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Parikh vectors
6RNS_1 1YVX_1 1HKX_1 Letter Amino acid
7 14 5 Q Glutamine
39 56 9 L Leucine
6 15 6 F Phenylalanine
5 34 8 P Proline
4 9 3 W Tryptophan
9 25 4 Y Tyrosine
13 43 9 R Arginine
4 14 5 N Asparagine
12 10 9 H Histidine
15 29 6 K Lycine
13 51 8 S Serine
9 40 11 T Threonine
15 38 7 V Valine
21 29 13 E Glutamic acid
13 33 10 G Glycine
6 23 12 I Isoleucine
38 54 8 A Alanine
15 25 8 D Aspartic acid
5 14 2 C Cysteine
3 14 4 M Methionine 

6RNS_1|Chains A, B|Gem-associated protein 5|Homo sapiens (9606)
>1YVX_1|Chain A|RNA dependent RNA polymerase|Hepatitis C virus (3052230)
>1HKX_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|CALCIUM/CALMODULIN-DEPENDENT PROTEIN KINASE TYPE II ALPHA CHAIN|MUS MUSCULUS (10090)
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 \(w=f(c)\)
6RNS , Knot 111 252 0.81 40 157 239 
ARSLLPLSTSLDHRSKEELHQDCLVLATAKHSRELNEDVSADVEERFHLGLFTDRATLYRMIDIEGKGHLENGHPELFHQLMLWKGDLKGVLQTAAERGELTDNLVAMAPAAGYHVWLWAVEAFAKQLCFQDQYVKAASHLLSIHKVYEAVELLKSNHFYREAIAIAKARLRPEDPVLKDLYLSWGTVLERDGHYAVAAKCYLGATCAYDAAKVLAKKGDAASLRTAAELAAIVGEDELSASLALRCAQELL
1YVX , Knot 227 570 0.84 40 257 530 
SMSYSWTGALITPCSPEEEKLPINPLSNSLLRYHNKVYCTTSKSASLRAKKVTFDRMQVLDAYYDSVLKDIKLAASKVSARLLTLEEACQLTPPHSARSKYGFGAKEVRSLSGRAVNHIKSVWKDLLEDSQTPIPTTIMAKNEVFCVDPTKGGKKAARLIVYPDLGVRVCEKMALYDVTQKLPQAVMGASYGFQYSPAQRVEFLLKAWAEKKDPMGFSYDTRCFDSTVTERDIRTEESIYQACSLPEEARTAIHSLTERLYVGGPMFNSKGQSCGYRRCRASGVLTTSMGNTITCYVKALAACKAAGIVAPTMLVCGDDLVVISESQGTEEDERNLRAFTEAMTRYSAPPGDPPRPEYDLELITSCSSNVSVALGPQGRRRYYLTRDPTTPIARAAWETVRHSPVNSWLGNIIQYAPTIWVRMVLMTHFFSILMAQDTLDQNLNFEMYGSVYSVSPLDLPAIIERLHGLDAFSLHTYTPHELTRVASALRKLGAPPLRAWKSRARAVRASLISRGGRAAVCGRYLFNWAVKTKLKLTPLPEARLLDLSSWFTVGAGGGDIYHSVSRARPR
1HKX , Knot 76 147 0.86 40 123 143 
GPHMTTIEDEDTKVRKQEIIKVTEQLIEAISNGDFESYTKMCDPGMTAFEPEALGNLVEGLDFHRFYFENLWSRNSKPVHTTILNPHIHLMGDESACIAYIRITQYLDAGGIPRTAQSEETRVWHRRDGKWQIVHFHRSGAPSVLPH

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 \(c\).

\(|P_{f(6RNS_1)}(2) \setminus P_{f(1YVX_1)}(2)|=38\), \(|P_{f(1YVX_1)}(2) \setminus P_{f(6RNS_1)}(2)|=138\). 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 1:100111100010000000100001111010000010001010100010111100010100110101010100101011001111010101110011001010001111111110011111101110010100001011001101001001101100001000111110101010011100101011011000100111100011100100110111001011010011011111100010101110010011
Pair \(Z_2\) Length of longest common subsequence
6RNS_1,1YVX_1 176 4
6RNS_1,1HKX_1 164 3
1YVX_1,1HKX_1 198 3

Newick tree

 
[
	1YVX_1:97.23,
	[
		6RNS_1:82,1HKX_1:82
	]:15.23
]

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 \((0.85)(0.8)(\frac{822 }{\log_{20} 822}-\frac{252}{\log_{20}252})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 6RNS_1 1YVX_1 196 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]