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Kim “Avrama” Blackwell
George Mason University
Modeling Calcium Concentration
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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
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Feedback Loops of Calcium DynamicsCalcium
Ca2+
Kinases
SK, BK
channelsMembrane
Potential
+
+
+
+
+
_
_
_
_
_
Potassium,
Sodium channels
Synaptic channels,
Calcium channels
Fast
Slow
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Control of Calcium Dynamics
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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)
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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
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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
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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
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• 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
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Calcium Release through Receptor
Channels
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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
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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
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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
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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
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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
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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
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Li and Rinzel Calcium Release
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Li and Rinzel Calcium Release
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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
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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
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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)
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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
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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
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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
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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.
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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
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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
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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
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Radial and Axial Diffusion
Methods in Neuronal Modeling, Koch and SegevChapter 6 by DeSchutter and Smolen
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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
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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
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Control of Calcium Dynamics
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For information on implementing calcium dynamics in Neuron, see:http://www.neuron.yale.edu/neuron/static/docs/rxd/index.html
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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
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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)
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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
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Morphology of Model Cell
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Calcium Dynamics in Model Cell
Ca2+
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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
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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
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Chemesis Release
• CICR implements calcium release states using Markov kinetic channel formalism
States
Forward
rate
constants
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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
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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)
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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
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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
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Calcium Release and SERCA
Cal4.g
Implements IICR from Cal4.g
Adds SERCA pump to remove calcium from cytosol
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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