# Spatial Simulations in Systems Biology: From Molecules to Cells

^{*}

## Abstract

**:**

## 1. Introduction

^{−10}to 10

^{−5}m and 10

^{−12}to 10

^{3}s [14].

## 2. The Mesoscale Level

#### 2.1. Diffusion in the Cell

**x**

_{j}) with mass M

_{j}in such a case based on [22]

**x**), at temperature T and subject to friction γ

_{j}with the Boltzmann constant k

_{B}and (zero mean) white noise vector

**R**(t), which represents the force induced by collisions with solvent molecules. According to the fluctuation-dissipation theorem the energy added by

**R**is dissipated by γ

_{j}such that the system reaches and fluctuates around T [22].

_{j}= γ

_{j}M

_{j}and the diffusion coefficient D

_{j}= k

_{B}T/ζ

_{j}. Especially if the particles are connected, e.g., because they represent sub-segments of a cellular filament, great care has to be taken on the definition of the corresponding interaction potential U between them [26,27,31,35].

**ξ**a three-dimensional zero mean Gaussian random variable with unit variance (the difference of the Wiener process R(t + Δt) − R(t) ~ (0,Δt); for convenience the Δt has been included in the square root in Equation (3) such that readily available standard normal random numbers can be used).

^{α}with α < 1 [40,42,61], a fact that can also be observed with Fluorescence Correlation Spectroscopy (FCS) [40] The degree of subdiffusion also is a measure of the crowding level in the cell [40,62]. Such studies with mobile crowding objects require much more computation time than simulations with static objects [63]. In addition, the obstacles hinder the test molecules, while the test molecules influence the mobility of the obstacles. This feedback requires to model molecular crowding as exactly as possible in order to analyze the effective diffusion in the cell, for instance by using an obstacle size and abundance distribution similar to the in vivo conditions [41,60]. Table 1 shows relations between molecular weight and hydrodynamic radius of the molecules as used in such detailed simulations. Thus mesoscale simulations can be used to calculate the mobility of biomolecules on the cellular level, although techniques like MDS [59], dissipative particle dynamics [64] or multiparticle collision dynamics [65] have been employed as well. Connected Brownian particles can also represent polymer chains like in the Brownmove package [34]. Such simulations allow the analysis of filament shapes and diffusion [26,66] and the rheology of biopolymer networks [27,29,35,67]. Simulations with motor proteins furthermore show how network patterns emerge [68].

**x**

_{j}) or by switching the transport mode if molecules bind to a cytoskeleton filament and moving the molecules linearly in their direction [74,75]. Of course also the action and properties of the motors have been studied extensively in detail using MDS [76,77]. Eventually, also the effect of the dynamic cytoskeleton and cell shape has to be included in the simulations [78].

#### 2.2. Reactions in the Cell

_{i}, r

_{j}and diffusion coefficients D

_{i}, D

_{j}):

_{i}= r

_{j}= 2.5 nm (corresponding to a molecular weight of about 25 kDa, cf. Table 1) and diffusion coefficient D

_{i}= D

_{j}= 80 μm

^{2}/s have a diffusion limit for their reaction of γ

_{ij}= 6.05 × 10

^{9}M

^{−1}s

^{−1}. Probably enzymes cannot react that efficiently, and thus this limit is in agreement with observed biomolecular reaction rate constants.

_{ij}denote the macroscopic (observable) reaction rate constant between molecule i and j, i.e., the rate constant which is also used in ODE/PDE models, assuming (locally) well mixed conditions. Obviously, the microscopic situation of two molecules being in contact is fundamentally different. The corresponding microscopic rate constant κ

_{ij}describes the fraction of collisions that lead to reactions. These microscopic rates can be observed e.g., in MDS or BD where the reaction dynamics is solely described by interaction potentials [82–84]. In three dimensions, the macroscopic rate constant related to the microscopic one is given by [85,86]

_{ij}→ 0), the microscopic rate equals the macroscopic one. The possible difference between these rates should however be considered, when comparing MDS and ODE models: while ODE models use the macroscopic descriptions, in MDS the simulation volume is so small that the molecules are basically always in contact, i.e., the microscopic description is more applicable [82].

_{i}and N

_{j}molecules in the cell with volume V and the reaction is described by mass action kinetics. Then the (ODE) reaction rate is given based on the concentrations c

_{m}= N

_{m}/V, where m is species i or j: r

_{ij}= k

_{ij}c

_{i}c

_{j}(the stochastic propensity is a

_{ij}= k

_{ij}N

_{i}N

_{j}/V respectively). This rate/propensity changes due to excluded volume effects, because only a smaller volume is accessible for the molecules. The molecules effectively have a higher concentration, but based on the original volume, the rate constant will appear to be increased (compared to the in vitro case in diluted conditions) [80,88].

_{ij}has different expressions in one, two, and three dimensions [97,98]. Thus it was for instance discussed whether signaling molecules are bound to the plasma membrane to make use of the resulting 2D kinetics [5,99]. Also the membranes can be crowded and structured, and in 2D caging effects occur much faster, leading to complex reaction diffusion behavior [100].

#### 2.3. Reactions in the Simulation: Implementation Issues

_{cat}and K

_{m}. This means that information is lost from the original description with three reactions (K

_{m}= (k

_{r}+ k

_{cat})/k

_{f}). With respect to particle based simulations, where reactions are triggered by collisions of molecules, this means that each colliding pair would have to know the current overall concentration [S] in order to determine the rate applicable to its own reaction, which is unphysical. Therefore the authors suggest to only use plain mass action kinetics in particle based simulations on the molecular level.

_{i}= k

_{i}c

_{i}or propensity a

_{i}= k

_{i}N

_{i}require to execute (on average) a

_{i}events per time unit on the population level. In a discrete time particle based simulation, each particle will react in the time interval (t, t+Δt] with probability [28]

_{1}has to be compared with P

_{i}. The particle will react if ξ

_{1}≤ P

_{i}with ξ

_{1}~ [0, 1] (and of course P

_{i}< 1).

_{i}for all molecules are exponentially distributed (Exp(k

_{i})), and in this context it is worth noting that the minimum time min (t

_{i}) ~ Exp(k

_{i}N

_{i}). Thus this description is compatible with the chemical master equation description on the population level. The trade-off is that all individual waiting times have to be stored and ordered, executed in their sequence, and especially updated if other processes interfere with the assigned reaction channel [104–106].

_{i}= k

_{i}

^{(0)}or propensity a

_{i}= k

_{i}can be implemented similarly. New molecules of the product species I are created (out of nothing) after the waiting time min (t

_{i}) ~ Exp(k

_{i}), where the position of the new molecule is drawn from a given spatial distribution. Alternatively they can be generated by a first order reaction based on a dummy species D with fixed concentration c

_{D}and the scheme D → D + I and rate constant k′ = k

_{i}

^{(0)}

^{/}c

_{D}[75]. Since the number of molecules is changing, a buffer of empty particles is required for the simulation [107].

**x**

_{i}(t) −

**x**

_{j}(t)||, microscopic rate constant κ

_{ij}and diffusion coefficient D

_{ij}= D

_{i}+ D

_{j}(see Figure 2). Note that in this framework the new particle position is not updated according to Equation (3) but according to the probability density distribution (pdf) such that the particles will never overlap. Obviously this description has to be applied only if the particles are within the interaction range. For separated particles, the pdf converges to the normal distribution.

**x**

_{i}(t) and j at

**x**

_{j}(t) and if so also the time t + τ

^{*}and position

**x**

^{*}of it can be deduced [110]. The resulting Green’s function reaction dynamics (GFRD) is in agreement with the corresponding analytical solutions for diffusion limited reactions [111–113].

_{ij}

^{*}(k

_{ij}, Δt) such that molecules will react if their distance is smaller than r

^{*}

_{ij}[28,80,116–118]. Such an algorithm will compute reactions faster than GFRD or the Fokker–Planck method because it does not need complex look-up tables. However, in order to reach the same accuracy, a shorter time step than in GFRD might have to be chosen, which requires more steps in total to complete the simulation [111]. In addition, the computation of a suitable critical radius r

^{*}

_{ij}(k

_{ij},Δt, . . . ) is not straightforward, requiring numerical calculations before the simulation can start and can be implementation dependent [28,39,118].

- set the critical reaction radius to the physical collision radius$${r}_{ij}^{*}={r}_{i}+{r}_{j}$$
- and execute reactions for particles with ||
**x**_{i}(t) −**x**_{j}(t)|| ≤ r^{*}_{ij}with probability$${P}_{ij}^{*}=\frac{{\kappa}_{ij}\mathrm{\Delta}t}{4\pi {({r}_{i}+{r}_{j})}^{3}/3}$$

^{*}

_{ij}< 1, which limits Δt. From tests we found that this approach works reliably up to P

^{*}

_{ij}< 0.2 even for significant degrees of diffusion control. Theoretical limits of this approach or respective correction factors for the reaction probability are calculated in [39].

