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Parikh vectors
8FSF_1 7UIY_1 1BKU_1 Letter Amino acid
11 50 1 R Arginine
14 32 1 N Asparagine
3 36 2 Q Glutamine
8 78 5 L Leucine
2 14 0 M Methionine
12 50 3 G Glycine
3 27 0 F Phenylalanine
2 25 2 P Proline
3 11 1 Y Tyrosine
7 44 1 D Aspartic acid
1 21 1 H Histidine
6 46 0 I Isoleucine
6 44 2 K Lycine
6 3 0 W Tryptophan
12 69 1 A Alanine
8 3 2 C Cysteine
2 64 1 E Glutamic acid
10 42 3 S Serine
7 39 4 T Threonine
6 60 2 V Valine 

8FSF_1|Chain A|Lysozyme C|Gallus gallus (9031)
>7UIY_1|Chains A, B, C, D, E, F|ATP-dependent Clp protease ATP-binding subunit ClpA|Escherichia coli (562)
>1BKU_1|Chain A|CALCITONIN|Anguilla japonica (7937)
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 \(w=f(c)\)
8FSF , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
7UIY , Knot 286 758 0.83 40 270 687 
MLNQELELSLNMAFARAREHRHEFMTVEHLLLALLSNPSAREALEACSVDLVALRQELEAFIEQTTPVLPASEEERDTQPTLSFQRVLQRAVFHVQSSGRNEVTGANVLVAIFSEQESQAAYLLRKHEVSRLDVVNFISHGTRKDEPTQSSDPGSQPNSEEQAGGEERTENFTTNLNQLARVGGIDPLIGREKELERAIQVLCRRRKNNPLLVGESGVGKTAIAEGLAWRIVQGDVPEVMADCTIYSLDIGSLLAGTKYRGDFEKRFKALLKQLEQDTNSILFIDEIHTIIGAGAASGGQVDAANLIKPLLSSGKIRVIGSTTYQEFSNIFEKDRALARRFQKIDITEPSIEETVQIINGLKPKYEAHHDVRYTAKAVRAAVELAVKYINDRHLPDKAIDVIDEAGARARLMPVSKRKKTVNVADIESVVARIARIPEKSVSQSDRDTLKNLGDRLKMLVFGQDKAIEALTEAIKMARAGLGHEHKPVGSFLFAGPTGVGKTEVTVQLSKALGIELLRFDMSEYMERHTVSRLIGAPPGYVGFDQGGLLTDAVIKHPHAVLLLDEIEKAHPDVFNILLQVMDNGTLTDNNGRKADFRNVVLVMTTNAGVRETERKSIGLIHQDNSTDAMEEIKKIFTPEFRNRLDNIIWFDHLSTDVIHQVVDKFIVELQVQLDQKGVSLEVSQEARNWLAEKGYDRAMGARPMARVIQDNLKKPLANELLFGSLVDGGQVTVALDKEKNELTYGFQSAQKHKAEAAH
1BKU , Knot 20 33 0.70 34 30 31 
CSNLSTCVLGKLSQELHKLQTYPRTDVGAGTPX

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 \(c\).

\(|P_{f(8FSF_1)}(2) \setminus P_{f(7UIY_1)}(2)|=23\), \(|P_{f(7UIY_1)}(2) \setminus P_{f(8FSF_1)}(2)|=189\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
8FSF_1,7UIY_1 212 4
8FSF_1,1BKU_1 108 3
7UIY_1,1BKU_1 254 3

Newick tree

 
[
	7UIY_1:13.42,
	[
		8FSF_1:54,1BKU_1:54
	]:77.42
]

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 \((0.85)(0.8)(\frac{887 }{\log_{20} 887}-\frac{129}{\log_{20}129})=212.\)
Status Protein1 Protein2 d d1/2
Query variables 8FSF_1 7UIY_1 264 153
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]