Dynamics of intracellular Ca 2+ oscillations in the presence of multisite Ca 2+ -binding proteins
aa r X i v : . [ q - b i o . S C ] S e p Dynamics of intracellular Ca oscillations in the presence ofmultisite Ca -binding proteins Roberto Chignola
Dipartimento di Biotecnologie, Universit`a di Verona,Strada Le Grazie 15 CV1, I-37134 Verona, Italy andIstituto Nazionale di Fisica Nucleare – Sezione di Trieste,Via Valerio 2, I-34127 Trieste, Italy
Alessio Del Fabbro and Edoardo Milotti
Universit`a di Trieste and Istituto Nazionale di Fisica Nucleare – Sezione di Trieste,Via Valerio 2, I-34127 Trieste, Italy (Dated: November 21, 2018)
Abstract
We study the dynamics of intracellular calcium oscillations in the presence of proteins that bindcalcium 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. Evenwith high concentrations of calcium-binding proteins, internal noise, which shows up spontaneouslyin cells in the process of calcium wave formation, can lead to self-oscillations. This produces oscilla-tory behaviors strikingly similar to those observed in real cells. In addition, for given intracellularconcentrations of both calcium and calcium-binding proteins the regularity of these oscillationschanges and reaches a maximum as a function noise variance, and the overall system dynamicsdisplays stochastic coherence. We conclude that calcium-binding proteins may have an importantand active role in cellular communication.
PACS numbers: 87.17.Aa, 87.18.Tt, 87.19.ln, 05.40.-a ) is the most versatile second messenger in living cellsand translates the information stored in the extracellular environment in time-dependentvariations of Ca intracellular concentration. These may take the form of waves, burstsand oscillations that propagate in time and space in the cell and through adjacent cells[1, 2, 3].Ca 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 sponta-neously or after stimulation by hormones, cytokines and neurotransmitters, and they driveimportant functions such as brain and cardiac activity, immune cell activation, hormonesecretion and cell death [1, 2].Two aspects of intracellular Ca oscillation have received little attention, with somenotable exceptions [4, 5]: Ca oscillations propagate in the cell in the presence of Ca -binding proteins that act as buffers, taking up to 99% of Ca in the cell; the mechanisms ofCa wave generation is intrinsically noisy. Here we address both aspects from a dynamicalperspective.Ca concentration within cells is controlled by reversible binding to specific classes ofproteins that act as Ca sensors and decode the information carried by FM and AM mod-ulation of the Ca oscillations [6]. Many intracellular proteins can bind Ca on multiplesites. For example, the activity of key enzymes such as CaM-kinase II is modulated eitherby direct binding to Ca or indirectly by chemical interaction with Ca -binding proteins[2]. Ca binding to proteins on multiple sites is not driven by enzymatic mechanisms andtherefore its chemistry obeys to the law of mass action. In addition, the binding sites onCa -binding proteins appear to be all equivalent in presence of the EF hand motif whichallows the specific and reversible binding of Ca ions [2].In this paper we investigate Ca dynamics in cells in the presence of Ca -binding proteinsand noise. To this end, the vitamin D-dependent protein calbindin-D29K (CaL) has beenchosen as an example of multisite Ca binding protein, representing a class of several Ca -binding proteins. To date, CaL has not yet beenfound to play a role in Ca -dependent regulation of enzyme activity although it is expectedto act as an intracellular Ca buffer. This seemingly minor role may nonetheless have aprofound effect on cell physiology, and indeed overexpression of CaL in cultured hippocam-pal pyramidal neurons affects synaptic plasticity and suppressed post-tetanic potentiation[10]. CaL has four reversible binding sites for Ca and, importantly, the on-off rates forthe binding reaction have been determined experimentally. Experiments have also shownthe presence of binding sites with two different affinities and a ratio of sites with high andlow affinity of 3 to 1 or 2 to 2 [11].We describe the dynamics of Ca reversible binding to CaL with the following differentialsystem, which is formally the same that we used in a previous study of multisite proteinmodification ([7, 8]; see also [9] for a derivation of these equations): d [ CaL ] dt = − k on [ CaL ][ Ca ] + k off [ CaL ] d [ CaL ] dt = − k off [ CaL ] − k on [ CaL ][ Ca ] +4 k on [ CaL ][ Ca ] + 2 k off [ CaL ] d [ CaL ] dt = − k off [ CaL ] − k on [ CaL ][ Ca ] +3 k on [ CaL ][ Ca ] + 3 k off [ CaL ] d [ CaL ] dt = − k off [ CaL ] − k on [ CaL ][ Ca ] +2 k on [ CaL ][ Ca ] + 4 k off [ CaL ] d [ CaL ] dt = − k off [ CaL ] + k on [ CaL ][ Ca ] d [ Ca ] dt = − X n =1 n d [ CaL n ] dt + f ([ Ca ] , t ) (1)where square brakets denote molar concentrations of each chemical species, CaL i with i =0 , , ..., i Ca bound ions, and f ([ Ca ] , t ) is a function describingthe time-dependent oscillations of Ca in the cell. We take the following values for themodel parameters [11]: k on =1 . · M − s − , k off =2 .
