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Parikh vectors
5RKG_1 4CKW_1 4XML_1 Letter Amino acid
8 1 7 F Phenylalanine
1 1 2 W Tryptophan
5 12 16 V Valine
3 1 4 C Cysteine
5 4 14 Q Glutamine
8 7 3 H Histidine
8 10 6 I Isoleucine
12 13 10 E Glutamic acid
3 13 14 G Glycine
17 9 30 S Serine
6 19 13 A Alanine
10 10 7 R Arginine
14 21 19 L Leucine
5 0 2 M Methionine
7 6 14 T Threonine
8 4 11 Y Tyrosine
5 4 7 N Asparagine
10 5 11 D Aspartic acid
8 1 12 K Lycine
6 5 13 P Proline 

5RKG_1|Chain A|PH-interacting protein|Homo sapiens (9606)
>4CKW_1|Chain A|3-DEHYDROQUINATE DEHYDRATASE|MYCOBACTERIUM TUBERCULOSIS (1773)
>4XML_1|Chain A[auth L]|Light chain of HIV-1 gp120 V3-specific human monoclonal antibody 2424|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 \(w=f(c)\)
5RKG , Knot 73 149 0.81 40 117 140 
MHHHHHHSSGVDLGTENLYFQSMSYDIQAWKKQCEELLNLIFQCEDSEPFRQPVDLLEYPDYRDIIDTPMDFATVRETLEAGNYESPMELCKDVRLIFSNSKAYTPSKRSRIYSMSLRLSAFFEEHISSVLSDYKSALRFHKRNTITKR
4CKW , Knot 66 146 0.75 38 102 138 
SELIVNVINGPSLGRLGRREPAVYGGTTHDELVALIEREAAELGLKAVVRQSDSEAQLLDWIHQAADAAEPVILNAGGLTHTSVALRDACAELSAPLIEVHISNVHAREEFRRHSYLSPIATGVIVGLGIQGYLLALRYLAEHLGT
4XML , Knot 100 215 0.83 40 154 206 
DIVMNQSPPSLAVTPGEPASISCRASQSLLYSDGHNYLDWYLQKPGQAPQLLIYLGSTRASGVPDRFSGSGSGTDFTLKISRVEAEDVGVYYCMQPLQSYTFGQGTKLEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNR

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 \(c\).

\(|P_{f(5RKG_1)}(2) \setminus P_{f(4CKW_1)}(2)|=77\), \(|P_{f(4CKW_1)}(2) \setminus P_{f(5RKG_1)}(2)|=62\). 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 1:10000000011011000101001000101100000011011100000011001101100100001100110110100010110000110100010111000010010000010010101011100010011000001101000001000
Pair \(Z_2\) Length of longest common subsequence
5RKG_1,4CKW_1 139 3
5RKG_1,4XML_1 151 3
4CKW_1,4XML_1 156 3

Newick tree

 
[
	4XML_1:79.03,
	[
		5RKG_1:69.5,4CKW_1:69.5
	]:9.53
]

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 \((0.85)(0.8)(\frac{295 }{\log_{20} 295}-\frac{146}{\log_{20}146})=45.9\)
Status Protein1 Protein2 d d1/2
Query variables 5RKG_1 4CKW_1 59 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]