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Parikh vectors
6KGV_1 7VYZ_1 8BAW_1 Letter Amino acid
38 24 18 D Aspartic acid
38 10 12 Q Glutamine
18 13 26 E Glutamic acid
13 4 6 M Methionine
40 14 12 P Proline
5 0 2 W Tryptophan
51 43 25 V Valine
57 32 18 A Alanine
22 17 17 N Asparagine
61 31 25 G Glycine
9 11 5 H Histidine
24 20 18 I Isoleucine
12 5 9 F Phenylalanine
3 0 1 C Cysteine
42 20 23 L Leucine
18 14 37 K Lycine
32 12 3 R Arginine
33 25 15 S Serine
31 18 15 T Threonine
15 1 13 Y Tyrosine 

6KGV_1|Chain A|Penicillin-binding protein PbpB|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>7VYZ_1|Chain A|Probable serine protease HtrA1|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>8BAW_1|Chains A, B|Siderophore ABC transporter substrate-binding protein|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
6KGV , Knot 218 562 0.81 40 240 513 
GPLGSGQLKVTDVQPAARGSIVDRNNDRLAFTIEARALTFQPKRIRRQLEEARKKTSAAPDPQQRLRDIAQEVAGKLNNKPDAAAVLKKLQSDETFVYLARAVDPAVASAICAKYPEVGAERQDLRQYPGGSLAANVVGGIDWDGHGLLGLEDSLDAVLAGTDGSVTYDRGSDGVVIPGSYRNRHKAVHGSTVVLTLDNDIQFYVQQQVQQAKNLSGAHNVSAVVLDAKTGEVLAMANDNTFDPSQDIGRQGDKQLGNPAVSSPFEPGSVNKIVAASAVIEHGLSSPDEVLQVPGSIQMGGVTVHDAWEHGVMPYTTTGVFGKSSNVGTLMLSQRVGPERYYDMLRKFGLGQRTGVGLPGESAGLVPPIDQWSGSTFANLPIGQGLSMTLLQMTGMYQAIANDGVRVPPRIIKATVAPDGSRTEEPRPDDIRVVSAQTAQTVRQMLRAVVQRDPMGYQQGTGPTAGVPGYQMAGKTGTAQQINPGCGCYFDDVYWITFAGIATADNPRYVIGIMLDNPARNSDGAPGHSAAPLFHNIAGWLMQRENVPLSPDPGPPLVLQAT
7VYZ , Knot 133 314 0.81 36 168 297 
MEPAGRFTKVAAAVADSVVTIESVSDQEGMQGSGVIVDGRGYIVTNNHVISEAANNPSQFKTTVVFNDGKEVPANLVGRDPKTDLAVLKVDNVDNLTVARLGDSSKVRVGDEVLAVGAPLGLRSTVTQGIVSALHRPVPLSGEGADTDTVIDAIQTDASINHGNSGGPLIDMDAQVIGINTAGKSLSDSASGLGFAIPVNEMKLVANSLIKDGKIVHPTLGISTRSVSNAIASGAQVANVKAGSPAQKGGILENDVIVKVGNRAVADSDEFVVAVRQLAIGQDAPIEVVREGRHVTLTVKPDPDSTLEHHHHHH
8BAW , Knot 130 300 0.82 40 172 285 
CGNKENASNGASGKNDEKKTEQAEEMTIKHQLGEAKVKKNPEKVVVFDFGVLDTLDKLGVKVTALPQMNVPKYLEKYKSSDYQNVGSLMEPDFEKLSEIKPDVIFISGRQANLYDKLKEIGPTVYIGIDTQHYWDSFTNNMKLIGQMFGKEKEVDEELANIEKQIEEVKTKAADKKALIILTTGGKVSAYGKGSRFGLIHDVLGVPAADPNLKVTNPHGQSVSFEYIAEKNPDYLFVIDRDAVVEGKPTAKQTIENALVKKTKAYQNGHIVYLDPNYWYLSGGGLTSVSEMIKQVEEGLK

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 \(c\).

\(|P_{f(6KGV_1)}(2) \setminus P_{f(7VYZ_1)}(2)|=108\), \(|P_{f(7VYZ_1)}(2) \setminus P_{f(6KGV_1)}(2)|=36\). 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 1:1111010101001011101011000000111010101101010010001001000001110100010011001110100010111110010000011011011011110110100101110000100011101110111110101011111000101111100101000010011111100000001101001110100010101000100100101100101111010010111110000101000110010001101110011011010011110111001100100110111010111101001100111100001111000011011100011100000110011110001111110011111110010100110111101101011010110011100110111011010111010000010100101101001001001101110001110001011011111001110010100101101001001011011111010010011111100110000111100111110011111100001110101111111010
Pair \(Z_2\) Length of longest common subsequence
6KGV_1,7VYZ_1 144 4
6KGV_1,8BAW_1 166 4
7VYZ_1,8BAW_1 136 3

Newick tree

 
[
	6KGV_1:80.66,
	[
		7VYZ_1:68,8BAW_1:68
	]:12.66
]

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 \((0.85)(0.8)(\frac{876 }{\log_{20} 876}-\frac{314}{\log_{20}314})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 6KGV_1 7VYZ_1 187 143
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]