Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1376 Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times LINA MEINECKE ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2016 ISSN ISBN urn:nbn:se:uu:diva

2 Dissertation presented at Uppsala University to be publicly examined in ITC 2446, Lägerhyddsvägen 2, Uppsala, Friday, 10 June 2016 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Reader Ramon Grima (University of Edinburgh). Abstract Meinecke, L Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. Analytical solutions can be derived only for the simplest models and numerical simulations are necessary in most cases to evaluate the models and their properties and to compare them with measured data. This thesis focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution. The second part of this thesis investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment. Keywords: computational systems biology, diffusion, first exit times, unstructured meshes, reaction-diffusion master equation, macromolecular crowding, excluded volume effects, finite element method, backward analysis, stochastic simulation Lina Meinecke, Department of Information Technology, Division of Scientific Computing, Box 337, Uppsala University, SE Uppsala, Sweden. Lina Meinecke 2016 ISSN ISBN urn:nbn:se:uu:diva (

3 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 Meinecke, L., and Lötstedt, P. Stochastic diffusion processes on Cartesian meshes. J. Comput. Appl. Math. 294, 1-11, Lötstedt, P., and Meinecke, L. Simulation of stochastic diffusion via first exit times. J. Comput. Phys. 300, , Meinecke, L., Engblom, S., Hellander, A., and Lötstedt, P. Analysis and design of jump coefficients in discrete stochastic diffusion models. SIAM J. Sci. Comput. 38(1), A55-A83, Meinecke L. Multiscale modeling of diffusion in a crowded environment. preprint arxiv: (Submitted) Meinecke, L., Eriksson, M. Excluded volume effects in on- and off-lattice reaction-diffusion models. preprint arxiv: (Submitted) Reprints were made with permission from the publishers.

4

5 Contents 1 Introduction Reaction-diffusion modeling Macroscopic model Mesoscopic model Well-mixed systems: the chemical master equation The reaction-diffusion master equation The stochastic simulation algorithm Microscopic model First passage kinetic Monte Carlo algorithms Cellular automata Connection between levels Jump coefficients on unstructured meshes FEM discretization The FET approach for mesoscopic jump rates Local FET Global FET Error analysis Backward error Computing γ Minimal backward error Macromolecular crowding Microscopic model CA and crowding Mesoscopic model Multiscale approach Macroscopic model Summary Authors contribution Summary in Swedish Acknowledgments References... 44

6

7 1. Introduction A cell is regarded as the true biological atom. Nothing is living but cells, or what can be directly traced back to cells. George Henry Lewes Cells are the building blocks of all living organisms, they contain the genetic material that defines which species we are. Prokaryotes, such as bacteria, are simple single cell organisms without a nucleus or membrane bound organelles. Mammals and other higher organisms on the other hand consist of trillions of complex eukaryotic cells, which have a nucleus containing the DNA, complicated membranes and specialized organelles such as mitochondria and the Golgi apparatus [1]. Although every cell in our body is genetically identical they are still able to specialize and perform very different tasks. The specialization during embryonic development or "simple" cell division and the effect of aging are still not fully understood and remain open problems in biology and medicine [81]. Imaging techniques have enabled biologists to study the cellular machinery and to understand how the genetic material is encoded on the DNA, how it is copied during cell division, and how it is first transcribed into mrna and then translated into proteins. Gene transcription is regulated by the binding of transcription factors to the DNA that either repress or promote the binding of RNA polymerase to start producing the mrna template for translation. The aim of systems biology is now to understand how gene regulation enables complex cell behavior such as cell division, stem cell differentiation and embryonic development. Mathematical modeling is a crucial tool to understand these complex mechanisms. The first requirement on the mathematical model is agreement with existing experimental data. But more importantly, the model is only useful if it allows us to obtain novel insight into cell behavior. These findings can then be validated by new biological experiments, which further refine the model. Accurate mathematical models are often too complex to analyze analytically and numerical simulations, so called in silico experiments, are needed. Due to the complexity of the reaction networks, we can in general only simulate a small subset of the molecules present inside cells [43], although a first attempt to simulate a complete cell - a bacterium with 525 genes - has been made [75]. The mathematical models in use today do not follow a unified mathematical framework. Instead, they span a wide range of accuracy and computational 7

8 cost. Many models assume the molecules to be well-mixed and equally distributed inside the cell and hence allow for the simplification that only reaction events have to be considered. Yet, cells are spatially organized objects with a cell membrane where extracellular particles bind, so that a signal is released into the cytosol and moves to the nucleus, or cells change their geometry during cell division. Hence space-dependent models are essential to capture many interesting cellular phenomena [39, 120]. Molecules can move e.g. by active transport or by Brownian motion. The latter is the random movement of particles due to their thermal energy and their multiple collisions with the smaller solvent molecules. As a result of this random walk molecules diffuse inside the cell from higher to lower concentrations. Reaction-diffusion processes are very prominent in modeling cellular systems, with the most famous example being the equations for pattern formation presented by Turing [125] in Another important detail of cellular processes are random fluctuations. As mentioned, the molecules diffusive movement is due to their random walk and both intrinsic and extrinsic noise render every reaction event random [119]. Since important molecules inside the cell are often present only at very low copy numbers (there is e.g. only one copy of DNA), the law of large numbers that holds for chemical systems with large molecule counts is not applicable inside cells and stochastic models are more accurate than simpler deterministic models [79, 90, 93]. The molecular fluctuations do not only lead to heterogeneity in the cell behavior, but cells also exploit the noise to stabilize their reaction networks by stochastic focusing [95, 103]. In this thesis we will present three levels of reaction-diffusion modeling: macroscopic models (deterministic), mesoscopic models (stochastic and discrete in space) and microscopic models (stochastic and continuous in space). The research presented in Papers I-V contributes to developing accurate and computationally efficient algorithms for the stochastic simulation of the molecular diffusion on the mesoscopic level. Diffusion is here modeled by a discrete jump process, and in order to accurately represent the complicated geometries inside cells an unstructured discretization of the domain is favorable. On these unstructured meshes, however, the traditional method of computing the jump coefficients might lead to negative and unphysical jump rates. Another shortcoming is that molecules are modeled as point particles, which changes the reaction-diffusion dynamics as compared to more physical models where the molecules occupy volume and interact through hard sphere repulsion. To address these two issues we: Derive the coefficients for the discrete stochastic jump process on an unstructured mesh such that they preserve the speed of diffusion (Papers I and II). Estimate and minimize the error introduced by different numerical techniques to obtain non-negative jump propensities (Paper III). Establish a multiscale model to efficiently simulate reaction-diffusion processes including the excluded volume effects inside cells (Paper IV). 8

