Dynamic modelling and tristability analysis of misfolded α-synuclein degraded via autophagy in Parkinson's disease

The widely-accepted hallmark pathology of Parkinson's disease (PD) is the presence of Lewy bodies with characteristic abnormal aggregated α-synuclein (αSyn). Growing physiological evidence suggests that there is a pivotal role for the autophagy-lysosome pathway (ALP) in the clearance of misfolded αSyn (αSyn∗). This work establishes a mathematical model for αSyn∗ degradation through the ALP. Qualitative simulations are used to uncover the tristable behavior of αSyn∗, i.e., the lower, medium, and upper steady states, which correspond to the healthy, critical, and disease stages of PD, respectively. Time series and codimension-1 bifurcation analysis suggest that the system shows tristability dynamics. Furthermore, variations in the key parameters influence the tristable dynamic behavior, and the distribution of tristable regions is exhibited more comprehensively in codimension-2 bifurcation diagrams. In addition, robustness analysis demonstrates that tristability is a robust property of the system. These results may be valuable in therapeutic strategies for the prevention and treatment of PD.

Dynamics of a model for the degradation mechanism of aggregated α-synuclein in Parkinson's disease

Accumulation of the misfolded synaptic protein α-synuclein (αSyn^*) is a hallmark of neurodegenerative disease in Parkinson's disease (PD). Recent studies suggest that the autophagy lysosome pathway (ALP) including both the Beclin1-associated and mTOR-signaling pathways is involved in the αSyn^* clearance mechanism. In this study, a mathematical model is proposed for the degradation of αSyn^* by ALP with the crosstalk element of mTOR. Using codimension-1 bifurcation analysis, the tri-stability of αSyn^* is surveyed under three different stress signals and, in addition, consideration is given to the regulatory mechanisms for the Beclin1- and mTOR-dependent rates on αSyn^* degradation using the codimension-1 and-2 bifurcation diagrams. It was found that, especially under internal and external oxidative stresses (S ₁), the bistable switch of the aggregation of αSyn^* can be transformed from an irreversible to a reversible condition through the ALP degradation pathways. Furthermore, the robustness of the tri-stable state for the stress S ₁ to the parameters related to mTOR-mediated ALP was probed. It was confirmed that mTOR-mediated ALP is important for maintaining the essential dynamic features of the tri-stable state. This study may provide a promising avenue for conducting further experiments and simulations of the degradation mechanism of dynamic modeling in PD.

Power Failure of Mitochondria and Oxidative Stress in Neurodegeneration and Its Computational Models

The brain needs more energy than other organs in the body. Mitochondria are the generator of vital power in the living organism. Not only do mitochondria sense signals from the outside of a cell, but they also orchestrate the cascade of subcellular events by supplying adenosine-5′-triphosphate (ATP), the biochemical energy. It is known that impaired mitochondrial function and oxidative stress contribute or lead to neuronal damage and degeneration of the brain. This mini-review focuses on addressing how mitochondrial dysfunction and oxidative stress are associated with the pathogenesis of neurodegenerative disorders including Alzheimer’s disease, amyotrophic lateral sclerosis, Huntington’s disease, and Parkinson’s disease. In addition, we discuss state-of-the-art computational models of mitochondrial functions in relation to oxidative stress and neurodegeneration. Together, a better understanding of brain disease-specific mitochondrial dysfunction and oxidative stress can pave the way to developing antioxidant therapeutic strategies to ameliorate neuronal activity and prevent neurodegeneration.

ROS networks: designs, aging, Parkinson's disease and precision therapies

How the network around ROS protects against oxidative stress and Parkinson's disease (PD), and how processes at the minutes timescale cause disease and aging after decades, remains enigmatic. Challenging whether the ROS network is as complex as it seems, we built a fairly comprehensive version thereof which we disentangled into a hierarchy of only five simpler subnetworks each delivering one type of robustness. The comprehensive dynamic model described in vitro data sets from two independent laboratories. Notwithstanding its five-fold robustness, it exhibited a relatively sudden breakdown, after some 80 years of virtually steady performance: it predicted aging. PD-related conditions such as lack of DJ-1 protein or increased α-synuclein accelerated the collapse, while antioxidants or caffeine retarded it. Introducing a new concept (aging-time-control coefficient), we found that as many as 25 out of 57 molecular processes controlled aging. We identified new targets for "life-extending interventions": mitochondrial synthesis, KEAP1 degradation, and p62 metabolism.

