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Parikh vectors
5VUU_1 5GML_1 7MIN_1 Letter Amino acid
25 24 33 G Glycine
36 24 105 L Leucine
13 3 12 W Tryptophan
20 11 54 A Alanine
26 10 43 D Aspartic acid
19 8 47 F Phenylalanine
26 23 51 S Serine
25 11 51 T Threonine
16 10 25 Y Tyrosine
11 3 13 C Cysteine
23 12 45 I Isoleucine
27 10 49 K Lycine
23 12 35 P Proline
22 10 41 R Arginine
18 13 39 N Asparagine
18 3 26 Q Glutamine
25 18 56 E Glutamic acid
14 12 15 H Histidine
11 3 25 M Methionine
24 15 43 V Valine 

5VUU_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>5GML_1|Chains A, B|Golgi reassembly-stacking protein 2|Mus musculus (10090)
>7MIN_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 3|Mus musculus (10090)
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 \(w=f(c)\)
5VUU , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
5GML , Knot 106 235 0.82 40 162 225 
MSYYHHHHHHLESTSLYKKAGFLVPRGSGSSQSVEIPGGGTEGYHVLRVQENSPGHRAGLEPFFDFIVSINGSRLNKDNDTLKDLLKANVEKPVKMLIYSSKTLELREASVTPSNLWGGQGLLGVSIRFCSFDGANENVWHVLEVESNSPAALAGLRPHSDYIIGADTVMNESEDLFSLIETHEAKPLKLYVYNTDTDNCREVIITPNSAWGGEGSLGCGIGYGYLHRIPTRPFE
7MIN , Knot 307 808 0.84 40 311 733 
MNAHSKEMAPLMGKRTTAPGGNPVVLTEKRPADLTPTKKSAHFFLEIEGFEPNPTVTKTSPPIFSKPMDSNIRQCLSGNCDDMDSPQSPQDDVTETPSNPNSPSANLAKEEQRQKKKRLKKRIFAAVSEGCVEELRELLQDLQDLCRRRRGLDVPDFLMHKLTASDTGKTCLMKALLNINPNTKEIVRILLAFAEENDILDRFINAEYTEEAYEGQTALNIAIERRQGDITAVLIAAGADVNAHAKGVFFNPKYQHEGFYFGETPLALAACTNQPEIVQLLMENEQTDITSQDSRGNNILHALVTVAEDFKTQNDFVKRMYDMILLRSGNWELETMRNNDGLTPLQLAAKMGKAEILKYILSREIKEKPLRSLSRKFTDWAYGPVSSSLYDLTNVDTTTDNSVLEIIVYNTNIDNRHEMLTLEPLHTLLHTKWKKFAKYMFFLSFCFYFFYNITLTLVSYYRPREDEDLPHPLALTHKMSWLQLLGRMFVLIWATCISVKEGIAIFLLRPSDLQSILSDAWFHFVFFVQAVLVILSVFLYLFAYKEYLACLVLAMALGWANMLYYTRGFQSMGMYSVMIQKVILHDVLKFLFVYILFLLGFGVALASLIEKCSKDKKDCSSYGSFSDAVLELFKLTIGLGDLNIQQNSTYPILFLFLLITYVILTFVLLLNMLIALMGETVENVSKESERIWRLQRARTILEFEKMLPEWLRSRFRMGELCKVADEDFRLCLRINEVKWTEWKTHVSFLNEDPGPIRRTADLNKIQDSSRSNSKTTLYAFDELDEFPETSVLVPRGSAAAWSHPQFEK

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 \(c\).

\(|P_{f(5VUU_1)}(2) \setminus P_{f(5GML_1)}(2)|=139\), \(|P_{f(5GML_1)}(2) \setminus P_{f(5VUU_1)}(2)|=39\). 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
5VUU_1,5GML_1 178 4
5VUU_1,7MIN_1 137 4
5GML_1,7MIN_1 187 6

Newick tree

 
[
	5GML_1:97.69,
	[
		5VUU_1:68.5,7MIN_1:68.5
	]:29.19
]

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 \((0.85)(0.8)(\frac{657 }{\log_{20} 657}-\frac{235}{\log_{20}235})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 5VUU_1 5GML_1 153 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]