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Parikh vectors
7RBL_1 6WLB_1 4ZOM_1 Letter Amino acid
26 80 30 L Leucine
7 24 6 M Methionine
14 56 3 P Proline
12 36 7 Y Tyrosine
15 30 11 H Histidine
23 61 10 I Isoleucine
36 57 11 K Lycine
12 46 6 N Asparagine
13 26 14 Q Glutamine
27 61 21 E Glutamic acid
22 65 10 G Glycine
19 47 7 T Threonine
18 76 13 V Valine
18 61 15 A Alanine
21 48 5 D Aspartic acid
3 27 8 C Cysteine
21 65 18 S Serine
19 49 17 R Arginine
14 53 11 F Phenylalanine
1 23 2 W Tryptophan 

7RBL_1|Chain A|DNA polymerase beta|Homo sapiens (9606)
>6WLB_1|Chains A, B, C|Cellulose synthase|Populus tremula x Populus tremuloides (47664)
>4ZOM_1|Chains A, B, C, D|Nuclear receptor ROR-gamma|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 \(w=f(c)\)
7RBL , Knot 147 341 0.83 40 200 324 
MSKRKAPQETLNGGITDMLTELANFEKNVSQAIHKYNAYRKAASVIAKYPHKIKSGAEAKKLPGVGTKIAEKIDEFLATGKLRKLEKIRQDDTSSSINFLTRVSGIGPSAARKFVDEGIKTLEDLRKNEDKLNHHQRIGLKYFGDFEKRIPREEMLQMQDIVLNEVKKVDSEYIATVCGSFRRGAESSGDMDVLLTHPSFTSESTKQPKLLHQVVEQLQKVHFITDTLSKGETKFMGVCQLPSKNDEKEYPHRRIDIRLIPKDQYYCGVLYFTGSDIFNKNMRAHALEKGFTINEYTIRPLGVTGVAGEPLPVDSEKDIFDYIQWKYREPKDRSEHHHHHH
6WLB , Knot 373 991 0.86 40 338 902 
MHHHHHHHHHHHHMMESGAPICHTCGEQVGHDANGDLFVACHECNYHICKSCFEYEIKEGRKVCLRCGSPYDENLLDDVEKKGSGNQSTMASHLNNSQDVGIHARHISSVSTVDSEMNDEYGNPIWKNRVESWKDKRNKKKKSNTKPETEPAQVPPEQQMENKPSAEASEPLSIVYPIPRNKLTPYRAVIIMRLIILGLFFHYRITNPVDSAFGLWLTSVICEIWFAFSWVLDQFPKWKPVNRETFIERLSARYEREGEPSQLAAVDFFVSTVDPLKEPPLITANTVLSILAVDYPVDKVSCYVSDDGAAMLTFESLVETAEFARKWVPFCKKFSIEPRAPEFYFSQKIDYLKDKVQPSFVKERRAMKRDYEEYKVRVNALVAKAQKTPDEGWTMQDGTPWPGNNTRDHPGMIQVFLGNTGARDIEGNELPRLVYVSREKRPGYQHHKKAGAENALVRVSAVLTNAPYILNLDCDHYVNNSKAVREAMCILMDPQVGRDVCYVQFPQRFDGIDRSDRYANRNIVFFDVNMKGLDGIQGPMYVGTGCVFNRQALYGYGPPSMPRLRKGKESSSCFSCCCPTKKKPAQDPAEVYRDAKREDLNAAIFNLTEIDNYDDYERSMLISQLSFEKTFGLSPVFIESTLMENGGVPESANSSTLIKEAIHVIGCGFEEKTEWGKEIGWIYGSVTEDILSGFKMHCRGWRSIYCMPVRPAFKGSAPINLSDRLHQVLRWALGSVEIFFSRHCPFWYGYGGGRLKWLQRLAYINTIVYPFTSLPLIAYCTIPAVCLLTGKFIIPTLSNLASMLFLGLFISIIVTAVLELRWSGVSIEDLWRNEQFWVIGGVSAHLFAVFQGFLKMLAGIDTNFTVTAKAADDTEFGELYMVKWTTLLIPPTTLLIINIVGVVAGFSDALNKGYEAWGPLFGKVFFAFWVILHLYPFLKGLMGRQNRTPTIVVLWSVLLTSVFSLVWVKINPFVNKVDNTLAGETCISIDC
4ZOM , Knot 105 225 0.84 40 153 219 
GSASLTEIEHLVQSVCKSYRETCQLRLEDLLRQRSNIFSREEVTGYQRKSMWEMWERCAHHLTEAIQYVVEFAKRLSGFMELCQNDQIVLLKAGAMEVVLVRMCRAYNADNRTVFFEGKYGGMELFRALGCSELISSIFDFSHSLSALHFSEDEIALYTALVLINAHRPGLQEKRKVEQLQYNLELAFHHHLCKTHRQSILAKLPPKGKLRSLCSQHVERLQIFQ

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 \(c\).

\(|P_{f(7RBL_1)}(2) \setminus P_{f(6WLB_1)}(2)|=24\), \(|P_{f(6WLB_1)}(2) \setminus P_{f(7RBL_1)}(2)|=162\). 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 1:10000110001011100110011010001001100001000110111001001001101001111100110010011101010010010000000010110010111101100110011001001000000100000111001101000110001101001110010010000110101010011000101011100101000000010110011001001011000100100011110011000000001000101011100000011101010011000101011001101000010111101111011110000011001010000100000000000
Pair \(Z_2\) Length of longest common subsequence
7RBL_1,6WLB_1 186 6
7RBL_1,4ZOM_1 159 5
6WLB_1,4ZOM_1 227 4

Newick tree

 
[
	6WLB_1:11.66,
	[
		7RBL_1:79.5,4ZOM_1:79.5
	]:31.16
]

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 \((0.85)(0.8)(\frac{1332 }{\log_{20} 1332}-\frac{341}{\log_{20}341})=258.\)
Status Protein1 Protein2 d d1/2
Query variables 7RBL_1 6WLB_1 329 218
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]