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PLoS One. 2012; 7(1): e30131.

Published online 2012 January 17. doi: 10.1371/journal.pone.0030131

PMCID: PMC3260218

Darren R. Flower, Editor^{}

Aston University, United Kingdom

* E-mail: ude.umc.werdna@sllessur

Conceived and designed the experiments: BL PRL RS. Performed the experiments: BL. Analyzed the data: BL PRL RS. Contributed reagents/materials/analysis tools: BL. Wrote the paper: BL PRL RS. Designed and implemented simulation software: BL.

Received 2011 August 10; Accepted 2011 December 14.

Copyright Lee et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This article has been cited by other articles in PMC.

Molecular crowding is one of the characteristic features of the intracellular environment, defined by a dense mixture of varying kinds of proteins and other molecules. Interaction with these molecules significantly alters the rates and equilibria of chemical reactions in the crowded environment. Numerous fundamental activities of a living cell are strongly influenced by the crowding effect, such as protein folding, protein assembly and disassembly, enzyme activity, and signal transduction. Quantitatively predicting how crowding will affect any particular process is, however, a very challenging problem because many physical and chemical parameters act synergistically in ways that defy easy analysis. To build a more realistic model for this problem, we extend a prior stochastic off-lattice model from two-dimensional (2D) to three-dimensional (3D) space and examine how the 3D results compare to those found in 2D. We show that both models exhibit qualitatively similar crowding effects and similar parameter dependence, particularly with respect to a set of parameters previously shown to act linearly on total reaction equilibrium. There are quantitative differences between 2D and 3D models, although with a generally gradual nonlinear interpolation as a system is extended from 2D to 3D. However, the additional freedom of movement allowed to particles as thickness of the simulation box increases can produce significant quantitative change as a system moves from 2D to 3D. Simulation results over broader parameter ranges further show that the impact of molecular crowding is highly dependent on the specific reaction system examined.

Chemistry in a living cell operates very differently than would be predicted from models of the same chemical reactions in an idealized *in vitro* environment, which is diluted and well-mixed [1]. Many features of a living cell make the intracellular environment distinctive, such as compartmentalization, active transport, the cytoskeleton network, and molecular crowding. More accurately addressing the effects on molecular interactions of these key features of cellular reaction systems is crucial to building more realistic models of reaction systems in the *in vivo* environment. Molecular crowding – i.e., the dense crowding of many kinds of macromolecules in a cell – directly influences many fundamental biological processes, such as protein folding [2], [3], protein aggregation and assembly [4]–[7], enzyme activity [8], [9], reaction kinetics [10], [11], and signal transduction [12]. Molecular crowding can hinder diffusion and provide strong steric hindrance to various reaction types, either inhibiting or enhancing chemical reactions based on many parameters of the system in question [13]–[15]. These complicated interrelated parameters make the strength and direction of the crowding effect extremely hard to accurately predict for any given model system.

Previously, we developed a two-dimensional stochastic off-lattice model (2DSOLM) [16] based on Green's function reaction dynamics [17]. The model was designed to better satisfy two constraints that confront all computational models: realism and efficiency. For example, continuum models such as ordinary differential equations and lattice Monte Carlo models [18]–[20] are very efficient but greatly simplify the actual system being modeled. Coarse-grained particle models, such as Brownian dynamics models using hard sphere particles with simplified interaction potentials [21], provide greater accuracy in exchange for increased computational cost. Full atomic resolution particle models [22], [23] provide even more realistic models of particles' behavior but at a high computational cost that makes it infeasible to simulate large systems or long time scales, especially in highly crowded conditions. Our prior stochastic off-lattice model (SOLM) uses Green's function reaction dynamics (GFRD) [17] to simulate realistic Brownian particle trajectories with reduced computational cost using discrete event models. In addition, SOLM uses a simple coarse-grained particle model to allow one to vary multiple parameters relevant to the crowding effect without the need for detailed and costly atomic structure calculations. Other similar approaches have proven successful for modeling reaction chemistry in crowded conditions. The virtual cytoplasm method also relies on a particle-based off-lattice model on a similar mesoscopic scale to address molecular crowding, but uses fixed time and space steps rather than the fully continuous time and space allowed by GFRD [24]. Kim and Yethiraj's reaction model uses Brownian dynamics combined with coarse-graining to similarly simulate the effect of crowding on a model of membrane receptor reactions [25]. In addition, Tsao, Minton, and Dokholyan's didactic model compares an analytical with a simulation method to uncover how the crowding effect alters protein folding and association using a toy model [26].

