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Parikh vectors
5LMX_1 8GPP_1 2JCS_1 Letter Amino acid
79 7 14 R Arginine
98 10 12 N Asparagine
105 12 8 D Aspartic acid
96 17 9 G Glycine
92 18 14 I Isoleucine
143 13 23 L Leucine
37 12 6 H Histidine
43 6 7 M Methionine
132 17 16 S Serine
104 8 13 T Threonine
103 14 13 A Alanine
65 10 10 Q Glutamine
130 9 15 K Lycine
64 9 8 P Proline
8 1 4 W Tryptophan
54 6 12 Y Tyrosine
24 4 5 C Cysteine
111 11 23 E Glutamic acid
61 5 7 F Phenylalanine
115 19 11 V Valine 

5LMX_1|Chain A|DNA-directed RNA polymerase I subunit RPA190|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>8GPP_1|Chains A, B, C|Carbonic anhydrase|Acinetobacter baumannii (470)
>2JCS_1|Chains A, B|DEOXYNUCLEOSIDE KINASE|DROSOPHILA MELANOGASTER (7227)
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 \(w=f(c)\)
5LMX , Knot 571 1664 0.84 40 354 1390 
MDISKPVGSEITSVDFGILTAKEIRNLSAKQITNPTVLDNLGHPVSGGLYDLALGAFLRNLCSTCGLDEKFCPGHQGHIELPVPCYNPLFFNQLYIYLRASCLFCHHFRLKSVEVHRYACKLRLLQYGLIDESYKLDEITLGSLNSSMYTDDEAIEDNEDEMDGEGSKQSKDISSTLLNELKSKRSEYVDMAIAKALSDGRTTERGSFTATVNDERKKLVHEFHKKLLSRGKCDNCGMFSPKFRKDGFTKIFETALNEKQITNNRVKGFIRQDMIKKQKQAKKLDGSNEASANDEESFDVGRNPTTRPKTGSTYILSTEVKNILDTVFRKEQCVLQYVFHSRPNLSRKLVKADSFFMDVLVVPPTRFRLPSKLGEEVHENSQNQLLSKVLTTSLLIRDLNDDLSKLQKDKVSLEDRRVIFSRLMNAFVTIQNDVNAFIDSTKAQGRTSGKVPIPGVKQALEKKEGLFRKHMMGKRVNYAARSVISPDPNIETNEIGVPPVFAVKLTYPEPVTAYNIAELRQAVINGPDKWPGATQIQNEDGSLVSLIGMSVEQRKALANQLLTPSSNVSTHTLNKKVYRHIKNRDVVLMNRQPTLHKASMMGHKVRVLPNEKTLRLHYANTGAYNADFDGDEMNMHFPQNENARAEALNLANTDSQYLTPTSGSPVRGLIQDHISAGVWLTSKDSFFTREQYQQYIYGCIRPEDGHTTRSKIVTLPPTIFKPYPLWTGKQIITTVLLNVTPPDMPGINLISKNKIKNEYWGKGSLENEVLFKDGALLCGILDKSQYGASKYGIVHSLHEVYGPEVAAKVLSVLGRLFTNYITATAFTCGMDDLRLTAEGNKWRTDILKTSVDTGREAAAEVTNLDKDTPADDPELLKRLQEILRDNNKSGILDAVTSSKVNAITSQVVSKCVPDGTMKKFPCNSMQAMALSGAKGSNVNVSQIMCLLGQQALEGRRVPVMVSGKTLPSFKPYETDAMAGGYVKGRFYSGIKPQEYYFHCMAGREGLIDTAVKTSRSGYLQRCLTKQLEGVHVSYDNSIRDADGTLVQFMYGGDAIDITKESHMTQFEFCLDNYYALLKKYNPSALIEHLDVESALKYSKKTLKYRKKHSKEPHYKQSVKYDPVLAKYNPAKYLGSVSENFQDKLESFLDKNSKLFKSSDGVNEKKFRALMQLKYMRSLINPGEAVGIIASQSVGEPSTQMTLNTFHFAGHGAANVTLGIPRLREIVMTASAAIKTPQMTLPIWNDVSDEQADTFCKSISKVLLSEVIDKVIVTETTGTSNTAGGNAARSYVIHMRFFDNNEYSEEYDVSKEELQNVISNQFIHLLEAAIVKEIKKQKRTTGPDIGVAVPRLQTDVANSSSNSKRLEEDNDEEQSHKKTKQAVSYDEPDEDEIETMREAEKSSDEEGIDSDKESDSDSEDEDVDMNEQINKSIVEANNNMNKVQRDRQSAIISHHRFITKYNFDDESGKWCEFKLELAADTEKLLMVNIVEEICRKSIIRQIPHIDRCVHPEPENGKRVLVTEGVNFQAMWDQEAFIDVDGITSNDVAAVLKTYGVEAARNTIVNEINNVFSRYAISVSFRHLDLIADMMTRQGTYLAFNRQGMETSTSSFMKMSYETTCQFLTKAVLDNEREQLDSPSARIVVGKLNNVGTGSFDVLAKVPNAA
8GPP , Knot 98 208 0.83 40 154 200 
MGHHHHHHPCYSIDGVIPVVSPDAFVHPTAVLIGDVIIEAGVYVGPFASLRADFGRIHINQNANIQDSCTVHGFPQSVTLVEEMGHIGHGAILHGCRIGKNVLVGMNSVILDYAEIGENTIIGANSLVKTKDIIPANVLAMGSPAKVARDLSEQEKKWKTRGTQEYMELAQRCLNSMQEVQPLSSESDDRLTYKDFSSSNYQIKQDSV
2JCS , Knot 106 230 0.83 40 162 221 
MAEAASCARKGTKYAEGTQPFTVLIEGNIGSGKTTYLNHFEKYKNDICLLTDPVEKWRNVNGVNLLELMYKDPKKWAMPFQSYVTLTMLQSHTAPTNKKLKIMERSIFSARYCFVENMRRNGSLEQGMYNTLEEWYKFIEESIHVQADLIIYLRTSPEVAYERIRQRARSEESCVPLKYLQELHELHEDWLIHQRRPQSCKVLVLDADLNLENIGTEYQRSESSIFDAIS

