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Parikh vectors
4DTO_1 8VGH_1 6ZAH_1 Letter Amino acid
52 22 19 G Glycine
14 18 10 H Histidine
74 11 15 I Isoleucine
70 9 20 K Lycine
12 9 5 W Tryptophan
54 15 28 A Alanine
57 19 24 D Aspartic acid
8 8 2 C Cysteine
25 4 4 M Methionine
54 11 20 S Serine
44 6 19 N Asparagine
31 13 17 Q Glutamine
70 9 18 E Glutamic acid
52 10 20 Y Tyrosine
48 25 22 V Valine
48 4 16 F Phenylalanine
43 23 20 P Proline
38 12 15 T Threonine
46 12 12 R Arginine
63 20 25 L Leucine 

4DTO_1|Chain A|DNA polymerase|Enterobacteria phage RB69 (12353)
>8VGH_1|Chains A, B, C, D|Tryptase alpha/beta-1|Homo sapiens (9606)
>6ZAH_1|Chain A|Isopenicillin N synthase|Emericella nidulans (strain FGSC A4 / ATCC 38163 / CBS 112.46 / NRRL 194 / M139) (227321)
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 \(w=f(c)\)
4DTO , Knot 338 903 0.85 40 319 827 
MKEFYLTVEQIGDSIFERYIDSNGRERTREVEYKPSLFAHCPESQATKYFDIYGKPCTRKLFANMRDASQWIKRMEDIGLEALGMDDFKLAYLSDTYNYEIKYDHTKIRVANFDIEVTSPDGFPEPSQAKHPIDAITHYDSIDDRFYVFDLLNSPYGNVEEWSIEIAAKLQEQGGDEVPSEIIDKIIYMPFDNEKELLMEYLNFWQQKTPVILTGWNVESFAIPYVYNRIKNIFGESTAKRLSPHRKTRVKVIENMYGSREIITLFGISVLDYIDLYKKFSFTNQPSYSLDYISEFELNVGKLKYDGPISKLRESNHQRYISYNIIAVYRVLQIDAKRQFINLSLDMGYYAKIQIQSVFSPIKTWDAIIFNSLKEQNKVIPQGRSHPVQPYPGAFVKEPIPNRYKYVMSFDLTSLYPSIIRQVNISPETIAGTFKVAPLHDYINAVAERPSDVYSCSPNGMMYYKDRDGVVPTEITKVFNQRKEHKGYMLAAQRNGEIIKEALHNPNLSVDEPLDVDYRFDFSDEIKEKIKKLSAKSLNEMLFRAQRTEVAGMTAQINRKALINGLAGALGNVWFRYYDLRNATAITTFGQMALQWIERKVNEYLNEVCGTEGEAFVLYGDTDSIYVSADKIIDKVGESKFRDTNHWVDFLDKFARERMEPAIDRGFREMCEYMNNKQHLMFMDREAIAGPPLGSKGIGGFWTGKKRYALNVWDMEGTRYAEPKLKIMGLETQKSSTPKAVQKALKECIRRMLQEGEESLQEYFKEFEKEFRQLNYISIASVSSANNIAKYDVGGFPGPKCPFHIRGILTYNRAIKGNIDAPQVVEGEKVYVLPLREGNPFGDKCIAWPSGTEITDLIKDDVLHWMDYTVLLEKTFIKPLEGFTSAAKLDYEKKASLFDMFDF
8VGH , Knot 116 260 0.82 40 186 251 
AGSTHHHHHHDDDDKIVGGQEAPRSKWPWQVSLRVHGPYWMHFCGGSLIHPQWVLTAAHCVGPDVKDLAALRVQLREQHLYYQDQLLPVSRIIVHPQFYTAQIGADIALLELEEPVNVSSHVHTVTLPPASETFPPGMPCWVTGWGDVDNDERLPPPFPLKQVKVPIMENHICDAKYHLGAYTGDDVRIVRDDMLCAGNTRRDSCQGDSGGPLVCKVNGTWLQAGVVSWGEGCAQPNRPGIYTRVTYYLDWIHHYVPKKP
6ZAH , Knot 149 331 0.87 40 206 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLSQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIHRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT

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 \(c\).

\(|P_{f(4DTO_1)}(2) \setminus P_{f(8VGH_1)}(2)|=167\), \(|P_{f(8VGH_1)}(2) \setminus P_{f(4DTO_1)}(2)|=34\). 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 1:100101010011001100010001000000100010111001000100010101010000111010010011001001110111100101101000000010000001011010101001011101001001101100000100010110110010101001010111010001100110011001101110000011100101100001111011010011110100010011100010010100000101100101000110111101100101000101000100010010010101101000111001000000001000111100110101000110101011001010100110110010111100100000111010001101011111001110000011010100101011001010100111010111100010111001001000010111000000111100100110000000101111000101100110010101001101000101000100010010100100111010000111101010001110111111101110000100101100110111011000100010010100101111010000101010011001100010000011011001100010111001100100010000011110001111111100111111010000110110101000101010111100000001011001100010011001000100010010001001001011010010011000111111100110101110000110101011011010010111100101110001111010010011000110110001110001101101100110100000101101101
Pair \(Z_2\) Length of longest common subsequence
4DTO_1,8VGH_1 201 4
4DTO_1,6ZAH_1 161 4
8VGH_1,6ZAH_1 198 3

Newick tree

 
[
	8VGH_1:10.39,
	[
		4DTO_1:80.5,6ZAH_1:80.5
	]:24.89
]

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 \((0.85)(0.8)(\frac{1163 }{\log_{20} 1163}-\frac{260}{\log_{20}260})=240.\)
Status Protein1 Protein2 d d1/2
Query variables 4DTO_1 8VGH_1 311 198.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]