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1. so that their activity can be modulated through allosteric interactions For example a binding site can be put in its active shape when a speci c modi cation site is phosphorylated and in its inactive shape when that modi cation site is unphosphorylated Enter network generation rules in your con guration le in a block of statements that starts with start_rules and ends with end_rules Within this block enter rules in the following categories 1 Under Modi cations declare all possible post translational modi cations For example name PP might rep resent double phosphorylation 2 Under Molecules de ne the mols their binding sites and their modi cation sites For example Fus3 ToSte5 ToSte12 inactive active PhosSite None P PP states that Fus3 is the name of a mol ToSte5 and ToSte12 are binding sites the ToSte12 532 S S Andrews binding site can adopt either an inactive or an active shape and PhosSite is a modi cation site that can have zero one or two phosphate groups Mols also need to be de ned as Smoldyn species Subheading 3 2 3 Under Explicit Species assign your own names to speci c multimeric complexes if desired 4 Under Explicit Species Class assign names to complexes that share speci c characteristics For example the class Fus3 ToSte12 1 Ste12 ToFus3 1 includes all molecular spe cies that include a Fus3 bou2. brate your model in one simulation save the result with savesim and then run other simulations that start from this equilibrated state All observation commands require that you explicitly declare the output le names with output_files and also with output_root if you do not want the les to be in your working directory The output le stdout sends the output to the terminal window To save data to a series of les number the 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 527 rst one with output_file_number and progress to subsequent ones with the incrementfile command Finally manipulation commands give you wide latitude to modify the system during the simulation and not necessarily in accordance with biochemical principles For example you can instruct the virtual experimenter to add molecules to the system with pointsource or volumesource remove molecules from inside spheres with killmolinsphere or replace a speci ed fraction of one molecular species in some volume with a different species using replacevolmol Also several commands allow you to create xed concentration inputs to your model such as fixmolcountonsurf and fixmolcountincmpt The set command is extremely powerful because you can follow it with essentially any Smoldyn statement This allows you to add species reactions or surfaces change the simulation time step or modify your model in many other ways all mid simula
3. complete installation instructions a user s manual a programmer s manual example les and several utility programs The Smoldyn User s Manual is much more thorough than this chapter and should be consulted for statement syntax available options and all other details Smol dyn is open source and runs on Macintosh Linux and Windows computers however the Windows version does not support rule based modeling described in Subheading 3 10 Faster computers run simulations faster of course although laptops or desktops up to about 5 years old are often adequate Smoldyn runs from a shell prompt which most operating systems supply with a Terminal or Command Prompt appli cation Installation on Mac or Linux computers is typically as easy as changing to the Smoldyn download directory and typing configure make and sudo make install at a shell prompt For Windows the download includes a prebuilt Smoldyn executable and all necessary dll les Smoldyn support is available at the on line forum http www smoldyn org forum or via e mail at support smoldyn org 3 Methods 3 1 Running Smoldyn Run Smoldyn by typing smoldyn myfile txt at a shell prompt where myfile txt is your con guration le name You can also append option ags to this command such as o to suppress text Fig 2 A model of carboxysome function in a cyanobacterium The wireframe outside surface depicts the cell membrane internally
4. then specify how molecules should interact with the boundaries The options are re ection transmission absorption or periodic periodic means that molecules that diffuse out of one side of the system are immediately diffused into the opposite side of the system which can avoid edge effects in effectively in nite systems On the other hand if your model includes any surfaces then the boundaries de ned here will only transmit molecules and you will have to use surfaces to con ne molecules Specify the simulation starting time stopping time and time step with time_start time_stop and time_step Smoldyn only supports xed length time steps in contrast to the adaptive time steps that MCell supports 12 or the event based time steps that the Green s Function Reaction Dynamics method uses 25 However you can change time steps midsimulation with the settimestep command Subheading 3 6 See Note 2 for advice on choosing time steps 3 4 Molecules Each Smoldyn molecule is a member of a chemical species and all molecules of a particular species are equivalent Enter individual species names with species statements Distinct forms of a molecule such as phosphorylated and unphosphory lated or monomeric and dimeric need to be de ned as separate species To simplify the de nitions for these multiple molecular forms Smoldyn can automatically enumerate both them and their chemical reactions using rule based modeling Subhe
5. Smoldyn represents it as a smooth continuous surface The four large green spheres are carboxysomes which are organelle like proteinaceous compartments that contain the carbon xation apparatus of cyanobacteria Smaller red dots are carbon dioxide molecules and larger light blue dots are 3 phosphoglycerate molecules Savage and coworkers are using this model to investigate the roles of compartmentalization and spatial organization in carbon xation 522 S S Andrews output p to just display parameters or t for text only opera tion Upon starting Smoldyn reads model parameters from your con guration le calculates and displays simulation parameters and runs the simulation As the simulation runs Smoldyn displays the simulated system to a graphics window and or saves quanti tative data to one or more output les Smoldyn stops when the simulation is complete Box 1 presents an example con guration le and its output 3 2 The Con guration File Smoldyn con guration les are plain text les The rst word of each line tells Smoldyn which parameters to set and the rest of the line lists those parameters separated by spaces or tabs Usually the statement sequence does not matter When it does it is usually obvious for example molecular species need to be de ned before their diffusion coef cients Smoldyn displays error messages and terminates if it cannot parse the con guration le Denote comments in y
