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Parikh vectors
6BAB_1 4UTX_1 6LEY_1 Letter Amino acid
12 13 9 F Phenylalanine
14 22 12 P Proline
15 15 22 T Threonine
8 6 19 N Asparagine
8 10 7 C Cysteine
12 11 14 H Histidine
29 19 53 L Leucine
17 18 20 V Valine
17 10 21 D Aspartic acid
13 7 8 Q Glutamine
21 21 25 E Glutamic acid
18 20 38 G Glycine
11 5 15 Y Tyrosine
24 26 38 A Alanine
17 19 17 R Arginine
22 13 28 K Lycine
6 5 16 M Methionine
18 13 17 I Isoleucine
15 18 37 S Serine
5 4 3 W Tryptophan 

6BAB_1|Chains A, B, C, D|Calcium/calmodulin-dependent protein kinase type II subunit delta|Mus musculus (10090)
>4UTX_1|Chains A, B|NAD-DEPENDENT PROTEIN DEACYLASE SIRTUIN-5, MITOCHONDRIAL|DANIO RERIO (7955)
>6LEY_1|Chains A, B|Protein fem-1 homolog C,Peptide from Nucleotide exchange factor SIL1|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 \(w=f(c)\)
6BAB , Knot 138 302 0.87 40 202 293 
GHMSTTTCTRFTDEYQLFEELGKGAFSVVRRCMKIPTGQEYAAKIINTKKLSARDHQKLEREARICRLLKHPNIVRLHDSISEEGFHYLVFDLVTGGELFEDIVAREYYSEADASHCIQQILESVNHCHLNGIVHRDLKPENLLLASKSKGAAVKLADFGLAIEVQGDQQAWFGFAGTPGYLSPEVLRKDPYGKPVDMWACGVILYILLVGYPPFWDEDQHRLYQQIKAGAYDFPSPEWDTVTPEAKDLINKMLTINPAKRITASEALKHPWICQRSTVASMMHRQETVDCLKKFNARRKLK
4UTX , Knot 124 275 0.84 40 191 268 
GIDPFTTRPSSDLTAFREHFAKAKHIAIITGAGVSAESGVPTFRGPGGFWRKWQAQDLATPEAFSRDPSLVWEFYHYRREVMRSKMPNPAHLAIAECEARLGQQGRSVVIITQNIDELHHRAGSKHVYEIHGSLFKTRCMSCGEVKANHKSPICPALDGKGAPDPNTKEARIPVELLPRCERKSCNGLLRPHVVWFGETLDSDILTAVERELEKCDLCLVVGTSSIVYPAAMFAPQVASRGVPVAEFNMECTPATQRFKYHFEGPCGSTLPPALE
6LEY , Knot 168 419 0.80 40 215 380 
GHMDLKTAVFNAARDGKLRLLTKLLASKSKEEVSSLISEKTNGATPLLMAARYGHLDMVEFLLEQCSASIEVGGSVNFDGETIEGAPPLWAASAAGHLKVVQSLLNHGASVNNTTLTNSTPLRAACFDGHLEIVKYLVEHKADLEVSNRHGHTCLMISCYKGHKEIAQYLLEKGADVNRKSVKGNTALHDCAESGSLDIMKMLLMYCAKMEKDGYGMTPLLSASVTGHTNIVDFLTHHAQTSKTERINALELLGATFVDKKRDLLGALKYWKKAMNMRYSDRTNIISKPVPQTLIMAYDYAKEVNSAEELEGLIADPDEMRMQALLIRERILGPSHPDTSYYIRYRGAVYADSGNFKRCINLWKYALDMQQSNLDPLSPMTASSLLSFAELFGGGSGGGSGGGSGGGSSVNSLLKELRG

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 \(c\).

\(|P_{f(6BAB_1)}(2) \setminus P_{f(4UTX_1)}(2)|=90\), \(|P_{f(4UTX_1)}(2) \setminus P_{f(6BAB_1)}(2)|=79\). 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 1:10100000001000001100110111011000101101000110110000101000001000101001100101101000100011001110110110110011100000010100010011001000010111000101001111000011110110111110101000111111101101010110001010110111011110111110111100000010001011100110101001010100110011010110010100110011100000110110000010010010100010
Pair \(Z_2\) Length of longest common subsequence
6BAB_1,4UTX_1 169 4
6BAB_1,6LEY_1 175 7
4UTX_1,6LEY_1 178 4

Newick tree

 
[
	6LEY_1:89.46,
	[
		6BAB_1:84.5,4UTX_1:84.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{577 }{\log_{20} 577}-\frac{275}{\log_{20}275})=85.1\)
Status Protein1 Protein2 d d1/2
Query variables 6BAB_1 4UTX_1 111 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]