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Parikh vectors
5SNY_1 3QQZ_1 7RSI_1 Letter Amino acid
36 21 73 L Leucine
8 15 42 K Lycine
34 26 27 S Serine
19 17 20 T Threonine
7 8 27 Y Tyrosine
51 16 57 A Alanine
4 1 12 C Cysteine
10 7 23 H Histidine
18 12 23 F Phenylalanine
17 8 27 P Proline
2 2 4 W Tryptophan
24 14 34 V Valine
22 18 77 E Glutamic acid
52 14 21 G Glycine
21 22 29 I Isoleucine
7 12 26 Q Glutamine
14 3 15 M Methionine
29 8 35 R Arginine
15 19 20 N Asparagine
24 12 29 D Aspartic acid 

5SNY_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 2|Pseudomonas aeruginosa (287)
>3QQZ_1|Chain A|Putative uncharacterized protein yjiK|Escherichia coli (655817)
>7RSI_1|Chain A[auth B]|KIF-binding protein|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 \(w=f(c)\)
5SNY , Knot 171 414 0.83 40 212 384 
MSRRRVVITGMGMLSPLGLDVPSSWEGILAGRSGIAPIEHMDLSAYSTRFGGSVKGFNVEEYLSAKEARKLDLFIQYGLAASFQAVRDSGLEVTDANRERIGVSMGSGIGGLTNIENNCRSLFEQGPRRISPFFVPGSIINMVSGFLSIHLGLQGPNYALTTAQTTGTHSIGMAARNIAYGEADVMVAGGSEMAACGLGLGGFGAARALSTRNDEPTRASRPWDRDRDGFVLSDGSGALVLEELEHARARGARIYAELVGFGMSGDAFHMTAPPEDGAGAARCMKNALRDAGLDPRQVDYINAHGTSTPAGDIAEIAAVKSVFGEHAHALSMSSTKSMTGHLLGAAGAVEAIFSVLALRDQVAPPTINLDNPDEGCDLDLVAHEAKPRKIDVALSNSFGFGGTNGTLVFRRFAD
3QQZ , Knot 114 255 0.82 40 172 245 
SNASNHAASFQNYHATIDGKEIAGITNNISSLTWSAQSNTLFSTINKPAAIVEMTTNGDLIRTIPLDFVKDLETIEYIGDNQFVISDERDYAIYVISLTPNSEVKILKKIKIPLQESPTNCGFEGLAYSRQDHTFWFFKEKNPIEVYKVNGLLSSNELHISKDKALQRQFTLDDVSGAEFNQQKNTLLVLSHESRALQEVTLVGEVIGEMSLTKGSRGLSHNIKQAEGVAMDASGNIYIVSEPNRFYRFTPQSSH
7RSI , Knot 243 621 0.84 40 263 575 
MANVPWAEVCEKFQAALALSRVELHKNPEKEPYKSKYSARALLEEVKALLGPAPEDEDERPEAEDGPGAGDHALGLPAEVVEPEGPVAQRAVRLAVIEFHLGVNHIDTEELSAGEEHLVKCLRLLRRYRLSHDCISLCIQAQNNLGILWSEREEIETAQAYLESSEALYNQYMKEVGSPPLDPTERFLPEEEKLTEQERSKRFEKVYTHNLYYLAQVYQHLEMFEKAAHYCHSTLKRQLEHNAYHPIEWAINAATLSQFYINKLCFMEARHCLSAANVIFGQTGKISATEDTPEAEGEVPELYHQRKGEIARCWIKYCLTLMQNAQLSMQDNIGELDLDKQSELRALRKKELDEEESIRKKAVQFGTGELCDAISAVEEKVSYLRPLDFEEARELFLLGQHYVFEAKEFFQIDGYVTDHIEVVQDHSALFKVLAFFETDMERRCKMHKRRIAMLEPLTVDLNPQYYLLVNRQIQFEIAHAYYDMMDLKVAIADRLRDPDSHIVKKINNLNKSALKYYQLFLDSLRDPNKVFPEHIGEDVLRPAMLAKFRVARLYGKIITADPKKELENLATSLEHYKFIVDYCEKHPEAAQEIEVELELSKEMVSLLPTKMERFRTKMALT

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 \(c\).

\(|P_{f(5SNY_1)}(2) \setminus P_{f(3QQZ_1)}(2)|=98\), \(|P_{f(3QQZ_1)}(2) \setminus P_{f(5SNY_1)}(2)|=58\). 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 1:100001110111110111101100101111100111110010101000011101011010001010010010111001111010110001101001000011101101111100100000011001100101111110110110111010111011001100100010001111100110101011111100111011111111110110000001001001100000111100101111100100101011010101111110101101011100111110010011001110100100101010001110110111100111001011010000010101111111101110111100011110101001001001011100101001011100011111001011100110
Pair \(Z_2\) Length of longest common subsequence
5SNY_1,3QQZ_1 156 5
5SNY_1,7RSI_1 159 4
3QQZ_1,7RSI_1 173 4

Newick tree

 
[
	7RSI_1:84.69,
	[
		5SNY_1:78,3QQZ_1:78
	]:6.69
]

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 \((0.85)(0.8)(\frac{669 }{\log_{20} 669}-\frac{255}{\log_{20}255})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 5SNY_1 3QQZ_1 144 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]