1. Kim “Avrama” Blackwell George Mason University Modeling Calcium Concentration

2. Importance of Calcium Calcium influences channel behaviour, and thereby spike dynamics Short term influences on calcium dependent potassium channels Long term influences such as potentiation and depression via kinases Electrical activity influences calcium concentration via I Ca Phosphorylation influences calcium concentration via kinetics of calcium permeable channels

3. Feedback Loops of Calcium Dynamics Calcium Ca 2+ Kinases SK, BK channels Membrane Potential ++++++++++ __________ Potassium, Sodium channels Synaptic channels, Calcium channels Fast Slow

4. Control of Calcium Dynamics

5. Calcium Sources – Calcium Currents Multiple types of voltage dependence calcium channels (L, N, P, Q, R, T) Calcium permeable synaptic channels (NMDA) – Release from Intracellular Stores (smooth endoplasmic reticulum) IP 3 Receptor Channel (IP 3 R) Ryanodine Receptor Channel (RyR)

6. Control of Calcium Dynamics Calcium Sinks – Pumps Smooth Endoplasmic Calcium ATPase (SERCA) Plasma Membrane Calcium ATPase (PMCA) Sodium-Calcium exchanger Source or Sink – Buffers - bind calcium when concentration is high, releases calcium as concentration decreases Calmodulin – active Calbindin - inactive – Diffusion – moves calcium from high concentration to low concentration regions

7. Calcium Currents L type (CaL1.x) – High threshold, Long lasting, no voltage dependent inactivation (except for CaL1.3) T type (CaL3.x) – Low threshold, Transient, prominent voltage dependent inactivation

8. Calcium Currents N type (Ca2.x) High threshold, moderate voltage dependent inactivation (Neither long lasting nor transient) P/Q type (Cal2.x) P type found in cerebellar Purkinje cells Properties similar to CaL R type (Cal2.x) Used to be “Residual” current Now subunit identified

10. Influx due to calcium current is calculated by: F is Faraday’s constant Software dependent negative sign: inward current is negative (physiologists convention) or positive (modeler’s convention) Flux has units of moles per unit time, converted to concentration using rxnpool, Ca_concen, diffshell, or pool object Calcium Current

11. Calcium Release through Receptor Channels

12. Calcium Release Calcium Release Receptor Channels are modeled as multi-state molecules – One state is the conducting state – For IP 3 receptor state transitions depend on calcium concentration and IP 3 concentration – For Ryanodine receptor, state transitions depend on calcium concentration

13. Dynamics of Release Channels Both IP 3 R and RyR have two calcium binding sites: – Fast binding to one site, causes channel opening – Slower binding to other site, causes slow channel closing

14. IP 3 Receptor Similar to RyR but with additional binding site for IP 3 8 state model of DeYoung and Keizer, 1992 Figure from Li and Rinzel, 1994

15. Calcium Release Release through the channel is proportional to concentration difference between ER and cytosol Release depends on fraction of channels in open state Ф R = P X n ([Ca 2+ ] ER – [Ca 2+ ]) P is permeability X is fraction of channels in open state n is number of independent subunits

16. Dynamics of Release Channels Dynamics similar to sodium channel Activation: – IP 3 plus calcium produces channel opening – Channel opening increases calcium concentration – Higher concentration causes more channels to open – Positive feed back produces calcium spike

17. Dynamics of Release Channels Inactivation – High calcium causes channels to close (inactivate) – Slow negative feedback SERCA pumps calcium back into ER – Analagous to repolarization – Calcium concentration returns to basal level

18. Li and Rinzel Calcium Release

20. Calcium Extrusion Mechanisms Plasma Membrane Calcium ATPase (PMCA) pump and sodium calcium exchanger (NCX) are the primary mechanism for re- equilibrating calcium in spines and thin dendrites (Scheuss et al. 2006) These mechanisms depress with high activity or calcium concentration – Decay of calcium transient is slower – Positive feedback elevates calcium in small compartments

21. Calcium ATPase Pumps Plasma membrane (PMCA) – Extrudes calcium to extracellular space – Binds one calcium ion for each ATP – Affinity ~300 -600 nM Smooth Endoplasmic Reticulum (SERCA) – Sequesters calcium in SER – Binds two calcium ions for each ATP – Affinity ~100 nM

22. Pump Equations Michaelis-Menten formulation Used for SERCA or PMCA pumps Implements the equation: K cat is the maximal pump capacity n is the Hill coefficient: number of calcium molecules bound K M is the affinity (Half maximal concentration)

24. Sodium Calcium Exchange (NCX) Stoichiometry – 3 sodium exchanged for 1 calcium Charge transfer – Unequal => electrogenic – One proton flows in for each transport cycle – Small current produces small depolarization Theoretical capacity ~50x greater than PMCA

25. Sodium Calcium Exchange (NCX) Depolarization may reverse pump direction Ion concentration change may reverse direction Increase in Na int or decrease in Na ext Increase in internal sodium may explain activity dependent depression Increase in Ca ext or decrease in Ca int

26. Other formulations in Campbell et al. 1988 J Physiol., DiFrancesco and Noble 1985 Philos Trans R Soc Lond B, Weber et al. 2001 J Gen Physiol

27. Calcium Buffers Calmodulin is a major calcium binding protein – Binds 4 calcium ions per molecule – High affinity for target enzymes Calcium-Calmodulin Dependent Protein Kinase (CaMKII, CaMKIV) Phosphodiesterase (PDE) Adenylyl Cyclase (AC) Protein Phosphatase 2B (PP2B = calcineurin) – K D1 = 1.5 uM, K D2 = 10 uM, – Recent estimates in Faas, Raghavachari, Lisman, Mody (2011) Nat Neurosci.

