLRIDVARDWTGFTLRNGQDFADTSAFSFRYEVEADGGALDGGTVDVAPVAPQSETVVELPGSVAALAAGLSDGRPAVLTVRAVLGADSAWADAGHEVAWGQSVREPGAPVPPAPVEPVQVQDSELTLGPVVFSRATGMPTSIGGVPVEKLGLTLWWAPTDNDLGREWGGADERPLATQWKDAGLNRLHTRLLGISANPGQDGGETLTVRTRVSAADKQYGVLVDYTWSTDGETVGLRTQVRRDGTWVNRGFEVEWARIGLEFVLGEETELVSWFGQGPHQSYPDTGQGARAGWFSLPLAKMDVEYVRPQECGARSGSRSAALQLGGRTLEICGDPFALTVRPYSQDVLDAAAHRPDLKADGRTYLYVDHALRGVGTAACGPGVLEQYRLKPRDADFILTLKVRS
7AAK , Knot 153 363 0.82 40 211 343 
GSMSGNGNAAATAEENSPKMRVIRVGTRKSQLARIQTDSVVATLKASYPGLQFEIIAMSTTGDKILDTALSKIGEKSLFTKELEHALEKNEVDLVVHSLKDLPTVLPPGFTIGAICKRENPHDAVVFHPKFVGKTLETLPEKSVVGTSSLRRAAQLQRKFPHLEFRSIRGNLNTWLRKLDEQQEFSAIILATAGLQRMGWHNRVGQILHPEECMYAVGQGALGVEVRAKDQDILDLVGVLHDPETLLRCIAERAFLRHLEGGCSVPVAVHTAMKDGQLYLTGGVWSLDGSDSIQETMQATIHVPAQHEDGPEDDPQLVGITARNIPRGPQLAAQNLGISLANLLLSKGAKNILDVARQLNDAH

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 \(c\).

\(|P_{f(5ZKY_1)}(2) \setminus P_{f(6ZJP_1)}(2)|=23\), \(|P_{f(6ZJP_1)}(2) \setminus P_{f(5ZKY_1)}(2)|=108\). 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 1:110000001001111011001101100111010100111010011011111011110001000111010110101010110000101011000110100111110011101111111001111111110010101001010010101101010101001000010100110110100010100111101001101001011010001110010010101110100100011111101001010001111101001110110010101100001000010110111101000000011100011000011011111001100101110101000011100100110100101011011110101101011000011001000011101101000110010000111001101101010111
Pair \(Z_2\) Length of longest common subsequence
5ZKY_1,6ZJP_1 131 5
5ZKY_1,7AAK_1 140 4
6ZJP_1,7AAK_1 141 4

Newick tree

 
[
	7AAK_1:71.76,
	[
		5ZKY_1:65.5,6ZJP_1:65.5
	]:6.26
]

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 \((0.85)(0.8)(\frac{1430 }{\log_{20} 1430}-\frac{420}{\log_{20}420})=259.\)
Status Protein1 Protein2 d d1/2
Query variables 5ZKY_1 6ZJP_1 321 226
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]