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Parikh vectors
9CQL_1 5QYF_1 9CSC_1 Letter Amino acid
18 18 18 A Alanine
5 1 1 C Cysteine
10 9 4 Q Glutamine
32 19 19 K Lycine
10 11 3 R Arginine
13 14 7 N Asparagine
6 8 8 M Methionine
28 18 7 S Serine
9 7 3 Y Tyrosine
25 14 15 V Valine
12 15 14 E Glutamic acid
14 7 22 G Glycine
11 23 21 I Isoleucine
21 29 24 L Leucine
13 10 6 F Phenylalanine
12 10 11 P Proline
11 14 11 D Aspartic acid
6 5 4 H Histidine
21 21 16 T Threonine
3 5 1 W Tryptophan 

9CQL_1|Chains A, E|9C2 TCR delta chain|Homo sapiens (9606)
>5QYF_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>9CSC_1|Chains A[auth B], C|3-oxoacid CoA-transferase, B subunit|Thermosipho melanesiensis (46541)
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 \(w=f(c)\)
9CQL , Knot 125 280 0.83 40 177 269 
AQKVTQAQSSVSMPVRKAVTLNCLYETSWWSYYIFWYKQLPSKEMIFLIRQGSDEQNAKSGRYSVNFKKAAKSVALTISALQLEDSAKYFCALGDPGGLNTDKLIFGKGTRVTVEPRSQPHTKPSVFVMKNGTNVACLVKEFYPKDIRINLVSSKKITEFDPAIVISPSGKYNAVKLGKYEDSNSVTCSVQHDNKTVHSTDFEVKTDSTDHVKPKETENTKQPSKSCHKPKAIVHTEKVNMMSLTVLGLRMLFAKTVAVNFLLTAKLFFLSRGRAKRGSG
5QYF , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
9CSC , Knot 95 215 0.79 40 133 205 
MNPKEKIAIRVAQELKKGQLVNLGIGLPTLVANYIPKDIHVFFQSENGIIGMGPAPKEGYENSDLTNAGASYITALPGAMTFDSAFSFGIIRGGHLDVTVLGGLQVDEEGHLANWMIPGKMIPGMGGAMDLVTGAKKVIVAMTHTAKGTPKIVKKCTLPLTSIRKVDLIVTELAVIEPTDEGLLLKEISKETTLDEVLKLTEAKLIIADDLKIFA

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 \(c\).

\(|P_{f(9CQL_1)}(2) \setminus P_{f(5QYF_1)}(2)|=85\), \(|P_{f(5QYF_1)}(2) \setminus P_{f(9CQL_1)}(2)|=83\). 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 1:1001001000101110011010010000110001110001100011111001000001001000101001100111010110100010010111011110000111101001010100010001011110010011011001010010101100001001011111010100011011000000010001000000100001010000000101000000001000000101110000101101011110111100111011101011110010100101
Pair \(Z_2\) Length of longest common subsequence
9CQL_1,5QYF_1 168 4
9CQL_1,9CSC_1 160 4
5QYF_1,9CSC_1 154 4

Newick tree

 
[
	9CQL_1:83.63,
	[
		9CSC_1:77,5QYF_1:77
	]:6.63
]

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 \((0.85)(0.8)(\frac{538 }{\log_{20} 538}-\frac{258}{\log_{20}258})=79.6\)
Status Protein1 Protein2 d d1/2
Query variables 9CQL_1 5QYF_1 100 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]