On the dynamical structure of oscillations

James Sneyda,1, Jung Min Hana, Liwei Wangb, Jun Chenc, Xueshan Yanga, Akihiko Tanimurad, Michael J. Sandersonc, Vivien Kirka, and David I. Yuleb

aDepartment of Mathematics, University of Auckland, Auckland 1142, New Zealand; bDepartment of Pharmacology and Physiology, University of Rochester, Rochester, NY 14642; cDepartment of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, MA 01655; and dDepartment of Pharmacology, School of Dentistry, Health Sciences University of Hokkaido, Ishikari-Tobetsu, Hokkaido 061-0293, Japan

Edited by Charles S. Peskin, New York University, New York, NY, and approved December 12, 2016 (received for review September 1, 2016) Oscillations in the concentration of free cytosolic Ca2+ are an ferent things, one of the most important of which is Ca2+ itself, important and ubiquitous control mechanism in many types. which both activates and inhibits the IPRs in a complex and It is thus correspondingly important to understand the mecha- not well understood network of positive- and negative-feedback nisms that underlie the control of these oscillations and how their reactions on different time scales. 2+ period is determined. We show that Class I Ca2+ oscillations (i.e., A concept similar to that of the Ca toolbox is important 2+ oscillations that can occur at a constant concentration of in the mathematical modeling of Ca dynamics. Models try to trisphosphate) have a common dynamical structure, irrespective extract fundamental mechanisms, omitting less important details of the oscillation period. This commonality allows the construc- so that the basic skeleton—the basic toolbox components—can tion of a simple canonical model that incorporates this underlying become clear. In the construction of such skeleton models, the dynamical behavior. Predictions from the model are tested, and concept of dynamical structure becomes important. The behav- confirmed, in three different cell types, with oscillation periods ior of a model can be qualitatively described by a set of bifur- ranging over an order of magnitude. The model also predicts that cations and attracting or repelling sets, and this description is Ca2+ oscillation period can be controlled by modulation of the essentially independent of the exact model equations and param- rate of activation by Ca2+ of the receptor. eters used to realize the underlying dynamical structure (in that Preliminary experimental evidence consistent with this hypothe- there can be many different equations and parameters that have sis is presented. Our canonical model has a structure similar to, but the same dynamical structure). not identical to, the classic FitzHugh–Nagumo model. The charac- One important question is how cells can generate Ca2+ oscil- terization of variables by speed of evolution, as either fast or slow lations of widely differing periods, even though they appear to variables, changes over the course of a typical oscillation, leading be using the same elements of the Ca2+ toolbox. Although it to a model without globally defined fast and slow variables. is mathematically trivial to generate oscillations with different periods from a single model (simply by rescaling time), this is inositol trisphosphate receptor | mathematical modeling | cytosolic calcium not, in general, an acceptable physiological explanation, because concentration modeling | multiple time scales it relies on the arbitrary selection of parameter values that are often nonphysiological. Here, we aim, firstly, to show that there is a common dynamical scillations in the concentration of free cytosolic calcium 2+ 2+ structure underlying Ca oscillations of widely varying period O([Ca ]) are a ubiquitous signaling mechanism, occurring in different cell types (such a common structure is by no means in many cell types and controlling a wide array of cellular func- guaranteed) and, secondly, to propose an experimentally testable tions (1–6). In many cases, the signal is carried by the oscilla- tion frequency; for example, Ca2+ oscillation frequency is known to control contraction of pulmonary and arteriole smooth mus- Significance cle (7, 8), as well as gene expression and differentiation (9–11). 