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Parikh vectors
6UBD_1 8IAV_1 3VGE_1 Letter Amino acid
9 39 42 I Isoleucine
11 37 47 K Lycine
15 14 21 P Proline
28 25 32 S Serine
16 24 28 N Asparagine
19 11 15 Q Glutamine
24 44 43 G Glycine
14 46 41 V Valine
7 45 48 E Glutamic acid
7 17 6 M Methionine
8 8 39 Y Tyrosine
4 28 23 R Arginine
19 35 40 D Aspartic acid
11 17 32 F Phenylalanine
20 33 39 L Leucine
13 35 18 T Threonine
12 2 7 W Tryptophan
23 49 21 A Alanine
4 1 3 C Cysteine
14 11 13 H Histidine 

6UBD_1|Chain A|Glyco_hydro_cc domain-containing protein|Trichoderma gamsii (398673)
>8IAV_1|Chains A, B, C, D|Pyruvate kinase|Streptococcus pneumoniae R6 (171101)
>3VGE_1|Chain A|Malto-oligosyltrehalose trehalohydrolase|Sulfolobus solfataricus (2287)
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 \(w=f(c)\)
6UBD , Knot 120 278 0.81 40 174 259 
MGSSHHHHHHSSGLVPAGSHMEFAQHPLTPNGRKAGSAGDTALAFWKDHLSWWHDWTPAPSSPKGAGLVPVSMLWGGGNNGQKDAQRLQQFEHLNSTPAYVMGFNEPDCSGADVSADIDVNTGVSLWNSLIAPMGQKGAALGSPAMCRQKDESWLKQFNQQQLTKSWDFTSIHIFKSDMTGVQADIDYYWNTYQKPLWVTEFACVFDQNNFTPCTDQNQINQWISDIVDLFEANEHVLAYAYTDGLGLGSVWPPVNSDGSLSQSGQAYLNAISKYHSR
8IAV , Knot 203 521 0.81 40 229 477 
MGSSHHHHHHSSGLVPRGSHMNKRVKIVATLGPAVEIRGGKKFGEDGYWGEKLDVEASAKNIAKLIEAGANTFRFNFSHGDHQEQGERMATVKLAEKIAGKKVGFLLDTKGPEIRTELFEGEAKEYSYKTGEKIRVATKQGIKSTREVIALNVAGALDIYDDVEVGRQVLVDDGKLGLRVVAKDDATREFEVEVENDGIIAKQKGVNIPNTKIPFPALAERDNDDIRFGLEQGINFIAISFVRTAKDVNEVRAICEETGNGHVQLFAKIENQQGIDNLDEIIEAADGIMIARGDMGIEVPFEMVPVYQKMIIKKVNAAGKVVITATNMLETMTEKPRATRSEVSDVFNAVIDGTDATMLSGESANGKYPLESVTTMATIDKNAQALLNEYGRLDSDSFERNSKTEVMASAVKDATSSMDIKLVVTLTKTGHTARLISKYRPNADILALTFDELTERGLMLNWGVIPMLTDAPSSTDDMFEIAERKAVEAGLVESGDDIVIVAGVPVGEAVRTNTMRIRTVR
3VGE , Knot 219 558 0.82 40 244 514 
TFAYKIDGNEVIFTLWAPYQKSVKLKVLEKGLYEMERDEKGYFTITLNNVKVRDRYKYVLDDASEIPDPASRYQPEGVHGPSQIIQESKEFNNETFLKKEDLIIYEIHVGTFTPEGTFEGVIRKLDYLKDLGITAIEIMPIAQFPGKRDWGYDGVYLYAVQNSYGGPEGFRKLVDEAHKKGLGVILDVVYNHVGPEGNYMVKLGPYFSQKYKTPWGLTFNFDDAESDEVRKFILENVEYWIKEYNVDGFRLSAVHAIIDTSPKHILEEIADVVHKYNRIVIAESDLNDPRVVNPKEKCGYNIDAQWVDDFHHSIHAYLTGERQGYYTDFGNLDDIVKSYKDVFVYDGKYSNFRRKTHGEPVGELDGCNFVVYIQNHDQVGNRGKGERIIKLVDRESYKIAAALYLLSPYIPMIFMGEEYGEENPFYFFSDFSDSKLIQGVREGRKKENGQDTDPQDESTFNASKLSWKIDEEIFSFYKILIKMRKELSIACDRRVNVVNGENWLIIKGREYFSLYVFSKSSIEVKYSGTLLLSSNNSFPQHIEEGKYEFDKGFALYKL

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 \(c\).

\(|P_{f(6UBD_1)}(2) \setminus P_{f(8IAV_1)}(2)|=68\), \(|P_{f(8IAV_1)}(2) \setminus P_{f(6UBD_1)}(2)|=123\). 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 1:11000000000011111100101100110101001101100111110001011001011100101111111011111100100010010010010001101111001000110101010100110110011111100111110111000000011001000010001010010110001011010100010000011110011011000010100000010011001101101000111010001111101111100010100010101011000000
Pair \(Z_2\) Length of longest common subsequence
6UBD_1,8IAV_1 191 16
6UBD_1,3VGE_1 188 3
8IAV_1,3VGE_1 147 4

Newick tree

 
[
	6UBD_1:10.84,
	[
		3VGE_1:73.5,8IAV_1:73.5
	]:27.34
]

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 \((0.85)(0.8)(\frac{799 }{\log_{20} 799}-\frac{278}{\log_{20}278})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 6UBD_1 8IAV_1 178 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]