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Parikh vectors
2JER_1 9CLN_1 7AKH_1 Letter Amino acid
36 53 23 V Valine
12 16 22 H Histidine
15 60 23 S Serine
32 61 34 G Glycine
21 42 19 I Isoleucine
22 74 45 L Leucine
21 41 32 K Lycine
16 47 17 F Phenylalanine
20 56 22 P Proline
20 62 20 A Alanine
24 41 21 Q Glutamine
31 61 20 D Aspartic acid
11 7 9 C Cysteine
10 30 16 M Methionine
18 66 16 T Threonine
15 44 28 R Arginine
19 73 16 N Asparagine
10 56 18 Y Tyrosine
28 49 24 E Glutamic acid
8 13 4 W Tryptophan 

2JER_1|Chains A, B, C, D, E, F, G, H|AGMATINE DEIMINASE|ENTEROCOCCUS FAECALIS (1351)
>9CLN_1|Chains A[auth J], B[auth K], C[auth L]|Hexon protein|Human adenovirus 5 (28285)
>7AKH_1|Chain A|Dual specificity tyrosine-phosphorylation-regulated kinase 2|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 \(w=f(c)\)
2JER , Knot 167 389 0.85 40 233 370 
MAKRIVGSTPKQDGFRMPGEFEPQEKVWMIWPERPDNWRDGGKPVQEAFTNVAKAISQFTPMNVVVSQQQFQNCRRQLPPEITVYEMSNNDAWVRDCGPSFVINDHGEIRGVDWTFNAWGGLVDGLYFPWDQDDLVAQKICEIEHVDSYRTDDFVLEGGSFHVDGQGTVLTTEMCLLSEGRNPQLSKEAIEQKLCDYLNVEKVLWLGDGIDPEETNGHVDDVACFIAPGEVACIYTEDQNSPFYEAAQDAYQRLLKMTDAKGRQLKVHKLCCPVKNVTIKGSFKIDFVEGTMPREDGDICIASYMNFLITNDGVIVPQYGDENDRLALEQVQTMFPDKKIVGVNTVEVVYGGGNIHCITQQEPKRVGGKQNSSSVDKLAAALEHHHHHH
9CLN , Knot 356 952 0.85 40 319 865 
MATPSMMPQWSYMHISGQDASEYLSPGLVQFARATETYFSLNNKFRNPTVAPTHDVTTDRSQRLTLRFIPVDREDTAYSYKARFTLAVGDNRVLDMASTYFDIRGVLDRGPTFKPYSGTAYNALAPKGAPNPCEWDEAATALEINLEEEDDDNEDEVDEQAEQQKTHVFGQAPYSGINITKEGIQIGVEGQTPKYADKTFQPEPQIGESQWYETEINHAAGRVLKKTTPMKPCYGSYAKPTNENGGQGILVKQQNGKLESQVEMQFFSTTEATAGNGDNLTPKVVLYSEDVDIETPDTHISYMPTIKEGNSRELMGQQSMPNRPNYIAFRDNFIGLMYYNSTGNMGVLAGQASQLNAVVDLQDRNTELSYQLLLDSIGDRTRYFSMWNQAVDSYDPDVRIIENHGTEDELPNYCFPLGGVINTETLTKVKPKTGQENGWEKDATEFSDKNEIRVGNNFAMEINLNANLWRNFLYSNIALYLPDKLKYSPSNVKISDNPNTYDYMNKRVVAPGLVDCYINLGARWSLDYMDNVNPFNHHRNAGLRYRSMLLGNGRYVPFHIQVPQKFFAIKNLLLLPGSYTYEWNFRKDVNMVLQSSLGNDLRVDGASIKFDSICLYATFFPMAHNTASTLEAMLRNDTNDQSFNDYLSAANMLYPIPANATNVPISIPSRNWAAFRGWAFTRLKTKETPSLGSGYDPYYTYSGSIPYLDGTFYLNHTFKKVAITFDSSVSWPGNDRLLTPNEFEIKRSVDGEGYNVAQCNMTKDWFLVQMLANYNIGYQGFYIPESYKDRMYSFFRNFQPMSRQVVDDTKYKDYQQVGILHQHNNSGFVGYLAPTMREGQAYPANFPYPLIGKTAVDSITQKKFLCDRTLWRIPFSSNFMSMGALTDLGQNLLYANSAHALDMTFEVDPMDEPTLLYVLFEVFDVVRVHRPHRGVIETVYLRTPFSAGNATT
7AKH , Knot 183 429 0.86 40 244 408 
MHHHHHHSSGVDLGTENLYFQSMGKVKATPMTPEQAMKQYMQKLTAFEHHEIFSYPEIYFLGLNAKKRQGMTGGPNNGGYDDDQGSYVQVPHDHVAYRYEVLKVIGKGSFGQVVKAYDHKVHQHVALKMVRNEKRFHRQAAEEIRILEHLRKQDKDNTMNVIHMLENFTFRNHICMTFELLSMNLYELIKKNKFQGFSLPLVRKFAHSILQCLDALHKNRIIHCDLKPENILLKQQGRSGIKVIDFGSSCYEHQRVYTYIQSRFYRAPEVILGARYGMPIDMWSLGCILAELLTGYPLLPGEDEGDQLACMIELLGMPSQKLLDASKRAKNFVSSKGYPRYCTVTTLSDGSVVLNGGRSRRGKLRGPPESREWGNALKGCDDPLFLDFLKQCLEWDPAVRMTPGQALRHPWLRRRLPKPPTGEKTSVKR

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 \(c\).

\(|P_{f(2JER_1)}(2) \setminus P_{f(9CLN_1)}(2)|=36\), \(|P_{f(9CLN_1)}(2) \setminus P_{f(2JER_1)}(2)|=122\). 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 1:11001110010001101110101000111111001001001101100110011011001011011100001000000111010100100001110001101110001010110101011111101101110000111001001001000000011101101010101011000101100100101000110001000101001111101101000010100110111110110100000001100110010001101001010010100100110010101010101101011000101011001011100011111001000001110010011100011110010110111010010000100111000000100111110000000
Pair \(Z_2\) Length of longest common subsequence
2JER_1,9CLN_1 158 4
2JER_1,7AKH_1 165 6
9CLN_1,7AKH_1 141 4

Newick tree

 
[
	2JER_1:83.91,
	[
		9CLN_1:70.5,7AKH_1:70.5
	]:13.41
]

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 \((0.85)(0.8)(\frac{1341 }{\log_{20} 1341}-\frac{389}{\log_{20}389})=246.\)
Status Protein1 Protein2 d d1/2
Query variables 2JER_1 9CLN_1 314 219.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]