Calculate the Number of Immobilized Proteins per Bead of Agarose Affinity Supports

A consideration of relative sizes and dimensions of agarose resin beads, antibodies, proteins, chemical modification groups, and affinity ligands

by Douglas A. Hayworth, Ph.D.; Greg T. Hermanson, B.S. - 03/03/14

Protein biology research as we know it would not be possible without the variety of chromatography resins, beads, membranes, plastic surfaces and other solid or porous materials to immobilize proteins and affinity ligands. Because these tools and methods are just a means to an end rather than the primary focus of research, it is easy for investigators to lose sight of the relative molecular, macromolecular and physical scales involved in the processes being used. In certain ways, those of us who communicate about the chemical reactions that form the basis for protein purification and detection are guilty of perpetuating false notions and perceptions relating to scale. For example, the following illustration (Figure 1) is at the same time both very informative concerning the chemical reactions that take place during the immobilization of an antibody and highly misleading regarding the relative dimensions of the components within the reaction.

For the purpose of communicating key features of the immobilization mechanism, such illustrations show the sizes of functional groups and beaded supports compressed or collapsed into one order of magnitude, even though the true molecular sizes of the functional elements actually span nearly six orders of magnitude (Table 1). Figure 2 provides a more accurate representation of the sizes involved when working with macromolecules having diameters in the low nanometer range as compared to porous resins having diameters in the micron range. In this illustration, individual functional groups and atoms cannot even be seen, as they are nearly one hundred times smaller than the macromolecules being immobilized.

Table 1. Relative sizes of functional groups in diagrams of immobilized affinity ligands.

Figure 2. Diagram of the relative sizes of agarose beads and antibodies (proteins).

In contrast to such diagrams are the actual specifications of ligand loading and binding capacities, which describe the concentration of immobilized ligands or proteins relative to the prepared affinity resins. Typically, these are stated as the mass of ligand coupled (milligrams, mg), or as the concentration of ligand coupled (micromoles, µmol or sometimes even nanomoles, nmol) of ligand per milliliter (mL) of settled agarose resin (Table 2).

Table 2. Example loading and/or binding capacity specifications of selected agarose affinity resins.

Values are product claims or test specifications and are expressed per milliliter of settled beads, as in a packed 1mL affinity column. See the respective product pages for details.

For the researcher who wishes to envision a more accurate picture of the underlying molecular and physical sizes contained in an immobilized affinity resin, it is instructive to calculate how the ligand loading values and binding capacities relate to the number and size of individual molecules. In this article, we present the calculations needed to answer the following sorts of questions:

• What is the volume of a single agarose bead?
• How many agarose beads are in one milliliter of settled resin?
  (What is the number of agarose beads per milliliter of packed resin?)
• How many functional groups are there per agarose bead?
  (What is the number of functional groups per bead?)
• How many antibody or protein molecules are coupled per agarose bead?
• How many antibody or protein molecules can bind to one agarose bead?

Number of agarose beads per milliliter of settled resin

The first step in making the conversion from concentration (e.g., µg ligand per mL of settled beads) to the number of ligand molecules per bead is to determine the number of beads per milliliter of packed resin. It would be ideal to determine this value empirically in a manner similar to the way cells are counted: make a known dilution of a measured volume of settled beads and then count the beads in a sample under a microscope. We were unable to find published results of this type of measurement. Therefore, our approach is to calculate the average bead volume and then use sphere-packing theory as the basis for estimating the number of those beads in one milliliter of resin.

1. Agarose beads have a diameter of 45 to 165µm.
   • Let’s assume the beads are spheres with a diameter of 100µm.
   • That means the average bead has a radius of 50µm = 0.05mm = 0.005cm
2. Volume of a single bead therefore is:
   • Volume of a sphere = (4/3)πr³
   • For radius = 0.005cm, the single bead volume = 5.236 x 10^-7cm³ = 5.236 x 10^-7mL
   • For those with a really tiny pipetter, that’s a volume of 0.524nL!
   • For the entire bead population, therefore, the individual bead volumes range from 0.047nL (for 45µm diameter beads) to 2.3nL (for 165µm diameter beads).
3. Sphere-packing theory:
   • Formulas indicate that spheres pack with 65 to 75% efficiency (http://en.wikipedia.org/wiki/Sphere_packing).
   • Beaded agarose is a mixture of bead sizes and the beads are neither perfectly spherical nor hard; as such, sphere-packing theory might provide only a rough estimate of the true value for the number of particles in a given volume. However, for our purposes, it is more than adequate.
   • In other words, assuming 75% packing efficiency, 0.75mL of actual bead volume will settle to form approximately 1mL of bed volume (e.g., 1mL of resin in a column).
4. Final calculation:
   • Number of beads per mL:
   • = 0.75(1 bead/5.236 x 10^-7mL)
   • = 1.432 x 10⁶ beads/mL of bed
   • = approx. 1.4 million beads per milliliter of resin

