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Parikh vectors
3PVZ_1 4MNK_1 3UVC_1 Letter Amino acid
21 10 2 Q Glutamine
30 23 3 E Glutamic acid
14 4 4 M Methionine
29 25 9 S Serine
2 8 3 W Tryptophan
25 35 13 A Alanine
16 17 7 R Arginine
28 15 13 D Aspartic acid
5 0 0 C Cysteine
18 16 10 T Threonine
7 4 14 H Histidine
35 29 9 L Leucine
10 14 7 Y Tyrosine
20 17 7 N Asparagine
22 28 19 G Glycine
30 14 11 I Isoleucine
29 17 7 K Lycine
28 23 11 F Phenylalanine
11 21 6 P Proline
19 28 9 V Valine 

3PVZ_1|Chains A, B, C, D|UDP-N-acetylglucosamine 4,6-dehydratase|Vibrio fischeri (312309)
>4MNK_1|Chain A|Chitinase A|Cycas revoluta (3396)
>3UVC_1|Chains A, B|Macrophage metalloelastase|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 \(w=f(c)\)
3PVZ , Knot 170 399 0.85 40 231 384 
SNAMNSILSLIGRDTELFHQDINANEKELQSVVSQSRFLVLGGAGSIGQAVTKEIFKRNPQKLHVVDISENNMVELVRDIRSSFGYINGDFQTFALDIGSIEYDAFIKADGQYDYVLNLSALKHVRSEKDPFTLMRMIDVNVFNTDKTIQQSIDAGAKKYFCVSTDKAANPVNMMGASKRIMEMFLMRKSEEIAISTARFANVAFSDGSLLHGFNQRIQKNQPIVAPNDIKRYFVTPQESGELCLMSCIFGENRDIFFPKLSEALHLISFADIAVKYLKQLGYEPHLCESEDEARELAKTLPAQGKWPCLFTSSDTTGEKDFEEFFTDKETLDMARFDNLGIIKNDSLYQQELLELFEQKIGQMKTDRQWTKEEIVQLFFIMIPDFGHKETGKYLDSKM
4MNK , Knot 145 348 0.81 38 196 330 
MQNALKVGFWPAYSVSEFPPSKINSRLFTHLYYAFAELNAPTFEVRVPPGSEKTAEDFTPTVRRLNPSVKTLISIGGGGSEVRDNFAKLNSDASARQRFVKSSIALARRYGFHGLDLDYEYPEPQLEMENFVKLVSELTAAIREEARTSGKPRLLLTEAVYFHQKLFPWEVVTEYPVQFIAAGLDWVNVMAYDFHGSWENFTGAPAALRDPNSKFTASVGIESFLAAGMPPEKLVLGIPLFGRSWLLKNNNEVGIGAPAVGAGPVDGALSFSEIQNFIRGGAREVFDTTTVSAYAYKDNVWVGYDNQQSVALKVQYAKEKRLGGYFFWSVNQDIDAILPKIASDTWGG
3UVC , Knot 80 164 0.83 38 120 157 
MGPVWRKHYITYRINNYTPDMNREDVDYAIRKAFQVWSNVTPLKFSKINTGMADILVVFARGAHGDFHAFDGKGGILAHAFGPGSGIGGDAHFDEDEFWTTHSGGTNLFLTAVHAIGHSLGLGHSSDPKAVMFPTYKYVDINTFRLSADDIRGIQSLYGHHHHH

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 \(c\).

\(|P_{f(3PVZ_1)}(2) \setminus P_{f(4MNK_1)}(2)|=87\), \(|P_{f(4MNK_1)}(2) \setminus P_{f(3PVZ_1)}(2)|=52\). 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 1:001100110111000011000101000010011000011111111011011000110001001011010000110110010001101010100111011010001110101000011010110010000011011011010110000010001011100010100001101101111000110111100000111001011011100101101100010000111110010001101000101011001110000111101001101101101110010011001010000001001100111010110110000001000100110000010110100111100001000011011000110100000100001101111111011000010010001
Pair \(Z_2\) Length of longest common subsequence
3PVZ_1,4MNK_1 139 4
3PVZ_1,3UVC_1 185 3
4MNK_1,3UVC_1 162 4

Newick tree

 
[
	3UVC_1:92.02,
	[
		3PVZ_1:69.5,4MNK_1:69.5
	]:22.52
]

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 \((0.85)(0.8)(\frac{747 }{\log_{20} 747}-\frac{348}{\log_{20}348})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 3PVZ_1 4MNK_1 137 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]