The complex physical structure of cytoplasm has been a long-standing topic of interest (Fulton, 1982). The physiological environment of intracellular biochemical reactants is not one of well-diluted, homogeneous space. This fact contradicts the basic assumption underlying the standard theories for reaction kinetics. The difference may make actual reaction processes deviate from those expected by the theories. If we reconcile the fundamental principle with experimental observations, reaction processes are mainly controlled by the movement of reactants. Consequently, when the environment affects molecular movement, the results would be reflected on the reaction process.

The most basic components that describe reaction kinetics are the concentrations of reactants and kinetic rate constants, which determine the probability of collision between reactants and of state transition of each reactant (Atkins and de Paula, 2006; Jackson, 2006).

Physiological events consist of quick responses in a molecular reaction level to environmental changes; i.e., a cell needs to store a portion of reaction species in a reaction environment to respond new sequential signals after current signals. This means a cell needs to quickly respond and to prevent exhaustion of reaction sources at the same time. Classical kinetic models which consist of the molecular concentration and constants fit well to in vitro reaction processes. However, it is too slow at the beginning of a reaction, or if we force to react quickly, the reaction system exhausts the entire reaction source in an early timing of a reaction process. Also, if we choose specific models such as Hill equation, Michaelis–Menten kinetics, etc., we can fit the equations to the shape of the curve of in vivo reaction data; however, the case we can find a satisfactory reason to adopt those equations is still rare even after a century of their development. Our results of this study provide a way to fit the model to in vivo experimental results, also the natural reason to use the model.

The inhomogeneous, crowded feature of the reaction space is the point we should concern to represent in vivo reaction processes precisely. Fractal kinetics has been studied as a candidate to represent in vivo reactions (Berry, 2002; Schnell and Turner, 2004). Kopelman et al. originally built their theory with reactions occurring on fractal structures, later confirmed the reality using a chemical species (naphthalene) on a considerable fractal structure (a filter membrane; Kopelman, 1988, 1991; Parus and Kopelman, 1989). The basic description of fractal kinetics is v = k(t)[A] instead of v = k[A], where v is the reaction rate, k and k(t) are the kinetic constant or function, and [A] is the concentration of a reactant; i.e., the reaction rate depends on t when it occurs on a fractal structure.

A space including fractal structure can be described quantitatively with a parameter named fractal dimension (d_f). The dynamics of particles in the fractal space is expressed by another parameter, which is spectral dimension d_s (Alexander and Orbach, 1982). The spectral dimension d_s is a quantity that can be translated into the kinetic function [k(t)]. Kopelman et al. succeeded to transform d_s into a kinetic function as the exponent in the transformation of the time variable t→tdsdr−1 (Kopelman, 1988, 1991; Parus and Kopelman, 1989). Here d_r is the fractal dimension, which has the value 2 in the case of Brownian particle motion in Euclidean space (Einstein, 1905). This means that if a reaction space has fractal structure, we can describe the dynamics of molecules by experimentally derived d_s and can reflect the spatial characteristics into the description. Deriving d_s let us investigate further effect of the intracellular environment on the form of the reaction.

As we mentioned above, the characteristics of reaction space can be quantified with d_f, and the dynamics of a particle or a molecule can be quantified with d_s. Now there remained a question if intracellular environment consists of fractal structures. Aon et al. (2004) performed experiments to analyze the fractality of mitochondria, and cytoskeletal protein-networks. They derived d_f of those organelle but did not use the value directly to calculate d_s.

In our study, we derived the value of d_s directly from our experimental results. We used the relationship between anomaly parameter of diffusion (α) and the above 2D;

It was originally defined by Alexander and Orbach (1982) and Berry (2002). By referring this relationship, we could calculate d_s by the values of α and d_f, which were experimentally obtained by fluorescence correlation spectroscopy (FCS) and transmission electron microscopy (TEM), respectively. Afterward we estimated the reaction kinetics in vivo using the transformed equation from the d_s.

Diffusion coefficient is the parameter to express molecular diffusion. The diffusion represented by α is a specific case which occurs in inhomogeneous space, named anomalous diffusion (Gefen et al., 1983). Anomalous diffusion is described with the mean-square displacement (MSD) as follows;

The case with α = 1 is for the normal diffusion. The case with α < 1 is for anomalous diffusion.

