# Models of calcium dynamics

Post-publication activity

Curator: James Sneyd

Figure 1: Diagram of the major fluxes involved in the control of calcium oscillations in non-excitable cells. Abbreviations: R (agonist receptor), G (G-protein), PLC (phospholipase C), SOC (store-operated channels), AAOC (arachadonic acid-operated channels), RyR (ryanodine receptor), Icrac (calcium release-activated channel).

In very many cell types, the concentration of free intracellular calcium oscillates, with a period ranging from a few seconds to a few minutes. These calcium oscillations are thought to control a wide variety of cellular processes, and are often organised into intracellular and intercellular calcium waves (Tsien and Tsien, 1990; Berridge, 1990, 1997; Berridge et al., 2003; Clapham, 1995; Sneyd et al., 1995; Thomas et al., 1996; Falcke, 2004; Schuster et al., 2002).

When constructing mathematical models of calcium oscillations and waves it is crucial to keep two points firmly in mind:

• Each cell type is different. The mechanisms that underlie calcium oscillations in one cell type might be, and usually are, quite different in detail from the mechanisms underlying oscillations in another cell type. Despite this, there is still a lot in common between cell types, and certain generalisations can be made. Carefully.
• It is a relatively simple matter to write down a system of equations that exhibits oscillations. It is even relatively simple to fiddle the model until the oscillations look like the data. It is, however, much more difficult to use the model to say something interesting and new about the cellular physiology.

## Calcium Oscillations

### Physiological Mechanisms

#### The resting state

At rest, the concentration of calcium in the cell cytoplasm is low, around 30 nM, while outside the cell, and in the internal compartments such as the endoplasmic reticulum (ER), sarcoplasmic reticulum (SR) and mitochondria it is orders of magnitude higher. These concentration gradients are maintained in part by calcium ATPase pumps, which use the energy of ATP to pump calcium against its gradient, either out of the cell or into internal compartments. Sodium-Calcium exchangers also play an important role in some cell types. These large concentration gradients allow for a very rapid increase in cytosolic calcium concentration when calcium channels are opened.

#### Different sources of calcium

The increased calcium concentration in an oscillation comes from one of two places:

• From internal stores such as the ER, the SR, the mitochondria, or calcium buffers.
• From outside the cell. Calcium channels in the plasma membrane can be controlled by a variety of things, including voltage, agonist stimulation and the amount of calcium in the ER.

As a general rule, voltage-gated calcium entry occurs in electrically excitable cells, while in non-excitable cells (such as epithelial cells, hepatocytes, or oocytes) calcium release is from internal stores.

#### Agonist stimulation

Typically in non-excitable cells, binding of an agonist such as a hormone or a neurotransmitter to cell-surface receptors initiates a series of reaction, linked through a G-protein, that ends in the production of the second messenger inositol trisphosphate (IP3), which diffuses through the cytoplasm of cell and binds to IP3 receptors (IPR) which are located on the membrane of the ER. IPR are also calcium channels, and when they bind IP3 they open, allowing for the fast release of calcium from the ER into the cytoplasm.

Figure 2: Schematic diagram of some of the interactions between calcium and IP3 dynamics. Calcium can increase the rate of IP3 formation and degradation.

#### Feedback mechanisms

Without some additional feedback mechanisms, simple release of calcium from the ER will not result in oscillations. There are a number of feedback pathways that operate in different cell types to mediate calcium oscillations. The two principal mechanisms are:

• Modulation by calcium of the IPR open probability. In most cell types, release of calcium through the IPR can stimulate the release of additional calcium from the ER, often by binding to the IPR and increasing its open probability. This leads to the autocatalytic release of calcium from the ER, in a process usually called calcium-induced calcium release, or CICR. Calcium can also bind to IPR at a different binding site to inhibit calcium release, leading to a slow negative feedback and termination of calcium release. Calcium can then be taken up into the ER by the ATPase pumps, leading to a decrease in cytoplasmic calcium concentration and resetting of the cycle.
• Modulation by calcium of IP3 production and degradation. Since the rates of IP3 production and degradation are calcium-dependent this can lead to either positive or negative feedback loops, with the resultant potential for creating oscillations.

The exact mechanisms operating in any particular cell type, to generate any particular oscillatory pattern, are extraordinarily difficult to determine. In most cases the oscillatory mechanism is not certain.

