Stochastic Simulation of Reaction-Diffusion Processes

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1042 Stochastic Simulation of Reaction-Diffusion Processes STEFAN HELLANDER ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2013 ISSN ISBN urn:nbn:se:uu:diva

2 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, Wednesday, June 5, 2013 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Abstract Hellander, S Stochastic Simulation of Reaction-Diffusion Processes. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala. ISBN Numerical simulation methods have become an important tool in the study of chemical reaction networks in living cells. Many systems can, with high accuracy, be modeled by deterministic ordinary differential equations, but other systems require a more detailed level of modeling. Stochastic models at either the mesoscopic level or the microscopic level can be used for cases when molecules are present in low copy numbers. In this thesis we develop efficient and flexible algorithms for simulating systems at the microscopic level. We propose an improvement to the Green's function reaction dynamics algorithm, an efficient microscale method. Furthermore, we describe how to simulate interactions with complex internal structures such as membranes and dynamic fibers. The mesoscopic level is related to the microscopic level through the reaction rates at the respective scale. We derive that relation in both two dimensions and three dimensions and show that the mesoscopic model breaks down if the discretization of space becomes too fine. For a simple model problem we can show exactly when this breakdown occurs. We show how to couple the microscopic scale with the mesoscopic scale in a hybrid method. Using the fact that some systems only display microscale behaviour in parts of the system, we can gain computational time by restricting the fine-grained microscopic simulations to only a part of the system. Finally, we have developed a mesoscopic method that couples simulations in three dimensions with simulations on general embedded lines. The accuracy of the method has been verified by comparing the results with purely microscopic simulations as well as with theoretical predictions. Keywords: stochastic simulation, microscale, mesoscale, Smoluchowski's equation, hybrid methods Stefan Hellander, Uppsala University, Department of Information Technology, Division of Scientific Computing, Box 337, SE Uppsala, Sweden. Department of Information Technology, Numerical Analysis, Box 337, SE Uppsala, Sweden. Stefan Hellander 2013 ISSN ISBN urn:nbn:se:uu:diva (

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5 List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III IV V VI VII S. Hellander and P. Lötstedt. Flexible single molecule simulation of reaction-diffusion processes. J. Comput. Phys., 230(10): , A. Hellander, S. Hellander, and P. Lötstedt. Coupled mesoscopic and microscopic simulation of stochastic reaction-diffusion processes in mixed dimensions. Multiscale Model. Simul., 10(2): , M. H. Bani-Hashemian, S. Hellander, and P. Lötstedt. Efficient sampling in event-driven algorithms for reaction-diffusion processes. Commun. Comput. Phys., 13(4): , S. Hellander, A. Hellander, and L. Petzold. Reaction-diffusion master equation in the microscopic limit. Phys. Rev. E., 85(4):042901, S. Wang, J. Elf, S. Hellander, and P. Lötstedt. Stochastic reaction-diffusion processes with embedded lower dimensional structures. Revision of Technical report , Department of Information Technology, Uppsala University, S. Hellander. Single molecule simulations in complex geometries with embedded dynamic one-dimensional structures. Technical report , Department of Information Technology, Uppsala University, M. B. Flegg, S. Hellander, and R. Erban. Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations. Technical report , Department of Information Technology, Uppsala University, Reprints were made with permission from the publishers.

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7 Contents 1 Introduction The microscopic level The Smoluchowski model Green s function reaction dynamics Updating pairs of molecules Operator splitting approach Interactions with lower dimensional structures Interaction between a molecule and a plane Interaction between a molecule and a straight line Dynamic lines and active transport Efficient sampling of random numbers Three dimensions Two dimensions Numerical example The mesoscopic level NSM in the limit of small subvolumes Mesh-dependent reaction rates Embedded one-dimensional structures Interaction with a polymer Hybrid methods Splitting of the system Numerical examples MAPK pathway Translocation into the cell nucleus Conclusions Summary in Swedish Author s contributions Acknowledgments References

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9 1. Introduction The use of numerical simulations to study complex biochemical reaction networks is a methodology that has gained a lot of popularity over the last couple of decades. Although the classical, deterministic ODE model is still the most widely used model, it has been demonstrated that a deterministic model can be unsuitable for systems where some species are present in low copy numbers [15, 16, 40, 52]. This is often the case for reaction networks in living cells, where certain macromolecules, for instance DNA, only exist in a few copies. For these cases discrete stochastic models are required. The focus of this thesis is on developing efficient and flexible methods for simulation of such stochastic models. Normally two levels of stochastic modeling are considered in systems biology: the mesoscopic level and the microscopic level. At the spatially homogeneous mesoscopic level, the position of every molecule is assumed to have a uniform distribution in the simulation volume. Thus we only need to consider the discrete number of molecules of each species in the system, rather than their position. The dynamics of a system at this level of approximation is governed by the chemical master equation (CME), and we can generate exact samples of the probability distribution that solves the CME with the stochastic simulation algorithm (SSA) by Gillespie [24]. At the spatially inhomogeneous mesoscopic level, the spatial domain is divided into subvolumes such that the system is close to spatially homogeneous in each subvolume. Each subvolume is therefore governed by the CME, but molecules are allowed to diffuse between subvolumes by discrete jumps on the lattice defined by the nodes of the subvolumes. The governing equation is now called the reaction-diffusion master equation (RDME), and exact trajectories of the system can be generated with the next subvolume method (NSM) [27]. The microscopic level includes models that track the continuous position of molecules in space rather than just the number of molecules in a subvolume. There are two microscopic models for the reactions, the Smoluchowski model [50] and the Doi model [10, 11], that have been studied in the context of numerical simulations. In this thesis we will only consider the Smoluchowski model, in which molecules are approximated by hard spheres rather than as pure point particles with a soft interaction radius as in the Doi model. In both models the motion of molecules is modeled with Brownian motion. Numerical simulations of the Smoluchowski model involve sampling random numbers from fairly complex probability density functions (PDFs), and part of this thesis deals with methods to simplify and make that particular step of 9

