A one-day workshop on the mathematical modelling of cellular calcium signals organised by Dr Stephen Coombes was held at the University of Nottingham on 28 April 2006. The goal of the meeting was to bring together leading experts in the field of experimental and theoretical analysis of cellular calcium dynamics and to provide a stimulating exchange among scientists working on various aspects of the cellular calcium toolbox.
|Prof Jim Keener (Mathematics, Utah)||Modelling Stochastic Calcium Oscillations|
|Dr Thomas Hoefer (Theoretical Biophysics, Berlin)||Coupled oscillations of calcium and IP3: theoretical analysis and experimental identification of underlying feedbacks|
|Prof Gregory D. Smith (Applied Mathematics, College of William and Mary)||Stochastic Modeling of Local and Global Intracellular Calcium Dynamics|
|Dr Martin Bootman (Molecular Signalling, Babraham institute)||The spatial properties of calcium signals modulate contraction in atrial myocytes|
|Dr Martin Falcke (Hahn-Meitner Institut, Berlin)||Mechanism of intracellular Ca2+ oscillations|
|Dr Alex Webb (Plant Sciences, Cambridge)||The Plant Circadian Signalling Networks|
Calcium oscillations are an important means of signalling in many cell types. These oscillations have been studied extensively using ordinary differential equation (ode) models, some with a high degree of complexity. These models generally show that the onset of oscillations is via a subcritical Hopf bifurcation.
The experimental data, however, show that calcium release is not nearly as regular as the ode models suggest, but is highly stochastic in nature. In fact, these data show that oscillations are initially very irregular with large average interspike interval and large variance, eventually settling into a regular oscillation as the bifurcation parameter (in this case [IP3]) is increased.
The reason for the discrepancy between data and models is that calcium release is via events that are fundamentally stochastic in time and discrete in space. Furthermore, the assumptions that permit a whole cell ode model, namely uniformly distributed calcium and the law of large numbers, do not apply under many physiological conditions.
In this talk, I will describe my attempt to develop and analyze mathematical models of the events underlying stochastic calcium oscillations, including a model for spark generation, a model for the spark to wave transition leading to a model of stochastic whole cell calcium release. The result of these is a description of the onset of calcium oscillations that shows improved agreement with experimental observations.
Hormones that act through the calcium-releasing messenger, inositol 1,4,5-trisphosphate (IP3), cause intracellular calcium oscillations. In a standard model, these oscillations originate from calcium feedbacks on the IP3 receptor. Recent experimental studies have shown that IP3 levels oscillate together with the calcium concentration. To investigate the functional significance of this phenomenon, we have developed mathematical models of the interaction of both second messengers. The models account for positive and negative feedbacks of calcium on IP3 metabolism, mediated by calcium activation of phospholipase C and IP3 3-kinase, respectively. The coupled IP3 and calcium oscillations have a greatly expanded frequency range compared to calcium fluctuations obtained with clamped IP3. Therefore the feedbacks can be physiologically important in supporting the efficient frequency encoding of hormone concentration observed in many cell types. This action of the feedback critically depends on the turnover rate of IP3. To shape the oscillations, positive feedback requires fast IP3 turnover, whereas negative feedback requires slow IP3 turnover. The ectopic expression of an IP3 binding protein has been used to decrease the rate of IP3 turnover experimentally, resulting in a dose-dependent slowing and eventual quenching of the Ca2+ oscillations. These results are consistent with a model based on positive feedback of Ca2+ on IP3 production.
Although there is consensus that calcium (Ca) puffs and sparks arise from the cooperative action of multiple intracellular Ca channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca excitability and oscillations is not well understood. Here we present and analyze a stochastic automata network model of instantaneously coupled Ca-regulated Ca channels that gives insight into how the stochastic dynamics of an individual Ca release site depends on channel density and the presence or absence of Ca inactivation. The relationship between such stochastic Ca release site dynamics and global Ca responses will then be discussed in the context of a novel probability density approach to modeling whole cell Ca dynamics. The method involves coupling ODEs for the bulk cytosolic and ER [Ca] to advection-reaction equations for the probability density of the [Ca] in cytosolic and lumenal domains associated with each channel and conditioned on channel state. The probability density approach is computationally more efficient than explicitly spatial Monte Carlo simulations and the representation of local Ca signals inherent in the probability density approach is more realistic than conventional stochastic compartmental models.
