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Parikh vectors
6FWJ_1 9NKI_1 7CMM_1 Letter Amino acid
19 31 10 R Arginine
2 7 6 C Cysteine
25 31 10 I Isoleucine
16 32 15 K Lycine
8 12 6 M Methionine
23 25 7 P Proline
11 1 1 W Tryptophan
24 13 14 N Asparagine
11 18 10 Q Glutamine
21 33 18 E Glutamic acid
32 28 14 G Glycine
14 4 9 H Histidine
21 37 20 L Leucine
26 25 11 T Threonine
13 27 17 V Valine
21 35 10 D Aspartic acid
17 16 16 F Phenylalanine
32 15 16 S Serine
26 10 11 Y Tyrosine
23 33 9 A Alanine 

6FWJ_1|Chain A|Glycosyl hydrolase family 71|Bacteroides xylanisolvens XB1A (657309)
>9NKI_1|Chain A|26S proteasome regulatory subunit 7|Homo sapiens (9606)
>7CMM_1|Chains A, B, C, D|Transcriptional enhancer factor TEF-1|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 \(w=f(c)\)
6FWJ , Knot 160 385 0.82 40 222 361 
MGSSHHHHHHSSGLVPRGSHMDDNNPSNSENNGGNNNLGTELDYDTFCFYYDWYGSEAIDGQYRHWAHAIAPDPNGGSGQNPGTIPGTQESIASNFYPQLGRYSSSDPNILTKHMDMFVMARTGVLALTWWNEQDETEAKRIGLILDAADKKKIKVCFHLEPYPSRNVQNLRENIVKLITRYGNHPAFYRKDGKPLFFIYDSYLIEPSEWEKLLSPGGSITIRNTAYDALMIGLWTSSPTVQRPFILNAHFDGFYTYFAATGFTYGSTPTNWVSMQKWAKENGKIFIPSVGPGYIDTRIRPWNGSVIRTRTDGQYYDAMYRKAIEAGVSAISITSFNEWHEGSQIEPAVPYTSSEFTYLDYENREPDYYLTRTAYWVGKFRESKQ
9NKI , Knot 180 433 0.84 40 224 419 
MPDYLGADQRKTKEDEKDDKPIRALDEGDIALLKTYGQSTYSRQIKQVEDDIQQLLKKINELTGIKESDTGLAPPALWDLAADKQTLQSEQPLQVARCTKIINADSEDPKYIINVKQFAKFVVDLSDQVAPTDIEEGMRVGVDRNKYQIHIPLPPKIDPTVTMMQVEEKPDVTYSDVGGCKEQIEKLREVVETPLLHPERFVNLGIEPPKGVLLFGPPGTGKTLCARAVANRTDACFIRVIGSELVQKYVGEGARMVRELFEMARTKKACLIFFDEIDAIGGARFDDGAGGDNEVQRTMLELINQLDGFDPRGNIKVLMATNRPDTLDPALMRPGRLDRKIEFSLPDLEGRTHIFKIHARSMSVERDIRFELLARLCPNSTGAEIRSVCTEAGMFAIRARRKIATEKDFLEAVNKVIKSYAKFSATPRYMTYN
7CMM , Knot 106 230 0.83 40 168 222 
GPLGSENLYFQGRSIGTTKLRLVEFSAFLEQQRDPDSYNKHLFVHIGHANHSYSDPLLESVDIRQIYDKFPEKKGGLKELFGKGPQNAFFLVKFWADLNCNIQDDAGAFYGVTSQYESSENMTVTCSTKVCSFGKQVVEKVETEYARFENGRFVYRINRSPMCEYMINFIHKLKHLPEKYMMNSVLENFTILLVVTNRDTQETLLCMACVFEVSNSEHGAQHHIYRLVKD

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 \(c\).

\(|P_{f(6FWJ_1)}(2) \setminus P_{f(9NKI_1)}(2)|=89\), \(|P_{f(9NKI_1)}(2) \setminus P_{f(6FWJ_1)}(2)|=91\). 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 1:1100000000001111010010000100000011000110010000101000101001101000011011110101101001101110000110010101100000010110001011111001111101100000001001111101100001010101010100010010001101100010011100001011111000011010010011011101010001001111111000101001111010101100011101100100100110100110001011110111101000101101011000001000011000110111011010010010010010111100000100100000010001000101110100000
Pair \(Z_2\) Length of longest common subsequence
6FWJ_1,9NKI_1 180 4
6FWJ_1,7CMM_1 184 4
9NKI_1,7CMM_1 182 3

Newick tree

 
[
	7CMM_1:91.99,
	[
		6FWJ_1:90,9NKI_1:90
	]:1.99
]

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 \((0.85)(0.8)(\frac{818 }{\log_{20} 818}-\frac{385}{\log_{20}385})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6FWJ_1 9NKI_1 150 141.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]