Kim “Avrama” Blackwell

George Mason University

Modeling Calcium Concentration

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 ICa

• Phosphorylation influences calcium concentration via kinetics of calcium permeable channels

Feedback Loops of Calcium DynamicsCalcium

Ca2+

Kinases

SK, BK

channelsMembrane

Potential

+

+

+

+

+

_

_

_

_

_

Potassium,

Sodium channels

Synaptic channels,

Calcium channels

Fast

Slow

Control of Calcium Dynamics

Control of Calcium Dynamics

• 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)

• IP3 Receptor Channel (IP3R)

• Ryanodine Receptor Channel (RyR)

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

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

Calcium CurrentsN 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

• 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

Calcium Release through Receptor

Channels

Calcium Release

• Calcium Release Receptor Channels are modeled as multi-state molecules

– One state is the conducting state

– For IP3 receptor state transitions depend on

calcium concentration and IP3

concentration

– For Ryanodine receptor, state transitions depend on calcium concentration

Dynamics of Release Channels

• Both IP3R and RyR have two calcium

binding sites:

– Fast binding to one site, causes channel opening

– Slower binding to other site, causes slow channel closing

IP3 Receptor

Similar to RyR but with additional binding site for IP3

8 state model of DeYoung and Keizer, 1992

Figure from Li and Rinzel, 1994

Calcium Release

• Release through the channel is proportional to concentration difference between ER and cytosol

• Release depends on fraction of channels in open state

• ФR = P Xn ([Ca2+]ER – [Ca2+])

• P is permeability

• X is fraction of channels in open state

• n is number of independent subunits

Dynamics of Release Channels

• Dynamics similar to sodium channel

• Activation:

– IP3 plus calcium produces channel opening

– Channel opening increases calcium concentration

– Higher concentration causes more channels to open

– Positive feed back produces calcium spike

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

Li and Rinzel Calcium Release

Li and Rinzel Calcium Release

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

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

Pump Equations

• Michaelis-Menten formulation

• Used for SERCA or PMCA pumps

• Implements the equation:

• Kcat is the maximal pump capacity

• n is the Hill coefficient: number of calcium molecules bound

• KM is the affinity (Half maximal concentration)

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

Sodium Calcium Exchange (NCX)Depolarization may reverse pump direction

Ion concentration change may reverse direction

Increase in Naint or decrease in Naext

Increase in internal sodium may explain activity dependent depression

Increase in Caext or decrease in Caint

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

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)

– KD1 = 1.5 uM, KD2 = 10 uM,

– Recent estimates in Faas, Raghavachari, Lisman, Mody (2011) Nat Neurosci.

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 m2/s

– KD = 700 nM, kon = 2.7 x107 /M-sec

• Parvalbumin

– In fast spiking interneurons

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

Kf

Kb

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

Radial and Axial Diffusion

Methods in Neuronal Modeling, Koch and SegevChapter 6 by DeSchutter and Smolen

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

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

Control of Calcium Dynamics

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

Calcium Objects

Ca_concen (genesis), CaConc (moose) Simplest implementation of calcium

Calcium current input converted to ion influxB = 1 / (z F vol): volume to produce

'reasonable' calcium concentration Calcium decays to minimum with single time

constant

• moose.doc(CaConc)

• showobject Ca_concen

Calcium Objects with DiffusionDifshell (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, usefixbuffer

Non-diffusible buffer (use with difshell)difbuffer

Diffusible buffer (use with difshell)

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

Morphology of Model Cell

Calcium Dynamics in Model Cell

Ca2+

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

Calcium Buffers and Diffusion

Cal2-SI.gTwo compartments: soma and dendrite

Calcium binding to buffer is implemented in function

Diffusion between soma and dendrite

Cal2difshell.gSame system, using difshell and difbuffer

Computationally more efficient

Chemesis Release

• CICR implements calcium release states using Markov kinetic channel formalism

States

Forward

rate

constants

Calcium Release Objects

• CICR implements calcium release states using Markov kinetic channel formalism

Create one element for each state, Rxx

• Parameters (Fields) 'Forward' rate constants,

State vector, e.g. 001 for 1 Ca++, 0 IP3

bound

Calculates fraction of receptors in state

• Inputs: calcium, IP3, other states

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*Xq (CaER-CaCyt)

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

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

Calcium Release and SERCA

Cal4.g

Implements IICR from Cal4.g

Adds SERCA pump to remove calcium from cytosol

Voltage Dependent Calcium ChannelsCal7.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