2+ Although there are cell types where the frequency of Ca oscil- Oscillations in the concentration of free cytosolic calcium are 2+ lation appears to be less important than the mean [Ca ] (12), an important control mechanism in many cell types. How- an understanding of how Ca2+ oscillation frequency is controlled ever, we still have little understanding of how some cells can remains critical to our understanding of many important cellular exhibit calcium oscillations with a period of less than a second, processes. Interestingly, it appears that the signal may not be car- whereas other cells have oscillations with a period of hun- ried by the absolute oscillation frequency but rather by a change dreds of seconds. Here, we show that one common type of cal- in frequency (13), leading to a signaling system that is robust to cium oscillation has a dynamic structure that is independent of intercellular variability, even within the same cell type. the period. We thus hypothesize that cells control their oscilla- Current understanding of Ca2+ oscillations is based on the tion period by varying the rate at which their critical internal concept of the Ca2+ “toolbox” (5). According to this concept, variables move around this common dynamic structure and cells can express a range of toolbox components [such as Ca2+ that this rate can be controlled by the rate at which calcium ATPase pumps, voltage-gated Ca2+ channels, or Ca2+ chan- activates calcium release from the endoplasmic/sarcoplasmic nels in the endoplasmic/ (ER/SR) mem- reticulum. brane]; by changing the spatial and temporal expression of these toolbox components, cells can control exactly where and when Author contributions: J.S., V.K., and D.I.Y. designed research; J.S., J.M.H., L.W., J.C., X.Y., 2+ A.T., M.J.S., V.K., and D.I.Y. performed research; L.W., J.C., M.J.S., and D.I.Y. performed [Ca ] is increased or decreased. new experiments; L.W., J.C., M.J.S., and D.I.Y. provided the data; J.S. analyzed data; and Here, the most important toolbox components are those that J.S. wrote the paper. act via the stimulation of C to make inositol The authors declare no conflict of interest. trisphosphate (IP3), which binds to IP3 receptors (IPRs) on the This article is a PNAS Direct Submission. membrane of the ER/SR. IPRs are also Ca2+ channels, and bind- 2+ Data deposition: The Matlab figure file corresponding to Fig. 1 B, D, and F can be down- ing of IP3 opens these channels to allow a rapid flow of Ca out loaded from https://figshare.com/s/87662da83a885447d6d8, and can be used by readers 2+ of the ER/SR, and a consequent rise in cytosolic [Ca ]. Pump- with Matlab to view Fig. 1 interactively. 2+ 2+ ing of Ca by Ca ATPases back into the ER/SR restarts the 1To whom correspondence should be addressed. Email: [email protected]. 2+ process, allowing for cyclical Ca release from the ER/SR and This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 2+ oscillations in cytosolic [Ca ]. IPRs are modulated by many dif- 1073/pnas.1614613114/-/DCSupplemental.

1456–1461 | PNAS | February 14, 2017 | vol. 114 | no. 7 www.pnas.org/cgi/doi/10.1073/pnas.1614613114 Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 ny tal. et Sneyd osqec,teIR pnadcoei eidcmanner, periodic a in slowly, close Ca and more to open then, As leading IPRs process). feedback the and negative consequence, slower process), a a (in feedback plateau a positive to decrease fast a (in [Ca Ca by Ca inhibited by activated slowly quickly are IPRs that assumption the structure. oscilla- dynamical this generate on based might periods, cells different of how tions for mechanism physiological h elo rme;w alti tutr Hp trumpet.” “Hopf of a structure shape this the call takes we trumpet; bifurcation, a Hopf of upper bell solu- the the periodic near unstable of that, surface tions the is importance particular of using and constructed tions diagram, bifurcation the responses of plotting model version by 3D different a compare of we part with assertion, oscillation. this unperturbed verify the of To phase the to relative perturbation IP of pulses exogenous model to oscillation the an use of response we the predict experimentally, to uncovered be can structure cal Predictions. Model iii) of rate the on operate IPRs. processes the these of