_{ij}Equation (5) in order to obtain the expected macroscopic rate constant. Note, that bimolecular rate constants (for reactions in 3D volumes) have units volume/time (conversion from M

^{−1}s

^{−1}to μm

^{3}/s using 10

^{15}/6.022 × 10

^{23}Mμm

^{3}). This can be interpreted as reaction volume per molecule and time [116]. The present approach uses the ratio of the microscopic reaction volume to the interaction volume as reaction probability given that a collision has happened, which is a mechanistic analogy to diffusion controlled reaction scheme where the microscopic rate constant describes how efficiently collisions are turned into reactions. Based on this formalism also reactions with other geometries can be defined, for instance binding/adsorption to membranes or cytoskeleton filaments [75]. If the reacting objects are not allowed to overlap, obviously the reaction volume has to be wrapped around the collision radius rather than being distributed within their collision radius. If the thickness of this reaction volume layer compared to the collision radius is negligible, the constraint that the critical reaction radius should match the collision radius in Equation (5) will be satisfied without using a reaction probability [75].

_{c}counts all dissociation events, while the macroscopic rate constant k

_{c}only includes the successful events where the molecules could escape from each other and reach a macroscopically observable distance. From Equation (10) follows that the backward first order reaction rate constant has the same scaling like the forward second order rate constant, which is determined by Equation (6) in 3D: κ

_{c}= k

_{c}× γ

_{ab}/(γ

_{ab}− k

_{ab}).

**x**

_{A}=

**x**

_{B}(=

**x**

_{C}from which they were created from) this delay term is

_{cat}and k

_{r}for the first order reactions and k

_{ES}instead of k

_{f}for the second order E +S → ES reaction [101]. If only k

_{cat}and K

_{m}are given, k

_{ES}can be set to k

_{ES}= k

_{cat}/K

_{m}[75,107].

#### 2.4. Performance and Accuracy

^{2}= 1. These random numbers can be obtained by several algorithms based on uniformly distributed pseudo random numbers [131]. From our observations, a normally distributed random number costs at least 20% more than a uniformly distributed one. However, the repeated application of the uniform distribution will lead to a normal distribution due to the central limit theorem [132], and the convergence sufficient for simulation occurs within 5 steps. The authors also are very faithful that a repeated rejection sampling of steps from the uniform distribution will converge to a reasonable distribution within a crowded intracellular environment, even without using complex reflection calculations [130].

^{2}= 1 uniform random variable the distribution has to have a width of $\sqrt{12}$. Taking into account the scaling factor of Equation (3), the maximum step length $\mathrm{\Delta}{x}_{max,u}=1/2\sqrt{12\times 2D\mathrm{\Delta}{t}_{u}}=\sqrt{6D\mathrm{\Delta}{t}_{u}}$. The maximum step length in space becomes important for instance if only the end point of a step is checked with all obstacles/boundaries. In order to not just jump across boundaries, Δx

_{max}has to be smaller than the smallest object (similar to microscopes, where the wavelength determines the spatial resolution). The normal distribution, in contrast, should not be truncated to less than 3σ, which means that $\mathrm{\Delta}{x}_{max,n}=3\sqrt{2D\mathrm{\Delta}{t}_{n}}=\sqrt{18D\mathrm{\Delta}{t}_{n}}$. If the step length decision is determined by the spatial aspects as indicated above, then a normal distribution requires Δt

_{n}≤ 1/3Δt

_{u}(depending on the truncation). Taking also the higher costs of normally distributed random numbers into account, a simulation with uniformly distributed random numbers could run about 4 times faster along the simulated time.

_{r}can be much bigger than the constraints from the random walk steps Δt

_{n}> Δt

_{u}. In order to save computation time on the expensive pair finding for bimolecular reactions, reactions can thus be executed with a lower frequency, while diffusion steps executed with uniformly distributed random walk steps at higher frequency can converge to the normal distribution and correctly sample the details of the spatial structures in the cell. Likewise this approach allows refilling of reaction volumes between two reaction steps to the local concentration, which is necessary to obtain the correct reaction rate [75].

## 3. Applications and Results of Spatial Simulations

#### 3.1. Current Spatial Simulation Frameworks for the Cellular Level

#### 3.2. The Reaction Diffusion Master Equation and Gillespie’s Algorithm

**k**= (k

_{1}, . . ., k

_{K})

^{T}[20]. The reactions take place in reaction compartment Ω. For a spatially resolved description Ω can be subdivided into U subvolumes of volume V

_{1}, . . ., V

_{U}[144,157,159–161]. Here we denote the number of particles in subvolume ν with

**N**

^{ν}(t) = (N

_{1}

^{ν}(t), . . ., N

_{M}

^{ν}(t))

^{T}. Time evolution on this level is driven by Markovian population dynamics. More specifically, the probability distribution P(

**N**

^{1}(t) =

**n**

^{1}, . . .,

**N**

^{U}(t) =

**n**

^{U}) = p(

**n**

^{1}, . . .,

**n**

^{U}, t) satisfies the RDME

**δ**

_{j}denotes the stoichiometric change vector associated with the j-th reaction. This change is performed by the shift operator E

_{ν}that applies to everything to its right-side. It is defined as

**E**

_{ν}

^{δj}f(

**n**

^{1}, . . .,

**n**

^{U}, t) = f(

**n**

^{1}, . . .,

**n**

^{ν}−

**δ**

_{j}, . . .,

**n**

^{U}, t) for any function f of appropriate domain—in particular f(

**n**

^{1}, . . .,

**n**

^{U}, t) = a

_{j}(

**n**

^{ν})p(

**n**

^{1}, . . .,

**n**

^{U}, t). Diffusion of species i with diffusion coefficient D

_{i}into another volume μ is translated into a first-order transport reactions with effective propensity ki

^{ν}

^{→}

^{μ}n

_{i}

^{ν}. The corresponding rate constant can be expressed as

_{ν,μ}is the surface/interface area of the cubic subvolumes. Thus, the diffusion operator reads

**Δ**

_{i}is the vector of change caused by the transport of a single molecule of type i from one volume to another one. The set (ν) denotes the volumes μ in the neighborhood of volume ν.

_{j}

^{ν}for every reaction j within volume ν is distributed exponentially with parameter a

_{j}(

**n**

^{ν}), called the propensity of reaction j within ν. The waiting time τ

^{ν}for any reaction to occur in ν is exponentially distributed according to parameter ${a}_{0}({\mathit{n}}^{\nu})={\sum}_{j=1}^{K}{a}_{j}({\mathit{n}}^{\nu})$. Starting from a given time t, the next event of reaction j in ν is according to Gillespie’s algorithm [20] at

**N**

^{μ}(t) →

**N**

^{μ}(t′), which according to Gillespie’s algorithm would require updating the precomputed waiting times t

_{−}

^{μ}in μ. Anderson [162] proved that the remaining fraction of the time to the next reaction can simply be stretched according to the changed propensity

_{0}or individually for a

_{j}

^{ν}for each subvolume and reaction channel. Note that the minimum waiting time of all individual waiting times will be distributed as Exp(a

_{0}) such that both descriptions are equivalent, and actually any partitioning/grouping of reaction channels is possible in order to improve the execution performance [155]. Individual reactions have to be executed in their order in time, while the resulting changes in

**N**

^{ν}(t) can require updates in other waiting times as in Equation (15). If the waiting time is calculated based on the cumulative a

_{0}instead, the reaction channel and compartment of the next reaction have to be found based on their individual probabilities a

_{j}

^{ν/}a

_{0}. Simulators based on Gillespie’s algorithm for the RDME are for example MesoRD [143] or STEPS [146] (cf. Table 2). The discretization into the subvolumes should not be too small such that the chance of two reactants being in the same subvolume goes to zero. In that case simulations of the RDME become diffusion limited [163–165].

^{th}species in ν all particles of that species with

**x**(t) ∈ V

^{ν}have to be counted. Conversely, the underlying assumption of the population dynamics model is that the N

_{i}

^{ν}(t) molecules are uniformly distributed in V

^{ν}.

#### 3.3. Rule-based Modeling

^{2}different reaction rate constants. In order to be able to generate and analyze detailed models of signal transduction despite this combinatorial explosion, rule based modeling strategies have been developed [166,167]. The syntax/input scheme of such models minimizes/structures the information that has to be entered by using state dependent rules. Based on these rules the models can also be further abstracted, decomposed or reduced for a comprehensive analysis of the complex system [168,169].

