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Parikh vectors
6NFZ_1 4WSJ_1 5TSQ_1 Letter Amino acid
5 3 7 C Cysteine
13 11 9 Q Glutamine
8 8 7 H Histidine
5 19 2 W Tryptophan
10 26 8 Y Tyrosine
18 31 35 V Valine
19 17 10 S Serine
11 27 25 T Threonine
14 20 15 R Arginine
8 20 11 N Asparagine
21 31 17 D Aspartic acid
15 29 20 E Glutamic acid
26 34 18 G Glycine
21 19 24 I Isoleucine
40 23 27 L Leucine
23 45 16 K Lycine
7 13 6 M Methionine
9 16 8 F Phenylalanine
15 21 20 P Proline
21 32 27 A Alanine 

6NFZ_1|Chains A, B|Mitogen-activated protein kinase kinase kinase kinase 1|Homo sapiens (9606)
>4WSJ_1|Chains A, B, C, D|Alpha-L-fucosidase|Bacteroides thetaiotaomicron (226186)
>5TSQ_1|Chain A|IUnH|Leishmania braziliensis braziliensis (1448350)
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 \(w=f(c)\)
6NFZ , Knot 135 309 0.83 40 185 295 
GSMDVVDPDIFNRDPRDHYDLLQRLGGGTYGEVFKARDKVSGDLVALKMVKMEPDDDVSTLQKEILILKTCRHANIVAYHGSYLWLQKLWICMEFCGAGSLQDIYQVTGSLSELQISYVCREVLQGLAYLHSQKKIHRDIKGANILINDAGEVRLADFGISAQIGATLARRLSFIGTPYWMAPEVAAVALKGGYNELCDIWSLGITAIELAELQPPLFDVHPLRVLFLMTKSGYQPPRLKEKGKWSAAFHNFIKVTLTKSPKKRPSATKMLSHQLVSQPGLNRGLILDLLDKLKNPGKGPSIGDIEDEE
4WSJ , Knot 187 445 0.85 40 245 420 
EIPLKYGATNEGKRQDPAMQKFRDNRLGAFIHWGLYAIPGGEWNGKVYGGAAEWLKSWAKVPADEWLKLMDQWNPTKFDAKKWAKMAKEMGTKYVKITTKHHEGFCLWPSKYTKYTVANTPYKRDILGELVKAYNDEGIDVHFYFSVMDWSNPDYRYDIKSKEDSIAFSRFLEFTDNQLKELATRYPTVKDFWFDGTWDASVKKNGWWTAHAEQMLKELVPGVAINSRLRADDKGKRHFDSNGRLMGDYESGYERRLPDPVKDLKVTQWDWEACMTIPENQWGYHKDWSLSYVKTPIEVIDRIVHAVSMGGNMVVNFGPQADGDFRPEEKAMATAIGKWMNRYGKAVYACDYAGFEKQDWGYYTRGKNDEVYMVVFNQPYSERLIVKTPKGITVEKATLLTTGEDITVVETTRNEYNVSVPKKNPGEPYVIQLKVRAAKGTKSIY
5TSQ , Knot 134 312 0.82 40 187 299 
PRKIILDCDPGIDDAVAILLAYGNPEIELLAITTVVGNQTLEKVTRNAQLVADVAGIVGVPIAAGCCKPLVRKVRTAPQIHGETGLGTVSYPSEFKTKLDKRHAVHLIIELIMSHEPKSITLVPTGGLTNIAMAARLEPRIVERVKEVVLMGGSCCIGNASPVAEFNIFVDPEAAHIVFNESWDVTMVGLDLTSQALATPEVLQRVKEVRTKPADFILKILEFYTKVYETQRNTYAKVHDPCAVAYVIDPTVMTTNRVPVNIELNGELTAGMTVTDFRYPRPEQCHTQVASKLDFSKYWDLVIDALQRIGDP

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 \(c\).

\(|P_{f(6NFZ_1)}(2) \setminus P_{f(4WSJ_1)}(2)|=52\), \(|P_{f(4WSJ_1)}(2) \setminus P_{f(6NFZ_1)}(2)|=112\). 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 1:101011010110001000001100111100101101000101011110110101000100100011110000010111001001110011101010111010010010101001010010001101110100000100010110111001101011011101011101100101110101111011111101100010011011101101101011110101101111100010011010001010111001101010001000101001100011001110011110110010011011011010000
Pair \(Z_2\) Length of longest common subsequence
6NFZ_1,4WSJ_1 164 4
6NFZ_1,5TSQ_1 154 4
4WSJ_1,5TSQ_1 160 3

Newick tree

 
[
	4WSJ_1:82.29,
	[
		6NFZ_1:77,5TSQ_1:77
	]: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 \((0.85)(0.8)(\frac{754 }{\log_{20} 754}-\frac{309}{\log_{20}309})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 6NFZ_1 4WSJ_1 154 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]