Modelling the transition from simple to complex Ca 2+ oscillations in pancreatic acinar cells

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1 Modelling the transition from simple to complex Ca 2+ oscillations in pancreatic acinar cells NEERAJ MANHAS 1, *, JAMES SNEYD 2 and KR PARDASANI 1 1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal , India 2 Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand *Corresponding author (Fax, ; , mannumanhas@gmail.com) A mathematical model is proposed which systematically investigates complex calcium oscillations in pancreatic acinar cells. This model is based on calcium-induced calcium release via inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) and includes calcium modulation of inositol (1,4,5) trisphosphate (IP 3 ) levels through feedback regulation of degradation and production. In our model, the apical and the basal regions are separated by a region containing mitochondria, which is capable of restricting Ca 2+ responses to the apical region. We were able to reproduce the observed oscillatory patterns, from baseline spikes to sinusoidal oscillations. The model predicts that calcium-dependent production and degradation of IP 3 is a key mechanism for complex calcium oscillations in pancreatic acinar cells. A partial bifurcation analysis is performed which explores the dynamic behaviour of the model in both apical and basal regions. [Manhas N, Sneyd J and Pardasani KR 2014 Modelling the transition from simple to complex Ca 2+ oscillations in pancreatic acinar cells. J. Biosci ] DOI /s Introduction Oscillations in the concentration of free intracellular calcium are an important control mechanism in many cell types (Berridge 1997; Dupont et al. 2007). The temporal and the spatial information encoded by these oscillations regulates many biological events including gene transcription, protein synthesis, protein degradation, apoptosis, necrosis and exocytosis (Nathanson et al. 1992; Feske et al. 2001; Berridge et al. 2003; Lee et al. 2005; Criddle et al. 2007; Petersen 2009; Low et al. 2010; Matveer et al. 2011). The exocrine cells of the pancreas are involved in producing pancreatic juice, and secrete a hydrolytic enzyme which enters into the duodenum for the digestion of food (Leung 2010). When stimulated by an extracellular agonist such as neurotransmitters or hormones, pancreatic acinar cells exhibit intracellular calcium oscillations (Tanimura 2009; Leite et al. 2010). The agonist binds to the cell membrane and initiates a chain of reactions that leads to the activation of phospholipase C (PLC), which then catalyses the formation of inositol (1,4,5) triphosphate (IP 3 ). IP 3 diffuses through the cytosol of the cell, binds to receptors (IPR) on the endoplasmic reticulum (ER) and opens them, leading to the release of large amounts of calcium into the cytoplasm (Petersen and Tepikin 2008). This release mechanism can affect the open probability of both (IPR) and ryanodine receptors (RyR), leading to the activation and deactivation of the receptors (Petersen et al. 1994; Ashby et al. 2002; Yule and Williams 2012). Then, the calcium is pumped back into the endoplasmic reticulum, or removed from the cell, and the cycle is repeated. This process leads to the formation of the different patterns of calcium responses in pancreatic acinar cell from baseline spikes to sinusoidal oscillations, upon a raised baseline (Yule et al. 1991; Tepikin and Petersen 1992), and can exhibit periodic global waves (Nathanson et al. 1992; Thorn et al. 1993; Kasai 1995; Yule et al. 1996). The acinar cell is polarized both morphologically and functionally (Petersen and Tepikin 2008).The apical and basal regions are separated by a region containing buffer mitochondria (Straub et al. 2000; Laude and Simpson Keywords. [IP 3 ]; calcium oscillations; chaos; FFT; IPR; pancreatic acinar cells; RyR , * Indian Academy of Sciences 463 Published online: 29 April 2014

2 464 N Manhas, J Sneyd and KR Pardasani 2009).The apical region is close to the central duct, and contains not only different densities of IPR and RyR than the basal region, but also different densities of calcium pumps and influx channels (Kasai et al. 1993; Thorn et al. 1993; Nathanson et al. 1994). However, since any differences in the densities of calcium pumps, influx channels, or other membrane-based calcium transport pathways are not well characterized, we focus here solely on the effects of varying densities of IPR and RyR from the apical and basal regions. When stimulated with different agonist concentrations, the acinar cell shows spatial heterogeneity; calcium oscillations in the apical region are significantly different from those in the basal region. Calcium waves that are initiated in the apical region can spread across the entire cell to form global intracellular waves, or can remain restricted to the apical region (Thorn 1993; Thorn et al. 1993; Toescu et al. 1994; Ashby and Tepikin 2002). Despite a great deal of experimental data, the exact mechanism underlying Ca 2+ wave propagation in a single pancreatic acinar cell remains unclear, as do the interactions between the IPRs and RyRs in wave propagation. A number of theoretical models have been developed previously to describe cytosolic Ca 2+ dynamics. A model proposed by Meyer and Stryer (1988) uses cooperativity and positive feedback between cytoplasmic Ca 2+ and IP 3 to give rise to repetitive Ca 2+ spiking. A two-pool model by Goldbeter et al. (1990) predicts that cytosol Ca 2+ oscillations are based on calcium-induced calcium release (CICR). Based on experimental data from Xenopus laevis oocytes, Atri et al. (1993) proposed a biphasic modulation of the IPR, where the Ca 2+ channel is opened by moderate concentrations of cytoplasmic Ca 2+ and inhibited by high cytoplasmic Ca 2+. LeBeau et al. (1999)show that agonist-dependent phosphorylation of the IPR could be important mechanism for agonist-specific calcium oscillations in pancreatic acinar cells. Sneyd et al. (2000) constructed and studied a model for intracellular wave propagation, with particular attention to pancreatic acinar cells. Subsequently, Romeo and Jones (2003) extended the work of Sneyd et al. and focused on the stability of travelling calcium pulses in the pancreatic acinar cell model. Based on the experimental work of Giovannucci et al. (2002), Sneyd and his coworkers constructed (Sneyd et al. 2003) a mathematical model of calcium waves in pancreatic and parotid acinar cells which supposes that concentration of IP 3 is fixed. Simpson et al. (2005) and Ventura and Sneyd (2006) performed a bifurcation analysis of this model by taking the concentration of IP 3 as the bifurcation parameter, and by varying the densities of IPR and RyR. These models are Class 1 models, as they assume that Ca 2+ oscillations are caused by sequential positive and negative feedback of Ca 2+ on the IPR; and that Ca 2+ oscillations occur at a constant value of IP 3 concentration. Class 2 models assume instead that Ca 2+ modulation of IP 3 levels, through feedback regulation of production and/or degradation, is the cause of calcium oscillations. Ca 2+ modulation of IP 3 production and degradation occurs in two principal ways: First, the activity of phospholipase C, and thus the rate of production, is an increasing function of cytoplasmic calcium, [Ca 2+ ]. Second, the activity of the 3-kinase that degrades IP 3 to IP 4 is an increasing function of [Ca 2+ ]. Recently, Sneyd et al. (2006) proposed an experimental method to distinguish between two classes of oscillatory mechanism. It was found that in pancreatic acinar cells, Ca 2+ -dependent IP 3 metabolism is the underlying mechanism driving the calcium oscillations (Sneyd et al. 2006), and thus the calcium oscillations in pancreatic acinar cells are of Class 2. However, there has yet been no detailed analysis of a Class 2 model for calcium oscillations in pancreatic cells. In view of the above, the main aim of the present study is to develop and analyse a model that can help us to understand the variety of Ca 2+ waves in pancreatic acinar cells by taking into account Ca 2+ stimulated production and degradation of IP The model construction 2.1 IPR model The present study is based on the model developed by Sneyd and Dufour (2002) for the type-ii IPR. The time-dependent data used in this model incorporated several ideas suggested by experiments: sequential binding of IP 3 and activating Ca 2+, modulation of IP 3 binding by Ca 2+, fitting of dynamic data instead of steady state data, time-dependent inactivation upon IP 3 binding and saturating binding rates of Ca 2+ (Sneyd and Dufour 2002; Sneyd and Falcke 2004). For completeness we summarize the IPR model here. The receptor is composed of four identical subunits each of which has four binding sites: one for IP 3, two for Ca 2+ activation, and one for Ca 2+ inactivation. The IPR receptor can exist in one of the six states:r (resting state), O (open state), S (shut state), A (activated state), I 1 and I 2 (inactivated states). Movement between the receptor states is sequential and the rates are saturating function of [Ca 2+ ]. In state R, neither Ca 2+ nor IP 3 is bound and the subunit can inactivate directly from this state, i.e. going to state I 1 by Ca 2+ binding. Before the subunit goes into activated state A, it has first to bind IP 3, and subsequently Ca 2+. From the activated state the subunit can inactivate by binding Ca 2+. From the open state O, the subunit can switch into the shut state S by binding IP 3. The parameters are determined by fitting to dynamic data on the time course of the IPR open probability. The simplified diagram of the model is shown in figure 1.