275 s − , k on =7 . · M − s − , and k off =39 .
501 s − . The system of equations (1) has no analytical solutions and thus it must3e solved with numerical methods.We still have to specify the function f ([ Ca ] , t ); there are many different models ofintracellular calcium oscillations, but this choice is not critical, and here we take the minimaland well-known model based on Ca -induced Ca release (CICR) as the basic model ofCa oscillations [12], which is sketched in fig.1 (For recent reviews, see also [3, 6] andreferences therein).The CICR model must be complemented by a stochastic term because the process ofCa wave generation is intrinsically noisy [13, 14]. Evidence of intracellular noise comesfrom direct inspection of sampled time-series in different cell types at the mesoscopic scaleas well as from experimental work which shows that, in the microscopic domain, Ca wavesoriginate from discrete random events, called puffs, occurring at specific sites in the cellsand composed of a small number of Ca ions. Several puffs cooperate to the formationof a supercritical nucleus that initiatiates a Ca wave. The probabilistic character ofnucleation introduces variability into the wave period, with a standard deviation which hasbeen experimentally estimated to reach values up to 40% [14]. Thus, internal noise in Ca dynamics cannot be neglected, and we introduce a stochastic term (as in [15, 17] that leadsto the following stochastic differential system for the driving function f = f ([ Ca ] , t ): f = d [ Ca ] dt = v + v β − v + v + k f [ Y ] − k [ Ca ] + ξ ( t ) d [ Y ] dt = v − v − k f [ Y ] (2)where v = V M [ Ca ] n ( K n + [ Ca ] n )and v = V M [ Y ] m ( K mR + [ Y ] m ) [ Ca ] p ( K pA + [ Ca ] p )and ξ ( t ) is Gaussian white noise with zero mean and variance D .In these equations (see also fig.1), [ Ca ] is the cytosolic Ca concentration, whereas[ Y ] denotes Ca concentration in the IP3-insensitive intracellular store. v = 10 − M s − is the input rate of Ca from the extracellular medium, v = 7 . · − M s − is the pa-rameter related to the IP3-modulated release of Ca from the IP3-sensitive store. V M =65 · − M s − denotes the maximum rate of Ca pumping into the IP3-insensitive store,4hereas V M = 500 · − M s − is the maximum rate of release of Ca from that storeinto the cytosol in a process activated by cytosolic Ca ; K = 10 − M, K R = 2 · − Mand K A = 0 . · − M are threshold constants for pumping, release and activation, respec-tively; k f = 1 s − is a rate constant that regulates the passive, linear leak of Ca from theIP3-insensitive store into the cytosol; k = 10 s − regulates the assumed linear transport ofcytosolic Ca into the extracellular medium. The exponents n = m = 2 and p = 4 denotethe Hill coefficients characterizing these processes.The parameter β regulates the saturation of the IP3 receptor and acts as the controlparameter which sets the level of the stimulus and varies from 0 to 1. We also introduce aparameter τ ≥ v , V , V M , V M , k f and k . In this way we can tunethe Ca period of the CICR oscillator to match oscillations observed in real experiments.We use equations (1) and (2) to investigate numerically the dynamic interplay betweenCa , Ca -binding proteins and noise in the cell. We integrate the stochastic differentialsystem with the Euler-Maruyama algorithm [16], and for stability reasons we take a shortintegration time step ∆ t = 0 . s .Before studying the influence of internal noise, we investigate the deterministic system’sdynamics both in the absence and in the presence of CaL. Simulations show that in theabsence of CaL, the system undergoes two Hopf bifurcations at β ≈ .
29 and β ≈ . oscillations, until they abruptly switch offat the critical concentration [ CaL ](0) / [ Ca ](0) ≈ . β = 0 .
