Approaches and tools for modeling signaling pathways and calcium dynamics in neurons

Differential behaviors of calcium-induced calcium release in one dimensional dendrite by Nernst-Planck equation, cable model and pure diffusion model

The source and dynamics of calcium is the key factor that regulates dendritic integration. Apart from the voltage-gated and ligand-gated calcium influx, an important source of calcium is from inner store of endoplasmic reticulum with a regenerative process of calcium-induced calcium release (CICR). To trigger this process, inositol 1,4,5-trisphosphate (IP3) and calcium are needed to satisfy certain requirements. The aim of our paper is to investigate how the CICR depends on the dynamics of membrane potential. We utilize one dimensional dendritic model to calculate membrane potential by Nernst-Planck Equation (NPE) and cable model and Pure Diffusion (PD) model, computational simulations are carried out to inject the calcium influx by synaptic stimulation and to predict subsequent CICR and calcium wave propagation. Our results demonstrate that CICR initiation and calcium wave propagation have much difference between electro-diffusion process of NPE and cable model. We find that cable model has lower threshold of IP3 stimulation to trigger CICR but is more difficult for calcium propagation than NPE, PD model requires even higher threshold of IP3 to initiate CICR process and calcium duration is shorter than NPE; the regenerative calcium wave propagates with faster speed in NPE than that in cable model and in PD model. Our work addresses the important role of electro-diffusion dynamics of charged ions in regulating CICR process in dendritic structure; and provides theoretical predictions for neurological process which requires sustaining calcium for downstream signaling processes.

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Reinforcing Interdisciplinary Collaborations to Unravel the Astrocyte "Calcium Code"

In this review article, we present the major insights from and challenges faced in the acquisition, analysis and modeling of astrocyte calcium activity, aiming at bridging the gap between those fields to crack the complex astrocyte “Calcium Code”. We then propose strategies to reinforce interdisciplinary collaborative projects to unravel astrocyte function in health and disease.

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Calcium (Ca²⁺) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP₃)-induced Ca²⁺ release. These Ca²⁺ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca²⁺, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca²⁺ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca²⁺ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca²⁺ transients in different dynamical situations. We test two different models of the IP₃ receptor (IP₃R). The model with nonlinear concentration response of binding of activating Ca²⁺ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca²⁺-dependent inhibition of the IP₃R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca²⁺ and IP₃ production sensitizing IP₃R for Ca²⁺-induced Ca²⁺ release. We also model ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca²⁺ transients. Finally, we propose possible ways of recovering Ca²⁺ transients under ataxia.

A multi-state model of the CaMKII dodecamer suggests a role for calmodulin in maintenance of autophosphorylation

Ca2+/calmodulin-dependent protein kinase II (CaMKII) accounts for up to 2 percent of all brain protein and is essential to memory function. CaMKII activity is known to regulate dynamic shifts in the size and signaling strength of neuronal connections, a process known as synaptic plasticity. Increasingly, computational models are used to explore synaptic plasticity and the mechanisms regulating CaMKII activity. Conventional modeling approaches may exclude biophysical detail due to the impractical number of state combinations that arise when explicitly monitoring the conformational changes, ligand binding, and phosphorylation events that occur on each of the CaMKII holoenzyme's subunits. To manage the combinatorial explosion without necessitating bias or loss in biological accuracy, we use a specialized syntax in the software MCell to create a rule-based model of a twelve-subunit CaMKII holoenzyme. Here we validate the rule-based model against previous experimental measures of CaMKII activity and investigate molecular mechanisms of CaMKII regulation. Specifically, we explore how Ca2+/CaM-binding may both stabilize CaMKII subunit activation and regulate maintenance of CaMKII autophosphorylation. Noting that Ca2+/CaM and protein phosphatases bind CaMKII at nearby or overlapping sites, we compare model scenarios in which Ca2+/CaM and protein phosphatase do or do not structurally exclude each other's binding to CaMKII. Our results suggest a functional mechanism for the so-called "CaM trapping" phenomenon, wherein Ca2+/CaM may structurally exclude phosphatase binding and thereby prolong CaMKII autophosphorylation. We conclude that structural protection of autophosphorylated CaMKII by Ca2+/CaM may be an important mechanism for regulation of synaptic plasticity.