Mathematical Biology Models of Parkinson's Disease

Parkinsons disease (PD) is a progressive neurodegenerative disease with substantial and growing socio-economic burden. In this multifactorial disease, aging, environmental, and genetic factors contribute to neurodegeneration and dopamine (DA) deficiency in the brain. Treatments aimed at DA restoration provide symptomatic relief, however, no disease modifying treatments are available, and PD remains incurable to date. Mathematical modeling can help understand such complex multifactorial neurological diseases. We review mathematical modeling efforts in PD with a focus on mechanistic models of pathogenic processes. We consider models of α-synuclein (Asyn) aggregation, feedbacks among Asyn, DA, and mitochondria and proteolytic systems, as well as pathology propagation through the brain. We hope that critical understanding of existing literature will pave the way to the development of quantitative systems pharmacology models to aid PD drug discovery and development.

How stimulation frequency and intensity impact on the long-lasting effects of coordinated reset stimulation

Several brain diseases are characterized by abnormally strong neuronal synchrony. Coordinated Reset (CR) stimulation was computationally designed to specifically counteract abnormal neuronal synchronization processes by desynchronization. In the presence of spike-timing-dependent plasticity (STDP) this may lead to a decrease of synaptic excitatory weights and ultimately to an anti-kindling, i.e. unlearning of abnormal synaptic connectivity and abnormal neuronal synchrony. The long-lasting desynchronizing impact of CR stimulation has been verified in pre-clinical and clinical proof of concept studies. However, as yet it is unclear how to optimally choose the CR stimulation frequency, i.e. the repetition rate at which the CR stimuli are delivered. This work presents the first computational study on the dependence of the acute and long-term outcome on the CR stimulation frequency in neuronal networks with STDP. For this purpose, CR stimulation was applied with Rapidly Varying Sequences (RVS) as well as with Slowly Varying Sequences (SVS) in a wide range of stimulation frequencies and intensities. Our findings demonstrate that acute desynchronization, achieved during stimulation, does not necessarily lead to long-term desynchronization after cessation of stimulation. By comparing the long-term effects of the two different CR protocols, the RVS CR stimulation turned out to be more robust against variations of the stimulation frequency. However, SVS CR stimulation can obtain stronger anti-kindling effects. We revealed specific parameter ranges that are favorable for long-term desynchronization. For instance, RVS CR stimulation at weak intensities and with stimulation frequencies in the range of the neuronal firing rates turned out to be effective and robust, in particular, if no closed loop adaptation of stimulation parameters is (technically) available. From a clinical standpoint, this may be relevant in the context of both invasive as well as non-invasive CR stimulation.

Bond graph modelling of chemoelectrical energy transduction

Energy‐based bond graph modelling of biomolecular systems is extended to include chemoelectrical transduction thus enabling integrated thermodynamically compliant modelling of chemoelectrical systems in general and excitable membranes in particular. Our general approach is illustrated by recreating a well‐known model of an excitable membrane. This model is used to investigate the energy consumed during a membrane action potential thus contributing to the current debate on the trade‐off between the speed of an action potential event and energy consumption. The influx of Na+ is often taken as a proxy for energy consumption; in contrast, this study presents an energy‐based model of action potentials. As the energy‐based approach avoids the assumptions underlying the proxy approach it can be directly used to compute energy consumption in both healthy and diseased neurons. These results are illustrated by comparing the energy consumption of healthy and degenerative retinal ganglion cells using both simulated and in vitro data.

Bond Graph Modeling of Chemiosmotic Biomolecular Energy Transduction

Engineering systems modeling and analysis based on the bond graph approach has been applied to biomolecular systems. In this context, the notion of a Faraday-equivalent chemical potential is introduced which allows chemical potential to be expressed in an analogous manner to electrical volts thus allowing engineering intuition to be applied to biomolecular systems. Redox reactions, and their representation by half-reactions, are key components of biological systems which involve both electrical and chemical domains. A bond graph interpretation of redox reactions is given which combines bond graphs with the Faraday-equivalent chemical potential. This approach is particularly relevant when the biomolecular system implements chemoelectrical transduction - for example chemiosmosis within the key metabolic pathway of mitochondria: oxidative phosphorylation. An alternative way of implementing computational modularity using bond graphs is introduced and used to give a physically based model of the mitochondrial electron transport chain To illustrate the overall approach, this model is analyzed using the Faraday-equivalent chemical potential approach and engineering intuition is used to guide affinity equalisation: a energy based analysis of the mitochondrial electron transport chain.

