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Parikh vectors
6SBI_1 7UVE_1 6QWO_1 Letter Amino acid
11 6 7 M Methionine
22 17 16 S Serine
4 13 6 Q Glutamine
10 12 16 D Aspartic acid
6 3 3 C Cysteine
11 5 10 H Histidine
7 8 17 F Phenylalanine
8 18 9 Y Tyrosine
5 10 11 N Asparagine
20 14 17 G Glycine
13 18 20 I Isoleucine
14 18 5 T Threonine
21 14 17 E Glutamic acid
12 17 10 R Arginine
22 13 26 L Leucine
20 14 19 K Lycine
17 15 9 P Proline
4 4 5 W Tryptophan
19 23 18 V Valine
19 15 12 A Alanine 

6SBI_1|Chains A, B, C, D|Acylpyruvase FAHD1, mitochondrial|Mus musculus (10090)
>7UVE_1|Chain A|Histone-lysine N-methyltransferase eggless|Drosophila melanogaster (7227)
>6QWO_1|Chains A, B|Oxygen-binding diiron protein|Thermotoga maritima (2336)
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 \(w=f(c)\)
6SBI , Knot 124 265 0.87 40 181 257 
MHHHHHHSSGLVPRGSGMKETAAAKFERQHMDSPDLGTMTQSCTMASTKPLSRFWEWGKNIVCVGRNYADHVKEMRSTVLSEPVLFLKPSTAYAPEGSPVLMPAYCRNLHHEVELGVLLGKRGEAIPEAAAMDYVAGYALCLDMTARDVQEECKKKGLPWTLAKSFTSSCPVSAFVPKEKIPDPHALRLWLKVNGELRQEGKTSSMIFSIPYIISYVSKIITLEEGDLILTGTPKGVGPIKENDEIEAGIDGVVSMRFKVKRSEY
7UVE , Knot 115 257 0.82 40 183 248 
SDYVHEVPPPGEIVRPPIQLGETYYAVKNKAIASWVSIKVIEFTESTAINGNTMKSYKIRYLNTPYQMIKTVTAKHIAYFEPPPVRLTIGTRVIAYFDGTTLSRGKDKGVVQSAFYPGIIAEPLKQANRYRYLIFYDDGYTQYVPHRDVRLVCQASEKVWEDVHAASRDFIQKYVEKYSVDRPMVQCTRGQSMTTESNGTWLYARVIDIDCSLVLMQFEGDKNHTEWIYRGSLRLGPVFRETQNNMNSSSAQQLRVP
6QWO , Knot 114 253 0.83 40 167 242 
MFENEILFDNGQHKFMFLGWEEKEEEIVQTNQYLILDGNEGILLDPGGAHVFPRVMSNVAEVVDLSSIRHIFYTHQDPDVTSGILLWLSICENAKIYISSLWVRFLPHFGIYDQKRIVPISDKGTKIKLLSGNELEILPAHFLHSTGNFVLYDPVAKILFSGDIGAAVFEKGKRYRYVDDFERHLPLMEAFHKRYMSSNAACKKWVDMVSKKKIDMIAPQHGAVFRGESVKKFLEWFRNLKCGVDLIDNLYSL

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 \(c\).

\(|P_{f(6SBI_1)}(2) \setminus P_{f(7UVE_1)}(2)|=91\), \(|P_{f(7UVE_1)}(2) \setminus P_{f(6SBI_1)}(2)|=93\). 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 1:1000000001111010110001110100001001011010000011000110011011001101100010010010001100111110100101101011111100001000101111110010111011110011101101010100100000001111011001000011011110001101011011101010100010000111011011001001101001011101010111110000010111011101010100000
Pair \(Z_2\) Length of longest common subsequence
6SBI_1,7UVE_1 184 3
6SBI_1,6QWO_1 170 3
7UVE_1,6QWO_1 180 4

Newick tree

 
[
	7UVE_1:92.92,
	[
		6SBI_1:85,6QWO_1:85
	]:7.92
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{257}{\log_{20}257})=75.5\)
Status Protein1 Protein2 d d1/2
Query variables 6SBI_1 7UVE_1 94 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]