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Parikh vectors
6ULW_1 2XZS_1 9FEV_1 Letter Amino acid
39 10 24 Y Tyrosine
125 20 21 A Alanine
62 12 12 Q Glutamine
109 29 18 E Glutamic acid
68 30 12 K Lycine
88 18 18 S Serine
88 15 30 T Threonine
67 17 24 D Aspartic acid
75 23 25 I Isoleucine
123 30 40 L Leucine
43 4 13 M Methionine
58 18 22 F Phenylalanine
54 16 26 R Arginine
77 16 25 G Glycine
57 6 7 H Histidine
51 11 18 P Proline
79 20 31 V Valine
34 13 17 N Asparagine
12 1 4 C Cysteine
18 3 8 W Tryptophan 

6ULW_1|Chains A, B, C, D|Amino acid adenylation domain-containing protein|Bacillus stratosphericus LAMA 585 (1236481)
>2XZS_1|Chains A, B|DEATH ASSOCIATED KINASE 1|HOMO SAPIENS (9606)
>9FEV_1|Chains A, B, C, D, E|Gamma-aminobutyric acid receptor subunit beta-3|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 \(w=f(c)\)
6ULW , Knot 471 1327 0.85 40 342 1136 
MKPSTINQTSRHESMDAILQDVKTTILQTNEIERFAVFSREKTIRPRPYHLSHLFLDKHIEQEAAAEAIQTQKAVPALADMPPALSYGGDLDVTQGAQTLQDVLTEAAKTTNGLTYIVNSKNELTQSYAQLQGDAERVLTGLRALGLKAGDPVFFQFSSNHAMVTAFWACVLGGFVPTLVSAAPTYREMNAAVKKLHHAWKLLEHPLILTDDSLIEEVQGLAFLWHTDQLRVAAVEPMLTLERDTAAHPAAPDDSVFFILTSGSTGMPKCVEHSHRSVLANVKGTVAANQFTQEDVSLDWMPLDHIGGIVMFHLVNVYTGCEQIRARTDDFIAQPLRWLDWMDRYRATKTWAPNFAFAMINDYEKEISSGSWDLSAMTCMINGAEAVVPKTIHRFLHLLAPHGLKGDVIRPAFGMSEISSAVVFSFAIERGDENSGVLTFEETSLTEQLRPAEARETGTVSFTELGKPIPGITIRIVNHQHELLPEDHIGRVQIKGPTTMKGYYRNDEANQEVFQADGWFHTGDLGFLHEGRLTLTGREKDMIIINGKNYHNYEIEAIAEEVPGVETSFVAACSVRMEASASDELILFFTPKLYEPAYIMRASQHIKSHIATKMGLSASRIIPVQKHAFPKTSSGKIERAQLKTRWQEGEFDEMIMEMDMRLENEHTLPNWSYQKKWIPAPLDAADKQIVGDVVIFADPLGLADQLKSFANSEQTYITVEPGQAFKQLTHTHFIIHPNRPSDYEQLFETLRSYGVSVKHIIHLWNYTNQEDKLYVSKAKQAQKTGSMSALFLTKAISVYEEQVAITFVSTHAMNVPEDHIHDVEKATTAGLMTAVQHEWPFIKVQCLDFSLAESVSSHSHAMAMMAELTHDQDCHVAAYRQGVRYIPFLAPLHLEKEQKRKEPPFESSGLYVITGGLGGIGRLVSEQLLTSYQAHLVLLGRKEREALSSDQQETLSLLEKQAKVHGGKVLYEKADITAEKEVEAIISRHEVAGNRRLDGIIHFAGIIQEELLADMTSVSLNDMYEAKVYGTIALHEAASKRRNVLFLSSSSARTLKGGMTVGAYCAANEFVEQFAHYQKLTSTVKPYCFSFSMWDEIGMSEGSMIKGMLKERGYLPISKQAGFQTMLACLMTDAPLIYTGLDRSKKEIHEMIHADAEMQTEQTFYVFFKTDQTKAVEERLYQSIHTCLQSRLSGEYAVHLFPEEYWKETEDGSPDEAYFHELIEESGMDADNRAPHTETEKLLAHIWSELLDVSDVSCGDHFFLLGGHSLKATQMLSAVRQKTGFDVPLAVLFEHTRLGELADWIDSNAKADEVIPMRAAAENLYFQ
2XZS , Knot 131 312 0.80 40 186 300 
GTVFRQENVDDYYDTGEELGSGQFAVVKKCREKSTGLQYAAKFIKKRRTKSSRRGVSREDIEREVSILKEIQHPNVITLHEVYENKTDVILILELVAGGELFDFLAEKESLTEEEATEFLKQILNGVYYLHSLQIAHFDLKPENIMLLDRNVPKPRIKIIDFGLAHKIDFGNEFKNIFGTPEFVAPEIVNYEPLGLEADMWSIGVITYILLSGASPFLGDTKQETLANVSAVNYEFEDEYFSNTSALAKDFIRRLLVKDPKKRMTIQDSLQHPWIKPKDTQQALSRKASAVNMEKFKKFAARKKWKQSVRLI
9FEV , Knot 171 395 0.86 40 232 379 
MDEKTTGWRGGHVVEGLAGELEQLRARLEHHPQGQREPDYDIPTTENLYFQGTGQSVNDPGNMSFVKETVDKLLKGYDIRLRPDFGGPPVCVGMNIDIASIDMVSEVNMDYTLTMYFQQYWRDKRLAYSGIPLNLTLDNRVADQLWVPDTYFLNDKKSFVHGVTVKNRMIRLHPDGTVLYGLRITTTAACMMDLRRYPLDEQNCTLEIESYGYTTDDIEFYWRGGDKAVTGVERIELPQFSIVEHRLVSRNVVFATGAYPRLSLSFRLKRNIGYFILQTYMPSILITILSWVSFWINYDASAARVALGITTVLTMTTINTHLRETLPKIPYVKAIDMYLMGCFVFVFLALLEYAFVNYIFFSQPARAAAIDRWSRIVFPFTFSLFNLVYWLYYVN

