1 Introduction

Living cells are spatially organized, e.g., eukaryotic cells have a confined nucleus containing the DNA and reaction complexes are often bound to the cell membrane. To simulate reaction networks accurately, it is therefore important to incorporate the molecules’ movement into the models and account for the time it takes for a signal to transmit, e.g., from the nucleus to the membrane.

Molecules move by diffusion through biological media such as the cytoplasm, which is a non-solute medium, where an estimated \(40\%\) (Luby-Phelps 2000; Schnell and Turner 2004) of the available space is occupied by macromolecules, such as proteins, ribosomes, RNA and the cytoskeleton. The environment is called crowded, meaning that the space is densely packed by molecules, but individual species are only present at very low concentrations. Macromolecular crowding is especially important on the cell membrane (Grasberger et al. 1986), where attaching actin filaments (Medalia et al. 2002) create barriers, that hinder the displacement of membrane bound molecules (Jin and Verkman 2007; Krapf 2015). In mitochondria, more than \(60\%\) of the matrix can be occupied by enzymes and proteins (Verkman 2002). Moreover, the extracellular space between, e.g., brain cells (Hrabe et al. 2004), is also considered crowded.

The steric repulsions between molecules in a crowded environment force diffusing molecules to move around obstacles, or “crowders,” this slows down diffusion. New techniques such as fluorescence fluctuation analysis (Rienzo et al. 2014) have shown that diffusion is not simply slowed down but that crowding can lead to anomalous diffusion, where the mean square displacement (MSD) of the molecule is no longer linear, but sublinear in time. As the crowder density increases, space is divided into subdomains and becomes inhomogeneous. For this fractal space, the dimension decreases to a non-integer and the MSD no longer follows the linear law applicable in integer dimensions (Ben-Avraham and Havlin 2000; Havlin and Ben-Avraham 2002).

The change in the diffusion rate in a crowded environment is a hydrodynamic effect. The excluded volume effect on the reaction rates is a thermodynamic consequence (Hall and Minton 2003) and can be both impeding and promoting. While diffusion-limited reaction rates are decreased due to the slower diffusion, transition-state-limited reactions and dimerizations are accelerated (Ellis 2001) since intermediate products reside longer in the vicinity of reaction complexes and dimers occupy less volume than two monomers. Hindered diffusion also leads to localized reactions and a heterogeneous distribution of products, which increases intrinsic noise (Hansen et al. 2015).

Scaled particle theory (SPT) has been used to describe the thermodynamic effect on the reaction rates in a crowded environment (Grima 2010; Hall and Minton 2003; Ridgway et al. 2008). Another approach is to perform Brownian dynamics (BD) simulations and fit the reaction rates to the microscopic results, (Lee et al. 2008; Smith et al. 2014). Berry (2002), Michaelis–Menten reaction dynamics are best fitted by fractal kinetics and the results are verified by microscopic cellular automata (CA) simulations. The fractal kinetics are modified in Schnell and Turner (2004) to a Zipf–Mandelbrot distribution of the reaction rates.

To better understand the effects of excluded volume on both diffusion and reactions, accurate reaction–diffusion simulations in the crowded cell environment are needed. The microscopic approaches mentioned above are computationally very expensive due to the high number of collisions in such a medium. In this paper we present a novel multiscale approach to simulate diffusion of a spherical particle surrounded by inert and inactive crowders of any size and shape. We resolve the microscopic positions and shape of the crowders initially to precompute jump rates for the moving molecules. The molecule follows a random walk on a coarse Cartesian grid that no longer resolves the multiple crowders for computationally more efficient simulations. With our approach we can connect a given distribution of obstacles to a space-dependent diffusion map which can be used to compute space-dependent reaction rates representing reactions in the crowded environment. The method can easily be extended to moving crowders, and an advantage over other techniques such as SPT is the versatility in the shape of the crowders. The upscaling to a coarse grid makes the stochastic simulations computationally much more efficient than BD and CA simulations.

In the next section we present existing models of spatial simulations in systems biology and how they incorporate crowding effects. We then present how the microscopic motion of a molecule can be used to calculate its first exit time (FET) from domains, which provides the jump rates in a coarse grained discrete jump process on the mesoscopic level. We continue by extending the FET approach to include macromolecular crowders. In Sect. 3.2, we use the jump coefficients and compute a space-dependent diffusion map for the macroscopic level and show in Sect. 3.3 how that affects the reaction rates in the crowded environment. We conclude with numerical experiments in the final section.

Vectors and matrices are written in boldface. A vector \(\mathbf {u}\) has the components \(u_i\), and the elements of a matrix \(\mathbf {A}\) are \(A_{ij}\). The derivative of a variable u with respect to time t is written \(u_t\).

2 Spatial Modeling in Systems Biology

In this section we first present existing models of diffusion simulations in systems biology and then describe how they can be adapted to include macromolecular crowding.

2.1 Models of Diffusion in Dilute Media

Molecules undergoing diffusion and reactions inside living cells are often modeled by the reaction–diffusion equations. These are continuous, deterministic partial differential equations (PDEs) describing the time evolution of the concentrations of molecules. For a diffusing molecule the concentration \(u(\mathbf {x},t)\) is described by the diffusion equation

$$\begin{aligned} u_t(\mathbf {x},t) = \gamma _0\Delta u(\mathbf {x},t),\quad \mathbf {x}\in \Omega \end{aligned}$$
(1)

with diffusion coefficient \(\gamma _0\) and suitable boundary conditions on \(\partial \Omega \). To include reactions corresponding terms are added. This macroscopic description is accurate in the limit of large molecule numbers, when stochastic fluctuations are small and the mean value is the quantity of interest. Important molecules such as DNA or transcription factors are, however, only present at very low copy numbers inside living cells. It has been observed in experiments (Elowitz et al. 2002; McAdams and Arkin 1997; Metzler 2001; Munsky et al. 2012; Raj and Oudenaarden 2008; Swain et al. 2002) and shown theoretically (Gardiner et al. 1976; McQuarrie 1967) that stochastic fluctuations play an important role and a discrete stochastic description is more accurate than the deterministic equations.

