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Spatial Stochastic Simulators

Spatial Stochastic Simulators. Kim “Avrama” Blackwell George Mason University Krasnow Institute of Advanced Studies. Diverse Numbers of Molecules Spatially Inhomogeneous. G protein coupled receptors Diffusion required for signal interaction. Glutamate receptors 1  M  60 molecules

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Spatial Stochastic Simulators

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  1. Spatial Stochastic Simulators Kim “Avrama” Blackwell George Mason University Krasnow Institute of Advanced Studies

  2. Diverse Numbers of MoleculesSpatially Inhomogeneous G protein coupled receptors Diffusion required for signal interaction • Glutamate receptors • 1 M  60 molecules • molecular interactions occur stochastically Small number of molecules in spines Large number of molecules in system Kotaleski and Blackwell 2010

  3. Spatial Stochastic Simulators • Particle based • Smoldyn, MCell, CDS • Individual molecules are represented as point-based particles, which diffuse random distance and random direction at each time step • If two reacting molecules pass near each other they may react • Computations increase with number of molecules Diffusion Association Dissociation sb Membrane

  4. q q q MCell Transparent • Geometry from volumetric imaging data using Blender (www.blender.org) • Mesh elements may be reflective, transparent, or absorptive • Surface or volume diffusion • Ray tracing determines whether molecules would have collided during (fixed) time step • Reaction rules depend on order of reaction, and whether surface or volume molecules involved (Kerr et al. SIAM J Sci Comput 2008) Reflective

  5. MCell Diffusion • Diffusion distance from probability density: • Radial distance from uniformly distributed random variable X: • Speed computations by storing values of X in look-up table • Direction: uniformly distributed random variable [0,2π)

  6. Mcell – STDP example • Calmodulin activation versus spike timing • Do NMDA receptors and VDCC produce different calmodulin profiles? • Neuron model to determine voltage-dependent open probability of VDCC and NMDA • MCell model with calmodulin, calbindin, NCX and PMCA • Model: Keller et al. PLoS One 2008, tutorial: http://www.mcell.org/tutorials/

  7. MCell Model NMDAR Pre-synaptic Terminal Spine Head Spine Neck Dendrite VDCC Calcium binding Proteins (cytosol) Pumps (Membrane)

  8. UnpairedStimuli • Calcium differs due to channel distribution Keller et al. PLoS One 2008

  9. Paired Stimuli EPSP-AP AP-EPSP • Calcium depends on timing of AP versus glutamate release Keller et al. PLoS One 2008

  10. CDS • Particle based simulator with event driven algorithm • All possible collisions are detected during short dt • If collision detected, the exact collision time is calculated • Earliest collision (or reaction events) are simulated one-by-one until dt • Particles have volume, thus can simulate crowding and volume exclusion • http://nba.uth.tmc.edu/cds/content/download.htm

  11. CDS Example • Morphology from triangular meshes • CaMKII diffusion out of spine depends on morphology (b) and also binding targets and F-actin • Byrne et al. J Comput Neuro 2011

  12. Stochastic (non-spatial) Simulators • Gillespie (Exact Stochastic Simulation Algorithm) Propensity of reaction aj Kf Np • Propensity of any reaction, a0 =  aj • Next reaction occurs with exponential distribution with mean a0: • Identity of reaction selected randomly, based on propensity • Computations increase with number of molecules

  13. ad2 a2 + ad1 ad4 a1 + ad3 Extensions to Gillespie Algorithms • Spatial Gillespie, e.g. Fange et al. 2010, PNAS • Morphology is subdivided into small compartments • Propensity of diffusion calculated from diffusion coefficient, ad D Nd • Diffusion considered as another reaction • Tau leap – non-spatial • Allow multiple reaction events, Kj, to occur for each reaction at each time step, t, according to Poisson:

  14. Hybrid Models • Partition the reaction-diffusion space into two or more sets of reactions (and diffusion) • Each set is simulated differently • Diffusion – deterministic, reactions – stochastic • Fast reactions - deterministic, slow reactions – stochastic • “Critical” reactions - exact stochastic, non-critical reactions – tau leap

  15. STEPS • Spatial extension of exact stochastic simulation algorithm • Tetrahedral meshes allows realistic geometries • Diffusion constant can vary between compartments • Simulations are specified in python, witih morphology, reactions and simulations specified independently (for ideal control of simulation experiments) • http://steps.sourceforge.net/STEPS/Home.html

  16. STEPS-Cerebellar LTD Calcium Buffers Calcium Pumps AMPA Receptor calcium Protein Phosphatase 5 PKC Raf Raf-act Protein Phosphatase 2A Arachidonic Acid Positive Feedback Loop MEK MapKinase Phosphatase 1 ERK cPLA2 Protein Phosphatase 1 Inactivation, dephosphorylation Activation, phosphorylation

  17. Single Spine Model • Average of multiple simulations reveals graded induction of LTD • Single runs reveals bistability at intermediate calcium Time (min) Antunes et al. J Neurosci 2012

  18. Model Limitations • All these model have either small volume (single spine) or small number of reactions (calmodulin+CaMKII) • Only MCell model uses voltage to determine calcium influx • Smoldyn • Particle simulation algorithm incorporated into Moose (Genesis 3) and VCell • No neuroscience examples yet

