EXAMORPHIC
Biotherapeutics · Scientific discovery
We have developed a physical model of protein aggregation at the concentrations used in antibody formulations. In this model, each aggregate has its own position, shape and surface chemistry. The collision efficiency between aggregates is computed from the properties of their binding sites, whereas conventional kinetic models obtain it by fitting to data.
Background
Monoclonal antibodies for subcutaneous injection are formulated at concentrations of 100 mg/mL and above. Under these conditions, partially unfolded proteins can bind to one another and form aggregates. Aggregation limits shelf life and is monitored as a critical quality attribute, because protein aggregates are associated with immunogenicity.
The stability of a formulation is still established largely by experiment. Candidate buffers, pH values and excipients are each stored for weeks or months at several temperatures. A model that predicts how aggregation depends on these conditions could reduce the number of such experiments.
Rate equations such as the Lumry–Eyring nucleated polymerization model describe the aggregate population by size alone. They do not include the positions or shapes of the aggregates. The collision efficiency between aggregates is fitted separately for each solution condition.
In this model, the dissolved protein is represented by a density field, and each aggregate is a region of space with a sharp boundary. Growth, merging and bonding between surface sites are described by conditions on these boundaries. The model therefore predicts the shapes of aggregates and their spatial arrangement. The collision efficiency is computed from the binding sites and is not fitted.
Method
Examorphic develops physical models of processes that are difficult to simulate with existing methods, either because the relevant time scales are too long for molecular dynamics or because averaged descriptions do not retain the spatial structure of the system. The Protein Aggregation Model was developed in six stages.
The growth of protein aggregates in solution is mathematically similar to the growth of islands on a crystal surface during epitaxy. A level set model of epitaxy already describes islands that nucleate, grow at their edges and coalesce, and this model was the starting point.
Monomers diffuse through the solution and attach to aggregate surfaces. Each surface has binding sites. The equilibrium fraction of unbonded sites is obtained from Wertheim’s thermodynamic perturbation theory for particles with attractive patches.
In the well-mixed limit, the spatial averages of the model should satisfy the established rate equations term by term. For two isolated aggregates, the merger rate should equal the Collins–Kimball rate. Both limits were derived analytically and then verified in computations.
Aggregates are represented by level sets on adaptive grids, which are refined near the interfaces. The computational cost therefore depends on the area of the interfaces and not on the number of protein molecules.
Before each validation computation, we recorded which result would count against the model. When a result disagreed with this expectation, we identified the cause of the disagreement before reporting the result.
The predictions were compared with independent published measurements on a monoclonal antibody. The paper reports where the model agrees with these data, where it does not, and which physical effects it omits.
Model
The film shows one computation in which 79 aggregates are seeded in a periodic box with a side of 28 protein diameters. During the computation, the amount of free monomer decreases to 74 per cent of its initial value. Mergers reduce the number of aggregates to 45, and 12 of these exceed the critical size and become insoluble. The four panels show the same state of the computation in four representations, which are described below the film.
Each aggregate surface is divided into sticky patches, on which monomer attaches, and inert patches, on which it does not. When an aggregate exceeds the critical size, it becomes insoluble and its surface no longer grows.
Each aggregate is the region in which a signed distance function is negative. Changes of shape follow from the evolution of this function. Two regions that come into contact merge without an additional rule.
The dissolved protein is described by a continuous density field that diffuses through the solution. A growing aggregate depletes the monomer in its surroundings. Neighbouring aggregates therefore compete for the same monomer.
The fraction of unbonded binding sites is a field defined on each aggregate surface. When two aggregates are in contact, they merge through a reaction between their unbonded sites. The collision efficiency is computed from the rate of this reaction.
Results
0.5806± 0.0013
The rate at which two aggregates merge depends on their sizes through a power law. In the model, the exponent of this law is determined by the geometry of the aggregates and by the availability of their binding sites. The computed value is 0.5806 ± 0.0013. Arosio and co-workers obtained 0.600 ± 0.010 by fitting measurements on a monoclonal antibody.
13/ 14
The stability ratio is the factor by which aggregation is slower than it would be if every collision formed a bond. We computed it with a bond strength of 12 kBT. The computed values reach the published values for 13 of 14 combinations of pH, salt and temperature when the bond rate is at the upper end of its estimated range. At the central estimate of the bond rate, 7 of the 14 values are reached.
2.2–3.8×
When the positions of the aggregates are resolved, rigid aggregates at a volume fraction of about 0.22 merge at 2.2 to 3.8 times the two-body Collins–Kimball rate. At this volume fraction, pairs of aggregates are in contact more often than in a dilute solution. A rate equation for a well-mixed solution does not include this effect.
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In the well-mixed limit, the computed monomer fraction agrees with the Lumry–Eyring nucleated polymerization model to within eight parts in ten thousand. The difference arises because the model identifies the size of an aggregate with its volume.
The model does not include hydrodynamic interactions between aggregates, and the aggregates do not rotate. The absolute value of the stability ratio depends on the bond rate between binding sites, which is known only to within a factor of ten.
Applications
In the model, solution conditions enter through a small number of properties of the binding sites. Once these properties are known for a molecule, buffers, pH values and temperatures could be compared by computation before material is committed to long stability studies.
Each aggregate in the model has a position and a shape. The model therefore gives aggregate morphologies, fractal dimensions and spatial correlations, which can be compared with subvisible particle imaging and light scattering. Rate equations do not provide these quantities.
The bond strength, the bond rate and the number of binding sites can be obtained from molecular simulations of a single antibody. The model then relates these molecular properties to aggregation in a formulation.
Origin
The mathematical formulation of the model was first developed for the growth of semiconductor crystals. The same free-boundary formulation is applied here to protein aggregation.
The level set formulation of island dynamics in epitaxial growth. Adatoms are described by a density field, and islands are level-set regions that nucleate, grow and coalesce.
Caflisch et al., Appl. Math. Lett. 12, 13–22An implementation of the island dynamics model on parallel adaptive quadtree grids.
Mistani et al., J. Comput. Phys. 361, 150–166The Protein Aggregation Model, in which three-dimensional aggregates have binding sites on their surfaces. In the well-mixed limit, the model reduces to the established kinetic models of protein aggregation.
Mistani, arXiv:2609.27185Preprint
Spatially Resolved Nucleated Polymerization: A Free-Boundary Model of Protein Aggregation in Concentrated Solutions
Contact
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