Reliable encoding of stimulus intensities within random sequences of intracellular Ca2+ spikes

Transient frequency preference responses in cell signaling systems

Ligand-receptor systems, covalent modification cycles, and transcriptional networks are the fundamental components of cell signaling and gene expression systems. While their behavior in reaching a steady-state regime under step-like stimulation is well understood, their response under repetitive stimulation, particularly at early time stages is poorly characterized. Yet, early-stage responses to external inputs are arguably as informative as late-stage ones. In simple systems, a periodic stimulation elicits an initial transient response, followed by periodic behavior. Transient responses are relevant when the stimulation has a limited time span, or when the stimulated component's timescale is slow as compared to the timescales of the downstream processes, in which case the latter processes may be capturing only those transients. In this study, we analyze the frequency response of simple motifs at different time stages. We use dose-conserved pulsatile input signals and consider different metrics versus frequency curves. We show that in ligand-receptor systems, there is a frequency preference response in some specific metrics during the transient stages, which is not present in the periodic regime. We suggest this is a general system-level mechanism that cells may use to filter input signals that have consequences for higher order circuits. In addition, we evaluate how the described behavior in isolated motifs is reflected in similar types of responses in cascades and pathways of which they are a part. Our studies suggest that transient frequency preferences are important dynamic features of cell signaling and gene expression systems, which have been overlooked.

Calcium oscillations optimize the energetic efficiency of mitochondrial metabolism

Energy transduction is central to living organisms, but the impact of enzyme regulation and signaling on its thermodynamic efficiency is generally overlooked. Here, we analyze the efficiency of ATP production by the tricarboxylic acid cycle and oxidative phosphorylation, which generate most of the chemical energy in eukaryotes. Calcium signaling regulates this pathway and can affect its energetic output, but the concrete energetic impact of this cross-talk remains elusive. Calcium enhances ATP production by activating key enzymes of the tricarboxylic acid cycle while calcium homeostasis is ATP-dependent. We propose a detailed kinetic model describing the calcium-mitochondria cross-talk and analyze it using nonequilibrium thermodynamics: after identifying the effective reactions driving mitochondrial metabolism out of equilibrium, we quantify the mitochondrial thermodynamic efficiency for different conditions. Calcium oscillations, triggered by extracellular stimulation or energy deficiency, boost the thermodynamic efficiency of mitochondrial metabolism, suggesting a compensatory role of calcium signaling in mitochondrial bioenergetics.

An integrate-and-fire approach to Ca2+ signaling. Part II: Cumulative refractoriness

Inositol 1,4,5-trisphosphate-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. The agonist concentration stimulating Ca2+ signals is encoded in the frequency of a Ca2+ concentration spike sequence. When a cell is stimulated, the interspike intervals (ISIs) often show a distinct transient during which they gradually increase, a system property we refer to as cumulative refractoriness. We extend a previously published stochastic model to include the Ca2+ concentration in the intracellular Ca2+ store as a slow adaptation variable. This model can reproduce both stationary and transient statistics of experimentally observed ISI sequences. We derive approximate expressions for the mean and coefficient of variation of the stationary ISIs. We also consider the response to the onset of a constant stimulus and estimate the length of the transient and the strength of the adaptation of the ISI. We show that the adaptation sets the coefficient of variation in agreement with current ideas derived from experiments. Moreover, we explain why, despite a pronounced transient behavior, ISI correlations can be weak, as often observed in experiments. Finally, we fit our model to reproduce the transient statistics of experimentally observed ISI sequences in stimulated HEK cells. The fitted model is able to qualitatively reproduce the relationship between the stationary interval correlations and the number of transient intervals, as well as the strength of the ISI adaptation. We also find positive correlations in the experimental sequence that cannot be explained by our model.

Maintenance of hematopoietic stem cell niche homeostasis requires gap junction-mediated calcium signaling

Regulation of stem cells requires coordination of the cells that make up the stem cell niche. Here, we describe a mechanism that allows communication between niche cells to coordinate their activity and shape the signaling environment surrounding resident stem cells. Using the Drosophila hematopoietic organ, the lymph gland, we show that cells of the hematopoietic niche, the posterior signaling center (PSC), communicate using gap junctions (GJs) and form a signaling network. This network allows PSC cells to exchange Ca2+ signals repetitively which regulate the hematopoietic niche. Disruption of Ca2+ signaling in the PSC or the GJ-mediated network connecting niche cells causes dysregulation of the PSC and blood progenitor differentiation. Analysis of PSC-derived cell signaling shows that the Hedgehog pathway acts downstream of GJ-mediated Ca2+ signaling to modulate the niche microenvironment. These data show that GJ-mediated communication between hematopoietic niche cells maintains their homeostasis and consequently controls blood progenitor behavior.

Structural titration reveals Ca2+-dependent conformational landscape of the IP3 receptor

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are endoplasmic reticulum Ca2+ channels whose biphasic dependence on cytosolic Ca2+ gives rise to Ca2+ oscillations that regulate fertilization, cell division and cell death. Despite the critical roles of IP3R-mediated Ca2+ responses, the structural underpinnings of the biphasic Ca2+ dependence that underlies Ca2+ oscillations are incompletely understood. Here, we collect cryo-EM images of an IP3R with Ca2+ concentrations spanning five orders of magnitude. Unbiased image analysis reveals that Ca2+ binding does not explicitly induce conformational changes but rather biases a complex conformational landscape consisting of resting, preactivated, activated, and inhibited states. Using particle counts as a proxy for relative conformational free energy, we demonstrate that Ca2+ binding at a high-affinity site allows IP3Rs to activate by escaping a low-energy resting state through an ensemble of preactivated states. At high Ca2+ concentrations, IP3Rs preferentially enter an inhibited state stabilized by a second, low-affinity Ca2+ binding site. Together, these studies provide a mechanistic basis for the biphasic Ca2+-dependence of IP3R channel activity.

