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Parikh vectors
9LYO_1 5JIA_1 3GYL_1 Letter Amino acid
5 5 8 C Cysteine
9 7 16 E Glutamic acid
3 9 9 H Histidine
13 10 8 K Lycine
12 10 17 Q Glutamine
15 14 23 L Leucine
1 3 1 M Methionine
10 14 20 P Proline
37 13 29 S Serine
17 11 21 V Valine
9 11 10 D Aspartic acid
6 10 12 I Isoleucine
3 3 7 W Tryptophan
10 13 10 Y Tyrosine
12 11 19 A Alanine
5 10 7 R Arginine
9 11 5 N Asparagine
18 26 23 G Glycine
6 8 4 F Phenylalanine
18 11 12 T Threonine 

9LYO_1|Chains A[auth g], F[auth i], H[auth l]|REGN10987 Fab homologue (Light chain)|Homo sapiens (9606)
>5JIA_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Ran-binding protein 10|Mus musculus (10090)
>3GYL_1|Chain A[auth B]|Prostasin|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 \(w=f(c)\)
9LYO , Knot 96 218 0.79 40 143 206 
QSALTQPASVSGSPGQSITISCTGTSSDVGGYNYVSWYQQHPGKAPKLMIYDVSKRPSGVSNRFSGSKSGNTASLTISGLQSEDEADYYCNSLTSISTWVFGGGTKLTVLGRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC
5JIA , Knot 97 210 0.82 40 156 203 
SPGEQELSRRLQRLYPAVNQHETPLPRSWSPKDKYNYIGLSQGNLRVHYKGHGKNHKDAASVRATHPIPAACGIYYFEVKIVSKGRDGYMGIGLSAQGVNMNRLPGWDKHSYGYHGDDGHSFCSSGTGQPYGPTFTTGDVIGCCVNLINGTCFYTKNGHSLGIAFTDLPANLYPTVGLQTPGEIVDANFGQQPFLFDIEDYMREWRAKVQ
3GYL , Knot 115 261 0.81 40 172 253 
ITGGSSAVAGQWPWQVSITYEGVHVCGGSLVSEQWVLSAAHCFPSEHHKEAYEVKLGAHQLDSYSEDAKVSTLKDIIPHPSYLQEGSQGDIALLQLSRPITFSRYIRPISLPAAQASFPNGLHCTVTGWGHVAPSVSLLTPKPLQQLEVPLISRETCNSLYNIDAKPEEPHFVQEDMVCAGYVEGGKDACQGDSGGPLSCPVEGLWYLTGIVSWGDACGARNRPGVYTLASSYASWIQSKVTELQPRVVPQTQESQPDSNL

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 \(c\).

\(|P_{f(9LYO_1)}(2) \setminus P_{f(5JIA_1)}(2)|=80\), \(|P_{f(5JIA_1)}(2) \setminus P_{f(9LYO_1)}(2)|=93\). 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 1:00110011010101100101000100001110001010000110110111001000101100010100010010101011000001000000100100111111001011100111101111110000100101011011001010010101010011001000001000000000001000101001000000101001000110011000100100
Pair \(Z_2\) Length of longest common subsequence
9LYO_1,5JIA_1 173 4
9LYO_1,3GYL_1 137 4
5JIA_1,3GYL_1 174 3

Newick tree

 
[
	5JIA_1:92.03,
	[
		9LYO_1:68.5,3GYL_1:68.5
	]:23.53
]

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 \((0.85)(0.8)(\frac{428 }{\log_{20} 428}-\frac{210}{\log_{20}210})=63.8\)
Status Protein1 Protein2 d d1/2
Query variables 9LYO_1 5JIA_1 81 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]