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

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

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 bb AssociationDissociation Diffusion Membrane

4. MCell 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)   Transparent 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 [0,2 π)

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

8. Unpaired Stimuli Calcium differs due to channel distribution Keller et al. PLoS One 2008

9. Paired Stimuli Calcium depends on timing of AP versus glutamate release EPSP-APAP-EPSP 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 a j  K f  N p –Propensity of any reaction, a 0 =  a j –Next reaction occurs with exponential distribution with mean a 0 : –Identity of reaction selected randomly, based on propensity –Computations increase with number of molecules

13. Extensions to Gillespie Algorithms Tau leap – non-spatial –Allow multiple reaction events, K j, to occur for each reaction at each time step, , according to Poisson: Spatial Gillespie, e.g. Fange et al. 2010, PNAS –Morphology is subdivided into small compartments –Propensity of diffusion calculated from diffusion coefficient, a d  D  N d –Diffusion considered as another reaction a1a1 a2a2 a d1 a d3 a d2 a d4 + +

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 PKC Arachidonic Acid cPLA 2 Protein Phosphatase 2A ERK MEK Raf Raf-act MapKinase Phosphatase 1 Protein Phosphatase 1 Protein Phosphatase 5 AMPA Receptor Calcium Buffers Calcium Pumps Inactivation, dephosphorylation Activation, phosphorylation Positive Feedback Loop

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 Oliveira et al. 2010, PLoS One Molecule A Molecule B

25. NeuroRD NeuroRD is up to 60 times faster than Smoldyn Computations increase linearly with number of compartments, but not molecules NeuroRDSmoldyn Simulation# initial molecules # injectedTime (h:mm:ss) Memory (kb)Time (h:mm:ss) Memory (kb) Diffusion020000:00:02.8616080:00:07.042344 Reaction2885300:00:05.9717640:08:03.5326524 Reaction & Diffusion I 66240000:00:04.5117640:02:48.9022168 Reaction & Diffusion II 6619400000:00:07.5817722:19:58.0023760 Oliveira et al. 2010, PLoS One

26. NeuroRD Development Biochemical 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 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) Protein quantity

29. Spatial Biochemical Oscillator DNA mRNA

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 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 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 5e-03 50e-03 0.2

34. NeuroRD – Initial Condition File Four types of initial conditions 1.General concentration of molecule in entire morphology, or 2.Region specific concentration Overrides general concentration 3.Surface Density of membrane molecules Overrides concentration specifications 4.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) Surface density if molecule is membrane bound Initial conditions for different parts of morphology followed by

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 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 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 Purkreactions Purkmorph Purkstim Purkic Purkio 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