In this thesis, we investigate interactions between conical inclusions in a lipid bilayer membrane and make predictions about the patterns they form. To find these patterns, we derive an expression for the energy of a membrane as a function of the inclusion locations and search numerically for the pattern
that gives minimum energy.

The energy of a membrane with conical inclusions can be derived using the point particles model with corresponding formalism developed by Dommersnes and Fournier [1]. In this thesis, we apply this formalism to the finite size particles model described by Weikl et al. [2]. We compare the results of both
models for a system of three inclusions, to validate the point particles model’s ability to accurately predict equilibrium patterns for conical inclusions. For most non-conical inclusions, however, the point particles model proves inadequate, leaving only the computationally intensive finite size particles model to be used for more complex inclusions.

We develop a new numerical method for finding equilibrium patterns: the gradient descent method. This method is several hundred times faster than the standard Metropolis algorithm, and gives acceptable results. For large systems of inclusions, the method is very sensitive to local minima and has difficulties
merging small groups. The addition of noise in the Brownian motion method proves to be unable to resolve the local minima sensitivity, but we speculate that small bursts of high noise or grouping stable inclusions structures and moving the groups as a whole may be more effective.

Using the point particles model, we found that four-inclusion square-shaped structures and six-inclusion butterfly-shaped structures are favored in all systems with more than six inclusions.

The phenomenon of cluster synchronization in ring networks is studied in the first part of this thesis by means of Representation Theory. The second part of this thesis is devoted to the determination of the size of the Basin of Attraction for a stable node.