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Parikh vectors
8XLA_1 6ZJS_1 8TBB_1 Letter Amino acid
19 37 2 Q Glutamine
28 106 9 G Glycine
11 14 1 M Methionine
3 5 6 C Cysteine
22 70 7 E Glutamic acid
7 24 6 Y Tyrosine
15 16 5 K Lycine
17 65 7 P Proline
1 26 3 W Tryptophan
23 89 11 V Valine
24 72 6 R Arginine
10 28 0 H Histidine
25 26 6 I Isoleucine
52 90 8 L Leucine
21 58 4 S Serine
17 47 5 T Threonine
30 111 6 A Alanine
19 22 8 N Asparagine
25 69 6 D Aspartic acid
18 35 4 F Phenylalanine 

8XLA_1|Chains A, B, C[auth E], D[auth F]|Beta sliding clamp|Neisseria gonorrhoeae FA 1090 (242231)
>6ZJS_1|Chain A|Beta-galactosidase|Arthrobacter sp. 32cB (1492190)
>8TBB_1|Chain A[auth D]|T-cell immunoglobulin mucin receptor 3|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 \(w=f(c)\)
8XLA , Knot 162 387 0.83 40 208 369 
MGSSHHHHHHSSGLVPRGSHMLILQAERDSLLKPLQAFTGIVERLHTLPILSNVLIEGRGGQTKLLATDLEIQIDTAGPEGGAGDFRITTNAKKFQDILRALPAGALVSLDWDDNRLTLKAGKSRFALQTLPAADFPMMNVGEDISATFSLGQERFKTMLSQVQYSMAVQDIRYYLNGLLMQVEGSQLRLVATDGHRLAYAACAIDADLPRAEVILPRKTVLELFKLLNNPDDPIQIELLDKQVRFQCNGTTIVSKVIDGKFPDFNRVIPLDNDKIFVLSRAELLGALERVSILANEKFRGARLFLQPGLLSVVCSNNEQEEAREEIEIAYQGGELEVGFNIGYLMDVLRNIHSDDMQLAFGDANRSTLFTVPNNPNFKYIVMPMRI
6ZJS , Knot 361 1010 0.82 40 292 869 
MSVETPSALADSSPHTAPGSAGRSLELGAADIQDLESFEAGRGALPARAYLQSDAPRLSLNGEWQFRLSPGSRVAPDDGWQLGEALNGFESLPVPSSWPMHGHGAPAYTNVQFPFAVEPPHVPEANPIGDHLVVFEAGPEFFPHALLRFDGIESAGTVWLNGVELGTTRGSRLAHEFDVSGILEQGENTLAVRVAQFSAASYVEDQDMWWLPGIFRDVTLQARPAAGIDDVFVHAGYDHITGEGILKVEASRGGQAIDAVVRVPELALELAAGTEVRVPAVEPWSAEVPKLYEAAVSAAGESVALQIGFRSIAIEDAQFKVNGRRILLRGVNRHEHHPRLGRVVPRDVVEAELRLMKQHNINAIRTSHYPPHPQFLALADQLGFYVVLECDLETHGFESAGWAQNPSDDPQWEDALVDRMRRTVERDKNHASVVMWSLGNQAGTGRNLAAMSRWTKDRDPSRPIHYEGDWSSEHVDVYSRMYASQAETALIGQGIEPALNDAALDARRRAMPFVLCEYVHAMGNGPGGMSEYQALFEKYPRLMGGFVWEWLEHGITVSTADGVDHYGYGGDFGEEVHDGNFVTDGLVDADRRPRPGLLDFKKVIEPLRIDVARDWTGFTLRNGQDFADTSAFSFRYEVEADGGALDGGTVDVAPVAPQSETVVELPGSVAALAAGLSDGRPAVLTVRAVLGADSAWADAGHEVAWGQSVREPGAPVPPAPVEPVQVQDSELTLGPVVFSRATGMPTSIGGVPVEKLGLTLWWAPTDNDLGREWGGADERPLATQWKDAGLNRLHTRLLGISANPGQDGGETLTVRTRVSAADKQYGVLVDYTWSTDGETVGLRTQVRRDGTWVNRGFEVEWARIGLEFVLGEETELVSWFGQGPHQSYPDTGQGARAGWFSLPLAKMDVEYVRPQECGARSGSRSAALQLGGRTLEICGDPFALTVRPYSQDVLDAAAHRPDLKADGRTYLYVDHALRGVGTAACGPGVLEQYRLKPRDADFILTLKVRS
8TBB , Knot 58 110 0.82 38 96 108 
SEVEYRAEVGQNAYLPCFYTPAAPGNLVPVCWGKGACPVFECGNVVLRTDERDVNYWTSRYWLNGDFRKGDVSLTIENVTLADSGIYCCRIQIPGIMNDEKFNLKLVIKP

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 \(c\).

\(|P_{f(8XLA_1)}(2) \setminus P_{f(6ZJS_1)}(2)|=36\), \(|P_{f(6ZJS_1)}(2) \setminus P_{f(8XLA_1)}(2)|=120\). 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 1:110000000000111101001111010000110110110111001001111001110101100011100101010011101111010100010010011011111111010100001010110001110011110111101100101010110001001100100011100100010111101010010111001001101101101011010111100011011011001001101011000101000100110011010110100111100001111001011111001011100010110111011110110000000010001011001101011101101101100100001011110100001101100101001111101
Pair \(Z_2\) Length of longest common subsequence
8XLA_1,6ZJS_1 156 4
8XLA_1,8TBB_1 186 4
6ZJS_1,8TBB_1 238 4

Newick tree

 
[
	8TBB_1:11.79,
	[
		8XLA_1:78,6ZJS_1:78
	]:36.79
]

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 \((0.85)(0.8)(\frac{1397 }{\log_{20} 1397}-\frac{387}{\log_{20}387})=260.\)
Status Protein1 Protein2 d d1/2
Query variables 8XLA_1 6ZJS_1 327 224
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]