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Parikh vectors
3MUR_1 7AJV_1 4USJ_1 Letter Amino acid
4 20 14 E Glutamic acid
11 30 17 K Lycine
5 13 9 M Methionine
6 16 4 F Phenylalanine
4 13 15 T Threonine
3 15 6 Y Tyrosine
7 17 29 A Alanine
5 15 10 N Asparagine
5 18 17 D Aspartic acid
7 40 31 L Leucine
6 20 14 R Arginine
0 6 3 C Cysteine
2 17 11 H Histidine
10 23 24 I Isoleucine
4 16 17 P Proline
4 18 31 V Valine
4 14 6 Q Glutamine
3 25 33 G Glycine
8 19 27 S Serine
0 4 0 W Tryptophan 

3MUR_1|Chain A[auth P]|U1 small nuclear ribonucleoprotein A|Homo sapiens (9606)
>7AJV_1|Chains A, B|Dual specificity tyrosine-phosphorylation-regulated kinase 1A|Homo sapiens (9606)
>4USJ_1|Chains A, B|ACETYLGLUTAMATE KINASE, CHLOROPLASTIC|ARABIDOPSIS THALIANA (3702)
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 \(w=f(c)\)
3MUR , Knot 53 98 0.82 36 84 96 
MAVPETRPNHTIYINNLNEKIKKDELKKSLHAIFSRFGQILDILVSRSLKMRGQAFVIFKEVSSATNALRSMQGFPFYDKPMRIQYAKTDSDIIAKMK
7AJV , Knot 153 359 0.83 40 221 343 
MGSSHHHHHHSSGLVPRGSHMIVKNGEKWMDRYEIDSLIGKGSFGQVVKAYDRVEQEWVAIKIIKNKKAFLNQAQIEVRLLELMNKHDTEMKYYIVHLKRHFMFRNHLCLVFEMLSYNLYDLLRNTNFRGVSLNLTRKFAQQMCTALLFLATPELSIIHCDLKPENILLCNPKRSAIKIVDFGSSCQLGQRIYQYIQSRFYRSPEVLLGMPYDLAIDMWSLGCILVEMHTGEPLFSGANEVDQMNKIVEVLGIPPAHILDQAPKARKFFEKLPDGTWNLKKTKDGKREYKPPGTRKLHNILGVETGGPGGRRAGESGHTVADYLKFKDLILRMLDYDPKTRIQPYYALQHSFFKKTADE
4USJ , Knot 134 318 0.81 38 179 301 
MGSSHHHHHHSSGLVPRGSHMTVSTPPSIATGNAPSPDYRVEILSESLPFIQKFRGKTIVVKYGGAAMTSPELKSSVVSDLVLLACVGLRPILVHGGGPDINRYLKQLNIPAEFRDGLRVTDATTMEIVSMVLVGKVNKNLVSLINAAGATAVGLSGHDGRLLTARPVPNSAQLGFVGEVARVDPSVLRPLVDYGYIPVIASVAADDSGQAYNINADTVAGELAAALGAEKLILLTDVAGILENKEDPSSLIKEIDIKGVKKMIEDGKVAGGMIPKVKCCIRSLAQGVKTASIIDGRRQHSLLHEIMSDEGAGTMITG

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 \(c\).

\(|P_{f(3MUR_1)}(2) \setminus P_{f(7AJV_1)}(2)|=29\), \(|P_{f(7AJV_1)}(2) \setminus P_{f(3MUR_1)}(2)|=166\). 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 1:11110001000101001000100001000101110011011011100010101011111001001001100101111000110100100000111010
Pair \(Z_2\) Length of longest common subsequence
3MUR_1,7AJV_1 195 3
3MUR_1,4USJ_1 165 3
7AJV_1,4USJ_1 172 21

Newick tree

 
[
	7AJV_1:94.86,
	[
		3MUR_1:82.5,4USJ_1:82.5
	]:12.36
]

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 \((0.85)(0.8)(\frac{457 }{\log_{20} 457}-\frac{98}{\log_{20}98})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 3MUR_1 7AJV_1 139 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]