Calcium

DanielHariprasad

Three pool model of calcium signaling

Daniel Hariprasad

College of William and Mary(adviser: Junping Shi, Jianjun Paul Tian, Megan McNulty)

Supported by NSF UBM grant and CSUMS grant

March 21, 20092009 GMU-WM CSUMS Workshop

Calcium

DanielHariprasad

Ca2+ Background

Majority of body’s Ca2+ is stored in the bones, where itcan be released by hormones for signaling.

Calcium

DanielHariprasad

Ca2+ Background

Majority of body’s Ca2+ is stored in the bones, where itcan be released by hormones for signaling.

Ca2+ is necessary for many cellular functions: musclemechanics, cardiac electrophysiology, hair cells, adaptationin photoreceptors.

Calcium

DanielHariprasad

Ca2+ Background

Prolonged high concentrations of Ca2+ is toxic. Constanthigh concentrations in muscles leads to a state of constanttension (rigor mortis).

Calcium

DanielHariprasad

Ca2+ Background

Prolonged high concentrations of Ca2+ is toxic. Constanthigh concentrations in muscles leads to a state of constanttension (rigor mortis).

Prolonged low concentrations of Ca2+ leads to theinability to perform the acts listed above. No heart beat ormuscle movement without Ca2+.

Calcium

DanielHariprasad

Calcium Dynamics

Extracellular [Ca2+] is generally kept at 1mM whileintracellular [Ca2+] is generally kept around .1µM.

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three Pool Model

t

U

Calcium

DanielHariprasad

Calcium Dynamics

Extracellular [Ca2+] is generally kept at 1mM whileintracellular [Ca2+] is generally kept around .1µM.This vast difference (4 orders of magnitude) allows for fastinflux of Ca2+ down the concentration gradient, but cellsneed to expend energy to keep this concentration disparity.

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three Pool Model

t

U

Calcium

DanielHariprasad

Calcium Dynamics

Extracellular [Ca2+] is generally kept at 1mM whileintracellular [Ca2+] is generally kept around .1µM.This vast difference (4 orders of magnitude) allows for fastinflux of Ca2+ down the concentration gradient, but cellsneed to expend energy to keep this concentration disparity.In response to hormones, cells have oscillations of [Ca2+]occur.

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three Pool Model

t

U

Calcium

DanielHariprasad

Calcium Dynamics

Extracellular [Ca2+] is generally kept at 1mM whileintracellular [Ca2+] is generally kept around .1µM.This vast difference (4 orders of magnitude) allows for fastinflux of Ca2+ down the concentration gradient, but cellsneed to expend energy to keep this concentration disparity.In response to hormones, cells have oscillations of [Ca2+]occur.Cells have mechanisms for this process.

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three Pool Model

t

U

Calcium

DanielHariprasad

Calcium Dynamics

There are two ways intracellular Ca2+ is effluxed: pumpedinto extracellular space, sequestered by internal stores suchas ER and mitochondria.

Calcium

DanielHariprasad

Calcium Dynamics

There are two ways intracellular Ca2+ is effluxed: pumpedinto extracellular space, sequestered by internal stores suchas ER and mitochondria.

There are two ways intracellular Ca2+ is influxed: Ca2+

travels through surface membrane channels, released frominternal stores.

Calcium

DanielHariprasad

Calcium Dynamics of the Plasma Membrane

The plasma membrane is the interface between the celland the outside world. It is a selectively permeablephospholipid bilayer that is interspersed with lipids andproteins that allow passage of selected molecules.

Calcium

DanielHariprasad

Calcium Dynamics of the Plasma Membrane

The plasma membrane is the interface between the celland the outside world. It is a selectively permeablephospholipid bilayer that is interspersed with lipids andproteins that allow passage of selected molecules.Plasma Membrane Ca2+ ATPase (PMCA) — Uses energyfrom ATP hydrolysis to pump Ca2+ out of the intracellularspace. Active transport that keeps intracellular [Ca2+] low.

Calcium

DanielHariprasad

Calcium Dynamics of the ER

The endoplasmic reticulum (ER) is responsible for proteintranslation, folding, and transport as well as sequesteringCa2+. Its basic structure and composition is similar to theplasma membrane.

Calcium

DanielHariprasad

Calcium Dynamics of the ER

IP3 receptor — Responds to IP3 concentrations andreleases Ca2+ from an internal store.

Calcium

DanielHariprasad

Calcium Dynamics of the ER

IP3 receptor — Responds to IP3 concentrations andreleases Ca2+ from an internal store.Sarco Endoplasmic Reticulum Ca2+ ATPase (SERCA) —Uses ATP hydrolysis to pump Ca2+ into the ER.

