Dynamics of intracellular Ca$^{2+}$ oscillations in the presence of multisite Ca$^{2+}$-binding proteins

We study the dynamics of intracellular calcium oscillations in the presence of proteins that bind calcium on multiple sites and that are generally believed to act as passive calcium buffers in cells. We find that multisite calcium-binding proteins set a sharp threshold for calcium oscillations. Even with high concentrations of calcium-binding proteins, internal noise, which shows up spontaneously in cells in the process of calcium wave formation, can lead to self-oscillations. This produces oscillatory behaviors strikingly similar to those observed in real cells. In addition, for given intracellular concentrations of both calcium and calcium-binding proteins the regularity of these oscillations changes and reaches a maximum as a function noise variance, and the overall system dynamics displays stochastic coherence. We conclude that calcium-binding proteins may have an important and active role in cellular communication.

Animal cells communicate through the exchange of molecules secreted in the extracellular environment and/or by cell-cell contacts through the chemical interactions of molecules expressed at the cell surface. Cell communication is at the basis of animal physiology and pathology, and the information stored in the environment must ultimately be decoded in the cell and propagate to the nucleus where target genes can either switch on or off in response to these stimuli. Calcium (Ca 2+ ) is the most versatile second messenger in living cells and translates the information stored in the extracellular environment in time-dependent variations of Ca 2+ intracellular concentration. These may take the form of waves, bursts and oscillations that propagate in time and space in the cell and through adjacent cells [1,2,3].
Ca 2+ oscillations occur in a large number of cell types such as excitable (e.g. neurons, cardiac cells) and non excitable (e.g. hepatocytes, endothelial cells) cells, either spontaneously or after stimulation by hormones, cytokines and neurotransmitters, and they drive important functions such as brain and cardiac activity, immune cell activation, hormone secretion and cell death [1,2].
Two aspects of intracellular Ca 2+ oscillation have received little attention, with some notable exceptions [4,5]: Ca 2+ oscillations propagate in the cell in the presence of Ca 2+binding proteins that act as buffers, taking up to 99% of Ca 2+ in the cell; the mechanisms of Ca 2+ wave generation is intrinsically noisy. Here we address both aspects from a dynamical perspective.
Ca 2+ concentration within cells is controlled by reversible binding to specific classes of proteins that act as Ca 2+ sensors and decode the information carried by FM and AM modulation of the Ca 2+ oscillations [6]. Many intracellular proteins can bind Ca 2+ on multiple sites. For example, the activity of key enzymes such as CaM-kinase II is modulated either by direct binding to Ca 2+ or indirectly by chemical interaction with Ca 2+ -binding proteins [2]. Ca 2+ binding to proteins on multiple sites is not driven by enzymatic mechanisms and therefore its chemistry obeys to the law of mass action. In addition, the binding sites on Ca 2+ -binding proteins appear to be all equivalent in presence of the EF hand motif which allows the specific and reversible binding of Ca 2+ ions [2].
In this paper we investigate Ca 2+ dynamics in cells in the presence of Ca 2+ -binding proteins and noise. To this end, the vitamin D-dependent protein calbindin-D29K (CaL) has been chosen as an example of multisite Ca 2+ binding protein, representing a class of several Ca 2+ binding proteins that share many biochemical traits with CaL [2]. CaL is a protein with a molecular weight of 30 kDa, that is expressed in the cytosol of specific cells of the central nervous system such as hippocampal granule cells and cerebellar Purkinje cells [2]. CaL is a member of the EF hand superfamily Ca 2+ -binding proteins. To date, CaL has not yet been found to play a role in Ca 2+ -dependent regulation of enzyme activity although it is expected to act as an intracellular Ca 2+ buffer. This seemingly minor role may nonetheless have a profound effect on cell physiology, and indeed overexpression of CaL in cultured hippocampal pyramidal neurons affects synaptic plasticity and suppressed post-tetanic potentiation [10]. CaL has four reversible binding sites for Ca 2+ and, importantly, the on-off rates for the binding reaction have been determined experimentally. Experiments have also shown the presence of binding sites with two different affinities and a ratio of sites with high and low affinity of 3 to 1 or 2 to 2 [11].