9 Compare on- and off-lattice particle based methods for reaction-diffusion simulations in the crowded cell environment (Paper V). The rest of this thesis is organized as follows. In Chapter 2 we review the different reaction-diffusion models existing on the three levels of accuracy, how they resolve space, how they relate to each other, and what software is available for their computation. The mesoscopic level with an unstructured space discretization is presented in more detail in Chapter 3, where we derive a first exit time approach to compute the jump rates such that the speed of diffusion is modeled accurately. We then perform backward analysis to estimate the error in unstructured jump processes and minimize it with a new set of jump coefficients. Chapter 4 is a further step towards more accurate models of cells, where we take molecular crowding effects in the cytosol or on the membrane into account, and present a multiscale model for stochastic reaction-diffusion simulations in such an environment. We summarize this thesis and the contributions of the papers in Chapter 5. The mathematical notation in this thesis is as follows: the time derivative of a quantity c(t) is denoted by c t (t), x R n is a n-dimensional vector with components x i, and capital bold face letters M denote matrices. For vectors x p denotes the vector norm in l p and M p its subordinate matrix norm. 9

10 2. Reaction-diffusion modeling In this chapter we present three levels of reaction-diffusion modeling: the deterministic macroscopic level and the more detailed stochastic mesoscopic and microscopic levels, along with how they relate to each other and when which model is applicable. 2.1 Macroscopic model The law of mass action [57] states that the rate of a chemical reaction is proportional to the masses of the reacting species. Assuming mass action we can derive the reaction rate equations (RREs), a system of deterministic ordinary differential equations (ODEs) describing the time evolution of the concentrations of well-mixed reacting species. Take e.g. the reversible binding with association rate k a and dissociation rate k d A + B k a k d C. (2.1) Let a(t) be the concentration of the A molecules at time t and equivalently for B and C, then by mass-action the change of concentration in time is governed by these RREs a t (t) = k a a(t)b(t)+k d c(t), b t (t) = k a a(t)b(t)+k d c(t), (2.2) c t (t) = k a a(t)b(t) k d c(t). A large set of analytic and computational tools exists to solve and analyze these types of ODE systems. If we assume that the cell is not a well-mixed system and we want to resolve the diffusive motion of the molecules inside the volume Ω, then the concentration of particles becomes a space-dependent quantity c(x, t) and its time evolution is governed by the diffusion equation c t (x,t) = γ 0 Δc(x,t), x Ω, (2.3) with initial distribution c 0 and reflecting or partially absorbing boundary conditions on Ω. Here, γ 0 denotes the diffusion coefficient in a dilute medium. 10

11 These diffusive terms are then added to the system (2.2) for a reaction diffusion model. To solve (2.3) numerically we semi-discretize the equation in space with a discretization matrix D c t (t)=dc (2.4) and then use an appropriate time stepping scheme. Possible space discretization methods are the finite difference method (FDM) for a Cartesian discretization or the finite volume method (FVM) and the finite element method (FEM) for discretizations with unstructured meshes. Describing the biological system by the RREs and diffusion equation is a valid model when the molecules are abundant. In that case the concentrations of molecules become continuous quantities and the system follows its deterministic mean behavior. But, inside living cells individual species are often only present at very low copy numbers, so that the discrete number of molecules becomes the quantity of interest, which is a stochastic variable. As a result, a stochastic model is often needed to capture cell biological processes accurately [90, 103, 130], where the diffusion of single particles is modeled as a random walk and reactions as random processes. In the limit of large molecule numbers these models should in return converge to the deterministic description in this section [83, 84]. We identify two levels of stochastic models, the mesoscopic (discrete in space, continuous in time) and the microscopic (continuous in both space and time). 2.2 Mesoscopic model In the mesoscopic model we describe the reaction-diffusion process by a continuous-time discrete-space Markov process. The state of the system is the random vector containing the discrete number of molecules of each species at time t. We first present the mesoscopic model for the well-mixed case and then proceed to the spatially resolved case Well-mixed systems: the chemical master equation Assume we have M different species that are well-mixed inside the domain Ω, meaning that a randomly chosen molecule has the same probability to be found in any equally sized subvolume [50]. The state vector of the system is then Y(t) N M 0, where Y i(t) represents the number of molecules of species i at time t. Let y(t) be one realization of the stochastic process. The species can undergo R different reactions with stoichiometry vectors n r, each firing of reaction r changes the state space by y w r(y) y n r. (2.5) 11

12 For the reversible bimolecular reaction (2.1) and y(t)=(a(t), B(t),C(t)), the two stoichiometry vectors are n 1 =(1,1, 1) and n 2 =( 1, 1,1) (2.6) for the forward and backward reactions respectively, and w r (y) are the propensity functions, describing the rate at which the association and dissociation reactions fire w 1 (y(t)) = k a A(t)B(t) and w 2 (y(t)) = k d C(t). (2.7) The state of the system at time t is then y(t)=y(0) r 1 (t)n 1 r 2 (t)n 2, (2.8) where r 1 (t) and r 2 (t) are counting processes, counting the number of forward and backward reactions that have happened until t. Let P be a unit rate Poisson process. The counting process is then given by the random time change representation [85] ( t ) r 1 (t)=p k a A(t)B(t)dt, (2.9) 0 and equivalently for r 2 (t). We can then show that the number of reactions fulfills the Markov property [5] p(r 1 (t + Δt) r 1 (t)=1 r 1 (s),s < t)=k a A(t)B(t)Δt + o(δt), (2.10) meaning that the probability for a reaction to happen only depends on the current state and not on the past of the system. The reaction probability is furthermore proportional to the propensity function and the length of the infinitesimal time interval. Consequently, the waiting time until the next reaction r occurs is exponentially distributed with propensity w r (y). Let p(y,t)=p(y,t y 0,t 0 ) be the probability density function (PDF) that the system is in state y at time t given that it started in y 0 at t 0. Using Dynkin s formula we can then show that its time evolution is governed by the forward Kolmogorov equation or chemical master equation (CME) p(y,t) t = R p(y,t)= R r=1 w r (y + n r )p(y + n r,t) w r (y)p(y,t). (2.11) This system of ODEs describes the probability distribution of the stochastic process. It can be solved analytically for monomolecular reactions [72] or simulated stochastically as presented in Section By choosing a fixed time step τ small enough, so that A(t) and B(t) can be considered constant in τ we can approximate the integral in (2.9), and (2.8) becomes 12 y(t + τ)=y(t) P(k a A(t)B(t)τ)n 1 P(k d C(t)τ)n 2. (2.12)