#### 3.4. Applications and Results

- Binding kinetics and binding sites: depending on the description level, protein-protein association can become quite complex [36]. For instance if multiple binding sites and diffusion-controlled reactions are considered. Biomolecules can have several binding sites for the same ligand, for instance receptors forming multimers or antibodies [128]. Kang et al. [173] analysed this and Park et al. [174] developed a theory for reversible reactions under these circumstances. For instance, two binding sites on a molecule would mean that the microscopic reaction rate constant κ
_{ij}is doubled, while the reaction radius is the same as for a molecule with just one binding site. Equation (6) shows that the macroscopic rate constant will not necessarily double under these circumstances. - Scaffolds and Channeling: Both in signaling and metabolic pathways co-localization of related molecules has been observed. Obviously co-localization has advantages because the local high concentration boosts the reaction rate [4,80,106,175,176]. Specific and even nonspecific binding interactions which modify the localization properties of molecules can thus enhance reactions [177]. Note, that the localization requires that molecules do not diffuse around/away, such that there is a trade-off between advantages due to co-localization and disadvantages due to the reduced mobility [75,80].
- Protein DNA interactions: Transcription factors have to find their target site on the DNA amongst millions of binding sites, and they do it surprisingly efficiently, e.g., by combining 1D sliding and 3D diffusion [178]. For instance nonspecific interactions could enhance association rates respectively [177]. Note that even DNA is well organized in space [6]. The spatial organization of DNA strands plays an important role, but long DNA strands can obviously not be modeled with full atomic detail in a MDS simulation such that multi-scale approaches have to be employed [25]. The observed bursting kinetics of transcription rates is likewise explained using open and closed chromatin states, which involve large-scale transitions of the DNA state [179].
- Assembly and fusion processes: Large polymer structures such as the cytoskeleton filaments play an important role for the spatial organization of the cell. Guo et al. modeled the actin assembly using Brownian dynamics [180]. Langevin dynamics have been used to simulate the assembly of virus polymers [181]. A rule-based description was likewise used to analyze the emergence of complex structures [172]. Likewise the fusion of membrane enclosed structures like vesicles is important for the functionality of the cell [64,151,182]. Note that the interplay of cytoskeleton filaments, motor proteins and vesicles can enhance their fusion process [150], while the cytoskeleton structure is for instance organized by the aforementioned growth processes but also motors pulling them together and creating spatial patterns [68].
- Non-uniform molecule distributions in space: In order to grow/move in specific directions cells have to polarize into front and back, which is associated with nonsymmetric particle distributions across the cell [134,183]. In addition receptors on the membrane can cluster together [9,184], and the output of spatial simulations shows the importance of the spatial organization in the cell [9]. Note that again reversible binding and/or unspecific binding interactions influence these reaction rates [177,184].