6. is the partition coef cient for the molecule into the membrane from the neighboring solution e g cytoplasm Dmem is the molecule s diffusion coef cient in the membrane and dmem is the membrane thickness Qualitatively this equation shows that hydrophobic molecules are transmitted faster than hydrophilic ones because they partition into the membrane more readily and that smaller molecules are transmitted faster than larger ones because they diffuse faster A paper by Paula et al 47 shows how to compute the necessary parameters It also lists some experimentally determined coef cients for trans mission through lipid bilayers including 3 5 108 mm s for potassium ions 0 014 mm s for urea 0 027 mm s for glycerol 5 mm s for protons and 150 mm s for water molecules these values are for 2 7 nm thick bilayers which is about the thickness of typical biological membranes Adsorption and transmission coef cients can also be esti mated using the respective characteristic times see Note 2 if these times can be inferred from experiments 6 Choosing reaction rate constants Experimental bimolecular reaction rates are limited to kmax 4p DA DB rAs rB by the rate at which reactants can diffuse together 48 Here kmax is the diffusion limited reaction rate constant DA and DB are the diffusion coef cients of the reactants and rA and rB are the reactant radii Highly reactive small molecules sometimes
7. reactions in the yeast phero mone response system A few representative rates from this wiki are pheromone binds Ste2 receptors at 1 8 105 M1 s1 3 1 104 mm3 s and unbinds at 103 s1 the Gpa1 G protein subunit exchanges GDP for GTP at 6 17 104 s1 and the Fus3 MAP kinase binds the Ste5 scaffold protein at 2 3 106 M1 s1 3 8 103 mm3 s and unbinds at 2 3 s1 While these are reasonably typical values for these types of interactions other similar reactions are often faster or slower by several orders of magnitude Note that bimolecular reactions that take place in one and two dimensionalsystems suchasalong lamentsor withinmem branes do not have reaction rate constants in the same sense as reactions in three dimensions Also current versions of Smoldyn cannot quantitatively simulate low dimensional reactions 7 Lowering bimolecular reaction accuracy for faster simulations Smoldyn partitions space into virtual boxes see Note 1 to reduce the number of potential molecule molecule and mole cule surface interactions that need checking at each time step Most molecule molecule interactions occur within single boxes in typical simulations However Smoldyn also has to check for interactions between molecules that are in adjacent boxes to achieve high accuracy There are typically about 50 times more potential interbox than intrabox interactions because 1 in three dimensional systems each box has 26 neighboring boxes or e
8. Chapter 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator Steven S Andrews Abstract This chapter describes how to use Smoldyn which is a computer program for modeling cellular systems with spatial and stochastic detail Smoldyn represents each molecule of interest as an individual point like particle These simulated molecules diffuse interact with surfaces e g biological membranes and undergo chemical reactions much as they would in real biochemical systems Smoldyn has been used to model signal transduction within bacterial cells pheromone signaling between yeast cells bacterial carboxysome function diffusion in crowded spaces and many other systems A new rule based model ing feature automatically generates chemical species and reactions as they arise in simulations due to protein modi cations and complexation Smoldyn is easy to use quantitatively accurate and computa tionally ef cient It is generally best for systems with length scales between nanometers and several microns time scales from tens of nanoseconds to tens of minutes and up to about 105 individual molecules Smoldyn runs on Macintosh Linux or Windows systems is open source and can be down loaded from http www smoldyn org Key words Computational biology Smoldyn Particle based simulation Spatial modeling Rule based modeling 1 Introduction Computational modeling is becoming an important cell biology research metho
9. M Rugman PA Birmingham J Garland PB 1986 Lateral 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 541 diffusion of proteins in the periplasm of Escher ichia coli J Bacteriol 165 787 794 44 Crane JM Verkman AS 2008 Long range nonanomalous diffusion of quantum dot labeled aquaporin 1 water channels in the cell plasma membrane Biophys J 94 702 713 45 Chou T D Orsogna MR 2007 Multistage adsorption of diffusing macromolecules and viruses J Chem Phys 127 105101 46 Huang KC Meir Y Wingreen NS 2003 Dynamic structures in Escherichia coli sponta neous formation of MinE rings and MinD polar zones Proc Natl Acad Sci U S A 100 12724 12728 47 Paula S Volkov AG van Hoek AN Haines TH Deamer DW 1996 Permeation of pro tons potassium ions and small polar mole cules through phospholipid bilayers as a function of membrane thickness Biophys J 70 339 348 48 von Smoluchowski M 1917 Versuch einer mathematischen Theorie der Koagulationski netik kolloider Lo sungen Z Phys Chem 92 129 168 49 Cohen B Huppert D Agmon N 2000 Non exponential Smoluchowski dynamics in fast acid base reaction J Am Chem Soc 122 9838 9839 50 Le Nove re N Bornstein B Broicher A Cour tot M Donizelli M Dharuri H Li L Sauro H Schilstra M Shapiro B Snoep JL Hucka M 2006 BioModels Database a free cen tralized database of curated published quan titative kinetic model
10. ad ing 3 10 Each molecule regardless of its species may be in any of ve states Solution state in which molecules are not bound to surfaces is the default Molecules can also bind surfaces in a front back up or down state The former two options represent molecules that bind a single side of a surface where typical uses include peripheral membrane proteins GPI anchored proteins or adsorbates The other two options up and down represent surface spanning molecules such as ion channels or trans membrane receptors The separate up and down states allow one to distinguish whether the protein s active side faces the surface s front or back 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 525 De ne the diffusion coef cient for each species and state with difc See Note 3 for advice on choosing diffusion coef cients Smoldyn can also diffuse molecules anisotropically difm state ment such as for con ning molecules to a plane or for simulating diffusion in anisotropic environments In addition molecular drift in a speci ed direction drift statement can represent molecules in ow systems or around motile cells For example you could simulate gel electrophoresis with a combination of diffusion and drift Internally Smoldyn keeps track of molecules with a dead list of inactive molecules and several live lists of simulated molecules See Note 4 for a
11. ally indistinguishable from those of the underlying model system Smoldyn achieves relatively high computational ef ciency through judicious approximations for simulating chemical reactions 22 and molecule surface interactions 23 and through data structures that reduce unnecessary computations Fig 1 A model of the effect of the Bar1 protease on yeast signaling The mesh surface on the outside of the system is a triangulated spherical boundary that absorbs molecules with Smoldyn s unbounded emitter method The central sphere is a receiver cell which is covered with 6 622 receptors red if bound to pheromone and blue if not in on line version light gray spheres are challenger cells which secrete pheromone slowly and the dark gray sphere on the right is a target cell which secretes pheromone quickly Green light gray in print version dots are Bar1 proteins and black dots are pheromone molecules This model showed that the pheromone degrading Bar1 prote ase improves yeast mating partner discrimination by sharpening the pheromone con centration gradient Republished with permission from ref 8 The supplementary information for the original publication presents the model parameter selection process unusually thoroughly 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 521 2 Materials Smoldyn is available at http www smoldyn org The distribu tion includes the program source code