10 the simulations more efficient. In Paper I we show how to solve the Smoluchowski equation for pairs of particles in two steps via operator splitting (in both two and three dimensions), hence introducing an error but ending up with two equations that are simpler to solve than the full equation. In Paper III we investigate different methods and strategies for sampling from the PDFs: Tabulating the PDFs, sampling from the analytical expressions on the fly, or solving the equations with a finite difference scheme rather than evaluating the analytical expressions. Eukaryotic cells are full of internal structure; two-dimensional membranes with complex topology and one-dimensional structures such as the cytoskeleton play an important role in many processes in the cell. Molecules in three dimensions may move first by pure diffusion, then bind to a fiber which is part of the cytoskeleton, and then get actively transported, e.g. towards the nucleus of the cell or towards the outer membrane of the cell. These internal structures are also highly dynamic, and the cytoskeleton changes configuration over time; actin and microtubules both grow and shrink and move around in space [41]. Simulation of such processes requires highly flexible algorithms. In Paper V we develop an algorithm for simulating reactions with embedded general one-dimensional structures at the mesoscopic level. In Paper II we show how to simulate reactions with general membranes at the microscopic level, and in Paper VI we consider the simulation of embedded general lines at the microscopic level. We show how to simulate reactions with the lines, how to couple this with molecules diffusing and reacting on the lines, and how to simulate active transport along the lines while the lines move around in space. Although there now exists efficient methods for simulations at the microscopic level [2, 3, 34, 57, 58], they are still significantly more expensive than methods that simulate systems at the mesoscopic level. However, some systems require a microscopic model to capture important macroscopic dynamics, and in Paper IV we show that the mesoscopic simulations cannot be made arbitrarily accurate by refining the mesh, since the model actually breaks down for small enough subvolumes. This is a problem also studied in [18, 20, 30, 32]. We derive new mesh-dependent reaction rates in two as well as three dimensions, and show how they relate to other reaction rates suggested in [18, 20]. Even though the mesoscopic model has limitations, we could consider hybrid methods that couple different levels of approximation. Some systems may not require the same accuracy everywhere, and splitting the system into a mesoscopic and a microscopic part, depending on the properties of the system, can be an attractive middle way. In Paper II we develop a method that couples microscopic and mesoscopic simulations, both by splitting the domain in two parts but also by splitting the species into mesoscopic and microscopic species. The system is then propagated using an operator splitting, where the mesoscopic degrees of freedom are advanced a time step followed by the propagation of the microscopic degrees of freedom. The method has 10

11 been successfully applied to a system that has microscale features that affect the macroscopic behavior. In Paper VII we compare the accuracy of several hybrid methods. This thesis is organized as follows. In Section 2 we review the Smoluchowski model and simulation methods at this scale. In Section 3 the mesoscopic model is reviewed, and we discuss how to simulate interactions between molecules and embedded lines at the mesoscopic level. In Section 4 we consider hybrid methods coupling the mesoscopic level with the microscopic level. 11

12 2. The microscopic level The microscopic level is usually defined to include models in which the continuous positions of individual molecules are tracked, in contrast to merely tracking the number of molecules at the mesoscopic level of approximation. Two such models that have received significant attention in the scientific literature in the context of numerical simulations are the Smoluchowski model [50] and the Doi model [10, 11]. The diffusive motion of molecules is modeled with Brownian motion in both the Doi model and the Smoluchowski model, but reactions are treated differently. In the Smoluchowski model molecules are approximated by hard spheres, and reactions are modeled with a mixed boundary condition at the reaction radius of the molecules. In the Doi model reactions instead occur with a prescribed probability per unit time when molecules are located within each others reaction radius. There exist several methods and software packages for simulations of biochemical systems at the microscopic level. Smoldyn [2] and MCell [34] both implement methods that approximate the Smoluchowski model by discretizing time, and Green s function reaction dynamics (GFRD) is an efficient method that is continuous in both time and space [57, 58]. In egfrd, the original GFRD algorithm has been combined with the first-passage kinetic Monte- Carlo (FPKMC) algorithm [47, 52]. The egfrd algorithm has been implemented in the software E-Cell [53]. Another approach is to discretize space. In Spatiocyte [3] molecules diffuse on a hexagonal packing of spheres and react with some probability when in neighboring spheres. This thesis has its main focus on the Smoluchowski model. In Paper I we devise an efficient and simple method to solve the Smoluchowski equation for pairs of molecules. Paper III also focuses on this problem, and investigates the efficiency of using analytical solutions of the Smoluchowski equation in simulations, compared to solving the equation using numerical methods or tabulating the solution of the equation. Paper VI concerns an algorithm combining simulations of diffusing and reacting molecules in space with diffusing and reacting molecules on moving, growing, and shrinking embedded onedimensional manifolds. 2.1 The Smoluchowski model Consider a system of N diffusing and reacting molecules with positions given by x 1n,...,x Nn at time t n inside some reaction volume. The full Smoluchowski 12

13 equation, though possible to write down, would be intractable to solve for N > 2 as the problem becomes an N-body problem with a combined PDF, p(x 1,...,x N,t x 1n,...,x Nn,t n ), in 3N dimensions plus time. The equation for a single molecule with diffusion constant D can be solved; the equation for the PDF p(x 1,t x 1n,t n ) is simply the diffusion equation p t = D p (2.1) with initial condition given by p(x 1,t n x 1n,t n ) = δ(x 1 x 1n ) and in the far-field p( x 1,t x 1n,t n ) = 0. The solution to this equation is a three-dimensional Gaussian p(x 1,t x 1n,t n ) = 1 exp (4πD t) 3/2 ( x 1 x 1n 2 4D t ), (2.2) where t = t t n. The Smoluchowski equation for two molecules is given by p t = D 1 x1 p + D 2 x2 p, (2.3) where D 1 and D 2 are the diffusion coefficients of the two molecules and the PDF p(x 1,x 2,t x 1n,x 2n,t n ) governs the positions x 1 and x 2 of the molecules at time t, given the positions x 1n and x 2n at time t n. As it turns out, also this equation can be solved analytically [8, 57]. To this end, we first introduce the new variables so that p can be written as Y = D 2 /D 1 x 1 + D 1 /D 2 x 2, y = x 2 x 1. (2.4) p(x 1,x 2,t x 1n,x 2n,t n ) = p Y (Y,t Y n,t n )p y (y,t y n,t n ). (2.5) Now (2.3) can be split into two independent equations t p Y = D Y p Y (2.6) t p y = D y p y, (2.7) where D is the sum of the diffusion coefficients D 1 and D 2 of the two molecules. The diffusion in the Y direction will be in free space, but the equation for the y variable will have a boundary condition at the sum of the reaction radius, σ, of the two molecules 4πσ 2 D p y n = k r p y ( y = σ,t y n,t n ) k d (1 S(t y n,t n )), (2.8) y =σ where k r is the association rate and k d the dissociation rate. For k r = k d = 0, the molecules will be non-reactive and will always be reflected upon collision. If k r > 0, k d = 0 the molecules react but never dissociate, and if k r > 0 and 13

