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Parikh vectors
2XCF_1 1CGQ_1 4NJP_1 Letter Amino acid
14 10 14 S Serine
5 5 6 Y Tyrosine
17 12 19 A Alanine
15 11 27 L Leucine
4 3 6 K Lycine
4 3 12 M Methionine
3 4 13 F Phenylalanine
4 8 3 N Asparagine
16 8 16 V Valine
7 4 5 D Aspartic acid
10 6 6 Q Glutamine
4 3 5 E Glutamic acid
23 9 23 G Glycine
12 8 6 P Proline
2 1 11 W Tryptophan
13 5 10 R Arginine
7 3 1 C Cysteine
9 2 11 H Histidine
10 6 13 I Isoleucine
19 4 4 T Threonine 

2XCF_1|Chains A, B|NS3 PROTEASE|HEPATITIS C VIRUS (3052230)
>1CGQ_1|Chains A, B, C|PROTEIN (MACROPHAGE MIGRATION INHIBITORY FACTOR)|Homo sapiens (9606)
>4NJP_1|Chain A|Rhomboid protease GlpG|Escherichia coli (562)
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 \(w=f(c)\)
2XCF , Knot 92 198 0.82 40 138 189 
ASMTGGQQMGAPITAYAQQTRGLLGCIITSLTGRDKNQVEGEVQIVSTATQTFLATCINGVCWTVYHGAGTRTIASPKGPVIQMYTNVDQDLVGWPAPQGSRSLTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPISYLKGSAGGPLLCPAGHAVGLFRAAVCTRGVAKAVDFIPVENLETTMRGSHHHHHH
1CGQ , Knot 60 115 0.82 40 99 112 
PAMFIVNTNVPRASVPDGFLSELTQQLAQATGKPPQYIAVHVVPDQLMAFGGSSEPCALCSLHSIGKIGGAQNRSYSKLLCGLLAERLRISPDRVYINYYDMNAANVGWNNSTFA
4NJP , Knot 94 211 0.79 40 138 201 
MGSSHHHHHHSSGLVPRGSHMAALRERAGPVTWVMMIACVVVFIAMQILGDQEVMLWLAWPFDPTLKFEFWRYFTHALMHFSLMHILFNLLWWWYLGGAVEKRLGSGKLIVITLISALLSGYVQQKFSGPWFGGLSGVVYALMGYVWLRGERDPQSGIYLQRGLIIFALIWIVAGWFDLFGMSMANGAHIAGLAVGLAMAFVDSLNARKRK

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 \(c\).

\(|P_{f(2XCF_1)}(2) \setminus P_{f(1CGQ_1)}(2)|=96\), \(|P_{f(1CGQ_1)}(2) \setminus P_{f(2XCF_1)}(2)|=57\). 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 1:101011001111101010000111101100101000001010101100100011100101101010011100011010111101000100011111110100010100010001011000101111000100010110101100101011111101110111110111000111011011110010001010000000
Pair \(Z_2\) Length of longest common subsequence
2XCF_1,1CGQ_1 153 3
2XCF_1,4NJP_1 160 7
1CGQ_1,4NJP_1 157 3

Newick tree

 
[
	4NJP_1:80.15,
	[
		2XCF_1:76.5,1CGQ_1:76.5
	]:3.65
]

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 \((0.85)(0.8)(\frac{313 }{\log_{20} 313}-\frac{115}{\log_{20}115})=61.5\)
Status Protein1 Protein2 d d1/2
Query variables 2XCF_1 1CGQ_1 75 59.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]