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Parikh vectors
8QWS_1 3ALW_1 6SXN_1 Letter Amino acid
10 19 13 Q Glutamine
11 49 19 G Glycine
22 59 34 L Leucine
9 16 7 M Methionine
17 54 9 S Serine
24 31 15 T Threonine
22 20 9 R Arginine
9 13 4 C Cysteine
12 26 10 Y Tyrosine
8 23 7 N Asparagine
11 31 10 P Proline
13 31 11 K Lycine
5 9 0 W Tryptophan
26 21 37 A Alanine
27 24 14 D Aspartic acid
15 27 17 I Isoleucine
20 16 4 F Phenylalanine
24 43 21 V Valine
21 27 21 E Glutamic acid
12 20 5 H Histidine 

8QWS_1|Chains A, B|Cyclooctat-9-en-7-ol synthase|Streptomyces melanosporofaciens (67327)
>3ALW_1|Chain A|Hemagglutinin, CDw150|Measles virus (11234)
>6SXN_1|Chain A|Geranylgeranyl pyrophosphate synthase|Synechococcus sp. PCC 7002 (32049)
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 \(w=f(c)\)
8QWS , Knot 139 318 0.84 40 202 299 
MTTGLSTAGAQDIGRSSVRPYLEECTRRFQEMFDRHVVTRPTKVELTDAELREVIDDCNAAVAPLGKTVSDERWISYVGLVLWSQSPRHIKDMEAFKAVCVLNCVTFVWDDMDPALHDFGLFLPQLRKICEKYYGPEDAEVAYEAARAFVTSDHMFRDSPIKAALCTTSPEQYFRFRVTDIGVDFWMKMSYPIYRHPEFTEHAKTSLAARMTTRGLTIVNDFYSYDREVSLGQITNCFRLCDVSDETAFKEFFQARLDDMIEDIECIKAFDQLTQDVFLDLIYGNFVWTTSNKRYKTAVNDVNSRIQAAALEHHHHHH
3ALW , Knot 216 559 0.81 40 271 511 
ETGSKGNCSGPTTIRGQFSNMSLSLLDLYLGRGYNVSSIVTMTSQGMYGGTYLVEKPNLSSKRSELSQLSMYRVFEVGVIRNPGLGAPVFHMTNYLEQPVSNDLSNCMVALGELKLAALCHGEDSITIPYQGSGKGVSFQLVKLGVWKSPTDMQSWVPLSTDDPVIDRLYLSSHRGVIADNQAKWAVPTTRTDDKLRMETCFQQACKGKIQALCENPEWAPLKDNRIPSYGVLSVDLSLTVELKIKIASGFGPLITHGSGMDLYKSNHNNVYWLTIPPMKNLALGVINTLEWIPRFKVSPYLFTVPIKEAGGDCHAPTYLPAEVDGDVKLSSNLVILPGQDLQYVLATYDTSRVEHAVVYYVYSPSRSFSYFYPFRLPIKGVPIELQVECFTWDQKLWCRHFCVLADSESGGHITHSGMVGMGVSCTGGGSGGGSGGGSMNCPKIVQQLGSDVLLPLTHERINTSMNKSIHIVVTMAKSLENSVENKIVSLDPSEAGPPRYLKDRYRFYLEHLSLAIYESTKKDEGWYFMTLEKNISVQRFCLHLKLYEQGTKHHHHHH
6SXN , Knot 117 267 0.81 38 159 253 
QGFSLAQYLQEQKTIVETALDQSLVITEPVTIYEAMRYSLLAGGKRLRPILCLAACEMLGGTAAMAMNTACALEMIHTMSLIHDDLPAMDNDDLRRGKPTNHKVYGEDIAILAGDALLSYAFEYVARTPDVPAERLLQVIVRLGQAVGAEGLVGGQVVDLESEGKDVAVETLNFIHTHKTGALLEVCVTAGAILAGAKPEEVQLLSRYAQNIGLAFQIVDDILDITVTYPKSQAEAQKLVAEAIASLEPYGEKANPLKALAEYIVNA

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 \(c\).

\(|P_{f(8QWS_1)}(2) \setminus P_{f(3ALW_1)}(2)|=53\), \(|P_{f(3ALW_1)}(2) \setminus P_{f(8QWS_1)}(2)|=122\). 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 1:100110011100110001010100000010011000110010010100101001100001111111001000011001111110001001001011011011001011100101110011111101001000001100101100110111000011000110111000010001010100111011101001100010100010001110100011011001000000101101000101001000011001101010011001001011001000111011010111000000000110010001011110000000
Pair \(Z_2\) Length of longest common subsequence
8QWS_1,3ALW_1 175 6
8QWS_1,6SXN_1 159 4
3ALW_1,6SXN_1 190 3

Newick tree

 
[
	3ALW_1:94.94,
	[
		8QWS_1:79.5,6SXN_1:79.5
	]:15.44
]

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 \((0.85)(0.8)(\frac{877 }{\log_{20} 877}-\frac{318}{\log_{20}318})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 8QWS_1 3ALW_1 188 147
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]