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Parikh vectors
5VXL_1 5FMZ_1 4JGS_1 Letter Amino acid
18 68 9 E Glutamic acid
6 16 2 H Histidine
3 31 4 F Phenylalanine
34 56 11 S Serine
16 45 7 V Valine
4 32 7 R Arginine
15 36 4 D Aspartic acid
15 41 1 I Isoleucine
33 68 18 L Leucine
7 26 0 P Proline
19 35 10 A Alanine
16 52 9 G Glycine
7 29 1 M Methionine
10 27 1 Y Tyrosine
0 12 2 C Cysteine
15 29 6 Q Glutamine
24 58 6 K Lycine
16 39 5 T Threonine
4 14 0 W Tryptophan
27 37 2 N Asparagine 

5VXL_1|Chain A|Invasin IpaD|Shigella flexneri (623)
>5FMZ_1|Chains A, D|POLYMERASE ACIDIC PROTEIN|INFLUENZA B VIRUS (B/MEMPHIS/13/2003) (1601067)
>4JGS_1|Chains A, B, C, D, E, F, G, H, I|MLV-related proviral Env polyprotein|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 \(w=f(c)\)
5VXL , Knot 122 289 0.79 38 164 269 
GSTGSHPVSSLTMLNDTLHNIRTTNQALKKELSQKTLTKTSLEEIALHSSQISMDVNKSAQLLDILSRNEYPINKDARELLHSAPKEAELDGDQMISHRELWAKIANSINDINEQYLKVYEHAVSSYTQMYQDFSAVLSSLAGWISPGGNDGNSVKLQVNSLKKALEELKEKYKDKPLYPANNTVSQEQANKWLTELGGTIGKVSQKNGGYVVSINMTPIDNMLKSLDNLGGNGEVVLDNAKYQAWNAGFSAEDETMKNNLQTLVQKYSNANSIFDNLVKVLSSTISSS
5FMZ , Knot 291 751 0.85 40 306 693 
GSHHHHHHHHGSGSMDTFITRNFQTTIIQKAKNTMAEFSEDPELQPAMLFNICVHLEVCYVISDMNFLDEEGKAYTALEGQGKEQNLRPQYEVIEGMPRTIAWMVQRSLAQEHGIETPKYLADLFDYKTKRFIEVGITKGLADDYFWKKKEKLGNSMELMIFSYNQDYSLSNESSLDEEGKGRVLSRLTELQAELSLKNLWQVLIGEEDVEKGIDFKLGQTISRLRDISVPAGFSNFEGMRSYIDNIDPKGAIERNLARMSPLVSVTPKKLTWEDLRPIGPHIYNHELPEVPYNAFLLMSDELGLANMTEGKSKKPKTLAKECLEKYSTLRDQTDPILIMKSEKANENFLWKLWRDCVNTISNEEMSNELQKTNYAKWATGDGLTYQKIMKEVAIDDETMCQEEPKIPNKCRVAAWVQTEMNLLSTLTSKRALDLPEIGPDVAPVEHVGSERRKYFVNEINYCKASTVMMKYVLFHTSLLNESNASMGKYKVIPITNRVVNEKGESFDMLYGLAVKGQSHLRGDTDVVTVVTFEFSSTDPRVDSGKWPKYTVFRIGSLFVSGREKSVYLYCRVNGTNKIQMKWGMEARRCLLQSMQQMEAIVEQESSIQGYDMTKACFKGDRVNSPKTFSIGTQEGKLVKGSFGKALRVIFTKCLMHYVFGNAQLEGFSAESRRLLLLIQALKDRKGPWVFDLEGMYSGIEECISNNPWVIQSAYWFNEWLGFEKEGSKVLESVDEIMDEGSGSGENLYFQ
4JGS , Knot 52 105 0.76 36 81 99 
SSGLVGSGTALVATKQFEQLQAAIHTDLGALEKSVSALEKSLTSLSEVVLQNRRGLDLLFLKEGGLCAALKEECSFYADHTGVVRDSMAKLRERLNQRQRLFESG

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 \(c\).

\(|P_{f(5VXL_1)}(2) \setminus P_{f(5FMZ_1)}(2)|=19\), \(|P_{f(5FMZ_1)}(2) \setminus P_{f(5VXL_1)}(2)|=161\). 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 1:1001001100101100010010000011000100001000010011100001010100010110110000011000100110011001010100110000111011001001000010100011000001000101110011111011100100101010010011001000000011011000100001001100111011010000110110101011001100100111010111001000110111010000100010011000001001100110110001000
Pair \(Z_2\) Length of longest common subsequence
5VXL_1,5FMZ_1 180 5
5VXL_1,4JGS_1 171 4
5FMZ_1,4JGS_1 249 4

Newick tree

 
[
	5FMZ_1:11.31,
	[
		5VXL_1:85.5,4JGS_1:85.5
	]:29.81
]

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 \((0.85)(0.8)(\frac{1040 }{\log_{20} 1040}-\frac{289}{\log_{20}289})=201.\)
Status Protein1 Protein2 d d1/2
Query variables 5VXL_1 5FMZ_1 255 172.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]