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Parikh vectors
4UAN_1 5THE_1 8WDY_1 Letter Amino acid
12 34 40 I Isoleucine
14 29 37 P Proline
12 44 54 S Serine
5 3 23 W Tryptophan
1 8 8 C Cysteine
6 24 38 Q Glutamine
28 9 16 H Histidine
21 26 39 T Threonine
9 41 54 N Asparagine
40 48 43 G Glycine
19 29 43 D Aspartic acid
17 25 56 E Glutamic acid
15 40 76 L Leucine
8 45 47 K Lycine
10 12 27 M Methionine
17 22 51 A Alanine
18 21 31 R Arginine
22 42 50 V Valine
8 24 39 F Phenylalanine
11 23 33 Y Tyrosine 

4UAN_1|Chains A, B, C|Copper oxidase|Streptomyces sviceus ATCC 29083 (463191)
>5THE_1|Chains A, C, E, G|Uncharacterized protein|Vanderwaltozyma polyspora (436907)
>8WDY_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 \(w=f(c)\)
4UAN , Knot 125 293 0.80 40 175 275 
MHHHHHHAPGGEVRRIKLYAERLADGQMGYGLEKGRATIPGPLIELNEGDTLHIEFENTMDVRASLQVHGLDYEVSSDGTTLNKSDVEPGGTRTYTWRTHAPGRRSDGTWRAGSAGYWHYHDHVVGTEHGTGGIRKGLYGPVIVRRKGDVLPDATHTIVFNDMLINNRPAHSGPNFEATVGDRVEFVMITHGEYYHTFHMHGHRWADNRTGMLTGPDDPSQVVDNKIVGPADSFGFQVIAGEGVGAGAWMYHCHVQSHSDMGMVGLFLVKKTDGTIPGYEPHEHSGQRAEHHH
5THE , Knot 219 549 0.83 40 252 511 
SIYKVENRHDYGTKGTKVDILTGSGRVPSRILDAPVVQFKESTFEYKDKSYGTKHEESKGNWNMKGHQFISTPAKQVNLRAIFINNANTAPPASMESELDISMDKFASDVKQLGVDFNVSGKPILINQFGPPIKKFQGGGRGGRGGRGSRGGRGGRGAPSGPPTFETSPGEISLLNLLENIPSNTYILYVLRRGNDSAVYDRLKYITDLKFGALNSCVVWDNFKKNSIQYNSNVVMKMNLKLLGSNHSLSIENNKLLIDKESNLPILVLGSDVTHYPEKDQNSIASLVGSYDDKFTQFPGDYMLQDGPGEEIITNVGSLMLNRLKIYQKHNNGKLPTKIMYFRDGVSVDQFSQVVKIEVKSIKESVRKFGPQLNGGNKYDPPVTCIATVKRNQVRFIPIQENAKNEKGEEVAVQSMGNVMPGTVVDRGITSVAHFDFFIQSHQALKGTGVPCHYWCLYDENQSTSDYLQEICNNLCYIFGRSTTSVKVPAPVYYADLLCTRATCFFKAGFELNMAQAPKEKGSKDQPTVSKNVLLPQVNDNIKSVMYYI
8WDY , Knot 311 805 0.86 40 316 752 
MSSSSWLLLSLVAVTAAQSTIEEQAKTFLDKFNHEAEDLFYQSSLASWNYNTNITEENVQNMNNAGDKWSAFLKEQSTLAQMYPLQEIQNLTVKLQLQALQQNGSSVLSEDKSKRLNTILNTMSTIYSTGKVCNPDNPQECLLLEPGLNEIMANSLDYNERLWAWESWRSEVGKQLRPLYEEYVVLKNEMARANHYEDYGDYWRGDYEVNGVDGYDYSRGQLIEDVEHTFEEIKPLYEHLHAYVRAKLMNAYPSYISPIGCLPAHLLGDMWGRFWTNLYSLTVPFGQKPNIDVTDAMVDQAWDAQRIFKEAEKFFVSVGLPNMTQGFWENSMLTDPGNVQKAVCHPTAWDLGKGDFRILMCTKVTMDDFLTAHHEMGHIQYDMAYAAQPFLLRNGANEGFHEAVGEIMSLSAATPKHLKSIGLLSPDFQEDNETEINFLLKQALTIVGTLPFTYMLEKWRWMVFKGEIPKDQWMKKWWEMKREIVGVVEPVPHDETYCDPASLFHVSNDYSFIRYYTRTLYQFQFQEALCQAAKHEGPLHKCDISNSTEAGQKLFNMLRLGKSEPWTLALENVVGAKNMNVRPLLNYFEPLFTWLKDQNKNSFVGWSTDWSPYADQSIKVRISLKSALGDKAYEWNDNEMYLFRSSVAYAMRQYFLKVKNQMILFGEEDVRVANLKPRISFNFFVTAPKNVSDIIPRTEVEKAIRMSRSRINDAFRLNDNSLEFLGIQPTLGPPNQPPVSIWLIVFGVVMGVIVVGIVILIFTGIRDRKKKNKARSGENPYASIDISKGENNPGFQNTDDVQTSF

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 \(c\).

\(|P_{f(4UAN_1)}(2) \setminus P_{f(5THE_1)}(2)|=62\), \(|P_{f(5THE_1)}(2) \setminus P_{f(4UAN_1)}(2)|=139\). 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 1:10000001111010010101001101011011001010111111010010010101000101010101011000100010010000101110000010001110000101011011010000011100010111001101111100010111010001110011100011001101010110010111100100000101010011000011101100100110001111100111011110111111110000100000111111111000010111001000010010000
Pair \(Z_2\) Length of longest common subsequence
4UAN_1,5THE_1 201 4
4UAN_1,8WDY_1 189 4
5THE_1,8WDY_1 144 4

Newick tree

 
[
	4UAN_1:10.68,
	[
		8WDY_1:72,5THE_1:72
	]:32.68
]

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 \((0.85)(0.8)(\frac{842 }{\log_{20} 842}-\frac{293}{\log_{20}293})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 4UAN_1 5THE_1 190 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]