14 k d > 0 the molecules both react and dissociate and thus undergo a reversible reaction. For k r the molecules always react upon collision. The function S in (2.8) is the survival probability, that is, the probability that the molecules are unbound at time t. It is given by (see [35, 36]) t S(t y n,t n ) = 1 4πσ 2 D p y dτ. (2.9) t n n y =σ Given the initial condition p y (y,t n y n,t n ) = δ(y y n ) (2.10) and with a condition at infinity given by p y ( y,t y n,t n ) = 0, (2.7) can be solved analytically for the case k d = 0, see [8, 58]. 2.2 Green s function reaction dynamics Given that the full N-body problem cannot be solved for N > 2, we realize that we need to reduce the problem into smaller and simpler problems. One approach is to approximate the full system with simple one-body problems by choosing a time step t such that each molecule moves only a short distance during this time step. If molecules collide, they would react with some probability or otherwise be reflected [2, 19, 34, 46]. However, in order to resolve collisions correctly, the time step has to be small and therefore this approach can be slow. Given that the analytical solution to the Smoluchowski equation for two molecules is available, a natural approach would be to not only consider one-body problems but rather divide the full problem into one- and two-body problems. This can be achieved in two steps. First, divide the molecules into subsets of one or two molecules by letting molecules that are each other s nearest neighbors be subsets consisting of two molecules, while the remaining molecules are subsets consisting of one molecule, and let the subsets be denoted by S 1,...,S M. Second, choose a time step t such that each subset is unlikely to interact with another subset during t. This is accomplished by first computing pairwise distances d i j between all the subsets, where d i j is the shortest distance between any of the molecules in the subset S i and the subset S j, and then by computing t i given by ( ) 2 min j, j i d i j t i =, (2.11) K 6D where K is a constant chosen large enough to make the probability for the subsets S i and S j to interact during the time step t i small. Note that (2.11) is related to the standard deviation of the three-dimensional normal distribution, 6D t. The global time step is chosen to be t = mini t i. The full N-body 14

15 problem can be approximated during t by the independent one- and two-body problems defined by the subsets S 1,...,S M. The approach outlined above is called the Green s function reaction dynamics [57, 58]. Similar algorithms have also been considered in [14, 48]. The GFRD algorithm is more efficient and accurate than the methods implemented in Smoldyn [2] and MCell [34] for systems that are not crowded. However, for dense systems the time steps in GFRD become small and GFRD will then be outperformed by Smoldyn and MCell, as the cost per time step is much smaller in those algorithms Updating pairs of molecules The position of single molecules are updated by sampling new positions from a normal distribution with the standard deviation 2D t in each direction (unless they are close to some internal structure, see Section 2.4, or dissociate during the time step). Pairs of molecules are updated according to the PDF obtained by solving the Smoluchowski equation for pairs of molecules (2.3). Given two molecules with positions x 1n and x 2n at time t n we sample the time t r when they react according to the flux over the boundary at the reaction radius, given by q(t y n,t n ) = k r p y ( y = σ,t y n,t n ), (2.12) see [57, 58]. If t r t, the reaction fires at t r and the product is updated as a single molecule for the remainder of the time step. If, on the other hand, t r > t we sample new positions from the distribution p. This is done in two steps. First we update Y by sampling from a normal distribution with standard deviation 2D t in all directions and next we update the y-coordinate by sampling random numbers from the random variable with PDF given by p y. Though the analytical solution for the case k d = 0 is available, it is fairly complicated and expensive to evaluate. In Paper I we solve the equation in two steps with an operator splitting scheme [38], introducing an error, but making each equation simpler than the full equation. This approach also makes it possible to consider the case k d > 0 which can speed up simulations in the case of molecules rebinding and dissociating quickly. 2.3 Operator splitting approach Consider (2.7) in a spherical coordinate system r = (r,θ,φ) rotated such that r n = (r n,0,0). Then the equation becomes (( p r 2 p r = D t r ) p r + 1 ( ( 1 r r r 2 sinθ p ) r )) p r sinθ θ θ sin 2 θ φ 2 (2.13) 15

16 with boundary condition 4πσ 2 D p r = k r p r (r = σ,θ,φ,t r n,t n ) k d (1 S(t r n,t n )), (2.14) r r=σ initial condition p r (r,θ,φ,t n r n,t n ) = δ(r r n)δ(θ) r 2, (2.15) sinθ and with p vanishing at infinity; p( r,t r n,t n ) = 0. This equation, as mentioned above, can be solved analytically for the case k d = 0: 1 p r (r,θ,φ,t r n,t n ) = 4π rr n n=0 where t = t t n, P n is a Legendre polynomial, and where 0 (2n + 1)P n (cosθ) (2.16) exp( 2u 2 t)f n+1/2 (ur)f n+1/2 (ur n )udu, (2.17) F n (ur) = (2σk r + 1)A 2uσB, (2.18) (C 2 + D 2 ) 1 2 A = J n (ur)y n (uσ) Y n (ur)j n (uσ) B = J n (ur)y n(uσ) Y n (ur)j n(uσ) C = (2σk r + 1)J n (uσ) 2uσJ n(uσ) D = (2σk r + 1)Y n (uσ) 2uσY n(uσ), (2.19) and where J n and Y n are Bessel functions of first and second kind. This expression is difficult and expensive to sample random numbers from. We could potentially precompute and tabulate p r, but since there are four parameters (r, θ, r n and t t n ) to tabulate for, it is hard to get a high accuracy without having very large tables. As an alternative we have suggested an approach where the equation is solved in two steps instead of solving the full equation directly. First we solve for the radial part of (2.13) p r t = D ( 2 p r r r p r r ), (2.20) with boundary condition given by (2.14), initial condition p r (r,t n r n,t n ) = δ(r r n ), and with p r (r,t r n,t n ) = 0. The analytical solution is derived in [35] and is given by p r (r,t r n,t n )4πrr n D = 1 4π t ( exp( R 2 1 ) + exp( R 2 2) ) + B(α,β,γ, t) B(β,γ,α, t) + B(γ,α,β, t), (2.21)

