^{1,2}and Martin Falcke

^{2}

### Abstract

The universality of as second messenger in living cells is achieved by a rich spectrum of spatiotemporal cellular concentration dynamics. release from internal storage compartments plays a key role in shaping cytosolic signals. Deciphering this signaling mechanism is essential for a deeper understanding of its physiological function and general concepts of cell signaling. Here, we review recent experimental findings demonstrating the stochasticity of oscillations and its relevance for modeling dynamics. The stochasticity arises by the hierarchical signal structure that carries molecular fluctuations of single channels onto the level of the cell leading to a stochastic medium as theoretically predicted. The result contradicts the current opinion of being a cellular oscillator. We demonstrate that cells use array enhanced coherence resonance to form rather regular spiking signals and that the “oscillations” carry information despite the involved stochasticity. The knowledge on the underlying mechanism also allows for determination of intrinsic properties from global observations. In the second part of the paper, we briefly survey different modeling approaches with regard to the experimental results. We focus on the dependence of the standard deviation on the mean period of the oscillations. It shows that limit cycle oscillations cannot describe the experimental data and that generic models have to include the spatial aspects of signaling.

Cytosolic oscillations are an extensively used mechanism in eukaryotic cells to translate extracellular signals into intracellular responses. Due to different spatiotemporal dynamics, cells can control a variety of cellular processes. Intracellular oscillations have served as a representative example of a cellular oscillator for the last two decades.

^{1,2}However, recent experimental findings have demonstrated that oscillations consist of sequences of random spikes.

^{3–5}Here, we review the experimental findings on how molecular fluctuations given by the random opening and closing of single channels determine the global behavior of the cytosolic concentration. It turns out that spikes occur by wave nucleation and that cells use array enhanced coherence resonance (AECR) to generate regular oscillations, a mechanism predicted theoretically. Moreover, we show that despite their stochasticity oscillations carry information and how cells can regulate the information content. In the second part of the paper, different modeling approaches are studied with respect to their capability to describe the experimentally observed variability. We find that noisy limit cycle models fail to explain the cellular dynamics. We demonstrate furthermore how extended dynamical properties of spatially resolved models can fit the experimental data.

We kindly thank R. D. Vilela and B. Lindner for providing the data of Fig. 12(d).

I. INTRODUCTION

II. EXPERIMENTAL INVESTIGATIONS

A. Theoretical predictions

B. Experimental evidence of a stochastic medium

C. Role of deterministic time

D. Information content

III. NEW MODELING APPROACH

A. Spatially homogeneous models

1. Goldbeter model

2. Hopf bifurcation

3. Saddle node bifurcation

4. Perfect leaky integrate and fire model

B. Spatially extended models

1. Green’s cell model

IV. CONCLUSION

### Key Topics

- Nucleation
- 23.0
- Intracellular signaling
- 21.0
- Entropy
- 18.0
- Poisson's equation
- 15.0
- Cluster dynamics
- 12.0

## Figures

Scheme of the pathway. If a plasma membrane receptor (a G-protein coupled receptor or a receptor tyrosin kinase) binds its specific agonist, it activates phospholipase C that produces at the plasma membrane. diffuses into the cytosol and can be detected by . If and are bound to an , it might open and diffuses from the ER into the cytosol. There it can be caught by buffers or open adjacent channels, which leads to more release (CICR). The pumps (SERCAs and plasma-membrane ATPases) remove from the cytosol.

Scheme of the pathway. If a plasma membrane receptor (a G-protein coupled receptor or a receptor tyrosin kinase) binds its specific agonist, it activates phospholipase C that produces at the plasma membrane. diffuses into the cytosol and can be detected by . If and are bound to an , it might open and diffuses from the ER into the cytosol. There it can be caught by buffers or open adjacent channels, which leads to more release (CICR). The pumps (SERCAs and plasma-membrane ATPases) remove from the cytosol.

