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Fluorescent Calcium Imaging and Subsequent In Situ Hybridization for Neuronal Precursor Characterization in <em>Xenopus laevis</em>

Spontaneous intracellular calcium activity can be observed in a variety of cell types and is proposed to play critical roles in a variety of physiological ...

College of William and Mary

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). We present a Kronecker structured representation for calcium release site models and perform benchmark stationary distribution calculations using numerical iterative solution techniques that leverage this structure. In this context we find multi-level methods and certain preconditioned projection methods superior to simple Gauss-Seidel type iterations. Response measures such as the number of channels in a particular state converge more quickly using these numerical iterative methods than occupation measures calculated via Monte Carlo simulation.

Markov chain models of coupled intracellular calcium channels: Kronecker structured representations and benchmark stationary distribution calculations.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. In order to overcome the state-space explosion that occurs in such compositionally defined calcium release site models, we have implemented several automated procedures for model reduction using fast/slow analysis. After categorizing rate constants in the single channel model as either fast or slow, groups of states in the expanded release site model that are connected by fast transitions are lumped, and transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. For small problems these conditional probability distributions can be numerically calculated from the full model without approximation. For large problems the conditional probability distributions can be approximated without the construction of the full model by assuming rapid mixing of states connected by fast transitions. Alternatively, iterative aggregation/disaggregation may be employed to obtain reduced calcium release site models in a memory-efficient fashion. Benchmarking of several different iterative aggregation/disaggregation-based fast/slow reduction schemes establishes the effectiveness of automated calcium release site reduction utilizing the Koury-McAllister-Stewart method.

Reduction of calcium release site models via fast/slow analysis and iterative aggregation/disaggregation.

Models of calcium (Ca(2 +)) release sites derived from continuous-time Markov chain (CTMC) models of intracellular Ca(2 +) channels exhibit collective gating reminiscent of the experimentally observed phenomenon of Ca(2 +) puffs and sparks. In order to overcome the state-space explosion that occurs in compositionally defined Ca(2 +) release site models, we have implemented an automated procedure for model reduction that replaces aggregated states of the full release site model with much simpler CTMCs that have similar within-group phase-type sojourn times and inter-group transitions. Error analysis based on comparison of full and reduced models validates the method when applied to release site models composed of 20 three-state channels that are both activated and inactivated by Ca(2 +). Although inspired by existing techniques for fitting moments of phase-type distributions, the automated reduction method for compositional Ca(2 +) release site models is unique in several respects and novel in this biophysical context.

Reduction of calcium release site models via moment fitting of phase-type distributions.

Methods to analyze the dynamics of calcium activity often rely on visually distinguishable features in time series data such as spikes, waves, or oscillations. However, systems such as the developing nervous system display a complex, irregular type of calcium activity which makes the use of such methods less appropriate. Instead, for such systems there exists a class of methods (including information theoretic, power spectral, and fractal analysis approaches) which use more fundamental properties of the time series to analyze the observed calcium dynamics. We present a new analysis method in this class, the Markovian Entropy measure, which is an easily implementable calcium time series analysis method which represents the observed calcium activity as a realization of a Markov Process and describes its dynamics in terms of the level of predictability underlying the transitions between the states of the process. We applied our and other commonly used calcium analysis methods on a dataset from Xenopus laevis neural progenitors which displays irregular calcium activity and a dataset from murine synaptic neurons which displays activity time series that are well-described by visually-distinguishable features. We find that the Markovian Entropy measure is able to distinguish between biologically distinct populations in both datasets, and that it can separate biologically distinct populations to a greater extent than other methods in the dataset exhibiting irregular calcium activity. These results support the benefit of using the Markovian Entropy measure to analyze calcium dynamics, particularly for studies using time series data which do not exhibit easily distinguishable features.

A Markovian Entropy Measure for the Analysis of Calcium Activity Time Series.

Calcium is a ubiquitous signaling molecule that plays a vital role in many physiological processes. Recent work has shown that calcium activity is especially critical in vertebrate neural development. Here, we investigated if calcium activity and neuronal phenotype are correlated only on a population level or on the level of single cells. Using  primary cell culture in which individual cells can be unambiguously identified and associated with a molecular phenotype, we correlated calcium activity with neuronal phenotype on the single-cell level. This analysis revealed that, at the neural plate stage, a high frequency of low-amplitude spiking activity correlates with an excitatory, glutamatergic phenotype, while high-amplitude spiking activity correlates with an inhibitory, GABAergic phenotype. Surprisingly, we also found that high-frequency, low-amplitude spiking activity correlates with neural progenitor cells and that differentiating cells exhibit higher spike amplitude. Additional methods of analysis suggested that differentiating marker -expressing cells exhibit relatively persistent and predictable calcium activity compared to the irregular activity of neural progenitor cells. Our study highlights the value of using a range of thresholds for analyzing calcium activity data and underscores the importance of employing multiple methods to characterize the often irregular, complex patterns of calcium activity during early neural development.

Peter Kemper

Department of Computer Science

College of William and Mary

Fluorescent Calcium Imaging and Subsequent In Situ Hybridization for Neuronal Precursor Characterization in Xenopus laevis

Peter Kemper

.css-o250i3{font-size:18px;font-family:Open Sans,sans-serif;line-height:21.6px;}Department of Computer Science

College of William and Mary

Fluorescent Calcium Imaging and Subsequent In Situ Hybridization for Neuronal Precursor Characterization in Xenopus laevis

Department of Computer Science