Differential regulation of proteins by bursting calcium oscillations--a theoretical study

An Information-Theoretical Approach for Calcium Signaling Specificity

Calcium is a key signaling agent in animals and plants. Its involvement in the regulation of a wide range of processes has led to the question of how calcium signals can activate stimulus-specific responses. We introduce a computational framework for studying intracellular calcium signaling using elements of information theory. We use mutual information to quantify the differential activation of proteins in response to different calcium signals to provide an operational definition of specificity. Using optimization procedures this framework allows us to explore the biochemical determinants of calcium decoding. We explore simple toy models and general binding kinetics approaches to demonstrate the utility and limitations of the proposed framework. Unravelling signaling specificity is key for understanding information processing within cells and for the future design of synthetic nanodevices for molecular communications.

Robustness of frequency vs. amplitude coding of calcium oscillations during changing temperatures

Intracellular calcium oscillations have been widely studied. It is assumed that information is conveyed in the frequency, amplitude and shape of these oscillations. In particular, calcium signalling in mammalian liver cells has repeatedly been reported to display frequency coding so that an increasing amount of stimulus is translated into an increasing frequency of the oscillations. However, recently, we have shown that calcium oscillations in fish liver cells rather exhibit amplitude coding with increasing stimuli being translated into increasing amplitudes. Practical consequences of this difference are unknown so far. Here we investigated advantages and disadvantages of frequency vs. amplitude coding, in particular in environments with substantially changing temperatures (e.g. 10-20 degrees). For this purpose, we use computational modelling and a new approach to generate a calcium model exactly displaying a specific frequency and/or amplitude. We conclude that despite the advantages in flexibility that frequencies might offer for the transmission of information in the cell, amplitude coding is obviously more robust with respect to changes in environmental temperatures. This potentially explains the observed differences between two classes of organisms, one operating at constant temperatures whereas the other is not.

Calcium Oscillation Frequency-Sensitive Gene Regulation and Homeostatic Compensation in Pancreatic β-Cells

Pancreatic islet [Formula: see text]-cells are electrically excitable cells that secrete insulin in an oscillatory fashion when the blood glucose concentration is at a stimulatory level. Insulin oscillations are the result of cytosolic [Formula: see text] oscillations that accompany bursting electrical activity of [Formula: see text]-cells and are physiologically important. ATP-sensitive [Formula: see text] channels (K(ATP) channels) play the key role in setting the overall activity of the cell and in driving bursting, by coupling cell metabolism to the membrane potential. In humans, when there is a defect in K(ATP) channel function, [Formula: see text]-cells fail to respond appropriately to changes in the blood glucose level, and electrical and [Formula: see text] oscillations are lost. However, mice compensate for K(ATP) channel defects in islet [Formula: see text]-cells by employing alternative mechanisms to maintain electrical and [Formula: see text] oscillations. In a recent study, we showed that in mice islets in which K(ATP) channels are genetically knocked out another [Formula: see text] current, provided by inward-rectifying [Formula: see text] channels, is increased. With mathematical modeling, we demonstrated that a sufficient upregulation in these channels can account for the paradoxical electrical bursting and [Formula: see text] oscillations observed in these [Formula: see text]-cells. However, the question of determining the correct level of upregulation that is necessary for this compensation remained unanswered, and this question motivates the current study. [Formula: see text] is a well-known regulator of gene expression, and several examples have been shown of genes that are sensitive to the frequency of the [Formula: see text] signal. In this mathematical modeling study, we demonstrate that a [Formula: see text] oscillation frequency-sensitive gene transcription network can adjust the gene expression level of a compensating [Formula: see text] channel so as to rescue electrical bursting and [Formula: see text] oscillations in a model [Formula: see text]-cell in which the key K(ATP) current is removed. This is done without the prescription of a target [Formula: see text] level, but evolves naturally as a consequence of the feedback between the [Formula: see text]-dependent enzymes and the cell's electrical activity. More generally, the study indicates how [Formula: see text] can provide the link between gene expression and cellular electrical activity that promotes wild-type behavior in a cell following gene knockout.

Combining modelling and experimental approaches to explain how calcium signatures are decoded by calmodulin-binding transcription activators (CAMTAs) to produce specific gene expression responses

Experimental data show that Arabidopsis thaliana is able to decode different calcium signatures to produce specific gene expression responses. It is also known that calmodulin-binding transcription activators (CAMTAs) have calmodulin (CaM)-binding domains. Therefore, the gene expression responses regulated by CAMTAs respond to calcium signals. However, little is known about how different calcium signatures are decoded by CAMTAs to produce specific gene expression responses. A dynamic model of Ca(2+) -CaM-CAMTA binding and gene expression responses is developed following thermodynamic and kinetic principles. The model is parameterized using experimental data. Then it is used to analyse how different calcium signatures are decoded by CAMTAs to produce specific gene expression responses. Modelling analysis reveals that: calcium signals in the form of cytosolic calcium concentration elevations are nonlinearly amplified by binding of Ca(2+) , CaM and CAMTAs; amplification of Ca(2+) signals enables calcium signatures to be decoded to give specific CAMTA-regulated gene expression responses; gene expression responses to a calcium signature depend upon its history and accumulate all the information during the lifetime of the calcium signature. Information flow from calcium signatures to CAMTA-regulated gene expression responses has been established by combining experimental data with mathematical modelling.

