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Parikh vectors
4NFD_1 2BSB_1 4HNR_1 Letter Amino acid
2 6 12 A Alanine
6 4 8 E Glutamic acid
7 18 12 V Valine
7 8 5 I Isoleucine
13 23 6 S Serine
7 12 11 T Threonine
4 3 1 H Histidine
2 4 6 M Methionine
6 6 5 P Proline
6 9 2 Y Tyrosine
3 8 6 D Aspartic acid
1 2 0 C Cysteine
7 7 7 Q Glutamine
9 22 6 G Glycine
7 6 5 F Phenylalanine
4 4 1 W Tryptophan
11 6 5 R Arginine
4 14 2 N Asparagine
9 11 10 L Leucine
5 3 10 K Lycine 

4NFD_1|Chain A|Paired immunoglobulin-like type 2 receptor beta|Homo sapiens (9606)
>2BSB_1|Chain A|F17G ADHESIN SUBUNIT|ESCHERICHIA COLI (562)
>4HNR_1|Chain A|Chemotaxis protein CheY|Vibrio cholerae (666)
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 \(w=f(c)\)
4NFD , Knot 64 120 0.85 40 103 116 
MLYGVTQPKHLSASMGGSVEIPFSFYYPWELAIVPNVRISWRRGHFHGQSFYSTRPPSIHKDYVNRLFLNWTEGQESGFLRISNLRKEDQSVYFCRVELDTRRSGRQQWQSIKGTKLTIT
2BSB , Knot 81 176 0.79 40 124 167 
AVSFIGSTENDVGPSQGSYSRTHAMDNLPFVYDTGYNIGYQNANVWHISGGFCVGLDGKVDLPVVGSLDGQSIYGLTEEVGLLIWMGDTNYSRGTAMSGNSWENVFSGWCVGNYVSTQGLSVHVRPVILKRNSSAQYSVQKTSIGSIRMRPYNGSSAGSVQTTVNFSLNPFTLNDT
4HNR , Knot 62 120 0.82 38 100 116 
TMAKVLAVDDSISIRQMVSHTLQDAGYEVETAADGREALAKAQKARFDVIISDVNMPVMTGFEFVKAVRMQSQYKFTPILMLTTETSPEKKQEGKAVGATGWLVKPFNPETLLKTLQRVL

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 \(c\).

\(|P_{f(4NFD_1)}(2) \setminus P_{f(2BSB_1)}(2)|=66\), \(|P_{f(2BSB_1)}(2) \setminus P_{f(4NFD_1)}(2)|=87\). 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 1:110110010010101110101110100110111110101010010101001000011010000100111010010001110100100000010100101000001000100101001010
Pair \(Z_2\) Length of longest common subsequence
4NFD_1,2BSB_1 153 3
4NFD_1,4HNR_1 169 2
2BSB_1,4HNR_1 142 3

Newick tree

 
[
	4NFD_1:83.55,
	[
		2BSB_1:71,4HNR_1:71
	]:12.55
]

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 \((0.85)(0.8)(\frac{296 }{\log_{20} 296}-\frac{120}{\log_{20}120})=54.9\)
Status Protein1 Protein2 d d1/2
Query variables 4NFD_1 2BSB_1 69 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]