Available online at www.sciencedirect.com
ScienceDirect Cellular and molecular structure as a unifying framework for whole-cell modeling Elijah Roberts Whole-cell modeling has the potential to play a major role in revolutionizing our understanding of cellular biology over the next few decades. A computational model of the entire cell would allow cellular biologists to integrate data from many disparate sources in a single consistent framework. Such a comprehensive model would be useful both for hypothesis testing and in the discovery of new behaviors that emerge from complex biological networks. Cellular and molecular structure can and should be a key organizing principle in a whole-cell model, connecting models across time and length scales in a multiscale approach. Here I present a summary of recent research centered around using molecular and cellular structure to model the behavior of cells. Addresses Department of Biophysics, Johns Hopkins University, Baltimore, MD 21218, USA Corresponding author: Roberts, Elijah ([email protected])
Current Opinion in Structural Biology 2014, 25:86–91 This review comes from a themed issue on Theory and simulation Edited by Rommie E Amaro and Manju Bansal
S0959-440X/$ – see front matter, # 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sbi.2014.01.005
Introduction In a recent opinion [1], Geyer expressed a viewpoint emerging among many computational biologists that molecular and systems-level models of the cell must be combined since ‘both the local details and the global behavior are equally important.’ The most comprehensive attempt to date at such a union has been that of Karr et al. [2], who presented a whole-cell model from a topdown perspective. In an impressive feat of computational biology, the authors showed that multiple independent modeling methods can be combined into a coherent simulation of cellular biochemistry. However, their efforts stopped short of including molecular-scale features in a detailed manner. One, arguably the best, scaffolding upon which to base a more complete molecular-systems fusion is the structure of the cell. Biomolecular systems in the cell use its structure to organize themselves into interacting units at multiple scales [3]. From the compartmentation of the Current Opinion in Structural Biology 2014, 25:86–91
cell into specialized subvolumes through the organization of the cytoskeleton and down to the assembly of macromolecular complexes, structure is critical to the function of the cell and can serve as a reference by which cellular models can be integrated. As such, here I use the term ‘whole-cell modeling’ in a broader sense to also imply a structural model of the cell. As a working definition of a whole-cell model, I propose the following criteria: – First, the model should account for the physical structure and organization of the cell. While of obvious importance in eukaryotic organisms, even in bacterial systems many phenomena cannot be accurately modeled without taking into account the threedimensional structure of the cell. Models should account for changing structure and organization during the cell cycle, including growth and division of the cell. Included in this criterion is the postulate the model should account for spatial localization of macromolecules within the cell. Many cellular processes require spatial gradients and cytoskeletal organization within the cell for proper function. – Second, the model should account for all known cellular processes, even if not at the level of every individual gene. Many models of individual biochemical processes have been studied in the context of a spatial model of the cell, but cannot be accurately described as whole-cell models. Models of individual pathways are useful for hypothesis testing of specific biological questions, but lack the complexity to capture emergent phenomena associated with a more discovery-driven approach. From a modeling perspective, a whole-cell model should be able to maintain cellular homeostasis without resorting to arbitrary sources and sinks of energy and mass. – Third, the model should account for cellular timescales. Much of cellular biology happens on the length scale of the cell cycle, typically measured in hours. A whole-cell model should be able to model processes for at least a cell cycle. – Fourth, the model should allow for varying levels of detail in model components. It is unreasonable to require that detailed atomic or kinetic data are available for every cellular component. At the same time, it is also unreasonable to expect that every component must be described using the lowest common denominator. A whole-cell model must therefore allow components where more information is known to be modeled at a higher level of detail. This will allow whole-cell models to be used in conjunction with common biological data www.sciencedirect.com
Cellular and molecular structure in whole-cell modeling Roberts 87
Figure 1
Regulatory Model
Systems-level - Data-driven modeling - All cellular genes and pathways
Data-driven network inference
Metabolic Model
Serial Execution Parameter Estimation MD
Metabolic flux
BD
Interaction energies
Potential-based Conformational sampling
RDME Nonspecific interactions
Particle diffusion
Potential-based
Probability-based
Simultaneous Execution State Exchange
Molecular-level - Physics-based modeling - Specific pathways and macromolecules
Reaction kinetics
Specific binding
re
Cytoskeleton Model M o le c u la r S t r u
ctu
Cytoskeletal dynamics
Cellular
Directed cargo & vesicle transport
Structure
Force-based
Current Opinion in Structural Biology
Schematic diagram showing a proposed hierarchical multiscale approach to a whole-cell model. Abbreviations: MD, molecular dynamics; BD, Brownian dynamics; RDME, reaction–diffusion master equation.
sets, such as the results from mutation and knockout experiments. I would like to emphasize that in the foreseeable future, any one computational method is unlikely to be able to meet all of the above criteria. A multiscale approach will be needed. Whether the most successful approach will be that of the Covert lab in which all of the models are run in parallel (simultaneous multiscale) or one in which different layers are modeled serially, feeding parameters from higher-to-lower resolution models (hierarchical multiscale; see Figure 1) is still a subject of debate. Although no whole-cell models have yet been developed that fit the above definition, many scientists who share this vision are making progress toward such a goal. In this minireview I focus on the first criterion enumerated above and present a summary of some recent methods for modeling the physical structure and organization of the cell, along with biological studies using these methods, that are pushing the envelope toward a comprehensive model of the cell. www.sciencedirect.com
Molecular models in a cellular context Structural biologists have been producing exquisite atomic scale structures of biological macromolecules for more than 50 years. Computational biologists have been using these structures for more than 30 years to study the dynamics of proteins and other macromolecules. Wholecell modelers would do themselves a disservice if they were unable to use the valuable data contained in molecular structures, and many researchers are looking for ways to bring this wealth of data to bear on cellular modeling. A primary difficulty is that extensive computational resources are needed to directly study cell-scale phenomena using molecular models. Nevertheless, several molecular approaches are potentially useful. Molecular dynamics
Because molecular dynamics is well-known and covered extensively in the literature, I mention it only briefly. For a recent review see [4]. While there have been a few efforts to simulate systems a substantial fraction of a cell in size using molecular dynamics, the computational resources necessary to reach cellular time scales are Current Opinion in Structural Biology 2014, 25:86–91
88 Theory and simulation
beyond our current capabilities. For a recent example of a large-scale molecular dynamics simulation, see the 64 million atom simulation of HIV-1 capsid [5]. Nevertheless, molecular dynamics modeling of the interactions of molecules can provide valuable insight. Continual improvements in computational power and also the development of coarse-grained models will make these methods critical to whole-cell modeling efforts, especially in hierarchical multiscale approaches.
distribution in the cytoplasm and did not study the effect of local structure of complexes or assemblies which are known to be prevalent in the cell. Future work will hopefully see Brownian dynamics simulations that are accurately able to capture specific and non-specific interactions from molecular structures scaling up to a significant volume of cytoplasm for biologically interesting timescales.
