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Parikh vectors
9LFL_1 8RLT_1 4NYO_1 Letter Amino acid
20 4 1 C Cysteine
8 15 0 Q Glutamine
6 21 10 L Leucine
7 15 7 V Valine
8 19 6 A Alanine
11 20 6 R Arginine
3 5 2 N Asparagine
7 17 8 D Aspartic acid
2 5 10 I Isoleucine
12 15 3 P Proline
3 9 5 W Tryptophan
1 14 2 H Histidine
6 9 11 K Lycine
1 3 1 M Methionine
3 9 2 F Phenylalanine
5 12 4 Y Tyrosine
8 26 16 E Glutamic acid
9 20 2 G Glycine
15 16 1 S Serine
17 22 5 T Threonine 

9LFL_1|Chain A|Tumor necrosis factor receptor superfamily member 1B|Homo sapiens (9606)
>8RLT_1|Chains A, F|HLA class I histocompatibility antigen, alpha chain E|Homo sapiens (9606)
>4NYO_1|Chains A, B, C, D, E, F|Divalent-cation tolerance protein CutA|Pyrococcus horikoshii (70601)
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 \(w=f(c)\)
9LFL , Knot 73 152 0.80 40 107 141 
TPYAPEPGSTCRLREYYDQTAQMCCSKCSPGQHAKVFCTKTSDTVCDSCEDSTYTQLWNWVPECLSCGSRCSSDQVETQACTREQNRICTCRPGWYCALSKQEGCRLCAPLRKCRPGFGVARPGTETSDVVCKPCAPGTFSNTTSSTDICRP
8RLT , Knot 120 276 0.81 40 179 263 
GSHSLKYFHTSVSRPGRGEPRFISVGYVDDTQFVRFDNDAASPRMVPRAPWMEQEGSEYWDRETRSARDTAQIFRVNLRTLRGYYNQSEAGSHTLQWMHGCELGPDGRFLRGYEQFAYDGKDYLTLNEDLRSWTAVDTAAQISEQKSNDASEAEHQRAYLEDTCVEWLHKYLEKGKETLLHLEPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQQDGEGHTQDTELVETRPAGDGTFQKWAAVVVPSGEEQRYTCHVQHEGLPEPVTLRWKP
4NYO , Knot 50 102 0.75 38 78 97 
MIIVYTTFPDWESAEKVVKTLLKERLIACANLREHRAFYWWEGKIEEDKEVGAILKTREDLWEELKERIKELHPYDVPAIIRIDVDDVNEDYLKWLIEETKK

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 \(c\).

\(|P_{f(9LFL_1)}(2) \setminus P_{f(8RLT_1)}(2)|=64\), \(|P_{f(8RLT_1)}(2) \setminus P_{f(9LFL_1)}(2)|=136\). 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 1:01011011000010000000101000000110010110000000100000000000110111001001000000010001000000010000111001100001001011100001111110110000011001011101000000001001
Pair \(Z_2\) Length of longest common subsequence
9LFL_1,8RLT_1 200 3
9LFL_1,4NYO_1 141 3
8RLT_1,4NYO_1 175 3

Newick tree

 
[
	8RLT_1:10.56,
	[
		9LFL_1:70.5,4NYO_1:70.5
	]:30.06
]

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 \((0.85)(0.8)(\frac{428 }{\log_{20} 428}-\frac{152}{\log_{20}152})=82.2\)
Status Protein1 Protein2 d d1/2
Query variables 9LFL_1 8RLT_1 105 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]