13 This approximate method is called τ-leaping and we will discuss in Section how it can be used to speed up stochastic simulations. Assuming that the propensity w r 1 we can further approximate the Poisson random numbers for the reaction count by a normal distribution N (w r,w r ). This leads to a stochastic differential equation (SDE) describing the reaction system, the so called chemical Langevin equation (CLE). Using van Kampen s system size expansion [126] we can linearize the noise term to obtain the linear noise approximation (LNA), which describes the fluctuations around the mean value with the variance scaling as the system size [56, 62]. Taking the full thermodynamic limit, meaning that the concentrations remain constant, while both volume and number of molecules tend towards infinity, the stochastic terms become negligible and the equations converge to the RREs describing the mean value and equations for the higher order moments exist [30]. A comprehensive summary of these models and the underlying physical assumptions is given in the review [50] The reaction-diffusion master equation In this section we extend the well-mixed mesoscopic model to resolve space. To this end, we first discretize space into N nodes x i with a space discretization parameter h, and introduce dual grid cells (so called voxels) V i with volume V i and centers x i, see Fig The state space is then extended to Y(t) N N M 0, where Y ij denotes the number of molecules of species i located inside voxel V j. Diffusion is modeled as a discrete jump process from a voxel V i to one of the neighboring voxels V j, which can be formulated analogously to a monomolecular reaction λ ij A i A j, (2.13) where A i denotes an A molecule inside voxel V i and λ ij denotes the jump coefficient for a jump from voxel V i to V j. In Chapter 3 we will present how to compute these jump coefficients, especially for unstructured meshes. The propensity function for jumps by molecules of species k is v ij (y ki (t)) = λ ij y ki (t). (2.14) Similar to the CME, we can state the diffusion master equation (DME) p(y,t) t = D p(y,t)= N N i=1 j=1 v ji (y+m ji )p(y+m ji,t) v ij (y)p(y,t), (2.15) where the transition vector m ji is zero except for m ji,i = 1 and m ji, j = 1. We assume that the voxels are small enough, so that the molecules are wellmixed inside and their positions are not resolved any further. Reactions are 13

14 Figure 2.1. The mesoscopic model for reaction-diffusion simulations on an unstructured mesh. The primal mesh (black) gives rise to the dual mesh, which creates the voxels (blue). The state of the system is the discrete number of molecules (red and grey) per voxel. The voxels are assumed to be small enough, so that the molecules are well-mixed inside and can react (green arrow) according to the CME. Diffusion is modeled by a discrete jump process from a voxel to a neighboring voxel (red arrows). then described by local CMEs (2.11) inside each voxel: R p(y,t)= R r=1 w r (y + n r )p(y + n r,t) w r (y)p(y,t), (2.16) with propensity functions w r (y i ) depending on the state y i of voxel V i. These local propensity functions are similiar to those in (2.7), but need to be rescaled according to the volume of the voxel and the type of the reaction: Birth process: /0 A w r (y i )=V i k b, Monomolecular reactions: C A + B w r (y i )=k d C(t), Bimolecular reaction: A + B C w r (y i )= k a A(t)B(t). V i Note that more complicated reactions can be decomposed into subsequent steps of these elementary reactions. Combining the CME for the space-dependent case (2.16) and the diffusion master equation (2.15) leads to the reactiondiffusion master equation (RDME): p(y,t) t = R p(y,t)+d p(y,t). (2.17) The solution of the DME converges towards that of the diffusion equation (2.3) for a decreasing space discretization h 0. A higher number of voxels, however, also means that two molecules are less likely to be located within the same voxel, where they can react. This leads to a decrease in the overall bimolecular reaction rate for finer discretizations, until they are completely neglected [69]. The CME furthermore assumes that that the molecules are dilute 14

15 in each voxel, which no longer holds for infinitesimally small voxels, invalidating the local CMEs. Different methods have been proposed to circumvent this model problem of the RDME. In [70] the convergent RDME (crdme) is presented, where the molecules are allowed to react within a h-independent reaction radius and do not need to be located in the same voxel. Other approaches include rescaling the reaction propensity to either accurately model the mean binding time for two molecules [66, 67], the robustness of the steady state [34] or matching the equilibration time of reversible reactions [38]. If there are only monomolecular reactions happening in the cell the CME (and RDME) can be solved analytically [72], but for bimolecular reactions the propensity functions w r are non-linear and there exists no analytic solution. A numerical solution is very costly and often unfeasible due to the high dimension (N M) of the state space. To overcome this curse of dimensionality, attempts to reduce the dimensions have been made by projecting the state space onto a smaller domain of interest [26, 31, 65, 71, 73, 99]. Instead of solving the master equation, a common method is to compute sample paths from the CME or RDME and then compute statistics by a Monte Carlo approach, as presented in the next section The stochastic simulation algorithm In this section we present how to sample realizations of the stochastic process Y(t). Using Monte Carlo methods we can then compute different moments, such as the mean behavior and variance, and Bayesian approaches allow us to regenerate the PDF. The algorithm to sample individual trajectories is generally known as the stochastic simulation algorithm (SSA), first proposed by Gillespie in the 1970s [47, 48] and therefore also called the Gillespie algorithm. A general guide to stochastic simulations and how to sample random numbers from any distribution by inverse transform sampling is presented in [33]. As mentioned above, the system is Markovian and therefore, the time until reaction r happens is exponentially distributed with parameter w r. In the direct method [48] we first compute the total propensity for any reaction to happen w 0 = r w r and then sample a time for the next reaction. The probability for a reaction of type r is w r /w 0, and the algorithm proceeds as follows. Algorithm 1 Stochastic Simulation Algorithm (SSA) 1: Initialize y at t = 0. 2: Evaluate w r (y) and compute w 0 (y)= r w r (y). 3: Sample the time step τ for the next reaction to occur from the probability density function w 0 e w 0(y)τ. 4: Sample which reaction r happens with probability w r w 0. 5: Update t := t + τ, y = y n r. 6: Go to 2. 15

