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Parikh vectors
5FPS_1 9IQE_1 5SWU_1 Letter Amino acid
33 40 17 I Isoleucine
11 17 8 M Methionine
34 43 15 R Arginine
13 11 10 N Asparagine
50 61 27 L Leucine
19 2 14 K Lycine
54 31 17 S Serine
63 37 11 T Threonine
61 65 17 V Valine
22 26 8 Q Glutamine
20 18 14 E Glutamic acid
43 21 12 P Proline
23 12 6 Y Tyrosine
31 37 19 D Aspartic acid
22 9 6 H Histidine
66 46 21 G Glycine
18 24 12 F Phenylalanine
6 8 3 W Tryptophan
56 77 24 A Alanine
21 5 2 C Cysteine 

5FPS_1|Chains A, B|HEPATITIS C VIRUS FULL-LENGTH NS3 COMPLEX|HEPATITIS C VIRUS (ISOLATE BK) (11105)
>9IQE_1|Chain A|ABC transporter, ATP-binding protein|Mycolicibacterium smegmatis MC2 155 (246196)
>5SWU_1|Chains A, B|Pentafunctional AROM polypeptide|Neosartorya fumigata (strain CEA10 / CBS 144.89 / FGSC A1163) (451804)
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 \(w=f(c)\)
5FPS , Knot 255 666 0.83 40 268 609 
MGSSHHHHHHSSGLVPRGSHMGSVVIVGRIILSGSGSITAYSQQTRGLLGCIITSLTGRDKNQVEGEVQVVSTATQSFLATCVNGVCWTVYHGAGSKTLAGPKGPITQMYTNVDQDLVGWQAPPGARSLTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPVSYLKGSSGGPLLCPSGHAVGIFRAAVCTRGVAKAVDFVPVESMETTMRSPVFTDNSSPPAVPQSFQVAHLHAPTGSGKSTKVPAAYAAQGYKVLVLNPSVAATLGFGAYMSKAHGIDPNIRTGVRTITTGAPVTYSTYGKFLADGGCSGGAYDIIICDECHSTDSTTILGIGTVLDQAETAGARLVVLATATPPGSVTVPHPNIEEVALSNTGEIPFYGKAIPIEAIRGGRHLIFCHSKKKCDELAAKLSGLGINAVAYYRGLDVSVIPTIGDVVVVATDALMTGYTGDFDSVIDCNTCVTQTVDFSLDPTFTIETTTVPQDAVSRSQRRGRTGRGRRGIYRFVTPGERPSGMFDSSVLCECYDAGCAWYELTPAETSVRLRAYLNTPGLPVCQDHLEFWESVFTGLTHIDAHFLSQTKQAGDNFPYLVAYQATVCARAQAPPPSWDQMWKCLIRLKPTLHGPTPLLYRLGAVQNEVTLTHPITKYIMACMSADLEVVT
9IQE , Knot 229 590 0.82 40 230 524 
MPEVVGSYFQSNMLWALLRQYVRPYRWLLAVVAVLQVISNMASLYLPTVNAAIIDDGVAKGDTARIVELGAVMLGVTALQVVCAVGAVFFGARAATGFGHDLRAAVFTHVTTFSAEEAGRFGAASLLTRTTNDVGHIQQLVQLTVTMLITAPIMSIGGIFMALHQDAGLSWLLLVSVPVLGLANYWIIRHLMPVFTRMQSLIDGINRVLRDQLSGIRVIRAFAREPLERVRFAEANQTLSDSALEAGRWQALMLPVTTLVINVSSVALIWFGGLRIDAGQMQVGSLIAFLAYFMQILMAVLMATFMLVIFPRAAVCADRIGEVLSTQTAITNPADPVRPAAIAGDIGVHDATFCYPGADRPVLQDVSFTVPRGTTTAVVGSTGSGKSTLISLICRLYDVTSGSLRIDGVDVRDLDIEQLWSAIGLVPQRGYLFSGTVAENLRYGRADATDDEMWEALRVAAAADFVRAHPQGLDMPVAQGGINFSGGQRQRLAIARAVIRRPAIYLFDDAFSALDVHTDARVRDALREVAADATVVIVSQRISTVIEADQVVVIDDGRVVGIGTHDTLLADCPIYAEFAESQALTAGEPR
5SWU , Knot 119 263 0.84 40 170 251 
FIKFVTGLEDSLGTIKKKQHSFFVSLTLPDVRGADQILEQACVGSDAVELRVDLLEDPDSSNGIPTVDFVADQISYLRSRITLPVIFTIRTKGQGGRFPDDAHAEAMQLYRLAVRSGCEFVDLEIAFPDEMLRAVTEMKGYSKIIASHHDPNGELSWANMSWMKYYNRALEYGDVIKLVGVARNLDDNTALRKFKNWAEEAHDVPLIAINMGGNGQLSRILNGFMTPVSHPALPFRAAPGQLSATDIRKGLSLMGEIKKGAAL

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 \(c\).

\(|P_{f(5FPS_1)}(2) \setminus P_{f(9IQE_1)}(2)|=94\), \(|P_{f(9IQE_1)}(2) \setminus P_{f(5FPS_1)}(2)|=56\). 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 1:110000000000111101001101111101110101010100000011110110010100000101010110010001110010110101001110001111011100100010001111011111001010001000101100010111100010001011010110010100111110101011111011100011101101111001000100111000001111100101101011010100001111011010011110101110111110100101101010011001001111000001011101100111001110000000000011111011001001110111110101110101101010011100010111010111101101100111000000000111010111101110001101011101101111100111010010100110000010001010101010100001100110000001001010011001101100101110001100000110110010110001010101001111100001011001101100101011000001100110111001010101011110100110011010101011011100111100010100110001110101010110
Pair \(Z_2\) Length of longest common subsequence
5FPS_1,9IQE_1 150 7
5FPS_1,5SWU_1 174 4
9IQE_1,5SWU_1 160 4

Newick tree

 
[
	5SWU_1:86.24,
	[
		5FPS_1:75,9IQE_1:75
	]:11.24
]

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 \((0.85)(0.8)(\frac{1256 }{\log_{20} 1256}-\frac{590}{\log_{20}590})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 5FPS_1 9IQE_1 214 200
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]