Recent advances in experimental techniques have allowed one to track microscopic particles while moving in different biological environments. A celebrated example is the real time tracking of single proteins as they diffuse on a cell membrane. It is of key importance in the field of biology to characterize the diffusion of such particles. To do so, it is crucial to understand how the particles interact with their surrounding environment, as this interaction affects vastly the diffusion of the particles. Due to the complexity of the systems studied, there is a lack of simplified microscopic models able to explain and predict the phenomena observed, hence being an exciting open research question.
In our work, we focus on finding new models, based on the Brownian motion theory, for the movement of biological entities across heterogeneous environments, e.g. cell membranes . Surprisingly, recent studies have found that some heterogeneous systems can be very well described by the Ising model, which is one of the oldest and simplest example of a condensed matter Hamiltonian. This model was first introduced in order to study ferromagnetic materials. However, due to its simplicity, it is now used in many other fields of physics and science in general. The Ising model consists on a set of spins (physical bodies that can take two values, generally +1/-1 or pointing up/down) occupying discrete regular or irregular positions in space coupled to its neighbours with certain strength. A very interesting property of this model is that, close to its critical temperature, the spins tend to align with theirs neighbours, forming domains of spins pointing on the same direction. Returning to the biological side, this domains have been shown to describe the distribution of heterogeneities in a cell membrane, hence our interest on it.
Once we have a simplified version of the environment, let us introduce the random walkers to such a system. In our recent work, we propose that each spin domain has different properties, due to the heterogeneity of the environment, which affect directly the motion of the walker. More precisely, we introduce a model in which the diffusivity of each domain is inversely proportional to its area to certain power. This parameter governs the strength of the interaction between the particle and the environment.
In our work we show how the motion of the particle is heavily affected by the strength of the interaction, but also to how fast the environment evolves with respect to the movement of the particle. We also show that when setting the Ising environment to its critical temperature, the particles moving through it perform a subdiffusive motion, which means that the diffusion process is non-linear with respect to time. However, when departing from criticality, the particles subdiffuse, only for a transient time, to recover a normal diffusion after it.
 G. Muñoz-Gil et al., Phys. Rev. E 96, 052140 (2017).