We distinguish two levels of accuracy of stochastic models. In the mesoscopic model the domain \(\Omega \) is partitioned into N non-overlapping voxels \(\mathcal {V}_i\) with nodes \(\mathbf {x}_i\) at the center. The state vector \(\mathbf {y}(t)\) contains the number of molecules \(y_i(t)\) in each voxel \(\mathcal {V}_i\) at time t. The voxels are small enough that the molecules can be considered well-mixed inside so that reactions can occur between molecules residing in the same voxel. An individual molecule can jump from a voxel \(\mathcal {V}_i\) to a neighboring voxel \(\mathcal {V}_j\) to model diffusion. The diffusion master equation (DME) describes the time evolution of the probability to be in state \(\mathbf {y}\) of a system with only diffusion

$$\begin{aligned} \frac{\partial p(\mathbf {y},t)}{\partial t} = \sum _{i=1}^N\sum _{j=1}^N\lambda _{ij}(\mathbf {y}-\varvec{\mu }_{ij})p(\mathbf {y}-\varvec{\mu }_{ij},t)-\lambda _{ii}(\mathbf {y})p(\mathbf {y},t), \end{aligned}$$
(2)

where \(\lambda _{ij}\) is the jump propensity from \(\mathcal {V}_i\) to \(\mathcal {V}_j\) and \(\lambda _{ii}=\sum _{j=1}^N\lambda _{ij}\) is the total propensity to leave voxel \(\mathcal {V}_i\). The transition vector \(\varvec{\mu }_{ij}\) is zero except for \(\mu _{ij,i}=-1\) and \(\mu _{ij,j}=1\). Let \(\theta _{ij}\) be the splitting probability, that a jump from \(\mathcal {V}_i\) goes to \(\mathcal {V}_j\), then

$$\begin{aligned} \lambda _{ij} = \theta _{ij}\lambda _{ii}. \end{aligned}$$
(3)

By including reaction terms in a similar manner, the DME can be extended to the reaction–diffusion master equation (RDME). In the presence of bimolecular reactions there exists no analytical solution, and a numerical solution is impossible due to the high dimension of \(\mathbf {y}\). Instead, one samples trajectories of the system with the stochastic simulation algorithm (SSA), first presented by Gillespie (1976) for only reactions and improved in Cao et al. (2005) and Gibson and Bruck (2000). The algorithm was extended to space-dependent problems with a Cartesian partitioning of the domain in Elf and Ehrenberg (2004) implemented in Hattne et al. (2005). The propensities \(\lambda _{ii}\) are here used to generate random numbers for the time until the next jump. To represent the complicated geometries present in cells the algorithm was extended to curved boundaries in Isaacson and Peskin (2006) and adapted for unstructured meshes in Engblom et al. (2009) with software in Drawert et al. (2012) and Hepburn et al. (2012).

In the more accurate microscopic model the molecules are tracked along their Brownian trajectories in a continuous space, continuous time Markov process. Brownian dynamics (BD) simulations discretize time and sample the positions of a particle at these discrete time points from a normal distribution. Versions of this method are for example implemented in the software packages Smoldyn (Andrews et al. 2010), ReaDDy (Schöneberg et al. 2014) and MCell (Kerr et al. 2008). Another approach is called Green’s function reaction dynamics (GFRD) (van Zon and ten Wolde 2005a, b), where protective domains are constructed around individual molecules in which they cannot interact with other molecules. This is an event-driven algorithm with an asynchronous time step that is chosen such that the probability to exit the protective domain is small. Takahashi et al. (2010) the algorithm is improved to eGFRD using the exact first exit times from Donev et al. (2010) and Oppelstrup et al. (2009) to make it exact and more efficient. The exit times and exit positions from these domains are sampled to propagate the system until molecules are close enough to interact without sampling all the intermediate jumps.

2.2 Include Macromolecular Crowding

The macroscopic, mesoscopic and microscopic models presented above are designed to simulate diffusion in a dilute medium. The microscopic model automatically incorporates crowding effects since the molecules are modeled as hard spheres with a given volume, but it becomes computationally very expensive in a densely packed space of inactive crowders, because the protective domains around molecules will be small and many short jumps will be simulated before meeting a potential reaction partner.

Cellular automata (CA) have been used in Berry (2002), Cianci et al. (2016), Schnell and Turner (2004) and Takahashi et al. (2005) to simulate diffusion in a crowded environment. This is a lattice-, or voxel-, based microscopic approach, where each site can hold one molecule and crowders are represented as already occupied lattice points. The jump length is here the size of a molecule which also leads to expensive simulations with many short jumps and the shape of the molecules corresponds to the shape of the chosen lattice. The choice of the lattice, moreover, influences the excluded volume effect (Grima and Schnell 2006; Meinecke and Eriksson 2016).

Roberts et al. (2013) a mesoscopic approach is used where each voxel can hold more than one molecule. After distributing immobile crowders it is decided which voxels are accessible and which are full. The crowders can move in Fanelli and McKane (2010) and Taylor et al. (2015), and the jump propensity to an adjacent node is scaled by the number of available spaces in the target voxel. Macroscopic nonlinear PDEs are then derived in Fanelli and McKane (2010) to model diffusion in the crowded cell environment, and the results are validated by physical experiments in Fanelli et al. (2013). This approach is extended in Penington et al. (2011) to derive nonlinear diffusion equations modeling more complicated interactions than steric repulsion between the molecules. Similarly, the averaged occupied volume in the whole domain is used in Landman and Fernando (2011) to rescale the jump propensities. These approaches at most take the averaged occupied volume in the target voxel into account and neglect the microscopic positions, the shape of the molecules and the surrounding medium. Hence, only an averaged behavior is observed and the MSD is linear like for normal diffusion but with a reduced diffusion constant (Phillips et al. 2008):

$$\begin{aligned} \langle \mathbf {x}^2(t)\rangle = (1-\phi )\gamma _0 t, \end{aligned}$$
(4)

where \(\phi \) is the occupied volume fraction.

In this paper we present a novel multiscale approach to simulate molecular crowding. We will use the microscopic information of the crowders’ positions to recalculate the jump propensities on an overlying mesoscopic mesh. Instead of simulating diffusion in the detailed environment, as in BD, we will have to solve many PDEs on small subdomains resolving the microscopic positions of the crowders. This is similar to homogenization techniques used, e.g., for flow simulations in porous media (Brown and Peterseim 2014; Målqvist and Peterseim 2014). The crowding molecules can have any shape, and this approach is especially useful when the crowded environment is stationary or evolves on a much slower timescale than the diffusing molecule, so that the jump coefficients can be precomputed and used for a long simulation time. This is reasonable since the macromolecules responsible for the majority of occupied volume are ribosomes, microtubules and actin filaments (Ellis 2001), which are large and hence diffuse on a slower timescale than for example transcription factors. Moreover, it was shown in Rienzo et al. (2014) that anomalous behavior is most likely to happen in a stationary environment, since otherwise averaging effects simply reduce the coefficient of normal diffusion. But, it is important to mention that by computing statistics our method can be made efficient also for moving crowders.

3 Upscaling of the Crowding Effects

We will now present a method of upscaling the detailed information about diffusion in a crowded environment on the microscopic level to the mesoscopic simulation framework and further to the macroscopic description. Moreover, we describe a way of accounting for crowding effects in the reaction dynamics on the mesoscopic and macroscopic levels.

3.1 Microscopic to Mesoscopic: First Exit Times

In this section we will use the methods developed for microscopic simulations of Brownian motion with protective domains (Oppelstrup et al. 2009), to derive the jump propensities \(\lambda _{ii}\) and splitting probabilities \(\theta _{ij}\) for the mesoscopic model. For simplicity the illustrations are given in dimension \(d=2\), but the method can be extended to \(d=3\) without modification.

