Long term depression (LTD) in the parallel fiber-Purkinje cell synapse is a well characterized form of synaptic plasticity. The post-synaptic components of LTD are localized in dendritic spines. Spines contain volumes < 1 fL which traditionally have been thought to be well-mixed. Under such assumptions changes in concentration are quickly equilibrated. However, the presence of organelles, spine shape and macro-molecular density could make spines a more tortuous environment for molecules to diffuse and react. We studied the effects of large concentration of non-reacting macro-molecules in the expression of an LTD model.

We translated a recently published well-mixed differential equation model of LTD into a Monte Carlo simulation [1]. We used known diffusion coefficients from the literature or calculated them based on a globular approximation using their molecular weight and the Stokes-Einstein relation. The simulations were implemented in MCell [2] and ran on a large cluster (http://www.cbi.utsa.edu).

The simulation includes all the molecules involved in the phosphorylation of AMPA receptors (AMPAR) after [Ca²⁺] increase. The simulation includes translocation of molecules to the plasma membrane and diffusion in the cytosol and membrane. The simulation was instantiated in a box of 0.17 µm³, the average volume of a Purkinje cell spine; the post-synaptic density (PSD) occupied one of the box faces. We ran each simulation for 150 seconds, with an increase in [Ca²⁺] at t = 15 sec. [Ca²⁺] increases range from 2-10 mM. The stimulus resulted in AMPAR accumulation in the PSD as a function of different levels of [Ca²⁺]. LTD expression was determined by calculating the percentage drop in AMPAR at t = 150 sec compared from the initial condition. LTD under normal conditions showed a smooth expression of LTD as a function of the stimulus. Molecular crowding in the cytosol was implemented with 120 identical cubes randomly distributed inside the box. The cubes occupied 30 % of the intracellular volume. A classic approach to this problem would suggest that crowding would result in a shift to the right of the LTD curve due to a slowdown in the diffusion of molecules. However, instead, our results show that the sensitivity of LTD to [Ca²⁺] increases, consistent with non-classical theories of reaction and diffusion due to molecular crowding. LTD under molecular crowding conditions resembles the switch-like response reported in experiments [1]. Overall, our results show that there is a strong influence of molecular crowding in the activation of biochemical signals in spines.