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Parikh vectors
5KWH_1 7WMJ_1 5XBF_1 Letter Amino acid
13 18 34 A Alanine
26 10 31 R Arginine
22 24 32 E Glutamic acid
14 14 21 G Glycine
16 10 17 H Histidine
10 9 9 M Methionine
14 13 39 S Serine
11 14 25 T Threonine
23 4 17 Y Tyrosine
10 8 25 Q Glutamine
5 3 12 W Tryptophan
26 17 28 V Valine
27 14 25 D Aspartic acid
2 7 11 C Cysteine
30 30 62 L Leucine
13 14 21 F Phenylalanine
15 6 14 N Asparagine
19 14 25 I Isoleucine
23 17 38 K Lycine
14 14 30 P Proline 

5KWH_1|Chains A, B|Casein kinase II subunit alpha|Homo sapiens (9606)
>7WMJ_1|Chain A|Isoform Beta-2 of Thyroid hormone receptor beta|Homo sapiens (9606)
>5XBF_1|Chain A|Unconventional myosin-VIIb|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 \(w=f(c)\)
5KWH , Knot 149 333 0.86 40 196 318 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQAR
7WMJ , Knot 121 260 0.86 40 183 255 
EELQKSIGHKPEPTDEEWELIKTVTEAHVATNAQGSHWKQKRKFLPEDIGQAPIVNAPEGGKVDLEAFSHFTKIITPAITRVVDFAKKLPMFCELPCEDQIILLKGCCMEIMSLRAAVRYDPESETLTLNGEMAVTRGQLKNGGLGVVSDAIFDLGMSLSSFNLDDTEVALLQAVLLMSSDRPGLACVERIEKYQDSFLLAFEHYINYRKHHVTHFWPKLLMKVTDLRMIGACHASRFLHMKVECPTELFPPLFLEVFED
5XBF , Knot 212 516 0.85 40 275 488 
EEPPKEKLHTLEEFSYEFFRAPEKDMVSMAVLPLARARGHLWAYSCEPLRQPLLKRVHANVDLWDIACQIFVAILRYMGDYPSRQAWPTLELTDQIFTLALQHPALQDEVYCQILKQLTHNSNRHSEERGWQLLWLCTGLFPPSKGLLPHAQKFIDTRRGKLLAPDCSRRIQKVLRTGPRKQPPHQVEVEAAEQNVSRICHKIYFPNDTSEMLEVVANTRVRDVCDSIATRLQLASWEGCSLFIKISDKVISQKEGDFFFDSLREVSDWVKKNKPQKEGAPVTLPYQVYFMRKLWLNISPGKDVNADTILHYHQELPKYLRGFHKCSREDAIHLAGLIYKAQFNNDRSQLASVPKILRELVPENLTRLMSSEEWKKSILLAYDKHKDKTVEEAKVAFLKWICRWPTFGSAFFEVKQTSEPSYPDVILIAINRHGVLLIHPKTKDLLTTYPFTKISSWSSGSTYFHMALGSLGRGSRLLCETSLGYKMDDLLTSYVQQLLSAMNKQRGSKAPALAST

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 \(c\).

\(|P_{f(5KWH_1)}(2) \setminus P_{f(7WMJ_1)}(2)|=93\), \(|P_{f(7WMJ_1)}(2) \setminus P_{f(5KWH_1)}(2)|=80\). 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 1:101111001010001000010001000001101100000011001101000011011010000011101101100001000101100101110110110110011000111110010000100100010000101010011011000001111000101001110000001011011110100110000101100010110111000100001011011011101110001110100000011011011100010001000010101010011100000010011000000110101101100110000000101001100101001100010
Pair \(Z_2\) Length of longest common subsequence
5KWH_1,7WMJ_1 173 4
5KWH_1,5XBF_1 181 4
7WMJ_1,5XBF_1 190 4

Newick tree

 
[
	5XBF_1:94.77,
	[
		5KWH_1:86.5,7WMJ_1:86.5
	]:8.27
]

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 \((0.85)(0.8)(\frac{593 }{\log_{20} 593}-\frac{260}{\log_{20}260})=93.9\)
Status Protein1 Protein2 d d1/2
Query variables 5KWH_1 7WMJ_1 119 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]