12. and you should also give the probability of gemi nate recombinations However these details are almost never known so I suggest using product_placement name pgemmax 0 2 where name is the reaction name This states that reaction rates were measured at equilibrium typically correct and that reaction products should have up to a 20 probability of recombin ing with each other This geminate recombination probability typically leads to ef cient simulations and physically sensible simulation parameters the binding and unbinding radii 18 22 Smoldyn uses the same parameters if you do not enter a product_placement statement but issues a warning 530 S S Andrews Smoldyn also uses the bimolecular reaction formalism for con formational spread and excluded volume interactions Use confor mational spread interactions to model mechanical coupling between stationary molecules set the maximum interaction distance with confspread_radius The prototypical conformational spread example occurs in the E coli agellar motor in which motor pro teins in active conformations induce activity in their neighbors and those with inactive conformations induce inactivity in their neighbors 26 this positive feedback makes the motor switch like Excluded volume interactions product_placement statement keep molecules spatially separate They are particularly useful for keeping molecules that are con ned to a channel from passi
13. biochemical networks in time and space J Chem Phys 123 234910 26 Duke TAJ LeNove re N Bray D 2001 Con formational spread in a ring of proteins a stochastic approach to allostery J Mol Biol 308 541 553 27 Slepchenko B Schaff J Macara I Loew LM 2003 Quantitative cell biology with the Vir tual Cell TRENDS Cell Biol 13 570 576 28 Ray S Deshpande R Dudani N Bhalla US 2008 A general biological simulator the Multiscale Object Oriented Simulation Environment MOOSE BMC Neurosci 9 Suppl 1 P93 29 Blinov ML Faeder JR Goldstein B Hlavacek WS 2004 BioNetGen software for rule based modeling of signal transduction based on the interactions of molecular domains Bioinformatics 20 3289 3291 30 Lok L Brent R 2005 Automatic generation of cellular reaction networks with Molecular izer 1 0 Nat Biotech 23 131 136 31 DePristo MA Chang L Vale RD Khan SM Lipkow K 2009 Introducing simulated cel lular architecture to the quantitative analysis of uorescent microscopy Prog Biophys Mol Biol 100 25 32 32 Dobrzynski M Rodr guez JV Kaandorp JA Blom JG 2007 Computational methods for diffusion in uenced biochemical reactions Bioinformatics 23 1967 1977 33 Palm MM Steijaert MN ten Eikelder HMM Hilbers PAJ 2009 Modeling molecule exchange at membranes In Proceedings of the Third International Conference on the Founda tions of Systems Biology in Engineering Den
14. d Modeling is useful for quantitatively testing hypotheses about systems of interest helping analyze and inter pret experimental data and developing a deeper understanding of how biochemical systems function In these and other roles simu lations have helped elucidate cellular systems such as bacterial che motaxis 1 the eukaryotic cell cycle 2 and actin based cellular protrusion 3 Active simulation algorithm research and frequent increases in computer power suggest that cellular modeling will Jacques van Helden et al eds Bacterial Molecular Networks Methods and Protocols Methods in Molecular Biology vol 804 DOI 10 1007 978 1 61779 361 5_26 Springer Science Business Media LLC 2012 519 continue gaining importance for many years to come Recent articles that review simulation methods and software include 4 7 This chapter describes how to use the Smoldyn simulation program which simulates biochemical processes with spatial and stochastic detail 8 Smoldyn represents each molecule of interest as an individual point like particle that has a location in continuous space Simulated molecules diffuse undergo chemical reactions and interact with membranes according to simple bio physical principles and user speci ed parameters Smoldyn is typically best for models with spatial scales from nanometers to microns temporal scales from tens of nanoseconds to tens of minutes and up to about 105 individual molecules The
15. d are often larger than physical molecular radii 22 If a molecular species is so concentrated that these binding radii frequently overlap each other which is especially problematic for clustered surface bound molecules e g the receptors in Fig 1 8 this decreases accuracy 4 For fast bimolecular reactions that are far from steady state Smoldyn s simulated results more closely agree with diffusion limited kinetics when short time steps are used and with activation limited kinetics when long time steps are used 22 This is unlikely to affect typical biochemical network models but may affect reaction biophysics models For all of these time step considerations note that Smoldyn displays all internal simulation parameters several characteristic times and any warnings for potential problems before it starts simulations It is prudent to check that this information seems reasonable In practice many researchers use time steps around 0 1 ms 16 17 19 20 31 32 Smoldyn has also been used with time steps as short as 0 06 ns in a test of its reaction algorithms 22 and with time steps up to 10 or 20 ms 8 33 The latter models one of which is shown in Fig 1 were able to use long time steps because they had relatively low molecular densities and large system geometries 3 Choosing diffusion coef cients In homogeneous solutions such as water or liquid growth media diffusion coef cients can often be approximated reas
16. dvice on arranging the live lists The following statements add molecules to the starting state of your model mol adds solution phase molecules to either speci c or random locations surface_mol enter after you de ne surfaces adds surface bound molecules to speci c surfaces or regions on surfaces and compartment_mol enter after you de ne compartments adds solution phase molecules to speci c spatial compartments 3 5 Graphics Smoldyn simulation results can only be visualized as they are computed While this can be inconvenient for generating publication quality gures the immediate output is very helpful for model development You can choose from three levels of rendering quality or no graphical output at all using the graphics statement Because graphical rendering is time con suming you can speed simulations up by using low quality ren dering or by using graphic_iter to only render graphics periodically Conversely you can slow simulations down with graphic_delay The default background color is white which you can modify with background_color For everything else the default color is black Color the boundary edges with frame_color spatial partitions with grid_color molecules with color and surfaces with surface name color You can also set boundary line weights with frame_thickness spatial partition line weights with grid_thickness molecule sizes with display_size surface edge weights with thickness and sur