16 This is an exact method and it is straightforward to extend it to sample the RDME by including the jumps as monomolecular reactions. But it becomes computationally expensive when computing many sample paths, since we have to generate two random numbers in every time step. The next reaction method (NRM) by Gibson and Bruck [45] initially generates the times when each reaction is supposed to fire and then updates the times for those reactions whose propensities were affected by previous reactions. Here, the reactions are ordered in a priority queue, often implemented as a binary heap, and one needs one new random number per reaction event. The next subvolume method (NSM) [27] is the extension to the space-dependent RDME, where reaction and diffusion events in a single subvolume are grouped together and the NRM is then applied to the subvolumes, so that only the random times of the affected subvolumes are resampled and reentered into the event queue. To further gain efficiency there also exist approximate methods. One of them is the τ-leaping method, first introduced in [49]. As mentioned in Section the idea is to keep the propensity functions constant for a sufficiently small fixed time step τ and use a constant rate Poisson process to determine how many reactions occurred. The risk is that τ is chosen too large and the method becomes inaccurate or that too many reactions fire, such that the resulting molecule number is negative, which is unphysical. On the other hand choosing τ too small makes the method inefficient (even more expensive than the SSA), since for a small τ many of the sampled Poisson random numbers will be zero. This issue has been resolved in [16] by changing τ adaptively and switching to the SSA for too small τ. Further improvements can be found e.g. in [2, 17, 123]. The other advantage of the τ-leaping method is that it allows us to introduce a nested grid of different time steps to perform a multilevel Monte Carlo (MLMC) simulation, first introduced by Giles [46] for SDEs in finance and adapted to the SSA in [3, 4]. The idea is to compute the mean value on different time discretization levels which are coupled in such a way that the total variance is reduced and one hence needs fewer sample paths for a given accuracy. This approach has been extended to adaptive time steps and nested levels of error, to efficiently simulate stiff reaction systems [18, 88]. 2.3 Microscopic model On the microscopic level we perform particle-based reaction-diffusion (PBRD) simulations, where we follow individual molecules along their Brownian trajectories and reactions happen with a certain probability when molecules come close to each other. This is the finest modeling level considered in this thesis and is inherently space-dependent. There are two different methods for advancing the molecules in time. First, Brownian dynamics (BD) simulations, where the Brownian trajectory of the 16

17 particles is discretized with a fixed time step Δt and random numbers are drawn at each time step to sample how far the molecules move in each of the Cartesian directions Δx = 2γ 0 Δtξ, (2.18) where ξ is a normally distributed random number N (0,1) and analogously for Δy and Δz, since the Brownian motion is independent along coordinate axes. Molecules that are within a predefined reaction radius σ of each other react with a given probability during the time step Δt. This operator splitting for the reaction and diffusion events introduces an error in the model. Examples of implementations of Brownian dynamics with a fixed time step Δt can be found in the software packages Smoldyn [6, 7] and MCell [77, 118]. In the second method, called Green s function reaction dynamics (GFRD) [128, 127], the many body problem is decomposed into one- and two-body problems. The probability distribution of the particles positions in these smaller subsystems are analytically computable with Green s functions. One can either choose a time step Δt small enough, so that each pair or single molecule has a very low probability to exit its surrounding protective domain and to interact with other pairs/singles before Δt. Alterantively one can sample the time until one of the protective domains is exited in a first passage kinetic Monte Carlo simulation (FPKMC) [24, 102, 120], an exact method implemented in the software ECell [124]. We will present the FPKMC approach in more detail in Section At each time step the positions and eventual reactions are updated, the molecules are regrouped, and after defining new protective domains a new next event time is sampled. This is an event-driven approach with an adaptive time step Δt, and is computationally advantageous for dilute systems with large protective domains, but when the system becomes dense BD simulations start to outperform the GFRD approach. Moreover, there are two models for how reaction events are handled. In the volume reactivity model by Doi [22, 23] molecules are modeled as points and react with each other with a prescribed probability, when they are in a reaction radius of each other. In the contact reactivity or Smoluchowski model [112] the particles are modeled as hard spheres and they react with a certain probability when two spheres collide, represented by a reactive boundary condition at the reaction radius σ for pair propagators in the GFRD approach. The different software packages available for particle based simulations on the microscopic scale are summarized and compared in [114] First passage kinetic Monte Carlo algorithms In this section we present how to compute the exit time distribution of a diffusing molecule from a given domain ω. This domain can either be the protective sphere or cube in the FPKMC algorithm (Fig. 2.2), or it can be used to compute the mesoscopic jump rates as we will present in Chapter 3. 17

18 Δt S(t) (a) t Figure 2.2. (a) Illustration of the FPKMC algorithm with spherical and rectangular protective domains. The FPKMC is an asynchronous event-driven algorithm where a new time step Δt is chosen in each iteration. The blue shaded region illustrates the solution to (2.19) for a given time t. (b) The survival probability S(t) of a diffusing particle inside a protective domain, which is used to sample the next jump time Δt. (b) Let c(x, t) be the probability distribution for a diffusing molecule starting in x 0 at time t = 0tobeinx without leaving ω until time t, then c t (x,t)=γ 0 Δc(x,t), x ω, (2.19) c(x,t)=0, x ω, c(x,0)=δ x0, where the homogeneous Dirichlet boundary condition models the particle s removal from the domain once it reaches the boundary, and δ x0 denotes the Dirac delta function centered at x 0. The survival probability of the particle inside ω until time t is S(t)= c(x, t)dx. (2.20) ω By Gauss formula the probability density p ω (t) that the particle leaves ω at t is given by p ω (t)= S(t) t = γ 0 ω n c(x(s),t)ds, (2.21) with the outward normal n. The flux out of the volume ω is γ 0 c n and the conditional probability that the molecule leaves ω at point x given that the exit occurs at time t is j(x,x 0,t)= γ 0n c(x,x 0,t) p ω (t x 0 ) = n c(x,x 0,t) ω n c(x(s),x 0,t)ds, (2.22) and consequently, the probability for the molecule to exit along the partial boundary ω i is ω j i (x 0,t)= j(x(s),x 0,t)ds = i n c(x(s),x 0,t)ds ω i ω n c(x(s),x 0,t)ds. (2.23) 18

19 In the FPKMC algorithm the protective domains ω are spheres or rectangles (Fig. 2.2) for which (2.19) has an analytic solution and after sampling the next jump time and new positions one redraws the protective domains. In this way the algorithm leaps over all the uninteresting jumps until the time when particles are potentially close together, which is efficient for dilute systems with large protective domains Cellular automata We will now present an approximate particle-based approach that reduces the computational cost compared to BD and GFRD simulations. In the cellular automata (CA) model we still follow individual particles [12], but the trajectories are now restricted to a discrete lattice or grid. Each grid cell can accommodate at most one particle, that means all particles have the same size and shape. At each time step the molecules are randomly moved to a neighboring lattice site. If this site is already occupied and the occupant is a potential reaction partner, a reaction happens with probability p r, otherwise the move is rejected and the molecule stays at its original position. That leads to the following algorithm for the example case of the bimolecular reaction A + B C. Algorithm 2 Cellular Automata 1: Place initial numbers of A, B and C molecules randomly on the grid. 2: while t < T do 3: Choose molecules in random order. 4: for each molecule do 5: Randomly choose a nearest neighbor site as target. 6: if target site is empty then 7: Move molecule. 8: else 9: if molecule is A(B) and target is occupied by B(A) then 10: Generate a random number ξ. 11: if ξ < p r then 12: Replace A and B with a C molecule at target site. 13: else 14: Reject the jump. 15: end if 16: else 17: Reject the jump. 18: end if 19: end if 20: end for 21: Update t := t + Δt. 22: end while 19

