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Parikh vectors
6DDO_1 3AZV_1 8PSU_1 Letter Amino acid
32 14 35 R Arginine
16 11 16 Q Glutamine
16 14 6 M Methionine
34 26 18 F Phenylalanine
5 8 4 W Tryptophan
31 28 15 Y Tyrosine
30 21 26 V Valine
42 27 24 D Aspartic acid
8 1 13 C Cysteine
61 30 43 L Leucine
25 10 31 A Alanine
18 54 8 N Asparagine
24 9 13 H Histidine
36 29 24 K Lycine
25 10 22 P Proline
38 19 32 E Glutamic acid
34 26 23 G Glycine
24 45 17 I Isoleucine
46 35 31 S Serine
34 26 18 T Threonine 

6DDO_1|Chains A, B|Cytosolic purine 5'-nucleotidase|Homo sapiens (9606)
>3AZV_1|Chains A, B|D/C mosaic neurotoxin|Clostridium botulinum (1491)
>8PSU_1|Chain A|Polymerase acidic protein (PA-like)|Tilapia lake virus (1549864)
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 \(w=f(c)\)
6DDO , Knot 231 579 0.84 40 282 534 
GSSHHHHHHSSGLVPRGSMSTSWSDRLQNAADMPANMDKHALKKYRREAYHRVFVNRSLAMEKIKCFGFNMDYTLAVYKSPEYESLGFELTVERLVSIGYPQELLSFAYDSTFPTRGLVFDTLYGNLLKVDAYGNLLVCAHGFNFIRGPETREQYPNKFIQRDDTERFYILNTLFNLPETYLLACLVDFFTNCPRYTSCETGFKDGDLFMSYRSMFQDVRDAVDWVHYKGSLKEKTVENLEKYVVKDGKLPLLLSRMKEVGKVFLATNSDYKYTDKIMTYLFDFPHGPKPGSSHRPWQSYFDLILVDARKPLFFGEGTVLRQVDTKTGKLKIGTYTGPLQHGIVYSGGSSDTICDLLGAKGKDILYIGDHIFGDILKSKKRQGWRTFLVIPELAQELHVWTDKSSLFEELQSLDIFLAELYKHLDSSSNERPDISSIQRRIKKVTHDMDMCYGMMGSLFRSGSRQTLFASQVMRYADLYAASFINLLYYPFSYLFRAAHVLMPHESTVEHTHVDINEMESPLATRNRTSVDFKDTDYKRHQLTRSISEIKPPNLFPLAPQEITHCHDEDDDEEEEEEEE
3AZV , Knot 173 443 0.79 40 212 410 
MEYFNNINEYFNSINDSKILSLQNKKNTLMDTSGYNAEVRVEGNVQLNPIFPFDFKLGSSGDDRGKVIVTQNENIVYNAMYESFSISFWIRINKWVSNLPGYTIIDSVKNNSGWSIGIISNFLVFTLKQNENSEQDINFSYDISKNAAGYNKWFFVTITTNMMGNMMIYINGKLIDTIKVKELTGINFSKTITFQMNKIPNTGLITSDSDNINMWIRDFYIFAKELDDKDINILFNSLQYTNVVKDYWGNDLRYDKEYYMINVNYMNRYMSKKGNGIVFNTRKNNNDFNEGYKIIIKRIRGNTNDTRVRGENVLYFNTTIDNKQYSLGMYKPSRNLGTDLVPLGALDQPMDEIRKYGSFIIQPCNTFDYYASQLFLSSNATTNRLGILSIGSYSFKLGDDYWFNHEYLIPVIKIEHYASLLESTSTHWVFVPASELEHHHHHH
8PSU , Knot 178 419 0.85 40 239 402 
MDSRFAQLTGVFCDDFTYSEGSRRFLSSYSTVERRPGVPVEGDCYDCLKNKWIAFELEGQPRKFPKATVRCILNNDATYVCSEQEYQQICKVQFKDYLEIDGVVKVGHKASYDAELRERLLELPHPKSGPKPRIEWVAPPRLADISKETAELKRQYGFFECSKFLACGEECGLDQEARELILNEYARDREFEFRNGGWIQRYTVASHKPATQKILPLPASAPLARELLMLIARSTTQAGKVLHSDNTSILAVPVMRDSGKHSKRRPTASTHHLVVGLSKPGCEHDFEFDGYRAAVHVMHLDPKQSANIGEQDFVSTREIYKLDMLELPPISRKGDLDRASGLETRWDVILLLECLDSTRVSQAVAQHFNRHRLALSVCKDEFRKGYQLASEIRGTIPLSSLYYSLCAVRLRMTVHPFAR

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 \(c\).

\(|P_{f(6DDO_1)}(2) \setminus P_{f(3AZV_1)}(2)|=111\), \(|P_{f(3AZV_1)}(2) \setminus P_{f(6DDO_1)}(2)|=41\). 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 1:100000000001111010100010001001101110100011000000100011100011100100111010001110001000011101010011011010011011000011001111001010110101010111010110110110000001001100000001011001101100011101101100010000000110010111000011001001101100010100001001000110010111110010011011110000000000110011011011011000011000101111010011111010110010000101011000111001110011000010011110100110110011101100000011001111101100101100000110010010111101000100000001010010001001000101001111011001000011100110010101101101100110011011011110000100001010010011100000010100000000010001001011011111100100000000000000000
Pair \(Z_2\) Length of longest common subsequence
6DDO_1,3AZV_1 152 6
6DDO_1,8PSU_1 153 4
3AZV_1,8PSU_1 173 4

Newick tree

 
[
	8PSU_1:83.45,
	[
		6DDO_1:76,3AZV_1:76
	]:7.45
]

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 \((0.85)(0.8)(\frac{1022 }{\log_{20} 1022}-\frac{443}{\log_{20}443})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 6DDO_1 3AZV_1 195 169.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]