3 Transition from simple to complex Ca 2+ oscillations 465 Figure 1. Schematic diagram of the IPR model. R,O,A,S,I 1 and I 2 denote the fraction of receptor in the respective states and R+O+A+I 1 +I 2 +S=1. Here all ϕ s are the functions of [Ca 2+ ] and are given in detail in Appendix A and all the parameter values are given in table 1. The equations for the transition between the receptor states are: dr ¼ ϕ 2O ϕ 2 ½IP 3 ŠR þ ðk 1 þ l 2 Þl 1 ϕ 1 R ð1þ is clearly seen that the more subunits are in state A, the greater the open probability of the IPR receptor. Thus, the open probability of IPR receptor is taken to be P IPR ¼ ð0:1o þ 0:9AÞ 4 ð6þ The flux through the IPR receptor is given by J IPR ¼ k IPR P IPR Ca 2þ ER Ca2þ ð7þ do ¼ ϕ 2 ½IP 3 ŠR ðϕ 2 þ ϕ 4 þ ϕ 3 ÞO þ ϕ 4 A þ k 3 S ð2þ da ¼ ϕ 4O ϕ 4 A ϕ 5 A þ ðk 1 þ l 2 Þl 2 ð3þ di 1 di 2 ¼ ϕ 1 R ðk 1 þ l 2 Þl 1 ð4þ ¼ ϕ 5 A ðk 1 þ l 2 ÞI 2 ð5þ It is assumed that the IPR receptor consists of four independent and identical subunits. It allows Ca 2+ current when all the four subunits are in open state O, or all four subunits are in activated state A, or some intermediate combination. It 2.2 RyR model In pancreatic acinar cells, theryanodine receptors (RyR) are distributed throughout the cell in both apical and basal regions (Leite et al. 1999; Fitzsimmons et al. 2000; Straub et al. 2000; Ashby et al. 2002; Ashby and Tepikin 2002; Ashby et al. 2003). In the present study we use the RyR model of Keizer and Levine (1996) which was originally developed for cardiac cells. This model represents the dynamic behavior of type-ii RyR. This model assumes that the RyR can exist in one of the four states, two open (O 1 and O 2 ) and two closed (C 1 and C 2 ), as shown in figure 2.By assuming that the transitions (O 1 C 1 ) and (O 1 O 2 ) are fast, the open probability can be expressed as w 1 þ ð½ca 2þ Š=K b Þ 3 P RyR ¼ ð8þ 1 þ ð½ca 2þ ŠÞ 4 þ ð½ca 2þ Š=K b Þ 3

4 466 N Manhas, J Sneyd and KR Pardasani Table 1. Parameters values for the model Receptors Densities. k IPR (apical) 0.71 s 1 k IPR (basal) 0.52 s 1 k IyR (apical) 0.32 s 1 k RyR (basal) 0.24 s 1 v MITO (buffer region) 100 s 1 IPR Receptor Parameters. k s 1 (μm) 1 k s 1 k s 1 (μm) 1 k s 1 k s 1 (μm) 1 k s 1 k s 1 (μm) 1 k s 1 L (μm) l s 1 L (μm) l s 1 (μm) L (μm) l s 1 RyR Receptor Parameters. + k a 1500 s 1 (μm) 4 k a 28.8 s 1 + k b + k c 1500 s 1 (μm) 3 k b 1.75 s 1 k c s s 1 IP 3 Parameters. J 0 1e-3 τ p 1/2.84 k (μm) k 3k 40 s 1 k 5P s 1 k deg 0.1 s 1 Transport parameter. J ER s 1 δ 0.1 γ D c 25 s 1 (μm) V SERCA 120 s 1 (μm) 2 K SERCA 0.18 (μm) V PM 28 s 1 (μm) K PM (μm) Figure 2. Schematic diagram of the RyR model taken from Keizer and Levine (1996).

