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In this paper we introduce a fully flexible coarse-grained model of immunoglobulin G (IgG) antibodies parametrized directly on cryo-EM data and simulate the binding dynamics of many IgGs to antigens adsorbed on a surface at increasing densities. Moreover, we work out a theoretical model that allows to explain all the features observed in the simulations. Our combined computational and theoretical framework is in excellent agreement with surface-plasmon resonance data and allows us to establish a number of important results. (i) Internal flexibility is key to maximize bivalent binding, flexible IgGs being able to explore the surface with their second arm in search for an available hapten. This is made clear by the strongly reduced ability to bind with both arms displayed by artificial IgGs designed to rigidly keep a prescribed shape. (ii) The large size of IgGs is instrumental to keep neighboring molecules at a certain distance (surface repulsion), which essentially makes antigens within reach of the second Fab always unoccupied on average. (iii) One needs to account independently for the thermodynamic and geometric factors that regulate the binding equilibrium. The key geometrical parameters, besides excluded-volume repulsion, describe the screening of free haptens by neighboring bound antibodies. We prove that the thermodynamic parameters govern the low-antigen-concentration regime, while the surface screening and repulsion only affect the binding at high hapten densities. Importantly, we prove that screening effects are concealed in relative measures, such as the fraction of bivalently bound antibodies. Overall, our model provides a valuable, accurate theoretical paradigm beyond existing frameworks to interpret experimental profiles of antibodies binding to multi-valent surfaces of different sorts in many contexts.

We investigate numerically pseudo-first-order irreversible bimolecular reactions of the type A + B → B between hard spheres undergoing event-driven Brownian dynamics. We study the encounter rate and the survival probability of A particles as functions of the packing fraction ϕ in the trapping (a single particle diffusing among static non-overlapping traps) and target (many traps diffusing in the presence of a single static target particle) settings, as well as in the case of diffusing traps and particles (full mobility). We show that, since inertial effects are accounted for in our simulation protocol, the standard Smoluchowski theory of coagulation of non-interacting colloids is recovered only at times greater than a characteristic time Δt, marking the transition from the under-damped to the over-damped regime. We show that the survival probability S(t) decays exponentially during this first stage, with a rate 1/τ0 is proportional to φ. Furthermore, we work out a simple analytical expression that is able to capture to an excellent extent the numerical results for t < Δt at low and intermediate densities. Moreover, we demonstrate that the time constant of the asymptotic exponential decay of S(t) for diffusing traps and particles is k(S)(-1), where kS = 4π(DA + DB)Rρ is the Smoluchowski rate. Detailed analyses of the effective decay exponent β = d [log(-logS(t))]/d (logt) and of the steady-state encounter rate reveal that the full mobility and trapping problem are characterized by very similar kinetics, rather different from the target problem. Our results do not allow one to ascertain whether the prediction S(t) is proportional to exp(-at(3/2)) (a = const.) as t → ∞ for the trapping problem in 3D is indeed recovered. In fact, at high density, S(t) is dominated by short encounter times, which makes it exceedingly hard to record the events corresponding to the exploration of large, trap-free regions. As a consequence, at high densities the steady-state rate simply tends to 1/τ0. Finally, we work out an analytical formula for the rate that shows a remarkable agreement with the numerics up φ = 0.4.

In the diffusion model of chemical reactions, the encounter (reaction) rate between reactant particles is governed by the Smoluchowski equation, which is a diffusion equation in a field of forces. We consider crowded environments where the particles diffuse through a "liquid" of like particles. Assuming that the liquid-like short-range structure around the reactant gives rise to an effective (osmotic) barrier leads us to map a complicated many-body problem to a one-dimensional problem. This allows us to describe theoretically such complex systems that are encountered in many applications where crowding, intermolecular interactions, and flow are simultaneously present. A particularly important effect is discovered which is due to the interplay between shear and crowding. This effect is responsible for unexpected peaks in the reactivity at low flow intensity which may explain, among other things, the bizarre colloidal stability behavior of concentrated protein suspensions.

We derive an analytical expression for the effective force between a pair of macrospheres immersed in a sea of microspheres, in the case where the interaction between the two unlike species is assumed to be a square well or a square shoulder of given range and depth (or height). This formula extends a similar one developed in the case of hard core interactions only. Qualitative features of such effective force and the resulting phase diagrams are then analyzed in the limit of no interaction between the small particles. Approximate force profiles are then obtained by means of integral equation theories (PY and HNC) combined with the superposition approximation and compared with exact ones from direct Monte Carlo simulations.