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Parikh vectors
5BNY_1 7NFK_1 5FYC_1 Letter Amino acid
23 4 31 E Glutamic acid
26 2 23 G Glycine
26 4 27 L Leucine
18 5 27 K Lycine
10 0 16 D Aspartic acid
7 2 11 Q Glutamine
21 3 20 I Isoleucine
3 0 12 M Methionine
22 0 23 T Threonine
12 0 21 Y Tyrosine
17 8 23 A Alanine
20 0 19 R Arginine
26 0 15 N Asparagine
9 0 5 C Cysteine
11 0 23 F Phenylalanine
24 1 24 S Serine
19 0 17 V Valine
4 0 15 H Histidine
18 0 21 P Proline
8 1 8 W Tryptophan 

5BNY_1|Chains A, C, E|Hemagglutinin|Influenza A virus (A/chicken/Guangdong/S1311/2010(H6N6)) (1454272)
>7NFK_1|Chains A, B|CC-Type2-(LaId)4-I24S|synthetic construct (32630)
>5FYC_1|Chains A, B|LYSINE-SPECIFIC DEMETHYLASE 4A|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 \(w=f(c)\)
5BNY , Knot 143 324 0.85 40 204 311 
DKICIGYHANNSTTKVDTILEKNVTVTHSVELLENQKEERFCKISNKAPLDLRDCTLEGWILGNPRCGILLADQSWSYIVERPNARNGICYPGTLNEAEELKALIGSGERVERFEMFPKSTWTGVNTESGVSSACPLGNGPSFYRNLLWIIKLKSSEYPVIRGTFNNTGDKSILYFWGVHHPPVTTEQNALYGSGDRYVRMGTESMNFARSPEIAARPAVNGQRGRIDYFWSILKPGETLNVESNGNLIAPWYAYRFVNKDSKGAIFRSNLPIENCDATCQTTEGVIRTNKTFQNVSPLWIGECPKYVKSKSLRLATGLRNVPQ
7NFK , Knot 13 32 0.46 20 17 20 
XGEIAQALKEIAKALKEIAWALKESAQALKGX
5FYC , Knot 167 381 0.86 40 238 367 
MHHHHHHSSGVDLGTENLYFQSMASESETLNPSARIMTFYPTMEEFRNFSRYIAYIESQGAHRAGLAKVVPPKEWKPRASYDDIDDLVIPAPIQQLVTGQSGLFTQYNIQKKAMTVREFRKIANSDKYCTPRYSEFEELERKYWKNLTFNPPIYGADVNGTLYEKHVDEWNIGRLRTILDLVEKESGITIEGVNTPYLYFGMWKTSFAWHTEDMDLYSINYLHFGEPKSWYSVPPEHGKRLERLAKGFFPGSAQSCEAFLRHKMTLISPLMLKKYGIPFDKVTQEAGEFMITFPYGYHAGFNHGFNCAESTNFATRRWIEYGKQAVLCSCRKDMVKISMDVFVRKFQPERYKLWKAGKDNTVIDHTLPTPEAAEFLKESEL

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 \(c\).

\(|P_{f(5BNY_1)}(2) \setminus P_{f(7NFK_1)}(2)|=194\), \(|P_{f(7NFK_1)}(2) \setminus P_{f(5BNY_1)}(2)|=7\). 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 1:001011001000000100110001010001011000000010010001110100001011111010011111000100110010100110011010010010111101001001011100010110000110010111011010001111101000001110101000100011011110011100000110101000101100010110010111011101001010011011011001010001011111010011000001111000111000010000001110000010010111110010010000101101100110
Pair \(Z_2\) Length of longest common subsequence
5BNY_1,7NFK_1 201 3
5BNY_1,5FYC_1 184 4
7NFK_1,5FYC_1 231 4

Newick tree

 
[
	7NFK_1:11.16,
	[
		5BNY_1:92,5FYC_1:92
	]:21.16
]

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 \((0.85)(0.8)(\frac{356 }{\log_{20} 356}-\frac{32}{\log_{20}32})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 5BNY_1 7NFK_1 139 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]