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Parikh vectors
3QEP_1 8XSM_1 6FEK_1 Letter Amino acid
70 5 24 K Lycine
25 6 11 M Methionine
48 19 13 F Phenylalanine
43 42 16 P Proline
54 58 18 A Alanine
57 6 15 D Aspartic acid
52 44 23 G Glycine
74 5 13 I Isoleucine
46 13 20 R Arginine
31 12 9 Q Glutamine
48 49 21 V Valine
63 81 35 L Leucine
54 34 18 S Serine
38 15 9 T Threonine
12 7 6 W Tryptophan
44 7 8 N Asparagine
8 13 3 C Cysteine
70 13 21 E Glutamic acid
14 9 6 H Histidine
52 7 10 Y Tyrosine 

3QEP_1|Chain A|DNA polymerase|Enterobacteria phage RB69 (12353)
>8XSM_1|Chain A|Solute carrier family 52, riboflavin transporter, member 2|Homo sapiens (9606)
>6FEK_1|Chain A|Proto-oncogene tyrosine-protein kinase receptor Ret|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 \(w=f(c)\)
3QEP , Knot 338 903 0.85 40 319 827 
MKEFYLTVEQIGDSIFERYIDSNGRERTREVEYKPSLFAHCPESQATKYFDIYGKPCTRKLFANMRDASQWIKRMEDIGLEALGMDDFKLAYLSDTYNYEIKYDHTKIRVANFDIEVTSPDGFPEPSQAKHPIDAITHYDSIDDRFYVFDLLNSPYGNVEEWSIEIAAKLQEQGGDEVPSEIIDKIIYMPFDNEKELLMEYLNFWQQKTPVILTGWNVESFAIPYVYNRIKNIFGESTAKRLSPHRKTRVKVIENMYGSREIITLFGISVLDYIDLYKKFSFTNQPSYSLDYISEFELNVGKLKYDGPISKLRESNHQRYISYNIIAVYRVLQIDAKRQFINLSLDMGYYAKIQIQSVFSPIKTWDAIIFNSLKEQNKVIPQGRSHPVQPYPGAFVKEPIPNRYKYVMSFDLTSLYPSIIRQVNISPETIAGTFKVAPLHDYINAVAERPSDVYSCSPNGMMYYKDRDGVVPTEITKVFNQRKEHKGYMLAAQRNGEIIKEALHNPNLSVDEPLDVDYRFDFSDEIKEKIKKLSAKSLNEMLFRAQRTEVAGMTAQINRKALINGLAGALGNVWFRYYDLRNATAITTFGQMALQWIERKVNEYLNEVCGTEGEAFVLYGDTDSIYVSADKIIDKVGESKFRDTNHWVDFLDKFARERMEPAIDRGFREMCEYMNNKQHLMFMDREAIAGPPLGSKGIGGFWTGKKRYALNVWDMEGTRYAEPKLKIMGLETQKSSTPKAVQKALKECIRRMLQEGEESLQEYFKEFEKEFRQLNYISIASVSSANNIAKYDVGGFPGPKCPFHIRGILTYNRAIKGNIDAPQVVEGEKVYVLPLREGNPFGDKCIAWPSGTEITDLIKDDVLHWMDYTVLLEKTFIKPLEGFTSAAKLDYEKKASLFDMFDF
8XSM , Knot 171 445 0.78 40 179 381 
MAAPTPARPVLTHLLVALFGMGSWAAVNGIWVELPVVVKELPEGWSLPSYVSVLVALGNLGLLVVTLWRRLAPGKDEQVPIRVVQVLGMVGTALLASLWHHVAPVAGQLHSVAFLALAFVLALACCASNVTFLPFLSHLPPRFLRSFFLGQGLSALLPCVLALVQGVGRLECPPAPINGTPGPPLDFLERFPASTFFWALTALLVASAAAFQGLLLLLPPPPSVPTGELGSGLQVGAPGAEEEVEESSPLQEPPSQAAGTTPGPDPKAYQLLSARSACLLGLLAATNALTNGVLPAVQSFSCLPYGRLAYHLAVVLGSAANPLACFLAMGVLCRSLAGLGGLSLLGVFCGGYLMALAVLSPCPPLVGTSAGVVLVVLSWVLCLGVFSYVKVAASSLLHGGGRPALLAAGVAIQVGSLLGAVAMFPPTSIYHVFHSRKDCADPCDS
6FEK , Knot 131 299 0.83 40 196 290 
GPLSLSVDAFKILAAPKWEFPRKNLVLGKTLGEGEFGKVVKATAFHLKGRAGYTTVAVKMLKENASPSELRDLLSEFNVLKQVNHPHVIKLYGACSQDGPLLLIVEYAKYGSLRGFLRESRKVGPGPDERALTMGDLISFAWQISQGMQYLAEMKLVHRDLAARNILVAEGRKMKISDFGLSRDVYEEDFYVKRSQGRIPVKWMAIESLFDHIYTTQSDVWSFGVLLWEIVTLGGNPYPGIPPERLFNLLKTGHRMERPDNCSEEMYRLMLQCWKQEPDKRPVFADISKDLEKMMVKRR

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 \(c\).

\(|P_{f(3QEP_1)}(2) \setminus P_{f(8XSM_1)}(2)|=170\), \(|P_{f(8XSM_1)}(2) \setminus P_{f(3QEP_1)}(2)|=30\). 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
3QEP_1,8XSM_1 200 4
3QEP_1,6FEK_1 163 4
8XSM_1,6FEK_1 163 4

Newick tree

 
[
	8XSM_1:94.23,
	[
		3QEP_1:81.5,6FEK_1:81.5
	]:12.73
]

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 \((0.85)(0.8)(\frac{1348 }{\log_{20} 1348}-\frac{445}{\log_{20}445})=232.\)
Status Protein1 Protein2 d d1/2
Query variables 3QEP_1 8XSM_1 303 220.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]