28. Calcium Buffers Calbindin – Binds 4 calcium ions per molecule – Not physiologically active – 40  M in CA1 pyramidal neurons (Muller et al. 2006) – Diffusion coefficient = 20 m 2 /s – K D = 700 nM, k on = 2.7 x10 7 /M-sec Parvalbumin – In fast spiking interneurons

29. Buffers Effect of buffers modeled using bimolecular reactions: Ca + Buf Ca.Buf Diffusible buffers require additional diffusion equations Cannot use conserve equations with diffusible molecules KfKf KbKb

30. Diffusion Calcium decay in spines exhibits fast and slow components (Majewska et al. 2000) – Fast component due to Buffered diffusion of calcium from spine to dendrite, which depends on spine neck geometry Pumps, which are independent of spine neck geometry – Slow component matches dendritic calcium decay Solely controlled by calcium extrusion mechanisms in the dendrite

31. Radial and Axial Diffusion Methods in Neuronal Modeling, Koch and Segev Chapter 6 by DeSchutter and Smolen

32. Derivation of Diffusion Equation Diffusion in a cylinder – Derive equation by looking at fluxes in and out of a slice of width  x Boundary Value Problems, Powers

33. Derivation of Diffusion Equation Flux into left side of slice is q(x,t) Flux out of right side is q(x+  x,t) – Fluxes may be negative if flow is in direction opposite to arrows Area for diffusional flux is A Boundary Value Problems, Powers

39. Control of Calcium Dynamics

40. For information on implementing calcium dynamics in Neuron, see: http://www.neuron.yale.edu/neuron/static/docs/rxd/index.html

41. Calcium Objects Ca_concen (genesis), CaConc (moose)  Simplest implementation of calcium  Calcium current input converted to ion influx B = 1 / (z F vol): volume to produce 'reasonable' calcium concentration  Calcium decays to minimum with single time constant moose.doc(CaConc) showobject Ca_concen

42. Calcium Objects with Diffusion Difshell (genesis) and DifShell (Moose)  concentration shell. Has ionic current flow, one-dimensional diffusion, first order buffering and pumps, store influx  Calculates volume and surface areas from diameter (dia), thick (length) and shape_mode (either slab or shell)  Combines rxnpool, reaction and diffusion into one object, thus must define kb, kf, diffusion constant  To store buffer concentrations, use fixbuffer  Non-diffusible buffer (use with difshell) difbuffer  Diffusible buffer (use with difshell)

43. Chemesis Calcium Objects Calcium and calcium buffers implemented using  rxnpool, which can take current influx as input  conservepool  Reaction  mmpump (genesis and chemesis) – Diffusion (chemesis) Uses geometry and concentration of two adjacent rxnpools to calculate flux between the rxnpools

44. Morphology of Model Cell

45. Calcium Dynamics in Model Cell Ca 2+

46. Calcium Buffer Demo CalTut.txt explains all tutorials step-by-step Cal1-SI.g Creates pools of buffer, calcium and calcium bound buffer Creates bimolecular reaction for buffering

47. Calcium Buffers and Diffusion Cal2-SI.g Two compartments: soma and dendrite Calcium binding to buffer is implemented in function Diffusion between soma and dendrite Cal2difshell.g Same system, using difshell and difbuffer Computationally more efficient

48. Chemesis Release CICR implements calcium release states using Markov kinetic channel formalism States Forward rate constants

49. Calcium Release Objects CICR implements calcium release states using Markov kinetic channel formalism  Create one element for each state, R xx Parameters (Fields)  'Forward' rate constants,       State vector, e.g. 001 for 1 Ca ++, 0 IP 3 bound  Calculates fraction of receptors in state Inputs: calcium, IP 3, other states

50. Calcium Release Objects CICRFLUX implements calcium release Messages (inputs) required: Calcium concentration of ER Calcium concentration of Cytosol Fraction of channels in open state, X Parameters (Fields) Permeability, P Number of independent subunits, q Calculates Ca flux = P*X q (Ca ER -Ca Cyt )

51. Calcium Release Demo Cal3-SI.g Illustrates how to set up calcium release using cicr object Requires ER compartment with calcium and buffers Calcium concentration increases, and then stays elevated due to lack of pumps

52. Calcium Pump Objects mmpump2 used for SERCA or PMCA Pump Parameters (fields) Affinity (half conc) Hill exponent (power) maximum rate (max_rate) Messages (inputs) Concentration Calculates flux due to pump dC/dt = max_rate*Ca^pow/(Ca^pow+half_conc^pow) Different than the mmpump in genesis Genesis mmpump has no hill coefficient

53. Calcium Release and SERCA Cal4.g Implements IICR from Cal4.g Adds SERCA pump to remove calcium from cytosol

54. Voltage Dependent Calcium Channels Cal7.g, Cal7difshell.g – Two concentration compartments, but no calcium release channels – Requires two voltage compartments – Uses the Goldman-Hodgkin-Katz formulation for driving potential – Depolarizes the cell with current injection to activate calcium channel Cal8.g – Investigate effect of mesh size on diffusion