Calcium

DanielHariprasad

Calcium Dynamics of the ER

IP3 receptor — Responds to IP3 concentrations andreleases Ca2+ from an internal store.Sarco Endoplasmic Reticulum Ca2+ ATPase (SERCA) —Uses ATP hydrolysis to pump Ca2+ into the ER.Ca2+ induced Ca2+ release (CICR) channels — Respondto increases in intracellular Ca2+ and release Ca2+ inresponse.

Calcium

DanielHariprasad

Calcium Diagram

Calcium

DanielHariprasad

Temporal dynamics of Ca2+

External stimulus releases Ca2+. This can either be astimulus that releases IP3, a depolarization of the cellmembrane, or any other stimulus that would increase ourintracellular[Ca2+]. For our purposes we will examine IP3

release.

Calcium

DanielHariprasad

Temporal dynamics of Ca2+

External stimulus releases Ca2+. This can either be astimulus that releases IP3, a depolarization of the cellmembrane, or any other stimulus that would increase ourintracellular[Ca2+]. For our purposes we will examine IP3

release.

The release of IP3 releases intracellular Ca2+ that opensup the CICR channel.

Calcium

DanielHariprasad

Temporal dynamics of Ca2+

External stimulus releases Ca2+. This can either be astimulus that releases IP3, a depolarization of the cellmembrane, or any other stimulus that would increase ourintracellular[Ca2+]. For our purposes we will examine IP3

release.

The release of IP3 releases intracellular Ca2+ that opensup the CICR channel.

After the increase in intracellular [Ca2+] other channels arestill operating that work to keep the [Ca2+] in the cell low.

Calcium

DanielHariprasad

Temporal dynamics of Ca2+

External stimulus releases Ca2+. This can either be astimulus that releases IP3, a depolarization of the cellmembrane, or any other stimulus that would increase ourintracellular[Ca2+]. For our purposes we will examine IP3

release.

The release of IP3 releases intracellular Ca2+ that opensup the CICR channel.

After the increase in intracellular [Ca2+] other channels arestill operating that work to keep the [Ca2+] in the cell low.

This leads to oscillations in the intracellular [Ca2+].

Calcium

DanielHariprasad

Temporal dynamics of Ca2+

External stimulus releases Ca2+. This can either be astimulus that releases IP3, a depolarization of the cellmembrane, or any other stimulus that would increase ourintracellular[Ca2+]. For our purposes we will examine IP3

release.

The release of IP3 releases intracellular Ca2+ that opensup the CICR channel.

After the increase in intracellular [Ca2+] other channels arestill operating that work to keep the [Ca2+] in the cell low.

This leads to oscillations in the intracellular [Ca2+].

Deactivating the [Ca2+] oscillations is dependent on theinitial stimulus. [Ca2+] oscillations could relax to a steadystate, the stimulus could detach, or a second stimuluscould bind to deactivate the oscillations.

Calcium

DanielHariprasad

Two Pool Schematic Diagram

Extracellular Space

Intracellular Space

6

Extrusion?

Influx

&%'$S Pool

?IP3 Dependent Release

6?

&%'$I Pool

?

Uptake

���

Leak

@@R

Calcium Dependent Release

Calcium

DanielHariprasad

Two Pool Model

dCai

dτ= r − kCai −

v1Cani

kn1 + Can

i

+v2Cam

s

km2 + Cam

s

Capi

kp3 + Ca

pi

+ kf Cas ,

dCas

dτ=

v1Cani

kn1 + Can

i

−v2Cam

s

km2 + Cam

s

Capi

kp3 + Ca

pi

− kf Cas ,

(1)

Model created by Goldbeter et al. [1990, PNAS].

Temporal model, does not show spatial effects.

Effectively models CICR, SERCA, PMCA, and constantIP3 release.

Produces effective [Ca2+] oscillations.

Calcium

DanielHariprasad

Two Pool Analysis

The r term correlates directly to the stimulus. If r is in acertain range, oscillations occur.

Use dimensional analysis to determine what parametersare meaningful.

We follow the analysis as presented by Sneyd et al. [1993,BMB].

Calcium

DanielHariprasad

Dimensionless Two Pool Model

du

dt= µ − u −

γ

ǫf (u, v),

dv

dt=

1

ǫf (u, v),

(2)

where

f (u, v) =βun

1 + un−

vm

1 + vm·

up

αp + up− δv . (3)

where the new variables and parameters are defined as:

u =Cai

k1, v =

Cas

k2, t = τk ,

α =k3

k1, β =

v1

v2, γ =

k2

k1,

δ =kf k2

v2, µ =

v

kk1, ǫ =

kk2

v1.

