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Parikh vectors
5ARK_1 8JOV_1 6OIA_1 Letter Amino acid
9 50 30 V Valine
6 53 7 Q Glutamine
11 10 8 H Histidine
2 52 43 L Leucine
4 43 23 P Proline
4 23 14 F Phenylalanine
9 30 18 N Asparagine
11 64 24 D Aspartic acid
8 2 6 C Cysteine
5 62 27 E Glutamic acid
7 84 32 A Alanine
13 61 19 R Arginine
6 40 35 G Glycine
8 25 6 M Methionine
1 11 6 W Tryptophan
4 17 18 Y Tyrosine
6 38 27 I Isoleucine
6 34 25 K Lycine
7 42 17 S Serine
7 40 19 T Threonine 

5ARK_1|Chains A, B, C, D|RIBONUCLEASE 4|SUS SCROFA (9823)
>8JOV_1|Chains A[auth 0], CA[auth 3], CB[auth 6], D[auth U], FB[auth Y], GA[auth g], G[auth c], KA[auth k], OA[auth o], SA[auth r], WA[auth u], ZA[auth x]|Portal protein|Ralstonia phage GP4 (2282904)
>6OIA_1|Chains A, B, C|Ornithine aminotransferase, mitochondrial|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 \(w=f(c)\)
5ARK , Knot 66 134 0.80 40 104 127 
MAHHHHHHVDDDDKMQDRMYQRFLRQHVDPDATGGNDAYCNLMMQRRKMTSHYCKRFNTFIHEDIWNIRSICSTSNIQCKNGQMNCHEGVVKVTDCRETGSSRAPNCRYRAMASTRRVVIACEGNPEVPVHFDK
8JOV , Knot 297 781 0.84 40 286 700 
MFDLRDDQNTRVIADHPEARMPTEEPGKDDAPPSHPLDSEEMKELHRRLRSYLHQELDRQAENRFQMAVDEDYYDSIQWTEADAQVLKERGQAPLVFNVIAQSVNWIIGSEKRGRTDFNILPRKKADAKPAEAKTKLLKYLSDVNRLPFHRSRSFEDTVKVGLGWIEDSYDDSTDGEPIYSRYESWRNVIFDSASTELDGTDMRYIFRPKWLDVDVACALVPDRADEIKKAAVAAERYGNYSEEDGDEAMDWAEFDRDTYSQSRTVSTHKRQRVRLIECWYRKPMRATKFLSGDLRGETLDESNPAHVEARDSGLYSLGERITMRVRVAIFTSRDLLFEGASPFRHNRFSLTPIWGFRRGRDNLPYGVIRWMRDIQDDINKRASKALYILSSNKVVMDEGAVEDIEEFREEVARPDAVLVKKPGKQIELNVDRELAAAHMDMMSRDIQMLQQVGGVTDELMGRSTNAVSGVAIQARQEQGTVATNKLFDNLRFAVQMQGEIQLSLIEQFVTEEKTFRITNERGKADFITVNDGLPENDIVRTKADFIIGESDWRATYRQAASEQLSQMIMKMPPQVGLVMLDLWADSTDLPNRDEIVKRIRQINGMRDPDATEPTPEELQQQQAAAEQAQAQKAMFMAELSEKQGKADKAQADAVAAQANADLRAAQAEQVRRQTVNTNVTSIAAAMEAATAIVTMPTIAKVGDAVLVEAGYENNGIAPAGGLHTPAPQQAVNPAAQGLPPQQPQPQQPQEPAVSPSPEQLNGAAPSNTGEPVTPDQTLQQ
6OIA , Knot 170 404 0.84 40 225 384 
GPPTSDDIFEREYKYGAHNYHPLPVALERGKGIYLWDVEGRKYFDFLSSYSAVNQGHCHPKIVNALKSQVDKLTLTSRAFYNNVLGEYEEYITKLFNYHKVLPMNTGVEAGETACKLARKWGYTVKGIQKYKAKIVFAAGNFWGRTLSAISSSTDPTSYDGFGPFMPGFDIIPYNDLPALERALQDPNVAAFMVEPIQGEAGVVVPDPGYLMGVRELCTRHQVLFIADEIQTGLARTGRWLAVDYENVRPDIVLLGKALSGGLYPVSAVLCDDDIMLTIKPGEHGSTYGGNPLGCRVAIAALEVLEEENLAENADKLGIILRNELMKLPSDVVTAVRGKGLLNAIVIKETKDWDAWKVCLRLRDNGLLAKPTHGDIIRFAPPLVIKEDELRESIEIINKTILSF

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 \(c\).

\(|P_{f(5ARK_1)}(2) \setminus P_{f(8JOV_1)}(2)|=25\), \(|P_{f(8JOV_1)}(2) \setminus P_{f(5ARK_1)}(2)|=207\). 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 1:11000000100000100010001100010101011001000111000010000000100110001101001000001000010100001110100000010001100000111000011110010101110100
Pair \(Z_2\) Length of longest common subsequence
5ARK_1,8JOV_1 232 5
5ARK_1,6OIA_1 213 3
8JOV_1,6OIA_1 137 4

Newick tree

 
[
	5ARK_1:12.34,
	[
		6OIA_1:68.5,8JOV_1:68.5
	]:53.84
]

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 \((0.85)(0.8)(\frac{915 }{\log_{20} 915}-\frac{134}{\log_{20}134})=217.\)
Status Protein1 Protein2 d d1/2
Query variables 5ARK_1 8JOV_1 273 159
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]