3.1.1 First Exit Times

Let \(c(\mathbf {x},t)\) be the probability distribution that a molecule in Brownian motion is at \(\mathbf {x}\) at time t and has not yet exited a domain \(\omega \). If \(\mathbf {x}_0\) is the starting position of the molecule diffusing with \(\gamma _0\), then \(c(\mathbf {x},t)\) fulfills

$$\begin{aligned} c_t(\mathbf {x},t)&= \gamma _0\Delta c(\mathbf {x},t),&\mathbf {x}&\in \omega ,\\ c(\mathbf {x},t)&= 0,&\mathbf {x}&\in \partial \omega ,\nonumber \\ c(\mathbf {x},0)&= \delta _{\mathbf {x}_0}.\nonumber \end{aligned}$$
(5)

The homogeneous Dirichlet boundary condition here models that the particle is removed once it hits the boundary. The survival probability of the particle inside \(\omega \) until time t is then

$$\begin{aligned} S(t) = \int _{\omega }c(\mathbf {x},t)\mathrm{d}\omega . \end{aligned}$$
(6)

By Gauss’ formula the probability density \(p_{\omega }(t)\) that the particle leaves \(\omega \) at t is

$$\begin{aligned} p_{\omega }(t) = -\frac{\partial S(t)}{\partial t} = -\gamma _0 \int _{\partial \omega }\mathbf {n}\cdot \nabla c(\mathbf {x},t)\mathrm{d}s, \end{aligned}$$
(7)

where \(\mathbf {n}\) is the outward normal. The expected time E for the molecule to leave \(\omega \) for the first time is given by

$$\begin{aligned} E = \int _0^\infty tp_{\omega }(t)\mathrm{d}t = \int _0^\infty S(t)\mathrm{d}t. \end{aligned}$$
(8)

We now use this FET approach to compute the jump propensities \(\lambda _{ii}\) in the space discrete mesoscopic model. We use a Cartesian grid with space discretization h and N nodes \(\mathbf {x}_i\) in the domain \(\Omega \). The voxels \(\mathcal {V}_i\) are here defined by the dual mesh, see Fig. 1b. Since the molecules are considered well-mixed inside the voxels, the domain \(\omega \) that diffusing particles have to leave to be well-mixed in the next voxel has to include the centers of the neighboring voxels. On a Cartesian grid we showed in Lötstedt and Meinecke (2015) that solving (5) on the circle \(\omega _i\) with center \(\mathbf {x}_i\) and radius h (similarly a line of length 2h in 1D or a sphere with radius h in 3D) and choosing \(\mathbf {x}_0 = \mathbf {x}_i\) gives the correct exit time from node i, see Fig. 1b. Observe that \(\omega _i\supsetneq \mathcal {V}_i\) and \(\omega _i\cap \omega _j\ne \emptyset \) for neighboring nodes i and j. Using (6) and (8) the expected exit time \(E_i\) from \(\mathcal {V}_i\) is

$$\begin{aligned} E_i = \frac{h^2}{2d\gamma _0}, \end{aligned}$$
(9)

see Gardiner (2004). Since the jump propensity is the inverse of the exit time this agrees with the mesoscopic rate on Cartesian grids

$$\begin{aligned} \lambda _{ii}=\frac{2d\gamma _0}{h^2}=E_i^{-1}. \end{aligned}$$
(10)

The probability to leave \(\omega _i\) through a given part of the boundary \(\partial \omega _{ij}\) at time t is given by the proportion of fluxes

$$\begin{aligned} \theta _{ij}(t)= & {} \frac{\int _{\partial \omega _{ij}}\mathbf {n}\cdot \nabla c(\mathbf {x},t)\mathrm{d}s}{\int _{\partial \omega _i}\mathbf {n}\cdot \nabla c(\mathbf {x},t)\mathrm{d}s}, \end{aligned}$$
(11)

and we can compute the expected probability to jump to a certain neighboring voxel by

$$\begin{aligned} \theta _{ij} = \int _0^\infty \theta _{ij}(t)p_{\omega _i}(t) dt = -\gamma _0\int _0^t\int _{\partial \omega _{ij}}\mathbf {n}\cdot \nabla c(\mathbf {x},t)\mathrm{d}s\mathrm{d}t. \end{aligned}$$
(12)

Choosing \(\partial \omega _{ij}\) to be the quarter segment of the boundary closest to \(\mathbf {x}_j\), see the blue line in Fig. 1b, yields the splitting probability \(\theta _{ij}=0.25\) as expected on a Cartesian grid. The method has been extended to a rectangular grid and possible jumps to the diagonal neighbors in Meinecke and Lötstedt (2016).

By conditioning on the first step, these time-dependent equations can be converted to equations describing directly the expected quantities (Redner 2001). The expected exit time for a molecule starting to diffuse in \(\mathbf {x}\) from the domain \(\omega _i\) fulfills the Poisson’s equation

$$\begin{aligned} \gamma _0\Delta E(\mathbf {x})&= -1,&\quad \mathbf {x}\in \omega _i,\nonumber \\ E(\mathbf {x})&= 0,&\quad \mathbf {x}\in \partial \omega _i. \end{aligned}$$
(13)

Equivalently, the expected splitting probability for this molecule to exit through the boundary \(\partial \omega _{ij}\) can be computed by the harmonic measure (Øksendal 2003, Ch. 7) and fulfills the Laplace equation

$$\begin{aligned} \Delta \theta _{ij}(\mathbf {x})&= 0,&\quad \mathbf {x}\in \omega ,\nonumber \\ \theta _{ij}(\mathbf {x})&= 1,&\quad \mathbf {x}\in \partial \omega _{ij},\nonumber \\ \theta _{ij}(\mathbf {x})&= 0,&\quad \mathbf {x}\in \partial \omega _i{\setminus }\partial \omega _{ij}. \end{aligned}$$
(14)

We solve Eqs. (13) and (14) and evaluate them at \(\mathbf {x}_i\) instead of solving the time-dependent equations and the integrals above. It is sufficient to solve (14) three times for each node since \(\sum _{j=1}^4\theta _{ij}=1\).

In the following, we will not simply use the circle \(\omega _i\) to compute the mesoscopic rates, but we will prohibit the molecule from diffusing where the crowders are located.

3.1.2 Include Crowding Molecules

The crowding molecules are represented as obstacles or holes in the domain \(\omega _i\) with reflecting boundary conditions. Equations (5),(13) and (14) describe the diffusion of point particles. To account for the volume of the diffusing molecule its radius is added to the excluded volume for the center of mass, see Fig 1a. We depict the crowder as a circle with radius R, but it is important to mention that any shape is possible for the crowding molecules, see Fig. 1c. The shape of the small (as compared to crowders) diffusing molecule is, however, restricted to circles or spheres with radius r.