20 Since molecules can only move in discrete space, this model is computationally less expensive than the off-lattice simulations in the previous section. It can be regarded as a coarse version of the so called cellular potts model [51], where the grid cells, or "pixels", are combined to form cellular components (e.g. the membrane, cytosol, nucleus or molecules). The pixels can then change their state with a probability proportional to the energy cost of the transformation allowing to model more complex elements and the morphogenesis of whole cells. 2.4 Connection between levels In this section we present how the three models presented above connect to each other. The microscopic model is the most accurate model we consider in this thesis for simulations in computational systems biology. Refining the microscopic level further leads to molecular dynamics simulations which resolve space on a much finer scale, so that simulations are restricted to either few atoms or to short time scales. Attempts to include detailed molecular dynamics information such as interaction potentials into the microscopic model are outlined in [114]. The spatial RDME is a coarse space model where the positions of individual particles are no longer resolved and we only keep track of how many molecules are located inside subdomains of size h at each time step. For a fine space discretization h, the RDME with particularly derived bimolecular reaction rates [34, 38, 67, 70] agrees with certain properties of microscopic Brownian dynamics. In a different limit, it is expected that the RDME converges to the well-mixed CME for infinitely fast diffusion γ 0. It has recently been shown [117] that this holds for the elementary reactions in Section 2.2.2, but not for more complex reactions such Michaelis-Menten dynamics. The discretized diffusion equation (2.4) and the RREs (2.2) are the deterministic approximations in the limit of large molecule numbers of the DME (2.15) and the CME (2.11) respectively [83, 84]. From the macroscopic to the microscopic level the models increase considerably in computational cost and the user needs some prior knowledge of the system to decide which level of accuracy is necessary for its simulation. To use the computational power only in the domains or for the species, where it is needed, hybrid methods have been developed, coupling the macroscopic and mesoscopic levels [14, 15, 40, 63], the mesoscopic and microscopic levels [41, 64, 80] and the macroscopic and microscopic levels [42]. 20

21 3. Jump coefficients on unstructured meshes In this chapter we present how to compute the jump rates in (2.13) for the RDME (2.17). The jump rates λ ij from V i to a neighboring V j have to be non-negative λ ij 0 (3.1) and we denote by λ i = λ ij (3.2) j the total propensity to leave voxel V i. We first compute the expected number of molecules in each voxel from the DME (2.15) for one species and then divide by the volume V i of each voxel and arrive at d dt c V j i = λ ji c j λ i c i, i = 1,...,N (3.3) j=1 V i where c i is the concentration of particles in voxel V i. If we interpret (3.3) as a semi-discretized version (2.4) of the diffusion equation (2.3), the discretization coeffcients D ij of the Laplacian Δ are related to the jump coefficients by λ ij = V j V i D ji and λ i = D ii. (3.4) On a Cartesian grid in d dimensions with grid size h, as used in the software MesoRD [60], a second-order finite difference (FDM) stencil leads to the jump rates λ ij = γ 0 h 2 and λ i = 2dγ 0 h 2. (3.5) Living cells have highly curved membranes. In order to represent these boundaries without using a very fine Cartesian mesh we will use an unstructured mesh (meaning triangular or tetrahedral). In [68] a finite volume (FVM) discretization is used to compute the jump coefficients between the triangles/tetrahedra and we will now present the method introduced in [32] and implemented in the software URDME [25], where the finite element method (FEM) is used to compute the jump rates between the dual voxels, see Fig

22 3.1 FEM discretization The FEM discretization of the diffusion equation (2.3) with linear test and basis functions is Mc t = Sc, (3.6) where M is the mass matrix and S the stiffness matrix. After masslumping in M, the mass matrix is a diagonal matrix with M ii = V i and we multiply by the inverse of the lumped matrix to obtain the discretization matrix D = M 1 S with D ij = γ 0 sin(α + β) (3.7) V i 2sin(α)sin(β) in 2D with α and β being the angles opposing the edge between x i and x j, see Fig A similar formula exists in 3D [131]. If α + β > π, (3.8) meaning we have a bad quality mesh with elongated triangles, the off-diagonal entry D ij is negative, violating (3.1) and the sufficient condition for the discrete maximum principle. The maximum principle bounds the solution of parabolic and elliptic PDEs by its boundary values and in particular guarantees the nonnegativity of solutions to the diffusion equation (2.3). To guarantee physical approximations of (2.3), the discrete solution is supposed to fulfill a discrete version of the maximum principle. It has been shown in [76] that it is impossible to construct a linear discretization of the Laplacian that is consistent and fulfills the maximum principle for any quadrilateral mesh. In the non-negative FEM (nnfem) approach we therefore relax the consistency condition in order to obtain jump coefficients fulfilling (3.1) by setting the negative jump rates to zero, as suggested in [32]. Denote the negative rates by λ ij with the corresponding corrected rate λ ij, and recompute the total propensity to leave voxel V i by j λ ij = λ i. This will lead to λ i < λ i, hence particles have a higher propensity to leave voxel V i and we simulate too fast diffusion [78]. In Papers I and II we address the problem of negative jump coefficients on unstructured meshes by proposing a method to compute them based on the first exit time (FET) introduced for particle based simulations in Section The FET approach for mesoscopic jump rates Local FET We will first use the FETs locally: the jump rate out of a domain ω is the inverse of the first exit time. In Paper II we show that in order to obtain the correct jump rate to leave V i on a Cartesian mesh, ω has to be chosen as the circle with radius h around x i, denoted ω i, see Fig The fact that 22

23 ω k ω ij ω i V k x k V j x j V i x i h x i α β x j (a) (b) Figure 3.1. The voxels V i around node x i and the exit time domain ω i on a Cartesian mesh (a) and an unstructured mesh (b). The red edge beween x i and x j in (b) leads to negative jump coefficients with a FEM discretization, since α + β > π. V i ω i accounts for the extra time the molecules need to become well-mixed inside the neighboring V j. One possibility is to solve (2.19) and compute the expected value of (2.21) to obtain λ i, but the expected local exit time e(x) of a diffusing molecule starting in x also fulfills the Poisson equation [101, 107] which we can solve and compute γ 0 Δe(x)= 1, x ω i (3.9) e(x)=0, x ω i, The jump propensity to a certain neighbor λ ij fulfills λ i = 1 e(x i ). (3.10) λ ij = θ ij λ i, (3.11) where θ ij is the expected splitting probability, which can be computed by taking the expected value in (2.23), or by computing the harmonic measure [101, 107] Δθ ij (x)=0, x ω i (3.12) θ ij (x)=1, x ω ij θ ij (x)=0, x Ω \ ω ij, with ω ij being the quarter segment closest to the neighbor x j, see Fig 3.1. In Paper I we extend this approach to a Cartesian mesh with a rectangular discretization where h x = κh y (3.13) and allow for the particles to jump along the diagonals as a first step towards an unstructured mesh. We compare the jump coefficients resulting from discretizations with FDM, FVM, FEM and FET in 2D. The FDM is here the 23

