column

salt gradient (NaCl)

sample

fraction collection

chromatogram

Shaded = salt range swept by the gradient. Each protein elutes where its k′ drops to roughly 0.1–1.

Directions

Load a three-protein mixture onto an anion-exchange column and elute it with a NaCl gradient. Adjust the column, flow rate, gradient and load, then choose a fraction-collection window to purify the target protein B.

How to Use

  1. Column – set the column length, particle diameter, and flow rate.
  2. Salt gradient (NaCl) – set the starting and ending NaCl concentration and how many column volumes (CV) the gradient runs over.
  3. Sample – set how much protein is loaded, relative to the column's binding capacity.
  4. Fraction collection – set the start and end time of the window used to collect protein B, or click "center on protein B" to center the window on its peak automatically.
  5. Click run to animate the separation, or show full result to jump straight to the finished chromatogram.
Challenge: Collect protein B at ≥ 95% purity and ≥ 90% yield in the shortest run time.

Guided Exploration Questions

  1. Set the flow rate to 5 mL/min. What happens to N and to the resolution between B and C? Use the van Deemter equation in Details to explain which term dominates for proteins in porous particles.
  2. With the default settings, make the gradient much steeper (shorter length, i.e. fewer CV). How do run time and resolution change? Why does a gentler gradient give better resolution but broader, more dilute peaks?
  3. Increase the sample load above 100% of column capacity. Where does the missing protein go, and what happens to peak width?
  4. Shrink the particle diameter from 90 µm to 15 µm. What do you gain and what is the practical cost (pressure drop, not modeled here)?
  5. Meet the challenge above. Then explain the trade-off between purity and yield as you widen or narrow the collection window.

Details

Equations

Retention factor (stoichiometric displacement model):

\[ k' = \left(\dfrac{c}{c^*}\right)^{-z} \]

Column dead time and volume:

\[ t_0 = \dfrac{\varepsilon_T V}{Q}, \qquad V = \dfrac{\pi}{4}D^2L \]

Peak standard deviation in gradient elution, with overload broadening:

\[ \sigma_t = \dfrac{t_0\left(1+k'_e\right)}{\sqrt{N}}\left(1+1.2\,\varphi\right), \qquad N=\dfrac{L}{H} \]

Van Deemter plate height:

\[ H = 2d_p + \dfrac{1.2\,D_m}{u} + \dfrac{d_p^{\,2}}{30\,D_{pore}}\,u, \qquad u=\dfrac{Q}{A} \]

Resolution between two adjacent peaks:

\[ R_s = \dfrac{t_{R,2}-t_{R,1}}{2\left(\sigma_1+\sigma_2\right)} \]

Mass of species $i$ collected between times $a$ and $b$ (peaks treated as Gaussian, $\Phi$ the standard normal CDF), and the resulting purity and yield of protein B:

\[ m_i(a,b) = m_i\left[\Phi\!\left(\dfrac{b-t_{R,i}}{\sigma_i}\right)-\Phi\!\left(\dfrac{a-t_{R,i}}{\sigma_i}\right)\right] \]
\[ \text{purity} = \dfrac{m_B(a,b)}{\sum_i m_i(a,b)}, \qquad \text{yield} = \dfrac{m_B(a,b)}{m_{B,\text{feed}}} \]

Symbols

  • $k'$ – retention factor
  • $c$ – NaCl concentration at the protein's position (M)
  • $c^*$ – characteristic salt concentration at which $k'=1$ (M)
  • $z$ – characteristic charge, the slope of log $k'$ vs. log $c$
  • $t_0$ – column dead time, the residence time of an unretained species (min)
  • $\varepsilon_T$ – total column porosity
  • $V$ – column volume (mL), $D$ – column diameter (cm), $L$ – column length (cm)
  • $Q$ – volumetric flow rate (mL/min)
  • $N$ – number of theoretical plates, $H$ – plate height (cm)
  • $d_p$ – particle diameter
  • $D_m$ – free (bulk) diffusivity of the protein
  • $D_{pore}$ – effective intraparticle (pore) diffusivity
  • $u$ – superficial velocity, $A$ – column cross-sectional area
  • $\varphi$ – sample load as a fraction of the column's binding capacity
  • $\sigma_t,\,\sigma_i$ – standard deviation (width) of a peak in time
  • $t_R$ – retention time, the time of the peak maximum
  • $R_s$ – resolution between two adjacent peaks
  • $m_i(a,b)$ – mass of species $i$ collected between times $a$ and $b$
  • $\Phi$ – cumulative distribution function of the standard normal distribution

Model Assumptions

Band migration is integrated through the salt gradient, with the salt front traveling at the mobile-phase velocity. Peaks are Gaussian. Overload is approximated: peaks broaden by $(1+1.2\varphi)$ and any load above 100% of capacity passes through unbound. Column: 1 cm i.d., total porosity 0.8, capacity 40 mg/mL resin. Pressure drop, dispersion outside the column, and non-linear isotherm effects are not modeled. Intended for teaching, not design.

Separation procedure

  1. Equilibration and load. The column is packed with anion-exchange resin and equilibrated with a low-salt buffer (the "start salt" value). The three-protein mixture is loaded at the inlet. At this low salt concentration, all three proteins bind tightly to the resin (high retention factor k′), so they stick near the top of the column rather than washing straight through – unless more protein is loaded than the resin's capacity, in which case the excess breaks through unbound (the "flow-through" peak).
  2. Low-salt wash (hold period). After loading, buffer continues to flow at the same low starting salt concentration for a short while before the gradient begins. This washes out anything not actually bound to the resin, without yet displacing the bound proteins.
  3. Salt gradient (elution). The feed salt concentration then ramps up linearly, from the start salt to the end salt, over however many column volumes are set as the gradient length. As salt increases, it competitively displaces the bound proteins – each protein's retention factor k′ falls as salt rises. A protein elutes once its k′ has dropped to roughly the 1 to 0.1 range.
  4. Differential elution. Because each protein has its own characteristic charge and binding strength, their k′ values fall off at different salt concentrations, so they elute at different times – that is the separation. The weaker-binding protein comes off first, at lower salt; the strongest binder comes off last.
  5. Detection and collection. A detector at the outlet records the elution profile as the chromatogram. A time window is chosen to "collect" – everything eluting outside that window goes to waste – and the purity and yield of protein B in the collected fraction are reported.

About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.