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Parikh vectors
5VTR_1 5MIQ_1 1UCJ_1 Letter Amino acid
16 14 5 R Arginine
12 11 5 Q Glutamine
22 29 5 I Isoleucine
20 10 6 P Proline
14 25 6 A Alanine
17 19 7 D Aspartic acid
22 24 5 V Valine
25 37 8 G Glycine
22 15 6 L Leucine
30 15 7 S Serine
30 19 14 T Threonine
6 2 0 W Tryptophan
10 12 8 Y Tyrosine
10 13 3 F Phenylalanine
24 9 2 N Asparagine
9 2 2 C Cysteine
9 30 5 E Glutamic acid
6 11 2 H Histidine
15 19 0 K Lycine
4 11 0 M Methionine 

5VTR_1|Chains A, C, E|Hemagglutinin HA1 chain|Influenza A virus (strain A/Hong Kong/1/1968 H3N2) (506350)
>5MIQ_1|Chain A|Thioredoxin reductase|Lactococcus lactis subsp. cremoris (1359)
>1UCJ_1|Chains A, B|Guanyl-specific ribonuclease Sa|Streptomyces aureofaciens (1894)
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 \(w=f(c)\)
5VTR , Knot 143 323 0.85 40 205 315 
ADPGATLCLGHHAVPNGTLVKTITDDQIEVTNATELVQSSSTGKICNNPHRILDGIDCTLIDALLGDPHCDVFQNETWDLFVERSKAFSNCYPYDVPDYASLRSLVASSGTLEFITEGFTWTGVTQNGGSNACKRGPGSGFFSRLNWLTKSGSTYPVLNVTMPNNDNFDKLYIWGVHHPSTNQEQTSLYVQASGRVTVSTRRSQQTIIPNIGSRPWVRLSSSRISIYWTIVKPGDVLVINSNGNLIAPRGYFKMRTGKSSIMRSDAPIDTCISECITPNGSIPNDKPFQNVNKITYGACPKYVKQNTLKLATGMRNVPEKQTR
5MIQ , Knot 140 327 0.82 40 188 308 
GSSHHHHHHSSGLVPRGSHMTEKKYDVVIIGSGPAGMTAAMYTARSEMKTLLLERGVPGGQMNNTAEIENYPGYETIMGPELSMKMAEPLEGLGVENAYGFVTGIEDHGDYKKIITEDDEFITKSIIIATGANHRKLEIPGEEEYGARGVSYCAVCDGAFFRNQEILVIGGGDSAVEEALYLTRFGQSVTIMHRRDKLRAQEIIQQRAFKEEKINFIWDSVPMEIKGDDKKIQSVVYKNVKTGEVTEKAFGGIFIYVGLDPVAEFVSDLGITDEAGWIITDDHMRTNIPGIFAVGDVRQKDFRQITTAVGDGAQAAQEAYKFVVELG
1UCJ , Knot 49 96 0.77 34 80 93 
DVSGTVCLSALPPEATDTLNLIASDGPFPYSQDGVTFQNRESVLPTQSYGYYHEYTVITPGARTRGTRRIITGEATQEDYYTGDHYATFSLIDQTC

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 \(c\).

\(|P_{f(5VTR_1)}(2) \setminus P_{f(5MIQ_1)}(2)|=95\), \(|P_{f(5MIQ_1)}(2) \setminus P_{f(5VTR_1)}(2)|=78\). 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 1:10111010110011101011001000010100100110000010100010011011000110111101000110000101110000110000100110010100111001010110011010110001100100011101110010110001000111010110000100101111001000000001010101010100000000111011001110100001010101101101111000101111010101001000110001110001000101010110001100100100110100100001011011001100000
Pair \(Z_2\) Length of longest common subsequence
5VTR_1,5MIQ_1 173 4
5VTR_1,1UCJ_1 175 4
5MIQ_1,1UCJ_1 184 3

Newick tree

 
[
	1UCJ_1:90.84,
	[
		5VTR_1:86.5,5MIQ_1:86.5
	]:4.34
]

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 \((0.85)(0.8)(\frac{650 }{\log_{20} 650}-\frac{323}{\log_{20}323})=90.5\)
Status Protein1 Protein2 d d1/2
Query variables 5VTR_1 5MIQ_1 118 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]