The probability density Eq. (4) for different parameters. For large regeneration rates the density reduces almost to a pure decaying exponential function except for a fast rise for small . The distribution exhibits pronounced peaks and smaller tails for small . This leads to smaller slopes in the relation (4) resulting from Eqs. (5) and (6) . The relation is obtained by shifting the lines to the right by .

The probability density Eq. (4) for different parameters. For large regeneration rates the density reduces almost to a pure decaying exponential function except for a fast rise for small . The distribution exhibits pronounced peaks and smaller tails for small . This leads to smaller slopes in the relation (4) resulting from Eqs. (5) and (6) . The relation is obtained by shifting the lines to the right by .

A typical experimental fluorescence signal of a PLA cell is shown in the upper panel. denotes the amount of bound -sensitive dye compared to the initial amount and corresponds to the cytosolic concentration. In the lower panel, for each spike the following ISI is shown by a dot, indicating that oscillations have a stochastic character, since the individual ISIs vary substantially in their length. (For more details see Ref. 5 .)

A typical experimental fluorescence signal of a PLA cell is shown in the upper panel. denotes the amount of bound -sensitive dye compared to the initial amount and corresponds to the cytosolic concentration. In the lower panel, for each spike the following ISI is shown by a dot, indicating that oscillations have a stochastic character, since the individual ISIs vary substantially in their length. (For more details see Ref. 5 .)

spikes occur randomly. The standard deviation of the ISIs is in the same range as their average . Black dots show results for astrocytes and red squares show stimulated HEK cells. The linear dependence is in accordance with the wave nucleation assumption.

spikes occur randomly. The standard deviation of the ISIs is in the same range as their average . Black dots show results for astrocytes and red squares show stimulated HEK cells. The linear dependence is in accordance with the wave nucleation assumption.

buffers render spiking even more unpredictable. Buffer loading influences both and , as shown by the representative example of a (a) PLA cell and a (b) HEK cell, where the reference measurements (red) exhibit faster and more regular oscillations compared to the behavior after loading (a) 130 EGTA-AM and (b) 1 BAPTA-AM for 5 min (blue), respectively, during the gap. The additional buffers lead to increased and (Ref. 5 ). (c) This is also reflected in the plot where the reference values before buffer loading (red dots) are shifted in and directions leading to blue crosses. From these shifts we determine the individual shifting slopes by drawing a line through the two data points corresponding to the cell. The intersection of this line with the axis is used to estimate the deterministic time indicated by the arrows. Cells are shifted approximately in the direction of the population by buffer loading, i.e., , which is confirmed in Table I .

buffers render spiking even more unpredictable. Buffer loading influences both and , as shown by the representative example of a (a) PLA cell and a (b) HEK cell, where the reference measurements (red) exhibit faster and more regular oscillations compared to the behavior after loading (a) 130 EGTA-AM and (b) 1 BAPTA-AM for 5 min (blue), respectively, during the gap. The additional buffers lead to increased and (Ref. 5 ). (c) This is also reflected in the plot where the reference values before buffer loading (red dots) are shifted in and directions leading to blue crosses. From these shifts we determine the individual shifting slopes by drawing a line through the two data points corresponding to the cell. The intersection of this line with the axis is used to estimate the deterministic time indicated by the arrows. Cells are shifted approximately in the direction of the population by buffer loading, i.e., , which is confirmed in Table I .

Dependence of CV on and . For a Poisson process with a recovery period the CV decreases with increasing and . The data taken from Ref. 5 exhibit such a behavior, as shown for the PLA cells here, where the dashed lines correspond to different and the red line is the best fit. This method can be used to determine the cell specific property .

Dependence of CV on and . For a Poisson process with a recovery period the CV decreases with increasing and . The data taken from Ref. 5 exhibit such a behavior, as shown for the PLA cells here, where the dashed lines correspond to different and the red line is the best fit. This method can be used to determine the cell specific property .