## 4. Towards Multi-Scale Simulations from Atoms to Cells

## Acknowledgments

## References

- Tomita, M. Whole-cell simulation: A grand challenge of the 21st century. Trends Biotechnol
**2001**, 19, 205–210. [Google Scholar] - Mathews, C.K. The cell-bag of enzymes or network of channels? J. Bacteriol
**1993**, 175, 6377–6381. [Google Scholar] - Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. Molecular Biology of the Cell; Garland Science: New York, NY, USA, 2002. [Google Scholar]
- Ovádi, J.; Srere, P.A. Macromolecular compartmentation and channeling. Int. Rev. Cyt
**1999**, 192, 255–280. [Google Scholar] - Bray, D. Signaling complexes: Biophysical constraints on intracellular communication. Ann. Rev. Biophys. Biomol. Struct
**1998**, 27, 59–75. [Google Scholar] - Lieberman-Aiden, E.; van Berkum, N.L.; Williams, L.; Imakaev, M.; Ragoczy, T.; Telling, A.; Amit, I.; Lajoie, B.R.; Sabo, P.J.; Dorschner, M.O.; et al. Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science
**2009**, 326, 289–293. [Google Scholar] - Kholodenko, B.N. Cell-signalling dynamics in time and space. Nat. Rev. Mol. Cell Biol
**2006**, 7, 165–176. [Google Scholar] - Kholodenko, B.N.; Hancock, J.B.; Kolch, W. Signalling ballet in space and time. Nat. Rev. Mol. Cell Biol
**2010**, 11, 414–426. [Google Scholar] - Costa, M.N.; Radhakrishnan, K.; Wilson, B.S.; Vlachos, D.G.; Edwards, J.S. Coupled stochastic spatial and non-spatial simulations of ErbB1 signaling pathways demonstrate the importance of spatial organization in signal transduction. PLoS One
**2009**, 4. [Google Scholar] [CrossRef] - ScienceVisuals. Available online: http://www.sciencevisuals.com accessed on 08 June 2012.
- De Heras Ciechomski, P.; Mange, R.; Peternier, A. Two-Phased Real-Time Rendering of Large Neuron Databases. Proceedings of the 2008 International Conference on Innovations in Information Technology, Al Ain, United Arab Emirates, 16–18 December 2008; pp. 712–716.
- Ando, T.; Skolnick, J. Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion. Proc. Natl. Acad. Sci. USA
**2010**, 107, 18457–18462. [Google Scholar] - Rafelski, S.M.; Marshall, W.F. Building the cell: Design principles of cellular architecture. Nat. Rev. Mol. Cell Biol
**2008**, 9, 593–602. [Google Scholar] - Bittig, A.T.; Uhrmacher, A.M. Spatial Modeling in Cell Biology at Multiple Levels. Proceedings of the Winter Simulation Conference (WSC), Baltimore MD, USA, 5–8 December 2010; pp. 608–619.
- Takahashi, K.; Arjunan, S.N.V.; Tomita, M. Space in systems biology of signaling pathways— Towards intracellular molecular crowding in silico. FEBS Lett
**2005**, 579, 1783–1788. [Google Scholar] - Tolle, D.P.; le Novere, N. Particle-based stochastic simulation in systems biology. Curr. Bioinf
**2006**, 1, 315–320. [Google Scholar] - Turner, T.E.; Schnell, S.; Burrage, K. Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem
**2004**, 28, 165–178. [Google Scholar] - Ridgway, D.; Broderick, G.; Ellison, M.J. Accommodating space, time and randomness in network simulation. Curr. Opin. Biotechnol
**2006**, 17, 493–498. [Google Scholar] - Burrage, K.; Burrage, P.; Leier, A.; Marquez-Lago, T.; Nicolau, D., Jr. Stochastic simulation for spatial modelling of dynamic process in a living cell. Des. Anal. Biomol. Circuits: Eng. Approaches Syst. Synth. Biol
**2011**, 43–62. [Google Scholar] - Gillespie, D.T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys
**1976**, 22, 403–434. [Google Scholar] - Leach, A.R. Molecular Modelling: Principles and Applications; Pearson Education Ltd: Harlow, England, 2001. [Google Scholar]
- Schlick, T. Molecular Modeling and Simulation: An Interdisciplinary Guide; Springer Verlag: Berlin, Germany, 2010. [Google Scholar]
- Bionumbers. Available online: http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=106198&ver=2 accessed on 8 June 2012.
- Tozzini, V. Coarse-grained models for proteins. Curr. Opin. Struct. Biol
**2005**, 15, 144–150. [Google Scholar] - Villa, E.; Balaeff, A.; Mahadevan, L.; Schulten, K. Multiscale method for simulating protein-DNA complexes. Multiscale Model. Simul
**2004**, 2, 527–553. [Google Scholar] - Chandran, P.L.; Mofrad, M.R.K. Averaged implicit hydrodynamic model of semiflexible filaments. Phys. Rev. E
**2010**, 81, 031920:1–031920:17. [Google Scholar] - Cyron, C.J.; Wall, W.A. Numerical method for the simulation of the Brownian dynamics of rod-like microstructures with three-dimensional nonlinear beam elements. Int. J. Numer. Methods Eng
**2012**, 90, 955–987. [Google Scholar] - Andrews, S.S.; Bray, D. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys. Biol
**2004**, 1, 137–151. [Google Scholar] - Sandersius, S.A.; Newman, T.J. Modeling cell rheology with the Subcellular Element Model. Phys. Biol
**2008**, 5. [Google Scholar] [CrossRef] - Sbalzarini, I.F.; Walther, J.H.; Bergdorf, M.; Hieber, S.E.; Kotsalis, E.M.; Koumoutsakos, P. PPM–A highly efficient parallel particle–mesh library for the simulation of continuum systems. J. Comp. Phys
**2006**, 215, 566–588. [Google Scholar] - Newman, T.J. Grid-free models of multicellular systems, with an application to large-scale vortices accompanying primitive streak formation. Curr. Topics Dev. Biol
**2008**, 81, 157–182. [Google Scholar] - Holcombe, M.; Adra, S.; Bicak, M.; Chin, S.; Coakley, S.; Graham, A.; Green, J.; Greenough, C.; Jackson, D.; Kiran, M.; et al. Modelling complex biological systems using an agent-based approach. Integr. Biol
**2012**, 4, 53–64. [Google Scholar] - Walker, D.C.; Southgate, J. The virtual cell a candidate co-ordinator for middle-outmodelling of biological systems. Brief. Bioinf
**2009**, 10, 450–461. [Google Scholar] - Geyer, T. Many-particle Brownian and Langevin Dynamics Simulations with the Brownmove package. BMC Biophys
**2011**, 4. [Google Scholar] [CrossRef] - Kim, T.; Hwang, W.; Lee, H.; Kamm, R.D. Computational analysis of viscoelastic properties of crosslinked actin networks. PLoS Comput. Biol
**2009**, 5. [Google Scholar] [CrossRef] - Gabdoulline, R.R.; Wade, R.C. Protein-protein association: Investigation of factors influencing association rates by Brownian dynamics simulations. J. Mol. Biol
**2001**, 306, 1139–1155. [Google Scholar] - Sun, J.; Weinstein, H. Toward realistic modeling of dynamic processes in cell signaling: Quantification of macromolecular crowding effects. J. Chem. Phys
**2007**, 127, 155105:1–155105:10. [Google Scholar] - Schmidt, R.R.; Cifre, J.G.H.; de la Torre, J.G. Comparison of Brownian dynamics algorithms with hydrodynamic interaction. J. Chem. Phys
**2011**, 135, 084116:1–084116:10. [Google Scholar] - Erban, R.; Chapman, S.J. Stochastic modelling of reaction–diffusion processes: Algorithms for bimolecular reactions. Phys. Biol
**2009**, 6, 046001. [Google Scholar] - Weiss, M.; Elsner, M.; Kartberg, F.; Nilsson, T. Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J
**2004**, 87, 3518–3524. [Google Scholar] - Ridgway, D.; Broderick, G.; Lopez-Campistrous, A.; Ru’aini, M.; Winter, P.; Hamilton, M.; Boulanger, P.; Kovalenko, A.; Ellison, M.J. Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm. Biophys. J
**2008**, 94, 3748–3759. [Google Scholar] - Klann, M.T.; Lapin, A.; Reuss, M. Stochastic simulation of signal transduction: Impact of the cellular architecture on diffusion. Biophys. J
**2009**, 96, 5122–5129. [Google Scholar] - Trinh, S.; Arce, P.; Locke, B.R. Effective diffusivities of point-like molecules in isotropic porous media by monte carlo simulation. Trans. Porous Media
**2000**, 38, 241–259. [Google Scholar] - Długosz, M.; Trylska, J. Diffusion in crowded biological environments: Applications of Brownian dynamics. BMC Biophys
**2011**, 4. [Google Scholar] [CrossRef] - Chang, R.; Jagannathan, K.; Yethiraj, A. Diffusion of hard sphere fluids in disordered media: A molecular dynamics simulation study. Phys. Rev. E
**2004**, 69. [Google Scholar] [CrossRef] - Ölveczky, B.P.; Verkman, A.S. Monte carlo analysis of obstructed diffusion in three dimensions: Application to molecular diffusion in organelles. Biophys. J
**1998**, 74, 2722–2730. [Google Scholar] - Verkman, A.S. Solute and macromolecule diffusion in cellular aqueous compartments. Trends Biochem. Sci
**2002**, 27, 27–33. [Google Scholar] - Lipkow, K.; Andrews, S.S.; Bray, D. Simulated diffusion of phosphorylated CheY through the cytoplasm of Escherichia coli. J. Bacteriol
**2005**, 187, 45–53. [Google Scholar] - Luby-Phelps, K. Cytoarchitecture and physical properties of cytoplasm: Volume, viscosity, diffusion, intracellular surface area. Int. Rev. Cytol
**2000**, 192, 189–221. [Google Scholar] - Jacobson, K.; Wojcieszyn, J. The translational mobility of substances within the cytoplasmic matrix. Proc. Natl. Acad. Sci. USA
**1984**, 81, 6747–6751. [Google Scholar] - Blum, J.J.; Lawler, G.; Reed, M.; Shin, I. Effect of cytoskeletal geometry on intracellular diffusion. Biophys. J
**1989**, 56, 995–1005. [Google Scholar] - Weissberg, H.L. Effective diffusion coefficient in porous media. J. Appl. Phys
**1963**, 34, 2636– 2639. [Google Scholar] - Whitaker, S. The Method of Volume Averaging; Springer: Berlin, Germany, 1998. [Google Scholar]
- Fan, T.H.; Dhont, J.K.G.; Tuinier, R. Motion of a sphere through a polymer solution. Phys. Rev. E