5 Transition from simple to complex Ca 2+ oscillations 467 where qffiffiffiffiffiffiffiffiffiffiffiffi w is the fraction qffiffiffiffiffiffiffiffiffiffiffiffi of RyR not in state C 2 and K a ¼ 4 k a =kþ 3 a ; K b ¼ k b =kþ b ; K c ¼ k c =kþ c ; Furthermore, w is governed by the differential equation dw ¼ k c ð w ½Ca 2þ Š wþ w ½Ca 2þ Š The function w [Ca 2+ ] takes the form w Ca 2þ ¼ 1 þ ðk a = ½Ca 2þ ŠÞ þ ð½ca 2þ Š=K b Þ 3 1 þ ð1=k c ÞþðK a = ½Ca 2þ ŠÞ þ ð½ca 2þ Š=K b Þ 3 ð9þ ð10þ Here all parameter values are shown in table 1.The flux through the RyR receptor is given by J RyR ¼ k RyR P RyR ca 2þ ER ca2þ ð11þ 2.3 SERCA and plasma membrane (PMCA) pumps Two calcium pumps, the sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA), and the plasma membrane calcium ATPase (PMCA) are located in the ER membrane and the plasma membrane, respectively. When [Ca 2+ ] increases, the SERCA pumps transfer Ca 2+ from cytoplasm into the ER. Meanwhile the PMCA transport Ca 2+ from the cytosol to the extracellular medium. Thus, [Ca 2+ ] returns to its resting state. The plasma membrane calcium pump is modelled by a Hill equation (Borghans et al. 1997; Perc and Marhl 2003; Sneyd et al. 2003; Gin et al. 2007; Ullah et al. 2007; Palk et al. 2010) with a Hill coefficient of 2. Thus, flux from the cytosol to the extracellular pool is given by J PM ¼ V PM ½Ca 2þ Š 2 K 2 PM þ ½ Ca2þ Š 2 ð12þ The SERCA pump is modulated by the calcium concentration in the ER (Favre et al. 1996; Mogami et al. 1998).As the calcium concentration in the ER decreases, the rate of SERCA pump increases. Hence the SERCA pump flux is described by a quasi-hill-form, and is valid only when [Ca 2+ ] ER is bounded away from zero. A more detailed description of this model is shown in Sneyd et al. (2003, 2004), Atanasova et al. 2005a, b; Ullah et al. 2007).Thus, the SERCA pump flux is ½ Š J SERCA ¼ V SERCA K SERCA þ ½Ca 2þ Š 1 ½Ca 2þ ð13þ Š ER Ca 2þ 2.4 Calcium leaks Intracellular Ca 2+ is also modified by the movement of Ca 2+ between the cytoplasm and the cell exterior. The entry of Ca 2+ from the outside (J IN ) is an increasing function of the maximum rate of IP 3 production. Hence (J IN ) is an increasing function of agonist concentration and is modeled as a constant leak (α 1 ) and agonist dependent influx (α 2 v) (Gin et al. 2007; Palk et al. 2010). J IN ¼ α 1 þ α 2 v ð14þ The other passive leak from the ER to cytosol is (J ER ), which is proportional to the difference in calcium concentrations. Such leaks are important to balance the pump fluxes in absence of any IPR or RyR fluxes. 2.5 Mitochondrial uptake Pancreatic acinar cells have an interesting distribution of active mitochondria (Tinel et al. 1999; Straub et al. 2000; Petersen 2005; Petersen and Tepikin 2008; Laude and Simpson 2009; Iino 2010), placed on the border between the apical region and the basal region. They block Ca 2+ diffusion from the Ca 2+ release sites in the trigger zone. We assume the mitochondria only uptakes calcium but do not release any calcium, so the Ca 2+ efflux from the mitochondria is insignificant. Thus, in our model the mitochondria simply as provide an additional Ca 2+ uptake term. This additional uptake term exists only in the mitochondrial buffer regions, situated between the apical and basal regions. Using the model of Colegrove et al. (2000), Sneyd et al. (2003), Atanasova et al. (2005b) we assume that the mitochondrial uptake (J MITO ) is given by ½Ca 2þ Š J MITO ¼ v MITO 1 þ ð1= ½Ca 2þ ŠÞ 2 ð15þ 2.6 IP 3 dynamics The activity of phospholipase C (PLC β ) depends upon [Ca 2+ ](Harootunianet al. 1991; Rebecche and Pentyala 2000; Rhee 2001) and so the rate of production of IP 3 is given by (Young and Keizer 1992; Ullah et al. 2006; Gin et al. 2007). J IP3prod ¼ J 0 ½ v Ca2þ Šþ ð1 αþk 5 τ P ½Ca 2þ ð16þ Šþ k 5 Note that the rate of IP 3 production also depends on the agonist dose v. The parameter k 5 is the dissociation constant for Ca 2+ stimulation of IP 3 production. Here α [0,1]. When