Fig. 1
figure 1

(Color figure online) a Excluded volume (gray and red) for the center of mass of the diffusing molecule (blue). b Cartesian mesh and protective domain \(\omega _i\) without crowding. c Solution of (13) with the effect of molecular crowding

Full size image

Solving (13) and (14) numerically on the perforated domain \(\omega _i\) means that the crowding molecules have to be resolved by a fine mesh. But, we have divided the global problem into N smaller local subproblems (one protective domain \(\omega _i\) around each node i). For each subproblem we have to solve 4 PDE’s ((13) and three equations of type (14)). These equations have to be solved only once in a precomputing step which is straight forward parallelizable. This is similar to the approach in Brown and Peterseim (2014) where deterministic local equations are solved on media with porous microstructures. The costly stochastic simulations of the spatial SSA are then performed on the coarse mesh with N nodes, that do no longer resolve the individual obstacles. The boundary conditions on the global domain \(\Omega \) (reflective or absorbing) are implemented by posing these conditions on the secants of the half or quarter circles, which are the protective domains for boundary nodes, see Fig. 2a. In Fig. 2b we briefly illustrate how the first exit time approach can be further used to compute the jump rates for a Cartesian grid, discretizing a domain \(\Omega \) with a curved boundary. Here, the part of \(\partial \omega _i\) originating from the circle is imposed with the boundary conditions in (13) and (14) and the part originating from \(\partial \Omega \) with the boundary condition valid on \(\Omega \), which is usually reflecting or (partially) absorbing. The circle is then divided in the same way as before to compute the splitting probabilities to the neighboring nodes remaining inside \(\Omega \).

The simultaneous interpretation of the moving molecules being well-mixed inside the voxels \(\mathcal {V}_i\) and jumping from node \(\mathbf {x}_i\) to \(\mathbf {x}_j\) leads to problems when including crowders. Consider the case where just the center \(\mathbf {x}_i\) is blocked, but voxel \(\mathcal {V}_i\) is sufficiently empty to be traversed, see Fig. 2c. In this case the jump into voxel \(\mathcal {V}_i\) is possible (\(\lambda _{ji}>0\)), but the expected time to leave \(\mathcal {V}_i\) is infinity and hence the molecules get trapped inside \(\mathcal {V}_i\). This does not agree with the microscopic situation, where molecules diffuse around \(\mathbf {x}_i\). To avoid this unrealistic trapping, all jump propensities \(\lambda _{ji}\) to voxels whose vertex is isolated are set to zero. In the case when a node \(\mathbf {x}_i\) is covered by the excluded volume (not pictured) Eqs. (13) and (14) cannot be evaluated. Setting all \(\lambda _{ji}\) to zero in this scenario would over estimate the crowding effect considerably, so we constrain the crowder distribution such that \(\mathbf {x}_i\) remains inside \(\omega _i\).

Fig. 2
figure 2

a Boundary treatment. b Using the first exit time approach to compute Cartesian jump rates on the boundary of a curved domain \(\Omega \). c Model error when interpreting molecules as well-mixed and jumping between nodes simultaneously

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By using the expected first exit time from \(\omega _i\) around \(\mathbf {x}_i\) to calculate the jump coefficients in a crowded environment it is the crowder distribution inside the whole circle \(\omega _i\) that affects the coefficients \(\lambda _{ii}\) and \(\theta _{ij}\). This differs from other approaches to simulate diffusion in a crowded environment with a discrete space jump process. Landman and Fernando (2011), Roberts et al. (2013) and Taylor et al. (2015) it is only the percentage of occupied volume in the target voxel \(\mathcal {V}_j\) and in Grima and Schnell (2007) the difference in occupancy between \(\mathcal {V}_j\) and the origin \(\mathcal {V}_i\) that affect the jump rate \(\lambda _{ij}=\theta _{ij}\lambda _{ii}\). In our approach the microscopic positions of all crowding molecules inside \(\omega _i\) are resolved and influence the jump coefficients. In the case of non-spherical crowding molecules also the orientation is taken into account and long thin molecules with small volume can have a significant effect on \(\lambda _{ij}\) and \(\theta _{ij}\), see Fig. 3. Note that in contrast to normal diffusion the jump propensities are no longer symmetric, i.e., in general \(\lambda _{ij}\ne \lambda _{ji}\) and \(\lambda _{ij}\ne \lambda _{im}\) for \(j\ne m\).

Fig. 3
figure 3

Not only the occupancy \(\phi \) but also the microscopic positions and orientations of the crowding molecules affect the jump propensities \(\lambda _{ij}=\theta _{ij}\lambda _{ii}\). a \(\theta _{ij}=0\). b \(\theta _{ij}> 0.25\)

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3.1.3 Statistics on the Mesoscopic Level

Solving N local problems where complicated geometries have to be resolved, see Fig. 1c, is computationally expensive and will be inefficient if the crowding molecules move and the coefficients \(\lambda _{ii}\) and \(\theta _{ij}\) have to be recomputed frequently. Since the crowders’ exact location is generally unknown we can compute statistics on a reference domain \(\omega _i\) for given a percentage of occupied volume, a given shape and size of the molecules and h. Instead of solving N PDEs of type (13) and 3N of type (14) at each time step, we can then sample the coefficients \(\lambda _{ii}\) and \(\theta _{ij}\) from these precomputed distributions. This will be especially applicable for moving crowders, where new coefficients can be drawn from the distributions on the timescale of their diffusion.

3.2 Mesoscopic to Macroscopic: A Space-Dependent Diffusion Map

In this section we derive a macroscopic diffusion equation with a space-dependent diffusion coefficient \(\gamma (\mathbf {x})\), representing the effect of macromolecular crowding. We approximate the mesoscopic jump process by Fickian diffusion with a constant diffusion coefficient \(\gamma _i\) inside each voxel \(\mathcal {V}_i\). The mesoscopic expected exit time \(E_i\) from a node \(\mathbf {x}_i\) or voxel \(\mathcal {V}_i\) is connected via  (9) to this diffusion coefficient \(\gamma _i\). So we obtain a modified version of the macroscopic deterministic diffusion equation (1)

$$\begin{aligned} u_t(\mathbf {x},t) = \nabla \cdot (\gamma (\mathbf {x})\nabla u(\mathbf {x},t)),\quad \mathbf {x}\in \Omega , \end{aligned}$$
(15)

where

$$\begin{aligned} \left. \gamma (\mathbf {x})\right| _{\mathcal {V}_i} =\gamma _i= \frac{h^2}{2dE_i}. \end{aligned}$$
(16)

For transferring the mesoscopic jump rate to the less detailed macroscopic level we only use \(\lambda _{ii}\) and the random walk becomes symmetric in each voxel, i.e., \(\theta _{ij}=\theta _{im}=0.25\) for \(j\ne m\), but the asymmetry between back and forth jumps is preserved, i.e., \(\lambda _{ij}\ne \lambda _{ji}\). Alternatively \(\gamma (\mathbf {x})\) can be defined on the edges \(\partial \mathcal {V}_{ij}\) (see Fig. 1b) of voxel \(\mathcal {V}_j\) by \(\gamma _{ij}=h^2/2(\lambda _{ij}+\lambda _{ji})\). This corresponds to a mesoscopic jump process with symmetry in \(\lambda _{ij}=\lambda _{ji}\) and non-symmetric jumps out of a box \(\lambda _{ij}\ne \lambda _{im}\) for \(j\ne m\).