17 where t = t t n R 1 = (r r n) 2 4D t R 2 = r+r n 2σ 4D t B(α,β,γ, t) = α(γ+α)(α+β) (γ β)(α β) exp(2αr 2 t + α 2 t)erfc(r 2 + α t), (2.22) and where α, β and γ solves σx 3 + ( D 1 + k ) r x 2 + σk d x + Dk d = 0. (2.23) 4πσD The second step is to solve for the angular part p θφ = D ( ( 1 t r 2 sinθ p ) θφ ) p θφ sinθ θ θ sin 2 θ φ 2 l=0 (2.24) with the initial condition p θφ (θ,φ,t n 0,0,t n ) = δ(θ)/(r 2 sinθ). As it turns out, p θφ will be independent of φ (see e.g. [54, 56]) ( ) 2l + 1 l(l + 1)D(t p θφ (θ,φ,t 0,0,t n ) = 4πr 2 exp tn ) r 2 P l (cosθ), (2.25) where P l is a Legendre function. Thus, φ will have a uniform distribution on the interval [0,2π). We can now sample y n+1 in five steps: 1. Transform the coordinate system to a spherical coordinate system with initial position given by r n = (r n,0,0). 2. Given r n at time t n, sample r n+1 from the PDF (2.21). 3. Given r n+1 at time t n+1, sample θ n+1 from the PDF (2.25) with r = r n Sample φ n+1 from a uniform distribution on the interval [0,2π). 5. Transform the coordinates back to obtain y n+1. Now, given y n+1 and Y n+1, we get x 1(n+1) and x 2(n+1) from solving (2.4). By solving the equation in two steps we simplify the computations, but we do introduce an error. In Paper I we show in several examples that the error is small. If, however, one would need higher accuracy than that given by a first order operator splitting scheme, a second order scheme, so called Strang splitting [51], is also discussed in Paper I. 2.4 Interactions with lower dimensional structures Eukaryotes have a complex internal structure; molecules can interact with two-dimensional membranes and with one-dimensional structures and polymers. To further complicate matters these structures are far from static [41] 17

18 and molecules in living cells are not only transported by pure diffusion, but are also actively transported on the cytoskeleton, a structure in the cell consisting of microtubules and actin cables. Molecules bind to the fibers of the cytoskeleton and are then transported in a specific direction by molecular motors. The cytoskeleton extends throughout the whole cytoplasm in eukaryotic cells, and is therefore an important structure to take into account when modeling complex reaction networks in cells. Furthermore, one-dimensional processes may take place on long polymers in the cell, such as DNA. The problem of coupling simulations in space with one-dimensional processes has also been studied in [39]. They develop an FPKMC method for simulating active transport due to chemical gradients along static lines. Methods for simulation of molecules interacting with surfaces have been developed in [34, 44]. In this section we will describe how to include reactive complex lowerdimensional structures in the GFRD algorithm. More specifically, we will summarize an algorithm for simulating complex and reactive surfaces (see Paper II) and an algorithm for simulating dynamic and reactive embedded one-dimensional manifolds (see Paper VI) Interaction between a molecule and a plane Consider a molecule in the vicinity of a plane. Without loss of generality we can assume that the plane is the x y plane and that the position of the molecule at time t n is given by x n = (0,0,z n ). The diffusion of the molecule along the x- and the y-axes is now independent of the interaction between the molecule and the plane, and the x- and y-coordinates are therefore updated by sampling new positions from a normal distribution with standard deviation 2D t in each direction. The interaction between the molecule and the plane is modeled by the one-dimensional Smoluchowski equation p z t = D 2 p z z 2, (2.26) with the boundary condition D p z z = k r p z (0,t z n,t n ). (2.27) and inititial condition p(z,t n z n,t n ) = δ(z z n ). Finally, p vanishes at infinity p(z,t z n,t n ) = 0. This equation is solved analytically in [8]. Similarly to when updating pairs of particles, we first sample the time t r when the molecule reacts with the surface. If t r t, the reaction fires at t r, and otherwise z n+1 is sampled from p z. 18

19 2.4.2 Interaction between a molecule and a straight line Now consider a straight, infinite line through some domain. We want to simulate the interaction between a molecule A and the line l. Without loss of generality, assume that the line is the z-axis and that the molecule has the initial position, in cylindrical coordinates, r n = (r n,θ n,z n ) = (r n,0,0) at time t n. The molecule has the reaction radius σ A and the line has the reaction radius σ l. Denote the sum of the reaction radii by σ and the diffusion coefficient of A by D. Note that the diffusion of the molecule in the z-direction is independent of the diffusion in the r- and θ-direction and the interaction with the line. The full equation is split into two equations using a first order operator splitting scheme. The equation for the radial part becomes p r t = D ( 2 p r r r p r r ), (2.28) with boundary condition at r = σ 2πσD p r = k r p r (r = σ,t r n,t n ) (2.29) r r=σ and initial condition given by p r (r,t n r n,t n ) = δ(r r n ). Finally, p r vanishes at infinity. This equation is solved analytically in [8] and the solution is given by p r (r,t r n,t n ) = 1 exp( Du 2 (t t n ))C(u,r,k,k r )C(u,r n,k,k r )udu, 2π 0 (2.30) where k = 2πσD and where the function C is defined by C(u,r,k,h) = J 0(ur)(kuY 1 (σu) + hy 0 (σu)) Y 0 (ur)(kuj 1 (σu) + hj 0 (σu)), ((kuy 1 (σu) + hy 0 (σu)) 2 + (kuj 1 (σu) + hj 0 (σu)) 2 ) 1 2 (2.31) with J 0 and J 1 being Bessel functions of the first kind and Y 0 and Y 1 Bessel functions of the second kind. Although the analytical solution is available, it is fairly complex and expensive to compute. Natural strategies to sample from the PDF p r are to either evaluate the analytical solution on the fly, compute a look-up table in the parameters r, r n, and t t n, or to use a numerical scheme to compute the solution. We investigate these three strategies in Paper III and the results are summarized in Section 2.6. Next we solve the equation for the θ and z directions p θz t = D ( 1 2 p θz r 2 θ p θz z 2 with initial condition p θz (θ,z,t n 0,0,t n ) = δ(θ)δ(z)/r. Gaussian in both θ and z. ), (2.32) The solution is a 19

