On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling

The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems, and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP3 induced Ca2+ signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system’s general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation

A Bayesian approach to modelling heterogeneous calcium responses in cell populations

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimuli time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environmen

Determining Ca²⁺ spike ISI statistics.

<p>Kolmogorov-Smirnov plots for Ca²⁺ spike sequences in HEK293T cells stimulated with (A) 100<i>μ</i>M and (B) 10<i>μ</i>M carbachol when the ISI statistics is assumed to be an IIG (blue), IP (red) and IG (grey) distribution. Box and whisker plots summarising the Kolmogorov-Smirnov plots for (C) 100<i>μ</i>M and (D) 10<i>μ</i>M carbachol stimulation for the IP, IIG and IG models. The results in (A) and (C) are based on the data shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g001" target="_blank">Fig 1</a>. In (C) and (D), the box extends from the first quartile (Q1) to the third quartile (Q3) with the red line at the median. The lower whisker corresponds to the smallest data point that is bigger than Q1−1.5×IQR, while the upper whisker extends to the largest value that is smaller than Q3+1.5×IQR, where IQR denotes the interquartile range Q3-Q1. We used 42 cells in (A), (C) and 21 cells in (B), (D).</p

Ca²⁺ spike sequences.

<p>(A-B) Raster plots of 40 out of 100 Ca²⁺ spike sequences simulated from <i>x</i>_det and <i>x</i>_GP, respectively. (C-D) Estimations of <i>x</i>_det and <i>x</i>_GP from all generated Ca²⁺ spike sequences based on a PSTH (beige) and KS (solid blue). The true values of <i>x</i>_det and <i>x</i>_GP are shown as a dashed red line. Ca²⁺ spike sequences were generated using inverse sampling. Parameter values are (A,C) <i>γ</i> = 5.9 and (B,D) <i>μ</i> = 2.1, <i>σ</i>_<i>f</i> = 1.5, <i>κ</i> = 0.5 and <i>γ</i> = 6.2.</p

Clustering of dynamic stimuli and cell positions.

<p>(A) Weights of the three leading principal components of the stimulus data for the experiment of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g007" target="_blank">Fig 7</a>. (B) Position of cells in the microfluidics chamber are shown by a plus sign. Cells that were included in the analysis, i.e. those that spiked more than 10 times, are identified by a circle. Stimuli (and hence cell positions) belonging to the same group as identified by the <i>k</i>-means algorithm are coloured identically and with the same colour as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g007" target="_blank">Fig 7</a>.</p

Gaussian process.

<p>Three realisations of a GP (red, green, purple) around a time-dependent mean (black line). The blue area delineates the 95% confidence interval. Note the changes in width, which reflect a time-dependent standard deviation.</p

Dynamically stimulated HEK293T cells.

<p>(A) Simulation of complex concentration surface generated inside the microfluidics chamber after transiently varying relative flow rates of agonist and buffer input streams (COMSOL Multiphysics, COMSOL Ltd, Cambridge, UK). (B) Agonist concentration (indicated by AF594 fluorescence; red lines) and normalised Fluo-5F fluorescence intensity traces (black lines) from ROIs centred on 4 cells across the width of the channel upon applying a single sine-wave stimulation regime. Stimulus fluorescence (C), raster plots (D) and GP Ca²⁺ spike rate estimations (E) for each cluster in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g008" target="_blank">Fig 8</a>. Black lines denote the mean stimulus (C) and the mean Ca²⁺ spike rate (E) in each group, respectively.</p

Surrogate Ca²⁺ spike sequences from experimentally determined Ca²⁺ spike rates.

<p>(A) Mean intensity functions as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g007" target="_blank">Fig 7E</a>. The colours correspond to the ones used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g007" target="_blank">Fig 7</a>. (B) Raster plot of recorded Ca²⁺ spike sequences as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005794#pcbi.1005794.g007" target="_blank">Fig 7D</a>. (C) Surrogate Ca²⁺ spike sequences generated from the mean intensity function depicted in (A). (D) Histograms of recorded and simulated Ca²⁺ spike sequences.</p