Amino Acid Form Factors

The form factor f_am of an amino acid, centered on the C^α atom position, is

where f_b is the scattering of the amino acid in a vacuum, f_ex is the adjustment due to the excluded volume of solvent, and ρ_ex is a constant. Each amino acid is assigned the same scattering function f_b(q), a five-factor exponential representation:

where {A_i, B_i}_i=1⁵ and C are empirically determined constants (a standard form used to fit molecular form factors³¹). The excluded volume effect is captured using an exponential model in the form

where r_w is the average atomic radius of the atom.^6,7,13 To calculate the excluded volume for amino acids, coordinates for all 20 amino acids³² and values of r_w for carbon, nitrogen, oxygen, hydrogen, and sulfur (e.g., ref (33)) were used to compute the excluded volume scattering, centered at the C^α, through

where r_i^α is the distance of atom i from the C^α molecule and N_am is the number of atoms in the amino acid. Since f_b does not discriminate individual amino acids, this value f_ex^am was averaged over all 20 amino acids, weighted by their abundance in globular proteins (see ref (34)). This averaged function, shown in Figure ​Figure55e, gives f_ex(q). Finally, eq 6 includes a constant ρ_ex which modulates the effect of the excluded volume scatter by comparison to f_b; this value is constrained to lie within 0.75 and 1.25 (similar constraints are used in refs (6, 7, and 13)). The scattering form for an individual water molecule in the hydration layer is

where f_hy and f_ox are the vacuum scatterings of hydrogen and oxygen, respectively.³¹ The constant ρ_h was empirically determined (as in ref (7)). A detailed description of the parameter determination method is given in section 3 of the Supporting Information.

Fits to scattering data for various molecules using appropriate C^α coordinates as a backbone model {c}_i=1ⁿ. (See section 3 of the Supporting Information for details.) In panels a–d, the data scattering data is shown overlaid by the smoothed data used for fitting (blue curve) and the model fit (red curve). Panel e is the averaged scattering function f_am^ex obtained by averaging the scattering parameters obtained from fits like those shown in (a)–(d).