Alexander and Orbach used δ¯ to express the index governing diffusion. The definition of anomalous diffusion in their description by using MSD is

By comparing Eqs. 2 and 3,

The definition of spectral dimension, in their terminology it is called fracton dimension d¯¯, is

where d¯ is fractal dimension (d_f).

Anomalous diffusion was also defined by using a random walker on percolation clusters (Gefen et al., 1983). Percolation theory deals with the number and properties of clusters which are formed as follows (Broadbent and Hammersley, 1957); each site of a very large lattice is occupied randomly with probability p, independent of its neighbors. In other words the group of neighboring occupied sites is the target of percolation theory (Stauffer and Aharony, 1994). When the probability p is over the critical value (p_c), the cluster reaches from a side to the opposite side. This p_c is the threshold to undergo phase transition, such as gelation of polymer sol, and so on. Anomalous diffusion in an inhomogeneous environment is observed when the reaction space is occupied inhomogeneously with obstacles until the relative volume of obstacles reaches close to the threshold. We can obtain each threshold value depending on the shape of the lattice by numerical simulation (Stauffer and Aharony, 1994). The value of p_c for the 3D cube is 0.312. In some of numerical simulation including our model, percolation lattice is used as a simple example of disordered medium. The mass of a cluster M and the linear size of the cluster L keep the relation

When p = 1, the cluster is the same with a regular lattice without disorder. In this case, the time dependency of MSD of the random walker is linear. On the other hand, if the random walker is confined at a specific place, then MSD is settled on a constant. The case between these two extreme cases was named anomalous diffusion by Gefen et al. (1983). The exponent representing the time dependency of MSD equals the anomaly parameter α.

Fluorescence correlation spectroscopy is a technique used to detect single fluorescent signals from molecules in a focal volume. The detected fluctuation can be translated into the diffusion time of the molecules via inverse Fourier transformation (Kohler et al., 2000). We used FCS to estimate α (Eq. 2) by analyzing the measured diffusion time for different pinhole diameters (Rigler et al., 1993).

We controlled intracellular environment with drugs which cause cytoskeletal disorder (for actin disruption: cytochalasin B, for microtubule disruption: colchicine). The aim to treat cells with these drugs is to build a reaction environment, which includes physiological components (such as cytoskeletal proteins, etc.) with original concentration, but the structures of polymers (e.g., actin filaments, microtubule) are not physiological. By using these cells, we could exclude the effect of the molecular concentration, also could avoid using artificial components to build a test environment. As the result, our experiments could investigate purely the meaning of the physiological structures of those filaments.

Transmission electron microscopy image analysis provides a direct check of structural self-similarity. Self-organizing molecules, such as titanium dioxide, and sol–gel powders, etc., are readily seen under TEM. We applied this method to determine the d_f of the intracellular environment using the procedure described by Dobrescu et al. (2004).

Generally, an image obtained by electron microscopy indicates the distribution of electron density of the section. If the electron density is high, the segment in an image appears dark. This means that the density of the electrons captured by cellular components is relatively low at faint-shadowed segment compared with the dark-shadowed segment. Also, there exist many images 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. Based on these facts, we estimated that cytoplasmic diffusion occurs between those components, where thus the faint-shadowed segment.

We produced two different d_fs for two different intracellular areas: one was for the immobile structures (Sd_f), and the other for the mobile area of a solute in a cell (Md_f). 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.

The objective of analyzing the parameters, two d_fs with α and d_s, was to estimate how the intracellular reaction space affected the reaction process using Md_f and α to calculate d_s, and to estimate environmental organization by comparing Sd_f obtained from TEM with those from the following four numerical models in Table 1, leading to estimation of the underlying mechanism for building the intracellular environment.

There exist some models in which growing random patterns can evolve into fractal structures. Of particular interest to us are the invasion percolation (IP; Wilkinson and Willemsem, 1983), diffusion-limited aggregation (DLA; Tolman and Meakin, 1989), diffusion-limited cluster–cluster aggregation (dlCCA; Meakin and Stanley, 1983), and reaction-limited CCA (rlCCA) model (Meakin and Djordjevi, 1985), all of which were used to compare our estimated values of d_f with experimental results.

