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Parikh vectors
1JPJ_1 8JTA_1 7DKD_1 Letter Amino acid
30 11 32 R Arginine
16 15 46 D Aspartic acid
0 9 4 C Cysteine
9 45 19 I Isoleucine
5 15 26 Y Tyrosine
33 43 103 A Alanine
2 14 33 N Asparagine
9 12 34 Q Glutamine
45 54 70 L Leucine
15 12 49 K Lycine
8 37 36 S Serine
31 10 32 E Glutamic acid
1 6 8 H Histidine
12 26 33 P Proline
1 4 10 W Tryptophan
25 33 49 V Valine
29 39 60 G Glycine
6 22 16 M Methionine
6 27 29 F Phenylalanine
13 23 31 T Threonine 

1JPJ_1|Chain A|SIGNAL RECOGNITION PARTICLE PROTEIN|Thermus aquaticus (271)
>8JTA_1|Chain A|Synaptic vesicular amine transporter|Homo sapiens (9606)
>7DKD_1|Chains A, B|Dipeptidyl-peptidase|Stenotrophomonas maltophilia (strain R551-3) (391008)
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 \(w=f(c)\)
1JPJ , Knot 122 296 0.78 38 146 263 
MFQQLSARLQEAIGRLRGRGRITEEDLKATLREIRRALMDADVNLEVARDFVERVREEALGKQVLESLTPAEVILATVYEALKEALGGEARLPVLKDRNLWFLVGLQGSGKTTTAAKLALYYKGKGRRPLLVAADTQRPAAREQLRLLGEKVGVPVLEVMDGESPESIRRRVEEKARLEARDLILVDTAGRLQIDEPLMGELARLKEVLGPDEVLLVLDAMTGQEALSVARAFDEKVGVTGLVLTKLDGDARGGAALSARHVTGKPIYFAGVSEKPEGLEPFYPERLAGRILGMGD
8JTA , Knot 185 457 0.82 40 217 425 
SRKLILFIVFLALLLDNMLLTVVVPIIPSYLYSIKHEKNATEIQTARPVHTASISDSFQSIFSYYDNSTMVTGNATRDLTLHQTATQHMVTNASAVPSDCPSEDKDLLNENVQVGLLFASKATVQLITNPFIGLLTNRIGYPIPIFAGFCIMFVSTIMFAFSSSYAFLLIARSLQGIGSSCSSVAGMGMLASVYTDDEERGNVMGIALGGLAMGVLVGPPFGSVLYEFVGKTAPFLVLAALVLLDGAIQLFVLQPSRVQPESQKGTPLTTLLKDPYILIAAGSICFANMGIAMLEPALPIWMMETMCSRKWQLGVAFLPASISYLIGTNIFGILAHKMGRWLCALLGMIIVGVSILCIPFAKNIYGLIAPNFGVGFAIGMVDSSMMPIMGYLVDLRHVSVYGSVYAIADVAFCMGYAIGPSAGGAIAKAIGFPWLMTIIGIIDILFAPLCFFLRSPP
7DKD , Knot 273 720 0.83 40 263 646 
MRSNLLAFSIVASLGLAQVAHAAEGMWVPQQLPEIAGPLQKAGLKLSPEQLANLTGDPMGAVVALGGCTASFVSPQGLVVTNHHCAYGAIQLNSTAQKNLIKDGFNAPTLKDELSAGPNARVFVLDQITDVTAQAKAAIAGAGNDPLARSRALDAFDKAQVAACEADAGFRCRLYSFSGGNTYRLFRNMEIKDVRLVYAPPGSVGKFGGDVDNWMWPRHTGDFSFYRAYVGKDGKPAAFAADNVPYQPKHFLKFADQPLGADDFVMVAGYPGRTNRYALAGEFNETASFTYPTIAKHYNAVLKMIADAGKADADVKVKYAATAASMNNVAKNYLGQLEGFKRIDAAGQKQAEEAAVLAWLKKQGAAGKPALAAHAQLLKHLDTSKSTRERDLFVGQFNNTSAVGAAITLYRLSIERSKPDAEREAGYQERDLTTIEGGLKQMDRRYVAKMDQQLQTYWLDQYVALPAAQRDNEVLNKWLAGSDAAAVKSLVNKLGGTELGSLDTRLKWFKADRAAFEASNDPAIQYAVAVMPALLKQEEQKKIREGESLTARPLYLQAVADYKKSQGEFVYPDANLSLRITFGNVMGYGKDGVKYTPFTTLEGVAAKETGEDPFDSPKALLDAVKAKRYGGLEDKRLGSVPVNFLSNLDITGGNSGSPVLDANGKLVGLAFDGNWESVSSNWVFDPVMTRMIAVDSRYMQWIMQEVAPAPQLLKELNLAK

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 \(c\).

\(|P_{f(1JPJ_1)}(2) \setminus P_{f(8JTA_1)}(2)|=43\), \(|P_{f(8JTA_1)}(2) \setminus P_{f(1JPJ_1)}(2)|=114\). 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 1:11001010100111010101010000101010010011101010101100110010001110011001011011110100110011110101111000011111110101000011011100010100111111000011100010111001111110110100100100010001010100111100110101001111011010011110011111011010011011011000111011110010101011111010010101101111000101101101001110111110
Pair \(Z_2\) Length of longest common subsequence
1JPJ_1,8JTA_1 157 4
1JPJ_1,7DKD_1 165 4
8JTA_1,7DKD_1 170 5

Newick tree

 
[
	7DKD_1:85.44,
	[
		1JPJ_1:78.5,8JTA_1:78.5
	]:6.94
]

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 \((0.85)(0.8)(\frac{753 }{\log_{20} 753}-\frac{296}{\log_{20}296})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 1JPJ_1 8JTA_1 157 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]