6 468 N Manhas, J Sneyd and KR Pardasani α=0, there is no Ca 2+ feedback on the production of IP 3.If α=1, Ca 2+ exerts positive feedback on IP 3 production. Degradation of IP 3 is also dependent on [Ca 2+ ] (Woodring and Garrison 1997; Nalaskowski and Mayr 2004) and the equation for IP 3 degradation (J IP3deg ) may be expressed as (Politi et al. 2006; Palk et al. 2010):! J IP3deg ¼ J 0 ½Ca 2þ Š 2 k 5p þ k 3k τ p ½Ca 2þ Š 2 ½IP þ k 2 3 Š ð17þ deg The rate of change of IP 3 concentration is given by dip ½ 3 Š ¼ J IP3prod J IP3deg 3. Calcium buffering ð18þ We assume all the calcium buffers are fast, immobile and unsaturated (Sneyd 1994; Wagner and Keizer 1994; Keener and Sneyd 2008). Thus, the calcium buffering is included implicitly in this model by treating all calcium fluxes as explicit fluxes. 4. Calcium model description Combining the above fluxes, and neglecting (for now) the diffusion of [Ca 2+ ], gives the following model: dca ½ 2þ Š ¼ J IPR þ J RyR þ J ER J SERCA þ δðj IN J PM Þ ð19þ dca ½ 2þ Š ER ¼ γ J IPR þ J RyR þ J ER þ J SERCA ð20þ dip ½ 3 Š ¼ J IP3prod J IP3deg ð21þ Here γ is the ratio of the cytoplasmic volume to the ER volume and δ is a parameter which controls the magnitude of the trans-membrane fluxes relative to the trans-er fluxes, without changing the resting calcium concentration. The parameters γ and δ are listed in table 1, and the systematic representations of all fluxes are shown in figure Temporal oscillations and bifurcation analysis To understand wave propagation in the cell, it is important first to solve the model in the absence of diffusion. We solve the system of nine differential equations from (A1 A9) as shown in the Appendix A and study calcium dynamics in the apical and basal regions separately. The rate of mitochondrial uptake (V MITO ) is assumed to be zero in both regions. Because (v) is the agonist-dependent rate of [IP 3 ] production, it will be used as the bifurcation parameter. All bifurcation diagrams in this paper were constructed numerically using Auto (Doedel et al. 2002) and Xpaut (Ermentrout 2002). When α=0, the rate of production ofip 3 is independent of calcium concentration. However, the IP 3 degradation rate still depends upon calcium. The behaviour of the model at different values of v and using the parameters of the apical region is summarized in figure 4A. As v increases, the steady state loses stability; oscillations arise via a supercritical Hopf bifurcation HB1 and disappear again by a subcritical Hopf bifurcation HB2. Thus, there is an intermediate range of v [1.82, 63.17]nMs 1 that supports oscillations. If the basal parameters are used, similar behaviour occurs. The supercritical Hopf Bifurcation occurs at v=2.517nms 1 and the system becomes oscillatory. At v=2.8nms 1, TR (torus point) exists and calcium oscillates with smaller amplitude and with period of around 15 s. As v is further increased, oscillations now occur with higher amplitude and frequency. Finally, the oscillation terminates and system becomes stable again after subcritical Hopf Bifurcation point HB2 at v=51nms 1 as shown in figure 4B. On comparing the diagrams in panels A and B of figure 4 we see that there is an intermediate range of v,i.e v 51.01nMs 1, in which both the apical and basal regions are oscillatory. However, [IP 3 ] oscillates approximately in the range (0.3 [IP 3 ] 10.2)μM for which the apical and basal regions are oscillatory. In both the regions as (v) increases, the oscillation first occurs with a large period and then this period of oscillations decreases. Figure 4 also shows time series at few values of v, for the apical and the basal regions. It also demonstrates the corresponding behaviour of [IP 3 ] in both the regions. In figure 4C, v=2.83nms 1,and[Ca 2+ ] shows sustained oscillations. When v=9.978nms 1,asinfigure4D, as expected, the apical region of the cell undergoes oscillations having constant amplitude but the frequency of oscillations is slightly higher than in figure 4C. On using the parameters of the basal region, at v=2.825nms 1, figure 4E, oscillations occur with smaller amplitude and higher frequency. On moving to v=5.15nms 1, the calcium profile in the basal region is quite different; the first spike is of highest amplitude followed by small amplitude spikes as shown in figure 4F. When α=1, calcium modulates the IP 3 production rate and is described by a Hill function of coefficient 1.The IP 3 degradation also depends upon calcium. When apical parameters are used the behaviour of the calcium response is quite different and complex. The stable focus becomes an unstable focus as (v) varies which supports complicated oscillatory behavior. This behavior is restricted between two Hopf Bifurcation points HB1 and HB2, where a cascade of period

7 Transition from simple to complex Ca 2+ oscillations 469 Figure 3. A simplified model of cellular Ca 2+ homeostasis in a pancreatic acinar cell. Calcium is released from the ER through two receptors J IPR, J RyR and a passive leak J ER. Both IPR and RyR exhibit CICR. Calcium is removed from the cytosol through SERCA pumps J SERCA, and enters into the cell through an agonist dependent influx J IN.Calcium is pumped back to the extracellular medium through PMCA J PM.Calcium modulates the production of IP 3 via PLC. IP 3 is degraded at constant rate k 5p, and at a calcium-dependent phosphorylation rate k 3k. doubling (PD) bifurcations, and several periodic orbits are found as shown in figure 5A. For small values of v, branches of unstable orbits combine with stable orbits in saddle nodes of periodic bifurcation. At each branch and point, an oscillation occurs with different amplitude and period. A more detailed description of branches is shown in figure 5B. The branch of stable oscillations (So) starts from v (HB2,PD1) (PD2), and unstable behaviour is described by various branches which are restricted to v (PD2,PD3) (PD4,PD1), as shown in figure 5A and B. The unstable branch b1 starts from the period doubling bifurcation point PD1. In this branch, the calcium oscillations have different amplitudes and periods ranging between [4.127,5.257] seconds. The branch of unstable oscillations b2, originating from the period doubling bifurcation point PD2, gives rise to oscillations having period ranging between [9.486,10.5]. Between two period doubling bifurcation points PD1 and PD2, there is an unstable branch as shown in figure 5A and B, and it oscillates between period [18.92, 18.97].There are two more unstable branches b3 and b4 originating from PD3 and PD4 respectively. The branch b3 gives rise to oscillations ranging between [18.97,21.02] seconds and unstable branch b4 oscillating with greater period ranging between [37.45,42.25] seconds. Figure 5 also shows time series of calcium and IP 3 oscillations for several values of v.in figure 5A,at (v=31.12nms 1 ), a period doubling bifurcation PD1occurs. As expected the time series is periodic containing repeated two different peaks as shown in figure 5C. At (v= 24.67nMs 1 ) there exists two distinct attractors in phase space. This strong attractor arises because the stable periodic orbit enters the period doubling bifurcation as shown in figure 5D. As v is increased to v=24.74nms 1,figure5E, the periodic orbit undergoes two more period doubling bifurcations, PD3 and PD4, that produce complicated attractors. Thus, different attractors merge and calcium oscillates with many sharp peaks having different amplitudes. When v is small the calcium oscillations in the apical region have different amplitudes and smaller frequencies. As v increases oscillations occur with higher frequency, and smaller amplitude. [IP 3 ] also oscillates with its peak lagging the calcium peak as shown in figure 5C E. The amplitude of [IP 3 ]is small and oscillates between [0.21,1.5]μM depending on the parameter values. When basal parameters are used calcium oscillations occur only between two Hopf bifurcation points HB1 and HB2.The subcritical point loses its stability at v= 20.94nMs 1 and becomes stable again after supercritical point at v=44.65nms 1.Figure5G shows the bifurcation diagram in the basal region of the cell. As seen in the diagram the amplitude of the oscillation is quite small ranging between [0.07,0.17]. Both [Ca 2+ ] and [IP 3 ] oscillate in phase and there is sinusoidal behaviour of [IP 3 ] and [Ca 2+ ] waves (figure 5F).