DeRemigio H and Smith GD. The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain. Cell Calcium. 38(2):73-86, 2005. (PDF)
Mazzag B, Tignanelli C and Smith GD. The effect of residual Ca2+ on the stochastic gating of Ca2+-regulated Ca2+ channel models. J. Theor. Biol. 235:121-150, 2005. (PDF)
Nguyen V, Mathias R, and Smith GD. A stochastic automata network descriptor for Markov chain models of instantaneously-coupled intracellular Ca2+ channels. Bull. Math. Bio. 67(3):393-432, 2005. (PDF)
Smith GD, Dai L, Miura R, and Sherman A. Asymptotic analysis of buffered Ca2+ diffusion near a point source. SIAM J. Applied Math. 61(5):1816-1838, 2001. (PDF)
Keizer J, Smith GD, Ponce-Dawson S, and Pearson J. Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J., 75(8):595-600, 1998. (PDF)
Keizer J and Smith GD. Spark-to-wave transition: saltatory transmission of Ca2+ waves in cardiac myocytes. Biophys. Chem., 72:87-100, 1998. (PDF)
Smith GD, Keizer J, Stern M, Lederer WJ, and Cheng H. A simple numerical model of Ca2+ spark formation and detection in cardiac myocytes. Biophys. J., 75(7):15-32, 1998. (PDF)
Smith GD. Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel. Biophys. J., 71(6):3064-3072, 1996. (PDF)
Smith GD, Pearson J, and Keizer J. Modeling intracellular Ca2+ waves and sparks. (PDF) In: Computational Cell Biology. Fall C, Marland E, Wagner J, Tyson J, editors. Pages 198-229. Springer-Verlag. 2002.
The spatial properties of calcium signals modulate contraction in atrial myocytes
We are interested in the processes that modulate cardiac calcium signals. In particular, we have been investigating the spatial pattern of calcium signalling during excitation-contraction coupling and how it is altered by 'inotropic' agonists that regulate blood flow. Atrial myocytes are an ideal model system for studying cellular Ca2+ signalling. The Ca2+ response that occurs following depolarisation is a product of the interplay of multiple processes that serve to increase or reduce cytosolic Ca2+ levels. The channels, pumps and transporters that impinge on atrial Ca2+ signalling are expressed in regular patterns, so that the Ca2+ signals in these cells are stereotypic. Depolarisation of an atrial myocyte triggers a Ca2+ signal that originates at the plasma membrane (sarcolemma) of the cells. The success or failure of this sarcolemmal Ca2+ signal to trigger a Ca2+ signal in the centre of the cells depends on the sensitivity of the Ca2+ channels (ryanodine receptors) that are located deeper within the cell. If the central ryanodine receptors are sufficiently sensitive, the Ca2+ signal can propagate in a saltatoric manner between Ca2+ spark sites. Since the Ca2+-sensitive contractile machinery is located in the centre of the cells, the centripetal propagation of the Ca2+ signal is necessary to trigger contraction. Positive inotropic stimuli enhance the centripetal propagation of the Ca2+ signal using mechanisms that enhance Ca2+-induced Ca2+ release. Negative inotropes, on the other hand, would be expected to reduce the centripetal propagation of the Ca2+ signals, and sharpen the subsarcolemmal ring of Ca2+. There is a roughly linear relationship between the centripetal movement of Ca2+ and atrial myocyte shortening. The extent of centripetal propagation of the Ca2+ signal therefore provides an analogue control over cellular contraction. The spatial continuum of Ca2+ signalling from peripherally-localised to global responses provides a mechanism by which atrial myocytes can precisely regulate their contribution to blood pumping in the heart. We will also present some preliminary data modelling atrial responses.
Mechanism of intracellular Ca2+ oscillations
Oscillations on a cellular level can arise by several fundamentally different mechanisms. The whole cell or parts of it may represent an oscillatory medium with oscillatory local dynamics or global events that may be initiated repetitively by non-periodic processes like repetitive wave initiation. The first mechanism implies regular interspike intervals (ISI) while the second one entails ISI distributions exhibiting a linear dependence of the standard deviation on the average ISI. We show that experimental results support the second mechanism.
This provides the framework for a theory of intracellular Ca2+ dynamics as a hierarchy of stochastic events with probabilities to be determined. We present first results for this theory.
The Plant Circadian Signalling Networks
The circadian clock is the internal timekeeper of plants. This clock regulates most aspects of plant physiology. This includes those cellular and ion transport processes controlling responses as diverse as stomatal, leaf and floral movements, hypocotyl elongation, nutrient uptake and inter- and intracellular transport.
The consensus model for plant circadian clock function is becoming refined with detail, especially at the level of molecular processes that underlie timekeeping (the circadian oscillator), and the synchronization of the oscillator with the external day-night cycle (entrainment). However, the signal transduction pathways that communicate endogenous estimates of external time from the molecular oscillator(s) to clock-controlled components of biochemistry and physiology remain poorly understood.
