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Parikh vectors
1TFP_1 1FUS_1 6NWA_1 Letter Amino acid
13 12 74 A Alanine
6 5 34 P Proline
2 0 26 W Tryptophan
11 11 40 T Threonine
5 9 23 Y Tyrosine
15 6 46 V Valine
1 4 4 C Cysteine
8 0 78 L Leucine
7 1 23 K Lycine
2 3 49 I Isoleucine
12 10 47 S Serine
3 2 21 R Arginine
2 10 27 N Asparagine
9 3 17 E Glutamic acid
5 2 43 H Histidine
1 0 24 M Methionine
8 4 49 F Phenylalanine
8 5 28 D Aspartic acid
3 5 24 Q Glutamine
9 14 74 G Glycine 

1TFP_1|Chains A, B|TRANSTHYRETIN|Gallus gallus (9031)
>1FUS_1|Chain A|RIBONUCLEASE F1|Gibberella fujikuroi (5127)
>6NWA_1|Chains A, DA[auth H], Q[auth a]|Photosystem I P700 chlorophyll a apoprotein A1|Synechocystis sp. PCC 6803 (1148)
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 \(w=f(c)\)
1TFP , Knot 63 130 0.78 40 102 124 
APLVSHGSVDSKCPLMVKVLDAVRGSPAANVAVKVFKKAADGTWQDFATGKTTEFGEIHELTTEEQFVEGVYRVEFDTSSYWKGLGLSPFHEYADVVFTANDSGHRHYTIAALLSPFSYSTTAVVSDPQE
1FUS , Knot 54 106 0.79 34 83 104 
QSATTCGSTNYSASQVRAAANAACQYYQNDDTAGSSTYPHTYNNYEGFDFPVDGPYQEFPIKSGGVYTGGSPGADRVVINTNCEYAGAITHTGASGNNFVGCSGTN
6NWA , Knot 291 751 0.85 40 303 685 
MTISPPEREAKAKVSVDNNPVPTSFEKWGKPGHFDRTLARGPKTTTWIWNLHANAHDFDSQTSDLEDVSRKIFSAHFGHLAVVFVWLSGMYFHGAKFSNYEGWLADPTHIKPSAQVVWPIVGQGILNGDVGGGFHGIQITSGLFYLWRASGFTDSYQLYCTAIGGLVMAALMLFAGWFHYHVKAPKLEWFQNVESMMNHHLAGLLGLGSLGWAGHQIHVSMPINKLLDAGVAPKDIPLPHEFILEPSKMAELYPSFAQGLTPFFTLNWGVYSDFLTFKGGLNPVTGGLWLSDTAHHHLAIAVLFIIAGHMYRTNWGIGHSMKEILEAHKGPFTGEGHKGLYEILTTSWHAQLAINLALLGSLTIIVAQHMYAMPPYPYQAIDYATQLSLFTHHMWIGGFLIVGAGAHGAIFMVRDYDPAKNVNNLLDRMLRHRDAIISHLNWVCIFLGFHSFGLYIHNDTMRALGRPQDMFSDTAIQLQPIFAQWVQHLHTLAPGATAPNALATASYAFGGETIAVAGKVAMMPITLGTADFMVHHIHAFTIHVTALILLKGVLYARSSRLVPDKANLGFRFPCDGPGRGGTCQVSGWDHVFLGLFWMYNSLSVVIFHFSWKMQSDVWGTVSPDGSVTHVTLGNFAQSAITINGWLRDFLWAQAANVINSYGSALSAYGIMFLAGHFVFAFSLMFLFSGRGYWQELIESIVWAHNKLNVAPAIQPRALSIIQGRAVGVAHYLLGGIVTTWAFFLARSLSIG

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 \(c\).

\(|P_{f(1TFP_1)}(2) \setminus P_{f(1FUS_1)}(2)|=70\), \(|P_{f(1FUS_1)}(2) \setminus P_{f(1TFP_1)}(2)|=51\). 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 1:1111001010000111101101101011101110110011010100110100001101001000001101100101000001011110110001011101000100000111110110000011100100
Pair \(Z_2\) Length of longest common subsequence
1TFP_1,1FUS_1 121 3
1TFP_1,6NWA_1 223 4
1FUS_1,6NWA_1 252 4

Newick tree

 
[
	6NWA_1:13.86,
	[
		1TFP_1:60.5,1FUS_1:60.5
	]:72.36
]

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 \((0.85)(0.8)(\frac{236 }{\log_{20} 236}-\frac{106}{\log_{20}106})=41.6\)
Status Protein1 Protein2 d d1/2
Query variables 1TFP_1 1FUS_1 51 46.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]