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Parikh vectors
1VTC_1 4KOT_1 6AUS_1 Letter Amino acid
0 12 22 R Arginine
0 16 36 L Leucine
0 6 11 M Methionine
0 23 20 A Alanine
0 6 23 I Isoleucine
0 4 27 K Lycine
0 4 25 T Threonine
0 7 16 Y Tyrosine
4 3 11 C Cysteine
0 8 25 E Glutamic acid
0 5 14 H Histidine
0 7 19 F Phenylalanine
0 9 23 P Proline
0 9 24 V Valine
0 5 18 N Asparagine
0 8 26 D Aspartic acid
0 10 18 Q Glutamine
4 14 25 G Glycine
0 4 26 S Serine
0 2 13 W Tryptophan 

1VTC_1|Chain A|DNA (5'-D(*GP*GP*CP*CP*GP*GP*CP*C)-3')|
>4KOT_1|Chain A|Uncharacterized protein|Pseudomonas aeruginosa (208964)
>6AUS_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
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 \(w=f(c)\)
1VTC , Knot 4 8 0.34 4 4 4 
GGCCGGCC
4KOT , Knot 79 162 0.82 40 117 156 
GHMQLSHRPAETGDLETVAGFPQDRDELFYCYPKAIWPFSVAQLAAAIAERRGSTVAVHDGQVLGFANFYQWQHGDFCALGNMMVAPAARGLGVARYLIGVMENLAREQYKARLMKISCFNANAAGLLLYTQLGYQPRAIAERHDPDGRRVALIQMDKPLEP
6AUS , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG

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 \(c\).

\(|P_{f(1VTC_1)}(2) \setminus P_{f(4KOT_1)}(2)|=4\), \(|P_{f(4KOT_1)}(2) \setminus P_{f(1VTC_1)}(2)|=117\). 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 1:11001100
Pair \(Z_2\) Length of longest common subsequence
1VTC_1,4KOT_1 121 1
1VTC_1,6AUS_1 262 3
4KOT_1,6AUS_1 235 3

Newick tree

 
[
	6AUS_1:13.37,
	[
		1VTC_1:60.5,4KOT_1:60.5
	]:78.87
]

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 \((0.85)(0.8)(\frac{170 }{\log_{20} 170}-\frac{8}{\log_{20}8})=59.5\)
Status Protein1 Protein2 d d1/2
Query variables 1VTC_1 4KOT_1 76 39.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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