3-n3ndeb6 (n=0-3)

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Supplementary Information 3.2:

Curve Fitting Procedure for Mixed Assemblies 2_3-n3_nDEB₆ (n=0-3).
The equilibrium between assemblies and individual building blocks is described by

3K_(n=0)=K_(n=1)=K_(n=2)=3K_(n=3), as a result of the three possibilities of forming an assembly with composition 223 (223, 232, and 322) and 233 (233, 323, and 332). It is assumed that free components 2 and 3 are not present and therefore K_(n) is assigned a value of 1E10.

Assemblies with P-chirality are only formed from combinations of building blocks with P-conformation, e.g. P-2 and P-3. The enthalpy of formation is identical for all assemblies (homomeric and heteromeric).

Based on these assumptions the mass balances for [2_tot] and [3_tot] are :
[2]_tot = 3[P-2₃DEB₆] + 2[P-2₂3DEB₆] + [P-23₂DEB₆] +

3[M-2₃DEB₆] + 2[M-2₂3DEB₆] + [M-23₂DEB₆].

[2]_tot= 3K_(n=3)[P-2]³ + 2K_(n=2)[P-2]²[P-3] + K_(n=1)[P-2][P-3]² +

3K_(n=3)[M-2]³ + 2K_(n=2)[M-2]²[M-3] + K_(n=1)[M-2][M-3]².

[2]_tot= 3K_(n=3)[P-2]³ + 6K_(n=3)[P-2]²[P-3] + 3K_(n=3)[P-2][P-3]² +

3K_(n=3)[M-2]³ + 6K_(n=3)[M-2]²[M-3] + 3K_(n=3)[M-2][M-3]².

[2]_tot/3K_(n=3)= [P-2]³ + 2[P-2]²[P-3] + [P-2][P-3]² +

[M-2]³ + 2[M-2]²[M-3] + [M-2][M-3]².

[M-2] = K₂ x [P-2] and

[M-3] = K₃ x [P-3] gives:

[2]_tot/3K_(n=3)= [P-2]³(1+K₂³) + 2[P-2]²[P-3](1+K₂²K₃) + [P-2][P-3]²(1+K₂K₃²).

With K₃= 1/K₂:

= [P-2]³(1+K₂³) + 2[P-2]²[P-3](1+K₂) + [P-2][P-3]²(1+1/K₂).
This equation expresses [P-3] as a function of K₂ and [P-2].
[3]_tot = 3[P-3₃DEB₆] + 2[P-23₂DEB₆] + [P-2₂3DEB₆] +

3[M-3₃DEB₆] + 2 [M-23₂DEB₆] + [M-2₂3DEB₆].

[3]_tot= 3K_(n=0)[P-3]³ + 2K_(n=1)[P-2][P-3]² + K_(n=2)[P-2]²[P-3] +

3K_(n=0)[M-3]³ + 2K_(n=1)[M-2][M-3]²+ K_(n=2)[M-2]²[M-3].

[3]_tot= 3K_(n=0)[P-3]³ + 6K_(n=0)[P-2][P-3]² + 3K_(n=0)[P-2]²[P-3] +

3K_(n=0)[M-3]³ + 6K_(n=0)[M-2][M-3]²+ 3K_(n=0)[M-2]²[M-3].

[3]_tot/3K_(n=0)= [P-3]³ + 2[P-2][P-3]² + [P-2]²[P-3] +

[M-3]³ + 2[M-2][M-3]²+ [M-2]²[M-3].

[M-2] = K₂ x [P-2] and

[M-3] = K₃ x [P-3] gives:

[3]_tot/3K_(n=0)= [P-3]³(1+K₃³) + 2[P-3]²[P-2](1+K₃²K₂) + [P-3][P-2]²(1+K₃K₂²).
With K₃= 1/K₂:

= [P-3]³(1+1/K₂³) + 2[P-3]²[P-2](1+1/K₂) + [P-3][P-2]²(1+K₂).
This equation expresses [P-2] as a function of K₂ and [P-3].
From these two equations [P-2] and [P-3] can be calculated numerically for a specific value of K₂.
From [P-2] and [P-3] a CD-signal is calculated using:
[P-2₃DEB₆] = K_(n=3)[P-2]³

[M-2₃DEB₆] = K_(n=3)[M-2]³ = K_(n=3)K₂³[P-2]³ = K₂³[P-2₃DEB₆]

[P-2₂3DEB₆] = 3K_(n=3)[P-2]²[P-3]

[M-2₂3DEB₆] = 3K_(n=3)[M-2]²[M-3] = 3K_(n=3)K₂²[P-2]²K₃[P-3] = K₂[P-2₂3DEB₆]

[P-23₂DEB₆] = 3K_(n=3)[P-2][P-3]²

[M-23₂DEB₆] = 3K_(n=3)[M-2][M-3]² = 3K_(n=3)K₂[P-2]K₃²[P-3]² = (1/K₂)[P-23₂DEB₆]

[P-3₃DEB₆] = K_(n=3)[P-3]³

[M-3₃DEB₆] = K_(n=3)[M-3]³ = K_(n=3)K₃³[P-3]³ = (1/K₂)³[P-3₃DEB₆]

resulting in:
_calc.= SF ([M-2₃DEB₆]-[P-2₃DEB₆]+[M-2₂3DEB₆]-[P-2₂3DEB₆]+

[M-23₂DEB₆]-[P-23₂DEB₆]+[M-3₃DEB₆]-[P-3₃DEB₆]+)

([M-2₃DEB₆]+[P-2₃DEB₆]+[M-2₂3DEB₆]+[P-2₂3DEB₆]+

[M-23₂DEB₆]+[P-23₂DEB₆]+[M-3₃DEB₆]+[P-3₃DEB₆])
= SF (d.e._6R[2₃DEB₆] + d.e._4R/2S[2₂3DEB₆] + d.e._2R/4S[23₂DEB₆]+ d.e._6S[3₃DEB₆]) x ₍₂₃__DEB6₎

(n=0-3) [2_3-n3_nDEB₆]

SF: Scaling Factor
The experimental CD-curve can now be fitted by variation of K₂. ([P-2] and [P-3] are numerically solved for each K₂ in a subroutine) using least rms fit analysis.