We have further shown that it is possible to use such simplified particle simulations to train multiparameter regression models, providing a method for predicting the effects of crowding on reaction systems that can combine the fast runtime of simple analytical methods with the greater versatility of particle models [27]. This simulation-based approach provides a more general and efficient algorithm to build a stochastic reaction model in various crowded conditions that can be expected to be easily extensible to more complicated models of particle structure and dynamics for which analytical models are unsuited. In addition, our stochastic models can easily investigate the crowding effect for different physical and system parameters by allowing us to alter the parameter value singly or in combination. In the present work, we extend our SOLM model from 2D to 3D, while retaining the GFRD and coarse-graining approach key to our model's efficiency, and compare the two model variants. Two-dimensional models have shown considerable value for exploring the theory of crowding, given their simplicity and relative computational tractability, but the question remains whether conclusions drawn from such models are of significant value in describing three-dimensional system. The question is particularly significant for “nearly” two-dimensional systems, such as diffusion in a membrane or at the leading edge of a migrating cell, where two-dimensional models have extra appeal. We specifically examine whether the parameter dependencies of crowding observed in our prior 2D models are qualitatively the same as those in 3D and how the quantitative behaviors vary as we interpolate between the two. This study is intended to help judge when one can rely on conclusions from 2D models and how well the two dimensional particle models and associated regression approach will extend to 3D systems. The work provides guidance for the degree to which we can rely on prior 2D models as descriptions of generic crowding phenomena and where 2D or 3D models can be trusted in modeling either 3D or pseudo-2D systems.

We characterized the effects of crowding on reaction chemistry across model types by examining the effects on a simple homodimerization test system for a variety of parameter sets. We examined two different homodimerization test cases for investigating the crowding effect on the model binding system: one using a varying reactant concentration (*C _{R}*: measured by the volume ratio of occupied reactant particles to the simulation box) without any inert crowding agent and the other using a fixed reactant concentration with additional varying inert crowding agent concentration (

Figure 2 shows simulation results for these two test cases. Figure 2(A) shows the reaction progress of a homodimerization reaction from 0 to 25 µs for both 0.1 *C _{R}*+0.35

We also calculated the estimated *K _{eq}* using statistical thermodynamics and scaled particle theory (SPT) [14], [28], [29] by assuming that the 1% pure reactant case was reasonably diluted condition so that non-ideal interactions among particles were negligible. Based on this assumption, we ran additional simulations for 1% pure reactant case, and used as the results of these low-concentration simulations to represent the reaction in the ideal state, explained in detail in Methods. These additional SPT data show how the particle simulations can deviate from expectations from an analytical model explicitly accounting for the excluded volume effect of macromolecular crowding. The estimation from SPT showed similar enhancement of

A major concern of our approach is how parameter changes in 3DSOLM affect binding chemistry, singly or in combination, for various crowded conditions. Here, we examine the following physical parameters of the homodimerization simulations: the probability of binding upon collision between two reactant monomers; the mean time between dissociation events, defined as the inverse of the dissociation rate constant; the diffusion coefficient for reactants and inert crowding particles; and the volume ratio of dimer to monomer, reflecting the relative compactness of a dimer. To simulate the various parameter values, we used a 50 nm×50 nm×50 nm cubic simulation box and tested a fixed reactant concentration with varying inert crowding agent concentrations. We simulated five binding probability values (*B*=0.1, 0.3, 0.5, 0.7, 0.9), five breaking mean time values (*M*=0.6, 0.8, 1.0, 1.2, 1.4 ns), five diffusion coefficient values (*D*=1.3, 4.63, 7.97, 11.3, 14.63×10^{−11} m^{2}s^{−1}), and five ratios of dimer to monomer volume (*α*=1.6, 1.8, 2.0, 2.2, 2.4). For each simulation, we simulated eight total concentration values (*C*=*C _{R}*+

