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Parikh vectors
7FNP_1 3NKP_1 4ONY_1 Letter Amino acid
5 13 13 W Tryptophan
14 49 37 V Valine
11 49 41 R Arginine
19 49 27 K Lycine
8 17 14 M Methionine
10 38 34 F Phenylalanine
18 60 38 S Serine
1 31 0 C Cysteine
5 28 15 H Histidine
23 34 24 I Isoleucine
10 59 41 P Proline
14 51 42 D Aspartic acid
9 20 23 Q Glutamine
15 53 31 E Glutamic acid
29 70 47 L Leucine
7 44 24 Y Tyrosine
18 25 51 A Alanine
14 41 23 N Asparagine
7 50 40 G Glycine
21 50 30 T Threonine 

7FNP_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>3NKP_1|Chain A|Ectonucleotide pyrophosphatase/phosphodiesterase family member 2|Mus musculus (10090)
>4ONY_1|Chains A, B|Extracellular solute-binding protein family 5|Brucella melitensis (546272)
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 \(w=f(c)\)
7FNP , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
3NKP , Knot 321 831 0.86 40 316 764 
AEWDEGPPTVLSDSPWTNTSGSCKGRCFELQEVGPPDCRCDNLCKSYSSCCHDFDELCLKTARGWECTKDRCGEVRNEENACHCSEDCLSRGDCCTNYQVVCKGESHWVDDDCEEIRVPECPAGFVRPPLIIFSVDGFRASYMKKGSKVMPNIEKLRSCGTHAPYMRPVYPTKTFPNLYTLATGLYPESHGIVGNSMYDPVFDATFHLRGREKFNHRWWGGQPLWITATKQGVRAGTFFWSVSIPHERRILTILQWLSLPDNERPSVYAFYSEQPDFSGHKYGPFGPEMTNPLREIDKTVGQLMDGLKQLKLHRCVNVIFVGDHGMEDVTCDRTEFLSNYLTNVDDITLVPGTLGRIRPKIPNNLKYDPKAIIANLTCKKPDQHFKPYMKQHLPKRLHYANNRRIEDLHLLVERRWHVARKPLDVYKKPSGKCFFQGDHGFDNKVNSMQTVFVGYGPTFKYRTKVPPFENIELYNVMCDLLGLKPAPNNGTHGSLNHLLRTNTFRPTLPEEVSRPNYPGIMYLQSDFDLGCTCDDKNKLEELNKRLHTKGSTEERHLLYGRPAVLYRTSYDILYHTDFESGYSEIFLMPLWTSYTISKQAEVSSIPEHLTNCVRPDVRVSPGFSQNCLAYKNDKQMSYGFLFPPYLSSSPEAKYDAFLVTNMVPMYPAFKRVWTYFQRVLVKKYASERNGVNVISGPIFDYNYNGLRDIEDEIKQYVEGSSIPVPTHYYSIITSCLDFTQPADKCDGPLSVSSFILPHRPDNDESCNSSEDESKWVEELMKMHTARVRDIEHLTGLDFYRKTSRSYSEILTLKTYLHTYESEISRENLYFQ
4ONY , Knot 230 595 0.82 38 259 561 
MAHHHHHHAANETAPDYALSMHGDVALPADYTHFPYTNPDAPKKGSLTVGVVGTFDSLNPFVLKSMRTTARGLYNDGEFGNMVYQTLMLRSRDEPFTLYSLLAEKVAIDPERKWVEFTLNPKAKWSDGQPVTVDDVLFTYDILTEKGRPPYNSRMSRVAKIEKTGERSVRFTFNEKSDREFPMLIAGSMPVLPKHAINRDTFGNSTLEPPIGSGPYVVASVQPGQRIVYKRNPDYWGKDLPSQRGFNNFDKISIEYYRNETSLFESFKKGILDIFIEGNPIRWEKLYDFPAVEQGKVIKDTFEKGTPADMLGFVFNTRRPIFADRRVRQALGLLFDFEWANSNLFAGQYRRTQSFWEGSQLSSVGRPADARERELLAPFPGAVREDVMNGTWHPPVTDGSGHDRVPAKKAYDLLSQAGFQFKDGMAIDPTAKPFAFEIMTRSPDEEKIALAYQRNLSRLGIAVEIHTVDDAQYQQRLQTFDYDMILGALASSLSPGNEQWLRWGSASRDVQGSFNFAGVADPAVDAMIEALLAARNRADFVSAVRALDRVLISGDYYVPLYHLPYQWVARWDRIEHPQKTPLSGYQLPAWWHTSQ

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 \(c\).

\(|P_{f(7FNP_1)}(2) \setminus P_{f(3NKP_1)}(2)|=26\), \(|P_{f(3NKP_1)}(2) \setminus P_{f(7FNP_1)}(2)|=167\). 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 1:111000001011000101110000100101000101011001101011010100101110110001111000100110100100101110011000010011100011101101011011011101001011101110100100111010010111101000110010000110010111011000000101111001010100001110100001101000100111000100001010110000100111100101
Pair \(Z_2\) Length of longest common subsequence
7FNP_1,3NKP_1 193 4
7FNP_1,4ONY_1 156 6
3NKP_1,4ONY_1 135 4

Newick tree

 
[
	7FNP_1:93.51,
	[
		4ONY_1:67.5,3NKP_1:67.5
	]:26.01
]

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 \((0.85)(0.8)(\frac{1089 }{\log_{20} 1089}-\frac{258}{\log_{20}258})=222.\)
Status Protein1 Protein2 d d1/2
Query variables 7FNP_1 3NKP_1 287 185.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]