We are investigating the hypothesis that the circadian clock regulates cellular physiology via a Ca2+-based signalling cascade. I will describe new data that demonstrate how the circadian clock encodes information in oscillations of cytosolic free calcium. I will also discuss our advances in identifying other components of the circadian signalling network using automated measurements of physiology and microarray analysis. Lastly, I will describe our entry in to mathematical approaches to understand the role of Ca2+ in the plant circadian signalling network.
Prof Jim Keener (Mathematics, Utah) opened the meeting with a solid introduction about the true nature of cellular calcium dynamics. He discussed the widespread deterministic approaches and opposed them to newly emerging stochastic methods. Prof Keener proposed a mathematical model for the generation of the fundamental building blocks of intracellular calcium dynamics, i.e. calcium sparks or puffs, and showed how the transition from sparks to waves can be understood in terms of a stochastic fire-diffuse-fire model. Dr Thomas Höfer (Theoretical Biophysics, Berlin) spoke next on modeling molecular steps of inositol-1,4,5-trisphosphate (IP3) induced calcium release. He demonstrated how positive and negative feedback of calcium on IP3 production shape the intracellular calcium patterns. He pointed out that such feedbacks may play a central role in frequency encoding. Focussing again on the stochastic character of intracellular calcium dynamics, Prof Gregory D. Smith (Applied Math, College of William and Mary) gave first an excellent overview of the impact of buffers on calcium concentration profiles. He then incorporated them into a stochastic automata network description for a single IP3 receptor cluster and showed the effect of time scale separation between channel dynamics and calcium dynamics on calcium oscillations. Dr Martin Bootman (Molecular Signalling, Babraham institute) addressed the role of calcium in excitation-contraction coupling and presented inspiring experimental findings for atrial myocytes. He brought home that the interplay between cellular components as e.g. calcium channels, mitochondria and SERCA pumps as well as their spatial arrangement are of pivotal relevance for understanding excitation-contraction coupling in atrial myocytes. Dr Martin Falcke (Hahn-Meitner Institut, Berlin) continued Prof Keener's discussion on the stochastic nature of intracellular calcium dynamics and presented thoroughly mathematical arguments on the absence of calcium oscillations in deterministic approaches. He supported these ideas with new experimental results in different cell types and presented a novel method to compute calcium puff frequencies. The meeting concluded with a talk by Dr Alex Webb (Plant Sciences, Cambridge) who gave fabulous insights into the role of calcium in plant circadian rhythm. His experimental data shed new light on the molecular network of the circadian clock and helped to identify new components in this network.
4th May 2006
Cripps Hall - the venue.
Greg Smith and Martin Hemberg.
Victoria Hotel Pub.
Duncan Barrack School of Mathematics, University of Nottingham Martin Bootman Molecular Signalling, Babraham institute John Bothwell Marine Biology, Plymouth Stephen Coombes School of Mathematics, University of Nottingham Neil Dalchau Plant Sciences, University of Cambridge Martin Falcke Hahn-Meitner-Institut, Berlin Nicola Fameli Department of Anesthesiology, Pharmacology and Therapeutics, University of British Columbia Martin Hemberg Department of Bioengineering, Imperial College London Thomas Hoefer Theoretical Biophysics, Berlin Rabab Jafro School of Mathematics, University of Nottingham Oliver Jensen School of Mathematics, University of Nottingham Katerina Kaouri School of Mathematics, University of Nottingham Jim Keener Department of Mathematics, University of Utah John King School of Mathematics, University of Nottingham J Krishnan Department of Chemical Engineering, Imperial College London Greg Lemon School of Mathematics, University of Nottingham Gabriel Lord Department of Mathematics, Heriot-Watt University John Love School of Biosciences, University of Exeter Nick Monk Genomic Medicine, University of Sheffield Grace Nabulo Plant Sciences, University of Nottingham Markus Owen School of Mathematics, University of Nottingham Matias Rantalainen Biological Chemistry, Imperial College london Yilei Ren School of Mathematics, University of Nottingham Vincent Rouilly Department of Bioengineering, Imperial College London Simon Scarle Kroto Research Institute, University of Sheffield Anna Sher Oxford University Computing Laboratory Greg Smith Applied Mathematics, College of William and Mary Jie Song Plant Sciences, University of Nottingham Ruediger Thul School of Mathematics, University of Nottingham Yulia Timofeeva School of Mathematics, University of Nottingham Sylvester Tumusiime Plant Sciences, University of Nottingham Ivan Valent Institute of Molecular Physiology & Genetics, Slovak Academy of Sciences, Bratislava Najl Valeyev Department of Biochemistry, University of Oxford Susheel Varma Kroto Research Institute, University of Sheffield Dawn Walker Computational Systems Biology, University of Sheffield Alex Webb Plant Sciences, University of Cambridge Philip White Warwick HRI Ron Yang Computer & Math Science, University of Exeter Hao Zhu School of Mathematics, University of Nottingham