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Parikh vectors
5WEI_1 4MXV_1 9CZF_1 Letter Amino acid
4 0 11 C Cysteine
10 10 60 G Glycine
12 7 4 H Histidine
12 7 28 T Threonine
10 12 50 A Alanine
10 11 48 D Aspartic acid
15 3 32 I Isoleucine
5 11 28 P Proline
2 2 5 W Tryptophan
7 8 26 Y Tyrosine
15 2 31 R Arginine
6 4 23 N Asparagine
3 10 31 Q Glutamine
14 7 28 K Lycine
15 20 44 S Serine
9 9 36 V Valine
20 2 23 E Glutamic acid
9 20 49 L Leucine
8 2 10 M Methionine
11 10 38 F Phenylalanine 

5WEI_1|Chain A|Polymerase acidic protein|Influenza A virus (641501)
>4MXV_1|Chains A, B, C[auth D]|Lymphotoxin-alpha|Homo sapiens (9606)
>9CZF_1|Chain A|Integrin alpha-V heavy chain|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 \(w=f(c)\)
5WEI , Knot 93 197 0.83 40 149 190 
MGSSHHHHHHSSGLVPRGSHMEDFVRQCFNPMIVELAEKAMKEYGEDPKIETNKFAAICTHLEVCFMYSDGGSKHRFEIIEGRDRIMAWTVVNSICNTTGVEKPKFLPDLYDYKENRFIDIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRSLWDSFRQSER
4MXV , Knot 76 157 0.81 38 112 151 
ADLGSDYKDDDDKKPAAHLIGDPSKQNSLLWRANTDRAFLQDGFSLSNNSLLVPTSGIYFVYSQVVFSGKAYSPKATSSPLYLAHEVQLFSSQYPFHVPLLSSQKMVYPGLQEPWLHSMYHGAAFQLTQGDQLSTHTDGIPHLVLSPSTVFFGAFAL
9CZF , Knot 241 605 0.85 40 263 564 
FNLDVDSPAEYSGPEGSYFGFAVDFFVPSASSRMFLLVGAPKANTTQPGIVEGGQVLKCDWSSTRRCQPIEFDATGNRDYAKDDPLEFKSHQWFGASVRSKQDKILACAPLYHWRTEMKQEREPVGTCFLQDGTKTVEYAPCRSQDIDADGQGFCQGGFSIDFTKADRVLLGGPGSFYWQGQLISDQVAEIVSKYDPNVYSIKYNNQLATRTAQAIFDDSYLGYSVAVGDFNGDGIDDFVSGVPRAARTLGMVYIYDGKNMSSLYNFTGEQMAAYFGFSVAATDINGDDYADVFIGAPLFMDRGSDGKLQEVGQVSVSLQRASGDFQTTKLNGFEVFARFGSAIAPLGDLDQDGFNDIAIAAPYGGEDKKGIVYIFNGRSTGLNAVPSQILEGQWAARSCPPSFGYSMKGATDIDKNGYPDLIVGAFGVDRAILYRARPVITVNAGLEVYPSILNQDNKTCSLPGTALKVSCFNVRFCLKADGKGVLPRKLNFQVELLLDKLKQKGAIRRALFLYSRSPSHSKNMTISRGGLMQCEELIAYLRDESEFRDKLTPITIFMEYRLDYRTAADTTGLQPILNQFTPANISRQAHILLDGGSLEVLFQG

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 \(c\).

\(|P_{f(5WEI_1)}(2) \setminus P_{f(4MXV_1)}(2)|=104\), \(|P_{f(4MXV_1)}(2) \setminus P_{f(5WEI_1)}(2)|=67\). 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 1:11000000000011110100100110001011110110011000100101000011110001010110001100001011010001111011001000011001011101000000011011100001010010010010000001011010100110010001000001010001101000110001100100000
Pair \(Z_2\) Length of longest common subsequence
5WEI_1,4MXV_1 171 3
5WEI_1,9CZF_1 192 4
4MXV_1,9CZF_1 205 5

Newick tree

 
[
	9CZF_1:10.49,
	[
		5WEI_1:85.5,4MXV_1:85.5
	]:17.99
]

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 \((0.85)(0.8)(\frac{354 }{\log_{20} 354}-\frac{157}{\log_{20}157})=59.6\)
Status Protein1 Protein2 d d1/2
Query variables 5WEI_1 4MXV_1 80 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]