20 2.5 Dynamic lines and active transport Consider a general curve Γ embedded in three-dimensional space. The curve can be approximated by a piecewise linear curve, with linear segments denoted by γ 1,...,γ M, to arbitrary accuracy. Consider a molecule close to one of the linear segments, say γ i. First choose a time step t such that the molecule and the line segment are unlikely to interact with any other object during the time step. Throughout the time step t the interaction between the line segment and the molecule can be approximated by the interaction between a molecule and an infinitely long line. Thus, the molecule can be updated according to the algorithm proposed in Section We simulate moving, growing, and shrinking lines with an operator splitting approach. First we update molecules with the lines kept constant, and then we update the lines with the molecules kept constant. More specifically, consider the line defined by the segments γ 1,...,γ M. The segment γ i is determined by a pair of endpoints (p 1 i,p2 i ), where the points are chosen such that the segments are connected. The line is now transformed by some transformation T (x, t) : R 3 R R 3, where the points at time t n+1 is given by applying T to all points at time t n. Thus, if the line is defined by the points p 1 1,p2 1,... at time t n, then the line is given by the points T (p 1 1,t n),t (p 2 1,t n),... at time t n+1. Active transport can be simulated by sampling new positions for molecules on the lines from a different distribution than the normal distribution, or even letting the molecules do a deterministic walk on the lines. Diffusion and reaction of molecules on the lines can be simulated with the GFRD algorithm in one dimension. In Paper VI we consider numerical examples with both dynamic lines and active transport. 2.6 Efficient sampling of random numbers Although the analytical solutions to many of the equations considered above are available, it is not obvious that they are efficient to use when sampling new positions of the molecules. For instance, the analytical solution to the radial part of the two-dimensional Smoluchowski equation is given as an integral of a complicated expression of Bessel functions, and evaluating that expression accurately and efficiently is not straight-forward. We compare several different approaches to sampling new positions in both three dimensions and two dimensions. The first approach is to use the analytical solutions to compute the PDF or the cumulative distribution function (CDF) on the fly, and then sample new positions by inverse transform sampling (ITS). ITS is based on sampling uniform random numbers and then transforming them with the inverse of the CDF. The second approach is to precompute the PDFs and tabulate the solutions. Computing the tables can be expensive, but is done only once before the simulation is started. However, the drawback of this approach is that in order to keep the interpolation error small we may 20

21 need very large three-dimensional tables consuming a lot of memory. This can make the look-up process slow and make the algorithm less suitable for multicore implementations. On the other hand, for cases when the table is not too large, this can be a very efficient method. The last approach that we consider is to compute the PDFs with a semi-implicit finite difference scheme. With the PDF, we determine the CDF numerically and sample from that Three dimensions We have compared sampling a new r and θ both using a look-up table and sampling by utilizing the analytical expressions for the PDF or the CDF when available. In three dimensions the analytical expression for p r is fairly simple, and as an analytical expression for the CDF is available, the CPU time for sampling by evaluating the analytical expression on the fly is on the same order as sampling from a look-up table. Solving the equation numerically turns out to be orders of magnitude slower, and thus it is clear that the preferred method is to sample random numbers using the analytical expression. The analytical expression for p θ is more complicated than that for p r, and involves summation of an infinite series. For the parameters that we considered in Paper III we found that sampling from the analytical PDF was expensive and that the CPU time per random number sampled was on the order of 0.1s, while sampling from a look-up table was 2-3 orders of magnitude faster, while still maintaining an error of around 0.01%. We implemented the methods in Matlab and the performance could probably be improved significantly if we were to write an optimized version in for instance C, but it is still unlikely that the main conclusion would change. Computing the look-up table was quite efficient (a CPU time of 17 seconds) and unless that time is significantly longer than the actual simulation time, the best method to sample a new θ would be to use the look-up table method (except for small values of the parameter where the method developed in [7] has a similar performance to the look-up table). Note that there is only one parameter to tabulate for in p θ, and the table can therefore be kept small (less than 500KB) Two dimensions In two dimensions the angular direction is straight-forward to update, while the analytical expression for p r in (2.30) is complicated. For this case we found that evaluating the analytical expression was the most expensive method and that sampling a new r using the numerical solution of the equation was about an order of magnitude faster. However, the look-up table was yet another 2-3 orders of magnitude faster, with an error of less than 5% for all parameters, and for most parameters significantly smaller than that (< 1%). The fastest way to generate the look-up table was to generate it by solving the 21

22 equation numerically, which took about 29 minutes of CPU time compared to approximately 7 hours using the analytical expression. 2.7 Numerical example In Paper VI we consider an example of active transport of molecules along moving microtubules in a spherical domain with radius 10 6 m. Molecules of species A diffuse in three dimensional space, react with microtubules, and are then actively transported towards the center of the sphere. We are thus considering the following reaction A +C k r kd A cyl +C (2.33) where C is a microtubule and A cyl is an A-molecule bound to the microtubule. The system is initialized with 100 A-molecules with uniform initial positions, and 25 microtubules, modeled by straight lines going through the center of the sphere. Each microtubule is therefore determined by a pair of points (p i, p i ), 1 i 25. The points p i are all sampled from a uniform distribution on the surface of the sphere. Molecules free in space move by pure diffusion with diffusion constant D = m 2 /s, while the molecules attached to a microtubule move deterministically towards the center of the sphere with speed t/40m/s. Two snapshots from the simulation are shown in Figure 2.1. We model the motion of the microtubules by rotation around the center of the sphere. Specifically, they move θ and φ radians per time step around the x-axis and the y-axis, where θ and φ are given by { θ = 2π tx 1 φ = 2π tx 2, (2.34) and where X 1 and X 2 are random numbers sampled from a normal distribution with standard deviation 1. Note that this is not meant as an accurate representation of the true biology, but rather as a demonstration of the flexibility of the algorithm. A trajectory of one of the microtubules as well as the radial distribution of molecules at the end of a simulation are plotted in Figure 2.2. As expected, the distribution of molecules in the sphere is shifted towards the center of the sphere compared to the radial distribution of molecules uniformly distributed on the sphere. 22

23 Figure 2.1. Two snapshots from the simulation. A cyl -molecules (blue dots) are bound to the microtubules (yellow lines). Once bound, they are transported deterministically towards the center of the sphere. Note that the free A-molecules are not shown in this figure Distribution at t f =1.0s Uniform distribution x 10 6 x(t) y(t) z(t) 5 Position Distance from origin x Time (s) Figure 2.2. To the left the radial distribution of the molecules at the final time (solid line) compared with the radial distribution at the initial time (dashed line). The initial positions of the molecules are sampled from a uniform distribution. To the right a trajectory of one of the microtubules. We plot the x-, y- and z-coordinate as functions of time. 23

