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Parikh vectors
2DAP_1 2IID_1 7GXR_1 Letter Amino acid
34 36 8 A Alanine
20 39 15 L Leucine
15 23 7 F Phenylalanine
22 26 7 T Threonine
2 5 0 W Tryptophan
14 21 3 P Proline
19 23 9 R Arginine
11 25 7 N Asparagine
3 6 5 C Cysteine
15 17 4 Q Glutamine
25 35 5 G Glycine
13 13 3 H Histidine
8 6 6 M Methionine
27 31 10 V Valine
29 35 8 D Aspartic acid
16 31 6 E Glutamic acid
15 32 9 I Isoleucine
11 36 4 K Lycine
13 31 9 S Serine
8 27 3 Y Tyrosine 

2DAP_1|Chain A|DIAMINOPIMELIC ACID DEHYDROGENASE|Corynebacterium glutamicum (1718)
>2IID_1|Chains A, B, C, D|L-amino-acid oxidase|Calloselasma rhodostoma (8717)
>7GXR_1|Chain A|B-cell lymphoma 6 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 \(w=f(c)\)
2DAP , Knot 145 320 0.87 40 209 312 
MTNIRVAIVGYGNLGRSVEKLIAKQPDMDLVGIFSRRATLDTKTPVFDVADVDKHADDVDVLFLCMGSATDIPEQAPKFAQFACTVDTYDNHRDIPRHRQVMNEAATAAGNVALVSTGWDPGMFSINRVYAAAVLAEHQQHTFWGPGLSQGHSDALRRIPGVQKAVQYTLPSEDALEKARRGEAGDLTGKQTHKRQCFVVADAADHERIENDIRTMPDYFVGYEVEVNFIDEATFDSEHTGMPHGGHVITTGDTGGFNHTVEYILKLDRNPDFTASSQIAFGRAAHRMKQQGQSGAFTVLEVAPYLLSPENLDDLIARDV
2IID , Knot 204 498 0.84 40 245 472 
ADDRNPLAECFQENDYEEFLEIARNGLKATSNPKHVVIVGAGMAGLSAAYVLAGAGHQVTVLEASERPGGRVRTYRNEEAGWYANLGPMRLPEKHRIVREYIRKFDLRLNEFSQENDNAWYFIKNIRKKVGEVKKDPGLLKYPVKPSEAGKSAGQLYEESLGKVVEELKRTNCSYILNKYDTYSTKEYLIKEGDLSPGAVDMIGDLLNEDSGYYVSFIESLKHDDIFAYEKRFDEIVDGMDKLPTAMYRDIQDKVHFNAQVIKIQQNDQKVTVVYETLSKETPSVTADYVIVCTTSRAVRLIKFNPPLLPKKAHALRSVHYRSGTKIFLTCTTKFWEDDGIHGGKSTTDLPSRFIYYPNHNFTNGVGVIIAYGIGDDANFFQALDFKDCADIVFNDLSLIHQLPKKDIQSFCYPSVIQKWSLDKYAMGGITTFTPYQFQHFSDPLTASQGRIYFAGEYTAQAHGWIDSTIKSGLRAARDVNLASENPSGIHLSNDNEL
7GXR , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE

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 \(c\).

\(|P_{f(2DAP_1)}(2) \setminus P_{f(2IID_1)}(2)|=55\), \(|P_{f(2IID_1)}(2) \setminus P_{f(2DAP_1)}(2)|=91\). 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 1:10010111110101100100111001010111110001010000111011010001001011110110100110011011011001000000001100001100110111011110011011110100101111110000001111110010001100111100110001100011001001011010100000000111101100001000100110011100101011001010000011101101100100111000100110100010101000111101100100010011101101110110100100111001
Pair \(Z_2\) Length of longest common subsequence
2DAP_1,2IID_1 146 4
2DAP_1,7GXR_1 175 3
2IID_1,7GXR_1 201 4

Newick tree

 
[
	7GXR_1:10.30,
	[
		2DAP_1:73,2IID_1:73
	]:27.30
]

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 \((0.85)(0.8)(\frac{818 }{\log_{20} 818}-\frac{320}{\log_{20}320})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 2DAP_1 2IID_1 168 138.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]