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Parikh vectors
8HSJ_1 4MMB_1 4DBH_1 Letter Amino acid
12 51 12 F Phenylalanine
22 25 22 P Proline
0 0 3 C Cysteine
14 6 5 Q Glutamine
42 24 19 E Glutamic acid
25 53 24 I Isoleucine
15 18 19 S Serine
25 26 18 T Threonine
39 21 11 R Arginine
19 13 20 D Aspartic acid
58 61 19 L Leucine
10 13 9 M Methionine
30 46 27 G Glycine
1 16 4 W Tryptophan
8 17 2 Y Tyrosine
24 37 19 V Valine
39 54 21 A Alanine
15 13 9 N Asparagine
7 6 9 H Histidine
23 19 16 K Lycine 

8HSJ_1|Chains A, B, C, D, E, F|Transcription termination factor Rho|Thermus thermophilus HB8 (300852)
>4MMB_1|Chain A|Transporter|Aquifex aeolicus (224324)
>4DBH_1|Chains A, B|2-HYDROXYHEPTA-2,4-DIENE-1,7-DIOATE ISOMERASE|Corynebacterium glutamicum (1718)
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 \(w=f(c)\)
8HSJ , Knot 178 428 0.84 38 208 406 
GPMRRKETLQETPLTYQELASKILPELHLLAQEAGIEGYKRMKKDQLIMALLERQTQGEGLRLVKGYLEISQDGYGFLTENLHNLESRVAIVSAGLIKQYALRAGDYVVGQARPPRENERYATLLKVEAVNNLDPEAAKNRPRFDELTPQFPDRQIRLETTPDELSTRVIDLLAPIGRGQRGLIVAPPKAGKTTLLKKIANAVLKNEPDIKVIVLLIDERPEEVTDFRESVQGAEVIASTFDEPPQNHIRVAEFVHERAKRIVEEGGHVMILLDSITRLARANNLVTPPTGRTLSGGLDSAALYFPKRFLGAARNIRGGGSLTILATALVETGSRMDDVIFEEFKGTGNMELHLSRRLEERRIFPAIDILKSGTRREELLLGEEVTHKMWLLRKVLADMDPAEAMEMLLARLARTKNNKEFLASLAAR
4MMB , Knot 204 519 0.82 38 222 458 
MEVKREHWATRLGLILAMAGYAVDLGNFLRFPVQAAENGGGAFMIPYIIAFLLVGIPLMWIEWAMGRYGGAQGHGTTPAIFYLLWRNRFAKILGVFGLWIPLVVAIYYVYIESWTLGFAIKFLVGLVPEPPPNATDPDSILRPFKEFLYSYIGVPKGDEPILKPSLFAYIVFLITMFINVSILIRGISKGIERFAKIAMPTLFILAVFLVIRVFLLETPNGTAADGLNFLWTPDFEKLKDPGVWIAAVGQIFFSLGLGFGAIITFASYVRKDQDIVLSGLTAATLNEKAEVILGGSISIPAAVAFFGVANAVAIAKAGAFNLGFITLPAIFSQTAGGTFLGFLWFFLLFFAGLTSSIAGMQPMIAFLEDELKLSRKHAVLWTAAIVFFSAHLVMFLNKSLDEMDFWAGTIGVVFFGLTELIIFFWIFGADKAWEEINRGGIIKVPRIYYYVMRYITPAFLAVLLVVWAREYIPKIMEETHWTVWITRFYIIGLFLFLTFLVFLAERRRNHESAGTLVPR
4DBH , Knot 130 288 0.85 40 182 276 
MGSSHHHHHHSSGLVPRGSHMRFGRIATPDGMCFCSIEGEGDDVANLTAREIEGTPFTEPKFTGREWPLKDVRLLAPMLPSKVVAIGRNYADHVAEVFKKSAESLPPTLFLKPPTAVTGPESPIRIPSFATKVEFEGELAVVIGKPCKNVKADDWKSVVLGFTIINDVSSRDLQFADGQWARAKGIDTFGPIGPWIETDINSIDLDNLPIKARLTHDGETQLKQDSNSNQMIMKMGEIIEFITASMTLLPGDVIATGSPAGTEAMVDGDYIEIEIPGIGKLGNPVVDA

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 \(c\).

\(|P_{f(8HSJ_1)}(2) \setminus P_{f(4MMB_1)}(2)|=57\), \(|P_{f(4MMB_1)}(2) \setminus P_{f(8HSJ_1)}(2)|=71\). 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 1:11100000100011000011001110101110011101000100001111110000010110110101010001011100010010001111011110001101100111010110000001011010110010101100010100101011000101000100100011011111101001111111011000110011011100010101111110001001001000101101110010011000101101100010011001101111100100110100110110100101110011101100111110010111010111011100100100111001010101010100010000111110110010000011110010001111001110101101101111011000000011101110
Pair \(Z_2\) Length of longest common subsequence
8HSJ_1,4MMB_1 128 4
8HSJ_1,4DBH_1 146 4
4MMB_1,4DBH_1 162 4

Newick tree

 
[
	4DBH_1:81.00,
	[
		8HSJ_1:64,4MMB_1:64
	]:17.00
]

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 \((0.85)(0.8)(\frac{947 }{\log_{20} 947}-\frac{428}{\log_{20}428})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 8HSJ_1 4MMB_1 168 154.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]