Whole-cell reaction–diffusion models Brownian dynamics
Perhaps the most detailed simulation methods capable of reaching to nearly cellular length and time scales are Brownian and Stokesian dynamics. These methods include one or more inter-molecular potential terms along with a random fluctuating force corresponding to the solvent interactions. See [6] for a more thorough description and review. Such methods are notable because they include to some degree the molecular details of the biomolecules. To date, Brownian dynamics studies at the upper limit of feasibility have provided valuable insight into the diffusion of macromolecules in the cytoplasm, an understanding of which will be critical for whole-cell modeling efforts. For example, it has recently been shown that changes in the diffusivity of the protein MEX-5 in different regions of the cytoplasm establish a concentration gradient involved in cell-fate determination in Caenorhabditis elegans [7]. McGuffee and Elcock [8] presented the first modern treatment of the cytoplasm in a Brownian dynamics simulation. Their model excluded hydrodynamic interactions, but they were able to recover the trend of decreasing long-time diffusion rates in the cytoplasm that have been experimentally observed. The authors made the case that nonspecific energetic interactions between proteins in the cytoplasm indeed play an important role in the molecules’ dynamics. Shortly thereafter, Ando and Skolnick [9] performed a study of macromolecular diffusion in a simulated cytoplasm using a Stokesian dynamics method accounting for hydrodynamic interactions. Their work showed that in the crowded cytoplasm intermolecular interactions through solvent fluid were of substantial importance. More recently, Mereghetti and Wade [10] investigated a mean-field approach to hydrodynamic interactions that was able to recover the change in diffusion due to hydrodynamic interactions, if not the local correlations, while using atomic structures and being substantially more efficient to calculate. While not an explicitly heterogeneous cytoplasm, their model of a concentrated protein solution likely represents a reasonable approximation to the conditions inside a cell.
In addition to methods for determining macromolecular structure, there are also new methods being developed for determining cellular structure. Primary examples are cryo-electron tomography [11,12] and soft X-ray tomography [13] of whole cells. It is also important to emphasize that structural information is also becoming available of subcellular structures, such as the nuclear pore complex [14,15]. These data all need to be integrated into whole cell models. With these new types of data, simulation methods must be used that can take advantage of experimentally determined cellular organization. Reaction–diffusion models represent a further simplification from molecular-scale models. They sacrifice molecular structure and interaction potentials and represent macromolecules as either dimensionless points or hard spheres. By abandoning molecular details reaction–diffusion methods can simulate time scales relevant for cells using a random walk model of Brownian motion and a Smoluchowski description of reactions. Several classes of reaction–diffusion models are in widespread use. PDE methods
Reaction–diffusion partial differential equation (PDE) methods use continuous and deterministic equations to calculate concentration flux due to diffusion and reaction processes. They provide spatial resolution and can account for compartments and other cellular organization. PDE models are very popular for modeling cellular systems, as exemplified by the wide use of the Virtual Cell project [16]. Since they are inherently continuous, they have difficulty dealing with macromolecules present in small copy numbers inside the cell and the inherent cell-to-cell variability that goes with small number fluctuations. Many gene regulatory interactions within the cell are likely to involve such features and it is unclear if PDE models by themselves can be adapted to such situations. However, they are computationally efficient and could prove especially useful in a multiscale approach to a whole-cell model [17]. Particle-based methods
The Brownian dynamics simulations described above typically account for a volume of 100 nm3 and time of 50 ms, so they are still relatively small scale compared to a full cell. Also, these studies assumed a random Current Opinion in Structural Biology 2014, 25:86–91
The next class of methods considered is generically known as particle-based. Molecules are treated as diffusing particles, usually as either a hard sphere or a dimensionless point. Reactions occur when two particles that www.sciencedirect.com
Cellular and molecular structure in whole-cell modeling Roberts 89
can undergo a reaction approach each other. The exact details vary from method to method but generally base their approach on the Smoluchowski equation. Particlebased methods forgo the idea of continuous deterministic equations in favor of a discrete view of the biomolecules in the cell. This allows the investigation of stochastic phenomena and particularly, the ability to characterize cell-to-cell variability. At the most microscopic level is the Green’s function reaction dynamics (GFRD) method [18]. This method decomposes each pair of reacting particles into a twobody formulation of the Smoluchowski equation that can be solved analytically using Green’s functions. Because of the exact solution of the Smoluchowski equation, GFRD is quite accurate, but can become slow for dense systems. It was recently used to study processivity during multisite phosphorylation in MAPK signaling [19]. Smoldyn [20] also solves the Smoluchowski equation but using a time step approach. Notably it pays particular attention to correctly calculating particle–surface interactions. Smoldyn has been used to investigate cellular scale models, including the effect of Bar1 on yeast mating response [20], the role of morphology on kinase dynamics in neural dendritic spines [21], and asymmetrical cell division in yeast [22]. MCell [23] was one of the first particle-based reaction–diffusion simulation methods and has been used by numerous researchers, many in the study of neurotransmitter release [24]. Particle-based simulation methods have proven to be a successful method for studying reaction–diffusion dynamics in a crowded cellular environment [25]. Researchers studying particlebased methods are still making progress toward faster and more accurate simulation methods, e.g., as in a recent study from Klann and Koeppl [26] describing a new method for dealing with geminate recombination. With such improvement, particle-based methods may achieve the speed-ups necessary to play a central role in wholecell models. Reaction–diffusion master equation methods
Demanding that a simulation model millions of particles for times on the order of the cell cycle presently requires further simplification. Especially, for very dense systems with many nearby players, particle-based methods cannot simulate out to the necessary time scales of cellular dynamics (hours). Instead, a further approximation of well-stirred local subvolumes can be made, with particles jumping between subvolumes and reacting freely with any other particle in its subvolume. This assumption greatly simplifies the calculations and the resultant model is known as the reaction–diffusion master equation (RDME). The first simulation code to use the RDME for cell scale simulation was MesoRD [27]. Initial work showed that complex dynamical processes, such as the oscillations of the Min division system in Escherichia coli, could be recovered even with a coarse www.sciencedirect.com
spatial discretization [28]. Recently, new theoretical developments from the Elf lab have shown that the RDME can be corrected for error introduced by spatial discretization [29], opening the possibility for the cell to be discretized at finer length scales. Roberts, Stone, and Luthey-Schulten introduced the idea of an in vivo version of the RDME, in which the reaction and diffusion propensities with the subvolumes could be controlled to reconstruct a particular cellular organization [30]. By including a static representation of the crowded cytoplasm, the authors were able to study deviations in the diffusion and reaction rates of particles and investigate the effects on gene expression for times on the order of the cell cycle [31]. Additionally, using cryo-electron tomograms of a single E. coli cell, the authors studied for the first time the effect of including experimentally determined cell geometry and organization on the dynamics of biochemical processes. Recently, the RDME has been used extensively to study cell-scale processes. Sturrock et al. applied the RDME to gene expression circuits in eukaryotic cells, reproducing stochastic variability in oscillations of Hes1 [32]. Flegg et al. recently [33] developed a hybrid method where particles in certain regions of interest are treated using a particle-based method while particles in the surrounding volumes are treated using the RDME. The authors show an example of the method’s utility for simulating large volumes studying the release of calcium from the endoplasmic reticulum [34]. Lawson et al. [35] recently published a study using the RDME to model how yeast track and respond to a pheromone gradient. In their stochastic simulations the authors recovered a much more tightly localized response than in their deterministic counterparts. Additionally, they were able to recover a mutant phenotype only with the stochastic simulations. This is likely to be a general principle in biology, where positive feedback loops can create bistable spatial systems with stochastic noise driving the populations to multiple states.
