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Parikh vectors
4FGH_1 2QNV_1 5CRB_1 Letter Amino acid
11 19 4 F Phenylalanine
12 16 12 T Threonine
2 17 12 A Alanine
8 14 6 H Histidine
12 15 10 I Isoleucine
11 17 6 K Lycine
4 14 8 M Methionine
8 22 8 S Serine
6 6 12 N Asparagine
11 23 10 E Glutamic acid
0 6 1 C Cysteine
4 22 9 Q Glutamine
11 20 27 G Glycine
16 40 19 L Leucine
9 15 12 P Proline
2 3 3 W Tryptophan
7 17 8 R Arginine
11 12 9 D Aspartic acid
3 7 8 Y Tyrosine
15 11 9 V Valine 

4FGH_1|Chain A|Dihydrofolate reductase|Staphylococcus aureus (1280)
>2QNV_1|Chain A|Orphan nuclear receptor PXR|Homo sapiens (9606)
>5CRB_1|Chains A, B|SdeA|Legionella pneumophila (446)
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 \(w=f(c)\)
4FGH , Knot 78 163 0.81 38 120 155 
MTLSILVAHDLQRVIGFENQLPWHLPNDLKHVKKLSTGHTLVMGRKTFESIGKPLPNRRNVVLTSDTSFNVEGVDVIHSIEDIYQLPGHVFIFGGQTLFEEMIDKVDDMYITVIEGKFRGDTFFPPYTFEDWEVASSVEGKLDEKNTIPHTFLHLIRKKLVPR
2QNV , Knot 139 316 0.84 40 201 300 
MKKGHHHHHHGSERTGTQPLGVQGLTEEQRMMIRELMDAQMKTFDTTFSHFKNFRLPGVLSSGCELPESLQAPSREEAAKWSQVRKDLCSLKVSLQLRGEDGSVWNYKPPADSGGKEIFSLLPHMADMSTYMFKGIISFAKVISYFRDLPIEDQISLLKGAAFELCQLRFNTVFNAETGTWECGRLSYCLEDTAGGFQQLLLEPMLKFHYMLKKLQLHEEEYVLMQAISLFSPDRPGVLQHRVVDQLQEQFAITLKSYIECNRPQPAHRFLFLKIMAMLTELRSINAQHTQRLLRIQDIHPFATPLMQELFGITGS
5CRB , Knot 93 193 0.84 40 143 188 
MPKYVEGVELTQEGMHAIFARMGYGDITSGSIYNGVPTIDTGALNRQGFMPVLTGVGPHRDSGHWIMLIKGPGNQYYLFDPLGKTSGEGYQNILAAQLPMGSTLSVIPNGSGLNMGLCGYWVASAGLRAHQALNQHNPPTLLNVGQTITNEMRNELDHDGYRKITGWLRAVADEFPEGDPQLDGKALRENTEK

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 \(c\).

\(|P_{f(4FGH_1)}(2) \setminus P_{f(2QNV_1)}(2)|=53\), \(|P_{f(2QNV_1)}(2) \setminus P_{f(4FGH_1)}(2)|=134\). 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 1:1010111100100111100011101100100100100100111100010011011100001110000010101101100100100111011111100110011001001010110101010011110010010110010101000001100110110001110
Pair \(Z_2\) Length of longest common subsequence
4FGH_1,2QNV_1 187 3
4FGH_1,5CRB_1 149 4
2QNV_1,5CRB_1 210 3

Newick tree

 
[
	2QNV_1:10.43,
	[
		4FGH_1:74.5,5CRB_1:74.5
	]:31.93
]

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 \((0.85)(0.8)(\frac{479 }{\log_{20} 479}-\frac{163}{\log_{20}163})=92.9\)
Status Protein1 Protein2 d d1/2
Query variables 4FGH_1 2QNV_1 120 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]