Biophysically inspired model for functionalized nanocarrier adhesion to cell surface: roles of protein expression and mechanical factors

In order to achieve selective targeting of affinity–ligand coated nanoparticles to the target tissue, it is essential to understand the key mechanisms that govern their capture by the target cell. Next-generation pharmacokinetic (PK) models that systematically account for proteomic and mechanical factors can accelerate the design, validation and translation of targeted nanocarriers (NCs) in the clinic. Towards this objective, we have developed a computational model to delineate the roles played by target protein expression and mechanical factors of the target cell membrane in determining the avidity of functionalized NCs to live cells. Model results show quantitative agreement with in vivo experiments when specific and non-specific contributions to NC binding are taken into account. The specific contributions are accounted for through extensive simulations of multivalent receptor–ligand interactions, membrane mechanics and entropic factors such as membrane undulations and receptor translation. The computed NC avidity is strongly dependent on ligand density, receptor expression, bending mechanics of the target cell membrane, as well as entropic factors associated with the membrane and the receptor motion. Our computational model can predict the in vivo targeting levels of the intracellular adhesion molecule-1 (ICAM1)-coated NCs targeted to the lung, heart, kidney, liver and spleen of mouse, when the contributions due to endothelial capture are accounted for. The effect of other cells (such as monocytes, etc.) do not improve the model predictions at steady state. We demonstrate the predictive utility of our model by predicting partitioning coefficients of functionalized NCs in mice and human tissues and report the statistical accuracy of our model predictions under different scenarios.

(1) NC translation: An attempt to randomly displace the center of mass of the NC within a cube of length δ centered around its current position. The value of δ is chosen (and also modifed during runtime) such that nearly 50% are the attempted moves are accepted [1,2].
(2) NC rotation: An attempt is made to change the orientation of the nanocarrier from  Fig. S1(c)) holding the triangulation fixed [3,4] (6) Link flip : A link shared by two neighboring faces of the triangulated surface is removed and reconnected to the two previously unconnected vertices as illustrated in is the weight factor to generate N independent configurations with energy H t,i and exp(−βH t,i ) is the weight factor to generate N −1 independent configurations starting with the initial configuration whose energy is given by H t, * .

MEMBRANE
Consider a spherical nanocarrier of radius a, expressing N ab antibodies on its surface, and is bound to a cell membrane, expressing N ant surface receptors. If X, M, and R be the shorthand notations for the nanocarrier, membrane, and receptor degrees of freedom, then we define the configurational integrals for the individual components as where x, y, and z denote the Cartesian coordinate system and φ, θ, and ψ denote the three Euler angles. The above equation describes the volume accessible to an unbound nanocarrier in the configurational space and can further be reduced to, If R i denotes the position vector of the i th receptor molecule, the configurational integral for the receptor degrees of freedom may be expressed as, and if M i denote the three dimensional position of the i th coarse grained bead on the membrane (i.e., the i th vertex of the triangulated surface) the configurational space for the membrane is defined as Following [7,8] the probability for the nanocarrier to be in an unbound state (i.e., multivalency m = 0) as, If we treat the membrane surface to be a flat substrate, its energy H sur = 0 and the contribution from the membrane degrees of freedom can be taken to be a constant, which allows us rewrite the above equation as, Here the integrations are carried out over all the unbounded states, A u N and A u R are the areas traversed by the nanocarrier and the receptor molecules, respectively, in their unbound state, and r * denotes the upper limit for the reaction coordinates beyond which bond formation is disallowed purely for geometric reasons. Now, consider the formation of the first receptor-ligand bond which is penalized by an energy cost given by the potential of mean force W(r), where r denotes a reaction coordinate. Since any given receptor molecule can in principle form bonds with any given antibody on the nanocarrier surface, the total number of microstates with a single receptor-ligand bond is given by N ab × N ant and the probability to find the nanocarrier with a multivalency of 1 can be expressed as, where the orientation of the nanocarrier is constrained to be fluctuating around the Euler angles φ 0 , θ 0 , and ψ 0 , when a specific antibody is engaged in bonding. A b R and A b N are the area traversed by a bound receptor and the center of mass of the bound nanocarrier.
The above equation can be generalized to n b receptor-ligand bonds as, As pointed in out in [7,8], the reversible work done to go from a unbounded state (with n b = 0) to state with n b simultaneous receptor-ligand bonds is the logarithm of the ratio, which is also related to the association constant as, with the nanocarrier concentration [L] = (A u N L z ) −1 for a system with exactly one nanocarrier. Using eqn. (9) in eqn. (10) we get the final expression for the association constant to be, combinatorial entropy for receptor-ligand bond formation ∆φ ∆θ ∆ψ 8π 2