24 convex combination of two five-point stencils with the combination parameter α, which we can choose such that the FDM is equivalent to any of the other methods. Choosing α such that the FDM agrees with the FET coefficients leads to a high jump propensity and fewer random numbers need to be generated. We then determine the boundary segments ω ij such that the FET approach agrees with FDM. In Paper II we examine the local first exit time approach on a truly unstructured mesh. There is no general definition of the exit domain ω i as for the Cartesian case and as an approximation we choose ω i to be the combination of the neigboring nodes x j. This domain is too small to produce the correct exit time (it corresponds to the smaller orange diamond for a Cartesian mesh (Fig. 3.1)) and hence too fast jump times are sampled which over-represents short jumps, such that the particles accumulate in the smallest voxel. To circumvent this shortcoming we instead use the FET times globally, as presented in the next section Global FET In the global approach (GFET) we instead try to correctly model the time it takes for a diffusing particle to reach the cell membrane, meaning the domain boundary Ω. By conditioning on the first step we can show that the expected time it takes for a molecule to travel from a node x i to the domain boundary is composed of the time it takes to leave a local environment plus the time it takes to reach the outer domain from the new position E(x i )=e(x i )+ θ ij E(x j ). (3.14) j Here, E(x i ) is the global expected time to leave Ω from x i and e(x i ) the local expected time to leave node x i to a neighboring node x j. Reorder (3.14) into j θ ij With (3.10) and (3.11) this becomes e(x i ) E(x j) E(x i) = 1. (3.15) e(x i ) λ ij E(x j ) λ i E(x i )= 1, (3.16) j which is a discretization of (3.9) on Ω just as the DME (2.15) is a discretization of the diffusion equation (2.3). Thus, the discretization matrix D of the Laplacian also fulfills the discrete exit time equation (3.16). We will exploit this fact, by correcting the negative coefficients λ ij in such a way that the new coefficients λ ij still fulfill (3.16) in order to model the time to reach the cell membrane correctly. The algorithm proceeds as follows. 24

25 Algorithm 3 GFET 1: Compute preliminary rates λ ij with a FEM discretization of γ 0 Δ. 2: Solve γ 0 ΔE(x)= 1 to obtain discrete global exit times E(x i ). 3: Find λ ij such that 4: Recompute λ i = j λ ij. min λij j (λ ij λ ij ) 2, j λ ij (E(x j ) E(x i )) = 1, λ ij 0, j. If the original coefficients resulting from a FEM discretization are already non-negative this algorithm preserves them. Otherwise if λ ij < 0wegiveup consistency with the Δ-discretization in favor of non-negative coefficients. The equality constraint in the minimization problem hereby preserves the expected first exit time. To show this, assume that the coefficients λ ij lead to different exit times Ẽ(x i ), then by conditioning on the first step Ẽ(x i )= 1 λ ij λ i + Ẽ(x j ). (3.17) j λ i After rearranging we see that (1,...,1) t = DẼ which has a unique solution (due to the Dirichlet boundary condition), so Ẽ = E. In numerical experiments we further confirm that the GFET approach preserves the global first exit times even for an extremely skewed mesh, see Fig. 3.2, while the nnfem results in too fast and the FVM in too slow diffusion. (1,1) (2,1) E nnfem GFET FVM E nnfem GFET FVM φ h x h y (0,0) h x (1,0) (a) Skewed mesh. (b) ϕ = 3/4π (c) ϕ = π/ Figure 3.2. (a) Illustration of the mesh with skewness parameter ϕ. In the simulations we use a finer mesh with n x = n y = 21. (b) & (c) The expected global exit time for a diffusing molecule starting along the diagonal (red line in (a)) for two ϕ. The reference line E is computed on an unstructured grid with 105 nodes leading to no negative edges. We also examine a simple signaling network where molecules of species A are created in the center (the cell nucleus) and diffuse through the domain (the cytoplasm) with γ 0. Once they reach the boundary (the membrane) they are transformed into molecules of type B which are degraded, see Table

26 /0 k 1 A at x 0 =(1,0.5) A μ B /0 on Ω inω B k 2 Table 3.1. A simple signaling network with parameters: k 1 = 50,k 2 = 1, μ = 200,γ 0 = The steady-state concentration of A depends inversely on the speed of diffusion, while the steady-state concentration of B is unaffected by a change in the diffusion rate. Simulating this system on the mesh in Fig. 3.3 shows that the faster nnfem results in a too low concentration of A at the final time, the slower FVM over-represents A, and the GFET agrees well with the reference solution. The concentration of B depends on γ 0 only by the time it takes to reach steady-state. In 2D, mesh generators are often able to generate meshes fulfilling the requirements for non-negative coefficients, whereas for very simple geometries (spheres and cubes) in 3D in average 17% of the edges have negative jump coefficients in our experience and the experiments in Paper II show the same performance of the methods on realistic three-dimensional meshes A B A B 150 A B Number of Molecules Number of Molecules Number of Molecules t t t (a) nnfem (b) GFET (c) FVM Figure 3.3. Average of 50 simulations of the system in Table 3.1 simulated on the mesh in Fig. 3.2(a) with n x = n y = 21 and ϕ = 3 4π until time T = 15. The dotted lines are the reference solutions calculated on an unstructured grid with 105 nodes and no negative edges. 26