inactiva- of activation slower both a but and process, process tion activation faster a contains model in in described given details briefly full is model The open Results an remains which hypothesis, this question. with consistent activation IPR dence Ca of which period at the trolling in rate the changes that to addi- predicts In responds magnitude. model of order our an than tion, with more oscillations over exhibiting vary types, that periods cell different three in mentally behavior dynamical essential Ca I simplified the Class captures a of model construct this we how Thus, show cases. Ca some of cap- in model to least sufficient at Ca be underlie lations, that might structures model dynamical important simple the ture a that suggest models different. constant quite a are Ca at with occur unchanged, that qualitatively is behavior tant of Ca result by the not Ca is Instead, feedback negative Ca slower the and feedback feedback, positive fast negative exhibits IPR the although that, Ca Ca are by oscilla- primarily neither II controlled and Class here, types. considered cell of not range a which like in in oscillations occur tions, and to (17), known models are I this Class called are models Such ii) i) oeerymdl fCa of models early Some h oe epnedpnsciial ntetmn fthe of timing the on critically depends response model The results: experimental following the reproduces model The experi- confirmed and tested are model the from Predictions recent more and older between similarities qualitative The shown have (26–30) IPR the of models detailed Subsequent 2+ 2+ h xsec foscillations. of existence the Ca h silto rqec nrae as Ca increases frequency oscillation The for so do they exist, constant oscillations these When disappears. then As 2+ h pnpoaiiyo h Pswl rtices quickly increase first will IPRs the of probability open the ], idn lwyt niatvtn idn ieo h IPR. the on site binding inactivating an to slowly binding 2+ 2+ [IP n oss natm-eedn anr h resul- The manner. time-dependent a in so does and nu increases. influx nu nune h rqec u sntncsayfor necessary not is but frequency the influences influx 3 c ] 2+ , and 2+ 2+ p p 2+ . ouae h aea hc h P sactivated is IPR the which at rate the modulates nrae,asal ii yl rtapasand appears first cycle limit stable a increases, , [IP p oscillations. siltos(aldte“aoia oe” and model”) “canonical the (called oscillations h siltosta a cu o constant a for occur can that oscillations stebfrainprmtr(i.1.Afeature A 1). (Fig. parameter bifurcation the as 3 odtriewehrteudryn dynami- underlying the whether determine To oriae o qiiru n eidcsolu- periodic and equilibrium for coordinates otipraty the importantly, Most Information. Supporting ] n Ca and 2+ 2+ hs nrsos oase nraein increase step a to response in Thus, . [Ca [IP 2+ 2+ siltos epoieidrc evi- indirect provide We oscillations. 2+ 2+ 3 nu rmotie(21–25). outside from influx u h ipyia mechanisms biophysical the but ], siltos(41)wr ae on based were (14–16) oscillations ] eesrl ohvr 1–0 are (17–20) vary both necessarily sacuilprmtrfrcon- for parameter crucial a is with , Methods and Materials 2+ p siltosta are that oscillations nrae ra the as or increases 2+ 2+ oscillations n more and 2+ [ oscil- IP 3 3 ]. . fIP of trumpet. Hopf the of end through plateau, the passes the a at trajectory point on the toward bifurcation once oscillations Hopf pushed oscillations small the larger is in to back resulting trajectory change states, which the steady downstroke, stable of the (E branch lower occur. on this can higher traverse spike must slightly next it whereupon the the surface, before green of surface the downstroke of the part of lower portions the lower the on occurs Ca solutions, (C pulse frequency. periodic the stable higher When of a D) surface with immediately, green occur the oscillations to whereupon dur- attracted occurs is that it where solution from model the IP of The the section states. ing steady the these indicates of line bifurcation black constant Hopf solid a for denotes HB model and constant the unstable), for for of model states the steady of cycles shows respectively, limit represent, surfaces stable green (B, and and line. blue unstable The blue space. the phase at the occurs in pulse tions The response. the of