### Mathematical Mechanisms

#### Basic model structure

Models of calcium oscillations describe the various calcium fluxes from and into the cytoplasm. Given the simple structure shown at right, conservation of calcium gives the equations ${dc\over dt} = J_{\rm release} - J_{\rm serca} + J_{\rm influx} - J_{\rm pm}$ ${dc_e\over dt} = \gamma(J_{\rm serca} - J_{\rm release}),$ where $$c$$ denotes the concentration of calcium in the cytoplasm, and $$c_e$$ denotes the concentration of calcium in the ER (or SR). Additional fluxes such as mitochondrial and buffer fluxes can be added in the same manner.

In general, the conservation equations for $$c$$ and $$c_e$$ are coupled to other equations that describe, for example, the states of the IPR, the states of the ATPase pumps, the calcium buffers, and the concentration of IP3 (Goldbeter et al., 1999; Atri et al., 1993; De Young and Keizer, 1992; Dupont et al., 1991, 2003; Falcke, 2003; Hofer, 1999).

The parameter $$\gamma$$ denotes the ratio of the cytoplasmic volume to the ER volume. This is necessary as the concentrations $$c$$ and $$c_e$$ refer to different volumes. Since the volume of the ER is less than that of the cytoplasm, the flow of calcium from the cytoplasm to the ER will cause a greater concentration change in the ER than it does in the cytoplasm.

This style of model assumes that the concentration of calcium is the same throughout the cell, i.e., that the cell is well-mixed.

Figure 3: Schematic diagram of the typical calcium fluxes in a simple model of calcium oscillations. Additional compartments, such as the mitochondria or calcium buffers, can be added in the same manner.

#### Modelling the fluxes

To construct a specific model of calcium oscillations it remains only to choose particular models of each of the fluxes and substitute them into the evolution equations. Practically any reasonable choice of model fluxes will lead to oscillations, of one type or another, for at least a range of parameter values.

Construction of models of each of these fluxes is an art form all to itself; models of the IPR range from simple two-variable models to complex models with thousands of variables (Li and Rinzel, 1994; Sneyd and Falcke, 2004), SERCA models range from a simple Hill function to multistate Markov models (Keener and Sneyd, 1998), while no good mechanistic model has yet been constructed of the calcium influx from outside the cell (as we just do not know enough about it, as yet).

#### Open-cell and closed-cell models

In some cases it is reasonably accurate to assume that there is little flux of calcium across the membrane of the cell, and thus the total amount of calcium in the cell remains constant. In this case $\gamma c + c_e = {\rm constant},$ and the model solutions live approximately on this line. Such models are called closed-cell models.

In cases where calcium fluxes across the plasma membrane are allowed no such constraint applies; such models are called open-cell models.

Often, the total amount of calcium in the cell can be treated as a slow parameter, as it is often the case that calcium transport to and from the outside of the cell is much slower than calcium transport to and from the internal compartments (Sneyd et al., 2004).

#### Stochastic models

In many cell types calcium oscillations do not occur with a regular period, but are clearly strongly influenced by stochastic processes (Marchant and Parker, 2001; Falcke, 2003). The exact nature of these stochastic processes is not always clear, but the most plausible explanation is that stochastic opening and closing of the IPR is the most important stochastic process causing irregularity in calcium oscillations. In fact, it can be shown that oscillations can occur in a stochastic model even when the underlying deterministic system is in a non-oscillatory state. This raises the possibility that many of the observed oscillations are actually a result of an underlying stochastic process rather than an underlying deterministic oscillatory process. Not surprisingly, it is difficult to distinguish between these hypotheses, and from an experimental point of view, the exact role of stochasticity remains to be determined.

## Calcium Waves

In some cell types, calcium oscillations occur practically uniformly across the cell. In such a situation, measurement of the calcium concentration at any point of the cell gives the same time course, and a well-mixed model is appropriate. More often, however, each oscillation takes the form of a wave moving across the cell; these intracellular "oscillations" are actually periodic intracellular waves. To model and understand such spatially distributed behavior, inclusion of calcium diffusion is vital.