27 3.3 Error analysis Backward error Above we have presented three methods for computing non-negative jump coefficients on unstructured meshes: (i) nnfem, which is no longer a consistent discretization of the Laplacian if (3.8) is fulfilled for any edge; (ii) FVM, which is only consistent with the Laplacian if the mesh is of Voronoi type [35, 96]; (iii) GFET, which loses consistency with the Laplacian as nnfem, but preserves exit times. In this section we summarize Paper III, where we perform backward analysis to quantify the error when diffusion is simulated with these methods. The coefficients resulting from these methods correspond to entries in a perturbed FEM discretization matrix D with only non-negative off-diagonal elements. From this perturbed matrix we go back to a diffusion equation and interpret D as the exact discretization matrix of a perturbed equation, with a spacedependent diffusion constant γ 0 (x). Original equation: c t (x,t)=γ 0 Δc(x,t), (3.18) Perturbed equation: c t (x,t)= ( γ 0 (x) c(x,t)), (3.19) for x Ω, with homogeneous Neumann boundary conditions c n = c n = 0 for x Ω, and initial data c 0 = c 0 at t = 0. We define the backward error as the error between the two diffusion coefficients γ 0 γ 0 2 := 1 γ Ω 0 γ 0 (x) 2 2 dω. (3.20) The analysis in Paper III shows that it bounds the forward error, the difference between the perturbed and unperturbed solutions Ω c(x,t) c(x,t) 2 L 2 Ct γ 0 γ 0 (3.21) for a constant C > 0. The linear growth in t is, however, pessimistic since both solutions start with c 0 and converge towards the same steady state solution, so there is a bounded maximum error. Using the Poincaré-Friedrich inequality we can also prove that the error between the first exit time equations corresponding to (3.18) and (3.19) is bounded by the backward error see Theorem 3.5 in Paper III. E(x) Ẽ(x) 2 L 2 C γ 0 γ 0, (3.22) Computing γ 0 To quantify the error we need to compute the perturbed diffusion coefficient γ 0 (x). The lumped mass matrix is the same for both equations, so D = M 1 S. 27

28 The perturbed coefficient γ 0 (x) then fulfills the FEM discretization formula S ij = ( ψ i, γ 0 (x) ψ j ). (3.23) Since we use a linear finite element discretization with linear basis functions ψ i it is only the mean value of γ 0 (x) on each triangle that contributes to (3.23) and we assume that γ 0 (x) is symmetric and constant on each triangle T k with mean value γ k. In a geometry with n e edges and n t triangles we hence have n e constraints of type (3.23) and 3n t (6n t ) degrees of freedom in 2D (3D), so γ 0 (x) is not uniquely defined. However, (3.21) and (3.22) hold for all γ 0 (x) fulfilling (3.23), so we compute the one minimizing (3.20) to obtain the sharpest bound. Local and global algorithms to compute γ 0 (x) are presented in Sections 4.2 and 4.3 in Paper III Minimal backward error The backward analysis can be extended by relaxing the constraint (3.23) to only require that the resulting coefficient is non-negative 0 ( ψ i, γ 0 (x) ψ j ). (3.24) This allows a larger search space to minimize the backward error (3.20) and we call this approach the minimal backward error (MBE). The local and global algorithms for minimizing (3.20) to compute the coefficients in MBE and γ 0 (x) < e 0.5 < e < e < e < e < e < e = e (a) FVM (b) GFET (c) nnfem (d) MBE ch ch L 2/ ch L FVM GFET nnfem MBE k t t γ 0 γ 0 FVM nnfem GFET MBE (f) (g) (h) Figure 3.4. First row: (a)-(d) show the local backwards error e on each triangle for FVM, GFET, nnfem and MBE. (f) The bad quality mesh with the edges leading to negative jump coefficients indicated in red. (g) The relative discrete forward error. (h) Table with the global backward errors defined in (3.20). 28

29 are presented in Sections 5.1 and 5.2 in Paper III. We perform numerical experiments on a bad quality mesh in 2D, see Fig. 3.4(f), where the red colored edges give rise to negative jump coefficients. The first row in Fig. 3.4 shows the local backward error e = γ k γ 0 2 on each triangle and the table in (h) shows the normalized sum, the global backward error (3.23). In Fig. 3.4(g) we see that the time-dependent bound in (3.21) is too pessimistic and that the backward error correctly ranks the performance of the different methods in the forward error. 29

30 4. Macromolecular crowding In the previous chapters we presented reaction-diffusion models for a dilute medium, which is a good approximation for test tube experiments resulting in the dilute diffusion coefficient γ 0. Living cells, however, are highly crowded environments, meaning that a large fraction of the volume is occupied by macromolecules, while individual species are only present at low concentrations. These molecular obstacles, e.g. proteins, ribosomes, RNA and the cytoskeleton occupy up to 40% [89, 113] of the cytoplasm and affect both the diffusion and reaction rates. The effect is increased on the cell membrane [52], where attaching actin filaments create barriers for the movement of membrane bound proteins [74, 82, 94] and inside the mitochondria where around 60% of the volume is occupied [129]. If the intracellular molecules are modeled as hard spheres the steric repulsions between a diffusing molecule and the numerous "crowders" slow down diffusion a hydrodynamic effect. Novel imaging techniques such as fluorescence-fluctuation analysis [21] have revealed that diffusion can further become anomalous, meaning that the mean square displacement (MSD) of a diffusing molecule no longer behaves linearly but sub-linearly in time. For high crowder densities space decomposes into inhomogeneous subdomains [59] and becomes a lower dimensional fractal on which diffusion behaves fundamentally different [9, 61]. Yet, many diffusive phenomena cannot be explained with steric repulsions alone and more complicated interactions, such as electrostatic forces or transient binding have to be considered for more accurate models [108, 114], where short-range attraction between obstacles and moving particles can even increase diffusion [106]. The thermodynamic effect of the excluded volume on the system is that reaction rates can be both increased and decreased. The impeded diffusion slows down diffusion-limited reactions, but transition state and rate-limited reactions and dimerizations are accelerated [29, 58], since the reaction partners reside in each others vicinity longer, a fact that cells use to increase their efficiency by clustering and colocalization [44, 98]. Since single-molecule imaging to track individual particles on short timescales is still very difficult it is crucial to develop simulation tools that accurately model the intracellular environment, taking crowding effects into account. In the following sections we present how excluded volume effects due to steric repulsions are modeled on the three levels of accuracy. 30

