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Parikh vectors
3ZWQ_1 9DNY_1 5IUI_1 Letter Amino acid
7 16 6 H Histidine
27 80 30 L Leucine
6 23 12 M Methionine
33 54 17 V Valine
10 14 12 Q Glutamine
23 63 24 G Glycine
10 44 14 F Phenylalanine
18 39 16 D Aspartic acid
1 18 10 C Cysteine
17 64 22 I Isoleucine
16 30 12 Y Tyrosine
37 69 17 A Alanine
9 27 12 N Asparagine
11 35 13 K Lycine
20 35 26 P Proline
18 56 22 S Serine
9 49 15 T Threonine
2 19 4 W Tryptophan
22 38 19 R Arginine
17 45 24 E Glutamic acid 

3ZWQ_1|Chains A, B|ALPHA/BETA HYDROLASE FOLD-3 DOMAIN PROTEIN|PYROBACULUM CALIDIFONTIS (410359)
>9DNY_1|Chains A, B[auth C]|H(+)/Cl(-) exchange transporter 3|Homo sapiens (9606)
>5IUI_1|Chain A|ALK tyrosine kinase receptor|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 \(w=f(c)\)
3ZWQ , Knot 131 313 0.80 40 183 297 
MPLSPILRQILQQLAAQLQFRPDMDVKTVREQFEKSSLILVKMANEPIHRVEDITIPGRGGPIRARVYRPRDGERLPAVVYYHGGGFVLGSVETHDHVCRRLANLSGAVVVSVDYRLAPEHKFPAAVEDAYDAAKWVADNYDKLGVDNGKIAVAGDSAGGNLAAVTAIMARDRGESFVKYQVLIYPAVNLTGSPTVSRVEYSGPEYVILTADLMAWFGRQYFSKPQDALSPYASPIFADLSNLPPALVITAEYDPLRDEGELYAHLLKTRGVRAVAVRYNGVIHGFVNFYPILEEGREAVSQIAASIKSMAVA
9DNY , Knot 311 818 0.85 40 304 751 
MESEQLFHRGYYRNSYNSITSASSDEELLDGAGVIMDFQTSEDDNLLDGDTAVGTHYTMTNGGSINSSTHLLDLLDEPIPGVGTYDDFHTIDWVREKCKDRERHRRINSKKKESAWEMTKSLYDAWSGWLVVTLTGLASGALAGLIDIAADWMTDLKEGICLSALWYNHEQCCWGSNETTFEERDKCPQWKTWAELIIGQAEGPGSYIMNYIMYIFWALSFAFLAVSLVKVFAPYACGSGIPEIKTILSGFIIRGYLGKWTLMIKTITLVLAVASGLSLGKEGPLVHVACCCGNIFSYLFPKYSTNEAKKREVLSAASAAGVSVAFGAPIGGVLFSLEEVSYYFPLKTLWRSFFAALVAAFVLRSINPFGNSRLVLFYVEYHTPWYLFELFPFILLGVFGGLWGAFFIRANIAWCRRRKSTKFGKYPVLEVIIVAAITAVIAFPNPYTRLNTSELIKELFTDCGPLESSSLCDYRNDMNASKIVDDIPDRPAGIGVYSAIWQLCLALIFKIIMTVFTFGIKVPSGLFIPSMAIGAIAGRIVGIAVEQLAYYHHDWFIFKEWCEVGADCITPGLYAMVGAAACLGGVTRMTVSLVVIVFELTGGLEYIVPLMAAVMTSKWVGDAFGREGIYEAHIRLNGYPFLDAKEEFTHTTLAADVMRPRRNDPPLAVLTQDNMTVDDIENMINETSYNGFPVIMSKESQRLVGFALRRDLTIAIESARKKQEGIVGSSRVCFAQHTPSLPAESPRPLKLRSILDMSPFTVTDHTPMEIVVDIFRKLGLRQCLVTHNGRLLGIITKKDILRHMAQTANQDPASIMFN
5IUI , Knot 138 327 0.81 40 205 314 
RTSTIMTDYNPNYSFAGKTSSISDLKEVPRKNITLIRGLGHGAFGEVYEGQVSGMPNDPSPLQVAVKTLPEVCSEQDELDFLMEALIISKFNHQNIVRCIGVSLQSLPRFILLELMAGGDLKSFLRETRPRPSQPSSLAMLDLLHVARDIACGCQYLEENHFIHRDIAARNCLLTCPGPGRVAKIGDFGMARDIYRASYYRKGGCAMLPVKWMPPEAFMEGIFTSKTDTWSFGVLLWEIFSLGYMPYPSKSNQEVLEFVTSGGRMDPPKNCPGPVYRIMTQCWQHQPEDRPNFAIILERIEYCTQDPDVINTALPIEYGPLVEEEEK

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 \(c\).

\(|P_{f(3ZWQ_1)}(2) \setminus P_{f(9DNY_1)}(2)|=20\), \(|P_{f(9DNY_1)}(2) \setminus P_{f(3ZWQ_1)}(2)|=141\). 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 1:1110111001100111010101010100100010000111101100110010010111011110101001001001111100011111110100000100011010111110100011100011111001001101110000011100101111100111011110111100010011000111011101010101001000110011101011111100010010011010101111010011111110100011000101010110001101111000111011101011100100110011101001111
Pair \(Z_2\) Length of longest common subsequence
3ZWQ_1,9DNY_1 161 3
3ZWQ_1,5IUI_1 160 4
9DNY_1,5IUI_1 167 4

Newick tree

 
[
	9DNY_1:82.67,
	[
		3ZWQ_1:80,5IUI_1:80
	]:2.67
]

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 \((0.85)(0.8)(\frac{1131 }{\log_{20} 1131}-\frac{313}{\log_{20}313})=216.\)
Status Protein1 Protein2 d d1/2
Query variables 3ZWQ_1 9DNY_1 272 186
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]