17. e con g uration le for Fig 2 and Roger Brent for encouragement and helpful discussions I also appreciate helpful comments and sug gestions from many Smoldyn users including Karen Lipkow and Shahid Khan in particular This work was supported by MITRE contract number 79729 awarded to Roger Brent References 1 Tindall MJ Porter SL Maini PK Gaglia G Armitage JP 2008 Overview of mathemati cal approaches used to model bacterial chemo taxis I the single cell Bull Math Biol 70 1525 1569 2 Tyson JJ Novak B 2008 Temporal organi zation of the cell cycle Curr Biol 18 R759 R768 3 Mogilner A 2006 On the edge modeling protrusion Curr Opin Cell Biol 18 32 39 4 Alves R Antunes F Salvador A 2006 Tools for kinetic modeling of biochemical networks Nat Biotechnol 24 667 672 5 Takahashi K Arjunan SNV Tomita M 2005 Space in systems biology of signaling pathways towards intracellular molecular crowding in silico FEBS Lett 579 1783 1788 6 Andrews SS Arkin AP 2006 Simulating cell biology Curr Biol 16 R523 R527 7 Andrews SS Dinh T Arkin AP 2009 Stochastic Models of Biological Processes In Encyclopedia of Complexity and System Sci ence Vol 9 Edited by Meyers RA Springer New York 8730 8749 8 Andrews SS Addy NJ Brent R Arkin AP 2010 Detailed simulation of cell biology with Smoldyn 2 1 PLoS Comp Biol 6 e1000705 9 Dayel MJ Hom EFY Verkman AS 1999 Dif
18. er the PhosSite is doubly phosphorylated Smoldyn s rule based modeling support is still very new so we are continuing to add functionality and improve the con gu ration le syntax 4 Notes 1 Partitioning space Smoldyn subdivides the system within its boundaries into a grid of uniformly spaced virtual boxes These boxes do not affect the simulated results Instead they make simulations more ef cient by reducing the number of potential molecule molecule and molecule surface interactions that Smoldyn needs to check at each time step The default partition spacing which yields an average of about four mole cules per box when the simulation starts is often good but can 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 533 usually be improved upon Improvement is especially important if the model starting state is not representative of its typical state or if molecules are not distributed homogenously Optimize the partition spacing by timing simulations Smol dyn displays the run time when it terminates that use a range of box sizes which you can set with molperbox or boxsize and choose the fastest size for which Smoldyn does not report any errors The errors to watch for are those that report that bimo lecular reaction binding radii or analogous distances are larger than the box widths When they arise they indicate that Smol dyn will not detect some bimolecular reactions which means tha
19. face drawing styles with polygon With Smoldyn s best rendering quality called opengl_better you can place light sources in the system including their ambient diffuse and specular colors using light To make full use of these lights set surface re ection parameters with shininess As always see the Smoldyn User s Manual for details Unfortunately Smoldyn renders partially transparent surfaces poorly This means that if you want to see 526 S S Andrews what s inside a surface such as a cell membrane it is usually best to render it with a wireframe You can manipulate Smoldyn s graphical display during your simulation using key strokes For example rotate the system with arrow keys pan with shift arrows zoom in with zoom out with or revert to the default view with 0 Also the space bar pauses the simulation T saves a TIFF image of the current graphics display and Q quits the simulation Smoldyn can save movies of simulations as stacks of TIFF les Again each image is a simple copy of the graphics window display To turn on recording set the number of simulation time steps per image with tiff_iter name the les with tiff_name and number the les with tiff_min and tiff_max ImageJ Quick Time Pro and other software programs can assemble TIFF stacks into self contained movies 3 6 Runtime Commands Smoldyn includes a virtual experimenter who can observe or mani
20. fusion of green uorescent protein in the aqueous phase lumen of endoplasmic reticu lum Biophys J 76 2843 2851 10 Partikian A O lveczky B Swaminathan R Li Y Verkman AS 1998 Rapid diffusion of green uorescent protein in the mitochondrial matrix J Cell Biol 140 821 829 11 Plimpton SJ Slepoy A 2005 Microbial cell modeling via reacting diffusive particles J Phys Conf Ser 16 305 309 12 Kerr RA Bartol TM Kaminsky B Dittrich M Chang J CJ Baden SB Sejnowski TJ Stiles JR 2008 Fast Monte Carlo simulation methods for biological reaction diffusion sys tems in solution and on surfaces SIAM J Sci Comput 30 3126 3149 13 Jin S Haggie PM Verkman AS 2007 Single particle tracking of membrane protein diffusion in a potential simulation detection and application to con ned diffusion of CFTR Cl channels Biophys J 93 1079 1088 14 Deich J Judd EM McAdams HH Moerner WE 2004 Visualization of the movement of single histidine kinase molecules in live Caulo bacter cells Proc Natl Acad Sci U S A 101 15921 15926 15 Coggan JS Bartol TM Esquenazi E Stiles JR Lamont S Martone ME Berg DK Ellisman MH Sejnowski TJ 2005 Evidence for ectopic neurotransmission at a neuronal synapse Science 309 446 451 540 S S Andrews 16 Grati Mh Schneider ME Lipkow K Strehler EE Wenthold RJ Kachar B 2006 Rapid turnover of stereocilia membrane proteins evidence from the t
21. g 3 4 mentions Smol dyn stores molecules in several lists By default Smoldyn stores all simulated molecules that do not diffuse in a xed list and 536 S S Andrews those that do diffuse in a diffusing list This scheme is more ef cient than a single list because Smoldyn does not need to diffuse or perform surface interactions for molecules in the xed list also Smoldyn does not need to check for bimolecular reactions between pairs of molecules in the xed list Adding more lists can speed simulations up further often by factors of ve or more typically by minimizing the number of potential bimolecular reactions that Smoldyn needs to check For example consider the chemical reaction A B C If Smoldyn stored all three species in the same list then when Smoldyn searched the list to see which AB molecule pairs could react it would also encounter many nonreactive AA AC BB BC and CC molecule pairs each of which would take a small amount of time to check and then ignore On the other hand Smoldyn would only encounter AB pairs if A B and C were stored in three separate lists Generalizing this example more molecule lists reduce the number of unnecessary checks so typically produce faster simulations This trend does not con tinue inde nitely though because each molecule list also requires some processing time this becomes important for lists with few molecules Thus overall simulation performance is
22. hat start with surface and the surface name but the block format is usually easier Within this block Within this block list each individual panel with panel list each individual panel with panel If you want surface bound molecules to be able to diffuse between neighbor ing panels whether they are on the same surface or different surfaces then list each panel s neighbors with neighbor Two utility programs included in the distribution help generate panel and neighbor data The rst wrl2smol reads Virtual Reality Modeling Language VRML les of triangle mesh data and 528 S S Andrews converts them to lists of triangle panels and their neighbors in Smoldyn format Mathematica MatLab ImageJ and other pro grams can generate VRML les The second utility program SmolCrowd generates random arrays of nonoverlapping circles or spheres which can be useful for modeling macromolecular crowding Several parameters characterize each surface These include graphical display parameters described in Subheading 3 5 and surface molecule interaction parameters Many types of interac tions are possible For example you can make a surface imperme able transmitting or irreversibly absorbing for molecules that diffuse into it and these interactions can differ for different molecular species and the two surface faces Specify these beha viors with action Alternatively specify coef cients for adsorp tion to desorption from or