Figure 3 shows *K _{eq}* values of SOLM and SPT for these four varying parameters (

One of the difficulties in accurately modeling crowded systems is the complex patterns of cross-dependency between distinct parameters. We chose to examine simultaneous changes in the *α* and *C*, parameters shown in 2D to exhibit cross-correlated non-linear effects on binding equilibria in crowded media [27]. Figure 3(D) shows a similar cross-dependency between *α* and *C* in 3D. A smaller *α* in highly crowded conditions exaggerates the crowding effect relative to that seen with larger *α* while the crowding effect is small across the range of *α* values examined at low to moderate levels of crowding.

To more quantitatively analyze the different parameter effects on equilibria for various crowded conditions in 2D and 3D, we built regression models for each parameter case. We first built a regression model for 3D using least-squares fitting. Figure 4(A) shows *K _{eq}* curves of the simulation and best fit regression models for varying degrees of polynomial from 0

(1)

As shown in the best-fit regression model of Eq (1), the total concentration nonlinearly altered the equilibrium of reaction system. This nonlinear effect of the total concentration parameter on the binding reaction has been observed in previous experiments [4]–[7], [14], [22] and in our previous 2D simulations [27], [30]. To reasonably compare between 2D and 3D, we built an additional regression model for 2D using the same degree as in the 3D case based on previous 2D simulation data [27], [30]. The best-fit regression model in 2D is given in Eq. (2).

(2)

Note that, because of the difference in simulation space in 2D (100 nm×100 nm) and 3D (50 nm×50 nm×50 nm) and the resulting different units of concentration, the absolute coefficients of the regression polynomials in 3D and 2D are not directly comparable.

Comparison between best-fit regression models in 2D and 3D, however, shows that the influence of total concentration on crowding is qualitatively similar between 2D and 3D. The other three parameters (*B*, *M*, and *D*) show a similar effect in 2D and 3D, both quantitatively and qualitatively. The parameters *B*, *M*, and *D* separately and linearly influence the equilibrium state of the model reaction system for both 2D and 3D. Figure 5(A), (C), (E), and (G) show the *K _{eq}* curves of simulation and best-fit regression models for 2D and figure 5(B), (D), (F), and (H) show the

(3)

(4)

(5)

(6)

(7)

(8)

Thus, binding probability upon collision between two reactants (*B*), mean time of dissociation reaction (*M*), and diffusion coefficient (*D*) independently and linearly altered the equilibrium constants and these parameter effects on *K _{eq}* can be accurately predicted by linear scaling for both 2D and 3D cases.

The volume ratio of dimer to monomer (*α*), however, shows a strong cross-dependency with the total concentration parameter (*C*) and must be fit in a multi-dimensional parameter space, similar to 2DSOLM [30]. Figure 4(C) shows the leave-one-out cross validation results for various degree of polynomial of *α* and *C* for 3DSOLM. The fifth-degree polynomial was selected as the best-fit regression model. Using the same polynomial least-square fitting method [30], the regression polynomial of *α* and *C* is

Figure 5(H) shows the *K _{eq}* curves from the average values of simulations and fit values from the regression polynomial of Eq. (9) for varying

The regression models in 3D and 2D show that the parameter effect of *α* is again nonlinear and cross-dependent with *C*, but can be accurately predicted by a high-order polynomial regression model, as shown in figure 5 (G,H).