8 470 N Manhas, J Sneyd and KR Pardasani Figure 4. At α=0 partial bifurcation diagrams of the model (with no diffusion) for both apical and basal regions corresponding to equations A1 A9. Panel A describes the bifurcation diagram of the model, using the parameters of the apical region. Panel B describes the response of the basal region. The blue dotted lines represent unstable branches, solid black lines represent stable periodic solutions, TR represents torus bifurcation point, and HB1 and HB2 are Hopf bifurcation points. The insets in panel A and panel B show the period of periodic orbit in the apical and basal regions respectively as a function of agonist v. Panels C and D show the time series from the apical region for two different values of v, i.e and Panel E shows the time series from basal region at v=2.82, (TR point) and panel F is the calcium profile at v=5.15nms 1.

9 Transition from simple to complex Ca 2+ oscillations 471 For apical and basal regions, there are intermediate values of v [20.94,44.65]nMs 1 for both the regions are oscillatory. Thus, the [IP 3 ] oscillates between [0.35,0.9]μM for which both the regions are oscillatory. Here, we discuss the bifurcation diagrams (figures 4 and 5) in some detail. Now it is important question to study: why the dynamic behaviour of calcium is quite different in panel A of figures 4 and 5 (Apical region), and panel B of figure 4 as well as panel G of figure 5 (Basal region), as the calcium behaviour in these regions are described by the same differential equation. In our model the apical and basal regions are only different because of the receptors densities. The apical region contains higher receptor densities than the basal region (the receptors parameter values are shown in table 1). We fix the receptors densities in such a way that both the regions are oscillatory. It is also known that continuously increasing or/ decreasing the receptor densities in such complex models give rise to the appearance and disappearance of oscillations (for a review, see Ventura and Sneyd 2006). Therefore, in this model the parameters are chosen in such a way that they give the prescribed dynamic behaviour. Moreover, ν is the agonist dependent rate of IP 3 production and parameter α shows the dependency of IP 3 production on [Ca 2+ ]. So slight change in these two parameters v and α give rise to different pattern of calcium oscillations in these regions. When α=1 (figure 5), it is worthwhile to see the behaviour of both the region as calcium modulates the IP 3 production and degradation. If α=0, calcium does not modulate the IP 3 production, and its degradation depends on calcium. Therefore, the calcium oscillation are caused by the Ca 2+ modulation of IP 3 production and degradation, and so, on changing the value of both the parameters v and α, the rate of change of IP 3 varies and thus the calcium dynamics in these regions. Hence, we get the different bifurcation diagrams in both the regions. It also clears the phenomenon that in pancreatic acinar cells, the oscillation are caused by the production or degradation of IP 3 through its regulation by calcium. Such type of behaviour is common to many previous models of [Ca 2+ ] oscillations (Romeo and Jones 2003; Simpson et al. 2005; Ventura and Sneyd 2006; Keener and Sneyd 2008) and also agrees with the experimental results. It is well known that the application of acetylcholine (Ach) gives rise to sinusoidal, raised-baseline, high-frequency spiking, while cholecystokinin (CCh) gives low-frequency baseline spiking (Petersen et al. 1991; Yule et al. 1991; Tepikin and Petersen 1992; Habara and Kanno 1993; Williams and Yule 2012). Thus, temporal oscillations in these model matches with experimental observations. 5. Numerical simulation and steady state results To understand the wave propagation in the whole of the cell, it is important to couple both of the regions. So we couple both the apical and basal region of the cell with diffusion. We assume the cell is 20μm in length. The cell is discretized by 40 grid points. The leftmost 8 grid points are assumed to be the apical region, whereas the next 6 grid points are consider the mitochondrial buffer region. Within the mitochondrial region there is additional calcium uptake through the mitochondrial uptake mechanism. Thus, this region of mitochondria acts solely as a buffer region separating the apical and basal region of the cell. The remaining grid points represent the basal region. The model equations, including diffusion, for ([Ca 2+ ],[Ca 2+ ] ER,[IP 3 ]) are solved by the backward time central space (BTCS) scheme in one spatial dimension. Hence, the concentration of all species ([Ca 2+ ],[Ca 2+ ] ER,[IP 3 ]) is dependent on the coordinates x (space) and t (time). No flux boundary conditions are applied at the edge of the integration area. A time step of Δt=0.005 and grid distance Δx=0.475 are used during the simulations. It is important to first find the steady state of the model variables. Hence the steady state of the model is a found by solving the nine differential equations (one with diffusion) as shown in (Appendix A) for v=0 until the approximate steady state is reached. In this model it is noted that the resting calcium concentration in ER and cytoplasm is spatially heterogeneous even when v=0. When this happens, first, there are different densities of RyR receptors in both the apical and basal regions that give different resting leaks from ER, resulting in the different resting ER calcium. Second, mitochondria uptake the increased calcium, which decreases the cytosolic calcium concentration. The steady state values of all variables are used as the starting point of the simulation. We assume that IP 3 diffuses so quickly that it remain essentially spatially homogeneous throughout the cell, and thus explicit diffusion of IP 3 does not appear in the model. 6. Coupling of regions The pancreatic acinar cell is a cell that transforms local calcium spikes to a global calcium transient. We simulate the diffusion model with different agonist concentrations by changing the IP 3 production rate (v). The different types of complex oscillations are illustrated in figures 6 8 both as a function of time and in phase space. Calcium oscillations are seen for (v) up to 180nMs 1. Any further increase in the IP 3 production rate results in no calcium oscillations. In this model we are able to reproduce sustained, periodic and chaotic oscillations. For small values of agonist (v), calcium oscillations are seen which are not spatially homogenous because of the different receptor densities in the apical and basal region. Moreover, the band of mitochondria also affects the propagation of waves from apical to basal region. In figure 6A we show simulations for v=10nms 1 in the absence of the mitochondrial buffer region, i.e. (V MITO =0s 1 ), the apical oscillations are transmitted to the basal region as an intracellular wave. Thus, calcium is actively propagated

