The study of diffusion in macromolecular solutions is important in many biomedical applications such as separations, drug delivery, and cell encapsulation, and key for many biological processes such as protein assembly and interstitial transport. Not surprisingly, multiple models for the a-priori prediction of diffusion in macromolecular environments have been proposed. However, most models include parameters that are not readily measurable, are specific to the polymer-solute-solvent system, or are fitted and do not have a physical meaning. Here, for the first time, we develop a homogenization theory framework for the prediction of effective solute diffusivity in macromolecular environments based on physical parameters that are easily measurable and not specific to the macromolecule-solute-solvent system. Homogenization theory is useful for situations where knowledge of fine-scale parameters is used to predict bulk system behavior. As a first approximation, we focus on a model where the solute is subjected to obstructed diffusion via stationary spherical obstacles. We find that the homogenization theory results agree well with computationally more expensive Monte Carlo simulations. Moreover, the homogenization theory agrees with effective diffusivities of a solute in dilute and semi-dilute polymer solutions measured using fluorescence correlation spectroscopy. Lastly, we provide a mathematical formula for the effective diffusivity in terms of a non-dimensional and easily measurable geometric system parameter.

We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in R d and the scaling limit of the associated continuous-time random walk Z(t) on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process ε 2 Z(t/ε 2 ) converges to a linear driftŪ t and the case of interest to us is that of null driftŪ = 0. In this case, we show that εZ(t/ε 2 ) converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our framework allows for the modeling of very general forms of interactions such as attraction, repulsion, and bonding. We provide some sufficient conditions for null drift that include certain symmetries of the graph. We also provide a formal asymptotic derivation of the effective diffusivity in analogy with homogenization theory for PDEs. For the case of reversible jump rates, we derive an equivalent variational formulation. This derivation involves developing notions of gradient for functions on the graph's nodes, divergence for R d -valued functions on the graph's edges, and a divergence theorem.

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reference to the experimental data, under the so-called maximum entropy principle. Recent practical formulations of this approach involve simulations carried out over multiple replicas or iterative ensemble-correction procedures based on the determination of several (Lagrange) parameters. Here, we present an alternative, self-learning approach to sample molecular ensembles compatible with experimental data with the minimal possible bias on the simulation trajectories. The method does not require multiple replicas and is based on adding an adaptive bias potential during the simulation that discourages the sampling of conformations that are not consistent with the experimental measurements. To illustrate this approach, we applied this novel simulation technique to spin-labeled T4-lysozyme, targeting a set of spin-spin distance distributions measured by DEER/EPR spectroscopy. We show how the proposed method is able to efficiently sample the experimental distance distributions without altering uncorrelated degrees of freedom. We anticipate that this new simulation approach will be widely useful to obtain conformational ensembles compatible with diverse types of experimental measurements of biomolecular dynamics.