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Parikh vectors
4FMO_1 4ROL_1 2LEA_1 Letter Amino acid
10 7 8 P Proline
12 3 6 Y Tyrosine
20 13 9 V Valine
13 3 16 R Arginine
5 10 10 H Histidine
40 18 6 L Leucine
12 7 4 F Phenylalanine
20 11 13 S Serine
5 9 6 T Threonine
28 8 12 D Aspartic acid
5 1 3 Q Glutamine
24 0 1 I Isoleucine
2 1 0 W Tryptophan
13 21 7 A Alanine
4 1 0 C Cysteine
4 2 9 M Methionine
23 11 4 K Lycine
15 4 1 N Asparagine
23 4 5 E Glutamic acid
10 7 15 G Glycine 

4FMO_1|Chain A|DNA mismatch repair protein MLH1|Saccharomyces cerevisiae (559292)
>4ROL_1|Chains A, C|Hemoglobin subunit alpha|Homo sapiens (9606)
>2LEA_1|Chain A|Serine/arginine-rich splicing factor 2|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 \(w=f(c)\)
4FMO , Knot 126 288 0.82 40 170 269 
GAMADDEKNALPISKDGYIRVPKERVNVNLTSIKKLREKVDDSIHRELTDIFANLNYVGVVDEERRLAAIQHDLKLFLIDYGSVCYELFYQIGLTDFANFGKINLQSTNVSDDIVLYNLLSEFDELNDDASKEKIISKIWDMSSMLNEYYSIELVNDGLDNDLKSVKLKSLPLLLKGYIPSLVKLPFFIYRLGKEVDWEDEQECLDGILREIALLYIPDMVPKVDTSDASLSEDEKAQFINRKEHISSLLEHVLFPCIKRRFLAPRHILKDVVEIANLPDLYKVFERC
4ROL , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
2LEA , Knot 61 135 0.73 36 92 120 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMSYGRPPPDVEGMTSLKVDNLTYRTSPDTLRRVFEKYGRVGDVYIPRDRYTKESRGFAFVRFHDKRDAEDAMDAMDGAVLDGRELRVQMARYGRPPDSHHS

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 \(c\).

\(|P_{f(4FMO_1)}(2) \setminus P_{f(4ROL_1)}(2)|=117\), \(|P_{f(4ROL_1)}(2) \setminus P_{f(4FMO_1)}(2)|=46\). 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 1:111100000111100010101100010101001001000100010001001110100111100000111100010111100101000110011100110110101000010001110011001001000100001100110100110000010110011000100101001111101011011011111001100101000000101110011110110111010000101000001011000001001100111101000111100110011011011010011000
Pair \(Z_2\) Length of longest common subsequence
4FMO_1,4ROL_1 163 3
4FMO_1,2LEA_1 166 3
4ROL_1,2LEA_1 139 3

Newick tree

 
[
	4FMO_1:86.08,
	[
		4ROL_1:69.5,2LEA_1:69.5
	]:16.58
]

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 \((0.85)(0.8)(\frac{429 }{\log_{20} 429}-\frac{141}{\log_{20}141})=86.1\)
Status Protein1 Protein2 d d1/2
Query variables 4FMO_1 4ROL_1 109 80
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]