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Parikh vectors
3QDK_1 9ITE_1 4DCX_1 Letter Amino acid
34 20 28 K Lycine
14 16 21 F Phenylalanine
10 1 9 W Tryptophan
19 15 21 R Arginine
6 4 0 C Cysteine
45 16 25 V Valine
57 7 52 A Alanine
30 17 28 D Aspartic acid
54 13 25 G Glycine
25 6 10 H Histidine
53 23 47 L Leucine
25 17 26 S Serine
30 12 29 T Threonine
20 4 19 Y Tyrosine
17 12 23 N Asparagine
19 9 27 Q Glutamine
46 16 32 E Glutamic acid
28 15 32 I Isoleucine
17 4 10 M Methionine
23 3 38 P Proline 

3QDK_1|Chains A, B, C, D|Ribulokinase|Bacillus halodurans (86665)
>9ITE_1|Chain A|engineered miniGa13|Homo sapiens (9606)
>4DCX_1|Chains A, B|Nickel-binding periplasmic protein|Escherichia coli (83333)
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 \(w=f(c)\)
3QDK , Knot 234 572 0.86 40 263 528 
MSLTKYTIGVDYGTESGRAVLIDLSNGQELADHVTPYRHGVIDQYLPNTNIKLGHEWALQHPLDYVEVLTTSVPAVMKESGVDADDVIGIGVDFTACTMLPVDEEGQPLCLLAQYKDNPHSWVKLWKHHAAQDKANAINEMAEKRGEAFLPRYGGKISSEWMIAKVWQILDEAEDVYNRTDQFLEATDWIVSQMTGKIVKNSCTAGYKAIWHKREGYPSNEFFKALDPRLEHLTTTKLRGDIVPLGERAGGLLPEMAEKMGLNPGIAVAVGNVDAHAAVPAVGVTTPGKLVMAMGTSICHMLLGEKEQEVEGMCGVVEDGIIPGYLGYEAGQSAVGDIFAWFVKHGVSAATFDEAQEKGVNVHALLEEKASQLRPGESGLLALDWWNGNRSILVDTELSGMLLGYTLQTKPEEIYRALLEATAFGTRAIVDAFHGRGVEVHELYACGGLPQKNHLLMQIFADVTNREIKVAASKQTPALGAAMFASVAAGSEVGGYDSIEEAAKKMGRVKDETFKPIPEHVAIYEKLYQEYVTLHDYFGRGANDVMKRLKALKSIQHRPSSLLTEGHHHHHH
9ITE , Knot 104 230 0.82 40 158 217 
MGSTVSAEDKAAAERSKEIDKCLSREKTYVKRLVKILLLGADNSGKSTFLKQMRIIHGGSGGSGGTKGIHEYDFEIKNVPFKMVDVGGQRSERKRWFECFDSVTSILFLVDSSDFNRLTESLNDFETIVNNRVFSNVSIILFLNKTDLLEEKVQIVSIKDYFLEFEGDPHCLRDVQKFLVECFRNKRRDQQQKPLYHHFTTAINTENARLIFRDVKDTILHDNLKQLMLQ
4DCX , Knot 202 502 0.83 38 246 472 
AAPDEITTAWPVNVGPLNPHLYTPNQMFAQSMVYEPLVKYQADGSVIPWLAKSWTHSEDGKTWTFTLRDDVKFSNGEPFDAEAAAENFRAVLDNRQRHAWLELANQIVDVKALSKTELQITLKSAYYPFLQELALPRPFRFIAPSQFKNHETMNGIKAPIGTGPWILQESKLNQYDVFVRNENYWGEKPAIKKITFNVIPDPTTRAVAFETGDIDLLYGNEGLLPLDTFARFSQNPAYHTQLSQPIETVMLALNTAKAPTNELAVREALNYAVNKKSLIDNALYGTQQVADTLFAPSVPYANLGLKPSQYDPQKAKALLEKAGWTLPAGKDIREKNGQPLRIELSFIGTDALSKSMAEIIQADMRQIGADVSLIGEEESSIYARQRDGRFGMIFHRTWGAPYDPHAFLSSMRVPSHADFQAQQGLADKPLIDKEIGEVLATHDETQRQALYRDILTRLHDEAVYLPISYISMMVVSKPELGNIPYAPIATEIPFEQIKPVKP

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 \(c\).

\(|P_{f(3QDK_1)}(2) \setminus P_{f(9ITE_1)}(2)|=142\), \(|P_{f(9ITE_1)}(2) \setminus P_{f(3QDK_1)}(2)|=37\). 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 1:10100001110010001011110100100110010100011100011000101100111001100101100011111000110100111111010100111100010110111000001001101100011000101100110001011110011010001111011011001001000000110100111001010110000011001110000101000110110101001000010101111100111111011001110111111110101011111111001101111110010011110000010110111001111101100110011101111110011011010010001101011100010010110011111011010001110001011111001000100100111010111001110110101101001010111100001110111010000101110000111111111011110011100010011001101000010111001110001000010100011011001100101100100010011001000000
Pair \(Z_2\) Length of longest common subsequence
3QDK_1,9ITE_1 179 5
3QDK_1,4DCX_1 141 4
9ITE_1,4DCX_1 178 4

Newick tree

 
[
	9ITE_1:94.67,
	[
		3QDK_1:70.5,4DCX_1:70.5
	]:24.17
]

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 \((0.85)(0.8)(\frac{802 }{\log_{20} 802}-\frac{230}{\log_{20}230})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 3QDK_1 9ITE_1 199 136
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]