Mathematical Modeling of Protein Misfolding Mechanisms in Neurological Diseases: A Historical Overview

In-vivo neuronal dysfunction by Aβ and tau overlaps with brain-wide inflammatory mechanisms in Alzheimer's disease

The molecular mechanisms underlying neuronal dysfunction in Alzheimer's disease (AD) remain uncharacterized. Here, we identify genes, molecular pathways and cellular components associated with whole-brain dysregulation caused by amyloid-beta (Aβ) and tau deposits in the living human brain. We obtained in-vivo resting-state functional MRI (rs-fMRI), Aβ- and tau-PET for 47 cognitively unimpaired and 16 AD participants from the Translational Biomarkers in Aging and Dementia cohort. Adverse neuronal activity impacts by Aβ and tau were quantified with personalized dynamical models by fitting pathology-mediated computational signals to the participant's real rs-fMRIs. Then, we detected robust brain-wide associations between the spatial profiles of Aβ-tau impacts and gene expression in the neurotypical transcriptome (Allen Human Brain Atlas). Within the obtained distinctive signature of in-vivo neuronal dysfunction, several genes have prominent roles in microglial activation and in interactions with Aβ and tau. Moreover, cellular vulnerability estimations revealed strong association of microglial expression patterns with Aβ and tau's synergistic impact on neuronal activity (q < 0.001). These results further support the central role of the immune system and neuroinflammatory pathways in AD pathogenesis. Neuronal dysregulation by AD pathologies also associated with neurotypical synaptic and developmental processes. In addition, we identified drug candidates from the vast LINCS library to halt or reduce the observed Aβ-tau effects on neuronal activity. Top-ranked pharmacological interventions target inflammatory, cancer and cardiovascular pathways, including specific medications undergoing clinical evaluation in AD. Our findings, based on the examination of molecular-pathological-functional interactions in humans, may accelerate the process of bringing effective therapies into clinical practice.

The role of A[Formula: see text] and Tau proteins in Alzheimer's disease: a mathematical model on graphs

In this Note we study a mathematical model for the progression of Alzheimer's Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as [Formula: see text] amyloid and Tau play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.

A Meta-Analysis Study of SOD1-Mutant Mouse Models of ALS to Analyse the Determinants of Disease Onset and Progression

A complex interaction between genetic and external factors determines the development of amyotrophic lateral sclerosis (ALS). Epidemiological studies on large patient cohorts have suggested that ALS is a multi-step disease, as symptom onset occurs only after exposure to a sequence of risk factors. Although the exact nature of these determinants remains to be clarified, it seems clear that: (i) genetic mutations may be responsible for one or more of these steps; (ii) other risk factors are probably linked to environment and/or to lifestyle, and (iii) compensatory plastic changes taking place during the ALS etiopathogenesis probably affect the timing of onset and progression of disease. Current knowledge on ALS mechanisms and therapeutic targets, derives mainly from studies involving superoxide dismutase 1 (SOD1) transgenic mice; therefore, it would be fundamental to verify whether a multi-step disease concept can also be applied to these animal models. With this aim, a meta-analysis study has been performed using a collection of primary studies (n = 137), selected according to the following criteria: (1) the studies should employ SOD1 transgenic mice; (2) the studies should entail the presence of a disease-modifying experimental manipulation; (3) the studies should make use of Kaplan-Meier plots showing the distribution of symptom onset and lifespan. Then, using a subset of this study collection (n = 94), the effects of treatments on key molecular mechanisms, as well as on the onset and progression of disease have been analysed in a large population of mice. The results are consistent with a multi-step etiopathogenesis of disease in ALS mice (including two to six steps, depending on the particular SOD1 mutation), closely resembling that observed in patient cohorts, and revealed an interesting relationship between molecular mechanisms and disease manifestation. Thus, SOD1 mouse models may be considered of high predictive value to understand the determinants of disease onset and progression, as well as to identify targets for therapeutic interventions.

A model-informed approach to assess the risk of immune checkpoint inhibitor-induced autoimmune myocarditis

Immune checkpoint inhibitors (ICIs), as a novel immunotherapy, are designed to modulate the immune system to attack malignancies. Despite their promising benefits, immune-related adverse events (IRAEs) may occur, and incidences are bound to increase with surging demand of this class of drugs in treating cancer. Myocarditis, although rare compared to other IRAEs, has a significantly higher fatal frequency. Due to the overwhelming complexity of the immune system, this condition is not well understood, despite the significant research efforts devoted to it. To better understand the development and progression of autoimmune myocarditis and the roles of ICIs therein, we suggest a new approach: mathematical modelling. Mathematical modelling of myocarditis has enormous potential to determine which parts of the immune system are critical to the development and progression of the disease, and therefore warrant further investigation. We provide the immunological background needed to develop a mathematical model of this disease and review relevant existing models of immunology that serve as the mathematical inspiration needed to develop this field.