3, see fig.2). Beyondthis critical CaL concentration no Ca oscillations are observed in the deterministic case.However, fluctuations of Ca concentration due to internal noise allow the system to crossthe 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 totime series sampled in real cells [18]. In these experiments, Ca spikes have been observedto occur randomly, and the average interspike interval h T i measured independently for hun-dreds of cells is correlated to the standard deviation σ in a rather broad range of h T i 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 noise-induced Ca spiking and spiking periodicity depend on noise variance, and moreover theobserved correlation between noise variance and oscillation frequency also suggest that whatwe are witnessing here is a form of stochastic coherence [17]. As in [17] we study this aspect5sing the regularity parameter R , which is the ratio between the average and standarddeviation of the interspike interval (the “period” of the oscillations): R = h T i σ (3)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 oursimulations (see fig.4) we find that regularity peaks at a given noise variance, and that theposition of the peak also depends on the CaL concentration. Thus the whole system dis-plays stochastic coherence, and proteins that bind Ca ions on multiple sites can tune Ca spiking in cells and may ultimately regulate cell communication. Remarkably, experimentswith engineered knock-out mice for CaL expression show severe impairment of motor coor-dination suggesting that CaL has an important role in cerebellar functions and intercellularcommunication [19]. We conclude that proteins, such as CaL, may not simply act as passiveCa buffers in cells, but be prime actors in the complex play of cellular communication. [1] M. J. Berridge, M. D. Bootman, P. Lipp, Nature , 645 (1989).[2] E. Carafoli, L. Santella, D. Branca, M. Brini, Crit. Rev. Biochem. Mol. Biol. , 107 (2001).[3] M. Falcke, Adv. Phys. , 255 (2004).[4] M. Falcke, New J. Phys. , 96.1 (2003).[5] M. Falcke, Biophys. J. , 28 (2003).[6] S. Schuster, M. Marhl, and T. H¨ofer, Eur. J. Biochem. , 1333 (2002).[7] R. Chignola, C. Dalla Pellegrina, A. Del Fabbro, E. Milotti, Physica A , 463 (2006).[8] E. Milotti, A. Del Fabbro, C. Dalla Pellegrina, R. Chignola, Physica A , 133 (2007).[9] S.I. Rubinow, Introduction to Mathematical Biology , pp. 71-80 (John Wiley & Sons, New York,1975, Dover Publications reprint 2003).[10] P. S. Chard, J. Jord´an, C. Marcuccini, R. J. Miller, J. M. Leiden, R. P. Roos, G. D. Ghadge,Proc. Natl. Acad. Sci. U.S.A. , 5144 (1995).[11] U. V. N¨ageri, D. Novo, I. Mody, J. L. Vergara, Biophys. J. , 3009 (2000).[12] A. Goldbeter, G. Dupont, M. J. Berridge, Proc. Natl. Acad. Sci. U.S.A. , 1461 (1990).[13] Y. Tao, J. Choi, I. Parker, J. Physiol. , 533 (1995).[14] J. Marchant, I. Parker, EMBO J. , 65 (2001)
15] H. Li, Z. Hou, H. Xin, Phys. Rev. E. , 061916 (2005).[16] D. J. Higham, SIAM Rev.
525 (2001).[17] R. C. Hilborn, R. J. Erwin, Phys. Rev. E , 031112 (2005).[18] A. Skupin, H. Kettenmann, U. Winkler, M. Wartenberg, H. Sauer, S. C. Tovey, C. W. Taylor,M. Falcke, Biophys. J. , 2404 (2008).[19] M. S. Airaksinen, J. Eilers, O. Garaschuk, H. Thoenen, A. Konnerth, M. Meyer, Proc. Natl.Acad. Sci. U.S.A. , 1488 (1997). IG. 1: Schematic representation of the CICR model. A signal S acts on receptor R and triggers theproduction of IP3 that stimulates the release of Ca ions from IP3-sensitive intracellular stores.IP3 regulates the constant flow of Ca into the cytosol ( v β ). Cytosolic Ca ions are pumped intoan IP3-insensitive store ( v ); Ca in this store (Y) is transported back to the cytosol in a processactivated by Ca itself ( v ). Rates v , k and k f denote replenishment of the IP3-sensitive storewith extracellular Ca , the efflux of cytosolic Ca from the cell and passive leak of Ca fromthe store Y into the cytosol. More details can be found in [3]. This classical scheme is modifiedhere to take into account the action of Ca protein buffers such as calbindin (CaL) that bindsCa ions on multiple sites. Binding follows the law of mass action and is described by rates k on and k off . Finally, in our model cytosolic Ca fluctuates because of internal noise. IG. 2: Upper panel: Bifurcation diagram for the deterministic dynamics without CaL: in theabsence of CaL the system displays two Hopf bifurcations as a function of the control parameter β . Lower panel: here β =0.3, which brings the system beyond the first Hopf bifurcation, into theoscillatory regime, however if CaL rises above a critical concentration ratio [ CaL ](0) / [ Ca ](0) ≈ . spiking. IG. 3: Simulation outputs that show Ca spiking in presence of the Ca -binding protein CaLand noise. In all panels the initial conditions are: [ Ca ](0) = 2 · − M, [
CaL ](0) / [ Ca ](0) = 2, β = 0 .
3. All the other parameters are as described in the text. Top panels: log D = −
16 M .Middle panels: log D = − . . Bottom panels: log D = − . . Time series are shownon the left whereas the plots on the right show the correlation between average interspike interval h T i and standard deviation σ . These simulation results are strikingly similar to those found inactual experiments [18] IG. 4: Plot of the regularity parameter R vs. noise variance for the indicated [ CaL ](0) / [ Ca ](0)ratios. Since both Ca and CaL concentrations vary greatly among different cell types, we use theratios [ CaL ] / [ Ca ] rather than concentrations. The presence of CaL deeply influences the natureof the calcium oscillator and leads to a stabilization of interspike intervals: this regularization ofthe Ca oscillations depends on noise variance and reaches a maximum for a nonvanishing valueof the noise variance.oscillations depends on noise variance and reaches a maximum for a nonvanishing valueof the noise variance.