Modeling IP3-induced Ca2+ signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

IP3 receptors: An "elementary" journey from structure to signals

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are large tetrameric channels which sit mostly in the membrane of the endoplasmic reticulum (ER) and mediate Ca2+ release from intracellular stores in response to extracellular stimuli in almost all cells. Dual regulation of IP3Rs by IP3 and Ca2+ itself, upstream "licensing", and the arrangement of IP3Rs into small clusters in the ER membrane, allow IP3Rs to generate spatially and temporally diverse Ca2+ signals. The characteristic biphasic regulation of IP3Rs by cytosolic Ca2+ concentration underpins regenerative Ca2+ signals by Ca2+-induced Ca2+-release, while also preventing uncontrolled explosive Ca2+ release. In this way, cells can harness a simple ion such as Ca2+ as a near-universal intracellular messenger to regulate diverse cellular functions, including those with conflicting outcomes such as cell survival and cell death. High-resolution structures of the IP3R bound to IP3 and Ca2+ in different combinations have together started to unravel the workings of this giant channel. Here we discuss, in the context of recently published structures, how the tight regulation of IP3Rs and their cellular geography lead to generation of "elementary" local Ca2+ signals known as Ca2+ "puffs", which form the fundamental bottleneck through which all IP3-mediated cytosolic Ca2+ signals must first pass.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

Computational investigation of IP3 diffusion

Inositol 1,4,5-trisphosphate (IP₃) plays a key role in calcium signaling. After stimulation, it diffuses from the plasma membrane where it is produced to the endoplasmic reticulum where its receptors are localized. Based on in vitro measurements, IP₃ was long thought to be a global messenger characterized by a diffusion coefficient of ~ 280 μm²s^-1. However, in vivo observations revealed that this value does not match with the timing of localized Ca²⁺ increases induced by the confined release of a non-metabolizable IP₃ analog. A theoretical analysis of these data concluded that in intact cells diffusion of IP₃ is strongly hindered, leading to a 30-fold reduction of the diffusion coefficient. Here, we performed a new computational analysis of the same observations using a stochastic model of Ca²⁺ puffs. Our simulations concluded that the value of the effective IP₃ diffusion coefficient is close to 100 μm²s^-1. Such moderate reduction with respect to in vitro estimations quantitatively agrees with a buffering effect by non-fully bound inactive IP₃ receptors. The model also reveals that IP₃ spreading is not much affected by the endoplasmic reticulum, which represents an obstacle to the free displacement of molecules, but can be significantly increased in cells displaying elongated, 1-dimensional like geometries.

Intravital calcium imaging in myeloid leukocytes identifies calcium frequency spectra as indicators of functional states

The assessment of leukocyte activation in vivo is mainly based on surrogate parameters, such as cell shape changes and migration patterns. Consequently, additional parameters are required to dissect the complex spatiotemporal activation of leukocytes during inflammation. Here, we showed that intravital microscopy of myeloid leukocyte Ca²⁺ signals with Ca²⁺ reporter mouse strains combined with bioinformatic signal analysis provided a tool to assess their activation in vivo. We demonstrated by two-photon microscopy that tissue-resident macrophages reacted to sterile inflammation in the cremaster muscle with Ca²⁺ transients in a distinct spatiotemporal pattern. Moreover, through high-resolution, intravital spinning disk confocal microscopy, we identified the intracellular Ca²⁺ signaling patterns of neutrophils during the migration cascade in vivo. These patterns were modulated by the Ca²⁺ channel Orai1 and Gα_i-coupled GPCRs, whose effects were evident through analysis of the range of frequencies of the Ca²⁺ signal (frequency spectra), which provided insights into the complex patterns of leukocyte Ca²⁺ oscillations. Together, these findings establish Ca²⁺ frequency spectra as an additional dimension to assess leukocyte activation and migration during inflammation in vivo.

Kramers Rate Theory of Pacemaker Dynamics in Noisy Excitable Media

Rhythmic activities, which are usually driven by pacemakers, are common in biological systems. In noisy excitable media, pacemakers are self-organized firing clusters, but the underlying dynamics remains to be elucidated. Here we develop a Kramers rate theory of coupled cells to describe the firing properties of pacemakers and their dependence on coupling strength and system size and dimension. The theory captures accurately the simulation results of tissue models with stochastic Hodgkin-Huxley equations except when transitions from pacemakers to spiral waves occur under weak coupling.

How noise can generate calcium spike-type oscillations in deterministic equilibrium modes

Stochastic excitability of spiking oscillatory regimes in the calcium kinetics is studied on the basis of the Li-Rinzel conceptual model. The probabilistic mechanisms of the noise-induced generation of large-amplitude oscillations in parametric zones, where the original deterministic model has only stable equilibria, are investigated numerically and analytically. A parametric statistical description of interspike intervals is curried out and the phenomenon of coherence resonance is discussed. For the analytical study of the stochastic excitement, the confidence domain method using a stochastic sensitivity technique is applied. In this analysis, a key role of mutual arrangement of the confidence ellipses and separatrices detaching the sub- and supercritical regions is demonstrated. It is shown that in the Li-Rinzel model such separatrices are the stable manifolds of the saddle equilibria and the transient semiattractors.

On the relation between input and output distributions of scRNA-seq experiments

MOTIVATION

Single-cell RNA sequencing determines RNA copy numbers per cell for a given gene. However, technical noise poses the question how observed distributions (output) are connected to their cellular distributions (input).