Mathematical Models to Shed Light on Amyloid-Beta and Tau Protein Dependent Pathologies in Alzheimer's Disease

The number of patients suffering from dementia due to Alzheimer's disease (AD) is constantly rising worldwide. This has accordingly resulted in huge burdens on the health systems and involved families. Lack of profound understanding of neural networking in normal brain and their interruption in AD makes the treatment of this neurodegenerative multifaceted disease a challenging issue. In recent years, mathematical and computational methods have paved the way towards a better understanding of the brain functional connectivity. Thus, much attention has been paid to this matter from both basic science researchers and clinicians with an interdisciplinary approach to determine what is not functioning properly in AD patients and how this malfunctioning can be addressed. In this review, a number of AD-related articles and well-studied pathophysiologic topics (e.g., amyloid-beta, neurofibrillary tangles, Ca²⁺ dysregulation, and synaptic plasticity alterations) has been literally surveyed from a computational and systems biology point of view. The neural networks were discussed from biological and mathematical point of views and their alterations in recent findings were further highlighted. Application of the graph theoretical analysis in the brain imaging was reviewed, depicting the relations between brain structure and function, without diving into mathematical details. Moreover, differential rate equations were briefly articulated, emphasizing the potential use of these equations in simplifying complex processes in relevance to pathologies of AD. Comprehensive insights were given into the AD progression from neural networks perspective, which may lead us towards potential strategies for early diagnosis and effective treatment of AD.

Neuroprotective Effects of Ferruginol, Jatrophone, and Junicedric Acid Against Amyloid-β Injury in Hippocampal Neurons

Soluble amyloid-β (Aβ) oligomers have been recognized as early neurotoxic intermediates with a key role in the synaptic dysfunction observed in Alzheimer's disease (AD). Aβ oligomers block hippocampal long-term potentiation (LTP) and impair rodent spatial memory. Additionally, the presence of Aβ oligomers is associated with imbalanced intracellular calcium levels and apoptosis in neurons. In this context, we evaluated the effects of three diterpenes (ferruginol, jatrophone, and junicedric acid) that are found in medicinal plants and have several forms of biological activity. The intracellular calcium levels in hippocampal neurons increased in the presence of ferruginol, jatrophone, and junicedric acid, a result that was consistent with the observed increase in CA1 synaptic transmission in mouse hippocampal slices. Additionally, assays using Aβ peptide demonstrated that diterpenes, particularly ferruginol, restore LTP and reduce apoptosis. Recovery of the Aβ oligomer-induced loss of the synaptic proteins PSD-95, synapsin, VGlut, and NMDA receptor subunit 2A was observed in mouse hippocampal slices treated with junicedric acid. This cascade of events may be associated with the regulation of kinases, e.g., protein kinase C (PKC) and calcium/calmodulin-dependent protein kinase II (CaMKII), in addition to the activation of the canonical Wnt signaling pathway and could thus provide protection against Aβ oligomers, which trigger synaptic dysfunction. Our results suggest a potential neuroprotective role for diterpenes against the Aβ oligomers-induced neurodegenerative alterations, which make them interesting molecules to be further studied in the context of AD.

Parallel Stochastic Discrete Event Simulation of Calcium Dynamics in Neuron

The intra-cellular calcium signaling pathways of a neuron depends on both biochemical reactions and diffusions. Some quasi-isolated compartments (e.g., spines) are so small and calcium concentrations are so low that one extra molecule diffusing in by chance can make a nontrivial difference in concentration (percentage-wise). These rare events can affect dynamics discretely in such a way that they cannot be evaluated by a deterministic and continuous simulation. Stochastic models of such a system provide a more detailed understanding of these systems than existing deterministic models because they capture their behavior at a molecular level. Our research focuses on the development of a high performance parallel discrete event simulation environment, Neuron Time Warp (NTW), which is intended for use in the parallel simulation of stochastic reaction-diffusion systems such as intra-calcium signaling. NTW is integrated with NEURON, a simulator which is widely used within the neuroscience community. We simulate two models, a calcium buffer and a calcium wave model. The calcium buffer model is employed in order to verify the correctness and performance of NTW by comparing it to a sequential deterministic simulation in NEURON. We also derived a discrete event calcium wave model from a deterministic model using the stochastic $\text{IP}_{3}\text{R}$IP3R structure.