Whole-cell modeling of the active cytoskeleton As a final example of the types of cellular organization that will need to be included in whole-cell models, consider the problem of modeling the dynamics of the cytoskeleton. A cytoskeletal model was missing from each of the models considered above, and yet is critical for many cellular functions. In a recent study using the Virtual Cell software, Ditlev et al. [36] modeled actin dynamics in response to the Nck protein, which induces actin polymerization. The authors quantitatively studied the dependence of actin polymerization on the local density of Nck, combining computation and experiment. Current Opinion in Structural Biology 2014, 25:86–91
90 Theory and simulation
Modeling the cytoskeleton involves the assembly and disassembly of long filaments and also the use of these cellular highways by motor proteins transporting various cargo. An important aspect of the cytoskeleton for wholecell modeling is the transport of vesicles. For instance, recent continuum modeling of yeast polarization has investigated the role of directed exocytosis and endocytosis of Cdc42 in the maintenance of a polarized cell state [37–39]. To integrate both reaction–diffusion and directed transport along discrete cytoskeletal filaments, Klann et al. introduced an agent-based modeling method for vesicle dynamics in which they walk along static cytoskeletal filaments [40]. The method is able to model vesicle budding, transport along cytoskeletal filaments, and fusion. The authors used the model to investigate receptor-mediated endocytosis, correctly recovering the polarization of the cell due to cytoskeletal asymmetry. Although the model lacks active remodeling of the cytoskeleton, it represents an important first step in creating a multiscale model of vesicle dynamics.
of any whole-cell model. Particularly, predicting how cellular fluctuations move individual cells between phenotypic states is key to understanding cellular decision making. RDME methods are probabilistic to capture such cell-to-cell variation, can calculate hours of simulation to capture dynamics on the time scales of the cell cycle, and can capture at a coarse-grained level cellular and subcellular structure. In Figure 1, I present a schematic diagram for a whole-cell modeling approach centered around the RDME. However, regardless of the actual approach that ultimately proves most successful, whole-cell modeling that integrates scales from the molecular to the system stands poised to help us quantitatively approach biological problems in new and exciting ways.
Another possible approach to directed transport has been explored by Alberts [41]. The author introduced a model in which cytoskeletal filaments are modeled as a series of rigid rods linked with translational and torsional springs. By tuning the parameters one can achieve biophysically realistic filament properties. The introduction of filament assembly and disassembly along with the modeling of force application by motor proteins carrying cargo would make this approach highly suitable for modeling active cytoskeleton in a whole-cell model.
References and recommended reading
Finally, transport in eukaryotic cells must also account for subcellular structures, such as the nuclear pore complex. A biophysical model of discrete filaments was recently used in a study of transport through the nuclear pore complex based on known structural features [42]. The combination of a gradient of RanGTP (a GTPase involved in cargo binding) and binding of FG filaments to localization sequences on the cargo provided a starting point for a comparison with experimental measurements, including single molecule experiments where molecular distributions can be used.
Conclusions In this minireview I presented some recent examples of the sorts of structure-based modeling methods that will be useful in a whole-cell model. Molecular and cellular structure can create a framework for simulating multiscale models of cellular processes. Several efforts are now beginning to be directed toward integrating different resolution methods into a multiscale approach. In my opinion, RDME methods are a leading candidate for a core method upon which to base a whole-cell model. It has become apparent that cell-to-cell variability is an important phenomena that must be an integral component Current Opinion in Structural Biology 2014, 25:86–91
Acknowledgement The author would like to acknowledge Zan Luthey-Schulten for many insightful conversations regarding most of the topics discussed in this review.
Papers of particular interest, published within the period of review, have been highlighted as: of special interest of outstanding interest 1.
Geyer T: Modeling metabolic processes between molecular and systems biology. Curr Opin Struct Biol 2013, 23:218-223.
2.
Karr JR, Sanghvi JC, Macklin DN, Gutschow MV, Jacobs JM, Bolival B, Assad-Garcia N, Glass JI, Covert MW: A whole-cell computational model predicts phenotype from genotype. Cell 2012, 150:389-401. A tour de force in systems biology modeling of cellular behavior. The authors present a multiscale model of Mycoplasma genitalium accounting for, among other things, metabolism, transcription, translation, and cellular growth.
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Betts MJ, Russell RB: The hard cell: from proteomics to a whole cell model. FEBS Lett 2007, 581:2870-2876.
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Feig M, Sugita Y: Reaching new levels of realism in modeling biological macromolecules in cellular environments. J Mol Graph Model 2013, 45C:144-156.
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Zhao G, Perilla JR, Yufenyuy EL, Meng X, Chen B, Ning J, Ahn J, Gronenborn AM, Schulten K, Aiken C, Zhang P: Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics. Nature 2013, 497:643-646.
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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.
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Griffin EE, Odde DJ, Seydoux G: Regulation of the MEX-5 gradient by a spatially segregated kinase/phosphatase cycle. Cell 2011, 146:955-968.
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McGuffee SR, Elcock AH: Diffusion, crowding & protein stability in a dynamic molecular model of the bacterial cytoplasm. PLoS Comput Biol 2010, 6:e1000694.