The densely crowded environment can impede diffusion and provide steric hindrance to reaction events for both reactants and products of a chemical reaction. The impact of crowding on diffusion would be expected to act differently in a 2D versus a 3D model, although it is not *prima facie* clear how the extra dimension will specifically alter the equilibrium and reaction rates of binding chemistry in one condition versus the other. To better understand the differences between 2D and 3D models, we conducted additional simulations varying the height of the simulation space while holding width and length fixed. These simulations were intended to examine the difference between 2D and 3D on a continuum between a purely 2D model and a full 3D model. We varied the height of the simulation box from 5.125 nm (the thickness of a single layer of particles, resulting in a pseudo-2D model that we call the x1 model) to 25.625 nm (five times of the single layer, x5) in increments of 5.125 nm. We note that the thickness here describes the volume in which the center of a particle can move, so the x1 model does allow some diffusion in the height dimension, but too little for particles to pass above or below one another. The width and length of the box is fixed at 50 nm×50 nm. For each test case, we simulated eight *C* values (0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45) for fixed reactant concentrations of 0.1 with varying additional inert crowding agent concentrations and varying pure reactant concentrations without inert crowding agents. Figure 1(D) and (E) show simulation snapshots of the quasi-equilibrium state (25 µs) for 0.1*C _{R}* at 5.125 nm thickness and 0.1

For tests varying the height of the simulation box, we used a different convention for labeling the concentration than elsewhere in the manuscript in order to more accurately describe boundary effects. Specifically, we calculated concentrations by accounting for the additional one particle-width beyond the bounding box that part of a particle can occupy. Although this correction was applied throughout the manuscript when calculating excluded volume effects, it is omitted elsewhere in labeling the axes of plots to improve readability. For example, the corrected volume of the single layer case (x1, 5.125 nm) is 55×55×10.125 nm^{3} and the corrected concentration for (*C*=0.1–0.45) is *C*=0.043–0.188. Figure 6 shows *K _{eq}* curves for both SOLM and SPT for these various height cases using the corrected volume and concentration. As with our previous test cases,

Molecular crowding can either enhance or inhibit reaction systems [1], [14] depending on complex interactions among many factors. As a further test of the realism of our model, we have conducted additional validation experiments over much broader parameter ranges to demonstrate the existence of domains in which crowding may enhance, inhibit, or show little effect on binding. We specifically excluded the parameters *α* and *β* shown in our 2D model to be able to modulate the net direction of the crowding effect [30], focusing instead on parameters *B* and *M*, which control the binding probability of collision and the dissociation rate, because these would be expected to interact only indirectly with crowding levels. We simulated homodimerization reactions for parameter variation over four orders of magnitude in *B* (0.1, 0.01, 0.001, 0.0001) and *M* (10 ns, 100 ns, 1 µs, 10 µs) in a 50 nm×50 nm×50 nm simulation box at four crowding levels: 0.1*C _{R}*+0.0

We have built our 3DSOLM model to explore how crowding and other simulation parameters alter the equilibrium state of a model reaction system in 3D, and compared the results with our previous models in 2D. Like the 2D case [16], the 3D model revealed a strong crowding effect, typically enhancing binding affinities by inhibiting dissociation events, across a range of physiologically realistic levels of crowding. This effect was observed in cases of both increasing concentrations of inert crowding agents and increasing reactant concentrations, although it is less pronounced for high reactant concentrations, as would be expected given the greater ability of the pure-reactant system to alter total volume through dimerization. Changes in the parameters *B*, *M*, and *D* in 3DSOLM showed a similar linear variation in binding equilibrium to that seen in 2DSOLM [27]. In addition, changes in the cross-dependent parameters *α* and *C* in 3DSOLM showed a similar nonlinear variation in binding equilibrium to that seen in 2DSOLM [30]. We would expect such effects to be more or less pronounced in different regions of the parameter space and a search across several orders of magnitude does indeed reveals that different parameter domains can lead to very different magnitudes of crowding effects and to either enhancement or suppression of net binding equilibrium. In experimental studies, additional possible interaction types lacking from our model have also been shown to modulate the crowding effect, e.g., the presence of repulsive interactions between particles [2]–[7], attractive interactions between reactants and crowding agents [31], or other nonspecific protein-protein interactions [32]. In other circumstances, crowding has been found to have no strong crowding effect on protein-protein interactions [33]. Our model considers only steric hindrance and the resulting excluded volume effect, and further work would therefore be needed to determine how our conclusions would be affected by the presence of other such long-distance interactions.

