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Parikh vectors
7OVU_1 3WGV_1 3ZST_1 Letter Amino acid
22 68 45 E Glutamic acid
13 73 51 G Glycine
9 45 57 P Proline
9 47 60 R Arginine
8 43 15 N Asparagine
7 16 39 H Histidine
23 96 57 L Leucine
15 54 13 K Lycine
18 55 33 S Serine
12 61 58 T Threonine
18 75 49 V Valine
13 82 69 A Alanine
0 23 3 C Cysteine
6 35 15 Q Glutamine
11 56 42 D Aspartic acid
9 82 24 I Isoleucine
10 24 9 M Methionine
6 45 21 F Phenylalanine
2 12 16 W Tryptophan
7 24 19 Y Tyrosine 

7OVU_1|Chain A|Acyl-CoA N-acyltransferases (NAT) superfamily protein|Arabidopsis thaliana (3702)
>3WGV_1|Chains A, D[auth C]|Sodium/potassium-transporting ATPase subunit alpha-1|Sus scrofa (9823)
>3ZST_1|Chains A, B|PUTATIVE GLUCANOHYDROLASE PEP1A GLGE ISOFORM 1|STREPTOMYCES COELICOLOR (1902)
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 \(w=f(c)\)
7OVU , Knot 104 218 0.85 38 153 213 
MVTRMATKVSLEGKRVVLVPYMAEHVPKYHQWMQDSALLEATGSEPLSLEQEYEMQLSWTQDPNKRTFIVLDKDFVKGDLAHGQPHVEAMTGDVNIYMNDVDDPKVAEVEIMIAEPRSRGKGLGKESVLIMMAYGVKNLEIHKFTAKIGESNTASLSLFRKLGFEESSYSGIFKEVTLEYPVTNLRREELLKLLDEVIRHTHSSNNPSDSLLSGEATA
3WGV , Knot 381 1016 0.86 40 316 912 
GRDKYEPAAVSEHGDKKKAKKERDMDELKKEVSMDDHKLSLDELHRKYGTDLSRGLTPARAAEILARDGPNALTPPPTTPEWVKFCRQLFGGFSMLLWIGAILCFLAYGIQAATEEEPQNDNLYLGVVLSAVVIITGCFSYYQEAKSSKIMESFKNMVPQQALVIRNGEKMSINAEEVVVGDLVEVKGGDRIPADLRIISANGCKVDNSSLTGESEPQTRSPDFTNENPLETRNIAFFSTNCVEGTARGIVVYTGDRTVMGRIATLASGLEGGQTPIAAEIEHFIHIITGVAVFLGVSFFILSLILEYTWLEAVIFLIGIIVANVPEGLLATVTVCLTLTAKRMARKNCLVKNLEAVETLGSTSTICSDKTGTLTQNRMTVAHMWFDNQIHEADTTENQSGVSFDKTSATWLALSRIAGLCNRAVFQANQENLPILKRAVAGDASESALLKCIELCCGSVKEMRERYTKIVEIPFNSTNKYQLSIHKNPNTAEPRHLLVMKGAPERILDRCSSILIHGKEQPLDEELKDAFQNAYLELGGLGERVLGFCHLFLPDEQFPEGFQFDTDDVNFPLDNLCFVGLISMIDPPRAAVPDAVGKCRSAGIKVIMVTGDHPITAKAIAKGVGIISEGNETVEDIAARLNIPVSQVNPRDAKACVVHGSDLKDMTSEQLDDILKYHTEIVFARTSPQQKLIIVEGCQRQGAIVAVTGDGVNDSPALKKADIGVAMGIAGSDVSKQAADMILLDDNFASIVTGVEEGRLIFDNLKKSIAYTLTSNIPEITPFLIFIIANIPLPLGTVTILCIDLGTDMVPAISLAYEQAESDIMKRQPRNPKTDKLVNERLISMAYGQIGMIQALGGFFTYFVILAENGFLPIHLLGLRVNWDDRWINDVEDSYGQQWTYEQRKIVEFTCHTAFFVSIVVVQWADLVICKTRRNSVFQQGMKNKILIFGLFEETALAAFLSYCPGMGVALRMYPLKPTWWFCAFPYSLLIFVYDEVRKLIIRRRPGGWVEKETYY
3ZST , Knot 266 695 0.83 40 262 620 
MGSSHHHHHHSSGLVPRGSHMPATHHSSATSAERPTVVGRIPVLDVRPVVQRGRRPAKAVTGESFEVSATVFREGHDAVGANVVLRDPRGRPGPWTPMRELAPGTDRWGATVTAGETGTWSYTVEAWGDPVTTWRHHARIKIPAGLDTDLVLEEGARLYERAAADVPGREDRRELLAAVDALRDESRPAASRLAAALTPQVDAVLARHPLRDLVTSSDPLPLLVERERALYGAWYEFFPRSEGTPHTPHGTFRTAARRLPAIAAMGFDVVYLPPIHPIGTTHRKGRNNTLSATGDDVGVPWAIGSPEGGHDSIHPALGTLDDFDHFVTEAGKLGLEIALDFALQCSPDHPWVHKHPEWFHHRPDGTIAHAENPPKKYQDIYPIAFDADPDGLATETVRILRHWMDHGVRIFRVDNPHTKPVAFWERVIADINGTDPDVIFLAEAFTRPAMMATLAQIGFQQSYTYFTWRNTKQELTEYLTELSGEAASYMRPNFFANTPDILHAYLQHGGRPAFEVRAVLAATLSPTWGIYSGYELCENTPLREGSEEYLDSEKYQLKPRDWTRAAREGTTIAPLVTRLNTIRRENPALRQLRDLHFHPTDKEEVIAYSKRQGSNTVLVVVNLDPRHTQEATVSLDMPQLGLDWHESVPVRDELTGETYHWGRANYVRLEPGRTPAHVCTVLRPSHPQIGGSHTT

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 \(c\).

\(|P_{f(7OVU_1)}(2) \setminus P_{f(3WGV_1)}(2)|=17\), \(|P_{f(3WGV_1)}(2) \setminus P_{f(7OVU_1)}(2)|=180\). 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 1:11001100101010011111011001100001100011101010011010000010101000100001111000110101101010101101010101001001011010111101000101110001111110110010100101011000010101100111000000111001010011001000011011001100000000100011010101
Pair \(Z_2\) Length of longest common subsequence
7OVU_1,3WGV_1 197 5
7OVU_1,3ZST_1 169 5
3WGV_1,3ZST_1 138 4

Newick tree

 
[
	7OVU_1:98.19,
	[
		3ZST_1:69,3WGV_1:69
	]:29.19
]

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 \((0.85)(0.8)(\frac{1234 }{\log_{20} 1234}-\frac{218}{\log_{20}218})=270.\)
Status Protein1 Protein2 d d1/2
Query variables 7OVU_1 3WGV_1 345 206
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]