What can computational modeling offer for studying the Ca2+ dysregulation in Alzheimer's disease: current research and future directions

Ca²⁺ dysregulation is an early event observed in Alzheimer’s disease (AD) patients preceding the presence of its clinical symptoms. Dysregulation of neuronal Ca²⁺ will cause synaptic loss and neuronal death, eventually leading to memory impairments and cognitive decline. Treatments targeting Ca²⁺ signaling pathways are potential therapeutic strategies against AD. The complicated interactions make it challenging and expensive to study the underlying mechanisms as to how Ca²⁺ signaling contributes to the pathogenesis of AD. Computational modeling offers new opportunities to study the signaling pathway and test proposed mechanisms. In this mini-review, we present some computational approaches that have been used to study Ca²⁺ dysregulation of AD by simulating Ca²⁺ signaling at various levels. We also pointed out the future directions that computational modeling can be done in studying the Ca²⁺ dysregulation in AD.

Challenges in Reproducibility, Replicability, and Comparability of Computational Models and Tools for Neuronal and Glial Networks, Cells, and Subcellular Structures

The possibility to replicate and reproduce published research results is one of the biggest challenges in all areas of science. In computational neuroscience, there are thousands of models available. However, it is rarely possible to reimplement the models based on the information in the original publication, let alone rerun the models just because the model implementations have not been made publicly available. We evaluate and discuss the comparability of a versatile choice of simulation tools: tools for biochemical reactions and spiking neuronal networks, and relatively new tools for growth in cell cultures. The replicability and reproducibility issues are considered for computational models that are equally diverse, including the models for intracellular signal transduction of neurons and glial cells, in addition to single glial cells, neuron-glia interactions, and selected examples of spiking neuronal networks. We also address the comparability of the simulation results with one another to comprehend if the studied models can be used to answer similar research questions. In addition to presenting the challenges in reproducibility and replicability of published results in computational neuroscience, we highlight the need for developing recommendations and good practices for publishing simulation tools and computational models. Model validation and flexible model description must be an integral part of the tool used to simulate and develop computational models. Constant improvement on experimental techniques and recording protocols leads to increasing knowledge about the biophysical mechanisms in neural systems. This poses new challenges for computational neuroscience: extended or completely new computational methods and models may be required. Careful evaluation and categorization of the existing models and tools provide a foundation for these future needs, for constructing multiscale models or extending the models to incorporate additional or more detailed biophysical mechanisms. Improving the quality of publications in computational neuroscience, enabling progressive building of advanced computational models and tools, can be achieved only through adopting publishing standards which underline replicability and reproducibility of research results.

Pleiotropic effects of schizophrenia-associated genetic variants in neuron firing and cardiac pacemaking revealed by computational modeling

Schizophrenia patients have an increased risk of cardiac dysfunction. A possible factor underlying this comorbidity are the common variants in the large set of genes that have recently been discovered in genome-wide association studies (GWASs) as risk genes of schizophrenia. Many of these genes control the cell electrogenesis and calcium homeostasis. We applied biophysically detailed models of layer V pyramidal cells and sinoatrial node cells to study the contribution of schizophrenia-associated genes on cellular excitability. By including data from functional genomics literature to simulate the effects of common variants of these genes, we showed that variants of voltage-gated Na⁺ channel or hyperpolarization-activated cation channel-encoding genes cause qualitatively similar effects on layer V pyramidal cell and sinoatrial node cell excitability. By contrast, variants of Ca²⁺ channel or transporter-encoding genes mostly have opposite effects on cellular excitability in the two cell types. We also show that the variants may crucially affect the propagation of the cardiac action potential in the sinus node. These results may help explain some of the cardiac comorbidity in schizophrenia, and may facilitate generation of effective antipsychotic medications without cardiac side-effects such as arrhythmia.