RESULTS

We model a single-cell RNA sequencing setup consisting of PCR amplification and sequencing, and derive probability distribution functions for the output distribution given an input distribution. We provide copy number distributions arising from single transcripts during PCR amplification with exact expressions for mean and variance. We prove that the coefficient of variation of the output of sequencing is always larger than that of the input distribution. Experimental data reveals the variance and mean of the input distribution to obey characteristic relations, which we specifically determine for a HeLa dataset. We can calculate as many moments of the input distribution as are known of the output distribution (up to all). This, in principle, completely determines the input from the output distribution.

AVAILABILITY AND IMPLEMENTATION

Source code freely available at https://github.com/danielschw188/InputOutputSCRNASeq.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

A gap-junction-mediated, calcium-signaling network controls blood progenitor fate decisions in hematopoiesis

Stem cell homeostasis requires coordinated fate decisions among stem cells that are often widely distributed within a tissue at varying distances from their stem cell niche. This requires a mechanism to ensure robust fate decisions within a population of stem cells. Here, we show that, in the Drosophila hematopoietic organ, the lymph gland (LG), gap junctions form a network that coordinates fate decisions between blood progenitors. Using live imaging of calcium signaling in intact LGs, we find that blood progenitors are connected through a signaling network. Blocking gap junction function disrupts this network, alters the pattern of encoded calcium signals, and leads to loss of progenitors and precocious blood cell differentiation. Ectopic and uniform activation of the calcium-signaling mediator CaMKII restores progenitor homeostasis when gap junctions are disrupted. Overall, these data show that gap junctions equilibrate cell signals between blood progenitors to coordinate fate decisions and maintain hematopoietic homeostasis.

Estimating Entropy Production from Waiting Time Distributions

Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication, or locomotion. Given time series observations of intra-, inter-, or multicellular processes, a key challenge is to detect nonequilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice, as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obviously violate detailed balance. Here, we introduce a novel method for bounding the entropy production of physical and living systems which uses only the waiting time statistics of hidden Markov processes and, hence, can be directly applied to experimental data. By determining a universal limiting curve, we infer entropy production bounds from experimental data for gene regulatory networks, mammalian behavioral dynamics, and numerous other biological processes. Further considering the asymptotic limit of increasingly precise biological timers, we estimate the necessary entropic cost of heartbeat regulation in humans, dogs, and mice.

High rates of calcium-free diffusion in the cytosol of living cells

Calcium (Ca²⁺) is a universal second messenger that participates in the regulation of innumerous physiological processes. The way in which local elevations of the cytosolic Ca²⁺ concentration spread in space and time is key for the versatility of the signals. Ca²⁺ diffusion in the cytosol is hindered by its interaction with proteins that act as buffers. Depending on the concentrations and the kinetics of the interactions, there is a large range of values at which Ca²⁺ diffusion can proceed. Having reliable estimates of this range, particularly of its highest end, which corresponds to the ions free diffusion, is key to understand how the signals propagate. In this work, we present the first experimental results with which the Ca²⁺-free diffusion coefficient is directly quantified in the cytosol of living cells. By means of fluorescence correlation spectroscopy experiments performed in Xenopus laevis oocytes and in cells of Saccharomyces cerevisiae, we show that the ions can freely diffuse in the cytosol at a higher rate than previously thought.

KRAP tethers IP3 receptors to actin and licenses them to evoke cytosolic Ca2+ signals

Regulation of IP₃ receptors (IP₃Rs) by IP₃ and Ca²⁺ allows regenerative Ca²⁺ signals, the smallest being Ca²⁺ puffs, which arise from coordinated openings of a few clustered IP₃Rs. Cells express thousands of mostly mobile IP₃Rs, yet Ca²⁺ puffs occur at a few immobile IP₃R clusters. By imaging cells with endogenous IP₃Rs tagged with EGFP, we show that KRas-induced actin-interacting protein (KRAP) tethers IP₃Rs to actin beneath the plasma membrane. Loss of KRAP abolishes Ca²⁺ puffs and the global increases in cytosolic Ca²⁺ concentration evoked by more intense stimulation. Over-expressing KRAP immobilizes additional IP₃R clusters and results in more Ca²⁺ puffs and larger global Ca²⁺ signals. Endogenous KRAP determines which IP₃Rs will respond: it tethers IP₃R clusters to actin alongside sites where store-operated Ca²⁺ entry occurs, licenses IP₃Rs to evoke Ca²⁺ puffs and global cytosolic Ca²⁺ signals, implicates the actin cytoskeleton in IP₃R regulation and may allow local activation of Ca²⁺ entry.

Highly localized intracellular Ca2+ signals promote optimal salivary gland fluid secretion

Salivary fluid secretion involves an intricate choreography of membrane transporters to result in the trans-epithelial movement of NaCl and water into the acinus lumen. Current models are largely based on experimental observations in enzymatically isolated cells where the Ca²⁺ signal invariably propagates globally and thus appears ideally suited to activate spatially separated Cl and K channels, present on the apical and basolateral plasma membrane, respectively. We monitored Ca²⁺ signals and salivary secretion in live mice expressing GCamp6F, following stimulation of the nerves innervating the submandibular gland. Consistent with in vitro studies, Ca²⁺ signals were initiated in the apical endoplasmic reticulum. In marked contrast to in vitro data, highly localized trains of Ca²⁺ transients that failed to fully propagate from the apical region were observed. Following stimuli optimum for secretion, large apical-basal gradients were elicited. A new mathematical model, incorporating these data was constructed to probe how salivary secretion can be optimally stimulated by apical Ca²⁺ signals.

