The complex physical structure of the cytoplasm has been a long-standing topic of interest [1, 2]. The physiological environment of intracellular biochemical reactants is not one of well diluted, homogeneous space. This fact is in contradiction with the basic assumption underlying the standard theories for reaction kinetics . The difference may render actual in vivo reaction processes deviate from those in vitro or in silico. Lately, we showed the results of a combined investigation of Fluorescence Correlation Spectroscopy (FCS) and Transmission Electron Microscopy (TEM) [4, 5]. We examined the effects of intracellular crowding and inhomogeneity on the mode of reactions in vivo by calculating the spectral dimension (d
) which can be translated into the reaction rate function. We compared estimates of the anomaly parameter, obtained from FCS data, with the fractal dimension from an analysis with transmission electron microscopy images. Therefrom we estimated a value of d
=1.34±0.27. This result suggests that the in vivo reactions run faster at initial times when compared to the reactions in a homogeneous space. The result is compatible with the result of our Monte Carlo simulation. Also, in our further investigation, we confirmed by the simulation that the above-mentioned in vivo like properties are different from those of homogeneously concentrated environments. Also other simulation results indicated that the crowding level of an environment affects the diffusion and reaction rate of reactants [6–9]. Such knowledge of the spatial condition enables us to construct realistic models for in vivo diffusion and reaction systems.
The novel points of this study are the following three:
we investigated the influence of the mobility of non-reactive obstacles (NRO) on the anomaly coefficient,
we investigated the influence of the size of the NROs, and
we reconstructed the static simulation space based on TEM images and run diffusion tests in these virtual volumes as well
in order to make the in silico simulation environment more realistic. The in vivo NROs have a wide size distribution and complex shapes. Based on our simulations we can suggest simpler systems with just one class of NROs which result in the same properties in the observed effective diffusion of the tracer molecules in the complex environment and experimental results.
While several projects investigated diffusion and reaction within compartments like the ER [10, 11], this study aims at resolving the diffusion and reaction of cytosolic proteins outside of these structures, for instance signaling molecules that have to travel from the plasma membrane to the nucleus [12, 13]. Cryoelectron tomography can be used to obtain a 3D reconstruction of only the scanned cell section [14, 15]. Statistical methods, in contrast, can be used to learn the properties of the 3D space and to generate many samples from it [16, 17]. In order to generate reaction volumes with the same properties like the TEM images, we therefore learned the image statistics. This enables us to test the influence of the structures such as mitochondria and membrane enclosed compartments on the diffusion and reaction of molecules in the cytosol. By using state-of-the-art volume visualization techniques we can also show the shape of the generated volumes.
The generated structures are used for a volumetric 3D pixel (voxel)-driven graphical representation, which was further filtered into a smooth analytic surface using the software package BioInspire [18, 19]. This analytic conversion for the visualization was done to better understand the properties of the 3D structure, which is not obvious from single 2D slices. The analytic surface is also the natural description of large intracellular objects like membrane enclosed compartments or mitochondria [11, 16] and avoids the discreteness of pixel/voxel-based approaches . The 3D ray tracing visualization package BioInspire is used to interactively sample the analytical surface to create the final image; therefore, never losing any details by going over some intermediate representation such as a triangle mesh as is common in literature [21, 22].
Generally, TEM images visualize the information of scattering/absorption or permeation of electron rays through a sample slice of the cell. The electron rays are detected by charge-coupled devices and converted to grey scale images. The part in a sample section where electrons have been scattered or absorbed appear darker on the image, while the parts permeating electron rays appear white. There exist many imaging studies which investigated intracellular structures by electron microscopy. In those images, organelle, such as nucleus, mitochondria, rough endoplasmic reticulum, zymogen granules, Golgi complex, etc., appear as clear shadows, resulting from scattered or absorbed electron rays.
Based on the above reasons, we assumed that the black segment in the TEM images consisted of solid structures comprising the non-reactive obstacle. Simultaneously, the non-reactive surface can provide anchorage for small mobile molecules. The faint segment areas in TEM images presumed to be made up of sol proteins, which formed the main reaction chamber for the intracellular reactants.
Besides the (at least temporarily) static structures the cytoplasm is known to be filled with all kinds of mobile-crowding molecules . Therefore, we added the mobility of the NRO and their size to the parameters that are investigated in this study.
In our former simulation, we used just one size of NRO, which could, e.g., represent single molecular obstacles [4, 5]. But in a cell, many of those molecules representing the NRO exist as complexes or polymers, for instance cytoskeletal proteins. In order to include this information, we analyzed if the overall radius of the obstacles would affect the diffusion and reaction processes. Especially, we checked the results obtained in such simulations for anomalous diffusion, which is a sensitive probe for crowding conditions .
Anomalous diffusion is a common phenomenon in cell biology  but was previously defined by using a random walker on percolation clusters . Percolation theory deals with the number and properties of clusters which are formed as follows ; each site of a very large lattice is occupied randomly with probability p, independent of its neighbors. The resulting network structure is the target of percolation theory . When the probability p is over the critical value (p
), the cluster reaches from one side to the opposite side of the lattice. This p
is the threshold to undergo phase transition like the gelation of polymer sol. Anomalous diffusion is observed when the reaction space is occupied inhomogeneously with obstacles until the relative volume of obstacles reaches close to the threshold. The value of p
for the 3D cube is 0.312 .
In several numerical simulations including our model, a percolation lattice is used as a simple example of the disordered medium [7, 28, 29] and we found that it is similar to the in vivo reaction space. Likewise the structured in vivo reaction space is similar to porous media [6, 30]. Such structures, which are often self-similar, can readily be seen under the TEM and are easily generated for instance by self-organizing molecules such as titanium dioxide and sol–gel powders.
=1, the cluster becomes a regular lattice without disorder. If the non-obstructed space in the cell forms such a regular lattice, the time dependency of the mean squared displacement (MSD) of a random walker on the lattice grows linear with time. On the other hand, if the random walker is confined at a specific volume, the MSD converges to a constant [31
]. The case between these two extreme cases was named anomalous diffusion by Gefen et al. [24
]. The exponent α
represents the anomaly of the MSD [23
We estimated diffusion constants of NRO based on simulation results in different environments. Our in silico models enables us to verify the consistency of the hypothesis that the intracellular component is built using a self-organization and that the structure provides a percolation cluster-like environment for soluble molecules. We computed α from the Monte Carlo simulations in these virtual environments, as well as D(t), and compared it with the experimental results from FCS measurements to find the parameters of the in silico models which match the in vivo results.