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Parikh vectors
6VSH_1 6KFW_1 2CHU_1 Letter Amino acid
6 3 1 W Tryptophan
14 9 24 N Asparagine
23 27 20 D Aspartic acid
10 9 8 Q Glutamine
17 16 2 H Histidine
11 18 9 T Threonine
24 30 4 R Arginine
7 5 2 M Methionine
14 29 20 S Serine
23 31 11 P Proline
37 40 20 A Alanine
7 3 0 C Cysteine
28 24 18 G Glycine
31 54 32 L Leucine
11 15 12 F Phenylalanine
23 30 18 E Glutamic acid
17 22 29 I Isoleucine
9 11 36 K Lycine
9 9 7 Y Tyrosine
28 25 23 V Valine 

6VSH_1|Chains A, B, C|Dicamba O-demethylase, oxygenase component|Stenotrophomonas maltophilia (40324)
>6KFW_1|Chain A|CxnD|Actinoplanes tsinanensis (2039464)
>2CHU_1|Chains A, B|ENTEROCHELIN UPTAKE PERIPLASMIC BINDING PROTEIN|CAMPYLOBACTER JEJUNI (197)
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 \(w=f(c)\)
6VSH , Knot 150 349 0.84 40 200 332 
MATFVRNAWYVAALPEELSEKPLGRTILDTPLALYRQPDGVVAALLDICPHRFAPLSDGILVNGHLQCPYHGLEFDGGGQCVHNPHGNGARPASLNVRSFPVVERDALIWIWPGDPALADPGAIPDFGCRVDPAYRTVGGYGHVDCNYKLLVDNLMDLGHAQYVHRANAQTDAFDRLEREVIVGDGEIQALMKIPGGTPSVLMAKFLRGANTPVDAWNDIRWNKVSAMLNFIAVAPEGTPKEQSIHSRGTHILTPETEASCHYFFGSSRNFGIDDPEMDGVLRSWQAQALVKEDKVVVEAIERRRAYVEANGIRPAMLSCDEAAVRVSREIEKLEQLEAARLEHHHHHH
6KFW , Knot 173 410 0.84 40 209 380 
MHHHHHHTDVIPTEFFTEPGSNPHATAAEYRSKCPVHRINVPPGADAYAVLGNKVVEEALGDSRLSKQVENLPARYRDKAVASSLLVVGNLGFADAPKHTRLKKPISRAFLPATVAQLRPRIQDIVDDLIDTFPENGEIDLLSSFALPMPLTVICEYLGIPVADRPLFLEWSYILSQDPLQHDEAELKAASEEFTDYFTKLVAERRTDLRDDLLSEIIRARDAGVYSETELLSTLLLLIIAGHKTVANMIGNGTALLLRHPQQLEMLRATPELIPSAIEEILRYEGSAAWASLRVAAEDMQLAGVDIPKGSFVHLSLSSAGRDPDVYDDPDGFDVTRSPNRHLSFGHGPHFCIGAPLGRLQGEIAFSTLLRRLPRFELAVPPEEVAWLSDSSLSRGLEALPIRVGERLPR
2CHU , Knot 126 296 0.80 38 163 270 
KTNTATVKVLPISMSDEGDSFLVKDSLGENKIPKNPSKVVILDLGILDTFDALKLNDKVVGVPAKNLPKYLQQFKNKPSVGGVQQVDFEAINALKPDLIIISGRQSKFYDKLKEIAPTLFVGLDNANFLSSFENNVLSVAKLYGLEKEALEKISDIKNEIEKAKSIVDEDKKALIILTNSNKISAFGPQSRFGIIHDVLGINAVDENIKVGTHGKSINSEFILEKNPDYIFVVDRNVILGNKERAQGILDNALVAKTKAAQNKKIIYLDPEYWYLASGNGLESLKTMILEIKNAVK

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 \(c\).

\(|P_{f(6VSH_1)}(2) \setminus P_{f(6KFW_1)}(2)|=70\), \(|P_{f(6KFW_1)}(2) \setminus P_{f(6VSH_1)}(2)|=79\). 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 1:1101100110111110010001110011001111000101111111010100111100111101010010011010111001001010110110101001111000111111110111101111101100101100011101010000011100110110100100101000110010001111010101110111101011110110110011011001010010111011111101010000100010011010001000011100001110010101110010101110000111011000010101011011110000111010001001001011010000000
Pair \(Z_2\) Length of longest common subsequence
6VSH_1,6KFW_1 149 6
6VSH_1,2CHU_1 149 4
6KFW_1,2CHU_1 146 4

Newick tree

 
[
	6VSH_1:74.99,
	[
		6KFW_1:73,2CHU_1:73
	]:1.99
]

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 \((0.85)(0.8)(\frac{759 }{\log_{20} 759}-\frac{349}{\log_{20}349})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 6VSH_1 6KFW_1 137 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]