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Parikh vectors
5ZLK_1 6XCJ_1 7QXN_1 Letter Amino acid
31 23 63 V Valine
28 11 31 R Arginine
31 27 57 I Isoleucine
42 19 99 L Leucine
33 33 47 T Threonine
8 9 29 M Methionine
26 10 29 Y Tyrosine
35 22 79 E Glutamic acid
22 9 21 H Histidine
25 25 70 K Lycine
46 16 42 P Proline
26 17 80 A Alanine
23 32 45 N Asparagine
32 12 54 D Aspartic acid
15 10 39 Q Glutamine
9 7 5 W Tryptophan
4 14 18 C Cysteine
32 26 57 G Glycine
16 11 32 F Phenylalanine
29 24 56 S Serine 

5ZLK_1|Chain A|Spore coat protein A|Bacillus subtilis subsp. subtilis str. 168 (224308)
>6XCJ_1|Chain A[auth G]|Envelope Glycoprotein gp120|Human immunodeficiency virus 1 (11676)
>7QXN_1|Chain A[auth U]|26S proteasome non-ATPase regulatory subunit 1|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 \(w=f(c)\)
5ZLK , Knot 212 513 0.86 40 271 486 
MTLEKFVDALPIPDTLKPVQQSKEKTYYEVTMEECTHQLHRDLPPTRLWGYNGLFPGPTIEVKRNENVYVKWMNNLPSTHFLPIDHTIHHSDSQHEEPEVKTVVHLHGGVTPDDSDGYPEAWFSKDFEQTGPYFKREVYHYPNQQRGAILWYHDHAMALTRLNVYAGLVGAYIIHDPKEKRLKLPSDEYDVPLLITDRTINEDGSLFYPSAPENPSPSLPNPSIVPAFCGETILVNGKVWPYLEVEPRKYRFRVINASNTRTYNLSLDNGGDFIQIGSDGGLLPRSVKLNSFSLAPAERYDIIIDFTAYEGESIILANSAGCGGDVNPETDANIMQFRVTKPLAQKDESRKPKYLASYPSVQHERIQNIRTLKLAGTQDEYGRPVLLLNNKRWHDPVTETPKVGTTEIWSIINPTRGTHPIHLHLVSFRVLDRRPFDIARYQESGELSYTGPAVPPPPSEKGWKDTIQAHAGEVLRIAATFGPYSGRYVWHCAILEHEDYDMMRPMDITDPHK
6XCJ , Knot 153 357 0.84 40 208 335 
VWKEAKTTLFCASDAKAYEKEVHNVWATHACVPTDPNPQEMVLKNVTENFNMWKNDMVDQMHEDVISLWDQSLKPCVKLTGGSVITQACPKVSFDPIPIHYCAPAGYAILKCNNKTFTGTGPCNNVSTVQCTHGIKPVVSTQLLLNGSLAEGEIIIRSENITNNVKTIIVHLNESVKIECTRPGNGGSGSGGDIREAYCNINESKWNETLQRVSKKLKEYFPHKNITFQPSSGGDLEITTHSFNCGGEFFYCNTSSLFNRTYMANSTDMANSTETNSTRTITIHCRIKQIINMWQEVGRAMYAPPIAGNITCISIITGLLLTRDGGKNNTETFRPGGGNMKDNWRSELYKYKVVEVK
7QXN , Knot 349 953 0.83 40 314 835 
MITSAAGIISLLDEDEPQLKEFALHKLNAVVNDFWAEISESVDKIEVLYEDEGFRSRQFAALVASKVFYHLGAFEESLNYALGAGDLFNVNDNSEYVETIIAKCIDHYTKQCVENADLPEGEKKPIDQRLEGIVNKMFQRCLDDHKYKQAIGIALETRRLDVFEKTILESNDVPGMLAYSLKLCMSLMQNKQFRNKVLRVLVKIYMNLEKPDFINVCQCLIFLDDPQAVSDILEKLVKEDNLLMAYQICFDLYESASQQFLSSVIQNLRTVGTPIASVPGSTNTGTVPGSEKDSDSMETEEKTSSAFVGKTPEASPEPKDQTLKMIKILSGEMAIELHLQFLIRNNNTDLMILKNTKDAVRNSVCHTATVIANSFMHCGTTSDQFLRDNLEWLARATNWAKFTATASLGVIHKGHEKEALQLMATYLPKDTSPGSAYQEGGGLYALGLIHANHGGDIIDYLLNQLKNASNDIVRHGGSLGLGLAAMGTARQDVYDLLKTNLYQDDAVTGEAAGLALGLVMLGSKNAQAIEDMVGYAQETQHEKILRGLAVGIALVMYGRMEEADALIESLCRDKDPILRRSGMYTVAMAYCGSGNNKAIRRLLHVAVSDVNDDVRRAAVESLGFILFRTPEQCPSVVSLLSESYNPHVRYGAAMALGICCAGTGNKEAINLLEPMTNDPVNYVRQGALIASALIMIQQTEITCPKVNQFRQLYSKVINDKHDDVMAKFGAILAQGILDAGGHNVTISLQSRTGHTHMPSVVGVLVFTQFWFWFPLSHFLSLAYTPTCVIGLNKDLKMPKVQYKSNCKPSTFAYPAPLEVPKEKEKEKVSTAVLSITAKAKKKEKEKEKKEEEKMEVDEAEKKEEKEKKKEPEPNFQLLDNPARVMPAQLKVLTMPETCRYQPFKPLSIGGIIILKDTSEDIEELVEPVAAHGPKIEEEEQEPEPPEPFEYIDD

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 \(c\).

\(|P_{f(5ZLK_1)}(2) \setminus P_{f(6XCJ_1)}(2)|=118\), \(|P_{f(6XCJ_1)}(2) \setminus P_{f(5ZLK_1)}(2)|=55\). 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 1:101001101111100101100000000001010000001000111001110011111101010000010101100110001111000100000000010100110101110100001010111000100011010001000100001111100001111001010111111011001000010110000011111000010001011010110010101101011111010011101011101010100001011010000000101001101101100111110010100101111000011101010010011110011011010100010110101001110000000100110010100001001001011100000101111100001001100010110001101101001001101011010110001101100000101000111111110001100010101101101110111001001100111000000110110100100
Pair \(Z_2\) Length of longest common subsequence
5ZLK_1,6XCJ_1 173 4
5ZLK_1,7QXN_1 139 4
6XCJ_1,7QXN_1 168 4

Newick tree

 
[
	6XCJ_1:89.90,
	[
		5ZLK_1:69.5,7QXN_1:69.5
	]:20.40
]

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 \((0.85)(0.8)(\frac{870 }{\log_{20} 870}-\frac{357}{\log_{20}357})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 5ZLK_1 6XCJ_1 178 148
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]