Comparison between SOLM and SPT in figure 3 shows that the calculated *K _{eq}* values from both methods are close to each other at low-to-moderate crowding levels, but the calculated

A key question in this study is how 2D and 3D crowding models differ. The question is relevant in part because of the many 2D studies already in the literature [34], [35] as well as the considerable computational advantages of 2D models over 3D for large systems. In addition, it is important for properly characterizing the crowding effect in genuinely 2D or nearly 2D environments, such as diffusing reaction systems within a membrane. Other examples of systems involving nearly 2D diffusion may include assembly of vesicles and sorting of cargo for intracellular transport [36] and migration of T cells, which can move on the surface of endothelial lining (2D) or interstitial space (3D) [37]. Migration of T cells from 2D to 3D involves specific signaling pathway, such as MEK-Cofilin [37], but it is still unknown how the crowding effect act in this condition. We examined the issue of whether crowding effects are qualitatively different in 2D versus 3D models by interpolating between a cubic 3D space and a pseudo-2D model produced by a simulation space too narrow to allow particles to pass one another in one dimension. Our model does show significant quantitative differences in *K _{eq}* as height varies, as shown in figure 6, primarily due to a much stronger barrier to diffusion, once the third dimension is effectively lost. In biological systems, various sizes and shapes of proteins contribute to the crowding effect [38], [39]. Our results suggest that the specific influence of these combinations of shapes and sizes in conjunction with the volumes in which they diffuse must be considered to judge whether a given system is effectively 2D or 3D for the purposes of accurately capturing the crowding effect. The results do, however, suggest that 2D and 3D models provide qualitatively consistent results across various parameters and that these effects do interpolate gradually between the two, indicating that fully 2D models can provide good matches to expected behaviors from nearly 2D systems. Likewise, the results show that the regression approach we developed in 2D as a way of accelerating multiparameter simulations of chemistry in crowded conditions work comparably in 3D as in 2D systems [30]. However, this consistency between 2D and 3D may be lost if we consider additional interactions with compartments or other large obstacles, such as cytoskeleton networks and nondiffusible polymers. Such observations may prove useful in guiding development of efficient crowding models for more complex and more realistic biological systems, and in particular in understanding how one can safely trade off model detail for improved computational tractability without compromising model accuracy.

The main algorithm of 3DSOLM is the same as that described for our prior 2DSOLM model [16]. Both stochastic off-lattice models use the GFRD discrete event simulation method [17] to efficiently simulate particle diffusion in continuous time and space. In 3DSOLM, we apply a hard reflective cubic or rectangular box boundary condition. The test binding reaction system is a homodimerization reaction. All particles in 3DSOLM are spheres. The radius of a diffusion limit sphere (*R _{diff}*) in 3DSOLM is set to:

(11)

where and *D* is diffusion coefficient from the Stokes-Einstein's diffusion equation ( ), shown in figure 9(A). Each Cartesian coordinate of the diffusion limit sphere is three times the standard deviation of the Gaussian distribution for a single coordinate in isolation. The square of the distance in which the particle has diffused in a three-dimensional space can be expressed as the sum of the squares of three independent normal distributions, a quantity that is chi-square distributed with three degrees of freedom. The probability that the particle will be confined to the radius of the diffusion limit sphere (*R _{diff}*) is then equivalent to the probability that the chi-square random variable is within (

(12)

where *d _{AB}* is the distance between two particles,

(13)

(14)

(15)

where . The parameter is the ratio of dimer volume to monomer volume (), and the dimer radius is therefore . The parameter is the ratio of inert particle volume to reactant monomer volume (), and the inert particle radius is therefore .

We can derive a more general equation by plugging Eqs. (13–15) into Eq. (12):

(16)

where , if the particle *A* and *B* are reactant monomers, , if the particle *A* and *B* are reactant dimers, and , if the particle *A* and *B* are inert particles.

