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Parikh vectors
1JMO_1 2POS_1 2CCR_1 Letter Amino acid
1 3 14 Q Glutamine
1 3 14 I Isoleucine
0 2 11 W Tryptophan
4 4 40 G Glycine
5 11 24 L Leucine
4 7 16 F Phenylalanine
5 6 25 T Threonine
3 1 11 R Arginine
3 4 26 D Aspartic acid
7 8 20 E Glutamic acid
0 2 5 M Methionine
2 8 14 P Proline
0 5 29 V Valine
2 9 44 A Alanine
1 6 0 C Cysteine
0 0 10 H Histidine
2 1 20 Y Tyrosine
1 4 24 N Asparagine
3 6 29 K Lycine
4 4 23 S Serine 

1JMO_1|Chain A[auth L]|Thrombin, light chain|Homo sapiens (9606)
>2POS_1|Chains A, B, C, D|SYLVATICIN|Pythium sylvaticum (82950)
>2CCR_1|Chains A, B|YVFO|BACILLUS LICHENIFORMIS (1402)
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 \(w=f(c)\)
1JMO , Knot 28 48 0.75 32 45 46 
TATSEYQTFFNPRTFGSGEADCGLRPLFEKKSLEDKTERELLESYIDG
2POS , Knot 48 94 0.77 38 78 91 
WEETKECAFTEFFKLAPLASNPALSVCQDASGWQMLPPAGYPTPEQLKLMCGTAECFTLIDAIKALNPNDCILVFGDVRLNVKKLVTEFEPSCF
2CCR , Knot 165 399 0.82 38 203 377 
AHRDSGTAKSGLYVEKVSGLRKDFIKGVDVSSIIALEESGVAFYNESGKKQDIFKTLKEAGVNYVRVRIWNDPYDANGNGYGGGNNDLEKAIQIGKRATANGMKLLADFHYSDFWADPAKQKAPKAWANLNFEDKKTALYQYTKQSLKAMKAAGIDIGMVQVGNETNGGLAGETDWAKMSQLFNAGSQAVRETDSNILVALHFTNPETSGRYAWIAETLHRHHVDYDVFASSYYPFWHGTLKNLTSVLTSVADTYGKKVMVAETSYTYTAEDGDGHGNTAPKNGQTLNNPVTVQGQANAVRDVIQAVSDVGEAGIGVFYWEPAWIPVGPAHRLEKNKALWETYGSGWATSYAAEYDPEDAGKWFGGSAVDNQALFDFKGRPLPSLHVFQYVDTGTPFKN

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 \(c\).

\(|P_{f(1JMO_1)}(2) \setminus P_{f(2POS_1)}(2)|=33\), \(|P_{f(2POS_1)}(2) \setminus P_{f(1JMO_1)}(2)|=66\). 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 1:010000001101001101010011011100001000000011000101
Pair \(Z_2\) Length of longest common subsequence
1JMO_1,2POS_1 99 2
1JMO_1,2CCR_1 180 3
2POS_1,2CCR_1 175 3

Newick tree

 
[
	2CCR_1:98.42,
	[
		1JMO_1:49.5,2POS_1:49.5
	]:48.92
]

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 \((0.85)(0.8)(\frac{142 }{\log_{20} 142}-\frac{48}{\log_{20}48})=33.1\)
Status Protein1 Protein2 d d1/2
Query variables 1JMO_1 2POS_1 42 31
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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