1. Kim “Avrama” Blackwell George Mason University Modelling 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 T type (CaL3.x) – Low threshold, Transient, prominent voltage dependent inactivation Vm

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

10. Flux has units of moles per unit time, converted to concentration using rxnpool, Ca_concen, diffshell, or pool object

11. Calcium Release through Receptor Channels

12. Calcium Release Calcium Release Receptor Channels are modelled 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: – Binding to one site is fast, causes fast channel opening – Binding to other site is slower, causes slow channel closing IP 3 R has an additional binding site for IP 3

14. IP 3 Receptor 8 state model of DeYoung and Keizer, 1992 Figure from Li and Rinzel, 1994

17. Dynamics of Release Channels Dynamics similar to sodium channel: – IP 3 with low calcium produces small channel opening – Channel opening increases calcium concentration – Higher concentration causes larger channel opening – Positive feed back produces calcium spike

18. Dynamics of Release Channels High calcium causes slower channel closing – Slow negative feedback – Channel inactivates – Inactivation analogous to sodium channel inactivation SERCA pumps calcium back into ER – Calcium concentration returns to basal level

21. 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

22. 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

25. 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

26. 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

27. 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

28. 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.

29. 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

31. 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

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

33. 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

34. 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

40. Control of Calcium Dynamics

42. Genesis Calcium Objects Ca_concen  Simplest implementation of calcium  Fields Time constant of decay Minimum calcium B = 1 / (z F vol): volume to produce 'reasonable' calcium concentration  Inputs Calcium current

43. Genesis Calcium Objects Code of all the following is in src/concen Concpool  Calcium concentration without diffusion  Fields: Shape and size  Inputs: Buffer rate constants, bound and free MMpump coefficients Influx and outflux of stores

44. Genesis Calcium Objects difshell  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)

45. Chemesis Calcium Objects Calcium buffers implemented using  rxnpool  conservepool  Reaction  Kinetikit:  Pools  reac

46. Morphology of Model Cell

47. Calcium Dynamics in Model Cell Ca 2+

48. Calcium Buffers 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

49. Chemesis Calcium Objects Diffusion  Parameters (Fields) Diffusion constant, D Units: 1 for SI, 1e-3 for mMole, etc. Dunits: 1 for meters, 1e-3 for mm, etc.  Messages (Inputs) Length, concentration, surface area from two reaction pools  Calculates Flux from one pool to another D SA Conc / len

50. 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

51. Chemesis Calcium Objects CICR implements calcium release states using Markov kinetic channel formalism States Forward rate constants

52. Chemesis Calcium Objects CICR implements calcium release states using Markov kinetic channel formalism  One element for each state, R xx  One of the elements may be conserved Parameters (Fields)  'Forward' rate constants,       State vector, e.g. 001 for 1 Ca ++ and 0 IP 3 bound  Fraction of receptors in this state (calculated)  Whether this element is conserved

53. Chemesis Calcium Objects CICR (cont.) Messages (Inputs) required: IP 3 concentration Cytosolic Ca ++ concentration fraction of molecules in states that can transition to this state rate constant governing transition from other states to this state Calculates Fraction of molecules in the state

54. Chemesis Calcium 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 Units: 1 for moles, 1e-3 for mmoles, etc Number of independent subunits, q Calculates Ca flux = P*X q (Ca ER -Ca Cyt )

55. Calcium Release Cal3.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

56. Chemesis Calcium objects MMPUMP2 used for SERCA or PMCA Pump  Fields Affinity (half _conc) Power (exponent) Maximum rate Units (1 for moles, 1e-3 for mmoles, etc)  Messages (inputs) Concentration  Calculates flux due to pump  Different than the mmpump in genesis  Genesis mmpump has no hill coefficient

57. Chemesis Calcium objects NCX (not in any tutorial)  Fields Affinity (kmhill), and hill coefficient (hill) Stoichiometry (ratio of sodium to calcium) Vunits (1 for volts, 1e-3 for mv) Gbar (maximal conductance) Gamma (partition coefficient) T (temp) Messages (inputs) Concentration of Na, Ca, both inside and outside Vm  Calculates current due to pump

58. Chemesis Calcium Objects Leak implemented using CICRFLUX Messages (inputs) required:  Calcium of cytosol  Calcium of ER or EC space  Value of 1.0 instead of open state Parameters (Fields)  Maximal Permeability (P L )  Hill coefficient (should be 1.0)

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

60. Integrating Calcium Mechanisms RXNPOOL takes flux messages from various calcium sources  VDCC sends message CURRENT, with fields current and charge  Diffusion and calcium release send message RXN2MOLES or RXN2, with fields difflux1 and difflux2, or fluxconc1 and fluxconc2, respectively  Mmpump sends message RXN0MOLES with field moles_out (to cytosol) or moles_in

61. Voltage Dependent Calcium Channels Cal7.g, Cal8.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