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Parikh vectors
1TMK_1 3EGZ_1 5CIE_1 Letter Amino acid
16 10 10 I Isoleucine
13 4 15 T Threonine
3 0 6 W Tryptophan
7 7 17 A Alanine
8 6 10 R Arginine
9 5 20 N Asparagine
14 4 20 E Glutamic acid
3 2 6 H Histidine
25 7 25 L Leucine
5 4 16 P Proline
11 8 15 S Serine
6 3 14 Y Tyrosine
19 5 24 D Aspartic acid
3 0 1 C Cysteine
10 4 9 Q Glutamine
21 11 23 K Lycine
10 6 18 F Phenylalanine
13 4 14 V Valine
15 3 26 G Glycine
5 5 5 M Methionine 

1TMK_1|Chains A, B|THYMIDYLATE KINASE|Saccharomyces cerevisiae (4932)
>3EGZ_1|Chain A|U1 small nuclear ribonucleoprotein A|Homo sapiens (9606)
>5CIE_1|Chains A, C|Cytochrome c peroxidase, mitochondrial|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
1TMK , Knot 98 216 0.81 40 151 208 
MMGRGKLILIEGLDRTGKTTQCNILYKKLQPNCKLLKFPERSTRIGGLINEYLTDDSFQLSDQAIHLLFSANRWEIVDKIKKDLLEGKNIVMDRYVYSGVAYSAAKGTNGMDLDWCLQPDVGLLKPDLTLFLSTQDVDNNAEKSGFGDERYETVKFQEKVKQTFMKLLDKEIRKGDESITIVDVTNKGIQEVEALIWQIVEPVLSTHIDHDKFSFF
3EGZ , Knot 53 98 0.82 36 84 96 
MAVPETRPNHTIYINNLNEKIKKDELKKSLHAIFSRFGQILDILVSRSLKMRGQAFVIFKEVSSATNALRSMQGFPFYDKPMRIQYAKTDSDIIAKMK
5CIE , Knot 134 294 0.86 40 194 287 
TTPLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPGGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL

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 \(c\).

\(|P_{f(1TMK_1)}(2) \setminus P_{f(3EGZ_1)}(2)|=108\), \(|P_{f(3EGZ_1)}(2) \setminus P_{f(1TMK_1)}(2)|=41\). 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 1:111010111101100010000001100010100011011000001111100010000101000110111010010110010001101001110001001110011010011010101010111101010111000010001000111000000101000100011011000100100010110100011001011110110111000100001011
Pair \(Z_2\) Length of longest common subsequence
1TMK_1,3EGZ_1 149 5
1TMK_1,5CIE_1 163 3
3EGZ_1,5CIE_1 188 3

Newick tree

 
[
	5CIE_1:92.02,
	[
		1TMK_1:74.5,3EGZ_1:74.5
	]:17.52
]

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 \((0.85)(0.8)(\frac{314 }{\log_{20} 314}-\frac{98}{\log_{20}98})=67.7\)
Status Protein1 Protein2 d d1/2
Query variables 1TMK_1 3EGZ_1 83 60
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]