(4)

Calcium

DanielHariprasad

Bifurcation of Two Pool

The µ parameter is the main parameter responsible for ouroscillations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

mu

u

H H

H H

H H

LPCLPC

Figure: Here the horizontal axis is µ, and the vertical axis is u.Parameters are ε = 0.04, α = 0.9, β = 0.13, γ = 2, δ = 0.004,m = 2, n = 2, p = 4. Hopf bifurcation points µ1 = 0.3109, µ2 = 0.70

Calcium

DanielHariprasad

Two Pool Phase Portrait

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.4

0.5

0.6

0.7

0.8

0.9

1Two Pool Model

U

V

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8Two Pool Model

U

V

Figure: Phase portraits of limit cycle. Here the horizontal axis is u,and the vertical axis is v . Parameters are ε = 0.04, α = 0.9,β = 0.13, γ = 2, δ = 0.004, m = 2, n = 2, p = 4. (left) µ = 0.4,(right) µ = 0.64.

Calcium

DanielHariprasad

Extension of Two Pool Model

The two pool model models Ca2+ oscillations effectively,but does not encompass all of the signaling pathways.

Calcium

DanielHariprasad

Extension of Two Pool Model

The two pool model models Ca2+ oscillations effectively,but does not encompass all of the signaling pathways.

As mentioned before, other organelles such as themitochondria are used in sequestering Ca2+.

Calcium

DanielHariprasad

Extension of Two Pool Model

The two pool model models Ca2+ oscillations effectively,but does not encompass all of the signaling pathways.

As mentioned before, other organelles such as themitochondria are used in sequestering Ca2+.

Create a model that extends the two pool to describe theother pathways in cellular Ca2+ signaling by add ing athird pool to model the mitochondria’s role in signaling.

Calcium

DanielHariprasad

Calcium Dynamics of Mitochondria

The mitochondria is responsible for producing energy forthe cell through the production of ATP. Cells can haveanywhere from one to thousands of mitochondrion.

Calcium

DanielHariprasad

Calcium Dynamics of Mitochondria

The mitochondria is responsible for producing energy forthe cell through the production of ATP. Cells can haveanywhere from one to thousands of mitochondrion.Uniporter — Influxes Ca2+ by using an electrochemicalgradient.

Calcium

DanielHariprasad

Calcium Dynamics of Mitochondria

The mitochondria is responsible for producing energy forthe cell through the production of ATP. Cells can haveanywhere from one to thousands of mitochondrion.Uniporter — Influxes Ca2+ by using an electrochemicalgradient.Na+-Ca2+ exchanger — Uses Na+ concentration gradientto influx Ca2+.

Calcium

DanielHariprasad

Three Pool Schematic

Extracellular Space

Intracellular Space

6

Extrusion?

Influx

&%'$S Pool

?IP3 Dependent Release

6?

&%'$I Pool

?

Uptake

���Leak

@@R

Calcium Dependent Release

&%'$M Pool-Uptake

?Leak

Calcium

DanielHariprasad

Three Pool Model

dCai

dτ= r − kCai −

v1Cani

kn1 +Can

i+ v2Cam

s

km2 +Cam

s

Capi

kp3 +Ca

pi

+kf Cas −v3Ca

qi

kq4 +Ca

qi

+ kmCam,

dCas

dτ=

v1Cani

kn1 +Can

i−

v2Cams

km2 +Cam

s

Capi

kp3 +Ca

pi

− kf Cas ,

dCam

dτ=

v3Caqi

kq4 +Ca

qi

− kmCam,

(5)

Adds a third equation to the Goldbeter model.

Keeps all of the two pool model features as well as addingthe functions of the mitochondria.

Mitochondrial pathways modeled are the uniporter and aleak channel.

Calcium

DanielHariprasad

Analysis of Three Pool Model

Use methods of Sneyd et al. [1993, BMB] to discoverdifferences from the two pool model.

Begin with dimensional analysis then use bifurcationtheory as well as other techniques.

Calcium

DanielHariprasad

Dimensional Analysis of Three Pool Model

du

dt= µ − u −

γ1

ǫ1f (u, v) −

γ2

ǫ2g(u, w),

dv

dt=

1

ǫ1f (u, v),

dw

dt=

1

ǫ2g(u, w),

(6)

where

f (u, v) =βun

1 + un−

vm

1 + vm·

up

αp1 + up

− δ1v ,

g(u, w) =uq

αq2 + uq

− δ2w .

(7)

New variables and parameters: u = Cai

k1, v = Cas

k2, w = Cam

k4,

t = τk , ǫ1 = kk2v2

, ǫ2 = kk4v3

, γ1 = k2k1

,γ2 = k4k1

, µ = rkk1

, β = v1v2

,

δ1 = kf k2v2

,δ2 = kmk4v3

, α1 = k3k1

, and α2 = k4k1

.