Fractional response analysis reveals logarithmic cytokine responses in cellular populations

Although we can now measure single-cell signaling responses with multivariate, high-throughput techniques our ability to interpret such measurements is still limited. Even interpretation of dose–response based on single-cell data is not straightforward: signaling responses can differ significantly between cells, encompass multiple signaling effectors, and have dynamic character. Here, we use probabilistic modeling and information-theory to introduce fractional response analysis (FRA), which quantifies changes in fractions of cells with given response levels. FRA can be universally performed for heterogeneous, multivariate, and dynamic measurements and, as we demonstrate, quantifies otherwise hidden patterns in single-cell data. In particular, we show that fractional responses to type I interferon in human peripheral blood mononuclear cells are very similar across different cell types, despite significant differences in mean or median responses and degrees of cell-to-cell heterogeneity. Further, we demonstrate that fractional responses to cytokines scale linearly with the log of the cytokine dose, which uncovers that heterogeneous cellular populations are sensitive to fold-changes in the dose, as opposed to additive changes.

Our ability to interpret single-cell multivariate signaling responses is still limited. Here the authors introduce fractional response analysis (FRA), involving fractional cell counting, capable of deconvoluting heterogeneous multivariate responses of cellular populations.

Improved bounds on entropy production in living systems

Living systems maintain or increase local order by working against the second law of thermodynamics. Thermodynamic consistency is restored as they consume free energy, thereby increasing the net entropy of their environment. Recently introduced estimators for the entropy production rate have provided major insights into the efficiency of important cellular processes. In experiments, however, many degrees of freedom typically remain hidden to the observer, and, in these cases, existing methods are not optimal. Here, by reformulating the problem within an optimization framework, we are able to infer improved bounds on the rate of entropy production from partial measurements of biological systems. Our approach yields provably optimal estimates given certain measurable transition statistics. In contrast to prevailing methods, the improved estimator reveals nonzero entropy production rates even when nonequilibrium processes appear time symmetric and therefore may pretend to obey detailed balance. We demonstrate the broad applicability of this framework by providing improved bounds on the energy consumption rates in a diverse range of biological systems including bacterial flagella motors, growing microtubules, and calcium oscillations within human embryonic kidney cells.

A Tale of two receptors

Calcium (Ca²⁺) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca²⁺ oscillations. We present a new model of Ca²⁺ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca²⁺ oscillations. The model accounts for Ca²⁺ regulation of the IP₃ receptor (IP₃R), the positive feedback from Ca²⁺ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca²⁺ oscillations in response to P2Y receptor activation.

Signaling pathways have an inherent need for noise to acquire information

BACKGROUND

All living systems acquire information about their environment. At the cellular level, they do so through signaling pathways. Such pathways rely on reversible binding interactions between molecules that detect and transmit the presence of an extracellular cue or signal to the cell's interior. These interactions are inherently stochastic and thus noisy. On the one hand, noise can cause a signaling pathway to produce the same response for different stimuli, which reduces the amount of information a pathway acquires. On the other hand, in processes such as stochastic resonance, noise can improve the detection of weak stimuli and thus the acquisition of information. It is not clear whether the kinetic parameters that determine a pathway's operation cause noise to reduce or increase the acquisition of information.

RESULTS

We analyze how the kinetic properties of the reversible binding interactions used by signaling pathways affect the relationship between noise, the response to a signal, and information acquisition. Our results show that, under a wide range of biologically sensible parameter values, a noisy dynamic of reversible binding interactions is necessary to produce distinct responses to different stimuli. As a consequence, noise is indispensable for the acquisition of information in signaling pathways.

CONCLUSIONS

Our observations go beyond previous work by showing that noise plays a positive role in signaling pathways, demonstrating that noise is essential when such pathways acquire information.

GPCR mediated control of calcium dynamics: A systems perspective

G-protein coupled receptor (GPCR) mediated calcium (Ca²⁺)-signaling transduction remains crucial in designing drugs for various complex diseases including neurodegeneration, chronic heart failure as well as respiratory diseases. Although there are several reviews detailing various aspects of Ca²⁺-signaling such as the role of IP₃ receptors and Ca²⁺-induced-Ca²⁺-release, none of them provide an integrated view of the mathematical descriptions of GPCR signal transduction and investigations on dose-response curves. This article is the first study in reviewing the network structures underlying GPCR signal transduction that control downstream [Ca_c²⁺]-oscillations. The central theme of this paper is to present the biochemical pathways, as well as molecular mechanisms underlying the GPCR-mediated Ca²⁺-dynamics in order to facilitate a better understanding of how agonist concentration is encoded in Ca²⁺-signals for G_αq, G_αs, and G_αi/o signaling pathways. Moreover, we present the GPCR targeting drugs that are relevant for treating cardiac, respiratory, and neuro-diseases. The current paper presents the ODE formulation for various models along with the detailed schematics of signaling networks. To provide a systems perspective, we present the network motifs that can provide readers an insight into the complex and intriguing science of agonist-mediated Ca²⁺-dynamics. One of the features of this review is to pinpoint the interplay between positive and negative feedback loops that are involved in controlling intracellular [Ca_c²⁺]-oscillations. Furthermore, we review several examples of dose-response curves obtained from [Ca_c²⁺]-spiking for various GPCR pathways. This paper is expected to be useful for pharmacologists and computational biologists for designing clinical applications of GPCR targeting drugs through modulation of Ca²⁺-dynamics.

