Avrama Blackwell George 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 I Ca

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 – IP 3 Receptor Channel (IP 3 R) – 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 Press Chapter 6 by DeSchutter and Smolen

Calcium Release through IP 3 R Levitan and Kaczmark, Oxford Press

Calcium Release ● Receptors are modeled as multi-state molecules – One state is the conducting state – For IP 3 Receptor state transitions depend on calcium concentration and IP 3 concentration – For Ryanodine Receptor, state transitions depend on calcium concentration

Dynamics of Release Channels ● Both IP 3 R 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 ● IP 3 R has an additional binding site for IP 3

IP 3 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: – IP 3 + 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 ~ 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 Na int or decrease in Na ext ● Increase in Ca ext or decrease in Ca int

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 -  C i  ● Quantity is remainder after all other forms of molecule accounted for. – pool (Kinetikit) ● dC/dt =  A -  B C ● Or C = Ctot -  C i  (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 V max 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 IP 3 bound – Fraction of receptors in this state – Whether this element is conserved

Chemesis Objects ● CICR (cont.) ● Messages (Inputs) required: – IP 3 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*X q (Ca ER -Ca Cyt )

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 (P L ) – 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