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 \(c\).

\(|P_{f(6ULW_1)}(2) \setminus P_{f(2XZS_1)}(2)|=168\), \(|P_{f(2XZS_1)}(2) \setminus P_{f(6ULW_1)}(2)|=12\). 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 1:1010010000000010111001000110000100111100000101010010011100010001110110000111111011111001101010011001001100110000110011000001000010101010011011011110110111101000011101111011111110110111000010111001001101100111100001100101111110000101111011101000011011110001111100100111001000000111010101110010000101011110011111110110100100010100001110110110110000100011101111110000001001010101100110110111100100110111101101011011111001001111011100100001110100001000101101000101010011011111010110000011100011010101100101000000100011010111001011110010101010000111101000000010111001111000111100101010100011111010100110110100010001100111010011110001110000101001010001001010011101010100000110100000111111011000111011111011111001001100000010101101100100001110100100000110010001101001101100000000101001001000101011110011010000111011000110110001001001001111011000111101001010110010000011111101000000011100011001111111010000000011100011011011111110110001100001011111000001100000001011000101011011000101010001011100001110001011101111100011101001010010010101011100110000011110000100101110111001100110011000010001010010101100111001011011100010111000111001110110011110011000000100110101010000010111000000110001000100010001010011011100010000010100101001100011010001100000011101100110100100100111111001010011011000011011111110000110110110001010011110111001010
Pair \(Z_2\) Length of longest common subsequence
6ULW_1,2XZS_1 180 4
6ULW_1,9FEV_1 148 6
2XZS_1,9FEV_1 162 3

Newick tree

 
[
	2XZS_1:89.15,
	[
		6ULW_1:74,9FEV_1:74
	]:15.15
]

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 \((0.85)(0.8)(\frac{1639 }{\log_{20} 1639}-\frac{312}{\log_{20}312})=340.\)
Status Protein1 Protein2 d d1/2
Query variables 6ULW_1 2XZS_1 432 264
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]