Circular Dichroism

Calculate CD parameters: ΔA, Δε, ellipticity, molar and mean-residue ellipticity and g-factor from the absorbances of left and right circularly polarized light.

Model: Exact

ΔA = A_L − A_R = Δε c l, tan θ = tanh(ln 10 · ΔA / 4)

θ (mdeg) ≈ 32 980 ΔA (small ΔA), [θ] = 3298.2 Δε (deg·cm²·dmol⁻¹, Δε in M⁻¹cm⁻¹)

g = 2ΔA / (A_L + A_R), [θ]_MRE = [θ] / (N − 1), S/N ∝ A·10^−A/2 (best at A = 2/ln 10)

Absorbance of left-circular light A_LAbsorbance of right-circular light A_RConcentration c of molecules (mM; for a protein, of chains)Path length l (mm)Residues N (0 = not a protein)

Δε needs c > 0 and l > 0. The mean-residue ellipticity is shown for a whole number of residues N ≥ 2 and uses N − 1 peptide bonds.

ΔA = A_L − A_R

0.01000

Ellipticity θ (exact)

329.814 mdeg

small-angle 329.821 mdeg

Δε

1000.000 M⁻¹cm⁻¹

Molar ellipticity [θ]

3.298e+6 deg·cm²·dmol⁻¹

Dissymmetry factor g

0.01258

2ΔA / (A_L + A_R)

Mean absorbance

0.7950

optimum for S/N: 0.869

Relative S/N

0.996

1 at the optimum (shot-noise limit)

Relative CD signal-to-noise against absorbance

At fixed g the shot-noise-limited S/N is proportional to A·10^−A/2: it peaks at A = 0.869 and falls on either side, so adjust c or l to bring the mean absorbance close to it.

Model and references

Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers

θ = atan(tanh(ln10·ΔA/4)) (≈ 32 980 ΔA mdeg); Δε = ΔA/(cl), c of molecules; [θ] = 3298.2 Δε, MRE = [θ]/(N − 1). The S/N optimum A = 2/ln10 assumes shot noise only.

Greenfield N. J. (2006). Using circular dichroism spectra to estimate protein secondary structure. Nat. Protoc. 1(6), 2876–2890: θ(mdeg) = 32 980 ΔA, [θ] = 3298.2 Δε. doi:10.1038/nprot.2006.202
Kelly S. M., Jess T. J., Price N. C. (2005). How to study proteins by circular dichroism. Biochim. Biophys. Acta 1751(2), 119–139: mean residue weight and ellipticity. doi:10.1016/j.bbapap.2005.06.005

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