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Parikh vectors
1IWM_1 2RQJ_1 4GTW_1 Letter Amino acid
10 4 25 A Alanine
10 0 37 R Arginine
0 0 33 C Cysteine
7 0 39 I Isoleucine
3 0 38 F Phenylalanine
13 0 40 T Threonine
11 0 42 N Asparagine
19 0 29 Q Glutamine
12 0 45 K Lycine
10 0 42 Y Tyrosine
14 0 46 D Aspartic acid
17 0 92 L Leucine
5 0 12 M Methionine
5 0 13 W Tryptophan
8 0 41 V Valine
6 0 50 E Glutamic acid
16 8 43 G Glycine
1 0 30 H Histidine
11 0 53 P Proline
8 0 73 S Serine 

1IWM_1|Chains A, B|Outer Membrane Lipoprotein LolB|Escherichia coli (562)
>2RQJ_1|Chains A, B|RNA (5'-R(*GP*GP*AP*GP*GP*AP*GP*GP*AP*GP*GP*A)-3')|
>4GTW_1|Chains A, B|Ectonucleotide pyrophosphatase/phosphodiesterase family member 2, Alkaline phosphodiesterase I|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 \(w=f(c)\)
1IWM , Knot 89 186 0.83 38 141 182 
ASVTTPKGPGKSPDSPQWRQHQQDVRNLNQYQTRGAFAYISDQQKVYARFFWQQTGQDRYRLLLTNPLGSTELELNAQPGNVQLVDNKGQRYTADDAEEMIGKLTGMPIPLNSLRQWILGLPGDATDYKLDDQYRLSEITYSQNGKNWKVVYGGYDTKTQPAMPANMELTDGGQRIKLKMDNWIVK
2RQJ , Knot 3 12 0.20 4 3 3 
GGAGGAGGAGGA
4GTW , Knot 313 823 0.85 40 321 754 
WTNTSGSCRGRCFERTFSNCRCDAACVSLGNCCLDFQETCVEPTHIWTCNKFRCGEKRLSRFVCSCADDCKTHNDCCINYSSVCQDKKSWVEETCESIDTPECPAEFESPPTLLFSLDGFRAEYLHTWGGLLPVISKLKNCGTYTKNMRPMYPTKTFPNHYSIVTGLYPESHGIIDNKMYDPKMNASFSLKSKEKFNPLWYKGQPIWVTANHQEVKSGTYFWPGSDVEIDGILPDIYKVYNGSVPFEERILAVLEWLQLPSHERPHFYTLYLEEPDSSGHSHGPVSSEVIKALQKVDRLVGMLMDGLKDLGLDKCLNLILISDHGMEQGSCKKYVYLNKYLGDVNNVKVVYGPAARLRPTDVPETYYSFNYEALAKNLSCREPNQHFRPYLKPFLPKRLHFAKSDRIEPLTFYLDPQWQLALNPSERKYCGSGFHGSDNLFSNMQALFIGYGPAFKHGAEVDSFENIEVYNLMCDLLGLIPAPNNGSHGSLNHLLKKPIYNPSHPKEEGFLSQCPIKSTSNDLGCTCDPWIVPIKDFEKQLNLTTEDDDIYHMTVPYGRPRILLKQHRVCLLQQQQFLTGYSLDLLMPLWASYTFLSNDQFSRDDFSNCLYQDLRIPLSPVHKCSYYKSNSKLSYGFLTPPRLNRVSNHIYSEALLTSNIVPMYQSFQVIWHYLHDTLLQRYAHERNGINVVSGPVFDFDYDGRYDSLEILKQNSRVIRSQEILIPTHFFIVLTSCKQLSETPLECSALESSAYILPHRPDNIESCTHGKRESSWVEELLTLHRARVTDVELITGLSFYQDRQESVSELLRLKTHLPIFSQED

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 \(c\).

\(|P_{f(1IWM_1)}(2) \setminus P_{f(2RQJ_1)}(2)|=139\), \(|P_{f(2RQJ_1)}(2) \setminus P_{f(1IWM_1)}(2)|=1\). 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 1:101001011100100101000000100100000011110100000101011100010000011100111000101010110101100010000100100111010111111001001111111010000100000100100000100101101100000011111010100110010101001110
Pair \(Z_2\) Length of longest common subsequence
1IWM_1,2RQJ_1 140 2
1IWM_1,4GTW_1 210 5
2RQJ_1,4GTW_1 320 2

Newick tree

 
[
	4GTW_1:15.94,
	[
		1IWM_1:70,2RQJ_1:70
	]:80.94
]

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 \((0.85)(0.8)(\frac{198 }{\log_{20} 198}-\frac{12}{\log_{20}12})=66.4\)
Status Protein1 Protein2 d d1/2
Query variables 1IWM_1 2RQJ_1 88 45
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]