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Parikh vectors
8JHY_1 3DWD_1 9DEF_1 Letter Amino acid
13 4 5 Q Glutamine
6 4 6 P Proline
8 5 0 W Tryptophan
22 11 18 R Arginine
17 5 4 N Asparagine
30 11 12 G Glycine
10 9 1 K Lycine
8 4 2 M Methionine
12 6 8 F Phenylalanine
30 9 19 A Alanine
30 6 13 D Aspartic acid
10 13 23 E Glutamic acid
18 3 8 I Isoleucine
32 11 16 L Leucine
30 15 7 S Serine
14 5 0 C Cysteine
12 8 4 H Histidine
28 3 6 T Threonine
8 3 0 Y Tyrosine
18 12 12 V Valine 

8JHY_1|Chain A[auth B]|Guanine nucleotide-binding protein G(I)/G(S)/G(T) subunit beta-1|Homo sapiens (9606)
>3DWD_1|Chains A, B|ADP-ribosylation factor GTPase-activating protein 1|Homo sapiens (9606)
>9DEF_1|Chain A|Win|synthetic construct (32630)
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 \(w=f(c)\)
8JHY , Knot 151 356 0.83 40 199 331 
MHHHHLEVLFQGPGSSGSELDQLRQEAEQLKNQIRDARKACADATLSQITNNIDPVGRIQMRTRRTLRGHLAKIYAMHWGTDSRLLVSASQDGKLIIWDSYTTNKVHAIPLRSSWVMTCAYAPSGNYVACGGLDNICSIYNLKTREGNVRVSRELAGHTGYLSCCRFLDDNQIVTSSGDTTCALWDIETGQQTTTFTGHTGDVMSLSLAPDTRLFVSGACDASAKLWDVREGMCRQTFTGHESDINAICFFPNGNAFATGSDDATCRLFDLRADQELMTYSHDNIICGITSVSFSKSGRLLLAGYDDFNCNVWDALKADRAGVLAGHDNRVSCLGVTDDGMAVATGSWDSFLKIWN
3DWD , Knot 73 147 0.82 40 117 139 
MGSSHHHHHHSSGLVPRGSMASPRTRKVLKEVRVQDENNVCFECGAFNPQWVSVTYGIWICLECSGRHRGLGVHLSFVRSVTMDKWKDIELEKMKAGGNAKFREFLESQEDYDPCWSLQEKYNSRAAALFRDKVVALAEGREWSLES
9DEF , Knot 74 164 0.76 34 101 152 
MSGMTEEELDQAVEDFLRVHSELVHRLAGDPPDELFQRLDRFVTDAIIEGNPERRDEIKADLARAARVFGEALERDITTPEDFNAFLRELGPEAVELVSTFTQQFVDVIRGDPQAVAEHLNISLEDVARLAEAGEAAIERGEGASLGVHRELRRIEARRNSGSG

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 \(c\).

\(|P_{f(8JHY_1)}(2) \setminus P_{f(3DWD_1)}(2)|=125\), \(|P_{f(3DWD_1)}(2) \setminus P_{f(8JHY_1)}(2)|=43\). 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 1:10000101110111001001001000100100010010010101010010001011101010000010101101011011000011101000101111000000010111100011100101101001101110010010010000101010001110010100001100001100010000111010010000010100101101011100011101100101011010011000010100001011011101011101000100011010100011000000110110010100010111110001000110110100111111000010011100011111010100110110
Pair \(Z_2\) Length of longest common subsequence
8JHY_1,3DWD_1 168 4
8JHY_1,9DEF_1 160 4
3DWD_1,9DEF_1 130 4

Newick tree

 
[
	8JHY_1:86.96,
	[
		9DEF_1:65,3DWD_1:65
	]:21.96
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{147}{\log_{20}147})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 8JHY_1 3DWD_1 131 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]