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Ando T, Skolnick J: Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion. Proc Natl Acad Sci U S A 2010, 107:18457-18462.
10. Mereghetti P, Wade RC: Atomic detail Brownian dynamics simulations of concentrated protein solutions with a mean field treatment of hydrodynamic interactions. J Phys Chem B 2012, 116:8523-8533. The authors present a mean field model of hydrodynamic interactions for Brownian dynamics simulations, while maintaining atomic-level detail in the molecular structures. www.sciencedirect.com
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11. Luˇciˇc V, Rigort A, Baumeister W: Cryo-electron tomography: the challenge of doing structural biology in situ. J Cell Biol 2013, 202:407-419. 12. Pilhofer M, Jensen GJ: The bacterial cytoskeleton: more than twisted filaments. Curr Opin Cell Biol 2013, 25:125-133. 13. Parkinson DY, Epperly LR, McDermott G, Le Gros MA, Boudreau RM, Larabell CA: Nanoimaging cells using soft X-ray tomography. Methods Mol Biol 2013, 950:457-481. 14. Hoelz A, Debler EW, Blobel G: The structure of the nuclear pore complex. Annu Rev Biochem 2011, 80:613-643. 15. Rigort A, Ba¨uerlein FJB, Villa E, Eibauer M, Laugks T, Baumeister W, Plitzko JM: Focused ion beam micromachining of eukaryotic cells for cryoelectron tomography. Proc Natl Acad Sci U S A 2012, 109:4449-4454. 16. Cowan AE, Moraru II, Schaff JC, Slepchenko BM, Loew LM: Spatial modeling of cell signaling networks. Methods Cell Biol 2012, 110:195-221. A very readable introduction to using PDEs to model cellular systems with spatial degrees of freedom, in the context of the Virtual Cell software. 17. Franz B, Flegg MB, Chapman SJ, Erban R: Multiscale reaction– diffusion algorithms: PDE-assisted Brownian dynamics. SIAM J Appl Math 2013, 73:1224-1247. 18. van Zon JS, ten Wolde PR: Green’s-function reaction dynamics: a particle-based approach for simulating biochemical networks in time and space. J Chem Phys 2005, 123:234910. 19. Takahashi K, Tanase-Nicola S, ten Wolde PR: Spatio-temporal correlations can drastically change the response of a MAPK pathway. Proc Natl Acad Sci U S A 2010, 107:2473-2478. 20. Andrews SS: Spatial and stochastic cellular modeling with the Smoldyn simulator. Methods Mol Biol 2012, 804:519-542. 21. Khan S, Reese TS, Rajpoot N, Shabbir A: Spatiotemporal maps of CaMKII in dendritic spines. J Comput Neurosci 2012, 33:123-139. 22. Boettcher B, Marquez-Lago TT, Bayer M, Weiss EL, Barral Y: Nuclear envelope morphology constrains diffusion and promotes asymmetric protein segregation in closed mitosis. J Cell Biol 2012, 197:921-937. 23. Kerr RA, Bartol TM, Kaminsky B, Dittrich M, Chang JCJ, Baden SB, Sejnowski TJ, Stiles JR: Fast Monte Carlo simulation methods for biological reaction–diffusion systems in solution and on surfaces. SIAM J Sci Comput 2008, 30:3126. 24. Dittrich M, Pattillo JM, King JD, Cho S, Stiles JR, Meriney SD: An excess-calcium-binding-site model predicts neurotransmitter release at the neuromuscular junction. Biophys J 2013, 104:2751-2763. 25. Ridgway D, Broderick G, Lopez-Campistrous A, Ru’aini M, Winter P, Hamilton M, Boulanger P, Kovalenko A, Ellison MJ: Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm. Biophys J 2008, 94:3748-3759.
An introduction to the popular MesoRD software for simulating cell-scale models using the RDME. 28. Hattne J, Fange D, Elf J: Stochastic reaction–diffusion simulation with MesoRD. Bioinformatics 2005, 21:2923-2924. 29. Fange D, Berg OG, Sj ¨oberg P, Elf J: Stochastic reaction–diffusion kinetics in the microscopic limit. Proc Natl Acad Sci U S A 2010, 107:19820-19825. 30. Roberts E, Stone JE, Luthey-Schulten Z: Lattice Microbes: highperformance stochastic simulation method for the reaction– diffusion master equation. J Comput Chem 2013, 34:245-255. 31. Roberts E, Magis A, Ortiz JO, Baumeister W, Luthey-Schulten Z: Noise contributions in an inducible genetic switch: a wholecell simulation study. PLoS Comput Biol 2011, 7:e1002010. The authors present a study using the Lattice Microbes software to study cell-to-cell variability of the lac genetic circuit. The authors use a cryoelectron tomogram of an individual E. coli cell to investigate the effect of cellular geometry. 32. Sturrock M, Hellander A, Matzavinos A, Chaplain MAJ: Spatial stochastic modelling of the Hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonic stem cell differentiation. J R Soc Interface 2013, 10:20120988. 33. Flegg MB, Chapman SJ, Erban R: The two-regime method for optimizing stochastic reaction–diffusion simulations. J R Soc Interface 2012, 9:859-868. 34. Flegg MB, R¨udiger S, Erban R: Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release. J Chem Phys 2013, 138:154103. 35. Lawson MJ, Drawert B, Khammash M, Petzold L, Yi TM: Spatial stochastic dynamics enable robust cell polarization. PLoS Comput Biol 2013, 9:e1003139. The authors present a model for cell polarization during the yeast mating response. By using the RDME, the authors are able to elucidate the role of stochasticity in the sharpness of the response. 36. Ditlev JA, Michalski PJ, Huber G, Rivera GM, Mohler WA, Loew LM, Mayer BJ: Stoichiometry of Nck dependent actin polymerization in living cells. J Cell Biol 2012, 197:643-658. 37. Layton AT, Savage NS, Howell AS, Carroll SY, Drubin DG, Lew DJ: Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment. Curr Biol 2011, 21:184-194. 38. Chou CS, Moore TI, Chang SD, Nie Q, Yi TM: Signaling regulated endocytosis and exocytosis lead to mating pheromone concentration dependent morphologies in yeast. FEBS Lett 2012, 586:4208-4214. 39. Slaughter BD, Unruh JR, Das A, Smith SE, Rubinstein B, Li R: Non-uniform membrane diffusion enables steady-state cell polarization via vesicular trafficking. Nat Commun 2013, 4:1380. 40. 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:e29645. A model for vesicle trafficking is presented that accounts for individual vesicles. The authors study the effect of various cytoskeletal layouts on overall vesicle statistics.
