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Parikh vectors
3KDS_1 1PRH_1 3LJR_1 Letter Amino acid
17 30 9 T Threonine
38 29 13 V Valine
33 31 14 R Arginine
26 23 11 D Aspartic acid
52 32 18 E Glutamic acid
44 61 37 L Leucine
32 25 12 K Lycine
15 27 11 S Serine
45 23 22 A Alanine
6 27 15 Q Glutamine
35 27 14 I Isoleucine
1 9 5 W Tryptophan
3 13 3 C Cysteine
39 42 16 G Glycine
9 15 5 M Methionine
16 37 9 F Phenylalanine
8 27 7 Y Tyrosine
16 18 3 N Asparagine
9 18 5 H Histidine
21 40 15 P Proline 

3KDS_1|Chains A[auth E], B[auth F], C[auth G]|Cell division protein FtsH|Thermotoga maritima (2336)
>1PRH_1|Chains A, B|PROSTAGLANDIN H2 SYNTHASE-1|Ovis aries (9940)
>3LJR_1|Chains A, B|GLUTATHIONE S-TRANSFERASE|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 \(w=f(c)\)
3KDS , Knot 187 465 0.82 40 221 432 
MATMYKPSGNKRVTFKDVGGAEEAIEELKEVVEFLKDPSKFNRIGARMPKGILLVGPPGTGATLLARAVAGEANVPFFHISGSDFVELFVGVGAARVRDLFAQAKAHAPCIVFIDEIDAVGRHRGAGLGGGHDEREQTLNQLLVEMDGFDSKEGIIVMAATNRPDILDPALLRPGRFDKKIVVDPPDMLGRKKILEIHTRNKPLAEDVNLEIIAKRTPGFVGADLENLVNEAALLAAREGRDKITMKDFEEAIDRVIAGPARKSLLISPAEKRIIAYHEAGHAVVSTVVPNGEPVHRISIIPRGYKALGYTLHLPEEDKYLVSRNELLDKLTALLGGRAAEEVVFGDVTSGAANDIERATEIARNMVCQLGMSEELGPLAWGKEEQEVFLGKEITRLRNYSEEVASKIDEEVKKIVTNCYERAKEIIRKYRKQLDNIVEILLEKETIEGDELRRILSEEFEKVVE
1PRH , Knot 232 554 0.88 40 277 537 
VNPCCYYPCQHQGICVRFGLDRYQCDCTRTGYSGPNCTIPEIWTWLRTTLRPSPSFIHFLLTHGRWLWDFVNATFIRDTLMRLVLTVRSNLIPSPPTYNIAHDYISWESFSNVSYYTRILPSVPRDCPTPMGTKGKKQLPDAEFLSRRFLLRRKFIPDPQGTNLMFAFFAQHFTHQFFKTSGKMGPGFTKALGHGVDLGHIYGDNLERQYQLRLFKDGKLKYQMLNGEVYPPSVEEAPVLMHYPRGIPPQSQMAVGQEVFGLLPGLMLYATIWLREHQRVCDLLKAEHPTWGDEQLFQTAKLILIGETIKIVIEEYVQQLSGYFLQLKFDPELLFGAQFQYRNRIAMEFNQLYHWHPLMPDSFRVGPQDYSYEQFLFNTSMLVDYGVEALVDAFSRQPAGRIGGGRNIDHHILHVAVDVIKESRVLRLQPFNEYRKRFGMKPYTSFQELTGEKEMAAELEELYGDIDALEFYPGLLLEKCHPNSIFGESMIEMGAPFSLKGLLGNPICSPEYWKASTFGGEVGFNLVKTATLKKLVCLNTKTCPYVSFHVPDPR
3LJR , Knot 110 244 0.82 40 165 233 
MGLELFLDLVSQPSRAVYIFAKKNGIPLELRTVDLVKGQHKSKEFLQINSLGKLPTLKDGDFILTESSAILIYLSCKYQTPDHWYPSDLQARARVHEYLGWHADCIRGTFGIPLWVQVLGPLIGVQVPEEKVERNRTAMDQALQWLEDKFLGDRPFLAGQQVTLADLMALEELMQPVALGYELFEGRPRLAAWRGRVEAFLGAELCQEAHSIILSILEQAAKKTLPTPSPEAYQAMLLRIARIP

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 \(c\).

\(|P_{f(3KDS_1)}(2) \setminus P_{f(1PRH_1)}(2)|=49\), \(|P_{f(1PRH_1)}(2) \setminus P_{f(3KDS_1)}(2)|=105\). 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 1:110100101000101001111001100100110110010010011101101111111110110111011110101111010100110111111110100111010101101111001011100011111110000000100111010110000111111100010110111101101000111011011100011010000011100101011100011111101001100111111001000101001001100111111000111011000111000110111001110101100101110100111001011000001100001100101111101100111101001110010010011001100111000111111100000111100100100000011001000100110000001001100000010011011100001010010011000100110
Pair \(Z_2\) Length of longest common subsequence
3KDS_1,1PRH_1 154 4
3KDS_1,3LJR_1 150 4
1PRH_1,3LJR_1 198 4

Newick tree

 
[
	1PRH_1:92.79,
	[
		3KDS_1:75,3LJR_1:75
	]:17.79
]

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 \((0.85)(0.8)(\frac{1019 }{\log_{20} 1019}-\frac{465}{\log_{20}465})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 3KDS_1 1PRH_1 189 169.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]