courses 1. Fig. iii) ii) EF CD AB i) µ µ µ 2+ hr r he ulttvl ifrn epne oapulse a to responses different qualitatively three are There ae h us a are h rjcoyt oiin with position, of a value to large trajectory this the relatively In carried a (17). has (ASMCs) pulse cells the oscillation muscle case, smooth in model airway rise in the a shown by peaks, 1 followed (Fig. oscillation spike frequency between immediate an occurs shows pulse the If h pk,tetaetr scridt oiinfrom position are steady a subsequently stable that to of fluctuations of branch small carried the in downstroke toward resulting is the quickly states, up trajectory moves it higher the which slightly spike, occurs the pulse the 1 frequency. increased (Fig. If an spike with reappear next still can oscillations it before surface and the In along solutions. slowly periodic attracting of region, surface this the position, of a part to lower carried is small trajectory relatively decreas- the the with spike, of the portions of lower the phase to ing close occurs pulse the of If function larger. increasing initially an is frequency periodic is because attracting frequency oscillating; of oscillation continues the surface thus the trajectory of The solutions. part upper the toward 3 pk,tetaetr spse oapsto hr ti trce to attracted is it where position a to pushed is trajectory the spike, . 0 us epne ftemdlwith model the of responses Pulse ,wihrslsi ea eoetenx ek u the but peak, next the before delay a in results which D), 3 us.(A pulse. h scagn lwyadtetaetr utmove must trajectory the and slowly changing is PNAS and A h IP The B) | and c eray1,2017 14, February rmwihi oe ucl othe to quickly moves it which from , .Ti euthspeiul been previously has result This B). c c 3 c c us uhsteslto oapoint a to solution the pushes pulse rmwihi oe quickly moves it which from , oscillation unperturbed and p p p s p | = p F HB hntepleoccurs pulse the When ) 2 (A, 0.12. o.114 vol. sldfrsal,dashed stable, for (solid trajectory post-pulse end ofthe IP h e curve red The p. 3 | pulse and C, and D, o 7 no. end ofthe IP stable limitcycles h 3 pulse unstable limit E F h state stable steady p end ofthe | Time ) Solu- ) h IP the , cycles 3 pulse 1457 and C

CELL BIOLOGY APPLIED MATHEMATICS attenuated to a decreasing plateau. Once p has decreased far A enough, the trajectory crosses the subcritical Hopf bifurca- tion and large-amplitude oscillations reappear. Whether or not the pulsed orbit initially moves toward the sur- face of attracting periodic solutions or toward the curve of stable steady states depends on the orbit’s position relative to a sepa- rating manifold which lies close to the Hopf trumpet. We discuss this manifold further in Discussion but, for now, use the Hopf B trumpet as a convenient approximation to the true separating manifold. Note that there is no general theoretical relationship between the phase of the pulse and whether or not the pulse takes the tra- jectory into the interior of the Hopf trumpet. The effect of the pulse depends crucially on the exact spatial relationship between C the unperturbed oscillation and the trumpet. Furthermore, the pulse response will depend on the magnitude of the pulse. A large enough pulse will always take the trajectory beyond the high p end of the Hopf trumpet, leading to oscillations on a raised baseline. However, these responses to large perturbations are independent of the phase of the pulse. If oscillations on a raised baseline occur only for some pulses, and depend on the Fig. 3. Responses of ASMCs. Ca2+ oscillations were initiated by metha- phase of the pulse, then this indicates the presence of a Hopf , and a pulse of exogenous IP3 was applied at the vertical red line. trumpet. The model also predicts the pulse responses as a func- For A–C, the left and right graphs are two examples of the same type of tion of phase and amplitude of the pulse. However, the difficulty response. The three different types of responses in A–C are as described in of determining the exact amount of IP3 released by the flash, the legend of Fig. 2. For more details of the experimental results, see Sup- combined with the difficulty of doing pulses at a precise phase, porting Information. make these predictions more difficult to test.

