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Parikh vectors
9JSX_1 5BVI_1 6ZAM_1 Letter Amino acid
1 8 16 F Phenylalanine
3 8 15 T Threonine
1 11 28 A Alanine
0 5 19 N Asparagine
6 9 19 G Glycine
1 7 10 H Histidine
3 26 25 L Leucine
1 7 20 K Lycine
1 1 4 M Methionine
3 10 19 S Serine
4 7 22 V Valine
0 12 24 D Aspartic acid
0 10 18 E Glutamic acid
2 8 20 Y Tyrosine
1 15 12 R Arginine
0 7 3 C Cysteine
2 14 17 Q Glutamine
1 7 15 I Isoleucine
1 11 20 P Proline
3 2 5 W Tryptophan 

9JSX_1|Chains A, B, C, D, E, F, G, H|M-alpha|Homo sapiens (9606)
>5BVI_1|Chains A, B|Interferon regulatory factor 4|Mus musculus (10090)
>6ZAM_1|Chain A|Isopenicillin N synthase|Emericella nidulans (strain FGSC A4 / ATCC 38163 / CBS 112.46 / NRRL 194 / M139) (227321)
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 \(w=f(c)\)
9JSX , Knot 22 34 0.76 32 31 32 
FVYVWKTWGQYWQVLGGPVSGLSIGTSRAMLGTH
5BVI , Knot 88 185 0.82 40 137 179 
QGSIRSAEALALSDCRLHICLYYRDILVKELTTTSPEGCRISHGHTYDVSNLDQVLFPYPDDNGQRKNIEKLLSHLERGLVLWMAPDGLYAKRLCQSRIYWDGPLALCSDRPNKLERDQTCKLFDTQQFLSELQVFAHHGRPAPRFQVTLCFGEEFPDPQRQRKLITAHVEPLLARQLYYFAQQN
6ZAM , Knot 150 331 0.87 40 207 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLCQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIHRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT

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 \(c\).

\(|P_{f(9JSX_1)}(2) \setminus P_{f(5BVI_1)}(2)|=20\), \(|P_{f(5BVI_1)}(2) \setminus P_{f(9JSX_1)}(2)|=126\). 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 1:1101100110010111111011011000111100
Pair \(Z_2\) Length of longest common subsequence
9JSX_1,5BVI_1 146 3
9JSX_1,6ZAM_1 204 3
5BVI_1,6ZAM_1 170 4

Newick tree

 
[
	6ZAM_1:99.88,
	[
		9JSX_1:73,5BVI_1:73
	]:26.88
]

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 \((0.85)(0.8)(\frac{219 }{\log_{20} 219}-\frac{34}{\log_{20}34})=63.1\)
Status Protein1 Protein2 d d1/2
Query variables 9JSX_1 5BVI_1 81 47.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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