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Parikh vectors
1JKI_1 1MOS_1 1YXO_1 Letter Amino acid
29 15 19 T Threonine
21 13 5 Y Tyrosine
6 3 6 C Cysteine
31 26 32 G Glycine
56 46 47 L Leucine
25 13 22 P Proline
35 28 11 S Serine
4 1 2 W Tryptophan
34 20 18 D Aspartic acid
19 14 10 Q Glutamine
32 26 17 I Isoleucine
11 9 6 M Methionine
20 8 7 F Phenylalanine
31 36 43 A Alanine
9 12 14 H Histidine
17 16 22 R Arginine
39 12 4 N Asparagine
32 28 17 E Glutamic acid
40 18 7 K Lycine
42 24 19 V Valine 

1JKI_1|Chains A, B|myo-inositol-1-phosphate synthase|Saccharomyces cerevisiae (4932)
>1MOS_1|Chain A|GLUCOSAMINE 6-PHOSPHATE SYNTHASE|Escherichia coli (562)
>1YXO_1|Chains A, B|4-hydroxythreonine-4-phosphate dehydrogenase 1|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
1JKI , Knot 216 533 0.84 40 255 498 
MTEDNIAPITSVKVVTDKCTYKDNELLTKYSYENAVVTKTASGRFDVTPTVQDYVFKLDLKKPEKLGIMLIGLGGNNGSTLVASVLANKHNVEFQTKEGVKQPNYFGSMTQCSTLKLGIDAEGNDVYAPFNSLLPMVSPNDFVVSGWDINNADLYEAMQRSQVLEYDLQQRLKAKMSLVKPLPSIYYPDFIAANQDERANNCINLDEKGNVTTRGKWTHLQRIRRDIQNFKEENALDKVIVLWTANTERYVEVSPGVNDTMENLLQSIKNDHEEIAPSTIFAAASILEGVPYINGSPQNTFVPGLVQLAEHEGTFIAGDDLKSGQTKLKSVLAQFLVDAGIKPVSIASYNHLGNNDGYNLSAPKQFRSKEISKSSVIDDIIASNDILYNDKLGKKVDHCIVIKYMKPVGDSKVAMDEYYSELMLGGHNRISIHNVCEDSLLATPLIIDLLVMTEFCTRVSYKKVDPVKEDAGKFENFYPVLTFLSYWLKAPLTRPGFHPVNGLNKQRTALENFLRLLIGLPSQNELRFEERLL
1MOS , Knot 158 368 0.84 40 209 352 
DAGDKGIYRHYMQKEIYEQPNAIKNTLTGRISHGQVDLSELGPNADELLSKVEHIQILACGTSYNSGMVSRYWFESLAGIPCDVEIASEFRYRKSAVRRNSLMITLSQSGETADTLAGLRLSKELGYLGSLAICNVPGSSLVRESDLALMTNAGTEIGVASTKAFTTQLTVLLMLVAKLSRLKGLDASIEHDIVHGLQALPSRIEQMLSQDKRIEALAEDFSDKHHALFLGRGDQYPIALEGALKLKEISYIHAEAYAAGELKHGPLALIDADMPVIVVAPNNELLEKLKSNIEEVRARGGQLYVFADQDAGFVSSDNMHIIEMPHVEEVIAPIFYTVPLQLLAYHVALIKGTDVDQPRNLAKSVTVE
1YXO , Knot 137 328 0.80 40 178 305 
MSLRFALTPGEPAGIGPDLCLLLARSAQPHPLIAIASRTLLQERAGQLGLAIDLKDVSPAAWPERPAKAGQLYVWDTPLAAPVRPGQLDRANAAYVLETLTRAGQGCLDGHFAGMITAPVHKGVINEAGIPFSGHTEFLADLTHTAQVVMMLATRGLRVALATTHLPLREVADAISDERLTRVARILHADLRDKFGIAHPRILVCGLNPHAGEGGHLGREEIEVIEPCLERLRGEGLDLIGPLPADTLFTPKHLEHCDAVLAMYHDQGLPVLKYKGFGAAVNVTLGLPIIRTSVDHGTALDLAGSGRIDSGSLQVALETAYQMAASRC

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 \(c\).

\(|P_{f(1JKI_1)}(2) \setminus P_{f(1MOS_1)}(2)|=106\), \(|P_{f(1MOS_1)}(2) \setminus P_{f(1JKI_1)}(2)|=60\). 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 1:10000111100101100000000001100000001110001010101010100011010100100111111111100100111011100001010000110010011010000010111010100101110011111010011101101001010011000011000100010101011011101001011110000010001010001010001010010010001001000011001111101000001010111000100110010000001110011111011011101010100011111101100010111100100100010011101110111011011000011000100101100100001000011001110001100001100100011100101110001110000001111100010100100001110111101111001000100001011000110100101110110011011100111011011000001100110111111000010100011
Pair \(Z_2\) Length of longest common subsequence
1JKI_1,1MOS_1 166 4
1JKI_1,1YXO_1 181 3
1MOS_1,1YXO_1 153 5

Newick tree

 
[
	1JKI_1:90.01,
	[
		1MOS_1:76.5,1YXO_1:76.5
	]:13.51
]

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 \((0.85)(0.8)(\frac{901 }{\log_{20} 901}-\frac{368}{\log_{20}368})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 1JKI_1 1MOS_1 179 151
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]