24 3. The mesoscopic level At the mesoscopic level of approximation we assume that the system, to high accuracy, can be approximated by a spatially homogeneous model inside some volume. In other words, the distribution of the position of a molecule should be close to uniform inside the volume. This condition is often referred to as the well-stirred condition. A system that is assumed to be well-stirred is therefore fully described by the discrete number of molecules present at a given time. Let x n = (x 1,...,x N ) denote an observation of the stochastic variable X(t) = (X 1,...,X N ) at time t n, where X j is the number of molecules of species j. Now let p(x,t x n,t n ) be the probability of the system to be in state x at time t, given that it was in state x n at time t n. The governing equation for p is the chemical master equation (CME) [22, 33], given by M d p dt = ω r (x n r )p(x n r,t) ω r (x)p(x,t), (3.1) r=1 where ω r is the propensity function of reaction r and n is the stoichiometry matrix. Thus, ω r describes the intensity with which the reaction r occurs, and column r of the matrix n, denoted by n r, describes how the state vector is affected by reaction r. As an example, consider a system with a single reversible reaction A + B k a kd C. (3.2) If we assume mass action kinetics, the propensity function for the association of A and B is given by ω 1 = k a ab, where a and b are the copy numbers of molecules A and B respectively, and the propensity function for the dissociation is given by ω 2 = k d c. The stoichiometry vectors are given by n 1 = [ 1, 1,1] and n 2 = [1,1, 1]. Exact trajectories of a well-stirred system can be generated with the stochastic simulation algorithm (SSA) by Gillespie [24]. Another exact method is the next reaction method (NRM) [23]. Approximate methods include the tauleaping method [25] and the finite state projection (FSP) algorithm [45]. Improved versions of tau-leaping have been developed in [1, 4, 5, 6]. Various methods for the well-mixed case have been implemented in the freely available software StochKit [49]. Some systems have spatial inhomogeneities that make them violate the well-stirred condition. If this is the case, the volume can be divided into 24

25 subvolumes, where in each subvolume the well-stirred condition is fulfilled. Diffusion is modeled by discrete jumps between the subvolumes and only molecules in the same subvolume can react. The governing equation is now called the reaction-diffusion master equation (RDME). The NRM algorithm has been extended to this setting in the next subvolume method (NSM) by Elf and Ehrenberg [15]. MesoRD [27] is a software that implements NSM on structured meshes. NSM can also be extended to unstructured meshes [17], and a software package that uses unstructured meshes instead of structured meshes is URDME [12]. STEPS [55] is another software for mesoscopic simulations on unstructured meshes. An approximate method that improve on the performance of NSM is the diffusive finite state projection algorithm (DFSP) [13]. The DFSP algorithm is implemented as a plugin solver for URDME. However, it has been shown that the RDME in fact becomes a poor model if the size of the subvolumes becomes too small [20, 29, 30, 32]. The model can be improved by allowing reaction rates to depend explicitly on the mesh [18, 20, 29], but such corrections to the model will inevitably be problemdependent. This problem is investigated in Paper IV, where we also derive mesh-dependent reaction rates in two dimensions as well as three dimensions for a model problem. These results are summarized in Section 3.1. In Section 3.2 we discuss how to simulate molecules in space interacting with onedimensional structures, summarizing the contents of Paper V. 3.1 NSM in the limit of small subvolumes A fundamental assumption that we have made is that the microscopic level provides a more accurate description of reality than the mesoscopic level. Therefore it is natural to verify the accuracy of a mesoscopic model by comparing it to a microscopic model. However, in order to do so we need to know how the mesoscopic reaction rates correspond to the microscopic, or intrinsic, reaction rates in a specific microscopic model. The relation between mesoscopic reaction rates and the corresponding reaction rates in the microscopic model is given by k meso = 4πσDk micro 4πσD + k micro (3.3) for sufficiently large subvolumes [9, 26]. This conversion formula is valid in three dimensions, but in two dimensions no such mesh-independent formula exists. It is easy to realize that the mesoscopic model becomes inaccurate if the size of the subvolumes becomes too small; if the subvolumes are so small that molecules no longer fit inside them we could not expect the model to accurately describe the system. However, it has been shown that the mesoscopic model does not agree with the microscopic model even for reasonably large subvolumes [18, 20, 29, 30]. One approach to resolve this problem would be 25

26 to extend the model to allow for reactions between molecules in different subvolumes, and this has been studied in [20, 31]. We will instead consider the case where only molecules in the same subvolume can react, and derive mesoscopic reaction rates that make the two models match for a model problem Mesh-dependent reaction rates Now, let us consider a system with only two molecules, a molecule A and a molecule B, that react according to A + B k r /0, (3.4) where k r is the intrinsic, microscopic reaction rate. If we assume that the molecule B is fixed at the origin and that the initial position of the molecule A is uniformly distributed in a square or cube with volume or area V, we can estimate the average time, τ micro, until the molecules react by (see [20]) τ micro = (1 + αf(λ))v k r, (3.5) where V is the volume or area of the domain, α = k r /4πDσ in three dimensions and α = k r /πd in two dimensions. Define R to be the radius of a sphere or disk with volume or area V. Then λ = σ/r, and F is given by 4ln(1/λ) (1 λ 2 )(3 λ 2 ) (2D) F(λ) = 4(1 λ 2 ) 2 (1 λ)(5+6λ+3λ 2 +λ 3 ) 5(1+λ+λ 2 ) 2 (3D). (3.6) The corresponding mesoscopic model, where the B molecule diffuses on a Cartesian mesh with subvolumes of width h and with periodic boundary conditions, can also be solved analytically, as demonstrated in Paper IV. The average time until the molecules react at the mesoscopic level, τ meso, depends on the mesh size h, the diffusion constant D, and the mesoscopic reaction rate k meso in the following way τ meso = τ D + N/k meso, (3.7) where N is the number of subvolumes and τ D is approximated by { ( ) V τ D 2πD ln V h D V, as h 0 (2D) Dh V, as h 0 (3D), (3.8) where V is the volume of the domain in three dimensions and the area of the domain in two dimensions. Note that (3.8) is a corollary that follows from results on random walks by Montroll and Weiss [42, 43]. 26