Network structure and transcriptomic vulnerability shape atrophy in frontotemporal dementia

Connections among brain regions allow pathological perturbations to spread from a single source region to multiple regions. Patterns of neurodegeneration in multiple diseases, including behavioural variant of frontotemporal dementia (bvFTD), resemble the large-scale functional systems, but how bvFTD-related atrophy patterns relate to structural network organization remains unknown. Here we investigate whether neurodegeneration patterns in sporadic and genetic bvFTD are conditioned by connectome architecture. Regional atrophy patterns were estimated in both genetic bvFTD (75 patients, 247 controls) and sporadic bvFTD (70 patients, 123 controls). First, we identified distributed atrophy patterns in bvFTD, mainly targeting areas associated with the limbic intrinsic network and insular cytoarchitectonic class. Regional atrophy was significantly correlated with atrophy of structurally- and functionally-connected neighbours, demonstrating that network structure shapes atrophy patterns. The anterior insula was identified as the predominant group epicentre of brain atrophy using data-driven and simulation-based methods, with some secondary regions in frontal ventromedial and antero-medial temporal areas. We found that FTD-related genes, namely C9orf72 and TARDBP, confer local transcriptomic vulnerability to the disease, modulating the propagation of pathology through the connectome. Collectively, our results demonstrate that atrophy patterns in sporadic and genetic bvFTD are jointly shaped by global connectome architecture and local transcriptomic vulnerability, providing an explanation as to how heterogenous pathological entities can lead to the same clinical syndrome.

Mathematical modelling of autoimmune myocarditis and the effects of immune checkpoint inhibitors

Autoimmune myocarditis is a rare, but frequently fatal, side effect of immune checkpoint inhibitors (ICIs), a class of cancer therapies. Despite extensive experimental work on the causes, development and progression of this disease, much still remains unknown about the importance of the different immunological pathways involved. We present a mathematical model of autoimmune myocarditis and the effects of ICIs on its development and progression to either resolution or chronic inflammation. From this, we gain a better understanding of the role of immune cells, cytokines and other components of the immune system in driving the cardiotoxicity of ICIs. We parameterise the model using existing data from the literature, and show that qualitative model behaviour is consistent with disease characteristics seen in patients in an ICI-free context. The bifurcation structures of the model show how the presence of ICIs increases the risk of developing autoimmune myocarditis. This predictive modelling approach is a first step towards determining treatment regimens that balance the benefits of treating cancer with the risk of developing autoimmune myocarditis.

Integrative Neuroinformatics for Precision Prognostication and Personalized Therapeutics in Moderate and Severe Traumatic Brain Injury

Despite changes in guideline-based management of moderate/severe traumatic brain injury (TBI) over the preceding decades, little impact on mortality and morbidity have been seen. This argues against the "one-treatment fits all" approach to such management strategies. With this, some preliminary advances in the area of personalized medicine in TBI care have displayed promising results. However, to continue transitioning toward individually-tailored care, we require integration of complex "-omics" data sets. The past few decades have seen dramatic increases in the volume of complex multi-modal data in moderate and severe TBI care. Such data includes serial high-fidelity multi-modal characterization of the cerebral physiome, serum/cerebrospinal fluid proteomics, admission genetic profiles, and serial advanced neuroimaging modalities. Integrating these complex and serially obtained data sets, with patient baseline demographics, treatment information and clinical outcomes over time, can be a daunting task for the treating clinician. Within this review, we highlight the current status of such multi-modal omics data sets in moderate/severe TBI, current limitations to the utilization of such data, and a potential path forward through employing integrative neuroinformatic approaches, which are applied in other neuropathologies. Such advances are positioned to facilitate the transition to precision prognostication and inform a top-down approach to the development of personalized therapeutics in moderate/severe TBI.

