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Parikh vectors
7EQB_1 6DDK_1 8UWR_1 Letter Amino acid
2 8 7 C Cysteine
1 24 14 H Histidine
4 61 31 L Leucine
0 34 20 T Threonine
11 36 17 K Lycine
6 16 10 M Methionine
0 25 11 P Proline
7 31 20 R Arginine
4 18 14 N Asparagine
5 17 13 Q Glutamine
10 38 16 E Glutamic acid
7 34 19 G Glycine
1 31 11 Y Tyrosine
2 30 25 V Valine
4 24 18 I Isoleucine
1 34 10 F Phenylalanine
7 46 24 S Serine
0 5 8 W Tryptophan
4 25 15 A Alanine
4 42 20 D Aspartic acid 

7EQB_1|Chains A, B|Kinesin-like protein|Caenorhabditis elegans (6239)
>6DDK_1|Chains A, B|Cytosolic purine 5'-nucleotidase|Homo sapiens (9606)
>8UWR_1|Chain A|Activin receptor type-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 \(w=f(c)\)
7EQB , Knot 42 80 0.76 34 67 77 
GHMGSSGGKDSLNDKDEEIKKLRGFCSRYKRENASMKERIASCEQGEQENALVMEKLMEQKMEDRKIIQSQKKAMRNVRG
6DDK , Knot 230 579 0.84 40 281 534 
GSSHHHHHHSSGLVPRGSMSTSWSDRLQNAADMPANMDKHALKKYRREAYHRVFVNRSLAMEKIKCFGFNMDYTLAVYKSPEYESLGFELTVERLVSIGYPQELLSFAYDSTFPTRGLVFDTLYGNLLKVDAYGNLLVCAHGFNFIRGPETREQYPNKFIQRDDTERFYILNTLFNLPETYLLACLVDFFTNCPRYTSCETGFKDGDLFMSYRSMFQDVRDAVDWVHYKGSLKEKTVENLEKYVVKDGKLPLLLSRMKEVGKVFLATNSDYKYTDKIMTYLFDFPHGPKPGSSHRPWQSYFDLILVDARKPLFFGEGTVLRQVDTKTGKLKIGTYTGPLQHGIVYSGGSSDTICDLLGAKGKDILYIGDHIFGDILKSKKRQGWQTFLVIPELAQELHVWTDKSSLFEELQSLDIFLAELYKHLDSSSNERPDISSIQRRIKKVTHDMDMCYGMMGSLFRSGSRQTLFASQVMRYADLYAASFINLLYYPFSYLFRAAHVLMPHESTVEHTHVDINEMESPLATRNRTSVDFKDTDYKRHQLTRSISEIKPPNLFPLAPQEITHCHDEDDDEEEEEEEE
8UWR , Knot 142 323 0.84 40 205 310 
MGHHHHHHSSGVDLGTENLYFQSMQRTVARDITLLECVGKGRYGEVWRGSWQGENVAVKIFSSRDEKSWFRETELYNTVMLRHENILGFIASDMTSRHSSTQLWLITHYHEMGSLYDYLQLTTLDTVSCLRIVLSIASGLAHLHIEIFGTQGKPAIAHRDLKSKNILVKKNGQCCIADLGLAVMHSQSTNQLDVGNNPRVGTKRYMAPEVLDETIQVDCFDSYKRVDIWAFGLVLWEVARRMVSNGIVEDYKPPFYDVVPNDPSFEDMRKVVCVDQQRPNIPNRWFSDPTLTSLAKLMKECWYQNPSARLTALRIKKTLTKID

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 \(c\).

\(|P_{f(7EQB_1)}(2) \setminus P_{f(6DDK_1)}(2)|=11\), \(|P_{f(6DDK_1)}(2) \setminus P_{f(7EQB_1)}(2)|=225\). 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 1:10110011000100000010010110000000010100011000010000111100110001000011000001100101
Pair \(Z_2\) Length of longest common subsequence
7EQB_1,6DDK_1 236 3
7EQB_1,8UWR_1 182 3
6DDK_1,8UWR_1 176 9

Newick tree

 
[
	7EQB_1:11.55,
	[
		8UWR_1:88,6DDK_1:88
	]:22.55
]

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 \((0.85)(0.8)(\frac{659 }{\log_{20} 659}-\frac{80}{\log_{20}80})=169.\)
Status Protein1 Protein2 d d1/2
Query variables 7EQB_1 6DDK_1 214 120
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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