The collision time () of two diffusion limit spheres, which is the analytical solution of Eq. (16), follows:

(17)

where , , .

We examine the crowding effect for various parameter conditions and different mixtures of reactants and inert crowding agents. To simplify analysis of the crowding effect specifically, we use a simple homodimerization reaction as our test system. The governing chemical equation of the test homodimerization reaction is:

(18)

where is the reactant monomer, is the reactant dimer, is the inert crowding particle, is the forward reaction rate, and is the reverse reaction rate. The equilibrium constant can be computed from Eq. (18) as follows:

(19)

where is the concentration of dimers at the quasi-equilibrium state, is the concentration of monomers at the quasi-equilibrium state, is the number of initial monomers, is the number of monomers at the quasi-equilibrium state, is the number of dimers at the quasi-equilibrium state, and *V* is the volume of simulation space. The concentration of a particle in Eq. (19) is determined by the empirically measured number of particles in the simulation divided by the total volume of the simulation space, similar to standard molar concentrations. The crowding effect of [I] drops out in Eq. (19), because this governing equation is based on the idealized mass-action model. We calculate the estimated *K _{eq}* of the binding reaction for various concentrations of [I] using Eq. (19) with the simulation results from 3DSOLM, based on the assumption that 3DSOLM appropriately represents the crowding effect of all particles in the various conditions. In addition, we can estimate the average number of dimers at the quasi-equilibrium state using the estimated

(20)

(21)

where .

We calculated the apparent equilibrium constant for various densities of reactants and inert crowding particles, and various volume ratio parameter (*α*) values using scaled particle theory [28], [29] and thermodynamic activity theory [14]. In thermodynamic theory, no interaction between particles occurs at the ideal gas state. The equilibrium constant at the ideal state (*K ^{o}*) is altered by the non-ideal interactions with increasing density of either reactants or inert crowding particles. The non-ideal interaction is approximately calculated using scaled particle theory and activity coefficients of reactants. The apparent equilibrium constant of various densities of particles is (22), where is a correction factor for the excluded volume effect. For our homodimerization reaction in Eq. (18), the correction factor is , where and are activity coefficients of reactant monomer and dimer, respectively. Based on scaled particle theory [28], [29], assuming all particles are hard spheres, the activity coefficients for reactant monomers and dimers are

where *ρ* (density)=number of particles/simulation volume and *R _{X}*=radius of a particle for each particle species

3DSOLM has seven different parameters: the total concentration (*C*), defined as the volume ratio of all particles to the investigated simulation space; the probability of binding upon a collision between two reactant monomers (*B*); the mean time for dissociation events (*M*), defined as the inverse of the dissociation rate constant; the diffusion coefficient (*D*); the volume ratio of a dimer to a reactant monomer (*α*); the volume ratio of an inert particle to a reactant monomer (*β*); and the threshold distance between two particles (*d _{th}*), describing the maximum distance at which two particles can interact with one another. We established a baseline simulation parameter set with default parameter values of

The 3DSOLM simulation program was implemented in C++ and run on a Linux Beowulf cluster. The collected data files were analyzed and plotted using Matlab (R2008a).

We created a movie file to demonstrate the simulation process in 3DSOLM and show the effect of molecular crowding. Video S1 presents a comparison of 0.1 *C _{R}* and 0.1

**Simulation movie of 3DSOLM in low and high crowding conditions.** The movie shows sample trajectories for two simulations, the left side representing a low crowding (0.1 *C _{R}*) case and the right side representing a high crowding (0.1

(MOV)

Click here for additional data file.^{(7.0M, mov)}

**Competing Interests: **The authors have declared that no competing interests exist.

**Funding: **This work was supported by National Institutes of Health NIAID award #1R01AI076318 (RS). This work was supported also in part by the National Science Foundation (CMMI-0856187 and CMMI-1013748) and the Office of Naval Research (N000140910215) (PRL). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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