Abstract

Diffusion over complex potential landscapes is a common problem in many areas of science. In this thesis, I consider the dynamics of single particle diffusion over various complex energy landscapes. Colloidal spheres in a thin layer are used to model diffusion over different potential landscapes and to obtain information about the potentials and dynamics of single particles simultaneously. In the first experiment, I studied colloidal diffusion over a surface with potential traps arranged on a quasicrystalline lattice. The particle trajectories can be in two distinct states: a trapped state when the particle is within a trap and a free diffusion state when it is over a flat portion of the substrate. Dynamic properties such as the mean square displacements (MSD), transition time between states and transition time between traps are measured from these trajectories. The measured long time diffusion coefficient DL is in good agreement with the predictions of two theoretical models which link DL to the characteristic local properties from different perspectives. In the second experiment, I studied colloidal diffusion over a quenched two-dimensional random potential field. The gravitational potential $U(x, y)$ is modulated by the rugged surface of a closely packed colloidal monolayer fixed on a glass substrate, consisting of a random mixture of bidisperse silica spheres. By measuring the population probability histogram $P(x, y)$ of diffusing particles on top of the rugged surface, we obtain the potential $U(x, y)$ by the Boltzmann distribution. The dynamic quantities including MSD and the escape time from traps are measured. The measured DL is in good agreement with theoretical predictions. The measured MSD reveals a wide region of subdiffusion caused by structural disorders. The crossover from subdiffusion to normal diffusion is explained by the Lorentz model for percolating systems. Based on my understanding of these well characterised systems, I used statistical tools to study the motion of proteins on live cell membrane. I investigated the anomalous diffusion of acetylcholine receptors on live cell membranes and compared it to the normal diffusion over a flat surface. From the measured distribution $P(\Delta x(\tau))$ of displacement $\Delta x(\tau)$ and the distribution $f(\delta)$ of the “instantaneous” diffusion coefficient $\delta$, I found that the dynamic heterogeneity of the AChRs is caused by the different local environment that each AChR experiences due to the partition of the cell membrane. The short-time subdiffusion of the mobile AChRs may result from the active agitation and viscoelasticity of the cell membrane, both of which may arise from the actin network beneath the membranes through the anchored proteins. It is usually difficult to obtain the transition probability density function (TPDF) of diffusion through an arbitrary potential. I analysed the measured TPDF for the special case of colloidal diffusion over a tilted periodic potential by fitting it to an empirical form. I found that the TPDF can be separated into two parts. One part represents the overall coarsegrained behaviour, which is described well by a shifted Gaussian function. The other represents detailed structures related to the periodic potential, resembling the steady state population distribution. The backward transition probabilities are suppressed exponentially in comparison with the forward transition probabilities. The first two works provide a comprehensive study of the effect of an imposed potential on the diffusion dynamics of particles. The statistical analysis also provides unique information on the function and structure in the more complex situation of a live cell.