Kim

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Modeling Calcium Concentration
Kim Avrama BlackwellGeorge Mason University
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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 Dynamics
Calcium
Slow
Kinases
Fast
Ca2
SK, BK channels
Membrane Potential
Potassium, Sodium channels
Synaptic channels, Calcium channels
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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 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
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Calcium Current

• Influx due to calcium current is calculated by
• F is Faradays constant
• Software dependent negative sign
• inward current is negative (physiologists
  convention) or positive (modelers convention)
• Flux has units of moles per unit time, converted
  to concentration using rxnpool, Ca_concen,
  diffshell, or pool object

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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
• ?R P Xn (Ca2ER 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 gt 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

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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
Segev Chapter 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 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

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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)

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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.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

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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 PXq (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_rateCapow/(Capowhalf_concpow)
• 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 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