31 4.1 Microscopic model As mentioned in Section 2.3, particles can be modeled as hard spheres with a radius r on the microscopic level and excluded volume effects are then inherent to this model. But the simulations become computationally very expensive [87], due to the many collisions that have to be modeled in a densely occupied domain. In [53, 87, 92, 116] microscopic simulations are performed to validate models for the reaction rates in the crowded environment. The review article [114] summarizes if and how the different off-lattice microscopic approaches presented in Section 2.3 account for excluded volume effects. Another approach is to perform computationally less expensive CA simulations, to investigate effective reaction rates in the crowded environment [10, 20, 54, 113, 121]. We will examine the performance of the CA model in a crowded environment in detail in the next section CA and crowding Generally, the CA model assumes that all molecules have the same size and shape, but in [28] the effect of differently sized and shaped crowding molecules (consisting of a combination of cubes) has been studied. The choice of the grid furthermore influences how strongly the excluded volume effect changes the reaction rates. In [54] it is shown that the CA model overestimates the crowding effect on fractal reaction dynamics. The discrepancy is due to the limited degrees of freedom of motion allowed in the on-lattice models. In Paper V we investigate this effect more closely by comparing BD simulations generated with the software Smoldyn [6, 7] to CA simulations on a Cartesian and on a hexagonal mesh. In a pure diffusion simulation we observe that the models behave differently than was observed in [54] for the reaction rates: for occupied volume fractions φ relevant for intracellular simulations, the BD diffusion is much more obstructed than the lattice-based approaches, see Fig. 4.1(a). A possible explanation is that the grid structure orders the molecules (just like cars on a parking lot), which effectively excludes less space as compared to when the particles can be placed randomly (imagine finding a parking spot if the other cars are parked randomly), as illustrated in Fig This makes the particles more mobile in the CA model than in the BD model. In other words, it is more probable to choose the free direction out of the restricted number of directions on a lattice, than the very small angle of possible movement ϕ out of 2π in BD, see Fig. 4.2(c). The hexagonal mesh here agrees less with the off-lattice simulation, since it allows for more possible jump directions and hence even faster diffusion than the Cartesian discretization. The excluded volume effect on the reaction rates can be modeled by: (i) a time-dependent diffusion constant k(t), following fractal kinetics [10, 54], or modified fractal kinetics [113]; or (ii) static corrections, such as a power-law 31

32 x 2 (t) Off-lattice Cartesian lattice Hexagonal lattice 4γ0t a(t) φ =0.0 :BD φ =0.0 :CA φ =0.1 :BD φ =0.1 :CA φ =0.2 :BD φ =0.2 :CA φ =0.3 :BD φ =0.3 :CA φ =0.4 :BD φ =0.4 :CA t (a) t (b) Figure 4.1. (a) The MSD x 2 (t) for φ = 0.4, simulated in 2D on a Cartesian and a hexagonal lattice and off-lattice with the software Smoldyn. The reference line 4γ 0 t corresponds to diffusion in a dilute medium. (b) Concentration a(t) of A molecules for the bimolecular diffusion-limited reaction A + B C, simulated on a Cartesian lattice (CA) and off-lattice (BD). p = 1 4 ϕ p = ϕ 2π (a) (b) Figure 4.2. The effectively excluded volume (grey) caused by the crowding molecules (red) for the center of mass (dark blue spot) of the moving molecule (blue). (a) CA. (b) BD. (c) The probability p that the possible direction of movement is chosen. (c) approximation of the RRE [111] or a rescaling of the reaction rates according to scaled particle theory (SPT) [52, 53, 58, 108]. In Fig. 4.1(b) we observe that crowding particles impede the diffusion-limited bimolecular reaction A + B C. Moreover, the effect of the artificial grid can both increase and decrease the excluded volume effect as compared to off-lattice simulations, but the effect is small when φ is small. The competing effects leading to this nonlinear behavior are illustrated in Fig

33 φ =0.4 :BD φ =0.4 :CA a(t) A B t B C A Phase I Phase II Figure 4.3. The concentration a(t) of A molecules in a reaction system A + B C at time t for φ = 0.4. Phase I: Initially close pairs of A and B cannot escape from each other easily in off-lattice (BD) simulations, hence they react faster such that a(t) decreases faster with BD than than with CA. Phase II: Distant pairs A and B encounter each other more rarely in BD and hence the BD reaction rate decreases compared to the CA reaction rate. 4.2 Mesoscopic model To account for the volume occupied by molecules each voxel can now only hold a finite number of particles on the mesoscopic level. The excluded volume effects can then be accounted for by rescaling the jump and reaction rates. In [109, 110] immobile crowders are randomly put on the mesh and then redistributed such that voxels are either full or empty. The crowders are mobile in [36, 122] and the jump propensity is reduced by the fraction of occupied spaces in the adjacent node. This crowded jump process can be approximated by non-linear PDEs on the macroscopic level [36] and the results are validated using physical experiments in [37]. The approach is extended to include other interactions than steric repulsions between the molecules in [104] and the overall fraction of occupied volume is used in [86] to rescale the jump propensities. These models only take the fraction φ of occupied volume in the target voxel into account and disregard the size, shape, and microscopic positions of the crowding molecules, as well as the configuration of the surrounding medium. As a result the mean square displacement (MSD) is linear as for diffusion in a dilute medium, but with a reduced diffusion constant x 2 (t) =(1 φ)γ 0 t. (4.1) In [55] the difference of occupancy in the original voxel and the target voxel is instead used in an exponential rescaling of the jump propensity. In the next section we will present the multiscale approach derived in Paper IV for recomputing the jump rates on the mesoscopic level in order to include detailed information from the microscopic level. 33

34 4.2.1 Multiscale approach In Paper II we showed that the local first exit time from a circle ω i with center x i and radius h can be used to compute the mesoscopic jump rate λ i on a Cartesian grid by solving (3.9) and (3.12). We extend this framework to include macromolecular crowding effects in Paper IV, where the crowding molecules are represented as stationary holes with reflecting boundary conditions in ω i, see Fig Since Equation (3.9) models the diffusion of a point particle we have to add the radius of the spherical diffusing particle to the excluded volume, see Fig. 4.4(b). In this way, we account for the microscopic positions and orientations of the crowding molecules, but compute the jump rates on an overlying Cartesian grid that does no longer resolve the numerous crowders. Consequently the up-scaled stochastic simulations are computationally cheaper than approaches that microscopically resolve the obstacles during the jump process. However, we have to numerically solve PDEs of type (3.9) and (3.12) on each perforated ω i in a pre-computing step, similar to multiscale methods for solving porous media flow [13, 91]. It is important to mention that the crowding particles can have any shape, contrary to previous models where they have to be spherical. (a) Crowder Moving molecule Excluded volume Figure 4.4. (a) Solution to (3.9) with excluded volume. (b) The excluded volume consists of the volume occupied by the crowder enlarged by the radius r of the moving molecule. In Fig. 4.5 we see that the jump rate depends severely on the size and shape of the crowding molecules as well as on the size of the diffusing molecule. The movement of larger particles is more obstructed than that of smaller ones and small crowding molecules with more reflective surfaces also have a stronger effect. This agrees with the experimental and theoretical results in [100]. We furthermore observe that the linear scaling with the occupancy φ underestimates crowding for larger molecules and that the exponential scaling with the occupancy proposed in [55] is an appropriate model when the diffusing and crowding particles are similar in size. Using (3.5), (3.10), and the crowded local first exit times e(x i ) we compute a space-dependent scalar diffusion map 34 γ(x) x Vi = (b) h2 2de(x i ), (4.2)