26. Klann M, Koeppl H: Reaction schemes, escape times and geminate recombinations in particle-based spatial simulations of biochemical reactions. Phys Biol 2013, 10:046005. A thorough explanation of theoretical basis for particle-based reaction– diffusion simulation methods. The authors then present new alternative approaches with improved performance characteristics.
41. Alberts JB: Biophysically realistic filament bending dynamics in agent-based biological simulation. PLoS ONE 2009, 4:e4748.
27. Fange D, Mahmutovic A, Elf J: MesoRD 10: stochastic reaction– diffusion simulations in the microscopic limit. Bioinformatics 2012, 28:3155-3157.
42. Mincer JS, Simon SM: Simulations of nuclear pore transport yield mechanistic insights and quantitative predictions. Proc Natl Acad Sci U S A 2011, 108:E351-E358.
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Current Opinion in Structural Biology 2014, 25:86–91
ScienceDirect Cellular and molecular structure as a unifying framework for whole-cell modeling Elijah Roberts Whole-cell modeling has the potential to play a major role in revolutionizing our understanding of cellular biology over the next few decades. A computational model of the entire cell would allow cellular biologists to integrate data from many disparate sources in a single consistent framework. Such a comprehensive model would be useful both for hypothesis testing and in the discovery of new behaviors that emerge from complex biological networks. Cellular and molecular structure can and should be a key organizing principle in a whole-cell model, connecting models across time and length scales in a multiscale approach. Here I present a summary of recent research centered around using molecular and cellular structure to model the behavior of cells. Addresses Department of Biophysics, Johns Hopkins University, Baltimore, MD 21218, USA Corresponding author: Roberts, Elijah ([email protected])
Current Opinion in Structural Biology 2014, 25:86–91 This review comes from a themed issue on Theory and simulation Edited by Rommie E Amaro and Manju Bansal
S0959-440X/$ – see front matter, # 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sbi.2014.01.005
Introduction In a recent opinion [1], Geyer expressed a viewpoint emerging among many computational biologists that molecular and systems-level models of the cell must be combined since ‘both the local details and the global behavior are equally important.’ The most comprehensive attempt to date at such a union has been that of Karr et al. [2], who presented a whole-cell model from a topdown perspective. In an impressive feat of computational biology, the authors showed that multiple independent modeling methods can be combined into a coherent simulation of cellular biochemistry. However, their efforts stopped short of including molecular-scale features in a detailed manner. One, arguably the best, scaffolding upon which to base a more complete molecular-systems fusion is the structure of the cell. Biomolecular systems in the cell use its structure to organize themselves into interacting units at multiple scales [3]. From the compartmentation of the Current Opinion in Structural Biology 2014, 25:86–91
cell into specialized subvolumes through the organization of the cytoskeleton and down to the assembly of macromolecular complexes, structure is critical to the function of the cell and can serve as a reference by which cellular models can be integrated. As such, here I use the term ‘whole-cell modeling’ in a broader sense to also imply a structural model of the cell. As a working definition of a whole-cell model, I propose the following criteria: – First, the model should account for the physical structure and organization of the cell. While of obvious importance in eukaryotic organisms, even in bacterial systems many phenomena cannot be accurately modeled without taking into account the threedimensional structure of the cell. Models should account for changing structure and organization during the cell cycle, including growth and division of the cell. Included in this criterion is the postulate the model should account for spatial localization of macromolecules within the cell. Many cellular processes require spatial gradients and cytoskeletal organization within the cell for proper function. – Second, the model should account for all known cellular processes, even if not at the level of every individual gene. Many models of individual biochemical processes have been studied in the context of a spatial model of the cell, but cannot be accurately described as whole-cell models. Models of individual pathways are useful for hypothesis testing of specific biological questions, but lack the complexity to capture emergent phenomena associated with a more discovery-driven approach. From a modeling perspective, a whole-cell model should be able to maintain cellular homeostasis without resorting to arbitrary sources and sinks of energy and mass. – Third, the model should account for cellular timescales. Much of cellular biology happens on the length scale of the cell cycle, typically measured in hours. A whole-cell model should be able to model processes for at least a cell cycle. – Fourth, the model should allow for varying levels of detail in model components. It is unreasonable to require that detailed atomic or kinetic data are available for every cellular component. At the same time, it is also unreasonable to expect that every component must be described using the lowest common denominator. A whole-cell model must therefore allow components where more information is known to be modeled at a higher level of detail. This will allow whole-cell models to be used in conjunction with common biological data www.sciencedirect.com
Cellular and molecular structure in whole-cell modeling Roberts 87
Figure 1
Regulatory Model
Systems-level - Data-driven modeling - All cellular genes and pathways
Data-driven network inference
Metabolic Model
Serial Execution Parameter Estimation MD
Metabolic flux
BD
Interaction energies
Potential-based Conformational sampling
RDME Nonspecific interactions
Particle diffusion
Potential-based
Probability-based
Simultaneous Execution State Exchange
Molecular-level - Physics-based modeling - Specific pathways and macromolecules
Reaction kinetics
Specific binding
re
Cytoskeleton Model M o le c u la r S t r u
ctu
Cytoskeletal dynamics
Cellular
Directed cargo & vesicle transport
Structure
Force-based
Current Opinion in Structural Biology
Schematic diagram showing a proposed hierarchical multiscale approach to a whole-cell model. Abbreviations: MD, molecular dynamics; BD, Brownian dynamics; RDME, reaction–diffusion master equation.
sets, such as the results from mutation and knockout experiments. I would like to emphasize that in the foreseeable future, any one computational method is unlikely to be able to meet all of the above criteria. A multiscale approach will be needed. Whether the most successful approach will be that of the Covert lab in which all of the models are run in parallel (simultaneous multiscale) or one in which different layers are modeled serially, feeding parameters from higher-to-lower resolution models (hierarchical multiscale; see Figure 1) is still a subject of debate. Although no whole-cell models have yet been developed that fit the above definition, many scientists who share this vision are making progress toward such a goal. In this minireview I focus on the first criterion enumerated above and present a summary of some recent methods for modeling the physical structure and organization of the cell, along with biological studies using these methods, that are pushing the envelope toward a comprehensive model of the cell. www.sciencedirect.com
Molecular models in a cellular context Structural biologists have been producing exquisite atomic scale structures of biological macromolecules for more than 50 years. Computational biologists have been using these structures for more than 30 years to study the dynamics of proteins and other macromolecules. Wholecell modelers would do themselves a disservice if they were unable to use the valuable data contained in molecular structures, and many researchers are looking for ways to bring this wealth of data to bear on cellular modeling. A primary difficulty is that extensive computational resources are needed to directly study cell-scale phenomena using molecular models. Nevertheless, several molecular approaches are potentially useful. Molecular dynamics
Because molecular dynamics is well-known and covered extensively in the literature, I mention it only briefly. For a recent review see [4]. While there have been a few efforts to simulate systems a substantial fraction of a cell in size using molecular dynamics, the computational resources necessary to reach cellular time scales are Current Opinion in Structural Biology 2014, 25:86–91
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beyond our current capabilities. For a recent example of a large-scale molecular dynamics simulation, see the 64 million atom simulation of HIV-1 capsid [5]. Nevertheless, molecular dynamics modeling of the interactions of molecules can provide valuable insight. Continual improvements in computational power and also the development of coarse-grained models will make these methods critical to whole-cell modeling efforts, especially in hierarchical multiscale approaches.