Number of functional or reactive groups per agarose bead

1. Activated resins are manufactured to have 10 to 100µmol of groups per mL of packed resin.
   • For example, CarboxyLink Resin is manufactured with 16-20µmol amines per mL of packed beads (resin bed)
   • We will do the calculation for just 1µmol per mL of resin bed (= 1 x 10^-6mol).
   • The number of molecules, then, is equal to the number of moles times Avogadro’s number: (1 x 10^-6mol) x (6.02 x 10²³) = 6.02 x 10¹⁷ molecules per mL of resin bed.
2. Final calculation:
   • 1µmol of affinity groups per mL of resin bed:
   • = (6.02 x 10¹⁷ groups per mL of resin bed) / (1.432 x 10⁶ beads per mL of resin bed)
   • = 4.204 x 10¹¹ groups per bead
   • = approx. 420 billion groups per agarose bead
3. Therefore:
   • We can safely say that, for typical activations where beads contain perhaps 20 to 40µmol of reactive groups per mL of resin:
   • There are trillions of active groups per agarose bead

Number of immobilized antibody molecules per agarose bead

1. Typically, the protein coupling capacities of activated resins are determined empirically with example experiments (often using mouse or rabbit whole IgG). Usually, these coupling capacities are expressed as micrograms or milligrams of protein per milliliter of resin.
   • For example, experiments indicate that 1mL of AminoLink Plus Coupling Resin (Part No. 20501) can immobilize 4.7mg mouse IgG with 97% efficiency. That translates into an effective coupling yield of 4.5mg IgG per mL of resin.
   • Let’s simplify and be conservative by doing the calculation for just 1mg of IgG actually coupled per mL of resin bed.
   • The MW of IgG = 150,000g/mol; therefore, 1mg of IgG = 6.67 x 10^-9mol
   • This mole quantity times Avogadro’s number is equal to 4.01 x 10¹⁵ IgG molecules per mL of resin bed.
2. Final calculation:
   • 1mg IgG per mL of resin bed:
   • = (4.01 x 10¹⁵ IgG molecules per mL of resin bed) / (1.432 x 10⁶ beads per mL of resin bed)
   • = 2.80 x 10⁹ IgG molecules per bead
   • = nearly 3 billion antibody molecules per agarose bead
3. Therefore:
   • We can safely say that, for typical protein immobilization procedures,
   • Billions of protein molecules couple per agarose bead

Number of bound protein molecules per agarose bead

1. Binding capacity, like covalent coupling capacity, is usually expressed as milligrams of protein target bound per milliliter of affinity resin.
   • For example, Thermo Scientific HisPur Ni-NTA and Cobalt Resins have static binding capacities for His-tagged proteins of 10 to 50mg/mL of resin. Testing is done using His-GFP (27kDa).
   • Let’s do the calculation for the 10mg of His-GFP fusion protein per mL of resin bed.
   • The His-GFP MW = 27,000g/mol; therefore, 1mg of His-GFP = 3.7 x 10^-8 mol
   • This mole quantity times Avogadro’s number is equal to 2.23 x 10¹⁷ His-GFP molecules per mL of resin bed.
2. Final calculation:
   • 10mg His-tagged fusion protein per mL of resin bed:
   • = (2.23 x 10¹⁷ His-GFP molecules per mL of resin bed) / (1.432 x 10⁶ beads per mL of resin bed)
   • = 1.56 x 10¹¹ His-GFP molecules per bead
   • = 156 billion His-GFP molecules per agarose bead
3. Therefore:
   • We can safely say that, for many protein purification procedures,
   • Many billions of protein molecules bind per agarose bead

There is another very important consideration when attempting to properly visualize the process of affinity purification at the scale of individual agarose beads. Agarose beads are not hard, solid, and impervious; rather, they are a highly porous aerogel with mesh-like structures composed of loosely interwoven polysaccharide strands in helical structures. Therefore, ligands and typical protein molecules can diffuse within the matrix and attach and bind throughout the entire volume of an agarose bead. In other words, attachment is not confined to the outer surface area.

Beads of standard 4% or 6% beaded agarose resins (available from many suppliers) range in diameter from 45 to 165µm, which corresponds to approximately 5000 to 50,000 times the globular protein length or width of antibodies and other proteins. Although diagrams of covalent immobilization methods are instructive for describing the chemical basis for coupling reactions, they are misleading with regard to the relative sizes and numbers of functional groups involved. With a few simple calculations, we have made the necessary unit conversions to express example concentration values (activation levels, loading levels, coupling and binding capacities) in terms of the number of molecules per individual agarose bead. Perhaps for most researchers, knowing these values satisfies merely academic curiosity and provides little practical value. However, we suggest that it is generally helpful for scientists to maintain accurate perceptions about the molecular dynamics of laboratory methods.

Summary of results:

• In 4% or 6% beaded agarose supports, individual agarose beads have an average spherical bead volume of about half a nanoliter (0.52nL).
• In one milliliter (1mL) of settled agarose resin there are approximately 1.5 million individual agarose beads.
• In typical activated affinity supports used for covalent immobilization, each individual agarose bead contains trillions (1 x 10¹²) of functional or reactive groups (e.g., aldehydes, amines, NHS esters, etc.).
• In typical affinity supports for protein purification, each individual agarose bead is capable of interacting with or binding to billions (1 x 10⁹) of target molecules (e.g., proteins, antibodies, etc.).

Table 3. Summary of calculated values for beaded agarose supports.