27 A reasonable assumption is that τ micro = τ meso should hold, and we can then solve this equation for the now mesh dependent mesoscopic reaction rates in two as well as three dimensions: k meso = N τ micro τ D. (3.9) Other mesh-dependent reaction rates in two and three dimensions have been derived by Fange et al. in [20], and similar reaction rates were derived by Erban and Chapman in [18] for the three-dimensional case. The reaction rates derived in [20] do not match the reaction rates that we get from studying the model problem above. This is not surprising as they study a different model problem, and we would then expect to get different mesoscopic reaction rates. We have compared how τ meso depends on the mesh and the different choices of reaction rates in Figure 3.1. Time (s) Fange et al. k meso Micro Conventional Erban and Chapman Fange et al. k meso Micro Grid size h (multiples of ρ) Grid size h (multiples of ρ) Figure 3.1. We have computed τ meso for different reaction rates in two dimensions (to the left) and in three dimensions (to the right). As we can see, the different reaction rates do not give the same results, and this is because they are derived from different model problems. The conventional reaction rates, as defined by (3.3), are good for larger subvolumes. In the figure above, ρ denotes the reaction radius. There is a lower bound on h, h, for when the equation τ meso = τ micro cannot be solved anymore. By solving τ micro = τ D for h, we obtain { h 3.1σ (3D) (3.10) 5.1σ (2D). This lower bound is an absolute lower bound for this problem and no mesoscopic reaction rate can make the two models match if h is below this critical lower bound. In Section we discuss how this problem has practical consequences when simulating a problem with fine-grained dynamics, and how we can resolve it for that model using a hybrid method coupling microscopic and mesoscopic simulations. 27

28 3.2 Embedded one-dimensional structures In Section 2.4 we described how to simulate interactions between molecules in three-dimensional space and embedded lines at the microscopic level. However, these simulations can be fairly expensive for large systems, so an accurate mesoscopic method is useful for the cases where a mesoscopic level of approximation is sufficient. The problem of coupling mesoscopic simulations in space with active transport on one-dimensional fibers was also considered in [28]. They derive how the fibers affect the mesoscopic jump rates, rather than simulating the fibers explicitly. In this section we describe how to simulate molecules reacting with and dissociating from general, embedded polymers. This section summarizes the contents of Paper V Interaction with a polymer We consider a Cartesian mesh of size h on which the molecules in space diffuse and react. The domain Ω is subdivided into cubical voxels Ω 1,...,Ω N. A general one-dimensional curve Γ(u) = (x(u),y(u),z(u)), u [0,u max ], goes through the mesh and is approximated by a polygon π(s), where the corners of the polygon are given by (x m,y m,z m ) = (x(u m ),y(u m ),z(u m )), m = 1,...,M 1 1 (3.11) u 0 = 0 < u 1 <... < u M1. (3.12) The segments of the polygon are surrounded by cylinders C 1,...,C M, where the radius of the cylinders, r c, is chosen such that πr 2 c = h 2. This way the area of the face of the cylinder is the same as the area of the face of the cubes in the mesh. Next we compute the overlap between the cylinders and the background mesh, and the reaction rates given by (3.9) are scaled by the fraction of the volume of the cylinder inside the volume of a voxel in the mesh. The system can now be simulated with a modified version of the NSM algorithm, as described in Paper V. Note that the three-dimensional simulation can be coupled with one-dimensional mesoscopic simulations on the fibers in a straight-forward way. Several such examples are considered in Paper V. 28

29 4. Hybrid methods It has been shown that reaction networks in living cells can have microscale features that affect the macroscopic behavior of the system [52]. These microscale features cannot be accurately captured even at the spatially heterogeneous mesoscopic scale, since they require a spatial resolution that is finer than the critical lower bound of the size of the subvolumes in (3.10). Thus we have to resort to the microscale to accurately simulate such a system. Unfortunately, even with efficient microscale methods, these simulations are slow compared to the corresponding mesoscopic simulations, and large systems with complicated outer boundaries and internal structures are difficult to simulate for long times if high accuracy is required. However, if the microscale behavior is restricted to only a part of the system, another strategy would be to simulate only this specific part with a microscopic method and the remaining part of the system at a coarser scale. In Paper II a method is developed for dividing the species into a microscopic and a mesoscopic part, as well as splitting space into a microscopic and a mesoscopic part. Other hybrid methods have been suggested in [21, 37]. In [21] they derive an accurate way of coupling the diffusion of molecules in a one-dimensional domain, where one part of the domain is simulated microscopically and one part is simulated mesoscopically. In [37] they also consider the possibility to split both species and space into a mesoscopic and a microscopic region. 4.1 Splitting of the system A domain Ω is covered with a primal mesh, consisting of non-overlapping tetrahedra in space and non-overlapping triangles on the boundary Ω. From the primal mesh we create the dual mesh. We have a set of species S = {S 1,...,S N } in our system. For each species we define a mesoscopic region and a microscopic region in space. Species S i is simulated at the mesoscopic level in the subvolumes ΩC i = {Ωi C1,...,Ωi CJ 1 } and at the microscopic level in the subvolumes Ω i M = {Ωi M1,...,Ωi MJ 2 }, where Ω = ΩC i Ωi M for all i, 1 i N. The system is now propagated with an operator splitting scheme. Choose a time step t split for the splitting. The first step is to freeze the microscopic part of the system and propagate the mesoscopic part with NSM on unstructured meshes for the full time step t split. The second step is to freeze all the 29

