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Parikh vectors
1YAJ_1 4PLJ_1 1CWL_1 Letter Amino acid
22 5 10 I Isoleucine
36 4 14 K Lycine
26 5 15 F Phenylalanine
39 13 9 V Valine
20 7 9 N Asparagine
44 11 23 G Glycine
59 13 7 L Leucine
36 8 6 P Proline
10 3 1 W Tryptophan
15 6 6 R Arginine
20 5 3 Q Glutamine
34 3 12 E Glutamic acid
15 1 5 M Methionine
33 17 8 S Serine
14 8 2 Y Tyrosine
40 12 9 A Alanine
23 6 7 D Aspartic acid
5 1 4 C Cysteine
11 2 4 H Histidine
30 18 11 T Threonine 

1YAJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|CES1 protein|Homo sapiens (9606)
>4PLJ_1|Chains A, B|Capsid protein|Hepatitis E virus (1678143)
>1CWL_1|Chain A|PEPTIDYL-PROLYL CIS-TRANS ISOMERASE A|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 \(w=f(c)\)
1YAJ , Knot 217 532 0.85 40 255 496 
SSPPVVDTVHGKVLGKFVSLEGFAQPVAIFLGIPFAKPPLGPLRFTPPQPAEPWSFVKNATSYPPMCTQDPKAGQLLSELFTNRKENIPLKLSEDCLYLNIYTPADLTKKNRLPVMVWIHGGGLMVGAASTYDGLALAAHENVVVVTIQYRLGIWGFFSTGDEHSRGNWGHLDQVAALRWVQDNIASFGGNPGSVTIFGESAGGESVSVLVLSPLAKNLFHRAISESGVALTSVLVKKGDVKPLAEQIAITAGCKTTTSAVMVHCLRQKTEEELLETTLKMKFLSLDLQGDPRESQPLLGTVIDGMLLLKTPEELQAERNFHTVPYMVGINKQEFGWLIPMLMSYPLSEGQLDQKTAMSLLWKSYPLVCIAKELIPEATEKYLGGTDDTVKKKDLFLDLIADVMFGVPSVIVARNHRDAGAPTYMYEFQYRPSFSSDMKPKTVIGDHGDELFSVFGAPFLKEGASEEEIRLSKMVMKFWANFARNGNPNGEGLPHWPEYNQKEGYLQIGANTQAAQKLKDKEVAFWTNLFAK
4PLJ , Knot 72 148 0.81 40 113 146 
SRPFSVLRANDVLWLSLTAAEYDQTTYGSSTNPMYVSDTVTFVNVATGAQGVSRSLDWSKVTLDGRPLTTIQQYSKTFFVLPLRGKLSFWEAGTTKAGYPYNYNTTASDQILIENAPGHRVCISTYTTNLGSGPVSISAVGVLAPHSA
1CWL , Knot 82 165 0.84 40 129 161 
MVNPTVFFDIAVDGEPLGRVSFELFADKVPKTAENFRALSTGEKGFGYKGSCFHRIIPGFMCQGGDFTRHNGTGGKSIYGEKFEDENFILKHTGPGILSMANAGPNTNGSQFFICTAKTEWLDGKHVVFGKVKEGMNIVEAMERFGSRNGKTSKKITIADCGQLE

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 \(c\).

\(|P_{f(1YAJ_1)}(2) \setminus P_{f(4PLJ_1)}(2)|=159\), \(|P_{f(4PLJ_1)}(2) \setminus P_{f(1YAJ_1)}(2)|=17\). 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 1:0011110010101110110101110111111111110111111010110110110110010001110000101101100110000001110100001010100110100000111111101111111110000111111000111101000111111100100000101101001111011000110111011010111001110010111101110011001100011110011100101011100111011000000111100100000001100010101101010101000011110110111110010010100010011011110000111111111001100101000011011100011101100111010000111000010000111011101111110111100000111100100100010100010100111001001101111111001100001010011101110110010101011101100000010101110001100100001111001110
Pair \(Z_2\) Length of longest common subsequence
1YAJ_1,4PLJ_1 176 3
1YAJ_1,1CWL_1 194 3
4PLJ_1,1CWL_1 156 3

Newick tree

 
[
	1YAJ_1:96.99,
	[
		4PLJ_1:78,1CWL_1:78
	]:18.99
]

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 \((0.85)(0.8)(\frac{680 }{\log_{20} 680}-\frac{148}{\log_{20}148})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 1YAJ_1 4PLJ_1 190 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]