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Parikh vectors
5KFQ_1 3QAG_1 5NDD_1 Letter Amino acid
19 11 29 D Aspartic acid
11 4 8 C Cysteine
44 29 74 L Leucine
8 13 25 Y Tyrosine
28 15 25 R Arginine
13 12 30 F Phenylalanine
25 12 33 T Threonine
21 13 42 I Isoleucine
13 10 30 N Asparagine
31 8 19 Q Glutamine
28 15 23 E Glutamic acid
29 12 24 G Glycine
25 12 41 K Lycine
12 6 15 M Methionine
31 17 32 S Serine
34 12 74 A Alanine
7 5 9 W Tryptophan
18 16 18 P Proline
30 11 47 V Valine
8 6 21 H Histidine 

5KFQ_1|Chain A|DNA polymerase eta|Homo sapiens (9606)
>3QAG_1|Chain A|Glutathione S-transferase omega-2|Homo sapiens (9606)
>5NDD_1|Chain A|Lysozyme,Proteinase-activated receptor 2,Soluble cytochrome b562,Proteinase-activated receptor 2|Enterobacteria phage T4 (10665)
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 \(w=f(c)\)
5KFQ , Knot 183 435 0.85 40 244 418 
GPHMATGQDRVVALVDMDCFFVQVEQRQNPHLRNKPCAVVQYKSWKGGGIIAVSYEARAFGVTRSMWADDAKKLCPDLLLAQVRESRGKANLTKYREASVEVMEIMSRFAVIERASIDEAYVDLTSAVQERLQKLQGQPISADLLPSTYIEGLPQGPTTAEETVQKEGMRKQGLFQWLDSLQIDNLTSPDLQLTVGAVIVEEMRAAIERETGFQCSAGISHNKVLAKLACGLNKPNRQTLVSHGSVPQLFSQMPIRKIRSLGGKLGASVIEILGIEYMGELTQFTESQLQSHFGEKNGSWLYAMCRGIEHDPVKPRQLPKTIGCSKNFPGKTALATREQVQWWLLQLAQELEERLTKDRNDNDRVATQLVVSIRVQGDKRLSSLRRCCALTRYDAHKMSHDAFTVIKNCNTSGIQTEWSPPLTMLFLCATKFSAS
3QAG , Knot 113 239 0.86 40 177 234 
MSGDATRTLGKGSQPPGPVPEGLIRIYSMRFCPYSHRTRLVLKAKDIRHEVVNINLRNKPEWYYTKHPFGHIPVLETSQSQLIYESVIACEYLDDAYPGRKLFPYDPYERARQKMLLELFSKVPHLTKECLVALRSGRESTNLKAALRQEFSNLEEILEYQNTTFFGGTSISMIDYLLWPWFERLDVYGILDCVSHTPALRLWISAMKWDPTVSALLMDKSIFQGFLNLYFQNNPNAFD
5NDD , Knot 239 619 0.82 40 267 555 
MVSAIVLYVLLAAAAHSAFAAASNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYIYEFFSVDEFSASVLTGKLTTVFLPIVYTIVFVVALPSNGMALWVFLFRTKKKAPAVIYMANLALADLLSVIWFPLKIAYHIHGNNWIYGEALCNVLIGFFYANMYCSILFLTCLSVQRAWEIVNPMGHSRKKANIAIGISLAIWLLILLVTIPLYVVKQTIFIPALQITTCHDVLPEQLLVGDMFNYFLSLAIGVFLFPAFLTASAYVLMIRALADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLENSEKKRKRAIKLAVTVAAMYLICFTPSNLLLVVHYFLIKSQGQSHVYALYIVALCLSTLNSCIDPFVYYFVSHDFRDHAKNALLCRSVRTVKQMQVSLTSKAAAHHHHHHHHHH

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 \(c\).

\(|P_{f(5KFQ_1)}(2) \setminus P_{f(3QAG_1)}(2)|=128\), \(|P_{f(3QAG_1)}(2) \setminus P_{f(5KFQ_1)}(2)|=61\). 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 1:110110100011111010011101000001010001011100001011111110001011110001110010010101111010000101010000010101101100111100101001010100110001001010110101110001011101100100010001100011101100101001001010101111110010111000011000111000011101101100100001100101101100111001001110111011011110011010010000100011000101101100110001101001100110000111001110000101111011001000100000000011001110101010001001000011000010010001101100000011000101110111101001010
Pair \(Z_2\) Length of longest common subsequence
5KFQ_1,3QAG_1 189 5
5KFQ_1,5NDD_1 169 4
3QAG_1,5NDD_1 186 3

Newick tree

 
[
	3QAG_1:96.64,
	[
		5KFQ_1:84.5,5NDD_1:84.5
	]:12.14
]

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 \((0.85)(0.8)(\frac{674 }{\log_{20} 674}-\frac{239}{\log_{20}239})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 5KFQ_1 3QAG_1 154 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]