Information transmission from NFkB signaling dynamics to gene expression

The dynamic signal encoding paradigm suggests that information flows from the extracellular environment into specific signaling patterns (encoding) that are then read by downstream effectors to control cellular behavior. Previous work empirically quantified the information content of dynamic signaling patterns. However, whether this information can be faithfully transmitted to the gene expression level is unclear. Here we used NFkB signaling as a model to understand the accuracy of information transmission from signaling dynamics into gene expression. Using a detailed mathematical model, we simulated realistic NFkB signaling patterns with different degrees of variability. The NFkB patterns were used as an input to a simple gene expression model. Analysis of information transmission between ligand and NFkB and ligand and gene expression allows us to determine information loss in transmission between receptors to dynamic signaling patterns and between signaling dynamics to gene expression. Information loss could occur due to biochemical noise or due to a lack of specificity. We found that noise-free gene expression has very little information loss suggesting that gene expression can preserve specificity in NFkB patterns. As expected, the addition of noise to the gene expression model results in information loss. Interestingly, this effect can be mitigated by a specific choice of parameters that can substantially reduce information loss due to biochemical noise during gene expression. Overall our results show that the cellular capacity for information transmission from dynamic signaling patterns to gene expression can be high enough to preserve ligand specificity and thereby the accuracy of cellular response to environmental cues.

Molecular and DNA Artificial Neural Networks via Fractional Coding

This article considers implementation of artificial neural networks (ANNs) using molecular computing and DNA based on fractional coding. Prior work had addressed molecular two-layer ANNs with binary inputs and arbitrary weights. In prior work using fractional coding, a simple molecular perceptron that computes sigmoid of scaled weighted sum of the inputs was presented where the inputs and the weights lie between [-1,1]. Even for computing the perceptron, the prior approach suffers from two major limitations. First, it cannot compute the sigmoid of the weighted sum, but only the sigmoid of the scaled weighted sum. Second, many machine learning applications require the coefficients to be arbitrarily positive and negative numbers that are not bounded between [-1,1]; such numbers cannot be handled by the prior perceptron using fractional coding. This paper makes four contributions. First molecular perceptrons that can handle arbitrary weights and can compute sigmoid of the weighted sums are presented. Thus, these molecular perceptrons are ideal for regression applications and multi-layer ANNs. A new molecular divider is introduced and is used to compute sigmoid(ax) where . Second, based on fractional coding, a molecular artificial neural network (ANN) with one hidden layer is presented. Third, a trained ANN classifier with one hidden layer from seizure prediction application from electroencephalogram is mapped to molecular reactions and DNA and their performances are presented. Fourth, molecular activation functions for rectified linear unit (ReLU) and softmax are also presented.

IP3-Dependent Ca2+ Oscillations Switch into a Dual Oscillator Mechanism in the Presence of PLC-Linked Hormones

Ca2+ oscillations that depend on inositol-1,4,5-trisphosphate (IP3) have been ascribed to biphasic Ca2+ regulation of the IP3 receptor (IP3R) or feedback mechanisms controlling IP3 levels in different cell types. IP3 uncaging in hepatocytes elicits Ca2+ transients that are often localized at the subcellular level and increase in magnitude with stimulus strength. However, this does not reproduce the broad baseline-separated global Ca2+ oscillations elicited by vasopressin. Addition of hormone to cells activated by IP3 uncaging initiates a qualitative transition from high-frequency spatially disorganized Ca2+ transients, to low-frequency, oscillatory Ca2+ waves that propagate throughout the cell. A mathematical model with dual coupled oscillators that integrates Ca2+-induced Ca2+ release at the IP3R and mutual feedback mechanisms of cross-coupling between Ca2+ and IP3 reproduces this behavior. Thus, multiple Ca2+ oscillation modes can coexist in the same cell, and hormonal stimulation can switch from the simpler to the more complex to yield robust signaling.

Endoplasmic Reticulum Calcium Pumps and Tumor Cell Differentiation

Endoplasmic reticulum (ER) calcium homeostasis plays an essential role in cellular calcium signaling, intra-ER protein chaperoning and maturation, as well as in the interaction of the ER with other organelles. Calcium is accumulated in the ER by sarco/endoplasmic reticulum calcium ATPases (SERCA enzymes) that generate by active, ATP-dependent transport, a several thousand-fold calcium ion concentration gradient between the cytosol (low nanomolar) and the ER lumen (high micromolar). SERCA enzymes are coded by three genes that by alternative splicing give rise to several isoforms, which can display isoform-specific calcium transport characteristics. SERCA expression levels and isoenzyme composition vary according to cell type, and this constitutes a mechanism whereby ER calcium homeostasis is adapted to the signaling and metabolic needs of the cell, depending on its phenotype, its state of activation and differentiation. As reviewed here, in several normal epithelial cell types including bronchial, mammary, gastric, colonic and choroid plexus epithelium, as well as in mature cells of hematopoietic origin such as pumps are simultaneously expressed, whereas in corresponding tumors and leukemias SERCA3 expression is selectively down-regulated. SERCA3 expression is restored during the pharmacologically induced differentiation of various cancer and leukemia cell types. SERCA3 is a useful marker for the study of cell differentiation, and the loss of SERCA3 expression constitutes a previously unrecognized example of the remodeling of calcium homeostasis in tumors.

A Statistical View on Calcium Oscillations

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

CaSiAn: a Calcium Signaling Analyzer tool

Summary

Ca²⁺ is a central second messenger in eukaryotic cells that regulates many cellular processes. Recently, we have indicated that typical Ca²⁺ signals are not purely oscillatory as widely assumed, but exhibit stochastic spiking with cell type and pathway specific characteristics. Here, we present the Calcium Signaling Analyzer (CaSiAn), an open source software tool that allows for quantifying these signal characteristics including individual spike properties and time course statistics in a semi-automated manner. CaSiAn provides an intuitive graphical user interface allowing experimentalists to easily process a large amount of Ca²⁺ signals, interactively tune peak detection, revise statistical measures and access the quantified signal properties as excel or text files.

Availability and implementation

CaSiAn is implemented in Java and available on Github (https://github.com/mmahsa/CaSiAn) as well as on the project page (http://r3lab.uni.lu/web/casa).