  19. NeuroRD Spatial extension to Gillespie tau leap Multiple reaction events and diffusion events can occur during each time step Morphology is subdivided into small compartments • Cuboidal meshes and cylindrical meshes possible

  20. NeuroRD – Mesoscopic • Subdivide dendrites and spines into sub-volumes • Pre-calculate the probability that one molecule leaves the compartment or reacts • Look-up tables store the probability that j out of N molecules leave a compartment or react • At each time step, for each molecule, choose a random number to determine the number, j, molecules out of N leaving or reacting

  21. NeuroRD

  22. Calculate number of molecules Calculate j reacting or k moving using Poisson distribution

  23. Determine destinations for diffusing molecules

  24. NeuroRD - Validation • An approximation, to allow large scale simulations • Agrees with Smoldyn, and deterministic solution for reaction-diffusion system Molecule A Molecule B Oliveira et al. 2010, PLoS One

  25. NeuroRD • NeuroRD is up to 60 times faster than Smoldyn • Computations increase linearly with number of compartments, but not molecules Oliveira et al. 2010, PLoS One

  26. NeuroRD DevelopmentBiochemical Oscillator Srivastava et al., J Chem Phys

  27. Spatial Gene Oscillator • mRNA is inactive in the nucleus, diffuses into cytosol • A diffuses to nucleus, binds to DNA • Effect of diffusion constant (2 cytosol compartment)

  28. Spatial Biochemical Oscillator Inactive mRNA in nucleus, activated by binding in cytosol compartment Vary number of compartments, and translation compartment Protein quantity mRNA production is faster when A binds to DNA mRNA production and degradation are faster for A than R Protein synthesis and degradation are faster for A than R R degrades A (at same catalytic rate that A spontaneously degrades)

  29. Spatial Biochemical Oscillator mRNA DNA

  30. NeuroRD Model specification allows good experimental design, with separate files for Reactions Spatial morphology Initial conditions Stimulation Output specification Top level file which specifies reactions, morphology, initial conditions, output specs, time step and spatial grid, random seed Tissue Experiment Simulation control

  31. NeuroRD – Morphology File • Specify start and end of each segment • Specification includes id, region type, location (x,y,z), radius, and optional label <Segment id="seg1" region="dendrite"> <start x="1.0" y="1.0" z="0.0" r="0.5" /> <end x="1.0" y="2.0" z="0.0" r="0.5" label="pointA"/> </Segment> • Additional segments start on a previous segment • Branching is possible – see branching.tar

  32. NeuroRD – Reaction File • Define each species that has either a reaction pool or conservepool • Include diffusion constant, which can be 0 <Specie name="mGluR" id="mGluR" kdiff="0" kdiffunit = "mu2/s"/> • Specify Reactions • First order – single reactant and product • Second order – two reactants or two products

  33. NeuroRD – Reaction File • Include forward and backward rate constants <Reaction name = "glu+mGluR--glu-mGluRreac" id="glu+mGluR--glumGluR_id"> <Reactant specieID="glu" /> <Reactant specieID="mGluR" /> <Product specieID="glu-mGluR" /> <forwardRate> 5e-03 </forwardRate> <reverseRate> 50e-03 </reverseRate> <Q10> 0.2 </Q10> </Reaction>

  34. NeuroRD – Initial Condition File • Four types of initial conditions • General concentration of molecule in entire morphology, or • Region specific concentration • Overrides general concentration • Surface Density of membrane molecules • Overrides concentration specifications • Surface Density of Membrane molecules in specific region • Overrides general surface density

  35. NeuroRD – Initial Condition File • General concentration of each molecule should be specified (zero otherwise) <NanoMolarity specieID="mGluR" value="5e3" /> • Surface density if molecule is membrane bound <PicoSD specieID="PLC" value="2.5" /> • Initial conditions for different parts of morphology <ConcentrationSet region="PSD" > followed by <NanoMolarity specieID=“IP3" value=“30" />

  36. NeuroRD – Stimulation File • Stimulation used to inject molecules • Temporary fix until software is integrated with software for simulating neuron electrical activity and ion channels • Specify molecule and injection site <InjectionStimspecieID="Ca" injectionSite="pointA"> • Repetitive trains can be created • Specify onset time, duration, rate (amplitude) • period and end used for train • InterTrain Interval to repeat train (e.g. For LTP)

  37. NeuroRD – Output Specification • Specify dt for output, species and compartment <OutputSet filename = "dt1" region="dendrite" dt="1.0"> <OutputSpecie name="glu" /> <OutputSpecie name="IP3" /> </OutputSet> • Multiple outputSets can be specified • Sample slowly changing molecules less frequently • Sample glutamate receptors from PSD only

  38. NeuroRD – Model file • Specify all the other files <reactionSchemeFile>Purkreactions</reactionSchemeFile> <morphologyFile>Purkmorph</morphologyFile> <stimulationFile>Purkstim</stimulationFile> <initialConditionsFile>Purkic</initialConditionsFile> <outputSchemeFile>Purkio</outputSchemeFile> • Specify some other parameters, such as algorithm variations and random seed • Indicate total simulation time, time step and largest compartment size

  39. NeuroRD – running simulation • Java -jar stochdiff.jar Purkmodel.xml • Morphology output file • Purkmodel.out-mesh.txt • Ascii output file • name of model file -- .out –output set name - conc.txt • Purkmodel.out-dt1-conc.txt

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