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensen's inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

Information transfer in signaling pathways: a study using coupled simulated and experimental data

BACKGROUND

The topology of signaling cascades has been studied in quite some detail. However, how information is processed exactly is still relatively unknown. Since quite diverse information has to be transported by one and the same signaling cascade (e.g. in case of different agonists), it is clear that the underlying mechanism is more complex than a simple binary switch which relies on the mere presence or absence of a particular species. Therefore, finding means to analyze the information transferred will help in deciphering how information is processed exactly in the cell. Using the information-theoretic measure transfer entropy, we studied the properties of information transfer in an example case, namely calcium signaling under different cellular conditions. Transfer entropy is an asymmetric and dynamic measure of the dependence of two (nonlinear) stochastic processes. We used calcium signaling since it is a well-studied example of complex cellular signaling. It has been suggested that specific information is encoded in the amplitude, frequency and waveform of the oscillatory Ca(2+)-signal.

RESULTS

We set up a computational framework to study information transfer, e.g. for calcium signaling at different levels of activation and different particle numbers in the system. We stochastically coupled simulated and experimentally measured calcium signals to simulated target proteins and used kernel density methods to estimate the transfer entropy from these bivariate time series. We found that, most of the time, the transfer entropy increases with increasing particle numbers. In systems with only few particles, faithful information transfer is hampered by random fluctuations. The transfer entropy also seems to be slightly correlated to the complexity (spiking, bursting or irregular oscillations) of the signal. Finally, we discuss a number of peculiarities of our approach in detail.

CONCLUSION

This study presents the first application of transfer entropy to biochemical signaling pathways. We could quantify the information transferred from simulated/experimentally measured calcium signals to a target enzyme under different cellular conditions. Our approach, comprising stochastic coupling and using the information-theoretic measure transfer entropy, could also be a valuable tool for the analysis of other signaling pathways.

Spatio-temporal modelling explains the effect of reduced plasma membrane Ca2+ efflux on intracellular Ca2+ oscillations in hepatocytes

In many non-excitable eukaryotic cells, including hepatocytes, Ca(2+) oscillations play a key role in intra- and intercellular signalling, thus regulating many cellular processes from fertilisation to death. Therefore, understanding the mechanisms underlying these oscillations, and consequently understanding how they may be regulated, is of great interest. In this paper, we study the influence of reduced Ca(2+) plasma membrane efflux on Ca(2+) oscillations in hepatocytes. Our previous experiments with carboxyeosin show that a reduced plasma membrane Ca(2+) efflux increases the frequency of Ca(2+) oscillations, but does not affect the duration of individual transients. This phenomenon can be best explained by taking into account not only the temporal, but also the spatial dynamics underlying the generation of Ca(2+) oscillations in the cell. Here we divide the cell into a grid of elements and treat the Ca(2+) dynamics as a spatio-temporal phenomenon. By converting an existing temporal model into a spatio-temporal one, we obtain theoretical predictions that are in much better agreement with the experimental observations.

Decoding of calcium oscillations by phosphorylation cycles: analytic results

Experimental studies have demonstrated that Ca(2+)-regulated proteins are sensitive to the frequency of Ca(2+) oscillations, and several mathematical models for specific proteins have provided insight into the mechanisms involved. Because of the large number of Ca(2+)-regulated proteins in signal transduction, metabolism and gene expression, it is desirable to establish in general terms which molecular properties shape the response to oscillatory Ca(2+) signals. Here we address this question by analyzing in detail a model of a prototypical Ca(2+)-decoding module, consisting of a target protein whose activity is controlled by a Ca(2+)-activated kinase and the counteracting phosphatase. We show that this module can decode the frequency of Ca(2+) oscillations, at constant average Ca(2+) signal, provided that the Ca(2+) spikes are narrow and the oscillation frequency is sufficiently low--of the order of the phosphatase rate constant or below. Moreover, Ca(2+) oscillations activate the target more efficiently than a constant signal when Ca(2+) is bound cooperatively and with low affinity. Thus, the rate constants and the Ca(2+) affinities of the target-modifying enzymes can be tuned in such a way that the module responds optimally to Ca(2+) spikes of a certain amplitude and frequency. Frequency sensitivity is further enhanced when the limited duration of the external stimulus driving Ca(2+) signaling is accounted for. Thus, our study identifies molecular parameters that may be involved in establishing the specificity of cellular responses downstream of Ca(2+) oscillations.

Role of cascades in converting oscillatory signals into stationary step-like responses

In biological signal transduction pathways intermediates are often oscillatory and need to be converted into smooth output signals at the end. We show by mathematical modelling that protein kinase cascades enable converting oscillatory signals into sharp stationary step-like outputs. The importance of this result is demonstrated for the switch-like protein activation by calcium oscillations, which is of biological importance for regulating different cellular processes. In addition, we found that protein kinase cascades cause memory effects in the protein activation, which might be of a physiological advantage since a smaller amount of calcium transported in the cell is required for an effective activation of cellular processes.

Selective regulation of cellular processes via protein cascades acting as band-pass filters for time-limited oscillations

We show by mathematical modelling that a two-level protein cascade can act as a band-pass filter for time-limited oscillations. The band-pass filters are then combined into a network of three-level signalling cascades that by filtering the frequency of time-limited oscillations selectively switches cellular processes on and off. The physiological relevance for the selective regulation of cellular processes is demonstrated for the case of regulation by time-limited calcium oscillations.