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Parikh vectors
3NLQ_1 2YUG_1 1QJH_1 Letter Amino acid
14 2 0 H Histidine
19 5 2 F Phenylalanine
18 3 6 Q Glutamine
25 14 9 E Glutamic acid
27 8 4 K Lycine
10 3 4 M Methionine
25 7 11 V Valine
19 6 7 N Asparagine
23 13 6 I Isoleucine
25 19 3 G Glycine
23 6 5 P Proline
25 6 0 T Threonine
13 3 1 W Tryptophan
16 3 5 Y Tyrosine
22 7 10 R Arginine
11 3 0 C Cysteine
36 10 12 L Leucine
26 15 2 S Serine
20 13 9 A Alanine
25 9 5 D Aspartic acid 

3NLQ_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>2YUG_1|Chain A|Protein FRG1|Mus musculus (10090)
>1QJH_1|Chain A|30S ribosomal protein S6|Thermus thermophilus (274)
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 \(w=f(c)\)
3NLQ , Knot 185 422 0.88 40 260 410 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIVLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRNYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
2YUG , Knot 73 155 0.79 40 116 151 
GSSGSSGLDIVGIWWTVSNFGEISGTIAIEMDKGAYIHALDNGLFTLGAPHREVDEGPSPPEQFTAVKLSDSRIALKSGYGKYLGINSDGLVVGRSDAIGPREQWEPVFQDGKMALLASNSCFIRCNEAGDIEAKNKTAGEEEMIKIRSCAERET
1QJH , Knot 53 101 0.80 34 84 98 
MRRYEVNIVLNPNLDQSQLALEKEIIQRALENYGARVEKVAILGLMVLAYPIAKDPQGYFLWYQVEMPEDRVNDLARELRIRDNVRRVMVVKSQEPFLANA

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 \(c\).

\(|P_{f(3NLQ_1)}(2) \setminus P_{f(2YUG_1)}(2)|=177\), \(|P_{f(2YUG_1)}(2) \setminus P_{f(3NLQ_1)}(2)|=33\). 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 1:01011010010001110001010001001000010110111100000010010000011111001100000010011001010010010001000000010000110110011001000110101001011010000010111001000100100010100110111000010001011000110011000101001101101010010100110110101011111101010010110111011101110010101100111010111110011101111010101101101100111000000000011001100101010000011000111010111100100001011000010001100100000001101101111111101010111000110001010100010110001101
Pair \(Z_2\) Length of longest common subsequence
3NLQ_1,2YUG_1 210 3
3NLQ_1,1QJH_1 208 3
2YUG_1,1QJH_1 150 2

Newick tree

 
[
	3NLQ_1:11.63,
	[
		1QJH_1:75,2YUG_1:75
	]:37.63
]

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 \((0.85)(0.8)(\frac{577 }{\log_{20} 577}-\frac{155}{\log_{20}155})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 3NLQ_1 2YUG_1 165 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]