**2007**, 75. [Google Scholar] [CrossRef] - Ogston, A.G.; Preston, B.N.; Wells, J.D. On the transport of compact particles through solutions of chain-polymers. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci
**1973**, 333, 297–316. [Google Scholar] - Cukier, R.I. Diffusion of Brownian spheres in semidilute polymer solutions. Macromolecules
**1984**, 17, 252–255. [Google Scholar] - Han, J.; Herzfeld, J. Macromolecular diffusion in crowded solutions. Biophys. J
**1993**, 65, 1155–1161. [Google Scholar] - Bruna, M.; Chapman, S.J. Excluded-volume effects in the diffusion of hard spheres. Phys. Rev. E
**2012**, 85. [Google Scholar] [CrossRef] - Sakha, F.; Fazli, H. Three-dimensional Brownian diffusion of rod-like macromolecules in the presence of randomly distributed spherical obstacles: Molecular dynamics simulation. J. Chem. Phys
**2010**, 133, 234904:1–234904:6. [Google Scholar] - Ando, T.; Skolnick, J. Brownian Dynamics Simulation of Macromolecule Diffusion in a Protocell. Proceedings of the International Conference of the Quantum Bio-Informatics IV, Tokyo, Japan, 10–13 March 2010; Georgia Institute of Technology and World Scientific Publishing: Atlanta, GA, USA, 2011; 28, pp. 413–426. [Google Scholar]
- Saxton, M.J. A biological interpretation of transient anomalous subdiffusion. I. Qualitative model. Biophys. J
**2007**, 92, 1178–1191. [Google Scholar] - Hiroi, N.; Lu, J.; Iba, K.; Tabira, A.; Yamashita, S.; Okada, Y.; Flamm, C.; Oka, K.; Köhler, G.; Funahashi, A. Physiological environment induces quick response–slow exhaustion reactions. Frontiers Physiol
**2011**, 2. [Google Scholar] [CrossRef] - Echeveria, C.; Tucci, K.; Kapral, R. Diffusion and reaction in crowded environments. J. Phys. Condens. Matter
**2007**, 19. [Google Scholar] [CrossRef] - Shillcock, J. Insight or illusion? Seeing inside the cell with mesoscopic simulations. HFSP J
**2008**, 2, 1–6. [Google Scholar] - Kapral, R. Multiparticle Collision Dynamics: Simulation of Complex Systems on Mesoscales. In Advances in Chemical Physics; Rice, S.A., Ed.; John Wiley and Sons, Inc: Hoboken, NJ, USA, 2008; Volume 140, pp. 89–146. [Google Scholar]
- Cyron, C.J.; Wall, W.A. Consistent finite-element approach to Brownian polymer dynamics with anisotropic friction. Phys. Rev. E
**2010**, 82, 66705:1–66705:12. [Google Scholar] - Lee, H.; Pelz, B.; Ferrer, J.M.; Kim, T.; Lang, M.J.; Kamm, R.D. Cytoskeletal deformation at high strains and the role of cross-link unfolding or unbinding. Cell. Mol. Bioeng
**2009**, 2, 28–38. [Google Scholar] - Karsenti, E.; Nédélec, F.; Surrey, T. Modelling microtubule patterns. Nat. Cell Biol
**2006**, 8, 1204–1211. [Google Scholar] - Renkin, E.M. Multiple pathways of capillary permeability. Circ. Res
**1977**, 41, 735–743. [Google Scholar] - Taylor, A.E.; Granger, D.N. Exchange of Macromolecules across the Microcirculation. In Handbook of Physiology: The Cardiovascular System: Microcirculation; American Physiological Society: Bethesda, MD, USA, 1984; Volume 4, pp. 467–520. [Google Scholar]
- Zimmerman, S.B.; Trach, S.O. Estimation of macromolecule concentrations and excluded volume effects for the cytoplasm of Escherichia coli. J. Mol. Biol
**1991**, 222, 599–620. [Google Scholar] - Niederalt, C. Bayer Technology Services. PK-Sim/MoBi from Bayer Technology Services. Personal communication, 2011. [Google Scholar]
- Vale, R.D. The molecular motor toolbox for intracellular transport. Cell
**2003**, 112, 467–480. [Google Scholar] - Falk, M.; Klann, M.; Reuss, M.; Ertl, T. Visualization of Signal Transduction Processes in the Crowded Environment of the Cell. Proceedings of IEEE Pacific Visualization Symposium 2009 (PacificVis ’09), Beijing, China, 20–23 April 2009; pp. 169–176.
- Klann, M. Development of a Stochastic Multi-Scale Simulation Method for the Analysis of Spatiotemporal Dynamics in Cellular Transport and Signaling Processes. Ph.D. Dissertation, Universität Stuttgart, Germany, November 2011. [Google Scholar]
- Li, G.; Cui, Q. Mechanochemical coupling in myosin: A theoretical analysis with molecular dynamics and combined QM/MM reaction path calculations. J. Phys. Chem. B
**2004**, 108, 3342–3357. [Google Scholar] - Kawakubo, T.; Okada, O.; Minami, T. Molecular dynamics simulations of evolved collective motions of atoms in the myosin motor domain upon perturbation of the ATPase pocket. Biophys. Chem
**2005**, 115, 77–85. [Google Scholar] - Otten, M.; Nandi, A.; Arcizet, D.; Gorelashvili, M.; Lindner, B.; Heinrich, D. Local motion analysis reveals impact of the dynamic cytoskeleton on intracellular subdiffusion. Biophys. J
**2012**, 102, 758–767. [Google Scholar] - Gershon, N.D.; Porter, K.R.; Trus, B.L. The cytoplasmic matrix: Its volume and surface area and the diffusion of molecules through it. Proc. Natl. Acad. Sci. USA
**1985**, 82, 5030–5034. [Google Scholar] - Klann, M.T.; Lapin, A.; Reuss, M. Agent-based simulation of reactions in the crowded and structured intracellular environment: Influence of mobility and location of the reactants. BMC Syst. Biol
**2011**, 5. [Google Scholar] [CrossRef] - Smoluchowski, M. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem
**1917**, 92, 129–168. [Google Scholar] - Zhang, Y.; McCammon, J.A. Studying the affinity and kinetics of molecular association with molecular-dynamics simulation. J. Chem. Phys
**2003**, 118, 1821:1–1821:7. [Google Scholar] - Gabdoulline, R.R.; Wade, R.C. Biomolecular diffusional association. Curr. Opin. Struct. Biol
**2002**, 12, 204–213. [Google Scholar] - Northrup, S.H.; Erickson, H.P. Kinetics of protein-protein association explained by Brownian dynamics computer simulation. Proc. Natl. Acad. Sci. USA
**1992**, 89, 3338–3342. [Google Scholar] - Rice, S.A. Diffusion-Limited Reactions; Elsevier: Amsterdam, The Netherlands, 1985. [Google Scholar]
- Collins, F.C.; Kimball, G.E. Diffusion-controlled reaction rates. J. Colloid. Sci
**1949**, 4, 425– 437. [Google Scholar] - Ellis, R.J. Macromolecular crowding: An important but neglected aspect of the intracellular environment. Curr. Opin. Struct. Biol
**2001**, 11, 114–119. [Google Scholar] - Minton, A.P. The influence of macromolecular crowding and macromolecular confinement on biochemical reactions in physiological media. J. Biol. Chem
**2001**, 276, 10577–10580. [Google Scholar] - Al-Habori, M. Microcompartmentation, metabolic channelling and carbohydrate metabolism. Int. J. Biochem. Cell Biol
**1995**, 27, 123–132. [Google Scholar] - Schnell, S.; Turner, T.E. Reaction kinetics in intracellular environments with macromolecular crowding: Simulations and rate laws. Prog. Biophys. Mol. Biol
**2004**, 85, 235–260. [Google Scholar] - Grima, R.; Schnell, S. A systematic investigation of the rate laws valid in intracellular environments. Biophys. Chem
**2006**, 124, 1–10. [Google Scholar] - Nicolau, D.V., Jr; Burrage, K. Stochastic simulation of chemical reactions in spatially complex media. Comput. Mathe. Appl
**2008**, 55, 1007–1018. [Google Scholar] - Means, S.; Smith, A.J.; Shepherd, J.; Shadid, J.; Fowler, J.; Wojcikiewicz, R.J.H.; Mazel, T.; Smith, G.D.; Wilson, B.S. Reaction diffusion modeling of calcium dynamics with realistic ER geometry. Biophys. J
**2006**, 91, 537–557. [Google Scholar] - Bergdorf, M.; Sbalzarini, I.F.; Koumoutsakos, P. A Lagrangian particle method for reaction–diffusion systems on deforming surfaces. J. Math. Biol
**2010**, 61, 649–663. [Google Scholar] - Loverdo, C.; Benichou, O.; Moreau, M.; Voituriez, R. Enhanced reaction kinetics in biological cells. Nat. Phys
**2008**, 4, 134–137. [Google Scholar] - Chaudhuri, A.; Bhattacharya, B.; Gowrishankar, K.; Mayor, S.; Rao, M. Spatiotemporal regulation of chemical reactions by active cytoskeletal remodeling. Proc. Natl. Acad. Sci. USA
**2011**, 108, 14825–14830. [Google Scholar] - Hardt, S.L. Rates of diffusion controlled reactions in one, two and three dimensions. Biophys. Chem
**1979**, 10, 239–243. [Google Scholar] - Torney, D.C.; McConnel, H.M. Diffusion-Limited reaction rate theory for two-dimensional systems. Proc. R. Soc. Lond. A
**1983**, 387, 147–170. [Google Scholar] - Kholodenko, B.N.; Hoek, J.B.; Westerhoff, H.V. Why cytoplasmic signalling proteins should be recruited to cell membranes. Trends Cell Biol
**2000**, 10, 173–178. [Google Scholar] - Lampoudi, S.; Gillespie, D.T.; Petzold, L.R. Effect of excluded volume on 2D discrete stochastic chemical kinetics. J. Comp. Phys
**2009**, 228, 3656–3668. [Google Scholar] - Bisswanger, H. Enzyme Kinetics; Wiley VCH: Chichester, England, 2002. [Google Scholar]
- Rohwer, J.; Hanekom, A.; Hofmeyr, J.H. A Universal Rate Equation for Systems Biology. Proceedings of the 2nd International Symposium on Experimental Standard Conditions of Enzyme Characterizations (ESEC 2006), Rüdesheim, Germany, 19–23 March 2006; pp. 175–187.
- Segel, L.A.; Slemrod, M. The quasi-steady-state assumption: A case study in perturbation. SIAM Rev
**1989**, 31, 446–477. [Google Scholar] - Byrne, M.J.; Waxham, M.N.; Kubota, Y. Cellular dynamic simulator: An event driven molecular simulation environment for cellular physiology. Neuroinformatics
**2010**, 8, 63–82. [Google Scholar] - Arjunan, S.N.V.; Tomita, M. A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation. M. Syst. Synth. Biol
**2010**, 4, 35–53. [Google Scholar] - Klann, M.; Ganguly, A.; Koeppl, H. Improved Reaction Scheme for Spatial Stochastic Simulations with Single Molecule Detail. Proceedings of the 8th International Workshop on Computational Systems Biology (WCSB 2011), Zurich, Switzerland, 6–8 June 2011.
- Falk, M.; Klann, M.; Ott, M.; Ertl, T.; Koeppl, H. Parallelized Agent-Based Simulation on CPU and Graphics Hardware for Spatial and Stochastic Models in Biology. Proceedings of the 9th International Conference on Computational Methods in Systems Biology, Paris, France, 21–23 September 2011; pp. 73–82.