Comparison of theoretical and experimental power spectra. (a) The power spectra defined by Eq. (19) with of the probability density (18) for parameter sets describing astrocytes (blue) and HEK cells (red) and . The peaks in the spectra occur due to as can be seen by the comparison of a spectrum with a vanishing deterministic time (black) with the corresponding spectrum of HEK cells and astrocytes. (b) Power spectral density derived from merged experimental spike trains normalized to their mean by Eq. (13) .

Comparison of theoretical and experimental power spectra. (a) The power spectra defined by Eq. (19) with of the probability density (18) for parameter sets describing astrocytes (blue) and HEK cells (red) and . The peaks in the spectra occur due to as can be seen by the comparison of a spectrum with a vanishing deterministic time (black) with the corresponding spectrum of HEK cells and astrocytes. (b) Power spectral density derived from merged experimental spike trains normalized to their mean by Eq. (13) .

Comparison of the Shannon information (solid line) and the Kullback entropy (dashed line) for two Poisson processes with rates and . We used natural units, i.e., .

Comparison of the Shannon information (solid line) and the Kullback entropy (dashed line) for two Poisson processes with rates and . We used natural units, i.e., .

Both the slope and the information gain depend only on the ratio . That defines the relation between and shown here with which we can estimate the detectable information in a signal by the experimental slope of the relation. We use natural units, i.e., .

Both the slope and the information gain depend only on the ratio . That defines the relation between and shown here with which we can estimate the detectable information in a signal by the experimental slope of the relation. We use natural units, i.e., .

(a) oscillation simulated by the deterministic Goldbeter model described by Eq. (28) with . The cytosolic (solid line) and lumenal (dashed line) concentration oscillate in phase. (b) The CV in the oscillatory regime does only increase up to 0.15 for large . In the excitable regime, the CV exhibits values comparable to the experiments. The minimum in the CV in dependence on indicates a coherent resonance. (c) The cytosolic oscillations become more irregular due to noise as shown for . (d) The relation for standard parameters in the oscillatory regime induced by different noise strengths exhibits a dependence different from those found in experiments.

(a) oscillation simulated by the deterministic Goldbeter model described by Eq. (28) with . The cytosolic (solid line) and lumenal (dashed line) concentration oscillate in phase. (b) The CV in the oscillatory regime does only increase up to 0.15 for large . In the excitable regime, the CV exhibits values comparable to the experiments. The minimum in the CV in dependence on indicates a coherent resonance. (c) The cytosolic oscillations become more irregular due to noise as shown for . (d) The relation for standard parameters in the oscillatory regime induced by different noise strengths exhibits a dependence different from those found in experiments.

Noisy Hopf limit cycle oscillator. (a) Spike trains containing ISIs each of a FHN neuron for , and different values of exhibit a relation comparable to those found in experiments only in the excitable regime indicated by the line . The oscillatory regime exhibits a linear dependence with a slope of only close to the bifurcation point. (b) The CV of the FHN model in dependence on and exhibits only in the excitable regime values comparable to the experiment also for large . Moreover, it demonstrates that the findings in (a) are rather independent of the parameter choice.

Noisy Hopf limit cycle oscillator. (a) Spike trains containing ISIs each of a FHN neuron for , and different values of exhibit a relation comparable to those found in experiments only in the excitable regime indicated by the line . The oscillatory regime exhibits a linear dependence with a slope of only close to the bifurcation point. (b) The CV of the FHN model in dependence on and exhibits only in the excitable regime values comparable to the experiment also for large . Moreover, it demonstrates that the findings in (a) are rather independent of the parameter choice.

