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Parikh vectors
1HNC_1 7PXH_1 6YYV_1 Letter Amino acid
12 47 21 P Proline
10 82 32 G Glycine
27 119 46 L Leucine
18 55 39 K Lycine
15 64 16 F Phenylalanine
9 55 18 N Asparagine
9 43 18 Q Glutamine
3 18 11 H Histidine
12 71 17 I Isoleucine
7 31 5 M Methionine
7 69 20 T Threonine
10 90 26 A Alanine
11 58 30 R Arginine
15 84 24 D Aspartic acid
3 29 7 C Cysteine
3 14 11 W Tryptophan
12 34 13 Y Tyrosine
5 79 22 V Valine
16 51 34 E Glutamic acid
13 87 19 S Serine 

1HNC_1|Chains A, B, C, D|GLUTATHIONE S-TRANSFERASE|Homo sapiens (9606)
>7PXH_1|Chains A, B, C, D|Isoform J of Calcium-activated potassium channel slowpoke|Drosophila melanogaster (7227)
>6YYV_1|Chain A|Aspartyl/asparaginyl beta-hydroxylase|Homo sapiens (9606)
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 \(w=f(c)\)
1HNC , Knot 101 217 0.83 40 153 209 
PMTLGYWNIRGLAHSIRLLLEYTDSSYEEKKYTMGDAPDYDRSQWLNEKFKLGLDFPNLPYLIDGTHKITQSNAILRYIARKHNLCGESEKEQIREDILENQFMDSRMQLAKLCYDPDFEKLKPEYLQALPEMLKLYSQFLGKQPWFLGDKITFVDFIAYDVLERNQVFEPSCLDAFPNLKDFISRFEGLEKISAYMKSSRFLPRPVFTKMAVFGNK
7PXH , Knot 429 1180 0.85 40 338 1034 
MASGLIDTNFSSTLANGMSGCDQSTVESLADDPTDSPFDADDCLKVRKYWCFLLSSIFTFLAGLLVVLLWRAFAFVCCRKEPDLGPNDPKQKEQKASRNKQEFEGTFMTEAKDWAGELISGQTTTGRILVVLVFILSIASLIIYFVDASSEEVERCQKWSNNITQQIDLAFNIFFMVYFFIRFIAASDKLWFMLEMYSFVDYFTIPPSFVSIYLDRTWIGLRFLRALRLMTVPDILQYLNVLKTSSSIRLAQLVSIFISVWLTAAGIIHLLENSGDPLDFDNAHRLSYWTCVYFLIVTMSTVGYGDVYCETVLGRTFLVFFLLVGLAIFASCIPEIIDLIGTRAKYGGTLKNEKGRRHIVVCGHITYESVSHFLKDFLHEDREDVDVEVVFLHRKPPDLELEGLFKRHFTTVEFFQGTIMNPIDLQRVKVHEADACLVLANKYCQDPDAEDAANIMRVISIKNYSDDIRVIIQLMQYHNKAYLLNIPSWDWKQGDDVICLAELKLGFIAQSCLAPGFSTMMANLFAMRSFKTSPDMQSWTNDYLRGTGMEMYTETLSPTFIGIPFAQATELCFSKLKLLLLAIEIKGAEEGADSKISINPRGAKIQANTQGFFIAQSADEVKRAWFYCKACHEDIKDETLIKKCKCKNLATFRKGVRAVQMVGRASDITRDREDTNLLNRNVRRPNGTGNGTGGMHHMNNTAAAAAAAAAAGKQVNKVKPTVNVSRQVEGQVISPSQYNRPTSRSSGTGTQNQNGGVSLPAGIADDQSKDFDFEKTEMKYDSTGMFHWSPAKSLEDCILDRNQAAMTVLNGHVVVCLFADPDSPLIGLRNLVMPLRASNFHYHELKHVVIVGSVDYIRREWKMLQNLPKISVLNGSPLSRADLRAVNVNLCDMCCILSAKVPSNDDPTLADKEAILASLNIKAMTFDDTIGVLSQRGPEFDNLSATAGSPIVLQRRGSVYGANVPMITELVNDGNVQFLDQDDDDDPDTELYLTQPFACGTAFAVSVLDSLMSTTYFNQNALTLIRSLITGGATPELELILAEGAGLRGGYSTVESLSNRDRCRVGQISLYDGPLAQFGECGKYGDLFVAALKSYGMLCIGLYRFRDTSSSCDASSKRYVITNPPDDFSLLPTDQVFVLMQFDPGLEYKPPAVRAPAGGRGTNTQGSGVGGGGSNKDDNS
6YYV , Knot 177 429 0.83 40 233 406 
KPKLLNKFDKTIKAELDAAEKLRKRGKIEEAVNAFKELVRKYPQSPRARYGKAQCEDDLAEKRRSNEVLRGAIETYQEVASLPDVPADLLKLSLKRRSDRQQFLGHMRGSLLTLQRLVQLFPNDTSLKNDLGVGYLLIGDNDNAKKVYEEVLSVTPNDGFAKVHYGFILKAQNKIAESIPYLKEGIESGDPGTDDGRFYFHLGDAMQRVGNKEAYKWYELGHKRGHFASVWQRSLYNVNGLKAQPWWTPKETGYTELVKSLERNWKLIRDEGLAVMDKAKGLFLPEDENLREKGDWSQFTLWQQGRRNENACKGAPKTCTLLEKFPETTGCRRGQIKYSIMHPGTHVWPHTGPTNCRLRMHLGLVIPKEGCKIRCANETKTWEEGKVLIFDDSFEHEVWQDASSFRLIFIVDVWHPELTPQQRRSLPAI

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 \(c\).

\(|P_{f(1HNC_1)}(2) \setminus P_{f(7PXH_1)}(2)|=17\), \(|P_{f(7PXH_1)}(2) \setminus P_{f(1HNC_1)}(2)|=202\). 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 1:1101101010111001011100000000000001101100000011000101110110110110100010000111001100001010000001000110001100010110100010100101001011101101000111001111100101101110011000011010010111010011001011001010100001110111001111100
Pair \(Z_2\) Length of longest common subsequence
1HNC_1,7PXH_1 219 4
1HNC_1,6YYV_1 192 4
7PXH_1,6YYV_1 161 4

Newick tree

 
[
	1HNC_1:10.44,
	[
		6YYV_1:80.5,7PXH_1:80.5
	]:28.94
]

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 \((0.85)(0.8)(\frac{1397 }{\log_{20} 1397}-\frac{217}{\log_{20}217})=310.\)
Status Protein1 Protein2 d d1/2
Query variables 1HNC_1 7PXH_1 397 232
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]