23. he model parameters that you wish to explore to the top of your con guration le with de ne statements and then refer to them later on by their token names see Box 1 3 3 Space and Time Specify whether your system is 1 2 or 3 dimensional with dim For example you could model protein motion along cytoskeletal laments in one dimension processes on cell membranes 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 523 Box 1 A sample con guration le and its output The sections of the con guration le match those in Methods The snapshot was saved from Smoldyn s graphical output at 2 seconds The dynamics were graphed with Microsoft Excel from data that Smoldyn saved to box1out txt Below S is substrate E is enzyme ES is enzyme substrate complex and P is product 524 S S Andrews with two dimensions or cellular systems with three dimensions Low dimensional models can also approximate higher dimen sional ones De ne the system s spatial extent with boundaries This does not limit the simulated space but provided that you keep all molecules within these boundaries helps Smoldyn run ef ciently Internally Smoldyn uses the boundaries to spatially parti tion the system volume so as to reduce the number of molecule molecule and molecule surface interactions that it has to investigate see Note 1 for partition optimization If your model does not include surfaces Subheading 3 7
24. ight neighbors in 2 D or two neighbors in 1 D and 2 there are n2 possible pairwise interactions for n molecules in each of two neighboring boxes but only about n2 2 possible pairwise interactions for n molecules within a box The result is that checking for interbox interactions is computationally costly rela tive to the few interactions that are actually detected Using the accuracy statement you may be able to speed Smoldyn up by instructing it to ignore some or all potential interbox interactions Of course this will cause Smoldyn to overlook some reactions that should occur so bimolecular reactions will simulate somewhat too slowly If desired this problem can be mitigated by determining reaction rate correc tion factors from trial simulations alternatively correction factors can be estimated from the total numbers of intrabox 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 539 and interbox reactions which Smoldyn reports upon termina tion In practice the simulation speed improvement is typi cally much less than the maximum possible factor of 50 and is often just a factor of 2 or less Thus the best approach is to try lowering Smoldyn s accuracy parameter and directly assessing how much the simulation speed improved and how much the actual accuracy decreased Acknowledgments I thank Nathan Addy for developing the Smoldyn build system and the libmoleculizer module David Savage for providing th
25. lly Smoldyn s slowest component Reversible association reactions of the form A B C involve an additional complication When a C molecule dissociates the new A and B molecules form in close proximity to each other which makes them especially likely to react with each other This reaction between dissociation products is called a geminate recom bination Nonspatial treatments of chemical kinetics including con ventional mass action theory and nonspatial simulations cannot account for geminate recombinations so they are often ignored However geminate recombinations affect reaction rates both in real systems and in particle based simulations For example the association reaction rate for A B C is faster if it is measured at equilibrium than if it is measured with all C removed as it is formed e g with the additional reaction C D E and a high concen tration of D because the former situation includes geminate recombinations and the latter does not So that Smoldyn can accu rately simulate the kinetics of reversible reactions and other reac tions where the products can react with each other you need to specify how Smoldyn should place dissociation products in the system with product_placement Ideally you should choose the placement method that represents whether the experimental rate constants were measured at equilibrium or with products removed as they were formed see Table 3 8 2 of the Smoldyn User s Manual
26. lower ends of these ranges arise from the fact that Smoldyn does not represent excluded volumes orientations or momenta of simulated molecules These limit Smoldyn s spatial resolution to a few molecular radii and temporal resolution to molecules rota tional diffusion time constants 9 10 The upper ends of the ranges arise from the fact that Smoldyn is too computationally intensive to simulate much larger systems conveniently on current desktop computers however recent work on parallelizing Smoldyn for graphics processing units is starting to enable larger simulations Smoldyn s level of simulation detail is ideally suited for mod eling intracellular organization More speci cally the so called particle based simulation method that Smoldyn uses along with ChemCell 11 and MCell 12 has proven useful for modeling single molecule tracking experiments 13 14 molecular diffusion in restricted environments 15 17 stochastic signaling noise that arises from spatial organization 6 8 18 and the causes and effects of protein localization 19 20 As some examples Lipkow and coworkers developed several Smoldyn simulations of intracellular signal transduction for Escher ichia coli chemotaxis Each model included about ten different proteins and about ten chemical reactions Using these simula tions they found that intracellular crowding accentuates signaling differences to different agellar motors 17 that the CheZ pho
27. mm2 s for lactose at 15C 38 For quick estimates the Stokes Einstein equation is usually correct to within a factor of two In eukaryotic cytoplasms nuclei and mitochondria experi ments by Verkman s group show that macromolecule diffusion coef cients are about 25 of their values in water for masses up to about 500 kDa 39 40 Larger molecules and lamen tous molecules such as DNA diffuse more slowly likely due to sieving by actin networks In the E coli bacterial cytoplasm GFP diffuses about another factor of 5 slower than it does in eukaryotes now with a diffusion coef cient around 3 8 mm2 s 41 42 Diffusion is much slower yet in the E coli periplasm where the 42 5 kDa maltose binding protein has a diffusion coef cient of about 0 009 mm2 s 43 Finally a couple of membrane bound protein diffusion coef cients are 0 09 mm2 s for aquaporin 1 about 30 kDa in nonpolarized broblast cell membranes 44 and 0 012 mm2 s for histidine kinase PleC 117 kDa including uorophore in Caulobacter membranes 14 From these data I suggest the rules of thumb use 80 of the Stokes Einstein value as calculated above for proteins in water and divide the aqueous diffusion coef cient by 4 for eukaryotic cell and organelle cytoplasms by 15 for bacterial cytoplasms by 1 000 for bacterial periplasms by 1 000 for eukaryotic membranes and by 4 000 for bacterial membranes 4 Optimizing molecule lists As Subheadin