10 472 N Manhas, J Sneyd and KR Pardasani

11 Transition from simple to complex Ca 2+ oscillations 473 Figure 5. Partial bifurcation diagrams and the time series of the model (with no diffusion) corresponding to equations A1 A9 for various values of agonist v by setting α=1. Panels A and G demonstrate the partial bifurcation diagrams of the apical and basal regions showing maximum and minimum of periodic orbits as a function of agonist v for both apical and basal regions, respectively. Panel B is the magnified view of the unstable branches in apical region. Panels C E show the dynamical response of apical region at v=31.12,24.67,24.74nms 1 respectively, and panel F is the time series of basal region which are represented as a plot of cytosolic calcium [Ca 2+ ] vs. time(s) at v=24.31nms 1. The insets in panels A and G show the period of periodic orbit in both apical and basal region respectively. The stable oscillations curves (dark black lines) are labelled as (So). Here HB1 and HB2 are the Hopf bifurcation points and PD denotes the period doubling point in the bifurcation diagrams. The various broken blue lines denote instability. by the diffusion of calcium from release sites by the process of calcium-induced calcium release (CICR). To see the behaviour more clearly, we present the magnified view of time series for one spike in the inset of Figure 6A. As seen the amplitude of basal region is high (blue line) and pulled towards the left (basal region). This entrainment of the basal region towards left shows that the apical response is propagated across the basal region. However, when the band of mitochondria is present, i.e. (V MITO =100s 1 ) (figure 6C), the apical calcium response is not converted to global waves. The inset in figure 6 C shows the amplitude of basal spike is small (blue line) and pulled towards the right (apical region), which means mitochondria play a significant role by restricting the propagation of calcium waves. This prediction Figure 6. Complex calcium responses of the spatially heterogeneous model based on the interplay between CICR and the calciumstimulated production and degradation of IP 3. Oscillatory responses are computed with two different agonist values. Time traces were taken from two places in the cell; one is in the apical region at x=6μmand the second is from basal region atx=17μm. Panel A represents the time series without mitochondria (V MITO =0s 1 ), and panel C denotes the calcium response with mitochondria (V MITO =100s 1 ) when stimulated with 10nMs 1 agonist concentration. Panel B demonstrates the heterogeneous calcium response of the cell in both the regions when stimulated with (v=50nms 1 ) agonist value. The black line shows the response of apical region and blue line shows the response of basal region. The insets in panel A C demonstrate the magnified view of the spikes. The corresponding attractor in phase space is shown in panel D. All the simulations are performed by settingα=0.98.

12 474 N Manhas, J Sneyd and KR Pardasani is consistent with the experimental data which suggests that at low agonist concentrations, the local apical calcium responses are restricted in pancreatic acinar cell (Kasai et al. 1993; Kasai 1995; Petersen 1995; Straub et al. 2000).Moreover, it is seen, at (v=50nms 1 ) (figure 6B) both the apical and basal regions can oscillate independently and the pancreatic acinar cells act as two coupled oscillators. The apical region oscillates with higher amplitudes than the basal region but the frequency of oscillations are same. The corresponding attractor is shown in figure 6D and [Ca 2+ ] ER shows a different calcium profile than cytosolic [Ca 2+ ]. Here it is important to note that for high value of agonists (without diffusion), the oscillations are seen in both the apical and basal regions. The basal region responds later than the apical region, and gives rise to the appearance of apparent waves. There intracellular waves are occurring due to the differing phase of oscillations in the apical and basal regions. More importantly the basal region is driven only because of the apical region. It means these regions show independent oscillations and behave like two couple oscillators. However, the entrainment between the apical and basal regions is the result of calcium diffusion. In our simulations the mitochondria appear to play little role in shaping the initial response of calcium. We varied the parameter V MITO up to 250 s 1 and found no significant change in the result of figures 6 and 7 for both the regions. Therefore, the spatially heterogeneous calcium response is due to inhomogeneous distribution of receptors rather than the distribution of mitochondria. The oscillations in the cytosol are due to the release of calcium from IPR receptors, RyR receptors, inward and outward fluxes through the process of CICR. It is seen experimentally (Thorn et al. 1993; Pozzanet al. 1994; Leiteet al. 1999; Fitzsimmons et al. 2000;Straubet al. 2000; Ashbyet al. 2002; Ashby and Tepikin 2002; Williams and Yule 2012)that calcium waves propagated in the cell are generally associated with the activation of RyR receptors. Thus, CICR depends on the cooperation between functional IPR and RyR receptors. To test whether RyR receptors are responsible for CICR, we blocked RyR receptors in the cell. We found that oscillations are dampened by removing RyR receptors (figure 7D) and oscillations are totally abolished on removing IPR receptors. Hence, the calcium responses in the apical and basal regions of the cell are partially dependent upon functional RyR receptors. Our results also suggest that initially calcium is released from IPR receptors in apical region because these receptors have an absolute requirement for IP 3 with calcium as an co-agonist for gating, but RyR receptors only require Ca 2+ to initiate CICR. An experimental trace also shows (Yule et al. 1991; Tepikin and Petersen 1992) that intracellular calcium does not show any oscillations when extracellular calcium is set to zero. We recreate this process by setting J IN = 0. The diffusion model does not show any oscillations while J IN =0. As J IN is set to zero, calcium is extracted from the cell through the leak J PM ;it decreases the calcium concentration in the cell and stops oscillations entirely. When J IN is restored to its nonzero value, calcium oscillations are seen again. Our results suggest that irregular wave results from quasi-periodic, aperiodic and chaotic behaviour of the oscillations. The different types of complex oscillations are illustrated in figures 7 and 8. As shown in figure 7A, on stimulation with agonist (v=130nms 1 ), the behaviour of both the regions are quite different. In apical region, the oscillatory behaviour is characterized by the existence of multiple frequencies. It shows the example of quasi-periodic oscillations. The corresponding relationship between cytosol calcium [Ca 2+ ] and ER calcium is shown in figure7b. It demonstrates that all the trajectories are concentrated to one point. Thus, the time series in the apical region greatly resembles the chaotic. As predicted the amplitude of oscillation in basal region is small and is quasi-periodic. The irregularity of oscillation shows up in both the amplitudes and in the time interval between successive calcium spikes. These oscillations are of reduced amplitude and does not undergo large excursion in phase space figure 7C. Thus, in this case the calcium oscillation in basal region is cast as low-dimension chaos. On comparing figures 7 and 8 we observed that oscillations are higher in figure 7 than those in figure 8 for apical and basal regions. There is minor change in the amplitude in both figures in the apical region but there is significant change in the amplitude of oscillations in figure 8 for the basal region as compared to that in figure 7. This implies that the frequency of oscillations increases with agonist. Also the amplitude of oscillations decreases slightly in the apical region and significantly in the basal region with increase in agonist concentration. For more clarification, we performed the fast Fourier analysis (FFT) of the full time series of 300 s for both the apical and basal regions as shown in figure 7. Figure9 shows the power spectra that have been obtained from the time series of apical region at x=6μm (blue lines), and basal regions at x=17μm (red lines). In apical region, we have two peaks and they are not multiples of each other. Therefore, the time series is quasiperiodic with two fundamental frequencies ω Hz and ω Hz. In the basal region, we have two sharp peaks at same positions, but there are three additional peaks (two at the left and one at the right); these three small peaks can be labelled by using two sharp peaks. The correct positions of all peaks are ω , ω , ω , ω , and ω Hz. The discrete lines in the power spectrum for the basal region are harmonic of the wave, and it is the component frequency of the signal that could be the integral multiple of fundamental frequency. For instance, ω 3 =0.55= =2 ω 1.The FFT analysis shows the time series for both the apical and basal regions should not show perfectly periodic nature. The power

