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Parikh vectors
5OEM_1 7NKA_1 8XNF_1 Letter Amino acid
11 30 31 V Valine
11 40 38 A Alanine
7 43 18 R Arginine
6 32 39 N Asparagine
24 64 60 L Leucine
11 36 27 F Phenylalanine
2 12 20 W Tryptophan
7 37 33 D Aspartic acid
17 77 47 E Glutamic acid
10 36 32 G Glycine
5 48 22 I Isoleucine
7 52 34 K Lycine
4 15 8 C Cysteine
13 18 30 Q Glutamine
7 13 22 H Histidine
19 31 27 P Proline
2 25 21 M Methionine
15 51 34 S Serine
7 36 32 T Threonine
4 20 28 Y Tyrosine 

5OEM_1|Chain A|Interferon regulatory factor 9|Mus musculus (10090)
>7NKA_1|Chain A|Polymerase acidic protein|Influenza A virus (A/Brevig Mission/1/1918(H1N1)) (88776)
>8XNF_1|Chain A|Angiotensin-converting enzyme 2|Homo sapiens (9606)
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 \(w=f(c)\)
5OEM , Knot 89 189 0.82 40 133 182 
EDPVFLEHQLPLNSDYSLLLTFIYGGRVVGKTQVHSLDCRLVAEASDSESSMEQVEFPKPDPLEPTQHLLNQLDRGVLVASNSRGLFVQRLCPIPISWNAPEAPPGPGPHLLPSNKCVELFKTTYFCRDLAQYFQGQGPPPKFQATLHFWEESPGSSHSQENLITVQMEQAFARHLLEKIPEEEKAALF
7NKA , Knot 279 716 0.85 40 294 663 
MEDFVRQCFNPMIVELAEKAMKEYGEDLKIETNKFAAICTHLEVCFMYSDFHFINERGESIIVESGDPNALLKHRFEIIEGRDRTMAWTVVNSICNTTGAEKPKFLPDLYDYKENRFIEIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRGLWDSFRQSERGEETIEERFEITGTMRRLADQSLPPNFSSLENFRAYVDGFEPNGYIEGKLSQMSKEVNARIEPFLKTTPRPLRLPDGPPCSQRSKFLLMDALKLSIEDPSHEGEGIPLYDAIKCMRTFFGWKEPNVVKPHEKGINPNYLLAWKQVLAELQDIENEEKIPKTKNMKKTSQLKWALGENMAPEKVDFDDCKDVSDLKQYDSDEPELRSLASWIQSEFNKACELTDSSWIELDEIGEDVAPIEHIASMRRNYFTAEVSHCRATEYIMKGVYINTALLNASCAAMDDFQLIPMISKCRTKEGRRKTNLYGFIIKGRSHLRNDTDVVNFVSMEFSLTDPRLEPHKWEKYCVLEIGDMLLRSAIGQVSRPMFLYVRTNGTSKIKMKWGMEMRRCLLQSLQQIESMIEAESSVKEKDMTKEFFENKSETWPIGESPKGVEEGSIGKVCRTLLAKSVFNSLYASPQLEGFSAESRKLLLIVQALRDNLEPGTFDLGGLYEAIEECLINDPWVLLNASWFNSFLTHALR
8XNF , Knot 248 603 0.87 40 290 573 
STIEEQAKTFLDKFNHEAEDLFYQSSLASWNYNTNITEENVQNMNNAGDKWSAFLKEQSTLAQMYPLQEIQNLTVKLQLQALQQNGSSVLSEDKSKRLNTILNTMSTIYSTGKVCNPDNPQECLLLEPGLNEIMANSLDYNERLWAWESWRSEVGKQLRPLYEEYVVLKNEMARANHYEDYGDYWRGDYEVNGVDGYDYSRGQLIEDVEHTFEEIKPLYEHLHAYVRAKLMNAYPSYISPIGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDAMVDQAWDAQRIFKEAEKFFVSVGLPNMTQGFWENSMLTDPGNVQKAVCHPTAWDLGKGDFRILMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKSIGLLSPDFQEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKDQWMKKWWEMKREIVGVVEPVPHDETYCDPASLFHVSNDYSFIRYYTRTLYQFQFQEALCQAAKHEGPLHKCDISNSTEAGQKLFNMLRLGKSEPWTLALENVVGAKNMNVRPLLNYFEPLFTWLKDQNKNSFVGWSTDWSPYADHHHHHH

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 \(c\).

\(|P_{f(5OEM_1)}(2) \setminus P_{f(7NKA_1)}(2)|=25\), \(|P_{f(7NKA_1)}(2) \setminus P_{f(5OEM_1)}(2)|=186\). 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 1:001111000111000001110110110111000100100011101000000100101101011010001100100111110000111100101111010110111111101110000101100001000110010101111010101011000110000000110101001110011001100001111
Pair \(Z_2\) Length of longest common subsequence
5OEM_1,7NKA_1 211 4
5OEM_1,8XNF_1 223 3
7NKA_1,8XNF_1 138 5

Newick tree

 
[
	5OEM_1:11.83,
	[
		7NKA_1:69,8XNF_1:69
	]:49.83
]

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 \((0.85)(0.8)(\frac{905 }{\log_{20} 905}-\frac{189}{\log_{20}189})=197.\)
Status Protein1 Protein2 d d1/2
Query variables 5OEM_1 7NKA_1 252 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]