Supplementary information

Supplementary data are available at Bioinformatics online.

Heterogeneity and emergent behaviour in the vascular endothelium

The endothelium is the single layer of cells lining all blood vessels, and it is a remarkable cardiovascular control centre. Each endothelial cell has only a small number (on average six) of interconnected neighbours. Yet this arrangement produces a large repertoire of behaviours, capable of controlling numerous cardiovascular functions in a flexible and dynamic way. The endothelium regulates the delivery of nutrients and removal of waste by regulating blood flow and vascular permeability. The endothelium regulates blood clotting, responses to infection and inflammation, the formation of new blood vessels, and remodelling of the blood vessel wall. To carry out these roles, the endothelium autonomously interprets a complex environment crammed with signals from hormones, neurotransmitters, pericytes, smooth muscle cells, various blood cells, viral or bacterial infection and proinflammatory cytokines. It is generally assumed that the endothelium responds to these instructions with coordinated responses in a homogeneous population of endothelial cells. Here, we highlight evidence that shows that neighbouring endothelial cells are highly heterogeneous and display different sensitivities to various activators. Cells with various sensitivities process different extracellular signals into distinct streams of information in parallel, like a vast switchboard. Communication occurs among cells and new ‘emergent’ signals are generated that are non-linear composites of the inputs. Emergent signals cannot be predicted or deduced from the properties of individual cells. Heterogeneity and emergent behaviour bestow capabilities on the endothelial collective that far exceed those of individual cells. The implications of heterogeneity and emergent behaviour for understanding vascular disease and drug discovery are discussed.

Structure and Function of IP3 Receptors

Inositol 1,4,5-trisphosphate receptors (IP₃Rs), by releasing Ca²⁺ from the endoplasmic reticulum (ER) of animal cells, allow Ca²⁺ to be redistributed from the ER to the cytosol or other organelles, and they initiate store-operated Ca²⁺ entry (SOCE). For all three IP₃R subtypes, binding of IP₃ primes them to bind Ca²⁺, which then triggers channel opening. We are now close to understanding the structural basis of IP₃R activation. Ca²⁺-induced Ca²⁺ release regulated by IP₃ allows IP₃Rs to regeneratively propagate Ca²⁺ signals. The smallest of these regenerative events is a Ca²⁺ puff, which arises from the nearly simultaneous opening of a small cluster of IP₃Rs. Ca²⁺ puffs are the basic building blocks for all IP₃-evoked Ca²⁺ signals, but only some IP₃ clusters, namely those parked alongside the ER-plasma membrane junctions where SOCE occurs, are licensed to respond. The location of these licensed IP₃Rs may allow them to selectively regulate SOCE.

Deterministic Limit of Intracellular Calcium Spikes

In nonexcitable cells, global Ca^{2+} spikes emerge from the collective dynamics of clusters of Ca^{2+} channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin, while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

A Two-Pulse Cellular Stimulation Test Elucidates Variability and Mechanisms in Signaling Pathways

Mammalian cells respond in a variable manner when provided with physiological pulses of ligand, such as low concentrations of acetylcholine present for just tens of seconds or TNFα for just tens of minutes. For a two-pulse stimulation, some cells respond to both pulses, some do not respond, and yet others respond to only one or the other pulse. Are these different response patterns the result of the small number of ligands being able to only stochastically activate the pathway at random times or an output pattern from a deterministic algorithm responding differently to different stimulation intervals? If the response is deterministic in nature, what parameters determine whether a response is generated or skipped? To answer these questions, we developed a two-pulse test that utilizes different rest periods between stimulation pulses. This "rest-period test" revealed that cells skip responses predictably as the rest period is shortened. By combining these experimental results with a mathematical model of the pathway, we further obtained mechanistic insight into potential sources of response variability. Our analysis indicates that in both intracellular calcium and NFκB signaling, response variability is consistent with extrinsic noise (cell-to-cell variability in protein levels), a short-term memory of stimulation, and high Hill coefficient processes. Furthermore, these results support recent works that have emphasized the role of deterministic processes for explaining apparently stochastic cellular response variability and indicate that even weak stimulations likely guide mammalian cells to appropriate fates rather than leaving outcomes to chance. We envision that the rest-period test can be applied to other signaling pathways to extract mechanistic insight.

Coding and decoding of oscillatory Ca2+ signals

About 30 years after their first observation, Ca²⁺ oscillations are now recognised as a universal mechanism of signal transduction. These oscillations are driven by periodic cycles of release and uptake of Ca²⁺ between the cytoplasm and the endoplasmic reticulum. Their frequency often increases with the level of stimulation, which can be decoded by some molecules. However, it is becoming increasingly evident that the widespread core oscillatory mechanism is modulated in many ways, depending on the cell type and on the physiological conditions. Interplay with inositol 1,4,5-trisphosphate metabolism and with other Ca²⁺ stores as the extracellular medium or mitochondria can much affect the properties of these oscillations. In many cases, these finely tuned characteristics of Ca²⁺ oscillations impact the physiological response that is triggered by the signal. Moreover, oscillations are intrinsically irregular. This randomness can also be exploited by the cell. In this review, we discuss evidences of these additional manifestations of the versatility of Ca²⁺ signalling.

