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Parikh vectors
1QIC_1 1YEU_1 7WEX_1 Letter Amino acid
11 7 33 P Proline
8 3 9 Y Tyrosine
13 21 50 A Alanine
6 4 8 N Asparagine
14 8 34 D Aspartic acid
8 10 18 H Histidine
8 0 10 I Isoleucine
12 7 29 G Glycine
16 18 66 L Leucine
12 9 20 T Threonine
3 1 2 W Tryptophan
8 13 26 V Valine
9 4 22 E Glutamic acid
6 11 5 K Lycine
2 2 4 M Methionine
8 7 10 F Phenylalanine
8 11 16 S Serine
6 3 41 R Arginine
0 1 2 C Cysteine
3 1 3 Q Glutamine 

1QIC_1|Chains A, B, C, D|PROTEIN (STROMELYSIN-1)|Homo sapiens (9606)
>1YEU_1|Chains A, C|Hemoglobin alpha chain|Homo sapiens (9606)
>7WEX_1|Chain A|Cytochrome P450 hydroxylase|Streptomyces avermitilis (strain ATCC 31267 / DSM 46492 / JCM 5070 / NBRC 14893 / NCIMB 12804 / NRRL 8165 / MA-4680) (227882)
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 \(w=f(c)\)
1QIC , Knot 79 161 0.83 38 123 156 
IPKWRKTHLTYRIVNYTPDLPKDAVDSAVEKALKVWEEVTPLTFSRLYEGEADIMISFAVREHGDFYPFDGPGNVLAHAYAPGPGINGDAHFDDDEQWTKDTTGTNLFLVAAHEIGHSLGLFHSANTEALMYPLYHSLTDLTRFRLSQDDINGIQSLYGPP
1YEU , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
7WEX , Knot 157 408 0.77 40 174 355 
MGSSHHHHHHSSGLVPRGSHMDPAEGLLADPYAVYDRLRDTAPVHRIAGTDGKPAWLVTRYDDVREGLANPLLSLDKKHALPGNYRGLALPPALDANLLNMDAPDHTRIRRLVGRAFTLRRVEQLREPVRETAHRLLDALGTHGSTDLIASYAAPLPITVICDLLGVPDEHRRDFRAWTDPLVTPDPARPDVARESVVSLLGFFTGLLADKRKNPADDLLSDLIAVQEEGDRLTEDELMSLAFLILFAGYENTVHLIGNAVLALLRHPEQLAALREDPARLPDAVGEFARYEGPALLAIRRFPVRDVTIGGVTVPAGETVLLSLSAANRDPSRFPDPDRLDLGRDAAGHLALGHGVHYCLGAPLARLETEVALAALLERFPDLALAETEPRRRPSLRARGLLALPVTY

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 \(c\).

\(|P_{f(1QIC_1)}(2) \setminus P_{f(1YEU_1)}(2)|=82\), \(|P_{f(1YEU_1)}(2) \setminus P_{f(1QIC_1)}(2)|=58\). 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 1:11010000100011000101100110011001101100101101001001010111011100010101101110111010111111010101000001000001001111110011001111001000111011000100100101000010110010111
Pair \(Z_2\) Length of longest common subsequence
1QIC_1,1YEU_1 140 3
1QIC_1,7WEX_1 151 3
1YEU_1,7WEX_1 155 4

Newick tree

 
[
	7WEX_1:78.55,
	[
		1QIC_1:70,1YEU_1:70
	]:8.55
]

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 \((0.85)(0.8)(\frac{302 }{\log_{20} 302}-\frac{141}{\log_{20}141})=49.6\)
Status Protein1 Protein2 d d1/2
Query variables 1QIC_1 1YEU_1 62 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]