13 Transition from simple to complex Ca2+oscillations 475 Figure 7. Oscillatory responces computed with (v=130nms 1) (with diffusion) corresponding to equations A1 A9 in Appendix A and α= 0.98.Time series were taken from two places in the cell. The oscillations with multiple frequences,and amplitutes are shown in panel A. The upper inset (black lines) represents the time series calculated for 300 s taken at x=6μmin the apical region, and the middle inset (black lines) denotes the time evolution at x=17μmin the basal region (300 s) of the cell. The lower inset of panel A demostrate the magnified view of time series for both the apical region (black lines) and basal regions (basal region). Panels B and C refer to the corresponding attactors in phase space in both apical and basal regions respectively. The damped calcium oscillations are shown in panel D on the removal of RyR receptors in both the regions.

14 476 N Manhas, J Sneyd and KR Pardasani Figure 8. The complex calcium response of the model (with diffusion) stimulated with v=170nms 1 and takingα=0.98. Panel A shows the calcium response in apical region (black solid lines) and also in basal region (blue solid lines). The corresponding relationship between ER calcium and cytosolic calcium [Ca 2+ ] in both apical and basal regions is shown in panel B and C respectively. The apical response is taken at x=6μmand basal response is taken at x=17μm. spectra show quasi-periodicity with two fundamental frequencies in both cases: (1) Hz and (2) Hz. 7. Discussion The hallmark of calcium signalling in the pancreatic acinar cell is the temporal spatial complexity. Stimulation by agonists leads to calcium oscillations in the cell, which can vary markedly in space, amplitude, frequency and time course. The present study was devoted to the analysis of temporally and spatially heterogeneous model of the acinar cell. It takes into account the dual action of cytoplasmic calcium on IPR and RyR receptors, such that the different channels can be linked by communication mediated by CICR, production of [IP 3 ] through the activation of PLC and degradation of [IP 3 ]through 3-kinase which is modulated by calcium. This model predicts that complex calcium oscillations in the form of chaos could be seen in pancreatic acinar cells. Our original motivation to study the temporal model was to identify the attracting solutions when both apical and basal regions are stimulated by agonist. It is also important to see how both the regions itself generate autonomous oscillations. Then taking α=[0,1] we modulate the [IP 3 ] production rate through calcium and [IP 3 ] is removed from the cytosol through calcium-dependent phosphorylation and de-phosphorylation with constant rate k 3k and k 5p respectively. At each value of (α) for both the regions, the results are shown in figures 4 5 with the help of partial bifurcation diagrams. All bifurcation diagrams are showing the steady states and oscillatory solutions as well as their stability when [IP 3 ] production is varied. It suggests that the model has a single steady state for small and large values of (v) and there is intermediate range of (v) between two Hopf bifurcation points (HB1 and HB2) where the steady state loses stability and gives rise to branches of oscillatory solutions. The Hopf bifurcation at HB1 is supercritical and at HB2 is subcritical. The unstable and stable branches corresponding to calcium oscillations with period,

15 Transition from simple to complex Ca 2+ oscillations 477 Figure 9. The upper panel blue line shows the FFT power spectra for the full-length time series (300 s) from apical region as shown in figure 7 (upper inset, panel A). The lower panel red lines shows the FFT power spectra for the full-length time series (300 s) from the basal region as shown in figure 7 (middle inset, panel A). The power spectra show the quasi-periodic and chaotic behaviour of time series. amplitude and frequency are in the physiologically relevant range. The cytosolic calcium oscillations occur with [IP 3 ] oscillations as shown in figures 4 5.These results agree with various studies which show that calcium oscillations in pancreatic acinar cell depend upon [IP 3 ] oscillations. The oscillations in the cytosolic [IP 3 ] concentration signify that cytosolic calcium [Ca 2+ ] has significant effect on [IP 3 ] production and degradation. The oscillations in [Ca 2+ ] and [IP 3 ] are not exactly in phase. The average value of [IP 3 ] concentration is dependent on the model parameter (v) which is the control parameter in our simulations. Thus, on changing the average concentration of [IP 3 ], the amplitude, frequency and period of cytosolic calcium change. These temporal model results are consistent with experimental studies (Petersen et al. 1990, 1991; Yule et al. 1991; Nathansonet al. 1992; Tepikin and Petersen 1992; Giovannucci et al. 2002; Williams and Yule 2012)that suggest that on agonist stimulation the cell can evoke varieties of different patterns like sinusoidal, short-lasting and transitions from sinusoidal to transient oscillations. Beside these temporal oscillations, in the pancreatic acinar cell an increase in the intracellular calcium concentrations starts locally, and then propagates within the cell as intracellular waves which are responsible for global calcium oscillations. It is well known that both the receptors (IPR and RyR) are heterogeneously distributed throughout the ER membrane in the cell. Thus, we introduced spatial heterogeneity in our model by assuming thatapical and basal regions have different concentrations of IPR and RyR receptors, and are separated by a highly buffered region with high mitochondrial density. In assuming the receptor densities, our model is consistent with experimental results (Kasai et al. 1993; Lee et al. 1997; Leite et al. 1999; Fogarty et al. 2000; Straubet al. 2000; Ashbyet al. 2002; Williams and Yule 2012) which suggests that the apical region has higher densities than the basal region. The dynamical behaviour of the spatially heterogeneous model is complex. The major outcomes from this model are as follows. 1. At low agonist concentration, mitochondrial uptake can eliminate the intracellular waves. This will happens only for a narrow range of agonist values for which calcium signals are confined to the apical region. 2. It is also observed that at low concentrations of agonist, the wave is propagated by an active mechanism, which means that Ca 2+ signals are transmitted from the apical to basal region by diffusion of calcium, coupled to exhibit release of Ca 2+ from IPR and RyR. 3. At high agonist concentrations both the apical and basal regions oscillates independently and behave like two coupled oscillators. Oscillations vary in amplitudes for both the regions but have the same frequency. 4. We also predict the apparent waves which are transmitted towards basal region occur due to time delay. These intracellular waves are the result of differing phase of