In HSY cells, the IP3 pulse results in three distinct types of Experimental Tests of the Predictions. We tested these predictions 2+ 2+ responses. When the pulse occurs between Ca spikes the result in three cell types exhibiting Ca oscillations: HSY cells (a is an immediate spike in c followed by oscillations with increased cell line derived from human parotid epithelial cells), DT40- frequency (Fig. 2A). When the pulse occurs on the downward 3KO cells (a cell line derived from chicken lymphocytes), and phase of the spike, there is a short delay before the spikes resume ASMC. In all three cell types, the Ca2+ oscillations occurred in 2+ with an increased frequency (Fig. 2B). When the pulse occurs the absence of Ca influx, with a frequency that is an increasing close to the peak of a spike the response is an oscillation on a p function of agonist stimulation and, thus, presumably, of . IP3 raised but decreasing plateau (Fig. 2C). was released by flash photolysis across multiple oscillating cells. In ASMCs, an IP3 pulse between oscillation peaks gives an immediate increase in oscillation frequency (Fig. 3A), a pulse on the downward phase of the oscillation gives a short delay before A oscillations resume with higher frequency (Fig. 3B), and a pulse close to the peak of a spike causes an oscillation on a raised but decreasing plateau (Fig. 3C). This third behavior is less obvious in ASMCs than in the other two cells types. The principal dif- ficulty is that the oscillations in ASMCs are an order of magni- tude faster than those in HSY or DT40 cells, making it difficult B to ensure the pulse is correctly timed or short enough compared with the oscillation frequency. In DT40-3KO cells, transfected with and stably expressing type II IPRs, all three behaviors can be clearly seen. Fig. 4A shows the increased frequency in response to a pulse of IP3 applied between Ca2+ spikes, Fig. 4B shows the response when the pulse occurs on the downward stroke of the spike, and Fig. 4C shows C the response when the pulse occurs right on the peak of the oscillation. The small increases immediately after the pulse in the responses in Fig. 4B (indicated by * in the figure) are due to a small signal contamination, from the responses of closely neighboring cells, in the region of interest (ROI). This portion of the signal demonstrates that the IP3 pulse elicits an immediate response in the neighboring cells but not in the cell under direct 2+ Fig. 2. Responses of HSY cells. Ca oscillations were initiated by ATP, observation, because the pulse occurred during the downward and a pulse of exogenous IP3 was applied at the vertical red line. For A– phase of the spike. C, the left and right graphs are two examples of the same type of response. (A) When the pulse occurs between Ca2+ spikes, it causes an immediate Discussion spike and an increase in oscillation frequency. (B) When the pulse occurs soon after a Ca2+ spike, there is no immediate response, but, after a delay, We have shown that three different cell types, with oscillation the oscillations resume with a higher frequency. (C) When the pulse occurs periods ranging from seconds to minutes, all respond to pulses close to the peak of the Ca2+ spike, it causes smaller amplitude oscillations of IP3 in a manner that is predicted by the geometric structure on a raised but decreasing baseline. For more details of the experimental shown in Fig. 1. In ASMCs, the oscillation moves around this results, see Supporting Information. structure quickly, giving an oscillation period of a few seconds,

1458 | www.pnas.org/cgi/doi/10.1073/pnas.1614613114 Sneyd et al. 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Ca periods robust with exhibit products oscillations peptides cleavage receptor In complementary to 33). corresponding the (31, expressing antibody cells anti-IgM Ca contrast, with support crosslinking receptor to B-cell frag- ability exhibit by IPRs little altered I type markedly very expressing is stably function cells DT40-3KO channel mentation. 5A, Fig. in IP shown of terms functional are in interrupted, which been obviously in has channels, continuity these peptide pore Remarkably, channel acids 1918–2749). the (amino acids containing protein (amino fragment the consist- C-terminal of fragment a fraction and cytosolic N-terminal 1–1917) the an of ter- in much C resulting of the ing IPR, toward the occurs of cleavage minus Calpain the calpain. to intracellular and encod- correspond the caspase cDNA by that from produced chains assembled products peptide be fragmentation can complementary reported IPRs two previously I have ing type We that mechanism. 32) a (31, such by controlled nrae rm0t 0.1 IPRs [cleaved to traces 0 experimental from the increased in s 100 (Left at (calpain) added was Anti-IgM IPRs. 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CELL BIOLOGY APPLIED MATHEMATICS relatively simple changes in IPR activation kinetics are a possible Hence, when τmax is large enough, a spike in c can lock the IPR into a explanation for the dramatic changes in Ca2+ dynamics caused closed state; the increase in c closes the IPR, which is then unable to escape by fragmentation. quickly from the closed state, even when c decreases again. In this way, changes in τ can effectively control the existence and the period of oscilla- Photoreleased IP3 degrades more slowly than native IP3 (34), h and thus the effects of the pulse persist long enough to be exper- tions (Fig. 5; also see Fig. S3). Our canonical model has a similar mathematical form to the earlier mod- imentally observable as an increase in oscillation period. Never- els of refs. 14–16, in that the open probability is a function of an auxiliary theless, due to the difficulty of measuring [IP3], the exact time variable, h, that, as c increases, decreases Po on a slow time scale. However, it course of the [IP3] after the pulse is uncertain. The model sim- is important to note that the biophysical interpretation is entirely different. ply assumes exponential decay of IP3 at a rate that is consistent In the earlier models, it was the inactivation of the IPRs that was a slow time- with the experimental observations. Thus, the model makes no dependent function of c. In contrast, in the model described here, the slow predictions about how long the oscillation period should remain time dependence on c, and subsequent slow decrease in Po, is controlled by elevated after the IP3 pulse. the rate of IPR activation.