It is widely believed intracellular calcium waves are driven by the diffusion of calcium between calcium release sites. According to this hypothesis, the calcium released from one group of release sites (usually either IPR or RyR) diffuses to neighboring release sites and initiates further calcium release from them. Repetition of this process can generate an advancing wave front of high calcium concentration, i.e., a calcium wave. Since they rely on the active release of calcium via a positive feedback mechanism, such waves are actively propagated.

However, when the underlying calcium kinetics are oscillatory (i.e., there is a stable limit cycle), waves can propagate by a kinematic, or phase wave, mechanism, in which the wave results from the spatially ordered firing of local oscillators. These waves do not depend on calcium diffusion for their existence, but merely on the fact that one end of the cell is oscillating with a different phase than the other end. In this case, calcium diffusion serves to synchronize the local oscillators, but the phase wave persists in the absence of calcium diffusive coupling.

It is usually (but not always) assumed that calcium diffuses with constant diffusion coefficient Dc, and that the cellular cytoplasm is isotropic and homogeneous. We also assume that $$c$$ and $$c_e$$ coexist at every point in space. Obviously this is not what really occurs. However, if the ER is "smeared out" sufficiently, it is reasonable to assume that every point in space is close enough to both the ER and the cytoplasm to allow for such a simplified model. This type of smeared model can be rigorously derived using homogenisation theory.

In this case, the reaction-diffusion equation for Ca2+ is ${\partial c\over \partial t} = D_c\nabla^2 c + J_{\rm release} - J_{\rm serca}.$ Note that the fluxes $$J_{\rm in}$$ and $$J_{\rm pm}$$ are omitted as they are fluxes only on the boundary of the cellular domain and must be treated differently. $$D_c$$ is an effective diffusion coefficient that can be derived using homogenisation.

### ER calcium

It is not known how well Ca2+ diffuses in the ER, or the extent to which the tortuosity of the ER plays a role in determining an effective diffusion coefficient for ER Ca2+. It is typical to assume either that Ca2+ does not diffuse in the ER, or that it does so with a restricted diffusion coefficient, $$D_e\ .$$ In either case, ${\partial c_e\over \partial t} = D_e\nabla^2 c_e - \gamma (J_{\rm release} - J_{\rm serca}),$ with $$0<D_e\le D_c\ .$$

### Boundary conditions

Cells are inherently three dimensional objects. When cells are viewed as three dimensional the boundary conditions are clear. At the boundary of the cell (i.e., the plasma membrane) the flux per unit area into the cell is $$J_{\rm in} - J_{\rm pm}\ .$$ Thus, $\begin{matrix} D_c \nabla c \cdot n &=& J_{\rm in} - J_{\rm pm}, \\ \nabla c_e \cdot n &=& 0, \end{matrix}$ where $$n$$ is the outward unit normal to the boundary.

However, if the cell is viewed as a long one dimensional object, with calcium homogeneously distributed in each cross-section, then the fluxes across the (cylindrical) wall must be included as source terms in the governing partial differential equation. For example, for a "one-dimensional" cell of length $$L$$ with no calcium flux at the ends, the boundary conditions are $${\partial c \over \partial x} = 0$$ at $$x=L$$ and $$x=0\ ,$$ while at each internal point we have ${\partial c\over \partial t} = D_c{\partial^2 c\over \partial x^2} + {\rho\over A}(J_{\rm in} - J_{\rm pm}) + J_{\rm release} - J_{\rm serca}.$ where $$\rho$$ is the cell perimeter and $$A$$ the cell cross-sectional area.

There are a number of ways to study reaction-diffusion models of this type, but the two most common are numerical simulation or bifurcation analysis of the traveling wave equations.

### Bifurcation analysis

If we introduce the traveling wave variable $$\xi=x+st\ ,$$ where $$s$$ is the wave speed we can write the equation for calcium diffusion as the pair of equations $\begin{matrix} c' &=& d, \\ D_cd' &=& sd - \sum J, \end{matrix}$ where $$\sum J$$ denotes all the calcium fluxes, and where a prime denotes differentiation with respect to $$\xi\ .$$ These two equations are coupled to the other equations for $$c_e\ ,$$ $$p\ ,$$ and the states of the various receptors. When all the unknown variables are written as functions of the variable $$\xi\ ,$$ these equations are called traveling wave equations.

