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Parikh vectors
1WYG_1 8CLT_1 1TLD_1 Letter Amino acid
73 23 8 P Proline
10 8 4 W Tryptophan
99 43 14 A Alanine
55 9 16 N Asparagine
46 14 10 Q Glutamine
68 16 3 F Phenylalanine
86 21 34 S Serine
94 13 10 T Threonine
28 8 10 Y Tyrosine
65 19 2 R Arginine
69 12 15 I Isoleucine
116 32 14 L Leucine
109 22 25 G Glycine
81 6 14 K Lycine
90 19 17 V Valine
54 21 6 D Aspartic acid
34 2 12 C Cysteine
93 21 4 E Glutamic acid
27 15 3 H Histidine
34 8 2 M Methionine 

1WYG_1|Chain A|Xanthine dehydrogenase/oxidase|Rattus norvegicus (10116)
>8CLT_1|Chains A, B|Zearalenone lactonase|Rhodococcus erythropolis (1833)
>1TLD_1|Chain A|BETA-TRYPSIN|Bos taurus (9913)
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 \(w=f(c)\)
1WYG , Knot 477 1331 0.86 40 336 1168 
MTADELVFFVNGKKVVEKNADPETTLLVYLRRKLGLCGTKLGCGEGGCGACTVMISKYDRLQNKIVHFSVNACLAPICSLHHVAVTTVEGIGNTQKLHPVQERIARSHGSQCGFCTPGIVMSMYTLLRNQPEPTVEEIENAFQGNLCRCTGYRPILQGFRTFAKDGGCCGGSGNNPNCCMNQTKDQTVSLSPSLFNPEDFKPLDPTQEPIFPPELLRLKDTPQKKLRFEGERVTWIQASTMEELLDLKAQHPDAKLVVGNTEIGIEMKFKNMLFPLIVCPAWIPELNSVVHGPEGISFGASCPLSLVESVLAEEIAKLPEQKTEVFRGVMEQLRWFAGKQVKSVASIGGNIITASPISDLNPVFMASGAKLTLVSRGTRRTVRMDHTFFPGYRKTLLRPEEILLSIEIPYSKEGEFFSAFKQASRREDDIAKVTSGMRVLFKPGTIEVQELSLCFGGMADRTISALKTTPKQLSKSWNEELLQSVCAGLAEELQLAPDAPGGMVEFRRTLTLSFFFKFYLTVLQKLGRADLEDMAGKLDPTFASATLLFQKDPPANVQLFQEVPKDQSEEDMVGRPLPHLAANMQASGEAVYCDDIPRYENELSLRLVTSTRAHAKITSIDTSEAKKVPGFVCFLTAEDVPNSNATGLFNDETVFAKDEVTCVGHIIGAVVADTPEHAQRAARGVKITYEDLPAIITIQDAINNNSFYGSEIKIEKGDLKKGFSEADNVVSGELYIGGQEHFYLETNCTIAVPKGEAGEMELFVSTQNTMKTQSFVAKMLGVPDNRIVVRVKRMGGGFGGKETRSTVVSTALALAAHKTGRPVRCMLDRDEDMLITGGRHPFLAKYKVGFMKTGTVVALEVAHFSNGGNTEDLSRSIMERALFHMDNAYKIPNIRGTGRICKTNLPSNTAFRGFGGPQGMLIAEYWMSEVAITCGLPAEEVRRKNMYKEGDLTHFNQKLEGFTLPRCWDECIASSQYLARKREVEKFNRENRWKKRGLCIIPTKFGISFTLPFLNQGGALVHVYTDGSVLLTHGGTEMGQGLHTKMVQVASRALKIPTSKIHISETSTNTVPNTSPTAASASADLNGQGVYEACQTILKRLEPFKKKKPTGPWEAWVMDAYTSAVSLSATGFYKTPNLGYSFETNSGNPFHYFSYGVACSEVEIDCLTGDHKNLRTDIVMDVGSSLNPAIDIGQVEGAFVQGLGLFTMEELHYSPEGSLHTRGPSTYKIPAFGSIPIEFRVSLLRDCPNKRAIYASKAVGEPPLFLASSIFFAIKDAIRAARAQHGDNAKQLFQLDSPATPEKIRNACVDQFTTLCVTGVPENSKSWSVRI
8CLT , Knot 145 332 0.84 40 208 320 
MAEEGTRSEAADAATQARQLPDSRNIFVSHRFPERQVDLGEVVMNFAEAGSPDNPALLLLPEQTGSWWSYEPVMGLLAENFHVFAVDIRGQGRSTWTPRRYSLDNFGNDLVRFIALVIKRPVVVAGNSSGGLLAAWLSAYAMPGQIRAALCEDAPFFASELVPAYGHSVLQAAGPAFELYRDFLGDQWSIGDWKGFVEAAKASPAKAMQLFPTPDEAPQNLKEYDPEWGRAFFEGTVALHCPHDRMLSQVKTPILITHHARTIDPETGELLGALSDLQAEHAQDIIRSAGVRVDYQSHPDALHMMHLFDPARYAEILTSWSATLPANDHHHH
1TLD , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN

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 \(c\).

\(|P_{f(1WYG_1)}(2) \setminus P_{f(8CLT_1)}(2)|=152\), \(|P_{f(8CLT_1)}(2) \setminus P_{f(1WYG_1)}(2)|=24\). 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 1:10100111110100110001010001110100011101001101011011001110000010001101010101111001001110010111000010110001100010001100111110100110001010100100110101000010011101100110011001101001000100000001010101101001011010001111101101000100010101001011010010011010100101011110001110101001111111011111010011011011011100110110011100110110000011011100101111001001101110110101100101111101101011001000010100011110000110100111010110000101101100100000011010011011101101010010101111100010110001001000100011001011110010111011111101000101011101010110011010100111010101101011100011101011001100000001110111011101010101100001100000101011000010101001000010011111011010011000101110000111000100110111111100100100110110100001111101001100001010010100101001100100110101011100010100000111101011010111000001000011101111100011101001111111000000110011111100010110011000001110110011110001111001011110110100110000100011001110100100110101010100001100011011111011111001100111001111001000010001010010001011011001000110000110000100100000100011011100111010111100111110100010111001100110110001101100110110001010000000110001011010101010110010001100101100001011101111010001101010110001011001000010110010011100010100101000010001110110010111011010111101111101001000101010001100001111101110101011000100011010011101111110011111001101101001001001101001101001001010010010101110000010101
Pair \(Z_2\) Length of longest common subsequence
1WYG_1,8CLT_1 176 4
1WYG_1,1TLD_1 228 4
8CLT_1,1TLD_1 196 4

Newick tree

 
[
	1TLD_1:11.73,
	[
		1WYG_1:88,8CLT_1:88
	]:23.73
]

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 \((0.85)(0.8)(\frac{1663 }{\log_{20} 1663}-\frac{332}{\log_{20}332})=340.\)
Status Protein1 Protein2 d d1/2
Query variables 1WYG_1 8CLT_1 437 270.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]