Circular Dichroism Mean Residue Ellipticity Calculator (CD [θ]MRE and Δε)
Formula and Units
The raw output of a CD spectrophotometer is ellipticity θ (millidegrees, mdeg), which depends simultaneously on protein concentration, path length, and protein size, and cannot be compared directly between samples. To compare with literature reference values or to feed deconvolution software, the raw signal must be converted to the normalized mean residue molar ellipticity [θ]_MRE (mean residue ellipticity).
Step 1: Mean Residue Weight (MRW)
MRW (g/mol) = Protein molecular weight (Da) / Number of amino acid residues
The MRW of a protein is typically 110–115 g/mol; the exact value depends on sequence.
Step 2: Mean Residue Molar Ellipticity
[θ]_MRE = θ_obs(mdeg) × MRW / (10 × c(mg/mL) × l(cm))
Units: deg·cm²·dmol⁻¹ (per mole of residues)
Step 3 (optional): Convert to Molar Circular Dichroism Δε
Δε (L·mol⁻¹·cm⁻¹) = [θ]_MRE / 3298.2
The conversion factor 3298.2 comes from the geometric relationship between ellipticity (in degrees) and absorbance difference: (180/π) × (ln10/4) × 100 (Greenfield, 2006, PMID: 17406547). Δε is the standard input format for quantum-chemical calculations and deconvolution software such as DICHROWEB.
Applicable Range
- Applicable to single-protein solutions at any single wavelength; the θ entered must be the net ellipticity after subtracting the buffer baseline
- Applicable to peptides, but MRW should be calculated from the actual sequence rather than using the 110 g/mol approximation
- Not applicable to multi-component mixtures or to raw instrument readings that have not yet been baseline-corrected
Path length unit confusion is the most common error in CD calculations: this tool requires path length in centimeters (cm). Far-UV CD (< 250 nm) typically uses a 1 mm quartz cuvette — enter 0.1; high-absorbance or micro-volume samples use a 0.1 mm ultra-thin cell — enter 0.01. A one-digit unit error shifts the result by one order of magnitude, yet the value still falls within a “plausible range” and is easily overlooked.
If you have only molar concentration, convert first:
c(mg/mL) = c(µM) × MW(Da) × 1×10⁻³
Worked Example
The parameters below can be verified by hand calculation; they correspond to a typical α-helical protein signal at 222 nm:
θ_obs = −30 mdeg, c = 0.1 mg/mL, l = 0.1 cm, MW = 11,000 Da, n_res = 100
MRW = 11000 / 100 = 110 g/mol
[θ]_MRE = (−30) × 110 / (10 × 0.1 × 0.1)
= −3300 / 0.1
= −33,000 deg·cm²·dmol⁻¹
Δε = −33,000 / 3298.2 ≈ −10.01 L·mol⁻¹·cm⁻¹
−10 L·mol⁻¹·cm⁻¹ is the canonical per-residue Δε for an α-helix at 222 nm (Greenfield, 2006).
FAQ
Why use mean residue ellipticity rather than reporting molar ellipticity directly?
The CD signal of a protein arises from every amide bond in the backbone. A 100-residue protein and a 1000-residue protein differ in molar ellipticity by a factor of ten simply because of size. Normalizing to the per-residue contribution, [θ]MRE, makes secondary-structure features comparable across proteins of different sizes and under different folding conditions; all α-helix and β-sheet reference values in the literature are given in [θ]MRE.
How large an error does a wrong path length unit introduce?
Path length must be entered in centimeters (cm): a 1 mm quartz cell = 0.1 cm, a 0.1 mm ultra-thin cell = 0.01 cm. If a 0.1 mm cell is entered as 0.1 instead of 0.01, the result is ten times too high; if millimeters are entered directly as centimeters, the result is ten times too low. This is the most common systematic error in CD conversion, and because the result still falls within a numerically plausible range, it rarely triggers an obvious alarm.
Which method gives the most accurate protein concentration?
The propagation coefficient of concentration error in [θ]MRE is 1: a 10% underestimate of concentration produces a 10% overestimate in the result. A280 with a known extinction coefficient is recommended (ε280 can be calculated from sequence using this site's protein-params tool), or amino acid analysis. Bradford and BCA assays can vary by 15–20% due to differential dye-binding across proteins, and should only be used when an accurate ε280 is unavailable.
Does a negative output indicate a measurement error?
No. Circular dichroism signals can be positive or negative depending on conformation and wavelength: α-helices give negative peaks at 208 nm and 222 nm, β-sheets give a negative band around 215–220 nm, and random coils give a strong negative peak near 200 nm. The sign is a structural fingerprint, not an indicator of measurement error.
Which is more commonly reported, Δε or [θ]MRE, and what is the difference?
[θ]MRE (deg·cm²·dmol⁻¹) is the standard format for experimental reports; nearly all published reference spectra use this unit. Δε (L·mol⁻¹·cm⁻¹) is used by quantum-chemical ab initio calculations and deconvolution software such as DICHROWEB and CDPro; the two quantities are interconverted exactly by [θ]MRE = 3298.2 × Δε and carry no additional information relative to each other.
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