Although our model is similar in many respects to the FHN 2+ model, there is one important difference. In the FHN model, Ca Signaling and Flash Photolysis of Caged-IP3 in ASMCs. Precision cut lung there is a clear separation of time scales between the fast slices (PCLSs) were prepared from female BALB/c mice (8–12 wk) as pre- 2+ viously described (7). HBSS was supplemented with 20 mM Hepes buffer and slow variables. However, in our canonical Ca oscillation (sHBSS) and adjusted to pH 7.4. PCLSs were initially incubated with sHBSS model, the time scale, τh , of the “slow” variable, h, is a decreas- containing 20 µM Oregon Green 488 1,2-bis(o-aminophenoxy)ethane- ing function of c. This feature, which is derived from the previ- N,N,N0,N0-tetraacetic acid tetra(acetoxymethyl) ester, 0.1% Pluronic F-127 ous modeling work of ref. 35, results in a model that gives more and 200 µM sulfobromophthalein (SB) in the dark at ≈30 ◦C for ≈1 h, 2+ realistic Ca spikes than does the traditional FHN model. In and subsequently incubated with sHBSS containing 4 µM caged-IP3, 0.1% this sense, our canonical model is similar to the models of refs. Pluronic F-127 and 200 µM SB for 1 h, followed by de-esterification in 2+ 36 and 37, both of which exhibit a similar time-scale fluidity. We sHBSS containing 200 µM SB for 30 min. The intracellular Ca signal- note, however, that the model here, being based on specific Ca2+ ing of ASMCs in PCLSs was examined with a custom-built, video-rate scan- transport mechanisms, can make experimentally testable predic- ning confocal microscope. To facilitate the flash photolysis, a pulse of 2+ UV light was generated from a mercury arc lamp with a band-pass fil- tions about Ca dynamics, whereas neither the FHN model nor ter (330 nm) and focused to a point into the custom-built confocal micro- the models of refs. 36 and 37 can do so, because they are based scope as described before (40). The fluorescence intensity in a ROI (8 × on quite different underlying biophysical mechanisms. 10 pixels) within an ASMC was determined using custom-written soft- 2+ In our discussion, so far, we have been treating Ca oscil- ware. Relative fluorescence intensity was expressed as Ft /F0, a ratio of the 2+ lations as deterministic. However, Ca oscillations are inher- fluorescence intensity at a particular time (Ft ) normalized to the initial ently stochastic. In stochastic models (13, 18, 38, 39), the period time (F0). Animal maintenance and experimental procedures complied with of the interspike interval is set by the probability that enough the requirements of the Animal Welfare Act, US Public Health Service Policy, IPRs will simultaneously release enough Ca2+ to form a global and NIH guidelines and were approved by the Institutional Animal Care and Ca2+ spike. Changes in interspike interval are then determined Use Committee of the University of Massachusetts Medical School. by the coupling within and between IPR clusters. According to DT40-3KO Cell Culture and Constructs. DT40-3KO cells, an IPR-null back- this approach, the difference between the short period of ASMCs ground cell line, were grown in Roswell Park Memorial Institute 1640 and the long period of HSY cells is simply that of diffusive cou- medium supplemented with 1% chicken serum, 10% (vol/vol) FBS, 100 ◦ pling between IPRs and not a result of changes in the rate of units/mL penicillin, 100 µg/mL streptomycin at 39 C with 5% (vol/vol) CO2. 