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Parikh vectors
5DYW_1 7WZF_1 4NUZ_1 Letter Amino acid
35 4 50 N Asparagine
21 15 72 D Aspartic acid
7 5 1 C Cysteine
22 18 32 R Arginine
43 27 60 G Glycine
48 27 77 L Leucine
37 3 33 F Phenylalanine
30 18 39 P Proline
32 18 53 A Alanine
7 16 13 H Histidine
26 9 49 I Isoleucine
8 5 16 M Methionine
35 19 54 S Serine
34 22 67 T Threonine
14 3 12 W Tryptophan
18 11 36 Y Tyrosine
19 6 31 Q Glutamine
31 21 51 V Valine
31 6 94 K Lycine
32 19 59 E Glutamic acid 

5DYW_1|Chains A, B|Cholinesterase|Homo sapiens (9606)
>7WZF_1|Chain A|YunM|Streptomyces yunnanensis (156453)
>4NUZ_1|Chain A|Endo-beta-N-acetylglucosaminidase F2|Streptococcus pyogenes serotype M1 (301447)
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 \(w=f(c)\)
5DYW , Knot 216 530 0.85 40 262 502 
TEDDIIIATKNGKVRGMNLTVFGGTVTAFLGIPYAQPPLGRLRFKKPQSLTKWSDIWNATKYANSCCQNIDQSFPGFHGSEMWNPNTDLSEDCLYLNVWIPAPKPKNATVLIWIYGGGFQTGTSSLHVYDGKFLARVERVIVVSMNYRVGALGFLALPGNPEAPGNMGLFDQQLALQWVQKNIAAFGGNPKSVTLFGESAGAASVSLHLLSPGSHSLFTRAILQSGSFNAPWAVTSLYEARNRTLNLAKLTGCSRENETEIIKCLRNKDPQEILLNEAFVVPYGTPLSVNFGPTVDGDFLTDMPDILLELGQFKKTQILVGVNKDEGTAFLVYGAPGFSKDNNSIITRKEFQEGLKIFFPGVSEFGKESILFHYTDWVDDQRPENYREALGDVVGDYNFICPALEFTKKFSEWGNNAFFYYFEHRSSKLPWPEWMGVMHGYEIEFVFGLPLERRDNYTKAEEILSRSIVKRWANFAKYGNPNETQNNSTSWPVFKSTEQKYLTLNTESTRIMTKLRAQQCRFWTSFFPKV
7WZF , Knot 120 272 0.82 40 172 260 
MGSSHHHHHHSSGLVPRGSHMTNGVEDIGYGRQFAGWYDRLIPDDEVTAEAVKGLLALHPAPEAGTIELGVGTGRIAVPLSESVGRVTGVDSSPEMLDAMRVKLKERGDVTPVHGDIRSYTSDSRYGLVYCVCATLSLLHTPEDQQLAVRRAADLLAPGGRLVIETHNKPPILELHEGRKRTTYFVPYPEPGTGVQTHSTLLDGDLWHCSHILYQANGTTRVGSELTRLTTPEEIDAYARAAGLEPETHLSRWDGTPYNEQSPMFVCCYVKS
4NUZ , Knot 334 899 0.84 40 289 796 
MKIPEKIPMKPLHGPLYGGYFRTWHDKTSDPTEKDKVNSMGELPKEVDLAFIFHDWTKDYSLFWKELATKHVPKLNKQGTRVIRTIPWRFLAGGDNSGIAEDTSKYPNTPEGNKALAKAIVDEYVYKYNLDGLDVQVEHDSIPKVDKKEDTAGVERSIQVFEEIGKLIGPKGVDKSRLFIMDSTYMADKNPLIERGAPYINLLLVQVYGSQGEKGGWEPVSNRPEKTMEERWQGYSKYIRPEQYMIGFSFYEENAQEGNLWYDINSRKDEDKANGINTDITGTRAERYARWQPKTGGVKGGIFSYAIDRDGVAHQPKKYAKQKEFKDATDNIFHSDYSVSKALKTVMLKDKSYDLIDEKDFPDKALREAVMAQVGTRKGDLERFNGTLRLDNPAIQSLEGLNKFKKLAQLDLIGLSRITKLDRSVLPANMKPGKDTLETVLETYKKDNKEEPATIPPVSLKVSGLTGLKELDLSGFDRETLAGLDAATLTSLEKVDISGNKLDLAPGTENRQIFDTMLSTISNHVGSNEQTVKFDKQKPTGHYPDTYGKTSLRLPVANEKVDLQSQLLFGTVTNQGTLINSEADYKAYQNHKIAGRSFVDSNYHYNNFKVSYENYTVKVTDSTLGTTTDKTLATDKEETYKVDFFSPADKTKAVHTAKVIVGDEKTMMVNLAEGATVIGGSADPVNARKVFDGQLGSETDNISLGWDSKQSIIFKLKEDGLIKHWRFFNDSARNPETTNKPIQEASLQIFNIKDYNLDNLLENPNKFDDEKYWITVDTYSAQGERATAFSNTLNNITSKYWRVVFDTKGDRYSSPVVPELQILGYPLPNADTIMKTVTTAKELSQQKDKFSQKMLDELKIKEMALETSLNSKIFDVTAINANAGVLKDCIEKRQLLKKL

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 \(c\).

\(|P_{f(5DYW_1)}(2) \setminus P_{f(7WZF_1)}(2)|=139\), \(|P_{f(7WZF_1)}(2) \setminus P_{f(5DYW_1)}(2)|=49\). 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 1:00001111000101011010111101011111101011110101001001001001101000100000010001111010011010001000010101111110100101111101111001000101001011101001111010001111111111101011101111000111011000111111010010111001111010101101100011001110010101111100100100001011010100000000110010000100111001111101011010111010101100110111011010000111110000101111011111000000110000100110111111001100011100001100001000001110111000110111010001001100111001000000111101111101001011111110000000010011000110011011001010000000001111000000010100000011001010000110011101
Pair \(Z_2\) Length of longest common subsequence
5DYW_1,7WZF_1 188 5
5DYW_1,4NUZ_1 127 4
7WZF_1,4NUZ_1 175 4

Newick tree

 
[
	7WZF_1:98.23,
	[
		5DYW_1:63.5,4NUZ_1:63.5
	]:34.73
]

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 \((0.85)(0.8)(\frac{802 }{\log_{20} 802}-\frac{272}{\log_{20}272})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 5DYW_1 7WZF_1 184 136.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]