28. mpartment surface and listing an interior de ning point near the center of the cell This compartment would behave as one would expect where any molecule within the bacterium including molecules in carboxysomes would be in the cell compartment and any molecule outside of the bacterial membrane would not be in the cell compartment We could de ne another compartment that included all of the carboxysome interiors called carboxysome by setting the compartment surface to the carboxysome surface which is disjoint and listing an interior de ning point at the center of each carboxysome The other way to de ne a compartment is with logical combina tions of previously de ned compartments using the compartment 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 531 statement Continuing with the previous examples we could de ne the bacterial cytoplasm in Fig 2 that is not within a carboxysome as a compartment called cytoplasm with compartment equal cell and then compartment andnot carboxysome Overall this compartment de nition method is somewhat nonintuitive but it is easy to use versatile and computationally ef cient 3 10 Rule Based Reaction Network Expansion Most proteins can adopt any of a very large number of different states such as by phosphorylation nucleotide binding multiple conformations or complexation with other proteins In Smoldyn as in mos
29. nd to a Ste12 using Fus3 s ToSte12 site and Ste12 s ToFus3 site this class leaves other modi cation states and other binding site occupancies unspeci ed Use species classes to set the graphical display of newly generated species using the Smoldyn statements i e not entered in the libmoleculizer rule block species_class_display_ size and species_class_color 5 Under Association Reactions de ne binding reactions between mols For example Fus3 ToSte12 active Ste12 ToFus3 gt Fus3 ToSte12 1 Ste12 ToFus3 1 de nes a possible binding between Fus3 and Ste12 and states that this association can only happen when Fus3 s ToSte12 binding site is in its active shape 6 Under Transformation Reactions list reactions in which mol modi cation states change such as between phosphorylated and unphosphorylated states For example Fus3 PhosSite None gt Fus3 PhosSite P is a reaction in which Fus3 is spontaneously phosphorylated 7 and 8 Under Alloste ric Complexes and Allosteric Omniplexes specify how binding site shapes should depend on the binding and modi cation states of a mol or complex Allosteric complexes are for speci c species and allosteric omniplexes are for classes of species An example of the latter is Fus3 ToSte12 active lt PhosSite PP which states that Fus3 s ToSte12 binding site should be in its active shape whenev
30. ng each other They also permit more general simulations of molecules with excluded volume although this aspect of Smoldyn is computation ally inef cient 3 9 Spatial Compartments Smoldyn compartments are regions of volume bounded by surfaces A cell s cytoplasm nucleus or extracellular space are typical exam ples Compartments are useful for specifying initial conditions of models for recording simulation results for de ning reactions that are only relevant in speci c regions and for compatibility with compartment based simulators e g Virtual Cell 27 and MOOSE 28 To use compartments de ne individual compart ments with blocks of statements bracketed by start_compart ment and end_compartment You can de ne a compartment with either of two methods or with a combination of them In the rst list the compartment s bounding surfaces the same surfaces that Subheading 3 7 describes with surface statements and list one or more inside de ning points with point statements These points de ne the compartment s volume as follows a molecule is de ned to be in a compartment if and only if a straight line can be drawn between it and an inside de ning point without crossing one of the compartment s bounding surfaces For example we could de ne a compartment for the model shown in Fig 2 that included the entire bacterial volume which we ll call cell by making the cell membrane the co
31. onably well with the Stokes Einstein equa tion It is D kBT 6pr where D is the diffusion coef cient kB is Boltzmann s constant T is the absolute temperature is the solution viscosity and r is the radius of the diffusing particle which is assumed to be spherical This particle radius can be easily calculated from its mass m and density r r 3m 4pr 3 s 0 0655 m 3p nm 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 535 In the latter equality m is measured in Daltons also it assumes an average density of 1 41 g cm which is reasonably accurate for proteins above 40 kDa and within 10 for smaller proteins 34 Combining this radius calculation with the Stokes Einstein equation yields the protein diffusion coef cient estimate D 3270 m 3p mm2 s This is for a temperature of 20C where water s viscosity is very nearly 1 mPa s Two examples are for lysozyme m is 14 6 kDa the experimental D is 111 mm2 s 35 and the computed D is 134 mm2 s and for green uorescent protein GFP m is 26 9 kDa the experimental D is 87 mm2 s 36 37 and the computed D is 109 mm2 s Of course theories that account for protein shape yield better predictions 35 For small molecules in water experimentally measured diffu sion coef cients are readily available Examples include 2 010 mm2 s for oxygen at 20C and 380
32. our con guration le with a symbol or bracket multiple comment lines with and Ideally your comments should list the model name its author or authors its creation and modi cation dates citations to relevant references and the model s distribution terms if so your model will obey the minimum information requested in the annotation of biochemical models MIRIAM 24 which aids model shar ing and reuse Your comments should also list the model units such as microns and seconds or nanometers and microseconds This is because Smoldyn does not assume any particular units but instead requires that all parameters use the same units For example if you choose microns and seconds then you will need to enter diffusion coef cients in mm2 s rst order reaction rate constants in s1 and second order reaction rate constants in mm3 s See Table 3 2 1 of the Smoldyn User s Manual Use read_file to separate your model description into multiple con guration les This can be useful for separating lists of surface or molecule information from the main model le or for separating xed model components from ones you wish to vary End each con guration le with end_file Several macro statements that substitute text tokens with replacement text can simplify model development These include define define_global undefine ifdefine ifunde fine else and endif In a typical use you would collect t
33. partial transmission through surfaces 23 with rate see Note 5 for advice on these coef cients Two molecular pseudo states fsoln and bsoln allow you to distinguish solution state molecules on the front side of a surface from solution state molecules on the back side of a surface You can also specify rates for transitions between surface bound states such as from the front state to the back state Additionally you can instruct molecules to change species when they cross a surface this is particularly useful for models with spatially variable diffu sion coef cients such as between extracellular and intracellular regions or nonraft and raft regions of membranes Jump surfaces de ned with action and jump magically transport molecules from one panel to another They are useful for adding holes to otherwise impermeable surfaces such as pores in a nuclear envelope and for creating periodic boundaries Finally unbounded emitter surfaces unbounded_emitter statement irreversibly absorb molecules with coef cients that make internal concentration pro les mimic those that would be seen for an unbounded system 23 The triangulated mesh in Fig 1 is an unbounded emitter surface 3 8 Chemical Reactions Smoldyn supports zeroth rst and second order chemical reac tions where the order is just the number of reactants Use zeroth order reactions to add molecules at random locations within the entire system vol
