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Parikh vectors
2ZJG_1 1TBG_1 9DVR_1 Letter Amino acid
10 8 19 W Tryptophan
27 30 38 D Aspartic acid
9 8 22 H Histidine
31 10 41 K Lycine
10 8 11 M Methionine
20 11 33 F Phenylalanine
25 5 34 P Proline
28 30 43 A Alanine
11 12 31 Q Glutamine
14 17 38 N Asparagine
8 14 11 C Cysteine
22 9 43 E Glutamic acid
27 18 51 I Isoleucine
34 30 52 L Leucine
30 17 45 V Valine
14 22 32 R Arginine
24 27 45 G Glycine
29 28 58 S Serine
22 28 40 T Threonine
15 8 60 Y Tyrosine 

2ZJG_1|Chains A, B|Kynurenine-oxoglutarate transaminase 3|Mus musculus (10090)
>1TBG_1|Chains A, C[auth B], E[auth C], G[auth D]|TRANSDUCIN|Bos taurus (9913)
>9DVR_1|Chains A, B|Antiplasmin-cleaving enzyme FAP, soluble form|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 \(w=f(c)\)
2ZJG , Knot 176 410 0.86 40 242 397 
NAKRIEGLDSNVWVEFTKLAADPSVVNLGQGFPDISPPSYVKEELSKAAFIDNMNQYTRGFGHPALVKALSCLYGKIYQRQIDPNEEILVAVGAYGSLFNSIQGLVDPGDEVIIMVPFYDCYEPMVRMAGAVPVFIPLRSKPTDGMKWTSSDWTFDPRELESKFSSKTKAIILNTPHNPLGKVYTRQELQVIADLCVKHDTLCISDEVYEWLVYTGHTHVKIATLPGMWERTITIGSAGKTFSVTGWKLGWSIGPAHLIKHLQTVQQNSFYTCATPLQAALAEAFWIDIKRMDDPECYFNSLPKELEVKRDRMVRLLNSVGLKPIVPDGGYFIIADVSSLGADLSDMNSDEPYDYKFVKWMTKHKKLTAIPVSAFCDSKSKPHFEKLVRFCFIKKDSTLDAAEEIFRAWN
1TBG , Knot 146 340 0.83 40 192 317 
MSELDQLRQEAEQLKNQIRDARKACADATLSQITNNIDPVGRIQMRTRRTLRGHLAKIYAMHWGTDSRLLVSASQDGKLIIWDSYTTNKVHAIPLRSSWVMTCAYAPSGNYVACGGLDNICSIYNLKTREGNVRVSRELAGHTGYLSCCRFLDDNQIVTSSGDTTCALWDIETGQQTTTFTGHTGDVMSLSLAPDTRLFVSGACDASAKLWDVREGMCRQTFTGHESDINAICFFPNGNAFATGSDDATCRLFDLRADQELMTYSHDNIICGITSVSFSKSGRLLLAGYDDFNCNVWDALKADRAGVLAGHDNRVSCLGVTDDGMAVATGSWDSFLKIWN
9DVR , Knot 293 747 0.86 40 310 694 
RPSRVHNSEENTMRALTLKDILNGTFSYKTFFPNWISGQEYLHQSADNNIVLYNIETGQSYTILSNRTMKSVNASNYGLSPDRQFVYLESDYSKLWRYSYTATYYIYDLSNGEFVRGNELPRPIQYLCWSPVGSKLAYVYQNNIYLKQRPGDPPFQITFNGRENKIFNGIPDWVYEEEMLATKYALWWSPNGKFLAYAEFNDTDIPVIAYSYYGDEQYPRTINIPYPKAGAKNPVVRIFIIDTTYPAYVGPQEVPVPAMIASSDYYFSWLTWVTDERVCLQWLKRVQNVSVLSICDFREDWQTWDCPKTQEHIEESRTGWAGGFFVSTPVFSYDAISYYKIFSDKDGYKHIHYIKDTVENAIQITSGKWEAINIFRVTQDSLFYSSNEFEEYPGRRNIYRISIGSYPPSKKCVTCHLRKERCQYYTASFSDYAKYYALVCYGPGIPISTLHDGRTDQEIKILEENKELENALKNIQLPKEEIKKLEVDEITLWYKMILPPQFDRSKKYPLLIQVYGGPCSQSVRSVFAVNWISYLASKEGMVIALVDGRGTAFQGDKLLYAVYRKLGVYEVEDQITAVRKFIEMGFIDEKRIAIWGWSYGGYVSSLALASGTGLFKCGIAVAPVSSWEYYASVYTERFMGLPTKDDNLEHYKNSTVMARAEYFRNVDYLLIHGTADDNVHFQNSAQIAKALVNAQVDFQAMWYSDQNHGLSGLSTNHLYTHMTHFLKQCFSLSDTGHHHHHHHHGGQ

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 \(c\).

\(|P_{f(2ZJG_1)}(2) \setminus P_{f(1TBG_1)}(2)|=117\), \(|P_{f(1TBG_1)}(2) \setminus P_{f(2ZJG_1)}(2)|=67\). 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 1:01001011000111010011101011011011101011001000100111100100000111011110110010101000010100011111110101100101110110011111110000011101111111111100010011010000101010010001000001111001001110100000101110101000010100010011100100010110111110001011011001010110111011110110010010000100010110111101111010010010001001100101000011011001110111101101111010011101001000010000110110000010111101100000010100110101100000101100110110
Pair \(Z_2\) Length of longest common subsequence
2ZJG_1,1TBG_1 184 5
2ZJG_1,9DVR_1 150 4
1TBG_1,9DVR_1 176 5

Newick tree

 
[
	1TBG_1:94.50,
	[
		2ZJG_1:75,9DVR_1:75
	]:19.50
]

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 \((0.85)(0.8)(\frac{750 }{\log_{20} 750}-\frac{340}{\log_{20}340})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZJG_1 1TBG_1 147 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]