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Parikh vectors
4MWL_1 5VSJ_1 6VGK_1 Letter Amino acid
18 69 22 A Alanine
31 46 13 G Glycine
38 27 14 S Serine
29 15 5 N Asparagine
5 8 7 M Methionine
18 19 6 Y Tyrosine
7 30 3 H Histidine
25 23 16 I Isoleucine
18 56 20 L Leucine
16 13 8 K Lycine
21 54 7 P Proline
23 59 11 R Arginine
21 42 15 D Aspartic acid
9 14 10 Q Glutamine
22 47 11 V Valine
28 56 13 T Threonine
14 16 0 W Tryptophan
18 3 1 C Cysteine
18 45 12 E Glutamic acid
9 21 6 F Phenylalanine 

4MWL_1|Chain A|Neuraminidase|Influenza A virus (11320)
>5VSJ_1|Chains A, B|Alpha-1,4-glucan:maltose-1-phosphate maltosyltransferase 1|Streptomyces coelicolor (strain ATCC BAA-471 / A3(2) / M145) (100226)
>6VGK_1|Chains A, B, C, D, E, F, G|ATP-dependent Clp protease proteolytic subunit 2|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
4MWL , Knot 165 388 0.84 40 225 370 
RNFNNLTKGLCTINSWHIYGKDNAVRIGESSDVLVTREPYVSCDPDECRFYALSQGTTIRGKHSNGTIHDRSQYRALISWPLSSPPTVYNSRVECIGWSSTSCHDGKSRMSICISGPNNNASAVVWYNRRPVAEINTWARNILRTQESECVCHNGVCPVVFTDGSATGPADTRIYYFKEGKILKWESLTGTAKHIEECSCYGERTGITCTCKDNWQGSNRPVIQIDPVAMTHTSQYICSPVLTDNPRPNDPNIGKCNDPYPGNNNNGVKGFSYLDGANTWLGRTISTASRSGYEMLKVPNALTDDRSKPIQGQTIVLNADWSGYSGSFMDYWAEGDCYRACFYVELIRGRPKEDKVWWTSNSIVSMCSSTEFLGQWNWPDGAKIEYFL
5VSJ , Knot 257 663 0.84 40 259 595 
MPATHHSSATSAERPTVVGRIPVLDVRPVVQRGRRPAKAVTGESFEVSATVFREGHDAVGANVVLRDPRGRPGPWTPMRELAPGTDRWGATVTAGETGTWSYTVEAWGDPVTTWRHHARIKIPAGLDTDLVLEEGARLYERAAADVPGREDRRELLAAVDALRDESRPAASRLAAALTPQVDAVLARHPLRDLVTSSDPLPLLVERERALYGAWYEFFPRSEGTPHTPHGTFRTAARRLPAIAAMGFDVVYLPPIHPIGTTHRKGRNNTLSATGDDVGSPWAIGSPEGGHDSIHPALGTLDDFDHFVTEAGKLGLEIALDFALQCSPDHPWVHKHPEWFHHRPDGTIAHAENPPKKYQDIYPIAFDADPDGLATETVRILRHWMDHGVRIFRVDNPHTKPVAFWERVIADINGTDPDVIFLAEAFTRPAMMATLAQIGFQQSYTYFTWRNTKQELTEYLTELSGEAASYMRPNFFANTPDILHAYLQHGGRPAFEVRAVLAATLSPTWGIYSGYELCENTPLREGSEEYLDSEKYQLKPRDWTRAAREGTTIAPLVTRLNTIRRENPALRQLRDLHFHPTDKEEVIAYSKRQGSNTVLVVVNLDPRHTQEATVSLDMPQLGLDWHESVPVRDELTGETYHWGRANYVRLEPGRTPAHVCTVLR
6VGK , Knot 93 200 0.82 38 136 193 
SLPSFIEHSSFGVKESNPYNKLFEERIIFLGVQVDDASANDIMAQLLVLESLDPDRDITMYINSPGGGFTSLMAIYDTMQYVRADIQTVCLGQAASAAAVLLAAGTPGKRMALPNARVLIHQPSLSGVIQGQFSDLEIQAAEIERMRTLMETTLARHTGKDAGVIRKDTDRDKILTAEEAKDYGIIDTVLEYRKLSAQTA

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 \(c\).

\(|P_{f(4MWL_1)}(2) \setminus P_{f(5VSJ_1)}(2)|=73\), \(|P_{f(5VSJ_1)}(2) \setminus P_{f(4MWL_1)}(2)|=107\). 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 1:0010010011001001010100011011000011100010100010000101100100101000010100000001110111001101000010011100000001000101010110001011110000111010011001100000001000110111100101011100010010010110100101010010000001000110000000101000111010111100000010011100010100101100001011000011011001011001110010010001001101101100000011010011101010100101100110100001010101101010000111000011010000011101011011010011
Pair \(Z_2\) Length of longest common subsequence
4MWL_1,5VSJ_1 180 4
4MWL_1,6VGK_1 185 4
5VSJ_1,6VGK_1 201 4

Newick tree

 
[
	6VGK_1:98.67,
	[
		4MWL_1:90,5VSJ_1:90
	]:8.67
]

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 \((0.85)(0.8)(\frac{1051 }{\log_{20} 1051}-\frac{388}{\log_{20}388})=175.\)
Status Protein1 Protein2 d d1/2
Query variables 4MWL_1 5VSJ_1 221 173
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]