Traveling pulses, traveling fronts, and periodic waves correspond to, respectively, homoclinic orbits, heteroclinic orbits and limit cycles of the traveling wave equations. However, such an approach does not readily give information about the stability of the wave solutions of the original reaction-diffusion equations.

## References

• A. Atri, J. Amundson, D. Clapham and J. Sneyd, (1993). A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte. Biophys J, 65, pp. 1727--1739
• M. J. Berridge, (1990). Calcium oscillations. Journal of Biological Chemistry, 265, pp. 9583--9586
• M. J. Berridge, (1997). Elementary and global aspects of calcium signalling. J Physiol, 499 ( Pt 2), pp. 291-306
• M. J. Berridge, M. D. Bootman and H. L. Roderick, (2003). Calcium signalling: dynamics, homeostasis and remodelling. Nat Rev Mol Cell Biol, 4, pp. 517-29
• D. Clapham, (1995). Calcium signaling. Cell, 80, pp. 259--268
• G. W. De Young and J. Keizer, (1992). A single pool IP3-receptor based model for agonist stimulated Ca2+ oscillations. Proc Natl Acad Sci U S A, 89, pp. 9895--9899
• G. Dupont, M. J. Berridge and A. Goldbeter, (1991). Signal-induced Ca2+ oscillations: properties of a model based on Ca2+-induced Ca2+ release. Cell Calcium, 12, pp. 73--85
• G. Dupont, O. Koukoui, C. Clair, C. Erneux, S. Swillens and L. Combettes, (2003). Ca2+ oscillations in hepatocytes do not require the modulation of InsP3 3-kinase activity by Ca2+. FEBS Lett, 534, pp. 101-5
• M. Falcke, (2003). Buffers and oscillations in intracellular Ca2+ dynamics. Biophys J, 84, pp. 28-41
• M. Falcke, (2003). On the role of stochastic channel behavior in intracellular Ca2+ dynamics. Biophys J, 84, pp. 42-56
• M. Falcke, (2004). Reading the patterns in living cells - the physics of Ca2+ signaling. Advances in Physics, 53, pp. 255-440
• M. Falcke, J. L. Hudson, P. Camacho and J. D. Lechleiter, (1999). Impact of mitochondrial Ca2+ cycling on pattern formation and stability. Biophys J, 77, pp. 37--44
• A. Goldbeter, G. Dupont and M. J. Berridge, (1990). Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation. Proc Natl Acad Sci U S A, 87, pp. 1461--1465
• T. Hofer, (1999). Model of intercellular calcium oscillations in hepatocytes: synchronization of heterogeneous cells. Biophys J, 77, pp. 1244-56
• J. Keener and J. Sneyd (1998). Mathematical Physiology. Springer-Verlag, New York.
• Y.-X. Li and J. Rinzel, (1994). Equations for InsP3 receptor-mediated Ca2+ oscillations derived from a detailed kinetic model: a Hodgkin-Huxley-like formalism. J theor Biol, 166, pp. 461--473
• J. Marchant and I. Parker (2001). Role of elementary Ca2+ puffs in generating repetitive Ca2+ oscillations. EMBO J., 20, pp. 65-71
• S. Schuster, M. Marhl and T. Hofer, (2002). Modelling of simple and complex calcium oscillations. From single-cell responses to intercellular signalling. Eur J Biochem, 269, pp. 1333-55
• J. Sneyd and M. Falcke, (2004). Models of the inositol trisphosphate receptor. Prog Biophys Mol Biol, pp. in press
• J. Sneyd, J. Keizer and M. J. Sanderson, (1995). Mechanisms of calcium oscillations and waves: a quantitative analysis. FASEB J, 9, pp. 1463--1472
• J. Sneyd, K. Tsaneva-Atanasova, D. I. Yule, J. L. Thompson and T. J. Shuttleworth, (2004). Control of calcium oscillations by membrane fluxes. Proc Natl Acad Sci U S A, 101, pp. 1392-6
• A. P. Thomas, G. S. J. Bird, G. Hajnóczky, L. D. Robb-Gaspers and J. W. J. Putney, (1996). Spatial and temporal aspects of cellular calcium signaling. FASEB J, 10, pp. 1505--1517
• R. W. Tsien and R. Y. Tsien, (1990). Calcium channels, stores, and oscillations. Annual Review of Cell Biology, 6, pp. 715--760

Internal references