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Parikh vectors
8TDT_1 4JJP_1 3QET_1 Letter Amino acid
0 27 70 K Lycine
0 8 48 F Phenylalanine
8 18 54 A Alanine
0 11 57 D Aspartic acid
0 19 70 E Glutamic acid
0 13 52 Y Tyrosine
4 1 8 C Cysteine
0 33 74 I Isoleucine
0 6 25 M Methionine
0 0 12 W Tryptophan
0 19 48 V Valine
0 12 44 N Asparagine
6 22 52 G Glycine
0 12 43 P Proline
0 20 63 L Leucine
0 17 54 S Serine
3 15 38 T Threonine
0 4 46 R Arginine
0 4 31 Q Glutamine
0 6 14 H Histidine 

8TDT_1|Chain A|DNA (5'-D(*GP*AP*GP*CP*AP*GP*AP*CP*CP*TP*GP*AP*CP*GP*GP*AP*AP*AP*TP*TP*A)-3')|synthetic construct (32630)
>4JJP_1|Chains A, B|Phosphomethylpyrimidine kinase|Clostridium difficile (272563)
>3QET_1|Chain A|DNA polymerase|Enterobacteria phage RB69 (12353)
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 \(w=f(c)\)
8TDT , Knot 10 21 0.48 8 14 18 
GAGCAGACCTGACGGAAATTA
4JJP , Knot 118 267 0.82 38 161 253 
SNAMSNYKIPTLTIAGSDSSGGAGIQADLKTFSAIGTYGMSVITAITAQNTKGVFAVEDLNKKIIKKQIEAVFEDIPPRAVKIGMVSSPEIILEIVENLKKYNPKYLVVDPVMISKSGYYLLKPEAKENLIKYLIPLAYIITPNIPEAEEITGIKIHNVDDMKRVGEEILQLGPKFVLMKGGHLDGEAVDILVGKNIFKVYKSERIDKKNTHGTGCTLSSAITSYLALGYEITEAVNLSKIYITEAIKRSFDIGHGVGPVHHFYKFE
3QET , Knot 338 903 0.85 40 319 827 
MKEFYLTVEQIGDSIFERYIDSNGRERTREVEYKPSLFAHCPESQATKYFDIYGKPCTRKLFANMRDASQWIKRMEDIGLEALGMDDFKLAYLSDTYNYEIKYDHTKIRVANFDIEVTSPDGFPEPSQAKHPIDAITHYDSIDDRFYVFDLLNSPYGNVEEWSIEIAAKLQEQGGDEVPSEIIDKIIYMPFDNEKELLMEYLNFWQQKTPVILTGWNVESFAIPYVYNRIKNIFGESTAKRLSPHRKTRVKVIENMYGSREIITLFGISVLDYIDLYKKFSFTNQPSYSLDYISEFELNVGKLKYDGPISKLRESNHQRYISYNIIAVYRVLQIDAKRQFINLSLDMGYYAKIQIQSVFSPIKTWDAIIFNSLKEQNKVIPQGRSHPVQPYPGAFVKEPIPNRYKYVMSFDLTSLYPSIIRQVNISPETIAGTFKVAPLHDYINAVAERPSDVYSCSPNGMMYYKDRDGVVPTEITKVFNQRKEHKGYMLAAQRNGEIIKEALHNPNLSVDEPLDVDYRFDFSDEIKEKIKKLSAKSLNEMLFRAQRTEVAGMTAQINRKALINGLAGALGNVWFRYYDLRNATAITTFGQMALQWIERKVNEYLNEVCGTEGEAFVLYGDTDSIYVSADKIIDKVGESKFRDTNHWVDFLDKFARERMEPAIDRGFREMCEYMNNKQHLMFMDREAIAGPPLGSKGIGGFWTGKKRYALNVWDMEGTRYAEPKLKIMGLETQKSSTPKAVQKALKECIRRMLQEGEESLQEYFKEFEKEFRQLNYISIASVSSANNIAKYDVGGFPGPKCPFHIRGILTYNRAIKGNIDAPQVVEGEKVYVLPLREGNPFGDKCIAWPSGTEITDLIKDDVLHWMDYTVLLEKTFIKPLEGFTSAAKLDYEKKASLFDMFDF

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 \(c\).

\(|P_{f(8TDT_1)}(2) \setminus P_{f(4JJP_1)}(2)|=7\), \(|P_{f(4JJP_1)}(2) \setminus P_{f(8TDT_1)}(2)|=154\). 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 1:111011100011011111001
Pair \(Z_2\) Length of longest common subsequence
8TDT_1,4JJP_1 161 3
8TDT_1,3QET_1 313 3
4JJP_1,3QET_1 188 4

Newick tree

 
[
	3QET_1:14.62,
	[
		8TDT_1:80.5,4JJP_1:80.5
	]:61.12
]

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 \((0.85)(0.8)(\frac{288 }{\log_{20} 288}-\frac{21}{\log_{20}21})=89.5\)
Status Protein1 Protein2 d d1/2
Query variables 8TDT_1 4JJP_1 114 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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