The anomalous diffusion is here modeled by a space-dependent diffusion coefficient. A different description of anomalous diffusion on the macroscopic level is fractional PDE’s (FPDE’s), which are used to derive a mesoscopic description in Blanc et al. (2016), where molecules change their internal state, i.e., their diffusion constant, spontaneously in time. This correlates to our model when the crowding macromolecules are moving and the diffusion constant hence also becomes time-dependent: \(\gamma (\mathbf {x},t)\), in Engblom et al. (2017) we connect statistics from our mesoscopic description to the internal states model in Blanc et al. (2016) and hence to a FPDE description on the macroscopic level.

In the next section we will use the diffusion map \(\gamma (\mathbf {x})\) to derive space-dependent reaction rates.

3.3 Reactions

Approaches to correct the reaction rates for crowding effects use either time-dependent reaction rates k(t) (Berry 2002; Schnell and Turner 2004) or static modifications (Ellis 2001; Grima 2010; Grima and Schnell 2007). Similarly to the latter we will use the space-dependent diffusion coefficient \(\gamma (\mathbf {x})\) from the previous section to compute mesoscopic reaction rates \(k_{i}\) inside each voxel \(\mathcal {V}_i\). This static rate becomes time-dependent \(k_i(t)\) if we model moving crowders. According to Hellander et al. (2015) the dilute mesoscopic rate \(k_0\) for bimolecular reactions in a three-dimensional cube of volume \(h^3\) is linked to the effective rate \(k_{CK}\) by Collins and Kimball (1949) by

$$\begin{aligned} k_{0} = k_{CK}/h^3,\quad \quad \quad k_{CK} = \frac{4\pi \sigma \gamma _0 k_r}{4\pi \sigma \gamma _0+k_r}, \end{aligned}$$
(17)

where \(k_r\) is the intrinsic reaction rate and \(\sigma \) the sum of the two reaction radii. In 2D there is no equivalent formula, but approximations are derived in Fange et al. (2010) and Hellander et al. (2012).

Assuming constant Fickian diffusion inside each voxel as in Sect. 3.2, we can now compute mesoscopic reaction rates \(k_i\) for each voxel by inserting the \(\gamma _i\) from (16) into (17)

$$\begin{aligned} k(\mathbf {x})= & {} \frac{1}{h^3}\frac{4\pi \sigma \gamma (\mathbf {x}) k_r}{4\pi \sigma \gamma (\mathbf {x})+k_r}\quad \text {and}\quad \left. k(\mathbf {x})\right| _{\mathcal {V}_i} =k_i. \end{aligned}$$
(18)

We can now add a space-dependent reaction term to the PDE (15) using these space-dependent reaction rates \(k_i\). To model reactions under excluded volume effects on the mesoscopic level the reactions inside each voxel \(\mathcal {V}_i\) have their specific reaction rate \(k_i\).

In this model only bimolecular associations are affected by macromolecular crowding, since the hindered diffusion changes the hitting time for the reaction partners. In the internal states model (Blanc et al. 2016) also birth-death processes and isomerizations become anomalous. It is, however, questionable if it is meaningful to talk about birth-death processes, when considering excluded volume effects, since all reactants and products also occupy space. With scaled particle theory (Grasberger et al. 1986) also dissociation events are affected by the excluded volume which is due to two spherical molecules having a different activity coefficient than one molecule with the same total area. In our model, dissociation is not affected either.

In Sect. 4.4 we perform numerical experiments to examine for which parameters crowding molecules enhance or decrease the rate of bimolecular reactions.

4 Numerical Experiments

In the following experiments we solve (13) and (14) in 2D with COMSOL Multiphysics on \(\omega _i,\,i=1\dots N\). To be able to evaluate the solutions at \(\mathbf {x}_i\) the crowders are randomly distributed such that the nodes \(\mathbf {x}_i\) remain inside the perforated domain \(\omega _i\) and are not cut out by the excluded volume.

4.1 Effect of Crowding on Jump Propensities

We first investigate how the jump propensity \(\lambda _{ii}\) changes in different crowding situations. We therefore compute \(\lambda _{ii}\) on a reference domain \(\omega _i\) with \(h = 1\) and different crowders and sizes of the moving molecule r and compare it to the jump rate \(\lambda _{0,ii}\) in dilute medium. In Fig. 4, we compare the mean value of the jump propensities \(\mathbb {E}[\lambda _{ii}]\) for different distributions of crowders and an increasing percentage of occupied volume \(\phi \) with the jump propensities when no crowders are present, where \(\mathbb {E}[\lambda _{0,ii}]=\lambda _{0,ii}=4\). We test two different sizes of crowding molecules for both rectangles and spheres. The reference line is the linear scaling where \(\lambda _{ii}=(1-\phi )\lambda _{0,ii}\) as in Landman and Fernando (2011) and Phillips et al. (2008).

Fig. 4
figure 4

(Color figure online) The mean value of the mesoscopic jump coefficients in the crowded environment \(\mathbb {E}[\lambda _{ii}]\) compared to \(\mathbb {E}[\lambda _{0,ii}]=4\) in dilute media and its dependence on the occupancy \(\phi \). Averages are taken over \(M = 100\) different crowder distributions. The obstacles are either spheres (blue and red) with radius R or rectangles (orange and green) with ratio of width to length equal to 20. The dashed reference line corresponds to Eq. (4) when only the averaged occupied volume in the target voxel is taken into account. The spherical moving molecule has radius r. In (e) we compare the mesoscopic coefficients with the results from a Brownian dynamics simulation, where we generate \(10^4\) trajectories for 10 different crowder distributions with the software Smoldyn