Finally we constructed a numerical model that enables us to verify the consistency of the hypothesis that the intracellular component is built using a kind of self-organization and that the structure provides a percolation cluster-like environment for soluble molecules. We computed α, derived a reaction rate function k(t), and compared experimental results with simulation. Additionally, we computed the MSD of a diffusing molecule to ascertain the effect of the non-reactive obstacles (NRO) in the reaction space.

Our results allow confirmation of the importance of the physiological intracellular structure, and provide new insights into causes of the argued point that a reaction or molecular movement exhibits a quick initial response, and later slows down to avoid exhaustion of the molecular source in physiological environments.

In this study, two separate biological experiments were conducted using mammalian cell lines and the results compared with the output of a single numerical model analysis. Experiments yielded values of the anomaly parameter α and fractal dimensions Sd_f and Md_f. Numerical simulations yielded an independent estimate of α, and also allowed us to observe the time course of in silico reactions. We compared the reaction form predicted by k(t), derived from

and produced by the simulation. We also simulated MSD for each different condition of the reaction space.

The intracellular environment is densely crowded and inhomogeneous as has been observed (Medalia et al., 2002). Diffusion of a molecule within a cell is restricted to the space between obstacles. To model diffusion-mediated processes, including modeling of kinetics, it is necessary to account for the effects of high concentrations of mobile reagents and immobile, non-reactant obstacles.

Diffusion occurring in such an environment is expected to differ from the normal diffusion of a pure random walker. In normal diffusion, the MSD is proportional to time, and the diffusion coefficient is constant. Anomalous diffusion is known to produce non-linear temporal behavior; the form of the time dependence is proportional to some power of time less than one (α). From this relationship, we infer a time-dependent diffusion coefficient for reactants in inhomogeneous, dense environments, of the form

which decreases to zero as t → ∞, where Γ is a constant. Mathematically anomalous diffusion is the result of deviations from the central limit theorem (Bouchaud and Georges, 1990); physiologically, it emerges from the presence of obstacles and traps.

We analyzed the anomaly parameter of GFP diffusion in cytoplasm (Figure 1). Our results indicated that the time-dependent form of the GFP diffusion is anomalous. Comparison with results from computational simulation (Table 5) yielded an estimated relative volume of NRO; it is approximately 60–70% (Figure 1B and Table 5) in the cytoplasm.

With regards to cellular physiology, this result is compatible with reliable reports of protein–water proportion in a cell, which comprises of 60% protein and 40% water (Fulton, 1982).

We investigated whether the total concentration of protein in a cell affects reaction kinetics in the intracellular environment, or whether organized cell structure plays a dominant role. We tested our ideas physiologically by subjecting CytB-treated cells and colchicine treated cells to FCS analyses (Figure 2, Table 2 and Figure A2 in Appendix), also by numerical simulation (Figure A3 in Appendix). Cytochalasin B disrupts the equilibrium condition between the actin polymer and the monomer. Colchicine inhibits tubulin polymerization into microtubule. The disruption of these cytoskeletal filaments while maintaining the total protein concentration changed the anomaly parameter, bringing it closer to the normal condition. This result suggests that not merely the concentration of intracellular proteins, but also their physiological organization profoundly affects diffusion of free molecules in a cell. And this conclusion is supported by our simulation results. When the same volume of NRO with randomly distributed NRO is shaped cube in the reaction space, reactants did not show anomaly characteristics.

There exists one interesting tendency among the condition of reaction space, which is represented by α values and their SD. In ideal in vitro space, α = 1 ± 0. In our results, the condition of cytoplasm of control cells is anomalous, at the same time widely varied the level of the anomaly which is represented by α. When we disordered cytoskeletal structures by chemical reagents, the condition of cytoplasm was still anomalous but the variety of anomaly was reduced. This seems natural if cytoskeletal disorganization changed intracellular environment from inhomogeneous space into homogeneous space. Further investigation is required to get clear conclusion from this point of view.

Fractals are self-similar, meaning that they exhibit similar fine-scale features at many magnifications. It was argued over 20 years ago that folded polymers, including chromatin, were fractals (Grosberg et al., 1988). Two recent reports have suggested that chromatin and the nucleoplasmic space surrounding it should be added to the fractal list (Bancaud et al., 2009; Lieberman-Aiden et al., 2009). In order to decide whether chromatin or its surrounding nucleoplasmic space is truly fractal, we need to visualize the structures directly.