distribution in the cytoplasm and did not study the effect of local structure of complexes or assemblies which are known to be prevalent in the cell. Future work will hopefully see Brownian dynamics simulations that are accurately able to capture specific and non-specific interactions from molecular structures scaling up to a significant volume of cytoplasm for biologically interesting timescales.
Whole-cell reaction–diffusion models Brownian dynamics
Perhaps the most detailed simulation methods capable of reaching to nearly cellular length and time scales are Brownian and Stokesian dynamics. These methods include one or more inter-molecular potential terms along with a random fluctuating force corresponding to the solvent interactions. See [6] for a more thorough description and review. Such methods are notable because they include to some degree the molecular details of the biomolecules. To date, Brownian dynamics studies at the upper limit of feasibility have provided valuable insight into the diffusion of macromolecules in the cytoplasm, an understanding of which will be critical for whole-cell modeling efforts. For example, it has recently been shown that changes in the diffusivity of the protein MEX-5 in different regions of the cytoplasm establish a concentration gradient involved in cell-fate determination in Caenorhabditis elegans [7]. McGuffee and Elcock [8] presented the first modern treatment of the cytoplasm in a Brownian dynamics simulation. Their model excluded hydrodynamic interactions, but they were able to recover the trend of decreasing long-time diffusion rates in the cytoplasm that have been experimentally observed. The authors made the case that nonspecific energetic interactions between proteins in the cytoplasm indeed play an important role in the molecules’ dynamics. Shortly thereafter, Ando and Skolnick [9] performed a study of macromolecular diffusion in a simulated cytoplasm using a Stokesian dynamics method accounting for hydrodynamic interactions. Their work showed that in the crowded cytoplasm intermolecular interactions through solvent fluid were of substantial importance. More recently, Mereghetti and Wade [10] investigated a mean-field approach to hydrodynamic interactions that was able to recover the change in diffusion due to hydrodynamic interactions, if not the local correlations, while using atomic structures and being substantially more efficient to calculate. While not an explicitly heterogeneous cytoplasm, their model of a concentrated protein solution likely represents a reasonable approximation to the conditions inside a cell.
In addition to methods for determining macromolecular structure, there are also new methods being developed for determining cellular structure. Primary examples are cryo-electron tomography [11,12] and soft X-ray tomography [13] of whole cells. It is also important to emphasize that structural information is also becoming available of subcellular structures, such as the nuclear pore complex [14,15]. These data all need to be integrated into whole cell models. With these new types of data, simulation methods must be used that can take advantage of experimentally determined cellular organization. Reaction–diffusion models represent a further simplification from molecular-scale models. They sacrifice molecular structure and interaction potentials and represent macromolecules as either dimensionless points or hard spheres. By abandoning molecular details reaction–diffusion methods can simulate time scales relevant for cells using a random walk model of Brownian motion and a Smoluchowski description of reactions. Several classes of reaction–diffusion models are in widespread use. PDE methods
Reaction–diffusion partial differential equation (PDE) methods use continuous and deterministic equations to calculate concentration flux due to diffusion and reaction processes. They provide spatial resolution and can account for compartments and other cellular organization. PDE models are very popular for modeling cellular systems, as exemplified by the wide use of the Virtual Cell project [16]. Since they are inherently continuous, they have difficulty dealing with macromolecules present in small copy numbers inside the cell and the inherent cell-to-cell variability that goes with small number fluctuations. Many gene regulatory interactions within the cell are likely to involve such features and it is unclear if PDE models by themselves can be adapted to such situations. However, they are computationally efficient and could prove especially useful in a multiscale approach to a whole-cell model [17]. Particle-based methods
The Brownian dynamics simulations described above typically account for a volume of 100 nm3 and time of 50 ms, so they are still relatively small scale compared to a full cell. Also, these studies assumed a random Current Opinion in Structural Biology 2014, 25:86–91
The next class of methods considered is generically known as particle-based. Molecules are treated as diffusing particles, usually as either a hard sphere or a dimensionless point. Reactions occur when two particles that www.sciencedirect.com
Cellular and molecular structure in whole-cell modeling Roberts 89
can undergo a reaction approach each other. The exact details vary from method to method but generally base their approach on the Smoluchowski equation. Particlebased methods forgo the idea of continuous deterministic equations in favor of a discrete view of the biomolecules in the cell. This allows the investigation of stochastic phenomena and particularly, the ability to characterize cell-to-cell variability. At the most microscopic level is the Green’s function reaction dynamics (GFRD) method [18]. This method decomposes each pair of reacting particles into a twobody formulation of the Smoluchowski equation that can be solved analytically using Green’s functions. Because of the exact solution of the Smoluchowski equation, GFRD is quite accurate, but can become slow for dense systems. It was recently used to study processivity during multisite phosphorylation in MAPK signaling [19]. Smoldyn [20] also solves the Smoluchowski equation but using a time step approach. Notably it pays particular attention to correctly calculating particle–surface interactions. Smoldyn has been used to investigate cellular scale models, including the effect of Bar1 on yeast mating response [20], the role of morphology on kinase dynamics in neural dendritic spines [21], and asymmetrical cell division in yeast [22]. MCell [23] was one of the first particle-based reaction–diffusion simulation methods and has been used by numerous researchers, many in the study of neurotransmitter release [24]. Particle-based simulation methods have proven to be a successful method for studying reaction–diffusion dynamics in a crowded cellular environment [25]. Researchers studying particlebased methods are still making progress toward faster and more accurate simulation methods, e.g., as in a recent study from Klann and Koeppl [26] describing a new method for dealing with geminate recombination. With such improvement, particle-based methods may achieve the speed-ups necessary to play a central role in wholecell models. Reaction–diffusion master equation methods