Calcium

DanielHariprasad

Equilibrium

The nondimensionalized equation has a unique equilibriumpoint (µ, v∗(µ), w∗(µ)), where (v∗(µ), w∗(µ)) satisfies

βµn

1 + µn−

vm

1 + vm·

µp

αp1 + µp

−δ1v = 0, w =µq

δ2(αq2 + µq)

. (8)

It is stable when µ is near 0 and near ∞. But it is possibleto have two Hopf bifurcation points in between, and thereis a curve of periodic orbits connecting the two Hopfbifurcation points.

We use Matlab package MatCont to plot the bifurcationof periodic orbits.

Calcium

DanielHariprasad

Bifurcation Diagram of Periodic Orbits

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

mu

v

H H

H H

LPCLPCLPC

Figure: Here the horizontal axis is µ, and the vertical axis is v .Parameters are ǫ1 = 0.04, ǫ2 = .2, α1 = 0.9, α2 = 10, β = 0.13,δ1 = 0.004, δ2 = .2, γ1 = 2, γ2 = 10, m = 2, n = 2, p = 4, andq = 2.

Calcium

DanielHariprasad

Phase Portrait

0

0.5

1

1.5

0.2

0.4

0.6

0.8

10

0.02

0.04

0.06

0.08

0.1

Three Pool Model

Figure: Phase portrait of the three pool system with u, v, and w.µ = .36.

Calcium

DanielHariprasad

Excitable Cycle

0 20 40 60 80 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Three Pool Model

t

U

Figure: Graph of u(t) when µ = .36

Calcium

DanielHariprasad

Subcritical Bifurcation

0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40

5

10

15

20

25

30

mu

Per

iod

LPC

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

mu

u

LPC

The first Hopf bifurcation is subcritical which leads to twostable solutions in the region µ ∈ [.3154, .3167].

Calcium

DanielHariprasad

Bistability

0.3 0.305 0.31 0.315 0.32 0.325 0.331

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045Three Pool Model

U

V

0.3

0.31

0.32

0.33

1

1.02

1.04

1.064

5

6

7

8

9

10

x 10−3

U

Three Pool Model

V

W

0.2 0.4 0.6 0.8 1 1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Three Pool Model

U

V

0.20.4

0.60.8

11.2

0

0.5

1

1.5

0

0.005

0.01

0.015

0.02

0.025

0.03

U

Three Pool Model

V

W

µ = 0.316. Top: spiral sink (u0, v0, w0) = (0.3, 1, 0.01);Bottom: limit cycle (u0, v0, w0) = (1, 0.4, 0.02)

Calcium

DanielHariprasad

Conclusion

The three pool model creates a region of bistability thatcould be useful if a cell wants to achieve an increasedCa2+ for a longer period of time.

Calcium

DanielHariprasad

Conclusion

The three pool model creates a region of bistability thatcould be useful if a cell wants to achieve an increasedCa2+ for a longer period of time.

Effectively added a third pool without degenerating thedynamics of the two pool model.

Calcium

DanielHariprasad

Conclusion

The three pool model creates a region of bistability thatcould be useful if a cell wants to achieve an increasedCa2+ for a longer period of time.

Effectively added a third pool without degenerating thedynamics of the two pool model.

Third pool slows kinetics slightly, but this can be adjustedby adjusting parameters.

Calcium

DanielHariprasad

Conclusion

The three pool model creates a region of bistability thatcould be useful if a cell wants to achieve an increasedCa2+ for a longer period of time.

Effectively added a third pool without degenerating thedynamics of the two pool model.

Third pool slows kinetics slightly, but this can be adjustedby adjusting parameters.

For further research and modeling, could use oscillating[IP3], could use spatial modeling, could include Na+-Ca2+

pathways, could include voltage gated pathways.

Calcium

DanielHariprasad

References

A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: aMATLAB package for numerical bifurcation analysis of ODEs.ACM Transactions of Mathematical Software 29: 141–164(2003).

A. Goldbeter, G. Dupont, and M. J. Berridge, Minimal model forsignal-induced Ca2+ oscillations and for their frequencyencoding through protein phosphorylation. Proceedings of the

National Academy of Science 87: 1461–1465 (1990).

J. Keener and J. Sneyd, Mathematical physiology.Interdisciplinary Applied Mathematics, 8. Springer-Verlag, NewYork, 1998.

J. Sneyd, S. Girard, and D. Clapham, Calcium wave propagationby calcium-induced calcium release: An unusual excitablesystem. Bulletin of Mathematical Biology 55: 315–344 (1993).