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 \(c\).

\(|P_{f(5LMX_1)}(2) \setminus P_{f(8GPP_1)}(2)|=204\), \(|P_{f(8GPP_1)}(2) \setminus P_{f(5LMX_1)}(2)|=4\). 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 1:10100111001001011110100100101001001011001101101110011111110010000110001011001010111100011110010101010011000101001010001001011001110000010010110100010000011000000101010000001000110010000000101111011001000001010101000000110010001100100000111010100011001100110000100001011100011000001001010001010000010110010001001000110001001100110000011001100010100011010011101111110010110011001000000011001100011100100010010000101000011100110111010001011100001010001011111100110000111000111001001100110101010000111111111010010110100110100111011001111001000010110111101000011100110100010000100010001000011110001010010111001011100001010010011001010100101011000010101101100000010100101101110001011111000001100000000101010100100000011011101101011101001100111010110111101100001000011010100011100111101110000011000111001001011011101101110110001010110011001010101001000110001001001110100100001100101100100110000001110110000101100011000110101001100010111101101001010011011100110100111110100110101000011111010101001101000010011100111001100000101000100010110100000100101011011011011010000010010101000011100001011100101001100000010000000001000001000111100011001101000100010011000001100001100001011101001001101101111110001101000101001011101110101111010011101011100101011110010000100100010011100110011100001000011101100011010110000000000100001001100011011011110010000000110111111010001100000000100000000000000011000010000100100100000001100000000000000101000100011010001001000000111000011000010000101001010111000011110110010000110011010001010100100111001101011100011101011000011111000110110001100100110001101010010111011000100111000110000001101000000011001110000001001010111101001101010111011011
Pair \(Z_2\) Length of longest common subsequence
5LMX_1,8GPP_1 208 4
5LMX_1,2JCS_1 214 4
8GPP_1,2JCS_1 158 3

Newick tree

 
[
	5LMX_1:11.97,
	[
		8GPP_1:79,2JCS_1:79
	]:33.97
]

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 \((0.85)(0.8)(\frac{1872 }{\log_{20} 1872}-\frac{208}{\log_{20}208})=426.\)
Status Protein1 Protein2 d d1/2
Query variables 5LMX_1 8GPP_1 545 303
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]