Demanding that a simulation model millions of particles for times on the order of the cell cycle presently requires further simplification. Especially, for very dense systems with many nearby players, particle-based methods cannot simulate out to the necessary time scales of cellular dynamics (hours). Instead, a further approximation of well-stirred local subvolumes can be made, with particles jumping between subvolumes and reacting freely with any other particle in its subvolume. This assumption greatly simplifies the calculations and the resultant model is known as the reaction–diffusion master equation (RDME). The first simulation code to use the RDME for cell scale simulation was MesoRD [27]. Initial work showed that complex dynamical processes, such as the oscillations of the Min division system in Escherichia coli, could be recovered even with a coarse www.sciencedirect.com
spatial discretization [28]. Recently, new theoretical developments from the Elf lab have shown that the RDME can be corrected for error introduced by spatial discretization [29], opening the possibility for the cell to be discretized at finer length scales. Roberts, Stone, and Luthey-Schulten introduced the idea of an in vivo version of the RDME, in which the reaction and diffusion propensities with the subvolumes could be controlled to reconstruct a particular cellular organization [30]. By including a static representation of the crowded cytoplasm, the authors were able to study deviations in the diffusion and reaction rates of particles and investigate the effects on gene expression for times on the order of the cell cycle [31]. Additionally, using cryo-electron tomograms of a single E. coli cell, the authors studied for the first time the effect of including experimentally determined cell geometry and organization on the dynamics of biochemical processes. Recently, the RDME has been used extensively to study cell-scale processes. Sturrock et al. applied the RDME to gene expression circuits in eukaryotic cells, reproducing stochastic variability in oscillations of Hes1 [32]. Flegg et al. recently [33] developed a hybrid method where particles in certain regions of interest are treated using a particle-based method while particles in the surrounding volumes are treated using the RDME. The authors show an example of the method’s utility for simulating large volumes studying the release of calcium from the endoplasmic reticulum [34]. Lawson et al. [35] recently published a study using the RDME to model how yeast track and respond to a pheromone gradient. In their stochastic simulations the authors recovered a much more tightly localized response than in their deterministic counterparts. Additionally, they were able to recover a mutant phenotype only with the stochastic simulations. This is likely to be a general principle in biology, where positive feedback loops can create bistable spatial systems with stochastic noise driving the populations to multiple states.
Whole-cell modeling of the active cytoskeleton As a final example of the types of cellular organization that will need to be included in whole-cell models, consider the problem of modeling the dynamics of the cytoskeleton. A cytoskeletal model was missing from each of the models considered above, and yet is critical for many cellular functions. In a recent study using the Virtual Cell software, Ditlev et al. [36] modeled actin dynamics in response to the Nck protein, which induces actin polymerization. The authors quantitatively studied the dependence of actin polymerization on the local density of Nck, combining computation and experiment. Current Opinion in Structural Biology 2014, 25:86–91
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Modeling the cytoskeleton involves the assembly and disassembly of long filaments and also the use of these cellular highways by motor proteins transporting various cargo. An important aspect of the cytoskeleton for wholecell modeling is the transport of vesicles. For instance, recent continuum modeling of yeast polarization has investigated the role of directed exocytosis and endocytosis of Cdc42 in the maintenance of a polarized cell state [37–39]. To integrate both reaction–diffusion and directed transport along discrete cytoskeletal filaments, Klann et al. introduced an agent-based modeling method for vesicle dynamics in which they walk along static cytoskeletal filaments [40]. The method is able to model vesicle budding, transport along cytoskeletal filaments, and fusion. The authors used the model to investigate receptor-mediated endocytosis, correctly recovering the polarization of the cell due to cytoskeletal asymmetry. Although the model lacks active remodeling of the cytoskeleton, it represents an important first step in creating a multiscale model of vesicle dynamics.
of any whole-cell model. Particularly, predicting how cellular fluctuations move individual cells between phenotypic states is key to understanding cellular decision making. RDME methods are probabilistic to capture such cell-to-cell variation, can calculate hours of simulation to capture dynamics on the time scales of the cell cycle, and can capture at a coarse-grained level cellular and subcellular structure. In Figure 1, I present a schematic diagram for a whole-cell modeling approach centered around the RDME. However, regardless of the actual approach that ultimately proves most successful, whole-cell modeling that integrates scales from the molecular to the system stands poised to help us quantitatively approach biological problems in new and exciting ways.
Another possible approach to directed transport has been explored by Alberts [41]. The author introduced a model in which cytoskeletal filaments are modeled as a series of rigid rods linked with translational and torsional springs. By tuning the parameters one can achieve biophysically realistic filament properties. The introduction of filament assembly and disassembly along with the modeling of force application by motor proteins carrying cargo would make this approach highly suitable for modeling active cytoskeleton in a whole-cell model.
References and recommended reading
Finally, transport in eukaryotic cells must also account for subcellular structures, such as the nuclear pore complex. A biophysical model of discrete filaments was recently used in a study of transport through the nuclear pore complex based on known structural features [42]. The combination of a gradient of RanGTP (a GTPase involved in cargo binding) and binding of FG filaments to localization sequences on the cargo provided a starting point for a comparison with experimental measurements, including single molecule experiments where molecular distributions can be used.
Conclusions In this minireview I presented some recent examples of the sorts of structure-based modeling methods that will be useful in a whole-cell model. Molecular and cellular structure can create a framework for simulating multiscale models of cellular processes. Several efforts are now beginning to be directed toward integrating different resolution methods into a multiscale approach. In my opinion, RDME methods are a leading candidate for a core method upon which to base a whole-cell model. It has become apparent that cell-to-cell variability is an important phenomena that must be an integral component Current Opinion in Structural Biology 2014, 25:86–91
Acknowledgement The author would like to acknowledge Zan Luthey-Schulten for many insightful conversations regarding most of the topics discussed in this review.
Papers of particular interest, published within the period of review, have been highlighted as: of special interest of outstanding interest 1.
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Zhao G, Perilla JR, Yufenyuy EL, Meng X, Chen B, Ning J, Ahn J, Gronenborn AM, Schulten K, Aiken C, Zhang P: Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics. Nature 2013, 497:643-646.