- Clifford, P.; Green, N.J.B. On the simulation of the Smoluchowski boundary condition and the interpolation of brownian paths. Mol. Phys
**1986**, 57, 123–128. [Google Scholar] - Lapin, A.; Klann, M.; Reuss, M. Stochastic Simulations of 4D Spatial Temporal Dynamics of Signal Transduction Processes. Proceedings of the FOSBE, Stuttgart, Germany, 9–12 September 2007; pp. 421–425.
- Van Zon, J.S.; ten Wolde, P.R. Simulating biochemical networks at the particle level and in time and space: Green’s function reaction dynamics. Phys. Rev Lett
**2005**, 94. [Google Scholar] [CrossRef] - Morelli, M.J.; ten Wolde, P.R. Reaction Brownian dynamics and the effect of spatial fluctuations on the gain of a push-pull network. J. Chem. Phys
**2008**, 129. [Google Scholar] [CrossRef] - Kim, H.; Shin, K.J. Exact solution of the reversible diffusion-influenced reaction for an isolated pair in three dimensions. Phys. Rev. Lett
**1999**, 82, 1578–1581. [Google Scholar] - Gopich, I.V.; Szabo, A. Kinetics of reversible diffusion influenced reactions: The self-consistent relaxation time approximation. J. Chem. Phys
**2002**, 117, 507:1–507:11. [Google Scholar] - Lapin, A.; Klann, M.; Reuss, M. Multi-Scale Spatio-Temporal Modeling: Lifelines of Microorganisms in Bioreactors and Tracking Molecules in Cells. In Biosystems Engineering II; Springer-Verlag: Berlin, Germany, 2010; Volume 121, pp. 23–43. [Google Scholar]
- Park, S.; Agmon, N. Theory and simulation of diffusion-controlled michaelis-menten kinetics for a static enzyme in solution. J. Phys. Chem. B
**2008**, 112, 5977–5987. [Google Scholar] - Pogson, M.; Smallwood, R.; Qwarnstrom, E.; Holcombe, M. Formal agent-based modelling of intracellular chemical interactions. Biosystems
**2006**, 85, 37–45. [Google Scholar] - Pogson, M.; Holcombe, M.; Smallwood, R.; Qwarnstrom, E. Introducing spatial information into predictive NF-κB modelling—An agent-based approach. PLoS One
**2008**, 3. [Google Scholar] [CrossRef] - Lipková, J.; Zygalakis, K.C.; Chapman, S.J.; Erban, R. Analysis of Brownian dynamics simulations of reversible bimolecular reactions. SIAM J. Appl. Math
**2011**, 71, 714–730. [Google Scholar] - Andrews, S.S.; Addy, N.J.; Brent, R.; Arkin, A.P. Detailed simulations of cell biology with Smoldyn 2.1. PLoS Comp. Biol
**2010**, 6. [Google Scholar] [CrossRef] - Andrews, S.S. Serial rebinding of ligands to clustered receptors as exemplified by bacterial chemotaxis. Phys. Biol
**2005**, 2, 111–122. [Google Scholar] - Berg, O.G. On diffusion-controlled dissociation. Chem. Phys
**1978**, 31, 47–57. [Google Scholar] - Klann, M.; Koeppl, H. Escape times and geminate recombinations in spatial simulations of chemical reactions. Biophys. J 2012.
- Wade, R.C.; Luty, B.A.; Demchuk, E.; Madura, J.D.; Davis, M.E.; Briggs, J.M.; McCammon, J.A. Simulation of enzyme–substrate encounter with gated active sites. Nat. Struct. Mol. Biol
**1994**, 1, 65–69. [Google Scholar] - Shoup, D.; Lipari, G.; Szabo, A. Diffusion-controlled bimolecular reaction rates. The effect of rotational diffusion and orientation constraints. Biophys. J
**1981**, 36, 697–714. [Google Scholar] - Dudko, O.K.; Berezhkovskii, A.M.; Weiss, G.H. Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface. J. Chem. Phys
**2004**, 121, 1562–1565. [Google Scholar] - Traytak, S.D. Diffusion-controlled reaction rate to an active site. Chem. Phys
**1995**, 192, 1–7. [Google Scholar] - Wu, Y.T.; Nitsche, J.M. On diffusion-limited site-specific association processes for spherical and nonspherical molecules. Chem. Eng. Sci
**1995**, 50, 1467–1487. [Google Scholar] - Bongini, L.; Fanelli, D.; Piazza, F.; de los Rios, P.; Sanner, M.; Skoglund, U. A dynamical study of antibody–antigen encounter reactions. Phys. Biol
**2007**, 4, 172–180. [Google Scholar] - Pettré, J.; Ciechomski, P.H.; Maïm, J.; Yersin, B.; Laumond, J.P.; Thalmann, D. Real-time navigating crowds: Scalable simulation and rendering. Comput. Animat. Virtual Worlds
**2006**, 17, 445–455. [Google Scholar] - Behringer, H.; Eichhorn, R. Hard-wall interactions in soft matter systems: Exact numerical treatment. Phys. Rev. E
**2011**, 83, 065701:1–065701:4. [Google Scholar] - Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; Vetterling, W.T. Numerical Recipes; Cambridge University Press: Cambridge, UK, 1986; Volume 547. [Google Scholar]
- Trotter, H.F. An elementary proof of the central limit theorem. Arch. Math
**1959**, 10, 226–234. [Google Scholar] - Dematte, L. Smoldyn on graphics processing units: Massively parallel brownian dynamics simulation. IEEE/ACM Trans. Comput. Biol. Bioinf
**2012**, 9, 655–667. [Google Scholar] - Jilkine, A.; Angenent, S.B.; Wu, L.F.; Altschuler, S.J. A density-dependent switch drives stochastic clustering and polarization of signaling molecules. PLoS Comp. Biol
**2011**, 7. [Google Scholar] [CrossRef] - Plimpton, S.; Slepoy, A. ChemCell: A Particle-Based Model of Protein Chemistry and Diffusion in Microbial Cells; Sandia National Laboratories Technical Report 2003-4509; Sandia National Laboratories: Albuquerque, USA, 2003. [Google Scholar]
- Plimpton, S.J.; Slepoy, A. Microbial cell modeling via reacting diffusive particles. J. Phys. Conf. Ser
**2005**, 16. [Google Scholar] [CrossRef] - Takahashi, K.; Ishikawa, N.; Sadamoto, Y.; Sasamoto, H.; Ohta, S.; Shiozawa, A.; Miyoshi, F.; Naito, Y.; Nakayama, Y.; Tomita, M. E-Cell 2: Multi-platform E-Cell simulation system. Bioinformatics
**2003**, 19, 1727–1729. [Google Scholar] - Tomita, M.; Hashimoto, K.; Takahashi, K.; Shimizu, T.S.; Matsuzaki, Y.; Miyoshi, F.; Saito, K.; Tanida, S.; Yugi, K.; Venter, J.C.; et al. E-CELL: Software environment for whole-cell simulation. Bioinformatics
**1999**, 15, 72–84. [Google Scholar] - Takahashi, K.; Kaizu, K.; Hu, B.; Tomita, M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics
**2004**, 20, 538–546. [Google Scholar] - Takahashi, K.; Tănase-Nicola, S.; ten Wolde, P.R. Spatio-temporal correlations can drastically change the response of a MAPK pathway. Proc. Natl. Acad. Sci. USA
**2010**, 107, 2473–2478. [Google Scholar] - Van Zon, J.S.; Morelli, M.J.; Tanase-Nicola, S.; tenWolde, P.R. Diffusion of transcription factors can drastically enhance the noise in gene expression. Biophys. J
**2006**, 91, 4350–4367. [Google Scholar] - Stiles, J.R.; Bart, T.M. Monte Carlo Methods for Simulating Realistic Synaptic Microphysiology Using MCell. In Computational Neuroscience—Realistic Modeling for Experimentalists; de Schutter, E., Ed.; CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Hattne, J.; Fange, D.; Elf, J. Stochastic reaction-diffusion simulation with MesoRD. Bioinformatics
**2005**, 21, 2923–2924. [Google Scholar] - Elf, J.; Doncic, A.; Ehrenberg, M. Mesoscopic reaction-diffusion in intracellular signaling. Proc. SPIE
**2003**, 5110, 114–124. [Google Scholar] - Ander, M.; Beltrao, P.; di Ventura, B.; Ferkinghoff-Borg, J.; Foglierini, M.; Kaplan, A.; Lemerle, C.; Tomas-Oliveira, I.; Serrano, L. SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: Analysis of simple networks. Syst. Biol
**2004**, 1, 129–138. [Google Scholar] - Wils, S.; de Schutter, E. STEPS: Modeling and simulating complex reaction-diffusion systems with Python. Frontiers Neuroinf
**2009**, 3. [Google Scholar] [CrossRef] - Stoma, S.; Fröhlich, M.; Gerber, S.; Klipp, E. STSE: Spatio-temporal simulation environment dedicated to biology. BMC Bioinf
**2011**, 12. [Google Scholar] [CrossRef] - Moraru, I.I.; Schaff, J.C.; Slepchenko, B.M.; Loew, L.M. The virtual cell. Ann. N. Y. Acad. Sci
**2002**, 971, 595–596. [Google Scholar] - Slepchenko, B.M.; Schaff, J.C.; Macara, I.; Loew, L.M. Quantitative cell biology with the virtual cell. Trends Cell Biol
**2003**, 13, 570–576. [Google Scholar] - Klann, M.; Koeppl, H.; Reuss, M. Spatial modeling of vesicle transport and the cytoskeleton: The challenge of hitting the right road. PLoS One
**2012**, 7. [Google Scholar] [CrossRef] - Shillcock, J.C.; Lipowsky, R. The computational route from bilayer membranes to vesicle fusion. J. Phys. Condens. Matt
**2006**, 18. [Google Scholar] [CrossRef] - Liou, S.H.; Chen, C.C. Cellular ability to sense spatial gradients in the presence of multiple competitive ligands. Phys. Rev. E
**2012**, 85, 011904:1–011904:5. [Google Scholar] - Angermann, B.R.; Klauschen, F.; Garcia, A.D.; Prustel, T.; Zhang, F.; Germain, R.N.; Meier-Schellersheim, M. Computational modeling of cellular signaling processes embedded into dynamic spatial contexts. Nat. Methods
**2012**, 9, 283–289. [Google Scholar] - Jeschke, M.; Uhrmacher, A.M. Multi-Resolution Spatial Simulation for Molecular Crowding. Proceedings of the 2008 Winter Simulation Conference, Miami, FL, USA, 7–10 December 2008; pp. 1384–1392.
- Pahle, J. Biochemical simulations: Stochastic, approximate stochastic and hybrid approaches. Brief. Bioinf
