The chromatographic technology is widely used to purify various biomolecules [[1], [2], [3], [4], [5]], due to its flexibility, scalability, robustness, rapid processing, and the ability to obtain target products with high purity [[6], [7], [8]]. Characterizing the adsorption behaviors of biomolecules on chromatography adsorbent is important for the design of suitable purification processes. For most small molecules and proteins, the pores in chromatography adsorbent are considered large enough to allow efficient entry of the adsorbates, as depicted in Fig. 1(a); while for some viral-based particles including virus and virus-like particles, which are important biomolecules used as vaccine antigens and gene therapy vectors [2], their chromatographic adsorption behaviors might be quite different. These bioparticles unusually have large sizes ranging from tens to hundreds of nanometers, which are close to or even larger than the pore diameters of most adsorbents. The bioparticles initially adsorbed on the adsorbent will cause physical blocking in the pore channels and hinder the diffusion of subsequent bioparticles toward the depth of the channels [[9], [10], [11]]. Therefore, unlike protein adsorption that can evenly distribute in the whole adsorbent, the adsorption of the bioparticles is usually limited in the outer layer or even on the surface of the adsorbent, as schematically illustrated in Fig. 1(b) and 1(c) [[12], [13], [14], [15]]. Establishing a suitable adsorption model to describe this adsorption behavior is important to understand the chromatographic process of bioparticles and design efficient separation strategies.

To date, many isotherm models have been developed and applied to describe the adsorption behavior of biomolecules on chromatography, which generally can be classified into the stoichiometric and non-stoichiometric adsorption models [[16], [17], [18], [19], [20]].

Among them, the steric mass action (SMA) model proposed by Brooks and Cramer [16] is one of the most typical stoichiometric models. According to model assumption, for ion-exchange adsorption, the counter-ions of the adsorbent are either displaced from the adsorbent or sterically shielded by the adsorbed biomolecules, thus the nonlinear adsorption behavior up to saturation of protein can be attributed to a reduction of available counter-ions. Compared with the classical Langmuir isotherm adsorption model, the SMA model explicitly accounts for the effect of the ionic strength on protein adsorption based on the mass action law [21]. More importantly, the SMA model explains the steric hindrance of salt counter-ions upon binding to proteins by introducing the steric factor, σ, which is the model parameter presenting the number of binding sites sterically shielded by each of the adsorbed protein molecules. Therefore, the SMA model is more suitable for describing the adsorption of biomolecules. With the development of thermodynamic and kinetic aspects, the applications of the SMA model have been extended from describing the adsorption isotherms to predicting the dynamic binding/eluting behavior in the chromatographic separation process [[22], [23], [24], [25]]. Meanwhile, besides the IEC process, the SMA model has also been applied in describing other types of chromatography processes, such as immobilized metal affinity chromatography (IMAC)[26,27], dye–ligand affinity chromatography[28], as well as hydrophobic chromatography of proteins[29].

The SMA model has not only made these significant advances in protein adsorption, but has also been applied to describe the adsorption process of very large bioparticles on chromatographic media [30] and anion-exchange membrane [7,31]. However, the model theory of SMA is all based on the assumption that all the ligands on the adsorbent are accessible for the free biomolecules, no matter whether the ligands are on the adsorbent surface or distributed inside the innermost of the porous adsorbent. For instance, when the SMA model was applied to develop the mechanistic model of adeno-associated virus (AAV) in ion exchange chromatography, it was assumed that all species had unrestricted access to all pores of the stationary phase particles and the steric factors to be 0 for all components [30]. Apparently, such assumptions are inconsistent with the real experimental phenomena, since pore-blocking characteristics for the adsorption of a large protein with diameters similar to AAV in IEC media have been well reported[32]. Therefore, as a consequence of such assumptions, the estimated isotherm parameter values differ from the fundamental values they represent [30]. For the adsorption of viral bioparticles in membrane chromatography the ligands bound to the membrane surface are accessible to bioparticles with almost no diffusivities, which is in contrast to typical resin bead that these large biologicals are excluded largely from the ligands in the intra-bead pores [7,31]. As a consequence, the pore-blocking phenomenon usually does not exist.

Based on above discussion, in case the pore-blocking phenomenon cannot be neglected, particularly for the adsorption of most bioparticles having a size close to or even significantly larger than the pore size of the resin adsorbent, as depicted in Fig. 1(b) and 1(c), a large fraction of the ligands in the deep channels might become physically inaccessible to the bioparticles due to the blocking of the pores, instead of shielding due to steric hindrance by adsorbed bioparticles. In such a case, the σ value derived from the SMA model will possibly be largely overestimated, since the ligands distributed in the pore-blocking region will also be regarded as being shielded. As a consequence, the accuracy of SMA model parameters will be affected by the pore-blocking phenomenon, and modification of the SMA model becomes necessary for improving the accuracy of the model.

To this end, we aimed to develop a pore-blocking steric mass action (PB-SMA) model for the adsorption of bioparticles. Here, the inactivated foot-and-mouth disease virus (iFMDV) was used as a model of bioparticles. The intact iFMDV is an important veterinary vaccine antigen against the highly contagious foot-and-mouth disease (FMD) in cloven-hoofed animals. It has a spherical shape with a diameter of about 30 nm [33], and there are about 1,200 histidine residues in each iFMDV particle enabling its adsorption on the IMAC adsorbent with high specificity [33]. Thus, its adsorption on the IMAC adsorbents with different pore sizes, either smaller than or about twice as large as the particle size of iFMDV, was studied. The typical pore-blocking phenomenon for iFMDV adsorption has been experimentally observed by confocal laser scanning microscopy analysis [12]. Therefore, it is a reprehensive system for developing the PB-SMA model describing the adsorption of viral-based particles.

To establish the PB-SMA model, an additional parameter reflecting the inaccessible ligand numbers (Lin) in the physically blocked pore area was introduced, and then the σ reflecting the actual ligands shielded by the adsorption of iFMDV was obtained by adsorption equilibrium experiments of iFMDV adsorbed on IMAC adsorbent with different pore sizes and ligand densities. The results showed that compared with the conventional SMA model, the σ derived from the PB-SMA model was more reasonable and closer to the maximum number of metal ion sites that could be shielded by a single viral particle. The PB-SMA model was further validated by studying the adsorption behaviors of hepatitis B surface antigen virus-like particles (HBsAg VLPs) on IEC adsorbent. The PB-SMA model compensates for the shortcomings of existing models in describing the adsorption behavior of bioparticles and provides a theoretical basis for guiding the design of bioparticle purification processes.