IP3 receptors – lessons from analyses ex cellula

Inositol 1,4,5-trisphosphate receptors (IP3Rs) are widely expressed intracellular channels that release Ca2+ from the endoplasmic reticulum (ER). We review how studies of IP3Rs removed from their intracellular environment (‘ex cellula’), alongside similar analyses of ryanodine receptors, have contributed to understanding IP3R behaviour. Analyses of permeabilized cells have demonstrated that the ER is the major intracellular Ca2+ store, and that IP3 stimulates Ca2+ release from this store. Radioligand binding confirmed that the 4,5-phosphates of IP3 are essential for activating IP3Rs, and facilitated IP3R purification and cloning, which paved the way for structural analyses. Reconstitution of IP3Rs into lipid bilayers and patch-clamp recording from the nuclear envelope have established that IP3Rs have a large conductance and select weakly between Ca2+ and other cations. Structural analyses are now revealing how IP3 binding to the N-terminus of the tetrameric IP3R opens the pore ∼7 nm away from the IP3-binding core (IBC). Communication between the IBC and pore passes through a nexus of interleaved domains contributed by structures associated with the pore and cytosolic domains, which together contribute to a Ca2+-binding site. These structural analyses provide evidence to support the suggestion that IP3 gates IP3Rs by first stimulating Ca2+ binding, which leads to pore opening and Ca2+ release.

IP3 receptors and Ca2+ entry

Inositol 1,4,5-trisphosphate receptors (IP₃R) are the most widely expressed intracellular Ca²⁺ release channels. Their activation by IP₃ and Ca²⁺ allows Ca²⁺ to pass rapidly from the ER lumen to the cytosol. The resulting increase in cytosolic [Ca²⁺] may directly regulate cytosolic effectors or fuel Ca²⁺ uptake by other organelles, while the decrease in ER luminal [Ca²⁺] stimulates store-operated Ca²⁺ entry (SOCE). We are close to understanding the structural basis of both IP₃R activation, and the interactions between the ER Ca²⁺-sensor, STIM, and the plasma membrane Ca²⁺ channel, Orai, that lead to SOCE. IP₃Rs are the usual means through which extracellular stimuli, through ER Ca²⁺ release, stimulate SOCE. Here, we review evidence that the IP₃Rs most likely to respond to IP₃ are optimally placed to allow regulation of SOCE. We also consider evidence that IP₃Rs may regulate SOCE downstream of their ability to deplete ER Ca²⁺ stores. Finally, we review evidence that IP₃Rs in the plasma membrane can also directly mediate Ca²⁺ entry in some cells.

Signaling pathways as linear transmitters

One challenge in biology is to make sense of the complexity of biological networks. A good system to approach this is signaling pathways, whose well-characterized molecular details allow us to relate the internal processes of each pathway to their input-output behavior. In this study, we analyzed mathematical models of three metazoan signaling pathways: the canonical Wnt, MAPK/ERK, and Tgfβ pathways. We find an unexpected convergence: the three pathways behave in some physiological contexts as linear signal transmitters. Testing the results experimentally, we present direct measurements of linear input-output behavior in the Wnt and ERK pathways. Analytics from each model further reveal that linearity arises through different means in each pathway, which we tested experimentally in the Wnt and ERK pathways. Linearity is a desired property in engineering where it facilitates fidelity and superposition in signal transmission. Our findings illustrate how cells tune different complex networks to converge on the same behavior.

Optogenetic Inhibition of Gαq Protein Signaling Reduces Calcium Oscillation Stochasticity

As fast terminators of G-protein coupled receptor (GPCR) signaling, regulators of G-protein signaling (RGS) serve critical roles in fine-tuning second messenger levels and, consequently, cellular responses to external stimuli. Here, we report the creation of an optogenetic RGS2 (opto-RGS2) that suppresses agonist-evoked calcium oscillations by the inactivation of Gαq protein. In this system, cryptochrome-mediated heterodimerization of the catalytic RGS2-box with its N-terminal amphipathic helix reconstitutes a functional membrane-localized complex that can dynamically suppress store-operated release of calcium. Engineered opto-RGS2 cell lines were used to establish the role of RGS2 as a key inhibitory feedback regulator of the stochasticity of the Gαq-mediated calcium spike timing. RGS2 reduced the stochasticity of carbachol-stimulated calcium oscillations, and the feedback inhibition was coupled to the global calcium elevation by calmodulin/RGS2 interactions. The identification of a critical negative feedback circuit exemplifies the utility of optogenetic approaches for interrogating RGS/GPCR biology and calcium encoding principles through temporally precise molecular gain-of-function.

Computing Mathematical Functions using DNA via Fractional Coding

This paper discusses the implementation of mathematical functions such as exponentials, trigonometric functions, the sigmoid function and the perceptron function with molecular reactions in general, and DNA strand displacement reactions in particular. The molecular constructs for these functions are predicated on a novel representation for input and output values: a fractional encoding, in which values are represented by the relative concentrations of two molecular types, denoted as type-1 and type-0. This representation is inspired by a technique from digital electronic design, termed stochastic logic, in which values are represented by the probability of 1's in a stream of randomly generated 0's and 1's. Research in the electronic realm has shown that a variety of complex functions can be computed with remarkably simple circuitry with this stochastic approach. This paper demonstrates how stochastic electronic designs can be translated to molecular circuits. It presents molecular implementations of mathematical functions that are considerably more complex than any shown to date. All designs are validated using mass-action simulations of the chemical kinetics of DNA strand displacement reactions.

The stretch to stray on time: Resonant length of random walks in a transient

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

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.

Modeling Cell-to-Cell Communication Networks Using Response-Time Distributions

Cell-to-cell communication networks have critical roles in coordinating diverse organismal processes, such as tissue development or immune cell response. However, compared with intracellular signal transduction networks, the function and engineering principles of cell-to-cell communication networks are far less understood. Major complications include: cells are themselves regulated by complex intracellular signaling networks; individual cells are heterogeneous; and output of any one cell can recursively become an additional input signal to other cells. Here, we make use of a framework that treats intracellular signal transduction networks as "black boxes" with characterized input-to-output response relationships. We study simple cell-to-cell communication circuit motifs and find conditions that generate bimodal responses in time, as well as mechanisms for independently controlling synchronization and delay of cell-population responses. We apply our modeling approach to explain otherwise puzzling data on cytokine secretion onset times in T cells. Our approach can be used to predict communication network structure using experimentally accessible input-to-output measurements and without detailed knowledge of intermediate steps.