Load balancing for multi-threaded PDES of stochastic reaction-diffusion in neurons

Stochastic simulation of chemical reactions and diffusion in a neuron helps to provide a realistic view of the molecular dynamics within a neuron. We developed a multi-threaded PDES simulator, Neuron Time Warp-Multi Thread, suitable for the stochastic simulation of reaction and diffusion in a neuron. In this paper we make use of Q-Learning and Simulated Annealing to determine the parameters for a dynamic load balancing algorithm and for dynamic window control. During the simulation, the runtime statistics of each thread are collected and used to determine the execution time of the simulation. Based upon this assessment, workload is migrated from the most overloaded threads to the most under-load ones. As the results for a calcium wave model show, both approaches can improve the execution time for small simulations by up to 31% (Q-Learning) and 19% (SA). The simulated annealing approach is more suitable for larger populations, decreasing execution time by 41%.

Multithreaded Stochastic PDES for Reactions and Diffusions in Neurons

Cells exhibit stochastic behavior when the number of molecules is small. Hence a stochastic reaction-diffusion simulator capable of working at scale can provide a more accurate view of molecular dynamics within the cell. This paper describes a parallel discrete event simulator, Neuron Time Warp-Multi Thread (NTW-MT), developed for the simulation of reaction diffusion models of neurons. To the best of our knowledge, this is the first parallel discrete event simulator oriented towards stochastic simulation of chemical reactions in a neuron. The simulator was developed as part of the NEURON project. NTW-MT is optimistic and thread-based, which attempts to capitalize on multi-core architectures used in high performance machines. It makes use of a multi-level queue for the pending event set and a single roll-back message in place of individual anti-messages to disperse contention and decrease the overhead of processing rollbacks. Global Virtual Time is computed asynchronously both within and among processes to get rid of the overhead for synchronizing threads. Memory usage is managed in order to avoid locking and unlocking when allocating and de-allocating memory and to maximize cache locality. We verified our simulator on a calcium buffer model. We examined its performance on a calcium wave model, comparing it to the performance of a process based optimistic simulator and a threaded simulator which uses a single priority queue for each thread. Our multi-threaded simulator is shown to achieve superior performance to these simulators. Finally, we demonstrated the scalability of our simulator on a larger CICR model and a more detailed CICR model.

Development of computational models of cAMP signaling

Despite the growing evidence defining the cAMP signaling network as a master regulator of cellular function in a number of tissues, regulatory feedback loops, signal compartmentalization, as well as cross-talk with other signaling pathways make understanding the emergent properties of cAMP cellular action a daunting task. Dynamical models of signaling that combine quantitative rigor with molecular details can contribute valuable mechanistic insight into the complexity of intracellular cAMP signaling by complementing and guiding experimental efforts. In this chapter, we review the development of cAMP computational models. We describe how features of the cAMP network can be represented and review the types of experimental data useful in modeling cAMP signaling. We also compile a list of published cAMP models that can aid in the development of novel dynamical models of cAMP signaling.

Multiscale model of dynamic neuromodulation integrating neuropeptide-induced signaling pathway activity with membrane electrophysiology

We developed a multiscale model to bridge neuropeptide receptor-activated signaling pathway activity with membrane electrophysiology. Typically, the neuromodulation of biochemical signaling and biophysics have been investigated separately in modeling studies. We studied the effects of Angiotensin II (AngII) on neuronal excitability changes mediated by signaling dynamics and downstream phosphorylation of ion channels. Experiments have shown that AngII binding to the AngII receptor type-1 elicits baseline-dependent regulation of cytosolic Ca(2+) signaling. Our model simulations revealed a baseline Ca(2+)-dependent response to AngII receptor type-1 activation by AngII. Consistent with experimental observations, AngII evoked a rise in Ca(2+) when starting at a low baseline Ca(2+) level, and a decrease in Ca(2+) when starting at a higher baseline. Our analysis predicted that the kinetics of Ca(2+) transport into the endoplasmic reticulum play a critical role in shaping the Ca(2+) response. The Ca(2+) baseline also influenced the AngII-induced excitability changes such that lower Ca(2+) levels were associated with a larger firing rate increase. We examined the relative contributions of signaling kinases protein kinase C and Ca(2+)/Calmodulin-dependent protein kinase II to AngII-mediated excitability changes by simulating activity blockade individually and in combination. We found that protein kinase C selectively controlled firing rate adaptation whereas Ca(2+)/Calmodulin-dependent protein kinase II induced a delayed effect on the firing rate increase. We tested whether signaling kinetics were necessary for the dynamic effects of AngII on excitability by simulating three scenarios of AngII-mediated KDR channel phosphorylation: (1), an increased steady state; (2), a step-change increase; and (3), dynamic modulation. Our results revealed that the kinetics emerging from neuromodulatory activation of the signaling network were required to account for the dynamical changes in excitability. In summary, our integrated multiscale model provides, to our knowledge, a new approach for quantitative investigation of neuromodulatory effects on signaling and electrophysiology.