2+ Ca inhibition of the IPR. It is unclear how much changes The method for constructing plasmid expressing calpain fragmented IPR in interspike interval result from purely stochastic mechanisms type I was described in ref. 31. In brief, IPR type I cDNA flanked by the NheI and how much from deterministic mechanisms. The most likely and NotI sites at the 50 and 30 ends in pcDNA3.1 were used as templates. answer is that changes in both are required. Thus, our model A short nucleotide sequence comprising, in the order of, a stop codon, points to a possible important change in deterministic behavior, NotI and NheI restriction sites, a Kozak sequence, and a start codon was inserted downstream of the sequence coding for glutamic acid 1917 by PCR. although accepting that changes in spatial structure and IPR cou- N- and C-terminal fragments were then extracted by digestions (NheI and pling is also very likely to be important. NotI for N-terminal fragment and NotI for C-terminal fragment) and sub- cloned under two separate promoters in a two-promoter vector. DT40-3KO Materials and Methods cells expressing mouse IPR type II, rat IPR type I, or rat calpain-fragmented Model Construction. We assume a spatially homogeneous cell, with four IPR type I were generated by stably transfecting the cells with correspond- Ca2+ fluxes: through the IPRs, SERCA pumps, a leak from outside, and ing constructs using Amaxa nucleofactor (Lonza) followed by selection with plasma membrane Ca2+ ATPases. Each of these fluxes is modeled in a stan- 2 mg/mL G418 (Life Technologies). dard way (full details are given in Supporting Information). The IPR model is based on the model of ref. 35. In this model α(c, p) and β(c, p, t) are, respec- DT40-3KO Single-Cell Imaging Assay. DT40 cells were first centrifuged at tively, the rates of inactivation and activation of the IPR, where c denotes 200 ×g for 5 min. For Fura-2AM calcium imaging experiments, cells were 2+ [Ca ] and p denotes [IP3]. The open probability, Po, of the IPR is given by resuspended in imaging buffer [10 mM Hepes, 5.5 mM glucose, 1.26 mM 2+ Po = β/(β + 0.4(α + β)). Ca , 137 mM NaCl, 0.56 mM MgCl2, 4.7 mM KCl, and 1 mM Na2HPO4 (pH dh 7.4)] containing 2 µM Fura-2AM (teflabs) for 20 min. Fura-2 loaded cells In addition, ref. 35 shows that β = m∞(c, p)h(c, t), where τh(c) dt = h∞(c) − h, and where τh and h∞ are decreasing functions of c. In particular, were perfused with imaging buffer to establish the recording baseline fol- 4 4 4 τh = τmaxKτ /(c + Kτ ). lowed by stimulation with imaging buffer containing anti-IgM (southern To understand intuitively how this model behaves, it is helpful to con- Biotech) to induce IPR activation. Calcium imaging was acquired by using a sider the response to a step increase in c. When c is increased and held Till Photonics imaging system. For uncaging assay, after centrifugation, cells fixed, m∞ increases immediately, so β increases immediately, so Po increases were resuspended in imaging buffer containing 1 µM Fluo-8 AM (teflabs) immediately. More slowly, h will decrease to h∞, thus decreasing Po. Hence, and 2 µM iso-Ins(1,4,5)P3/PM (caged) (enzolife) for 30 min; 50 nM trypsin in response to a step increase in c, Po will first increase and then decrease was used to induce calcium response followed by a UV flash to uncage the more slowly to a lower plateau. Note that the decrease in Po will occur with IP3 at the indicated time points. rate proportional to 1/τh, which increases when c increases. 2+ The response to a decrease in c depends on the value of τmax. A decrease Ca Signaling and Flash Photolysis of Caged-IP3 in HSY Cells. Media and cell 2+ in c leads to an immediate decrease in m∞ and thus a decrease in Po. If τmax cultures were prepared as previously described (41). IP3-induced Ca oscil- is not too large, h then recovers to its high resting value, allowing the IPR lations were examined with cell-permeable caged IP3, iso-Ins(1,4,5)P3/PM to open in response to a subsequent increase in c. However, if τmax is large (caged) (Alexks Biochemicals). In these experiments, cells were incubated enough, h cannot recover quickly to its resting value and so remains low for with cell-permeable caged IP3 (2–10 µM) and Fluo-3 (2 µM) in HBSS-H con- 2+ a long time, preventing any subsequent increase in Po. taining 1% BSA for 30 min at room temperature. During monitoring Ca

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