Ca2+ signals initiate at immobile IP3 receptors adjacent to ER-plasma membrane junctions

IP₃ receptors (IP₃Rs) release Ca²⁺ from the ER when they bind IP₃ and Ca²⁺. The spatial organization of IP₃Rs determines both the propagation of Ca²⁺ signals between IP₃Rs and the selective regulation of cellular responses. Here we use gene editing to fluorescently tag endogenous IP₃Rs, and super-resolution microscopy to determine the geography of IP₃Rs and Ca²⁺ signals within living cells. We show that native IP₃Rs cluster within ER membranes. Most IP₃R clusters are mobile, moved by diffusion and microtubule motors. Ca²⁺ signals are generated by a small population of immobile IP₃Rs. These IP₃Rs are licensed to respond, but they do not readily mix with mobile IP₃Rs. The licensed IP₃Rs reside alongside ER-plasma membrane junctions where STIM1, which regulates store-operated Ca²⁺ entry, accumulates after depletion of Ca²⁺ stores. IP₃Rs tethered close to ER-plasma membrane junctions are licensed to respond and optimally placed to be activated by endogenous IP₃ and to regulate Ca²⁺ entry.

IP₃ receptors mediate Ca²⁺ release from the endoplasmic reticulum. Here the authors show that only a small fraction of IP₃ receptors initiate Ca²⁺ signals; these immobile IP₃ receptors adjacent to the plasma membrane are optimally placed to control STIM1-dependent Ca²⁺ entry.

Temporal Bias: Time-Encoded Dynamic GPCR Signaling

Evidence suggests that cells can time-encode signals for secure transport and perception of information, and it appears that this dynamic signaling is a common principle of nature to code information in time. G-protein-coupled receptor (GPCR) signaling networks are no exception as their composition and signal transduction appear temporally flexible. In this review, we discuss the potential mechanisms by which GPCRs code biological information in time to create 'temporal bias.' We highlight dynamic signaling patterns from the second messenger to the receptor-ligand level and shed light on the dynamics of G-protein cycles, the kinetics of ligand-receptor interaction, and the occurrence of distinct signaling waves within the cell. A dynamic feature such as temporal bias adds to the complexity of GPCR signaling bias and gives rise to the question whether this trait could be exploited to gain control over time-encoded cell physiology.

Cross-hierarchy systems principles

One driving motivation of systems biology is the search for general principles that govern the design of biological systems. But questions often arise as to what kind of general principles biology could have. Concepts from engineering such as robustness and modularity are indeed becoming a regular way of describing biological systems. Another source of potential general principles is the emerging similarities found in processes across biological hierarchies. In this piece, I describe several emerging cross-hierarchy similarities. Identification of more cross-hierarchy principles, and understanding the implications these convergence have on the construction of biological systems, I believe, present exciting challenges for systems biology in the decades to come.

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 stimulus 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 environment.

Upon stimulation a large number of cell types respond with transient increases of the intracellular calcium concentration, which often take the form of repetitive spikes. It is therefore believed that calcium spikes play a central role in cellular signal transduction. A critical feature of these calcium spikes is that they occur randomly, which raises the question of how we can predict the timing of calcium spikes. We here show that by using Bayesian ideas and concepts from stochastic processes, we can quantitatively compute the calcium spike rate for a given stimulus. Our analysis also demonstrates that traditional methods for spike rate estimation perform less favourably compared to a Bayesian approach when small numbers of cells are investigated. To test our methodology under conditions that closely mimic those experienced in vivo we challenged cells with agonist concentrations that vary both in space and time. We find that cells that experience similar stimulus profiles are described by similar calcium spike rates. This suggests that calcium spike rates may constitute a quantitative description of whole-cell calcium spiking that reflects both the randomness and the spatiotemporal organisation of the calcium signalling machinery.

Endogenous signalling pathways and caged IP3 evoke Ca2+ puffs at the same abundant immobile intracellular sites

The building blocks of intracellular Ca²⁺ signals evoked by inositol 1,4,5-trisphosphate receptors (IP₃Rs) are Ca²⁺ puffs, transient focal increases in Ca²⁺ concentration that reflect the opening of small clusters of IP₃Rs. We use total internal reflection fluorescence microscopy and automated analyses to detect Ca²⁺ puffs evoked by photolysis of caged IP₃ or activation of endogenous muscarinic receptors with carbachol in human embryonic kidney 293 cells. Ca²⁺ puffs evoked by carbachol initiated at an estimated 65±7 sites/cell, and the sites remained immobile for many minutes. Photolysis of caged IP₃ evoked Ca²⁺ puffs at a similar number of sites (100±35). Increasing the carbachol concentration increased the frequency of Ca²⁺ puffs without unmasking additional Ca²⁺ release sites. By measuring responses to sequential stimulation with carbachol or photolysed caged IP₃, we established that the two stimuli evoked Ca²⁺ puffs at the same sites. We conclude that IP₃-evoked Ca²⁺ puffs initiate at numerous immobile sites and the sites become more likely to fire as the IP₃ concentration increases; there is no evidence that endogenous signalling pathways selectively deliver IP₃ to specific sites.

Summary: Ca²⁺ puffs are the building blocks for IP₃-evoked Ca²⁺ signals. Ca²⁺ puffs evoked by caged IP₃ or via endogenous signalling pathways initiate at the same fixed intracellular sites.