We describe the dynamics of Ca 2+ reversible binding to CaL with the following differential system, which is formally the same that we used in a previous study of multisite protein modification ( [7,8]; see also [9] for a derivation of these equations): We still have to specify the function f ([Ca 2+ ], t); there are many different models of intracellular calcium oscillations, but this choice is not critical, and here we take the minimal and well-known model based on Ca 2+ -induced Ca 2+ release (CICR) as the basic model of Ca 2+ oscillations [12], which is sketched in fig.1 (For recent reviews, see also [3,6] and references therein).
The CICR model must be complemented by a stochastic term because the process of Ca 2+ wave generation is intrinsically noisy [13,14]. dynamics cannot be neglected, and we introduce a stochastic term (as in [15,17] that leads to the following stochastic differential system for the driving function f = f ([Ca 2+ ], t): where ] p ) and ξ(t) is Gaussian white noise with zero mean and variance D.
In these equations (see also fig.1), [Ca 2+ ] is the cytosolic Ca 2+ concentration, whereas [Y ] denotes Ca 2+ concentration in the IP3-insensitive intracellular store. v 0 = 10 −6 M s −1 is the input rate of Ca 2+ from the extracellular medium, v 1 = 7.3 · 10 −6 M s −1 is the parameter related to the IP3-modulated release of Ca 2+ from the IP3-sensitive store. V M 2 = whereas V M 3 = 500 · 10 −6 M s −1 is the maximum rate of release of Ca 2+ from that store into the cytosol in a process activated by cytosolic Ca 2+ ; K 2 = 10 −6 M, K R = 2 · 10 −6 M and K A = 0.9 · 10 −6 M are threshold constants for pumping, release and activation, respectively; k f = 1 s −1 is a rate constant that regulates the passive, linear leak of Ca 2+ from the IP3-insensitive store into the cytosol; k = 10 s −1 regulates the assumed linear transport of cytosolic Ca 2+ into the extracellular medium. The exponents n = m = 2 and p = 4 denote the Hill coefficients characterizing these processes.
The parameter β regulates the saturation of the IP3 receptor and acts as the control parameter which sets the level of the stimulus and varies from 0 to 1. We also introduce a parameter τ ≥ 1 s that scales the rates v 0 , V 1 , V M 2 , V M 3 , k f and k. In this way we can tune the Ca 2+ period of the CICR oscillator to match oscillations observed in real experiments.
We use equations (1) and (2) to investigate numerically the dynamic interplay between Ca 2+ , Ca 2+ -binding proteins and noise in the cell. We integrate the stochastic differential system with the Euler-Maruyama algorithm [16], and for stability reasons we take a short integration time step ∆t = 0.001s.
Before studying the influence of internal noise, we investigate the deterministic system's However, fluctuations of Ca 2+ concentration due to internal noise allow the system to cross the threshold again and oscillate even in the presence of supercritical CaL concentrations ( fig.3). One feature of the simulation outputs in fig.3 is that they are strikingly similar to time series sampled in real cells [18]. In these experiments, Ca 2+ spikes have been observed to occur randomly, and the average interspike interval T measured independently for hundreds of cells is correlated to the standard deviation σ in a rather broad range of T values [18]. We find the same result in our simulations ( fig.3).
Thus, both experiments and the numerical results shown in fig.3, suggest that noiseinduced Ca 2+ spiking and spiking periodicity depend on noise variance, and moreover the observed correlation between noise variance and oscillation frequency also suggest that what we are witnessing here is a form of stochastic coherence [17]. As in [17] we study this aspect using the regularity parameter R, which is the ratio between the average and standard deviation of the interspike interval (the "period" of the oscillations): As already noted in [17], the reciprocal of this quantity is just the "coefficient of variation" often used in neuroscience as an estimator of the regularity interspike intervals. In our simulations (see fig.4) we find that regularity peaks at a given noise variance, and that the position of the peak also depends on the CaL concentration. Thus the whole system dis-  [19]. We conclude that proteins, such as CaL, may not simply act as passive Ca 2+ buffers in cells, but be prime actors in the complex play of cellular communication. Rates v 0 , k and k f denote replenishment of the IP3-sensitive store with extracellular Ca 2+ , the efflux of cytosolic Ca 2+ from the cell and passive leak of Ca 2+ from the store Y into the cytosol. More details can be found in [3]. This classical scheme is modified here to take into account the action of Ca 2+ protein buffers such as calbindin (CaL) that binds Ca 2+ ions on multiple sites. Binding follows the law of mass action and is described by rates k on and k of f . Finally, in our model cytosolic Ca 2+ fluctuates because of internal noise.