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We observe that small obstacles (blue and orange in Fig. 4) hinder diffusion more than big crowders for the same percentage of occupied volume, since they have more reflecting surfaces than larger crowders. The same holds for elongated rectangular crowders (green and orange in Fig. 4), as they create long barriers without occupying a lot of volume. In three space dimensions the crowding effect is weaker and higher occupation fractions \(\phi \) would be necessary for a similar slow down. Since multiple small spheres or a cylinder of the same size as a large sphere also has larger surface areas in three dimensions, the same relative effects of crowders of different shapes are expected. An increasing size r of the diffusing molecule leads to an increase in the crowding effect, which is intuitive, since a bigger molecule finds less holes through which to escape. These results agree with the findings in Muramatsu and Minton (1988) and Ellery et al. (2015). We see that the averaged linear reduction in the jump propensity can be a good model when the diffusing molecule is about a tenth of the size of the crowders, but over- or underestimates the effect of occupied volume when the diffusing species is smaller or bigger, respectively. Since an average protein has a radius of ca. 2 nm (Phillips et al. 2008) and the biggest macromolecules in the cell, the ribosomes, have a radius of up to 15nm the linear correction is a good approximation for many scenarios. The reduction in jump propensity, however, starts to behave exponentially, as in Grima and Schnell (2007), for large diffusing molecules. The case \(r=0\) corresponds to a point particle which is irrelevant when simulating excluded volume effects, but we include it to show the limit for very small particles. To confirm these mesoscopic jump rates we compare them to the inverse of the expected exit time computed by a Brownian dynamics simulation. We simulate \(10^4\) trajectories with the open source software Smoldyn (Andrews et al. 2010; Andrews and Bray 2004) for 10 different crowder distributions and equally sized crowding and moving molecules with \(R=r=0.1\) and see in Fig. 4e that the computationally expensive microscopic results agree well with the mesoscopic coefficients. In our previous work (Meinecke and Eriksson 2016) we have compared the computational time for detailed Brownian dynamics simulations on the full domain \(\Omega \) with microscopic grid-based methods and measured a 42-fold speedup. Coarser mesoscopic simulations on \(\Omega \) will be even faster than the CA simulations and our method hence promises a great safe in computational time compared to detailed BD simulations.

Fig. 5
figure 5

(Color figure online) Standard deviation \(\sigma (\lambda _{ii}/\lambda _{0,ii})\) around the mean value of the change in jump coefficients in (a) and (d) in Fig. 4

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In Fig. 5, the standard deviation \(\sigma (\lambda _{ii}/\lambda _{0,ii})\) initially increases as more and more crowders are added but converges toward zero when the system approaches the state where no escape to the boundary is possible.

Simply rescaling the jump propensity \(\lambda _{ii}\) for each node i by a constant factor will lead to normal diffusion at reduced rate. To observe anomalous diffusion in the crowded environment it is helpful to investigate the mean square displacement (MSD).

4.2 The Mean Square Displacement

As mentioned in Sect. 2.2 the MSD is linear in time for normal diffusion

$$\begin{aligned} \langle \mathbf {x}^2(t)\rangle = 2d\gamma _0t, \end{aligned}$$
(19)

but for anomalous diffusion the relation is no longer linear

$$\begin{aligned} \langle \mathbf {x}^2(t)\rangle = 2d\gamma _0t^\alpha , \end{aligned}$$
(20)

where \(\alpha <1\) for subdiffusion. Rienzo et al. (2014), Galanti et al. (2014) and Mommer and Lebiedz (2009) it was shown that diffusion in a crowded environment can be modeled by a temporal change in the diffusion constant. First the molecule diffuses normally with rate \(\gamma _0\) for very short timescales, before it undergoes a transient anomalous phase with a changing diffusion coefficient \(\gamma \) and finally stagnates into normal diffusion at a lower diffusion rate \(\gamma _\infty \). The initial normal diffusion represents the time the molecule diffuses in the solution before it encounters the first adjacent macromolecule and is slowed down by collisions. On a large timescale the molecule appears to diffuse in a denser medium instead of around obstacles, hence the reduced diffusion rate \(\gamma _\infty \), see Fig. 6. If the crowding macromolecules are distributed evenly the MSD decays monotonically between \(\gamma _0\) and \(\gamma _\infty \) (pale line), but due to stochastic variations in the medium it fluctuates before converging to \(\gamma _\infty \) (dark line). In Fig. 6a we only depict collisions with the macromolecules responsible for excluded volume effects, but note that there are many collisions with the much smaller solvent molecules responsible for the Brownian motion.

Since we choose \(h> R\) a jump in the mesoscopic model spans a number of macromolecules and the initial free diffusion phase is not resolved and we start to observe diffusion after the first jump of length h that occurs after a critical time \(t_c\) which can be approximated by

$$\begin{aligned} t_c\sim \frac{h^2}{2d\gamma _0}. \end{aligned}$$
(21)
Fig. 6
figure 6

(Color figure online) a Diffusion in the crowded cell environment: initial free diffusion with \(\gamma _0\) (green). After colliding with the first macromolecules the observed diffusion is slowed down and the molecule’s diffusion coefficient decays (orange) to the longtime behavior of slower diffusion with constant \(\gamma _\infty \) in a dense medium (red). b MSD curve for diffusion in a crowded medium (solid line) and as reference normal diffusion (dashed line). The pale line corresponds to an ideal well-mixed medium and the dark line to a realistic medium with stochastic fluctuations in the positions of the crowders

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In the following we will plot the \(\langle \mathbf {x}^2\rangle /2d\gamma _0t\) in log-log-scale for different crowding situations to examine when anomalous behavior occurs. Let \(\mathbf {p}(t)\in \mathbb {R}^N\) be the probability vector for a diffusing molecule, such that \(p_i(t)\) is the probability that the molecule is in voxel \(\mathcal {V}_i\) at time t. As described in Sect. 2 \(\mathbf {p}(t)\) evolves by the master equation

$$\begin{aligned} \mathbf {p}_t = \mathbf {D}\mathbf {p}(t), \quad \mathbf {p}(0) = \mathbf {p}_0, \end{aligned}$$
(22)

where \(D_{ij}=\lambda _{ji}\) for \(i\ne j\) and \(D_{ii}=-\lambda _{ii}\). The initial probability distribution \(\mathbf {p}_0\) is \((0,\dots ,1,\dots ,0)^\mathrm{T}\) with one at the starting node \(\mathbf {x}_0\). We choose the discretization such that \(\mathbf {D}\) is small enough to solve (22) numerically and compute the mean square displacement by

$$\begin{aligned} \langle (\mathbf {x}(t)-\mathbf {x}_0)^2\rangle = \sum _{i=1}^Np_i(t)(\mathbf {x}_i-\mathbf {x}_0)^2. \end{aligned}$$
(23)

In the following experiments we discretize the square \([0,1]\times [0,1]\) into \(N=n^2\) voxels with space discretization \(h=1/(n-1)\). If not mentioned otherwise we release the molecules in [0.5, 0.5] at time \(t=0\) and choose \(n=41\). To avoid boundary effects we set homogeneous Dirichlet boundary conditions on \(\partial \Omega \) and show the solutions as long as more than \(99\%\) of the mass is preserved, i.e., \(\sum _{i=1}^Np_i(t)>0.99\).