30 mesoscopic variables and propagate the microscopic variables according to the algorithm outlined in Section 2 for the full time step. At the end of each time step the state vector is updated by converting microscopic molecules that have ended up in a mesoscopic region to mesoscopic molecules, and vice versa. Note that during a time step all variables remain either mesoscopic or microscopic, meaning that a mesoscopic molecule diffusing into a microscopic region of space will not be treated as a microscopic variable until the next time step, and vice versa. Therefore we should choose the time step such that a molecule does not diffuse through several subvolumes during that time step, or the accuracy will suffer. On the other hand, if the time step is too small relative to the size of the subvolumes, the spatial distribution of molecules will be biased towards the mesoscopic regions of space. This can be realized by imagining a microscopic molecule close to the mesoscopic region. If the time step is really small, it may diffuse into the mesoscopic region, but it is highly likely that it will still be close to the microscopic region. However, at the next time step it will be treated as a mesoscopic molecule and is therefore assumed to be uniformly distributed in the subvolume, and we get a shift in the distribution. In Paper VII we develop another hybrid method for coupling of a mesoscopic and a microscopic region, the ghost cell method (GCM), and compare the error of the GCM to the error of the method outlined above. In the GCM method we only consider splitting of space, and not splitting of species, and therefore, for simplicity, we study the problem of pure diffusion in a cube with side length 10 6 m and diffusion constant m 2 /s. The cube is first discretized with an unstructured mesh, and then divided into a microscopic region, which is defined to be all subvolumes left of x = m, and a mesoscopic region which is defined to be all remaining subvolumes. We initialize the system by sampling molecules uniformly on the domain, and then propagate the system for 0.1s. As the initial condition is a uniform distribution, we expect the distribution to be uniform also at the end of the simulation. We compute the error E defined by E = 10 i=1 N i N 0 /10 N 0, (4.1) where N 0 is the total number of molecules, and N i, 1 i 10 is the number of molecules with x-coordinate in the interval [(i 1)/ ,i/ ] at the end of the simulation. Molecules that are in the mesoscopic region are assigned to an interval by sampling a continuous position from a uniform distribution on their respective subvolume. The error of the two methods are compared in Figure 4.1. We find that the error of the GCM method decreases with decreasing time step, while the error of the method in Paper II increases if the time step becomes too small, but that the error is smaller than that of the GCM method for time steps such that 2D t h. Here h is defined to be the cubic root of the average volume of a subvolume in the mesh. A good 30

31 approach would therefore be to choose method based on the time step; for really small time steps we would use the GCM method and for larger time steps we would switch to the method in Paper II voxels voxels voxels 10 1 CPM method GCM method h 2 /(2D) E E t t Figure 4.1. On the left we have plotted the error E of the GCM method for different resolutions of the mesh and different time steps. We see that the error decreases with the time step. On the right we compare the error of the GCM method with the error of the method proposed in Paper II (referred to as the CPM method in the legend). The vertical dashed line marks t = h 2 /(2D). 4.2 Numerical examples In the first example we study a model of one layer of a mitogen-activated protein kinase (MAPK), first proposed in [52], and in the second example we simulate a model of translocation of molecules into the cell nucleus. The MAPK system has microscale dynamics that can be captured with the hybrid method by a splitting of species into a mesoscopic and a microscopic part, while the translocation model will be simulated by splitting the domain into a mesoscopic and a microscopic part. These two examples were also studied in Paper II MAPK pathway To verify the accuracy and applicability of the method we first apply it to a problem studied in [52]. They consider a model of a MAPK pathway KK + K k 1 k2 KK K k 3 KK + K p, KK + K p k 4 k5 KK K p k 6 KK + K pp, (4.2) P + K pp k 1 k2 P K pp k 3 P + K p, P + K p k 4 k5 P K p k 6 P + K, (4.3) KK k 7 KK, P k 7 P. (4.4) 31

32 First, in (4.2), K is phosphorylated in two steps by the kinase KK to become K p and K pp. In (4.3) K is dephosphorylated in two steps from K pp by the phosphatase P. After dephosphorylation the enzymes are inactive, KK and P, and are activated again through the reactions in (4.4). If k 7 is large, the inactive enzymes KK and P are reactivated quickly, and the spatial correlations that exist for a short time after a dissociation become important. The response time τ res of the system is defined as the time until we have 50% of the steady state level of doubly phosphorylated substrate K pp. The spatial correlations due to fast rebindings cannot be captured at the macroscopic scale or the mesoscopic scale and the macroscopic behavior of this system is only captured accurately at the microscopic level. Since we know that the critical reactions are the dissociations of KK K and P KK with fast rebindings due to the quickly reactivated enzymes, it seems natural to split the system into a mesocopic part and a microscopic part, where the microscopic part consists of the species KK K and P KK and where the mesoscopic part will be the rest of the system. Since microscopic molecules are simulated as microscopic during a full time step, we would, with a correctly chosen time step, capture the dissociations and fast rebindings during that time step. The time step for the splitting scheme should thus be chosen such that the molecules have enough time to get well-mixed in a subvolume during that time step. The results are shown in Figure 4.2. We find that the hybrid method can attain an accuracy close to the accuracy obtained with purely microscopic simulations, but with a significant reduction of the molecules simulated at the microscopic level. Since the computational cost scales almost with the square of the number of microscopic molecules, also the gain in computational time is significant even though the overhead of the method can be up to 30% of the total computational cost. τ res Hybrid method GFRD Mean field Average number of molecules Microscale Mesoscale D (µm 2 /s) D (µm 2 /s) Figure 4.2. To the left we compare τ res computed with the hybrid method, with GFRD simulations from [52] and the corresponding ODE model. The response time is significantly smaller at the microscopic level as compared to the ODE description. With the hybrid method we are able to accurately simulate the system, with a reduction of the average number of molecules simulated on the microscale, as shown to the right. 32

33 4.2.2 Translocation into the cell nucleus In Paper II we also considered a model of translocation of molecules from the cytoplasm to the nucleus of the cell. The full system is described in Table 4.1. (1) /0 µ 1 A (2) /0 µ 2 B Reactions in the bulk (3,4) A + B k 1 k2 C Reversible adsorption of C to the membrane (5,6) C k a kd C m Translocation of C into nucleus (7,8) C m + P k p C m P k p (9) C m P k r P +C n Table 4.1. A and B undergo a reversible reaction in the cytosol, where the product C in turn can react with the membrane to form C m. C m -molecules diffuse on the membrane and can react with nuclear pores located on the membrane, and are then transported into the nucleus via the pore. The interaction between the molecules and the membrane of the nucleus is not easily modeled at the mesoscopic level, while the reactions in the bulk do not require a particularly fine-grained model. Therefore we consider a partitioning of space where we have a few layers of microscopic voxels around the membrane of the nucleus while the rest of the domain is kept mesoscopic. The partitioning of space and the time evolution of the system are both plotted in Figure

34 14 Average number of molecules B C C m P C m P C n Time (s) Figure 4.3. The partitioned domain is plotted to the left, with red illustrating microscopic subvolumes and blue mesoscopic subvolumes. The green sphere is the nucleus. To the right we have plotted the time evolution of the system. 34