16 478 N Manhas, J Sneyd and KR Pardasani oscillations in the apical and basal region and are referred as kinematic waves. 5. The diffusion model also reveals that the regenerative global calcium waves in the cell are irregular in both amplitude and frequency. Such type of irregularities in the oscillations is the evidence of chaos. These complex calcium waves are only seen under appropriate agonist concentration. The irregular response of cellular oscillations is absorbed as quasi-periodic. 6. When the wave is propagated by an active mechanism it travels between 5 and 35μm/s, which are speeds in the physiological range. However, the kinematic waves can travel much faster, around μm/s. These faster speeds are not usually observed in pancreatic acinar cells, although similar speeds are seen in parotid acinar cells, which are structurally very similar. It is not currently known what effect the wave speed is an emergent property of the system of calcium release and reuptake, without any particular physiological significant at all. In these simulations the spatially heterogeneous Ca 2+ response is due to the inhomogeneous distribution of IPR and RyR receptors in the apical and basal regions rather than the localization of mitochondrial distribution. However, as long as agonist lies in a relatively narrow range, the increased calcium in the cytosol is blocked by the mitochondrial buffer region which prevents the spread of global signals. Our results suggest that at super-maximal agonist concentration, there is a complex interplay between the two oscillatory regions even when the mitochondrial buffer region is present. It is because of high IPR density in apical region. This region tends to have shorter period than the basal region, and the combined behaviour of both the regions is complex. The waves which start from the apical region can spread through the basal region, but the basal response is not completely entrained by the apical region. Thus, not only the global waves that initiate from the apical region transmit across the basal region but the waves which start from basal region, and not entrained in apical region, also propagate across the basal region. Such types of waves are suggested (Xu et al. 1996) in his experiments. They observed that although the calcium waves in pancreatic acinar cell are usually initiated in apical region, they can also originate from the basal region. In this model the chaotic oscillations may only occur because an increase in cytosolic Ca 2+ has two opposite effects. In one hand, due to CICR calcium is released from internal store and it increases the calcium concentration. This increase in the calcium concentration boosts the [IP 3 ] production rate. On the other hand the 3-kinase which is also modulated by Ca 2+ degrades the [IP 3 ] concentration. It reduces the rate of calcium release into the cytoplasm. These counteracting effects of Ca 2+ are the source of chaos in present model. The apical region behaves as a forced oscillator and the oscillations occur with multiple frequencies. Moreover the basal region is also attached with the apical region which gives rise to aperiodic oscillations. This seems only because the quasi-periodic waves from the apical region are transmitted towards the basal region, but the basal region itself produces autonomous oscillations. This merging of two waves gives rise to aperiodic oscillations in the basal region. It is also important to note that this type of chaotic dynamics is hard to distinguish from periodic oscillations. One approach which is powerful tool for analysing the irregularities in cytosolic calcium oscillation is the return map reconstruction method. Based on these return map approaches the analysis of some time series in pancreatic acinar cell (Strizhak et al. 1995) led to the conclusion that these time series could result from chaotic dynamics. Our results are consistent with these observations. We have observed non-periodic propagation at both apical and basal areas at high values of agonist. However, to check the exact nature of the time series, which we have recorded the representative points at each region, we looked at them in the frequency domain; we used fast Fourier transformation (FFT) (Pushpavanam 1998; Shajhan et al. 2009, Nayak et al. 2013) technique to see their behaviour in the frequency domain. We obtained two incommensurate frequencies (0.28 and 0.38 Hz) at the apical region; therefore, it indicates quasi-periodic behaviour (Pushpavanam 1998; Shajhan et al. 2009; Nayak et al. 2013). We also obtained quasi-periodic behaviour with two fundamental frequencies at the basal region. Since the frequencies appear at the same positions in the both cases, we therefore expect the propagation speed of calcium wave from apical to basal areas to not change. Moreover, in various experiments (Yule and Gallacher 1988; Petersen et al. 1990; Nathanson et al. 1992; Tepikin and Petersen 1992; Yule et al. 1996; LeBeau et al. 1999; Straub et al. 2000, Petersen 2005) on pancreatic acinar cells it was found that the calcium oscillations have both periodic and irregular periodicity (Yule and Gallacher 1988; Strizhak et al. 1995) which is always between 0.1 and 2 Hz. It was also found that the pancreatic acinar cells show up to 17 oscillations per minute and our results are consistent with these observations. The frequency of calcium oscillations/waves modulates with the agonist concentrations. In construction of this type of model, it is important to consider the assumption carefully. Currently it is not possible to construct the model using parameters only from pancreatic acinar cells. Thus, to understand wave propagation in the cell, it is important to take the data from other sources. Three isoforms of IPR receptors (types I, II and III) have been identified in pancreatic acinar cells. These three subtypes are heterogeneously distributed throughout the cytoplasm. It is now clear that these subtypes are differently regulated by Ca 2+ and IP 3. In particular, different open probabilities at