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Parikh vectors
3LCT_1 6FKU_1 9IJM_1 Letter Amino acid
16 16 20 D Aspartic acid
6 13 9 H Histidine
13 8 16 M Methionine
15 31 13 T Threonine
19 39 20 R Arginine
12 19 10 N Asparagine
11 1 3 C Cysteine
14 28 13 F Phenylalanine
4 13 2 W Tryptophan
13 10 25 Q Glutamine
26 47 31 E Glutamic acid
33 47 26 L Leucine
17 43 19 V Valine
23 18 23 S Serine
13 14 2 Y Tyrosine
18 47 17 A Alanine
27 57 21 G Glycine
22 20 17 I Isoleucine
15 20 15 K Lycine
27 30 19 P Proline 

3LCT_1|Chain A|ALK tyrosine kinase receptor|Homo sapiens (9606)
>6FKU_1|Chains A, B|Aldehyde dehydrogenase|Thermus thermophilus (strain HB27 / ATCC BAA-163 / DSM 7039) (262724)
>9IJM_1|Chains A, B|PomB|Vibrio alginolyticus (663)
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 \(w=f(c)\)
3LCT , Knot 147 344 0.83 40 208 330 
GAMGSELQSPEYKLSKLRTSTIMTDYNPNYCFAGKTSSISDLKEVPRKNITLIRGLGHGAFGEVYEGQVSGMPNDPSPLQVAVKTLPEVCSEQDELDFLMEALIISKFNHQNIVRCIGVSLQSLPRFILLELMAGGDLKSFLRETRPRPSQPSSLAMLDLLHVARDIACGCQYLEENHFIHRDIAARNCLLTCPGPGRVAKIGDFGMARDIYRAGYYRKGGCAMLPVKWMPPEAFMEGIFTSKTDTWSFGVLLWEIFSLGYMPYPSKSNQEVLEFVTSGGRMDPPKNCPGPVYRIMTQCWQHQPEDRPNFAIILERIEYCTQDPDVINTALPIEYGPLVEEEEK
6FKU , Knot 206 521 0.82 40 233 475 
MHHHHHHRKAAGKYGNTLEFGHLVGGEEVLEGPLLERRNPSDREDVVARFPEADKDLVRKAALKAREAFAEWSRTPAPIRGQVLFNLVKILEREKPTLTRLMVREVGKTPKEAAGDVQEAIDTALFFASEGRRLYGQTVPSEMRDKELFTFRRPLGVVGIITAGNFPIAVPSWKLIPAVLTGNTVVWKPSEDAPTLSFVFAKLFEEAGLPPGVLNVVFGGGKGSTGQWMVELMDEGLFQKFAFTGSTQVGRWIGEVAGRNLIRPTLELGGKNPLVVMRDADLDLAVEGAWWSAFATGGQRCTSAGNILVDAPIYEEFKRRFLERVEATLVGNPLLHPEVTYGPFINERFFARWQEHYRVGEAEGARLLFGRGRITRENPYPRFLGDPEAGLYGWPTVWEVRPGTRLFTEEVFGPTINLVKVDGIEEAIAVANSTPYGLSSAIYTNHRHWAYLFKVGIRAGMTSINNATVGAEAHLPFGGVKASGNGGRESGIWVLEEYTYWHAVNEEYSGRLQLAQMDTGY
9IJM , Knot 138 321 0.82 40 193 302 
MDDEDNKCDCPPPGLPLWMGTFADLMSLLMCFFVLLLSFSEMDVLKFKQIAGSMKFAFGVQNQLEVKDIPKGTSIIAQEFRPGRPEPTPIDVIMQQTMDITQQTLEFHEGESDRAGGTKRDEGKLTGGQSPATSTQNNESAEADMQQQQSKEMSQEMETLMESIKKALEREIEQGAIEVENLGQQIVIRMREKGAFPEGSAFLQPKFRPLVRQIAELVKDVPGIVRVSGHTDNRPLDSELYRSNWDLSSQRAVSVAQEMEKVRGFSHQRLRVRGMADTEPLLPNDSDDNRALNRRVEISIMQGEPLYSEEVPVIQHHHHHH

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 \(c\).

\(|P_{f(3LCT_1)}(2) \setminus P_{f(6FKU_1)}(2)|=70\), \(|P_{f(6FKU_1)}(2) \setminus P_{f(3LCT_1)}(2)|=95\). 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 1:11110010010001001000011000010001110000100100110001011011101111010010101110010110111001101000000101110111100100001100111010011011110111110100110000101001001111011011001101000100001100011100011001111011011011110010011000011011111011110111011100000010111111011011011010000001101100110101100011110011000100010001011111001000000101100111100111100000
Pair \(Z_2\) Length of longest common subsequence
3LCT_1,6FKU_1 165 4
3LCT_1,9IJM_1 159 3
6FKU_1,9IJM_1 174 6

Newick tree

 
[
	6FKU_1:86.46,
	[
		3LCT_1:79.5,9IJM_1:79.5
	]:6.96
]

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 \((0.85)(0.8)(\frac{865 }{\log_{20} 865}-\frac{344}{\log_{20}344})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 3LCT_1 6FKU_1 175 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]