34. pulate the simulated system You can instruct this virtual exper imenter to perform tasks before during or after the simulation including at periodic intervals see Table 3 6 1 of the Smoldyn User s Manual Issue commands using the cmd statement the timing parameters the name of the speci c task and any task speci c parameters The rst class of commands are simulation control com mands Examples are stop and pause which stop or pause the simulation Also keypress mimics the behavior of the user pressing a key see Subheading 3 5 which can be useful for automating the graphical display These control commands are particularly useful when combined with conditional commands For example the statement cmd E ifno substrate stop tells the virtual experimenter to count the number of substrate mole cules at every time step and to stop the simulation if none remain The second class observation commands save information about the system to text les For example molcount records the number of molecules for each species and molcountspace records histograms of molecule counts along a line in the system from which concentration pro les can be calculated A particularly powerful observation command is savesim which saves the entire simulation state to a Smoldyn readable con guration le By saving the simulation state regularly you can abort a simulation mid run and then restart it later on Alternatively you can equili
35. raf cking and mobility of plasma membrane Ca2 ATPase 2 J Neurosci 26 6386 6395 17 Lipkow K Andrews SS Bray D 2005 Simulated diffusion of CheYp through the cytoplasm of E coli J Bact 187 45 53 18 Andrews SS 2005 Serial rebinding of ligands to clustered receptors as exempli ed by bacterial chemotaxis Phys Biol 2 111 122 19 Lipkow K 2006 Changing cellular location of CheZ predicted by molecular simulations PLoS Comp Biol 2 e39 20 Lipkow K Odde DJ 2008 Model for pro tein concentration gradients in the cytoplasm Cell Mol Bioeng 1 84 92 21 Jackson CL Hartwell LH 1990 Courtship in S cerevisiae both cell types choose mating partners by responding to the strongest pher omone signal Cell 63 1039 1051 22 Andrews SS Bray D 2004 Stochastic simu lation of chemical reactions with spatial reso lution and single molecule detail Phys Biol 1 137 151 23 Andrews SS 2009 Accurate particle based simulation of adsorption desorption and par tial transmission Phys Biol 6 46015 24 Le Nove re N Finney A Hucka M Bhalla US Campagne F Julio C V Crampin EJ Halstead M Klipp E Mendes P Nielsen P Sauro H Shapiro B Snoep JL Spence HD Wanner BL 2005 Minimum information requested in the annotation of biochemical models MIR IAM Nat Biotechnol 23 1509 1515 25 van Zon JS ten Wolde PR 2005 Green s function reaction dynamics a particle based approach for simulating
36. react with rates that are close to this maximum 49 but typical biochemical reaction rate constants are much slower You can verify that your simulated rates will be well below this maxi mum by comparing Smoldyn s output parameter labeled binding radius if dt were 0 to the sum of the physical reactant radii They will be equal if your simulated reaction is at the maximum rate and proportionately less for slower reactions Other than this one qualitative check reaction rates usually have to found from the scienti c literature that is relevant to your model Typically some reaction rates will have been measured others can be calculated from published data e g association reaction rates can be calculated from dissociation constants and dissociation reaction rates others can be estimated from characteristic reaction times and published gures and yet others simply have to be guessed Previously developed 538 S S Andrews models whether of your system or similar systems are often a good source of reaction rates and literature references For this the BioModels database http www ebi ac uk biomodels main 50 is a particularly good place to start Finally curated literature summaries that list quantitative data are rarely available but can be excellent resources when they are For example the Yeast Pheromone Model wiki http yeastpheromonemodel org wiki Main_page lists reaction rates and their references for most
37. s phatase is likely to change its intracellular location upon cellular stimulation 19 and that stable concentration gradients are likely to arise for the CheY signal transducer 20 In another example Fig 1 several colleagues and I used Smoldyn to help understand why haploid yeast cells that secrete the pheromone degrading pro tease Bar1 are better able to discriminate between potential mating partners than yeast cells that do not secrete Bar1 21 We found that Bar1 sharpens the local pheromone concentration gradient because pheromone is progressively inactivated as it diffuses through a Bar1 cloud 8 This model included four proteins six chemical reactions and about 2 105 individual molecules it also covered about 7 000 mm3 of volume simulated about 75 min of time and took 520 S S Andrews about 10 h to run As a nal example Savage is using Smoldyn to investigate carbon transport and xation in cyanobacteria which takes place in organelle like carboxysomes Fig 2 His model includes several partially transmitting surfaces Within the eld of particle based simulation Smoldyn is nota ble for its highaccuracy and good computational ef ciency 8 Each Smoldyn algorithm is either exact or nearly exact for any length simulation time step and complete Smoldyn simulations approach exactness as time steps are reduced toward zero 18 22 23 by de nition exact simulation methods produce results that are theo retic
38. s of biochemical and cellular systems Nucleic Acids Res 34 D689 D691 542 S S Andrews
39. t simulators each of these states needs to be modeled as a distinct species with distinct chemical reactions Manually listing these species and their reactions is often impractical though so modelers typically simplify their models to only include the most essential states while ignoring the rest The drawbacks of this approach are of course that it can overlook important biological interactions and it precludes the study of realistic reaction networks Thus several groups have pursued a different approach in which the simulator generates the species and chemical reactions automatically from interaction rules 29 30 Smoldyn supports this so called rule based modeling with a module called libmoleculizer A notable libmoleculizer feature is that it only generates species and their reactions as they arise this means that Smoldyn does not need to keep track of multimeric complexes that could theoretically exist but that never actually form This speeds simulations reduces com puter memory use and simpli es models In the libmoleculizer formalism chemical species are assem bled from building blocks called mols Both individual mols and multimeric complexes of mols are chemical species Each mol can have modi cation sites such as for phosphorylation methylation or nucleotide binding Also each mol can have binding sites with which it can reversibly bind other mols to form multimeric complexes These binding sites have shapes
