On the dynamical structure of calcium 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 cell 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 inositol 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 inositol trisphosphate 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 j mathematical modeling j cytosolic calcium not, in general, an acceptable physiological explanation, because concentration modeling j 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/sarcoplasmic reticulum (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 phospholipase 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 j PNAS j February 14, 2017 j vol. 114 j no. 7 www.pnas.org/cgi/doi/10.1073/pnas.1614613114 Downloaded by guest on September 29, 2021 physiological mechanism for how cells might generate oscilla- ABunperturbed post-pulse tions of different periods, based on this dynamical structure. oscillation trajectory Some early models of Ca2+ oscillations (14–16) were based on stable limit cycles µ 2+ unstable limit the assumption that IPRs are quickly activated by Ca and more c HB cycles 2+ slowly inhibited by Ca . Thus, in response to a step increase in end of the IP pulse [Ca2+], the open probability of the IPRs will first increase quickly 3 (in a fast positive feedback process), and then, more slowly, 0 stable steady h state decrease to a plateau (in a slower negative feedback process). As p a consequence, the IPRs open and close in a periodic manner, CD 2+ leading to Ca oscillations that can occur for a constant [IP3]. Such models are called Class I models (17), and oscillations like µ c this are known to occur in a range of cell types. Class II oscilla- 2+ tions, in which [IP3] and Ca necessarily both vary (17–20) are not considered here, and neither are Ca2+ oscillations that are 2+ controlled primarily by Ca influx from outside (21–25). h p end of the IP pulse Subsequent detailed models of the IPR (26–30) have shown 3 that, although the IPR exhibits fast positive feedback and slower EF negative feedback, the negative feedback is not the result of 2+ Ca binding slowly to an inactivating binding site on the IPR. µ 2+ Instead, Ca modulates the rate at which the IPR is activated c by Ca2+ and does so in a time-dependent manner. The resul- tant behavior is qualitatively unchanged, with Ca2+ oscillations that occur at a constant [IP3], but the biophysical mechanisms end of the IP pulse are quite different. p 3 h The qualitative similarities between older and more recent models suggest that a simple model might be sufficient to cap- Fig. 1. Pulse responses of the model with ps = 0:12. (A, C, and E) Time ture the important dynamical structures that underlie Ca2+ oscil- courses of the response. The pulse occurs at the blue line. (B, D, and F) Solu- APPLIED lations, at least in some cases. Thus, we construct a simplified tions in the phase space.