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Parikh vectors
2GPN_1 4QAM_1 3NDY_1 Letter Amino acid
46 26 33 N Asparagine
22 14 7 H Histidine
36 17 12 P Proline
12 5 11 W Tryptophan
62 31 24 V Valine
63 20 34 A Alanine
49 12 16 D Aspartic acid
9 11 0 C Cysteine
30 13 7 Q Glutamine
21 6 14 M Methionine
65 29 18 E Glutamic acid
48 46 27 G Glycine
48 22 19 K Lycine
38 25 13 F Phenylalanine
36 28 26 T Threonine
63 9 12 R Arginine
51 17 22 I Isoleucine
78 35 15 L Leucine
29 26 20 S Serine
36 8 15 Y Tyrosine 

2GPN_1|Chain A|GLYCOGEN PHOSPHORYLASE B|Oryctolagus cuniculus (9986)
>4QAM_1|Chain A|X-linked retinitis pigmentosa GTPase regulator|Homo sapiens (9606)
>3NDY_1|Chains A, C[auth B], E[auth C], G[auth D]|Endoglucanase D|Clostridium cellulovorans (1493)
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 \(w=f(c)\)
2GPN , Knot 321 842 0.85 40 316 771 
SRPLSDQEKRKQISVRGLAGVENVTELKKNFNRHLHFTLVKDRNVATPRDYYFALAHTVRDHLVGRWIRTQQHYYEKDPKRIYYLSLEFYMGRTLQNTMVNLALENACDEATYQLGLDMEELEEIEEDAGLGNGGLGRLAACFLDSMATLGLAAYGYGIRYEFGIFNQKICGGWQMEEADDWLRYGNPWEKARPEFTLPVHFYGRVEHTSQGAKWVDTQVVLAMPYDTPVPGYRNNVVNTMRLWSAKAPNDFNLKDFNVGGYIQAVLDRNLAENISRVLYPNDNFFEGKELRLKQEYFVVAATLQDIIRRFKSSKFGCRDPVRTNFDAFPDKVAIQLNDTHPSLAIPELMRVLVDLERLDWDKAWEVTVKTCAYTNHTVIPEALERWPVHLLETLLPRHLQIIYEINQRFLNRVAAAFPGDVDRLRRMSLVEEGAVKRINMAHLCIAGSHAVNGVARIHSEILKKTIFKDFYELEPHKFQNKTNGITPRRWLVLCNPGLAEIIAERIGEEYISDLDQLRKLLSYVDDEAFIRDVAKVKQENKLKFAAYLEREYKVHINPNSLFDVQVKRIHEYKRQLLNCLHVITLYNRIKKEPNKFVVPRTVMIGGKAAPGYHMAKMIIKLITAIGDVVNHDPVVGDRLRVIFLENYRVSLAEKVIPAADLSEQISTAGTEASGTGNMKFMLNGALTIGTMDGANVEMAEEAGEENFFIFGMRVEDVDRLDQRGYNAQEYYDRIPELRQIIEQLSSGFFSPKQPDLFKDIVNMLMHHDRFKVFADYEEYVKCQERVSALYKNPREWTRMVIRNIATSGKFSSDRTIAQYAREIWGVEPSRQRLPAPDEKIP
4QAM , Knot 160 400 0.80 40 201 357 
MREPEELMPDSGAVFTFGKSKFAENNPGKFWFKNDVPVHLSCGDEHSAVVTGNNKLYMFGSNNWGQLGLGSKSAISKPTCVKALKPEKVKLAACGRNHTLVSTEGGNVYATGGNNEGQLGLGDTEERNTFHVISFFTSEHKIKQLSAGSNTSAALTEDGRLFMWGDNSEGQIGLKNVSNVCVPQQVTIGKPVSWVSCGYYHSAFVTTDGELYVFGEPENGKLGLPNQLLGNHRTPQLVSEIPEKVIQVACGGEHTVVLTENAVYTFGLGQFGQLGLGTFLFETSEPKVIENIRDQTISYISCGENHTALITDIGLMYTFGDGRHGKLGLGLENFTNHFIPTLCSNFLRFIVKLVACGGCHMVVFAAPHRGVAKEIEFDEINDTCLSVATFLPWSHPQFEK
3NDY , Knot 151 345 0.85 38 211 334 
STAFTGVRDVPAQQIVNEMKVGWNLGNTMDAIGGETNWGNPMTTHAMINKIKEAGFNTLRLPVTWDGHMGAAPEYTIDQTWMKRVEEIANYAFDNDMYVIINLHHENEWLKPFYANEAQVKAQLTKVWTQIANNFKKYGDHLIFETMNEPRPVGASLQWTGGSYENREVVNRYNLTAVNAIRATGGNNATRYIMVPTLAASAMSTTINDLVIPNNDSKVIVSLHMYSPYFFAMDINGTSSWGSDYDKSSLDSEFDAVYNKFVKNGRAVVIGEMGSINKNNTAARVTHAEYYAKSAKARGLTPIWWDNGYSVAGKAETFGIFNRSNLTWDAPEVMKAFIKGIGGSS

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 \(c\).

\(|P_{f(2GPN_1)}(2) \setminus P_{f(4QAM_1)}(2)|=143\), \(|P_{f(4QAM_1)}(2) \setminus P_{f(2GPN_1)}(2)|=28\). 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 1:00110000000010101111100100100010001010110000110100001111001000111011000000000010010010101011001000110111001000100011101001001000111101111011101100110111110101100011110001011101001001100101100101010111010101000001101100011111100011110000110010110101100101001011101011100011001001101000110100101000011111010011001000011000110001011100111010000101111011011101001010011010100010000011101100111011001110010110010001100111111101001001011001110010110101110011011101000110001100100101001000001101001111001111011100110001001001001100100011100110100000101110100000101010011010100100000011001011010001000100111100111110111100110111011011101100011110010111100001011001111101000100110010101010111011101101011010110011000111111010010010001001000000110100110010011101001011001101110000101110000010000010110001001001110011001010000011001001111010000111100011
Pair \(Z_2\) Length of longest common subsequence
2GPN_1,4QAM_1 171 4
2GPN_1,3NDY_1 169 5
4QAM_1,3NDY_1 168 6

Newick tree

 
[
	2GPN_1:85.33,
	[
		3NDY_1:84,4QAM_1:84
	]:1.33
]

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 \((0.85)(0.8)(\frac{1242 }{\log_{20} 1242}-\frac{400}{\log_{20}400})=219.\)
Status Protein1 Protein2 d d1/2
Query variables 2GPN_1 4QAM_1 278 200
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]