Avrama BlackwellGeorge Mason University

Modeling Calcium Concentration and

Biochemical Reactions

Objectives● Explain importance of and relation

between biochemical reactions and calcium dynamics

● Present equations describing biochemical reactions and calcium dynamics

● Describe mechanisms modulating calcium concentration

● Demonstrate dynamics using small simulations

Importance of Calcium

● Calcium influences channel behavior, 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

Importance of Biochemical Reactions

● Some mechanisms of calcium dynamics are modeled as biochemical reactions

● Second messengers, .e.g Dopamine, modulate channel behavior

● Second messenger pathways are modeled as biochemical reactions

Biochemical Reactions

● Bimolecular Reactions– Stoichiometric interactions between

substrate molecules to form product molecule

● Formation of bond between the substrate molecules

● Stoichiometric implies that the reaction specifies the number of each molecule required for reaction

● Molecules are consumed in order to make product

Biochemical Reactions

● Bimolecular Reactions– Reaction order is the number of

simultaneously interacting molecules● First order reaction: single substrate becomes

product● Rate constants: rate (units of per sec) at which

substrate becomes product● Ratio of rate constants gives concentration of

substrates and products at equilibrium

When to Model Biochemical Reactions

● Metabotropic Receptors– Protein does not form channel– Protein is linked to GTP binding protein– Effect mediated by

● Activated G protein subunits● Down stream second messengers

Ionotropic vs Metabotropic

From Nicholls et al. Sinauer

Activation of GTP Binding Protein

From Nicholls et al. Sinauer

Direct Modulation of Channel via Active G Protein Subunits

From Nicholls et al. Sinauer

From Nicholls et al. Sinauer

From Nicholls et al. Sinauer

Importance of Calcium Dynamics

Control of Calcium Dynamics● Calcium Current● Pumps

– Smooth Endoplasmic Calcium ATPase (SERCA)

– Plasma Membrane Calcium ATPase (PMCA)– Sodium-Calcium exchanger

Control of Calcium Dynamics● Release from Intracellular Stores

– IP3 Receptor Channel (IP3R)

– Ryanodine Receptor Channel (RyR)● Buffers● Diffusion

Calcium Current

High Threshold, Persistent

Derivation of Diffusion Equation

● Diffusion in a cylinder– Derive equation by looking at fluxes in and

out of a slice of width x

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

Radial and Axial Diffusion

From Koch and Segev, MIT PressChapter 6 by DeSchutter and Smolen

Calcium Release through IP3R

Levitan and Kaczmark, Oxford Press

Calcium Release

● Receptors 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:– 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

● Figure from Li and Rinzel

Flow of calcium ions through release channels

Levitan and Kaczmark, Oxford Press

Dynamics of Release Channels

● Dynamics similar to sodium channel:– IP3 + 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 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 (maybe 4) 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 Caext or decrease in Caint

GENESIS objects

● Compartment-like objects– Keep track of molecule quantities and

concentrations● Similar to compartment calculating voltage

– Requires geometry/morphology values● length● radius● area of outer surface● area of inner surface (can be zero)● area of side surface● volume

GENESIS objects● Keep track of molecule quantities and

concentrations– rxnpool (Chemesis)

● dC/dt = A - B C● A = change in quantity independent of present

quantity● B = rate of change● Receives messages with quantities A and B

from other objects (enzymes, reactions, also calcium influx)

GENESIS objects● Keep track of molecule quantities and

concentrations– conservepool (Chemesis)

● C = Ctot - Ci● Quantity is remainder after all other forms of

molecule accounted for.– pool (Kinetikit)

● dC/dt = A - B C ● Or C = Ctot - Ci(if flag is set to conserve)

● Can also implement stochastic reactions

GENESIS objects

● Calculate changes due to reactions– mmenz (Chemesis)

● Use if MM assumptions are met● Fields: Km and Vmax● Inputs: Enzyme, substrate concentration● Calculates Vmax times [Enzyme] times substrate

● Empirical feedback modification of enzyme activity can be added.

GENESIS objects

● Calculate changes due to reactions– Enzyme (Chemesis)

● Fields: Kcat, Kf, Kb● Inputs: enzyme, substrate quantity● Calculates amount of Enzyme-Substrate

complex● Calculates change in product, enzyme,

substrate

– Enz (kinetikit)● Fields: Kcat, Kf, Kb● Inputs: enzyme, substrate quantity● Can implement stochastic reactions

GENESIS objects

● Calculate changes due to reactions– reaction (Chemesis) or reac (kinetikit)

● Fields: kf, kb● Inputs: substrates and products● Calculates:

– forward rate constant times substrate molecules– backward rate constant times product molecules

Chemesis Objects● CICR implements calcium release

states– One element for each state– 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– Whether this element is conserved

Chemesis 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 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)

Chemesis Objects

● Diffusion● Parameters (Fields)

– Diffusion constant, D● Messages (Inputs)

– Length, concentration, surface area from two reaction pools

● Calculates– Flux from one pool to another– D SA Conc / len

Chemesis Objects

● 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)

Chemesis objects

● MMPUMP used for SERCA or PMCA Pump● Fields

– Affinity– Power (exponent)– Maximum rate

● Messages (inputs)– Concentration

● Calculates flux due to pump

Integrating Calcium Mechanisms

● RXNPOOL takes flux messages from various calcium sources– VDCC sends message CURRENT, with fields

current and charge– Diffusion and 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

– Reactions send messages RXN0 - RXN2

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

● fixbuffer– Non-diffusible buffer (use with difshell)

● Difbuffer– Diffusible buffer (use with difshell)

Morphology of Model Cell

Calcium Dynamics in Model Cell