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Parikh vectors
7KNT_1 9BHV_1 2KTE_1 Letter Amino acid
10 28 5 D Aspartic acid
7 3 1 C Cysteine
9 21 7 I Isoleucine
5 7 11 K Lycine
5 5 6 W Tryptophan
10 43 3 R Arginine
4 21 7 Q Glutamine
8 39 18 E Glutamic acid
9 36 12 G Glycine
14 50 11 L Leucine
3 7 3 M Methionine
13 29 11 V Valine
2 13 7 N Asparagine
8 14 7 F Phenylalanine
6 22 8 P Proline
7 26 9 S Serine
6 10 0 Y Tyrosine
11 50 8 A Alanine
3 13 8 H Histidine
9 33 10 T Threonine 

7KNT_1|Chain A[auth E]|Receptor activity-modifying protein 1|Homo sapiens (9606)
>9BHV_1|Chain A|Nucleotide-binding protein|Streptomyces griseus subsp. griseus (67263)
>2KTE_1|Chain A|Uncharacterized protein yndB|Bacillus subtilis (1423)
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 \(w=f(c)\)
7KNT , Knot 77 149 0.86 40 117 145 
MKTIIALSYIFCLVFADYKDDDDKHGSCQEANYGALLRELCLTQFQVDMEAVGETLWCDWGRTIRSYRELADCTWHMAEKLGCFWPNAEVDRFFLAVHGRYFRSCPISGRAVRDPPGSILYPFIVVPITVTLLVTALVVWQSKRTEGIV
9BHV , Knot 190 470 0.83 40 228 447 
MTVDSTSEARLEVPRQSSLGTAAARNLASTTKSAPQMQEITSRWLLRMLPWVETKGGAYRVNRRLTFTVGDGRVEFVQDGSTVRVIPQELGELALLRDFDDAEVLSAIADRCVQRDFRAGETLVERGTAADELHLIAHGRIGQASAGSYGDEVTLDVLADGDRFGEHALLDEDARWSQTATAETSGTLLTLSRADFAAVVANSPALRSHLAAFTARSEQRQNHRGEAEIAMSAGHVGEHELPGAFADYELKPREYELSVAQTILRVHTRVADLYNGPMNQTEEQLRLTIEALRERQEHELINNREFGLLHNADFKQRIQTHSGPPTPDDLDELLCRRRGTKFFLAHPRTIAAMGREFNARGLYPDHTDLGGQQVPAWRGVPILPCGKIPITPERTSSILALRTGEEDQGVIGLRQTGLPDEYEPGLSVRFMNIDEKAIISYLVSTYYSAAILVPDAVGVLENVQIANWPR
2KTE , Knot 74 152 0.81 38 119 146 
MAQNNENALPDITKSITLEAPIQKVWETVSTSEGIAKWFMPNDFQLKEGQEFHLQSPFGPSPCKVLAVQAPTELSFEWDTEGWVVTFQLEDLGEKTGFTLIHSGWKEPNEVIGKANEKSSVVRGKMDGGWTGIVNERLRKAVEELEHHHHHH

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 \(c\).

\(|P_{f(7KNT_1)}(2) \setminus P_{f(9BHV_1)}(2)|=49\), \(|P_{f(9BHV_1)}(2) \setminus P_{f(7KNT_1)}(2)|=160\). 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 1:10011110011011110000000001000010011110010100101010111001100110010000011000101100110111010100111110100100011010110011101101111111010111011111000000111
Pair \(Z_2\) Length of longest common subsequence
7KNT_1,9BHV_1 209 4
7KNT_1,2KTE_1 164 3
9BHV_1,2KTE_1 185 4

Newick tree

 
[
	9BHV_1:10.64,
	[
		7KNT_1:82,2KTE_1:82
	]:21.64
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{149}{\log_{20}149})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 7KNT_1 9BHV_1 171 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]