Kim “Avrama” Blackwell

George Mason University

Modelling 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

• T type (CaL3.x)– Low threshold, Transient, prominent voltage

dependent inactivation

CaL1.2

-10

-8

-6

-4

-2

0

0 5 10 15 20

Time

Cu

rren

t (n

A) 0

-20

20

CaT

-0.8

-0.6

-0.4

-0.2

0

0.00 5.00 10.00 15.00 20.00

Time (ms)

Cu

rren

t (n

A)

0

-20

20

VmVm

Calcium Currents

00.10.20.30.40.50.60.70.80.9

1

-80 -30 20

Membrane Potential (mV)

Ste

ady

Sta

te

CaR

CaL1.2

CaT

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

•Flux has units of moles per unit time, converted to concentration

using rxnpool, Ca_concen, diffshell, or pool object

Calcium Release through Receptor

Channels

Calcium Release

• Calcium Release Receptor Channels are modelled 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:

– Binding to one site is fast, causes fast channel opening

– Binding to other site is slower, causes slow channel closing

• IP3R has an additional binding site for

IP3

IP3 Receptor

8 state model of DeYoung and Keizer, 1992

Figure from Li and Rinzel, 1994

Dynamics of Release Channels

• Dynamics similar to sodium channel:

– IP3 with low calcium produces small

channel opening

– Channel opening increases calcium concentration

– Higher concentration causes larger channel opening

– Positive feed back produces calcium spike

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

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

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

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

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

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

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)

Chemesis Calcium Objects

Calcium buffers implemented using rxnpool conservepool Reaction

Kinetikit: Pools reac

Morphology of Model Cell

Calcium Dynamics in Model Cell

Ca2+

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

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

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

Chemesis Calcium Objects

• CICR implements calcium release states using Markov kinetic channel formalism

States

Forward

rate

constants

Chemesis Calcium Objects

• CICR implements calcium release states using Markov kinetic channel formalism

One element for each state, Rxx

One of the elements may be conserved

• Parameters (Fields) 'Forward' rate constants,

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

bound

Fraction of receptors in this state (calculated)

Whether this element is conserved

Chemesis Calcium Objects• CICR (cont.)

• Messages (Inputs) required:

• IP3 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

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*Xq (CaER-CaCyt)

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

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

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

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 (PL)

Hill coefficient (should be 1.0)

Calcium Release and SERCA

Cal4.g Implements IICR from Cal4.g Adds SERCA pump to remove calcium

from cytosol

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

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