Fig. 7
figure 7

(Color figure online) The MSD on a mesoscopic grid with \(h = 0.025\) for different distributions of spherical crowders with \(R=5\times 10^{-3}\) and a moving molecule with \(r=5\times 10^{-4}\). a Two different crowder distributions (red and blue), where we vary the starting position of diffusion from the center [0.5, 0.5] (solid line) to one voxel to the right/left and up/down \([0.5\pm h,0.5\pm h]\) (dashed lines) to show the sensibility of the MSD plot to the local environment. The crowding coefficient is \(\phi =0.4\), and the diffusion constant is \(\gamma _0=1\). b By varying the diffusion coefficient \(\gamma _0\) for the two extreme distributions highlighted in gray in (a) we see that the diffusion coefficient only affects when the molecules undergo anomalous diffusion. c Changing the crowding percentage \(\phi \) for \(\gamma _0 = 1\) for the same two distributions as in (b) affects both the longtime reduced diffusion and the duration of the anomalous phase. d Varying the space discretization h for the red distribution in (a) for all 5 starting positions. We see that a finer discretization better resolves the transient regime

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4.2.1 The Effect of \(\gamma _0\) and \(\phi \)

In Fig. 7a we plot \(MSD/(4\gamma _0t)\) in the crowded environment for different distributions of crowders. Lines in the same color show diffusion in the same environment with starting positions [0.5, 0.5] (solid) and \([0.5\pm h, 0.5\pm h]\) (dashed). We clearly observe the anomalous behavior since \(MSD/(4\gamma _0t)\) is not constant in time and the fluctuations due to the variations of the local environment around the starting position, but for longer times they converge toward the same longtime behavior, before the boundary effects become apparent.

We choose the distributions and starting positions of the curves highlighted in gray to examine the effect of \(\gamma _0\) and \(\phi \) on the MSD. In Fig. 7b we observe that the diffusion constant \(\gamma _0\) only affects when the molecule undergoes anomalous diffusion, but the length of the anomalous phase and the longtime behavior are independent of \(\gamma _0\). The percentage of occupied volume \(\phi \) on the other hand changes both the average diffusion constant in the longtime behavior and the duration of the transient regime of anomalous diffusion, Fig 7c.

4.2.2 Dependence on the Space Discretization h

The mesoscopic model is designed for a voxel size h considerably larger than the molecular radius in order to save computational effort compared to a microscopic simulation. For \(h\rightarrow 0\) the dilute and well-mixed assumptions in each voxel do no longer hold, and the mesoscopic model is known to break down for the simulation of bimolecular reactions (Isaacson 2009). Different corrections to the reaction rates have been suggested (Gillespie et al. 2013) and the references therein, but a minimal \(h_{min}>R\) remains and space cannot be resolved any finer in the mesoscopic model. For a finer resolution one has to switch to microscopic models, such as BD or CA, and we examine the effect of h only for \(h\gg R\). A larger h shifts the critical time \(t_c\) after which we start to observe the molecule’s motion to the right in Fig. 6b, so for very large h we will only see the longtime behavior. In Fig. 7d we see that the initial faster diffusion with \(\gamma \sim \gamma _0\) is less resolved for large h where the trajectories start at a much later time. Furthermore, the boundary effects become apparent earlier for a larger h, but all discretizations are expected to converge toward the same longtime behavior.

4.3 Comparison of Mesoscopic and Macroscopic Simulations

The MSD is only one quantity of interest to examine, but since it is a mean not all features are captured and we will now compare the distributions of molecules resulting from either a mesoscopic or macroscopic simulation. Again, we discretize the square \([0,1]\times [0,1]\) with homogeneous Dirichlet boundary conditions into 41 nodes in each direction and let molecules start diffusing in (0.5, 0.5) in an environment with rectangular crowders of different sizes. We solve (22) once with the mesoscopic \(\mathbf {D}\) and once with \(\tilde{\mathbf {D}}\), where the off-diagonal elements are all equal to \(\lambda _{ii}/4\), corresponding to a finite difference approximation of the macroscopic equation with the space-dependent diffusion constant \(\gamma (\mathbf {x})\) derived from the \(\lambda _{ii}\)’s. In Fig. 8 the macroscopic model agrees with the mesoscopic results for small and evenly distributed crowding molecules, whereas for long barriers only the mesoscopic approach simulates the expected behavior. This is due to the symmetrization of \(\tilde{\mathbf {D}}\), so that only \(\mathbf {D}\) can capture the asymmetric diffusion close to the barriers. The diagonal barriers are not completely impermeable in the mesoscopic model since a small part of the boundary \(\partial \omega _{ij}\) remains.

Fig. 8
figure 8

(Color figure online) Mesoscopic and macroscopic simulations with crowders of different sizes and a moving molecule with radius \(r = 10^{-3}\) starting in the center of the domain (red dot). The heat maps show the probability distribution \(\mathbf {p}(0.5)\) of the location of the diffusing molecule at \(t=0.5\), which is the solution to Eq. (22) with homogeneous Dirichlet boundary conditions. a Rectangles of size \(0.0005\times 0.01\) and \(\phi =0.01\). b, c The mesoscopic (\(\mathbf {D}\)) and symmetrized macroscopic (\(\tilde{\mathbf {D}}\)) simulation results with the distribution in a, respectively. d 5 rectangles of size \(0.004\times 0.8\). e, f The mesoscopic (\(\mathbf {D}\)) and symmetrized macroscopic (\(\tilde{\mathbf {D}}\)) simulation results with the distribution in (d), respectively

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4.4 Reaction Rates

Due to the reduction in diffusion in a crowded environment and (18) the overall reaction rate is decreased. It has, however, been shown (Grasberger et al. 1986; Schnell and Turner 2004), that protein associations can also be enhanced. We examine the mean time \(\mathbb {E}^{r}_j\) until the bimolecular reaction

$$\begin{aligned} A+B\xrightarrow {k_r} C \end{aligned}$$
(24)

happens, where reactant A is confined to voxel \(\mathcal {V}_i\), and molecule B starts diffusing in voxel \(\mathcal {V}_j\) at time \(t=0\), see Fig. 9a. Due to molecular crowding we assume a simplified space-dependent diffusion map with \(\gamma _i<\gamma _0\) inside \(\mathcal {V}_i\) and \(\gamma <\gamma _0\) in the rest of the domain. With \(k_i\) given by (18) and \(\lambda _{ii}\) by (10) we can use conditioning on the first step to compute the expected time until the reaction happens:

$$\begin{aligned} \mathbb {E}^r_i= & {} \frac{k_i}{k_i+\lambda _{ii}}\frac{1}{k_i+\lambda _{ii}}+\frac{\lambda _{ii}}{k_i+\lambda _{ii}}\left[ \frac{1}{k_i+\lambda _{ii}}+\sum _m\theta _{im}E(\mathbf {x}_m)+\mathbb {E}^r_i\right] \end{aligned}$$
(25)
$$\begin{aligned} \mathbb {E}^r_j= & {} E(\mathbf {x}_j)+\frac{h^2+4\gamma _i\sum _m\theta _{im}E(\mathbf {x}_m)}{h^2k_i}, \end{aligned}$$
(26)

where \(E(\mathbf {x}_m)\) is the expected time it takes a molecule located in the neighboring voxel \(\mathcal {V}_m\) to jump into voxel \(\mathcal {V}_i\) and can be computed by solving

$$\begin{aligned} \mathbf {p}_t(t)= & {} (\mathbf {D}-\mathbf {K}_i)\mathbf {p}(t),\quad \mathbf {p}(0)=\mathbf {p}_0, \end{aligned}$$
(27)
$$\begin{aligned} E(\mathbf {x}_m)= & {} \int _0^\infty \sum _{k=1}^N|\mathcal {V}_k|p_{k}(t)dt, \end{aligned}$$
(28)

