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Parikh vectors
6QGB_1 7KRX_1 1UAJ_1 Letter Amino acid
19 11 13 I Isoleucine
20 8 8 M Methionine
44 12 14 P Proline
17 23 5 Y Tyrosine
25 18 11 Q Glutamine
66 19 26 G Glycine
23 12 7 F Phenylalanine
41 17 13 S Serine
14 2 5 W Tryptophan
28 7 16 R Arginine
14 24 14 K Lycine
31 33 13 T Threonine
37 13 14 D Aspartic acid
10 10 3 C Cysteine
15 16 23 E Glutamic acid
14 9 20 H Histidine
38 26 31 L Leucine
26 20 16 V Valine
91 22 18 A Alanine
23 16 4 N Asparagine 

6QGB_1|Chains A, B, C, D, E, F|Mono(2-hydroxyethyl) terephthalate hydrolase|Ideonella sakaiensis (1547922)
>7KRX_1|Chain A|Papain-like protease|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1UAJ_1|Chain A|tRNA (Guanine-N(1)-)-methyltransferase|Haemophilus influenzae (727)
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 \(w=f(c)\)
6QGB , Knot 229 596 0.81 40 254 537 
MNHKVHHHHHHMGGGSTPLPLPQQQPPQQEPPPPPVPLASRAACEALKDGNGDMVWPNAATVVEVAAWRDAAPATASAAALPEHCEVSGAIAKRTGIDGYPYEIKFRLRMPAEWNGRFFMEGGSGTNGSLSAATGSIGGGQIASALSRNFATIATDGGHDNAVNDNPDALGTVAFGLDPQARLDMGYNSYDQVTQAGKAAVARFYGRAADKSYFIGCSEGGREGMMLSQRFPSHYDGIVAGAPGYQLPKAGISGAWTTQSLAPAAVGLDAQGVPLINKSFSDADLHLLSQAILGTCDALDGLADGIVDNYRACQAAFDPATAANPANGQALQCVGAKTADCLSPVQVTAIKRAMAGPVNSAGTPLYNRWAWDAGMSGLSGTTYNQGWRSWWLGSFNSSANNAQRVSGFSARSWLVDFATPPEPMPMTQVAARMMKFDFDIDPLKIWATSGQFTQSSMDWHGATSTDLAAFRDRGGKMILYHGMSDAAFSALDTADYYERLGAAMPGAAGFARLFLVPGMNHCSGGPGTDRFDMLTPLVAWVERGEAPDQISAWSGTPGYFGVAARTRPLCPYPQIARYKGSGDINTEANFACAAPP
7KRX , Knot 141 318 0.85 40 201 302 
SNAEVRTIKVFTTVDNINLHTQVVDMSMTYGQQFGPTYLDGADVTKIKPHNSHEGKTFYVLPNDDTLRVEAFEYYHTTDPSFLGRYMSALNHTKKWKYPQVNGLTSIKWADNNSYLATALLTLQQIELKFNPPALQDAYYRARAGEAANFCALILAYCNKTVGELGDVRETMSYLFQHANLDSCKRVLNVVCKTCGQQQTTLKGVEAVMYMGTLSYEQFKKGVQIPCTCGKQATKYLVQQESPFVMMSAPPAQYELKHGTFTCASEYTGNYQCGHYKHITSKETLYCIDGALLTKSSEYKGPITDVFYKENSYTTTIK
1UAJ , Knot 121 274 0.82 40 178 258 
MGSSHHHHHHSSGLVPRGSHMWIGVISLFPEMFKAITEFGVTGRAVKHNLLKVECWNPRDFTFDKHKTVDDRPYGGGPGMLMMVQPLRDAIHTAKAAAGEGAKVIYLSPQGRKLDQGGVTELAQNQKLILVCGRYEGIDERLIQTEIDEEWSIGDYVLTGGELPAMTLIDAVARFIPGVLGKQASAEEDSFADGLLDCPHYTRPEVLEGLTVPPVLMSGHHEEIRKWRLKQSLQRTWLRRPELLEGLALTDEQRKLLKEAQAEHNSLEHHHHHH

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 \(c\).

\(|P_{f(6QGB_1)}(2) \setminus P_{f(7KRX_1)}(2)|=120\), \(|P_{f(7KRX_1)}(2) \setminus P_{f(6QGB_1)}(2)|=67\). 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 1:10001000000111100111110001100011111111100110011001010111101101101111001111010111110000101111000110101001010101110101011101101001010110101111011011000110110011000110001011101111101010101100000010011011110101011000011100011001111000110000111111110011011101110000111111110101111100010010101100111100011011101110000100111011011011010110011100100101101011001111110011011000111011101101000001100111101000100100101101001110110110111100111011010101011011100101000010101100001111000110111001100111011001000001111111111110111111100001111000101101111110010110010110101101111100011010101100010101000101101111
Pair \(Z_2\) Length of longest common subsequence
6QGB_1,7KRX_1 187 3
6QGB_1,1UAJ_1 178 6
7KRX_1,1UAJ_1 185 4

Newick tree

 
[
	7KRX_1:94.29,
	[
		6QGB_1:89,1UAJ_1:89
	]:5.29
]

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 \((0.85)(0.8)(\frac{914 }{\log_{20} 914}-\frac{318}{\log_{20}318})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 6QGB_1 7KRX_1 205 159
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]