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10. Mereghetti P, Wade RC: Atomic detail Brownian dynamics simulations of concentrated protein solutions with a mean field treatment of hydrodynamic interactions. J Phys Chem B 2012, 116:8523-8533. The authors present a mean field model of hydrodynamic interactions for Brownian dynamics simulations, while maintaining atomic-level detail in the molecular structures. www.sciencedirect.com
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11. Luˇciˇc V, Rigort A, Baumeister W: Cryo-electron tomography: the challenge of doing structural biology in situ. J Cell Biol 2013, 202:407-419. 12. Pilhofer M, Jensen GJ: The bacterial cytoskeleton: more than twisted filaments. Curr Opin Cell Biol 2013, 25:125-133. 13. Parkinson DY, Epperly LR, McDermott G, Le Gros MA, Boudreau RM, Larabell CA: Nanoimaging cells using soft X-ray tomography. Methods Mol Biol 2013, 950:457-481. 14. Hoelz A, Debler EW, Blobel G: The structure of the nuclear pore complex. Annu Rev Biochem 2011, 80:613-643. 15. Rigort A, Ba¨uerlein FJB, Villa E, Eibauer M, Laugks T, Baumeister W, Plitzko JM: Focused ion beam micromachining of eukaryotic cells for cryoelectron tomography. Proc Natl Acad Sci U S A 2012, 109:4449-4454. 16. Cowan AE, Moraru II, Schaff JC, Slepchenko BM, Loew LM: Spatial modeling of cell signaling networks. Methods Cell Biol 2012, 110:195-221. A very readable introduction to using PDEs to model cellular systems with spatial degrees of freedom, in the context of the Virtual Cell software. 17. Franz B, Flegg MB, Chapman SJ, Erban R: Multiscale reaction– diffusion algorithms: PDE-assisted Brownian dynamics. SIAM J Appl Math 2013, 73:1224-1247. 18. van Zon JS, ten Wolde PR: Green’s-function reaction dynamics: a particle-based approach for simulating biochemical networks in time and space. J Chem Phys 2005, 123:234910. 19. Takahashi K, Tanase-Nicola S, ten Wolde PR: Spatio-temporal correlations can drastically change the response of a MAPK pathway. Proc Natl Acad Sci U S A 2010, 107:2473-2478. 20. Andrews SS: Spatial and stochastic cellular modeling with the Smoldyn simulator. Methods Mol Biol 2012, 804:519-542. 21. Khan S, Reese TS, Rajpoot N, Shabbir A: Spatiotemporal maps of CaMKII in dendritic spines. J Comput Neurosci 2012, 33:123-139. 22. Boettcher B, Marquez-Lago TT, Bayer M, Weiss EL, Barral Y: Nuclear envelope morphology constrains diffusion and promotes asymmetric protein segregation in closed mitosis. J Cell Biol 2012, 197:921-937. 23. Kerr RA, Bartol TM, Kaminsky B, Dittrich M, Chang JCJ, Baden SB, Sejnowski TJ, Stiles JR: Fast Monte Carlo simulation methods for biological reaction–diffusion systems in solution and on surfaces. SIAM J Sci Comput 2008, 30:3126. 24. Dittrich M, Pattillo JM, King JD, Cho S, Stiles JR, Meriney SD: An excess-calcium-binding-site model predicts neurotransmitter release at the neuromuscular junction. Biophys J 2013, 104:2751-2763. 25. Ridgway D, Broderick G, Lopez-Campistrous A, Ru’aini M, Winter P, Hamilton M, Boulanger P, Kovalenko A, Ellison MJ: Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm. Biophys J 2008, 94:3748-3759.
An introduction to the popular MesoRD software for simulating cell-scale models using the RDME. 28. Hattne J, Fange D, Elf J: Stochastic reaction–diffusion simulation with MesoRD. Bioinformatics 2005, 21:2923-2924. 29. Fange D, Berg OG, Sj ¨oberg P, Elf J: Stochastic reaction–diffusion kinetics in the microscopic limit. Proc Natl Acad Sci U S A 2010, 107:19820-19825. 30. Roberts E, Stone JE, Luthey-Schulten Z: Lattice Microbes: highperformance stochastic simulation method for the reaction– diffusion master equation. J Comput Chem 2013, 34:245-255. 31. Roberts E, Magis A, Ortiz JO, Baumeister W, Luthey-Schulten Z: Noise contributions in an inducible genetic switch: a wholecell simulation study. PLoS Comput Biol 2011, 7:e1002010. The authors present a study using the Lattice Microbes software to study cell-to-cell variability of the lac genetic circuit. The authors use a cryoelectron tomogram of an individual E. coli cell to investigate the effect of cellular geometry. 32. Sturrock M, Hellander A, Matzavinos A, Chaplain MAJ: Spatial stochastic modelling of the Hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonic stem cell differentiation. J R Soc Interface 2013, 10:20120988. 33. Flegg MB, Chapman SJ, Erban R: The two-regime method for optimizing stochastic reaction–diffusion simulations. J R Soc Interface 2012, 9:859-868. 34. Flegg MB, R¨udiger S, Erban R: Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release. J Chem Phys 2013, 138:154103. 35. Lawson MJ, Drawert B, Khammash M, Petzold L, Yi TM: Spatial stochastic dynamics enable robust cell polarization. PLoS Comput Biol 2013, 9:e1003139. The authors present a model for cell polarization during the yeast mating response. By using the RDME, the authors are able to elucidate the role of stochasticity in the sharpness of the response. 36. Ditlev JA, Michalski PJ, Huber G, Rivera GM, Mohler WA, Loew LM, Mayer BJ: Stoichiometry of Nck dependent actin polymerization in living cells. J Cell Biol 2012, 197:643-658. 37. Layton AT, Savage NS, Howell AS, Carroll SY, Drubin DG, Lew DJ: Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment. Curr Biol 2011, 21:184-194. 38. Chou CS, Moore TI, Chang SD, Nie Q, Yi TM: Signaling regulated endocytosis and exocytosis lead to mating pheromone concentration dependent morphologies in yeast. FEBS Lett 2012, 586:4208-4214. 39. Slaughter BD, Unruh JR, Das A, Smith SE, Rubinstein B, Li R: Non-uniform membrane diffusion enables steady-state cell polarization via vesicular trafficking. Nat Commun 2013, 4:1380. 40. 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:e29645. A model for vesicle trafficking is presented that accounts for individual vesicles. The authors study the effect of various cytoskeletal layouts on overall vesicle statistics.
26. Klann M, Koeppl H: Reaction schemes, escape times and geminate recombinations in particle-based spatial simulations of biochemical reactions. Phys Biol 2013, 10:046005. A thorough explanation of theoretical basis for particle-based reaction– diffusion simulation methods. The authors then present new alternative approaches with improved performance characteristics.
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Current Opinion in Structural Biology 2014, 25:86–91