**2009**, 10, 53–64. [Google Scholar] - Chatterjee, A.; Vlachos, D.G. Multiscale spatial monte carlo simulations: Multigriding, computational singular perturbation, and hierarchical stochastic closures. J. Chem. Phys
**2006**, 124, 064110. [Google Scholar] - Jeschke, M.; Ewald, R.; Park, A.; Fujimoto, R.; Uhrmacher, A.M. A parallel and distributed discrete event approach for spatial cell-biological simulations. ACM SIGMETRICS Perform. Eval. Rev
**2008**, 35, 22–31. [Google Scholar] - Xing, F.; Yao, Y.P.; Jiang, Z.W.; Wang, B. Fine-grained parallel and distributed spatial stochastic simulation of biological reactions. Adv. Mater. Res
**2012**, 345, 104–112. [Google Scholar] - Rodríguez, J.V.; Kaandorp, J.A.; Dobrzyński, M.; Blom, J.G. Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli. Bioinformatics
**2006**, 22, 1895–1901. [Google Scholar] - Lampoudi, S.; Gillespie, D.T.; Petzold, L.R. The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems. J. Chem. Phys
**2009**, 130, 094104. [Google Scholar] - Stundzia, A.B.; Lumsden, C.J. Stochastic simulation of coupled reaction-diffusion processes. J. Comp. Phys
**1996**, 127, 196–207. [Google Scholar] - Anderson, D.F. A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys
**2007**, 127, 214107:1–214107:10. [Google Scholar] - Gillespie, D.T. A diffusional bimolecular propensity function. J. Chem. Phys
**2009**, 131, 164109:1–164109:13. [Google Scholar] - Fange, D.; Berg, O.G.; Sjöberg, P.; Elf, J. Stochastic reaction-diffusion kinetics in the microscopic limit. Proc. Natl. Acad. Sci. USA
**2010**, 107, 19820–19825. [Google Scholar] - Gillespie, D.T. Deterministic limit of stochastic chemical kinetics. J. Phys. Chem. B
**2009**, 113, 1640–1644. [Google Scholar] - Danos, V.; Feret, J.; Fontana, W.; Harmer, R.; Krivine, J. Rule-based Modelling of Cellular Signalling. Proceedings of the Eighteenth International Conference on Concurrency Theory, CONCUR’07, Lisbon, Portugal, 3–8 September 2007, Caires, L., Vasconcelos, V.T., Eds.; Springer: Berlin, Germany, 2007; Volume 4703, pp. 17–41. [Google Scholar]
- Faeder, J.R.; Blinov, M.L.; Hlavacek, W.S. Rule-based modeling of biochemical systems with BioNetGen. Methods Mol. Biol
**2009**, 500, 113–167. [Google Scholar] - Camporesi, F.; Feret, J.; Koeppl, H.; Petrov, T. Automatic Reduction of Stochastic Rules-Based Models in a Nutshell. AIP Conference Proceedings. International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2010), Rhodes, Greece, 19–25 September 2010; 1281, pp. 1330–1334.
- Petrov, T.; Ganguly, A.; Koeppl, H. Model decomposition and stochastic fragments. Theor. Comput. Sci
**2012**, (in press). [Google Scholar] - Tolle, D.P.; le Novère, N. Meredys, a multi-compartment reaction-diffusion simulator using multistate realistic molecular complexes. BMC Syst. Biol
**2010**, 4. [Google Scholar] [CrossRef] - Yang, J.; Meng, X.; Hlavacek, W.S. Rule-based modelling and simulation of biochemical systems with molecular finite automata. Syst. Biol. IET
**2010**, 4, 453–466. [Google Scholar] - Gruenert, G.; Ibrahim, B.; Lenser, T.; Lohel, M.; Hinze, T.; Dittrich, P. Rule-based spatial modeling with diffusing, geometrically constrained molecules. BMC Bioinf
**2010**, 11. [Google Scholar] [CrossRef] - Kang, A.; Kim, J.H.; Lee, S.; Park, H. Diffusion-influenced reactions involving a reactant with two active sites. J. Chem. Phys
**2009**, 130, 094507:1–094507:8. [Google Scholar] - Park, S.; Agmon, N. Multisite reversible geminate reaction. J. Chem. Phys
**2009**, 130, 074507:1–074507:11. [Google Scholar] - Bauler, P.; Huber, G.; Leyh, T.; McCammon, J.A. Channeling by proximity: The catalytic advantages of active site colocalization using Brownian dynamics. J. Phys. Chem. Lett
**2010**, 1, 1332–1335. [Google Scholar] - Locasale, J.W.; Shaw, A.S.; Chakraborty, A.K. Scaffold proteins confer diverse regulatory properties to protein kinase cascades. Proc. Natl. Acad. Sci. USA
**2007**, 104. [Google Scholar] [CrossRef] - Zhou, H.X.; Szabo, A. Enhancement of association rates by nonspecific binding to DNA and cell membranes. Phys. Rev. Lett.
**2004**, 93, 178101:1–178101:4. [Google Scholar] - Halford, S.E. An end to 40 years of mistakes in DNA-protein association kinetics? Biochem. Soc. Trans
**2009**, 37, 343–348. [Google Scholar] - Zechner, C.; Ruess, J.; Krenn, P.; Pelet, S.; Peter, M.; Lygeros, J.; Koeppl, H. Moment-based inference predicts bimodality in transient gene expression. Proc. Natl. Acad. Sci. USA
**2012**, (in press). [Google Scholar] - Guo, K.; Shillcock, J.; Lipowsky, R. Treadmilling of actin filaments via Brownian dynamics simulations. J. Chem. Phys
**2010**, 133. [Google Scholar] [CrossRef] - Mahalik, J.P.; Muthukumar, M. Langevin dynamics simulation of polymer-assisted virus-like assembly. J. Chem. Phys
**2012**, 136, 135101:1–135101:13. [Google Scholar] - Noguchi, H.; Takasu, M. Fusion pathways of vesicles: A Brownian dynamics simulation. J. Chem. Phys
**2001**, 115, 9547–9551. [Google Scholar] - Mogilner, A.; Allard, J.; Wollman, R. Cell polarity: Quantitative modeling as a tool in cell biology. Science
**2012**, 336, 175–179. [Google Scholar] - Mugler, A.; Bailey, A.G.; Takahashi, K.; Wolde, P.R. Membrane clustering and the role of rebinding in biochemical signaling. Biophys. J
**2012**, 102, 1069–1078. [Google Scholar] - Cichocki, B.; Ekiel-Jezewska, M.L.; Wajnryb, E. Communication: Translational Brownian motion for particles of arbitrary shape. J. Chem. Phys
**2012**, 136, 071102:1–071102:4. [Google Scholar] - Lee, D.; Redfern, O.; Orengo, C. Predicting protein function from sequence and structure. Nat. Rev. Mol. Cell Biol
**2007**, 8, 995–1005. [Google Scholar] - Zhou, H.; Skolnick, J. Protein structure prediction by pro-Sp3-TASSER. Biophys. J
**2009**, 96, 2119–2127. [Google Scholar] - Accelrys Discovery Studio. Available online: http://accelrys.com/products/discovery-studio/index.html accessed on 8 June 2012.
- Molsoft. Available online: http://www.molsoft.com accessed on 8 June 2012.
- García de la Torre, J.; Huertas, M.L.; Carrasco, B. Calculation of hydrodynamic properties of globular proteins from their atomic-level structure. Biophys. J
**2000**, 78, 719–730. [Google Scholar] - Moal, I.H.; Bates, P.A. Kinetic rate constant prediction supports the conformational selection mechanism of protein binding. PLoS Comp. Biol
**2012**, 8. [Google Scholar] [CrossRef] - De Jong, D.H.; Schäfer, L.V.; de Vries, A.H.; Marrink, S.J.; Berendsen, H.J.C.; Grubmüller, H. Determining equilibrium constants for dimerization reactions from molecular dynamics simulations. J. Comp. Chem
**2011**, 32. [Google Scholar] [CrossRef] - Lee, J.; Yang, S.; Kim, J.; Lee, S. An efficient molecular dynamics simulation method for calculating the diffusion-influenced reaction rates. J. Chem. Phys
**2004**, 120, 7564–7575. [Google Scholar] - Thomas, A.S.; Elcock, A.H. Direct measurement of the kinetics and thermodynamics of association of hydrophobic molecules from molecular dynamics simulations. J. Phys. Chem. Lett
**2011**, 2, 19–24. [Google Scholar] - Thomas, A.S.; Elcock, A.H. Direct observation of salt effects on molecular interactions through explicit-solvent molecular dynamics simulations: Differential effects on electrostatic and hydrophobic interactions and comparisons to Poisson-Boltzmann theory. J. Am. Chem. Soc
**2006**, 128, 7796–7806. [Google Scholar] - Garcia-Viloca, M.; Gao, J.; Karplus, M.; Truhlar, D.G. How enzymes work: Analysis by modern rate theory and computer simulations. Science
**2004**, 303, 186–195. [Google Scholar] - Gabdoulline, R.R.; Wade, R.C. Simulation of the diffusional association of barnase and barstar. Biophys. J
**1997**, 72, 1917–1929. [Google Scholar] - Schlosshauer, M.; Baker, D. Realistic protein–protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness. Protein Sci
**2004**, 13, 1660–1669. [Google Scholar] - Peter, E.; Dick, B.; Baeurle, S.A. A novel computer simulation method for simulating the multiscale transduction dynamics of signal proteins. J. Chem. Phys
**2012**, 136, 124112:1–124112:14. [Google Scholar] - Ghaemmaghami, S.; Huh, W.K.; Bower, K.; Howson, R.W.; Belle, A.; Dephoure, N.; O’Shea, E.K.; Weissman, J.S. Global analysis of protein expression in yeast. Nature
**2003**, 425, 737–741. [Google Scholar] - Fujioka, A.; Terai, K.; Itoh, R.E.; Aoki, K.; Nakamura, T.; Kuroda, S.; Nishida, E.; Matsuda, M. Dynamics of the Ras/ERK MAPK cascade as monitored by fluorescent probes. J. Biol. Chem
**2006**, 281, 8917–8926. [Google Scholar] - Gutenkunst, R.N.; Waterfall, J.J.; Casey, F.P.; Brown, K.S.; Myers, C.R.; Sethna, J.P. Universally sloppy parameter sensitivities in systems biology models. PLoS Comp. Biol
**2007**, 3. [Google Scholar] [CrossRef] - Chen, W.W.; Schoeberl, B.; Jasper, P.J.; Niepel, M.; Nielsen, U.B.; Lauffenburger, D.A.; Sorger, P.K. Input–output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data. Mol. Syst. Biol
**2009**, 5. [Google Scholar] [CrossRef] - Hengl, S.; Kreutz, C.; Timmer, J.; Maiwald, T. Data-based identifiability analysis of non-linear dynamical models. Bioinformatics
**2007**, 23, 2612–2618. [Google Scholar] - Koh, G.; Hsu, D.; Thiagarajan, P.S. Component-based construction of bio-pathway models: The parameter estimation problem. Theor. Comput. Sci
**2011**. [Google Scholar] [CrossRef] - Gopalakrishnan, M.; Forsten-Williams, K.; Nugent, M.A.; Täuber, U.C. Effects of receptor clustering on ligand dissociation kinetics: Theory and simulations. Biophys. J
**2005**, 89, 3686–3700. [Google Scholar] - Falk, M.; Klann, M.; Reuss, M.; Ertl, T. 3D Visualization of Concentrations from Stochastic Agent-based Signal Transduction Simulations. Proceedings of the IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2010, Rotterdam, The Netherlands, 14–17 April 2010; pp. 1301–1304.