40. t those reactions will simulate too slowly and simulation accuracy will be reduced see Note 7 Other Smoldyn warnings about box sizes such as those that comment on unusually many or few molecules per box are simply suggestions that different partition spacings may improve performance 2 Choosing simulation time steps When choosing the time step for a simulation there is an unavoidable trade off between using shorter time steps to get more accurate results and using longer time steps for faster simulations Thus the succinct answer to the question of what time step to use is that you should choose the longest time step that yields suf ciently accurate results You can nd it by running trial simulations with a wide range of time steps and graphing representative simulation results e g the maximum amount of a product concentration time until a substrate concentration is halved steady state receptor occu pancy etc against the log of the time step Typically this plot will show results that are independent of time step lengths for short time steps that errors increase log linearly for long time steps and that these regions are separated by a cross over region that is about a factor of 10 in width A judgment call is required at this point to decide how much accuracy if any you are willing to sacri ce for faster simulations In addition to this heuristic method it is worth considering how the time step length affects individ
41. tion 3 7 Surfaces Unlike biological membranes or other real surfaces Smoldyn sur faces are in nitely thin This means that molecules in solution phase or in front or back surface bound states are always on the front or back side of a surface also molecules in up or down states are always exactly at the surface Each Smoldyn surface is composed of geometric primitives called panels which can be triangular spher ical cylindrical or other shapes Along with being convenient and computationally ef cient this representation method is also quite versatile because it allows arbitrarily complex surface geometries disjoint surfaces e g a collection of vesicles and or open surfaces e g the fragmented membrane of a lysed cell For example the model shown in Fig 1 includes four sur faces 1 a mesh of 480 triangles that surrounds the entire system 2 a spherical receiver cell in the middle 3 a spherical tar get cell shown toward the right side in dark gray and 4 ve spherical challenger cells shown in light gray Note that this last surface is disjoint The model shown in Fig 2 includes two surfaces 1 the cell membrane assembled from two hemispheres and a cylinder and 2 four spherical carboxysome protein shells De ne each surface with a block of statements that starts with start surface and ends with end_surface you can also enter surface parameters with statements t
42. ual simulation algo rithms In particular 1 Smoldyn displaces each diffusing molecule by about s 2DDt 1 2 where D is the molecule s diffusion coef cient and Dt is the time step at each time step The average displacement for the fastest diffusing species smax is the simulation s spatial resolution In general smax should be signi cantly smaller than important geometrical features sur face curvature radii and distances between xed molecules 2 Characteristic transition times for unimolecular reactions molecular desorption and transitions between surface bound states are all t 1 k where k is the appropriate rate constant Also characteristic times for bimolecular reactions with well mixed reactants are t A B k A B where k is the 534 S S Andrews reaction rate constant and A and B are the reactant concentrations And characteristic times for molecular adsorp tion and permeability are t d k where d is the distance between surfaces e g the cell length and k is the adsorption or transmission coef cient To simulate these dynamics accu rately the time step should be smaller than these characteristic times 3 Smoldyn performs bimolecular reactions between pairs of reactant molecules that are closer than their binding radius Binding radii which Smoldyn computes from reaction rates diffusion coef cients and the simulation time step increase as time steps are made longer an
43. ume or within a compartment Subheading 3 9 at a roughly constant rate These violate mass balance and are thus unphysical however they are particularly useful for including protein or mRNA synthesis in models but not the respective synthesis machinery Use rst order or unimolecular reactions for spontaneous molecular changes such as protein conforma tion dissociation and decay processes These are also useful for pseudo rst order reactions in which one of two reactants e g ATP is unmodeled and assumed to be constant and uniformly distributed Use second order or bimolecular 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 529 reactions for interactions between pairs of molecules such as association and enzymatic reactions Enter reactions with the syntax reaction name reac tants gt products rate where name is the reaction name reactants and products are lists of species and states that are separated by symbols use 0 if a list is empty and use the fsoln and bsoln pseudo states as necessary and rate is the reaction rate constant Alternatively use reaction_surface or reaction_cmpt for reactions that should only occur on speci c surfaces or in speci c compartments Smoldyn allows up to 16 products for each reaction Note 6 offers advice on choosing reaction rate constants and Note 7 describes ways to speed up bimolecular reaction simulation which is typica
44. usually best when each abundant species has its own list and when sparsely populated species share lists Further list optimization by considering the potential bimolecular and molecule surface interactions that Smoldyn has to check at each time step can produce additional gains Create molecule lists with molecule_lists and assign species to them with mol_list 5 Choosing molecule surface interaction coef cients Adsorption coef cients are limited to about the thermal velocity of the adsorbent which is 45 kmax kBT m r 1 56 109 m p mm s The latter equality assumes that m is measured in Daltons and the temperature is 20C For example the largest possible adsorp tion coef cient for a 50 kDa protein is about 7 106 mm s Real adsorption coef cients are likely to be vastly smaller than this maximum For example Huang and coworkers chose an adsorp tion coef cient of 0 025 mm s for the adsorption of MinD 29 5 kDa to the inside of the E coli cell membrane in a biochemical model 46 Unfortunately remarkably few experi mental papers present quantitative data on protein adsorption to or desorption from lipid bilayers despite an extensive literature 26 Spatial and Stochastic Cellular Modeling with the Smoldyn Simulator 537 Transmission coef cients for molecules through membranes can be calculated using the equation 47 k KmemDmem dmem where Kmem
45. ver Colorado 34 Fischer H Polikarpov I Craievich AF 2004 Average protein density is a molecular weight dependent function Protein Sci 13 2825 2828 35 Brune D Kim S 1993 Predicting protein diffusion coef cients Proc Natl Acad Sci U S A 90 3835 3839 36 Swaminathan R Hoang CP Verkman AS 1997 Photobleaching recovery and aniso tropic decay of green uorescent protein GFP S65T in solution and cells cytoplasmic viscosity probed by green uorescent protein translational and rotational diffusion Biophys J 72 1900 1907 37 Brown EB Wu ES Zipfel W Webb WW 1999 Measurement of molecular diffusion in solution by multiphoton uorescence photobleaching recovery Biophys J 77 2837 2849 38 Lide DR Ed 2004 CRC Handbook of Chemistry and Physics CRC Press Boca Raton FL 39 Verkman AS 2002 Solute and macromole cule diffusion in cellular aqueous compart ments Trends Biochem Sci 27 27 33 40 Dix JA Verkman AS 2008 Crowding effects on diffusion in solutions and cells Annu Rev Biophys 37 247 263 41 Elowitz MB Surette MG Wolf P E Stock JB Leibler S 1999 Protein mobility in the cyto plasm of Escherichia coli J Bacteriol 181 197 203 42 van den Bogaart G Hermans N Karasnikov V Poolman B 2007 Protein mobility and dif fusive barriers in Escherichia coli consequences of osmotic stress Mol Microbiol 64 858 871 43 Brass JM Higgins CF Foley
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