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Parikh vectors
6APD_1 4OLB_1 6CZX_1 Letter Amino acid
11 32 6 H Histidine
44 53 33 K Lycine
20 39 19 F Phenylalanine
19 58 16 P Proline
20 28 12 Y Tyrosine
15 22 7 C Cysteine
22 57 13 Q Glutamine
29 41 28 E Glutamic acid
44 54 13 T Threonine
5 6 3 W Tryptophan
21 40 17 D Aspartic acid
55 61 31 L Leucine
10 21 11 M Methionine
56 43 25 S Serine
46 70 29 V Valine
28 54 28 A Alanine
18 57 10 R Arginine
42 22 18 N Asparagine
30 53 26 G Glycine
38 48 17 I Isoleucine 

6APD_1|Chains A, B, C|Fusion glycoprotein F0,Envelope glycoprotein|Respiratory syncytial virus (12814)
>4OLB_1|Chain A|Protein argonaute-2|Homo sapiens (9606)
>6CZX_1|Chains A, B, C, D|Phosphoserine aminotransferase 1, chloroplastic|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
6APD , Knot 228 573 0.84 40 254 527 
MELLILKANAITTILTAVTFCFASGQNITEEFYQSTCSAVSKGYLSALRTGWYTSVITIELSNIKEIKCNGTDAKVKLIKQELDKYKNAVTELQLLMQSTPATNNRARRELPRFMNYTLNNAKKTNVTLSKKRKRRFLGFLLGVGSAIASGVAVSKVLHLEGEVNKIKSALLSTNKAVVSLSNGVSVLTSKVLDLKNYIDKQLLPIVNKQSCSIPNIETVIEFQQKNNRLLEITREFSVNAGVTTPVSTYMLTNSELLSLINDMPITNDQKKLMSNNVQIVRQQSYSIMSIIKEEVLAYVVQLPLYGVIDTPCWKLHTSPLCTTNTKEGSNICLTRTDRGWYCDNAGSVSFFPQAETCKVQSNRVFCDTMNSLTLPSEVNLCNVDIFNPKYDCKIMTSKTDVSSSVITSLGAIVSCYGKTKCTASNKNRGIIKTFSNGCDYVSNKGVDTVSVGNTLYYVNKQEGKSLYVKGEPIINFYDPLVFPSDEFDASISQVNEKINQSLAFIRKSDELLSAIGGYIPEAPRDGQAYVRKDGEWVLLSTFLGSMEVLFQGPGHHHHHHHHSAWSHPQFEK
4OLB , Knot 326 859 0.85 40 325 795 
MYSGAGPALAPPAPPPPIQGYAFKPPPRPDFGTSGRTIKLQANFFEMDIPKIDIYHYELDIKPEKCPRRVNREIVEHMVQHFKTQIFGDRKPVFDGRKNLYTAMPLPIGRDKVELEVTLPGEGKDRIFKVSIKWVSCVSLQALHDALSGRLPSVPFETIQALDVVMRHLPSMRYTPVGRSFFTASEGCSNPLGGGREVWFGFHQSVRPSLWKMMLNIDVSATAFYKAQPVIEFVCEVLDFKSIEEQQKPLTDSQRVKFTKEIKGLKVEITHCGQMKRKYRVCNVTRRPASHQTFPLQQESGQTVECTVAQYFKDRHKLVLRYPHLPCLQVGQEQKHTYLPLEVCNIVAGQRCIKKLTDNQTSTMIRATARSAPDRQEEISKLMRSADFNTDPYVREFGIMVKDEMTDVTGRVLQPPSILYGGRNKAIATPVQGVWDMRNKQFHTGIEIKVWAIACFAPQRQCTEVHLKSFTEQLRKISRDAGMPIQGQPCFCKYAQGADSVEPMFRHLKNTYAGLQLVVVILPGKTPVYAEVKRVGDTVLGMATQCVQMKNVQRTTPQTLSNLCLKINVKLGGVNNILLPQGRPPVFQQPVIFLGADVTHPPAGDGKKPSIAAVVGSMDAHPNRYCATVRVQQHRQEIIQDLAAMVRELLIQFYKSTRFKPTRIIFYRDGVSEGQFQQVLHHELLAIREACIKLEKDYQPGITFIVVQKRHHTRLFCTDKNERVGKSGNIPAGTTVDTKITHPTEFDFYLCSHAGIQGTSRPSHYHVLWDDNRFSSDELQILTYQLCHTYVRCTRSVSIPAPAYYAHLVAFRARYHLVDKEHDSAEGSHTSGQSNGRDHQALAKAVQVHQDTLRTMYFA
6CZX , Knot 157 362 0.85 40 218 341 
SNARVFNFAAGPATLPENVLLKAQADLYNWRGSGMSVMEMSHRGKEFLSIIQKAESDLRQLLEIPQEYSVLFLQGGATTQFAALPLNLCKSDDTVDFVVTGSWGDKAVKEAKKYCKTNVIWSGKSEKYTKVPSFEELEQTPDAKYLHICANETIHGVEFKDYPVPKNGFLVADMSSNFCSKPVDVSKFGVIYGGAQKNVGPSGVTIVIIRKDLIGNAQDITPVMLDYKIHDENSSLYNTPPCFGIYMCGLVFEDLLEQGGLKEVEKKNQRKADLLYNAIEESNGFFRCPVEKSVRSLMNVPFTLEKSELEAEFIKEAAKEKMVQLKGHRSVGGMRASIYNAMPLAGVEKLVAFMKDFQAKHA

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 \(c\).

\(|P_{f(6APD_1)}(2) \setminus P_{f(4OLB_1)}(2)|=31\), \(|P_{f(4OLB_1)}(2) \setminus P_{f(6APD_1)}(2)|=102\). 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 1:101111010110011011010110100100010000001100101011001100011010100100100010010101100010000011001011100011000010001101100010010000101000000011111111101110111100110101010010011100001110100110110001101000100011111000000110100110100000011010001010111001100011000011011001110000001100010110000001101100011101101110111001010100011000000010010100000110000110101110100001000011000100101100101001011010000011000001000110011111000100000100000111001001000100011001011001001000010010101011101001111100010101001000100011110000011011110110110010101000101111001110101110111000000000110010100
Pair \(Z_2\) Length of longest common subsequence
6APD_1,4OLB_1 133 4
6APD_1,6CZX_1 152 4
4OLB_1,6CZX_1 157 4

Newick tree

 
[
	6CZX_1:80.52,
	[
		6APD_1:66.5,4OLB_1:66.5
	]:14.02
]

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 \((0.85)(0.8)(\frac{1432 }{\log_{20} 1432}-\frac{573}{\log_{20}573})=217.\)
Status Protein1 Protein2 d d1/2
Query variables 6APD_1 4OLB_1 278 229
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]