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Parikh vectors
4IGF_1 3HWR_1 3QPL_1 Letter Amino acid
12 15 10 H Histidine
11 9 6 M Methionine
10 5 9 F Phenylalanine
15 11 19 T Threonine
19 27 12 G Glycine
26 32 22 L Leucine
8 11 11 P Proline
20 6 4 Y Tyrosine
21 14 13 D Aspartic acid
31 15 8 I Isoleucine
29 10 4 K Lycine
28 18 18 S Serine
2 1 5 W Tryptophan
13 31 14 V Valine
22 46 33 A Alanine
12 20 19 R Arginine
39 9 8 N Asparagine
2 4 0 C Cysteine
9 14 6 Q Glutamine
16 20 15 E Glutamic acid 

4IGF_1|Chains A, B|Enoyl-acyl carrier reductase|Plasmodium falciparum (36329)
>3HWR_1|Chains A, B|2-dehydropantoate 2-reductase|Ralstonia eutropha (264198)
>3QPL_1|Chain A|HTH-type transcriptional regulator EthR|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
4IGF , Knot 145 345 0.81 40 194 325 
MVHHHHHHNEDICFIAGIGDTNGYGWGIAKELSKRNVKIIFGIWPPVYNIFMKNYKNGKFDNDMIIDKDKKMNILDMLPFDASFDTANDIDEETKNNKRYNMLQNYTIEDVANLIHQKYGKINMLVHSLANAKEVQKDLLNTSRKGYLDALSKSSYSLISLCKYFVNIMKPQSSIISLTYHASQKVVPGYGGGMSSAKAALESDTRVLAYHLGRNYNIRINTISAGPLKSRAATAINKLNNTYENNTNQNKNRNSHDVHNIMNNSGEKEEKKNSASQNYTFIDYAIEYSEKYAPLRQKLLSTDIGSVASFLLSRESRAITGQTIYVDNGLNIMFLPDDIYRNENE
3HWR , Knot 134 318 0.81 40 177 300 
MGSDKIHHHHHHENLYFQGMKVAIMGAGAVGCYYGGMLARAGHEVILIARPQHVQAIEATGLRLETQSFDEQVKVSASSDPSAVQGADLVLFCVKSTDTQSAALAMKPALAKSALVLSLQNGVENADTLRSLLEQEVAAAVVYVATEMAGPGHVRHHGRGELVIEPTSHGANLAAIFAAAGVPVETSDNVRGALWAKLILNCAYNALSAITQLPYGRLVRGEGVEAVMRDVMEECFAVARAEGVKLPDDVALAIRRIAETMPRQSSSTAQDLARGKRSEIDHLNGLIVRRGDALGIPVPANRVLHALVRLIEDKQQHG
3QPL , Knot 102 236 0.78 38 142 222 
MGSSHHHHHHSSGLVPRGSHMTTSAASQASLPRGRRTARPSGDDRELAILATAENLLEDRPLADISVDDLAKGAGISRPTFYFYFPSKEAVLLTLLDRVVNQADMALQTLAENPADTDRENMWRTWINVFFETFGSHKAVTRAGQAARATSVEVAELWSTFMQKWIAYTAAVIDAERDRGAAPRTLPAHELATALNLMNERTLFASFAGEQPSVPEARVLDTLVHIWVTSIYGENR

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 \(c\).

\(|P_{f(4IGF_1)}(2) \setminus P_{f(3HWR_1)}(2)|=90\), \(|P_{f(3HWR_1)}(2) \setminus P_{f(4IGF_1)}(2)|=73\). 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 1:110000000001011111100010111110010000101111111110011100000101000111000001011011110101001001000000000001100001001101100001010111001101001000110000010101100000011010001101101000110100010001111011110010111000001110011000010100101111000110110010000000000000000001001100010000000010000011001100000011100011000110110111000001101001010011011111001000000
Pair \(Z_2\) Length of longest common subsequence
4IGF_1,3HWR_1 163 6
4IGF_1,3QPL_1 156 6
3HWR_1,3QPL_1 161 6

Newick tree

 
[
	3HWR_1:81.97,
	[
		4IGF_1:78,3QPL_1:78
	]:3.97
]

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 \((0.85)(0.8)(\frac{663 }{\log_{20} 663}-\frac{318}{\log_{20}318})=95.4\)
Status Protein1 Protein2 d d1/2
Query variables 4IGF_1 3HWR_1 119 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]