**Figure 1.**(

**a**) Visualization of the cytoplasm from a Brownian dynamics simulation including cytoskeleton filaments and just the signaling molecules of one pathway. For visualization, all molecules and cytoskeleton filaments have been replaced by their atomic structure and rendered by raytracing (ScienceVisuals [10,11]); (

**b**) Physiological level of crowding, i.e., a representative molecular size distribution and abundance, modeled either with spheres or real molecule shapes by Ando and Skolnick [12] in a cubic subvolume of the cytoplasm. Reproduced with permission of PNAS. Due to the high density of molecules it is impossible to see through the cytoplasm. These crowding conditions affect diffusion and reactions in the cell.

**Figure 2.**(

**a**) Two diffusing molecules can collide and will be reflected if they do not react; (

**b**) Corresponding probability density function (pdf) for the distance of j relative to i as described by the Fokker–Planck equation; (

**c**) Reaction probability depending on the initial distance; (

**d**)–(

**f**) Fokker–Planck equation and boundary conditions. The pdf for the distance r between two diffusing molecules as described by (

**d**) starting from W(0) = δ(

**x**

_{i}−

**x**

_{j}) is shown in (

**b**). In this description, particles react if they diffuse “into” the reaction partner, which is accounted for by the flux across the collision surface and depends on the microscopic rate constant κ. Therefore the boundary condition (

**e**) is partially reflecting. For κ = 0 (

**e**) becomes completely reflective, describing two inert particles. Note the “blister” in (

**b**) which deforms the normal distribution at the boundary caused by the reflection. Due to incomplete reflection, the total probability ∫ WdV < 1. The loss corresponds to the reaction probability. Hence the reaction probability (

**c**) for a given initial distance is found as ${P}_{FP}^{reaction}=1-{\int}_{{r}^{*}}^{\infty}WdV$ [75,109,114].

**Table 1.**Empiric approximations for the hydrodynamic radius r

_{h}based on the molecular weight MW in kDa. (i) is a fit to experimental data, e.g., from [69,70]. The other equations assume that the mass is (re-)distributed in a sphere, for instance with a specific volume of 1 cm

^{3}/g in (ii) [41,71]. Due to the in general nonspherical shape and the “holes” of the molecule, an exponent larger than 1/3 as in (i) is reasonable. The hydrodynamic radii reported by [60] fall between (i) and (ii).

Hydrodynamic Radius [nm] | Reference | |
---|---|---|

(i) | r_{h} = 0.6169 × MW^{0.4226} | [72] |

(ii) | r_{h} = 0.7468 × MW^{1}^{/}^{3} | [41] |

(iii) | r_{h} = 0.5429 × MW^{1}^{/}^{3} | [40] |

Name/Authors | Features | Website/References |
---|---|---|

Smoldyn S. Andrews et al. | Particle based simulator for reaction diffusion processes in arbitrarily shaped compartments. (point particles, no crowding). | www.smoldyn.org [28,48,119,120,133,134] |

ChemCell | Particle based simulator within realistic cell shapes. | chemcell.sandia.gov [93,135,136] |

E-Cell | Complete software environment for simulations on several levels. Contains further analysis tools. | www.e-cell.org [137–139] |

(GFRD,eGFRD) ten Wolde et al. | Green’s function reaction dynamics will be included in the E-Cell project | [110,111,140,141] |

FLAME | Agent-based multi-scale simulation (also beyond the cellular level). | www.flame.ac.uk [32,116,117] |

MCell | Monte Carlo simulator of reaction diffusion processes. Reactions can only happen at membranes | www.mcell.cnl.salk.edu [142] |

MesoRD | Spatial derivative of Gillespie’s algorithm to solve the Reaction-Diffusion Master Equation (RDME) with the “next subvolume method” | mesord.sourceforge.net [143,144] |

SmartCell Serrano et al. | Spatial derivative of Gillespie’s algorithm in arbitrarily shaped compartments. | software.crg.es/smartcell [145] |

STEPS | Tetrahedral mesh based spatial derivative of Gillespie’s algorithm | steps.sourceforge.net/STEPS [146] |

STSE S.Stoma | PDE based simulator with compartments and direct linking to microscope images. | www.stse-software.org [147] |

V Cell | ODE/PDE or SDE based simulator within realistic cell shapes. | www.nrcam.uchc.edu [148,149] |

M. Klann et al. | Agent-based Brownian dynamics including cytoskeleton, crowding and vesicle transport. | www.bison.ethz.ch/research/spatial simulations [75,80,106,107,122,150] |

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Klann, M.; Koeppl, H.
Spatial Simulations in Systems Biology: From Molecules to Cells. *Int. J. Mol. Sci.* **2012**, *13*, 7798-7827.
https://doi.org/10.3390/ijms13067798

**AMA Style**

Klann M, Koeppl H.
Spatial Simulations in Systems Biology: From Molecules to Cells. *International Journal of Molecular Sciences*. 2012; 13(6):7798-7827.
https://doi.org/10.3390/ijms13067798

**Chicago/Turabian Style**

Klann, Michael, and Heinz Koeppl.
2012. "Spatial Simulations in Systems Biology: From Molecules to Cells" *International Journal of Molecular Sciences* 13, no. 6: 7798-7827.
https://doi.org/10.3390/ijms13067798