We used TEM to directly obtain values of cytoplasmic d_f (Figure 3). The Sd_f is the fractal dimension of the surface of the intracellular structures (Figure 3B). Our Sd_f value is similar to those of DLA structures. This result suggests that intracellular structures may be organized in a diffusion-limited manner.

During the 1980s, numerical analyses were performed to estimate the d_s value of DLA. The estimated values are between d_s = 1.2–1.4 (Meakin and Djordjevi, 1985). Also theoretical analysis estimated d_s value as 1.32 (Alexander and Orbach, 1982). From our experimentally obtained results, we deduced a d_s of Sd_f (equaling the d_f of a DLA like intracellular structures), it is 1.34 ± 0.084. Even there remains the possibility to be argued about the noise and threshold effects, the derived d_s value is supportive to estimate our assumption is proper for this analysis. Our calculation procedure for d_s is modified to accommodate experiments with real cell materials. In the numerical analysis, d_w and d_f are both used. Here d_w is the dimension of a random walker on a fractal structure. On the other hand, since we can readily analyze α experimentally (e.g., by FCS): we chose to use α and d_f. Parameters derived from experimental techniques were compatible with those obtained from previous numerical simulations and theoretical works. We also confirmed the compatibility of our experimental results with our original numerical derivation of α. We therefore conclude that our procedure is appropriate for experimental estimates of d_s, and applicable further investigation of modeling reaction kinetics.

The Md_f is the fractal dimension of the space between intracellular structures, supposedly the space available to a diffusing molecule. The value of Md_f is similar to the d_f of an IP cluster. This result suggests that a molecule diffusing within intracellular space follows the rules of percolation.

The inference from our Md_f result does not contradict our further investigation of reaction form and the numerical simulation results. The both reaction form estimated from the d_s value and stochastic model, which included NRO in its reaction space, exhibited the same characteristics; that is quick response at the early stages of a reaction and slow exhaustion after a long times (Figures 4 and 5). The ratio of NRO in the reaction space in our simulation was close to the percolation threshold. All these results are consistent and support each other.

The results of our numerical simulation of MSD are of special interest. The curvature of the MSD versus reaction time plots depends on the relative volume of NRO in the reaction space (Figure 5C). When NRO is absent from the reaction space, the MSD of the reagent is linear in time. This case indicates the case of normal diffusion. While the spatial occupancy of NRO is below 40%, the curve appeared linear but the slope become smaller, indicating hindered diffusion. As the occupancy of NRO approaches the percolation threshold, the curve develops a concave form. Among the diffusion models, directed diffusion (MSD = 2dD₀t + v²t², where d is the Euclidean dimension, D₀ is the diffusion coefficient of normal diffusion, and v is the velocity of a flow in the reaction space) appears to best describe this phenomenon. When NRO occupancy exceeds the percolation threshold, the curve developed signs anomalous or confined diffusion.

Therefore, such an environment may boost the diffusion velocity of a molecule if our result is correct in environments supporting NRO occupancies slightly less than the percolation threshold. Since diffusion velocity affects the reaction rate, such an environment may accelerate or enhance biochemical reactions. This phenomenon may explain the quick response at the first stage of a reaction in such a crowded space.

In conclusion, the d_f value of static structures (Sd_f) in a cell is close to the d_f of diffusion-limited self-organizing aggregation. Furthermore, estimating the behavior of reactants between the aggregations is found to resemble the behaviors of molecules in the vicinity of critical percolation, as evidenced by d_f of the reactant space being close to d_fs of IP.

Similar results have been obtained by comparing numerical simulation and in vitro self-organization of fibronectin, a component of the extracellular matrix (Berry, 2008). This in vitro observation suggests that physiological molecules self-organize into a fractal structure. Our results suggest that actual intracellular structures organize similarly. Additionally, such an environment renders the reaction form time-dependent, favoring a quick response – slow exhaustion process over a slow response – quick exhaustion by constant production process, which continues until all the resources have been exhausted. The cause of both the initial quick response and remaining reaction sources could be the percolation cluster-like environment itself.

This property of in vivo reaction processes potentially affects the characteristics of entire biochemical networks.

We are planning to analyze a system level dynamics of biochemical network which includes such in vivo reaction processes and compare the dynamics with equivalent in vitro network. This investigation might suggest the characteristics of a living system.

Our results are applicable to aid the construction of precise in vivo oriented models.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.