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Parikh vectors
8XCI_1 6VVS_1 6FEK_1 Letter Amino acid
86 26 18 A Alanine
59 27 21 E Glutamic acid
56 22 13 I Isoleucine
40 8 13 F Phenylalanine
102 33 21 V Valine
34 9 10 Y Tyrosine
51 11 8 N Asparagine
15 7 6 H Histidine
19 3 11 M Methionine
83 25 18 S Serine
104 25 9 T Threonine
21 1 6 W Tryptophan
65 18 20 R Arginine
70 30 15 D Aspartic acid
11 1 3 C Cysteine
48 8 9 Q Glutamine
53 19 16 P Proline
88 27 23 G Glycine
78 35 35 L Leucine
49 15 24 K Lycine 

8XCI_1|Chains A[auth F], B[auth J], C[auth Z]|Tip attachment protein J|Escherichia phage Lambda (2681611)
>6VVS_1|Chains A, B, K[auth T]|DNA-directed RNA polymerase subunit alpha|Mycolicibacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
>6FEK_1|Chain A|Proto-oncogene tyrosine-protein kinase receptor Ret|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 \(w=f(c)\)
8XCI , Knot 414 1132 0.85 40 333 999 
MGKGSSKGHTPREAKDNLKSTQLLSVIDAISEGPIEGPVDGLKSVLLNSTPVLDTEGNTNISGVTVVFRAGEQEQTPPEGFESSGSETVLGTEVKYDTPITRTITSANIDRLRFTFGVQALVETTSKGDRNPSEVRLLVQIQRNGGWVTEKDITIKGKTTSQYLASVVMGNLPPRPFNIRMRRMTPDSTTDQLQNKTLWSSYTEIIDVKQCYPNTALVGVQVDSEQFGSQQVSRNYHLRGRILQVPSNYNPQTRQYSGIWDGTFKPAYSNNMAWCLWDMLTHPRYGMGKRLGAADVDKWALYVIGQYCDQSVPDGFGGTEPRITCNAYLTTQRKAWDVLSDFCSAMRCMPVWNGQTLTFVQDRPSDKTWTYNRSNVVMPDDGAPFRYSFSALKDRHNAVEVNWIDPNNGWETATELVEDTQAIARYGRNVTKMDAFGCTSRGQAHRAGLWLIKTELLETQTVDFSVGAEGLRHVPGDVIEICDDDYAGISTGGRVLAVNSQTRTLTLDREITLPSSGTALISLVDGSGNPVSVEVQSVTDGVKVKVSRVPDGVAEYSVWELKLPTLRQRLFRCVSIRENDDGTYAITAVQHVPEKEAIVDNGAHFDGEQSGTVNGVTPPAVQHLTAEVTADSGEYQVLARWDTPKVVKGVSFLLRLTVTADDGSERLVSTARTTETTYRFTQLALGNYRLTVRAVNAWGQQGDPASVSFRIAAPAAPSRIELTPGYFQITATPHLAVYDPTVQFEFWFSEKQIADIRQVETSTRYLGTALYWIAASINIKPGHDYYFYIRSVNTVGKSAFVEAVGRASDDAEGYLDFFKGKITESHLGKELLEKVELTEDNASRLEEFSKEWKDASDKWNAMWAVKIEQTKDGKHYVAGIGLSMEDTEEGKLSQFLVAANRIAFIDPANGNETPMFVAQGNQIFMNDVFLKRLTAPTITSGGNPPAFSLTPDGKLTAKNADISGSVNANSGTLSNVTIAENCTINGTLRAEKIVGDIVKAASAAFPRQRESSVDWPSGTRTVTVTDDHPFDRQIVVLPLTFRGSKRTVSGRTTYSMCYLKVLMNGAVIYDGAANEAVQVFSRIVDMPAGRGNVILTFTLTSTRHSADIPPYTFASDVQVMVIKKQALGISVV
6VVS , Knot 148 350 0.82 40 192 338 
MLISQRPTLSEETVAENRSRFVIEPLEPGFGYTLGNSLRRTLLSSIPGAAVTSIRIDGVLHEFTTVPGVKEDVTDIILNLKGLVVSSDDDEPVTMYLRKQGPGVVTAGDIVPPAGVTVHNPDMHIATLNDKGKLEVELVVERGRGYVPAVQNKASGAEIGRIPVDSIYSPVLKVTYKVEATRVEQRTDFDKLIIDVETKNSISPRDALASAGGTLVELFGLARELNADSEHIEIGPSPAEADHIASFALPIDDLDLTVRSYNCLKREGVHTVGELVARTESDLLDIRNFGQKSIDEVKIKLHQLGLSLKDSPATFDPSEVAGYDAATGTWTSDAGYDLDDNQDYAETEQL
6FEK , Knot 131 299 0.83 40 196 290 
GPLSLSVDAFKILAAPKWEFPRKNLVLGKTLGEGEFGKVVKATAFHLKGRAGYTTVAVKMLKENASPSELRDLLSEFNVLKQVNHPHVIKLYGACSQDGPLLLIVEYAKYGSLRGFLRESRKVGPGPDERALTMGDLISFAWQISQGMQYLAEMKLVHRDLAARNILVAEGRKMKISDFGLSRDVYEEDFYVKRSQGRIPVKWMAIESLFDHIYTTQSDVWSFGVLLWEIVTLGGNPYPGIPPERLFNLLKTGHRMERPDNCSEEMYRLMLQCWKQEPDKRPVFADISKDLEKMMVKRR

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 \(c\).

\(|P_{f(8XCI_1)}(2) \setminus P_{f(6VVS_1)}(2)|=157\), \(|P_{f(6VVS_1)}(2) \setminus P_{f(8XCI_1)}(2)|=16\). 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 1:1101000100100100010000110110110011101110110011100011100010001011011101100000110110001000111001000011000100101001010111011100000100010010111010001111000010101000000110111101110110101001010000001000011000001101000010011111010000110001000001010110110000100000011101010110000111011011001001110011110100111011100000011011110010100010100000110110010011001111010010110001000010000001111001111000101100000110101101001100100110000111001001001011100001010011111100011000010101110110011101101000001110011011110000001010001011001011101101010110101001001101010011011100011010110100011001010000010011011001100011100110101000101011011110010101010010001110100101101101110101010010001100100000000100111100010101101110010110101011111110010101101010101011100101010111000011010010000001101101111010101100001010010011001110111010001010101101010000110011001010000100100100010010001011111010000010001111110100000101001111100111101101000111110100111001110010110100110111101010101010010101010100101001011000010101010011101101101111000000101101000101000011000111111010100001010000010010111011110011100110110011011110101110101000000101110011001011110001111011
Pair \(Z_2\) Length of longest common subsequence
8XCI_1,6VVS_1 173 4
8XCI_1,6FEK_1 175 4
6VVS_1,6FEK_1 140 5

Newick tree

 
[
	8XCI_1:91.97,
	[
		6VVS_1:70,6FEK_1:70
	]:21.97
]

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 \((0.85)(0.8)(\frac{1482 }{\log_{20} 1482}-\frac{350}{\log_{20}350})=291.\)
Status Protein1 Protein2 d d1/2
Query variables 8XCI_1 6VVS_1 373 240.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]