The Impact of Mathematical Modeling in Understanding the Mechanisms Underlying Neurodegeneration: Evolving Dimensions and Future Directions

Neurodegenerative diseases are a heterogeneous group of disorders that are characterized by the progressive dysfunction and loss of neurons. Here, we distil and discuss the current state of modeling in the area of neurodegeneration, and objectively compare the gaps between existing clinical knowledge and the mechanistic understanding of the major pathological processes implicated in neurodegenerative disorders. We also discuss new directions in the field of neurodegeneration that hold potential for furthering therapeutic interventions and strategies.

Modular bond-graph modelling and analysis of biomolecular systems

Bond graphs can be used to build thermodynamically-compliant hierarchical models of biomolecular systems. As bond graphs have been widely used to model, analyse and synthesise engineering systems, this study suggests that they can play the same rôle in the modelling, analysis and synthesis of biomolecular systems. The particular structure of bond graphs arising from biomolecular systems is established and used to elucidate the relation between thermodynamically closed and open systems. Block diagram representations of the dynamics implied by these bond graphs are used to reveal implicit feedback structures and are linearised to allow the application of control-theoretical methods. Two concepts of modularity are examined: computational modularity where physical correctness is retained and behavioural modularity where module behaviour (such as ultrasensitivity) is retained. As well as providing computational modularity, bond graphs provide a natural formulation of behavioural modularity and reveal the sources of retroactivity. A bond graph approach to reducing retroactivity, and thus inter-module interaction, is shown to require a power supply such as that provided by the ATP ⇌ ADP + Pi reaction. The mitogen-activated protein kinase cascade (Raf-MEK-ERK pathway) is used as an illustrative example.

The spacing principle for unlearning abnormal neuronal synchrony

Desynchronizing stimulation techniques were developed to specifically counteract abnormal neuronal synchronization relevant to several neurological and psychiatric disorders. The goal of our approach is to achieve an anti-kindling, where the affected neural networks unlearn abnormal synaptic connectivity and, hence, abnormal neuronal synchrony, by means of desynchronizing stimulation, in particular, Coordinated Reset (CR) stimulation. As known from neuroscience, psychology and education, learning effects can be enhanced by means of the spacing principle, i.e. by delivering repeated stimuli spaced by pauses as opposed to delivering a massed stimulus (in a single long stimulation session). To illustrate that the spacing principle may boost the anti-kindling effect of CR neuromodulation, in this computational study we carry this approach to extremes. To this end, we deliver spaced CR neuromodulation at particularly weak intensities which render permanently delivered CR neuromodulation ineffective. Intriguingly, spaced CR neuromodulation at these particularly weak intensities effectively induces an anti-kindling. In fact, the spacing principle enables the neuronal population to successively hop from one attractor to another one, finally approaching attractors characterized by down-regulated synaptic connectivity and synchrony. Our computational results might open up novel opportunities to effectively induce sustained desynchronization at particularly weak stimulation intensities, thereby avoiding side effects, e.g., in the case of deep brain stimulation.

Metabolomics and in-silico analysis reveal critical energy deregulations in animal models of Parkinson's disease

Parkinson's disease (PD) is a multifactorial disease known to result from a variety of factors. Although age is the principal risk factor, other etiological mechanisms have been identified, including gene mutations and exposure to toxins. Deregulation of energy metabolism, mostly through the loss of complex I efficiency, is involved in disease progression in both the genetic and sporadic forms of the disease. In this study, we investigated energy deregulation in the cerebral tissue of animal models (genetic and toxin induced) of PD using an approach that combines metabolomics and mathematical modelling. In a first step, quantitative measurements of energy-related metabolites in mouse brain slices revealed most affected pathways. A genetic model of PD, the Park2 knockout, was compared to the effect of CCCP, a mitochondrial uncoupler [corrected]. Model simulated and experimental results revealed a significant and sustained decrease in ATP after CCCP exposure, but not in the genetic mice model. In support to data analysis, a mathematical model of the relevant metabolic pathways was developed and calibrated onto experimental data. In this work, we show that a short-term stress response in nucleotide scavenging is most probably induced by the toxin exposure. In turn, the robustness of energy-related pathways in the model explains how genetic perturbations, at least in young animals, are not sufficient to induce significant changes at the metabolite level.