where \(\mathbf {K}_i\) models a sink at node \(\mathbf {x}_i\) and is the zero matrix except for a suitably large \(K_{ii}\) (e.g., \(K_{i,i}=10^9\)) and \(\mathbf {p}_0\) is the zero vector except for \(p_{0,m}=1/h^3\), see Meinecke et al. (2016) for a derivation. We solve these equations numerically in 3D for a cube with length \(L=1\) and a uniform discretization with \(h=0.1\) in space and reflecting boundary conditions. The voxel \(\mathcal {V}_i\), where the reaction happens is chosen to be the center voxel, such that \(E(\mathbf {x}_m)\) are equal for all 6 neighbors. The diffusing molecule B starts in \(\mathbf {x}_j=(0.7,0.5,0.5)\). In Fig. 9 we compare the mean binding time in the crowded \(\mathbb {E}_j^r\) environment with different \(\gamma _i\) and \(\gamma \) to the time \(\mathbb {E}^r_{j,0}\) it takes to react in a dilute solution where \(\gamma (\mathbf {x})=\gamma _0=1\). The data points with scaled error bars (\(\pm (\frac{\sigma }{\sqrt{M}\mathbb {E}^r_j}+\frac{\sigma _0}{\sqrt{M}\mathbb {E}^r_{j,0}})\)) are from a SSA simulation of the reaction–diffusion process with \(M=200\) trajectories for \(k_r=10^{-4}\) and \(k_r=10^{-3}\) and \(M=500\) for \(k_r=5\times 10^{-3}\) and \(k_r=10^{-2}\).

Fig. 9
figure 9

(Color figure online) a Two-dimensional projection of the three-dimensional experimental setting where a B molecule (blue) starts diffusing in \(\mathbf {x}_j=(0.7,0.5,0.5)\) and reacts with A (red) that is confined to voxel \(\mathcal {V}_i\) with \(\mathbf {x}_i=(0.5,0.5,0.5)\) (red shaded region). Due to an uneven distribution of crowders we assume that the diffusion rate is \(\gamma _i\) inside \(\mathcal {V}_i\) and \(\gamma \) [vertical dashed lines in (b, c)] in the rest of the domain. In (b, c) we compare the time it takes to react in this crowded environment \(\mathbf {E}^r_j\) to the time it takes in an uncrowded environment \(\mathbf {E}^r_{j,0}\) with \(\gamma _0=1\) for different intrinsic reaction rates \(k_r\). b \(\gamma = 0.7\). c \(\gamma =0.9\)

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An overall slower diffusion rate \(\gamma <\gamma _0\) as a result of obstacles reduces the rate of bimolecular reactions (18) in each voxel. But, due to an uneven distribution of crowding agents compartmentalization with locally differing diffusion rates can occur inside the cells. In this case Fig. 9 shows that our model of reaction–diffusion dynamics in a crowded environment can model the increase in reaction rates due to crowders. For certain intrinsic reaction parameters the overall reaction rate increases for a compartmentalization where \(\gamma _i\) is smaller than \(\gamma \), despite the locally slower reaction. The intuitive explanation for this is that once the diffusing molecule enters voxel \(\mathcal {V}_i\) it escapes slower and hence gets trapped close to its reaction partner, which increases the chance of collisions. Like this, cells can boost their efficiency by locating important reaction complexes in areas of slower diffusion, which will be especially productive for reaction cascades where intermediate products are already produced inside the compartment and have a low chance of escaping before being processed further. The fact that macromolecular crowding and locally slower diffusion rates can change the qualitative behavior of a multistep reaction has also been shown by microscopic simulations (Takahashi et al. 2010) and experimentally (Aoki et al. 2011). In the limit when \(\gamma _i\rightarrow 0\) the binding time goes toward infinity, since then both reactants are immobile inside \(\mathcal {V}_i\) and never collide. For the case \(\gamma _i =1\) the reaction dynamics inside voxel i behave as in the dilute case with \(\gamma _0=1\) and the time until reaction is hence only governed by the expected time to reach voxel i, which is inversely proportional to the diffusion rates \(\gamma \) and independent of \(k_r\).

5 Conclusion

We have presented a multiscale framework to model diffusion and reactions in a crowded environment, which is an important feature for realistic simulations inside living cells and on their membranes. First, we homogenize the system by solving a set of PDEs on local domains resolving the microscopic positions and shapes of the crowding molecules. This precomputing step is perfectly parallelizable and yields local first exit times, which can be transformed into the jump rates on an overlying Cartesian grid at the mesoscopic level. We then use these local first exit times to compute a space-dependent diffusion coefficient for the macroscopic diffusion equation, which corresponds to space-dependent reaction rates according to the formula by Collins and Kimball.

Our approach is general in the sense that the crowding molecules can have arbitrary shapes and can be located anywhere inside the domain. We indicate how to adapt our method to moving crowders by computing statistics which is presented in more detail in Engblom et al. (2017). As the jump process is simulated on a coarse Cartesian mesh, no longer resolving the numerous crowders, the stochastic simulation is computationally much more efficient than a microscopic simulation capturing all the collisions.

In numerical experiments we foremost observe that shape and size considerably affect how strongly diffusion is impeded: small crowders have more reflective surface and hence hinder diffusion more severely than bigger obstacles, so do elongated crowders, which create long barriers. The effect is also stronger for larger diffusing molecules than for smaller ones, since the former need bigger gaps to pass through. This gives some new insight into how non-idealized (non-spherical) macromolecules affect the diffusion, since most existing models either assume that all particles are spheres or only consider the percentage of occupied volume.

Comparing the mesoscopic and macroscopic models for diffusion in the crowded environment we note that the former captures the asymmetries created by long barriers better and that they both behave similarly for small crowding molecules compared to the grid size.

The space-dependent diffusion rate can be interpreted as a compartmentalization effect, which has been observed in cells. In a simplified example we see that reactions located inside a compartment with high crowding/low diffusivity can be enhanced since the reaction partners reside longer in the vicinity of each other. Hence, the concentration of reaction complexes in an area with slow diffusion as compared to the rest of the cytoplasm or cell membrane can increase the reaction turn-over, an effect that has been capitalized by cells through colocalization of reaction complexes and scaffolding. Otherwise, reactions between initially distant molecules are impeded by excluded volume since it takes longer time for the reactants to find each other.

Hard sphere reflections on obstacles are not the sole cause of anomalous diffusion (Ridgway et al. 2008), but there are other interactions between macromolecules, such as transient binding or electrostatic repulsion, which have been modeled by a continuous time random walk (Barkai et al. 2012; Schulz et al. 2014) and fractional or multifractional Brownian motion (Marquez-Lago et al. 2012). We can include these types of interactions, by modifying boundary conditions on the crowders from reflecting to partially absorbing or adding potential barriers.