Characteristics of a saddle node bifurcation. (a) Potential for different values of . A minimum occurs for and a spike can only be initiated by noise. (b) Strong noise approximation for CV derived from Ref. 49 . For large noise the system does not notice the different potentials, whereas for low noise the oscillatory regime exhibits small CVs, and the spike trains of the excitable system have a large variability. (c) Strong noise approximation for CV in dependence of . In contrast to the Hopf bifurcation system, the saddle node bifurcation leads to rather high CV in the oscillatory regime . (d) Exact values for the relations demonstrate the discrepancy of the experimental data with the noisy limit cycle assumption. Only for small the relation exhibits a slope of one (solid line). For large the slope decreases to 0.5 (dashed line).

Characteristics of a saddle node bifurcation. (a) Potential for different values of . A minimum occurs for and a spike can only be initiated by noise. (b) Strong noise approximation for CV derived from Ref. 49 . For large noise the system does not notice the different potentials, whereas for low noise the oscillatory regime exhibits small CVs, and the spike trains of the excitable system have a large variability. (c) Strong noise approximation for CV in dependence of . In contrast to the Hopf bifurcation system, the saddle node bifurcation leads to rather high CV in the oscillatory regime . (d) Exact values for the relations demonstrate the discrepancy of the experimental data with the noisy limit cycle assumption. Only for small the relation exhibits a slope of one (solid line). For large the slope decreases to 0.5 (dashed line).

relation of perfect integrate and fire dynamics for and varying values of and . The data points are generated by numerical integration of Eq. (33) where each curve corresponds to fixed and varying , whereas the set of curves corresponds to different . The results demonstrate that noise can lead to a rather regular regime for a specific choice of the two parameters.

relation of perfect integrate and fire dynamics for and varying values of and . The data points are generated by numerical integration of Eq. (33) where each curve corresponds to fixed and varying , whereas the set of curves corresponds to different . The results demonstrate that noise can lead to a rather regular regime for a specific choice of the two parameters.

The simulated influence of buffers on the spiking behavior is in accordance with the experiment shown in Fig. 5 . The upper panel exhibits the corresponding theoretical fluorescent signal . The lower panel shows the individual interspike interval following each spike. An increasing EGTA concentration from in the first measuring period (red) to (blue) in the second measuring period leads to an increase in from 78 to 288 s.

The simulated influence of buffers on the spiking behavior is in accordance with the experiment shown in Fig. 5 . The upper panel exhibits the corresponding theoretical fluorescent signal . The lower panel shows the individual interspike interval following each spike. An increasing EGTA concentration from in the first measuring period (red) to (blue) in the second measuring period leads to an increase in from 78 to 288 s.

The linear relation between and obtained from simulations is in accordance with the experimental findings in Fig. 4 and points to wave nucleation.

The linear relation between and obtained from simulations is in accordance with the experimental findings in Fig. 4 and points to wave nucleation.

Influence of clustering and decreased open times for 32 lone separated by and (upper row, red), clustered with (middle row, blue), and (lower row, black). (a) The number of open channels demonstrates that cells can generate spikes due to clustering and that the decreased open time leads to less frequent spikes. (b) This is also shown in the average cytosolic concentrations, where we observe a decrease in the spike height for shorter open times (middle row, blue) compared to the clustered with larger (lower row, black).

Influence of clustering and decreased open times for 32 lone separated by and (upper row, red), clustered with (middle row, blue), and (lower row, black). (a) The number of open channels demonstrates that cells can generate spikes due to clustering and that the decreased open time leads to less frequent spikes. (b) This is also shown in the average cytosolic concentrations, where we observe a decrease in the spike height for shorter open times (middle row, blue) compared to the clustered with larger (lower row, black).

## Tables

Comparison between the population slopes before and after buffer application and the average shifting slope . The consistency between the slopes and the rather clear separation between astrocytes and HEK cells demonstrates that the different cell types exhibit different regeneration rates .

Comparison between the population slopes before and after buffer application and the average shifting slope . The consistency between the slopes and the rather clear separation between astrocytes and HEK cells demonstrates that the different cell types exhibit different regeneration rates .

Abbreviations.

Abbreviations.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content