NC rotational entropy
a.
Computing spatial maps for the bound receptors: The spatial map of the bound receptors, for a given conformation of the membrane and the nanocarrier, is computed as follows. The mean orientation of the bound receptor-ligand bondsn defines a tangent plane that is shown in Fig. S2. A vector in the cartesian coordinate system can be  the bound ligands onto this plane, as shown in Fig. S2, to obtain spatial maps which are used to estimate the area traversed by a bound nanocarrier, and a bound receptor. We represent these patterns in terms of [t 1 ,t 2 ], the orthonormal unit vectors in a coordinate system attached to this plane.

S3. FORMULATION OF THE PHARMACO-KINETIC (PK) MODEL: RELA-TION TO STANDARDIZED UPTAKE VALUES
Conventional PK model for non-targeted delivery: In classical perfusion limited pharmacokinetic models [9,10], with C T denotes the concentration of the drug in a tissue compartment whose volume is V T , the outlet concentration C out at steady-state is expressed as: with K p being the partitioning coefficient which denotes the ratio of the probability for the drug to be associated with the tissue to that in the blood stream.

A. Targeted uptake
In considering NC targeting, the tissue volume is divided into two compartments, namely the endothelial cells lining the vasculature and the bulk tissue. The NC concentration at the endothelial cell surface and in the tissue are given by C * and C T , as shown in Fig. S3. As before, the volume of the tissue is V T . In the absence of any functional groups the injected NCs are absorbed from the blood stream and distributed into the tissue primarily through non-specific mechanisms such as diffusion, permeation, and facilitated transport, that proceed via both intracellular and extracellular routes. In such a scenario, the concentration of the nanoparticles on the endothelial surface is given by C * = C out .
Targeted NCs, on the other hand, adhere preferentially to the endothelial cells and as a result the concentration of the nanoparticle at the boundary of the tissue differs from C out .
The concentration of the NC is enhanced at the endothelial cells boundaries such that the enhanced value C * = C out , and in this case the partitioning of the NC concentration in the tissue is given by: In order to proceed with the development of the PK model for targeted carriers, it is necessary to establish a relation between C * and the bulk concentration C out . It should be noted that C * is the concentration of the NCs in the endothelial cells computed with respect to the volume of the tissue V T . Hence, the total number of NCs bound to an endothelial cell (with cross sectional area l 2 EC ), when the NC concentration is C * , is: where L EC,b is the length of the boundary layer within which the NC concentration is enhanced due to binding; i.e., L EC,b = r * in Eq. 11 for EC. N bound can also be expressed as C out l 2 EC L cap P b , where L cap is the size of the cell free layer in the capillary in which the NC marginates and P b is the probability of NC binding. Using the relationship for P b given by [11]: or alternatively, we obtain in the limit of K EC C out 1, Assuming that the entire NC fraction bound to the EC ends up in the tissue, (i.e., we assume the NC enternalization is neither rate-limiting nor saturates at steady state, and that the bound NC fraction on the EC also contributes to the tissue targeting in vivo), the concentration of the NC which includes contributions from both the tissue and from those adhered to the endothelial cells needs to be accounted for in the biodistribution.
Hence, the total number of nanoparticles harvested in the tissue N tot is given by: Experiments measure the volume concentration of endothelial cells as where the endothelial cell volume is approximated as V EC = l 2 EC D EC with D EC being the EC diameter. The above equation may be rewritten as: This framework can also account for additional contributions such as the where, K M is the association constant for targeted binding of nanoparticles to other cells.
The above equation can be related to the standardized uptake value, usually measured in percentage injected dose per gram of tissue, as Experiments based on targeting specific receptors have shown that other cells (e.g., monocytes) and endothelial cells are found with nearly similar compositions in a tissue, with typical values to be around ϕ EC ∼ 3−30%, ϕ M ∼ 3−10% [12][13][14]. The references [15,16] provide estimates of the diameters of endothelial cells and other cells.