Ca2+ dysregulation in the endoplasmic reticulum related to Alzheimer's disease: A review on experimental progress and computational modeling

Alzheimer's disease (AD) is a devastating, incurable neurodegenerative disease affecting millions of people worldwide. Dysregulation of intracellular Ca(2+) signaling has been observed as an early event prior to the presence of clinical symptoms of AD and is believed to be a crucial factor contributing to its pathogenesis. The progressive and sustaining increase in the resting level of cytosolic Ca(2+) will affect downstream activities and neural functions. This review focuses on the issues relating to the increasing Ca(2+) release from the endoplasmic reticulum (ER) observed in AD neurons. Numerous research papers have suggested that the dysregulation of ER Ca(2+) homeostasis is associated with mutations in the presenilin genes and amyloid-β oligomers. These disturbances could happen at many different points in the signaling process, directly affecting ER Ca(2+) channels or interfering with related pathways, which makes it harder to reveal the underlying mechanisms. This review paper also shows that computational modeling is a powerful tool in Ca(2+) signaling studies and discusses the progress in modeling related to Ca(2+) dysregulation in AD research.

Neuronal calcium wave propagation varies with changes in endoplasmic reticulum parameters: a computer model

Calcium (Ca²⁺) waves provide a complement to neuronal electrical signaling, forming a key part of a neuron's second messenger system. We developed a reaction-diffusion model of an apical dendrite with diffusible inositol triphosphate (IP₃), diffusible Ca²⁺, IP₃ receptors (IP₃Rs), endoplasmic reticulum (ER) Ca²⁺ leak, and ER pump (SERCA) on ER. Ca²⁺ is released from ER stores via IP₃Rs upon binding of IP₃ and Ca²⁺. This results in Ca²⁺-induced-Ca²⁺-release (CICR) and increases Ca²⁺ spread. At least two modes of Ca²⁺ wave spread have been suggested: a continuous mode based on presumed relative homogeneity of ER within the cell and a pseudo-saltatory model where Ca²⁺ regeneration occurs at discrete points with diffusion between them. We compared the effects of three patterns of hypothesized IP₃R distribution: (1) continuous homogeneous ER, (2) hotspots with increased IP₃R density (IP₃R hotspots), and (3) areas of increased ER density (ER stacks). All three modes produced Ca²⁺ waves with velocities similar to those measured in vitro (approximately 50-90 μm /sec). Continuous ER showed high sensitivity to IP₃R density increases, with time to onset reduced and speed increased. Increases in SERCA density resulted in opposite effects. The measures were sensitive to changes in density and spacing of IP₃R hotspots and stacks. Increasing the apparent diffusion coefficient of Ca²⁺ substantially increased wave speed. An extended electrochemical model, including voltage-gated calcium channels and AMPA synapses, demonstrated that membrane priming via AMPA stimulation enhances subsequent Ca²⁺ wave amplitude and duration. Our modeling suggests that pharmacological targeting of IP₃Rs and SERCA could allow modulation of Ca²⁺ wave propagation in diseases where Ca²⁺ dysregulation has been implicated.

Calcium: amplitude, duration, or location?

Calcium plays a role in long-term plasticity by triggering postsynaptic signaling pathways for both the strengthening (LTP) and weakening (LTD) of synapses. Since these are opposing processes, several hypotheses have been developed to explain how calcium can trigger LTP in some situations and LTD in others. These hypotheses fall broadly into three categories, based on the amplitude of calcium concentration, the duration of the calcium elevation, and the location of the calcium influx. Here we review the experimental evidence for and against each of these hypotheses and the recent computational models utilizing each. We argue that with new experimental techniques for the precise visualization of calcium and new computational techniques for the modeling of calcium diffusion, it is time to take a new look at the location hypothesis.

On the dynamics of the adenylate energy system: homeorhesis vs homeostasis

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life.