Combined Model of Aggregation and Network Diffusion Recapitulates Alzheimer's Regional Tau-Positron Emission Tomography

Background: Alzheimer's disease involves widespread and progressive deposition of misfolded protein tau (τ), first appearing in the entorhinal cortex, coagulating in longer polymers and insoluble fibrils. There is mounting evidence for "prion-like" trans-neuronal transmission, whereby misfolded proteins cascade along neuronal pathways, giving rise to networked spread. However, the cause-effect mechanisms by which various oligomeric τ species are produced, aggregate, and disseminate are unknown. The question of how protein aggregation and subsequent spread lead to stereotyped progression in the Alzheimer brain remains unresolved. Materials and Methods: We address these questions by using mathematically precise parsimonious modeling of these pathophysiological processes, extrapolated to the whole brain. We model three key processes: τ monomer production; aggregation into oligomers and then into tangles; and the spatiotemporal progression of misfolded τ as it ramifies into neural circuits via the brain connectome. We model monomer seeding and production at the entorhinal cortex, aggregation using Smoluchowski equations; and networked spread using our prior Network-Diffusion model. Results: This combined aggregation-network-diffusion model exhibits all hallmarks of τ progression seen in human patients. Unlike previous theoretical studies of protein aggregation, we present here an empirical validation on in vivo imaging and fluid τ measurements from large datasets. The model accurately captures not just the spatial distribution of empirical regional τ and atrophy but also patients' cerebrospinal fluid phosphorylated τ profiles as a function of disease progression. Conclusion: This unified quantitative and testable model has the potential to explain observed phenomena and serve as a test-bed for future hypothesis generation and testing in silico.

Graph Models of Pathology Spread in Alzheimer's Disease: An Alternative to Conventional Graph Theoretic Analysis

Background: Graph theory and connectomics are new techniques for uncovering disease-induced changes in the brain's structural network. Most prior studied have focused on network statistics as biomarkers of disease. However, an emerging body of work involves exploring how the network serves as a conduit for the propagation of disease factors in the brain and has successfully mapped the functional and pathological consequences of disease propagation. In Alzheimer's disease (AD), progressive deposition of misfolded proteins amyloid and tau is well-known to follow fiber projections, under a "prion-like" trans-neuronal transmission mechanism, through which misfolded proteins cascade along neuronal pathways, giving rise to network spread. Methods: In this review, we survey the state of the art in mathematical modeling of connectome-mediated pathology spread in AD. Then we address several open questions that are amenable to mathematically precise parsimonious modeling of pathophysiological processes, extrapolated to the whole brain. We specifically identify current formal models of how misfolded proteins are produced, aggregate, and disseminate in brain circuits, and attempt to understand how this process leads to stereotyped progression in Alzheimer's and other related diseases. Conclusion: This review serves to unify current efforts in modeling of AD progression that together have the potential to explain observed phenomena and serve as a test-bed for future hypothesis generation and testing in silico. Impact statement Graph theory is a powerful new approach that is transforming the study of brain processes. There do not exist many focused reviews of the subfield of graph modeling of how Alzheimer's and other dementias propagate within the brain network, and how these processes can be mapped mathematically. By providing timely and topical review of this subfield, we fill a critical gap in the community and present a unified view that can serve as an in silico test-bed for future hypothesis generation and testing. We also point to several open and unaddressed questions and controversies that future practitioners can tackle.

Integrating molecular, histopathological, neuroimaging and clinical neuroscience data with NeuroPM-box

Understanding and treating heterogeneous brain disorders requires specialized techniques spanning genetics, proteomics, and neuroimaging. Designed to meet this need, NeuroPM-box is a user-friendly, open-access, multi-tool cross-platform software capable of characterizing multiscale and multifactorial neuropathological mechanisms. Using advanced analytical modeling for molecular, histopathological, brain-imaging and/or clinical evaluations, this framework has multiple applications, validated here with synthetic (N > 2900), in-vivo (N = 911) and post-mortem (N = 736) neurodegenerative data, and including the ability to characterize: (i) the series of sequential states (genetic, histopathological, imaging or clinical alterations) covering decades of disease progression, (ii) concurrent intra-brain spreading of pathological factors (e.g., amyloid, tau and alpha-synuclein proteins), (iii) synergistic interactions between multiple biological factors (e.g., toxic tau effects on brain atrophy), and (iv) biologically-defined patient stratification based on disease heterogeneity and/or therapeutic needs. This freely available toolbox ( neuropm-lab.com/neuropm-box.html ) could contribute significantly to a better understanding of complex brain processes and accelerating the implementation of Precision Medicine in Neurology.

A reaction-diffusion model of spatial propagation of A[Formula: see text] oligomers in early stage Alzheimer's disease

The misconformation and aggregation of the protein Amyloid-Beta (A[Formula: see text]) is a key event in the propagation of Alzheimer's Disease (AD). Different types of assemblies are identified, with long fibrils and plaques deposing during the late stages of AD. In the earlier stages, the disease spread is driven by the formation and the spatial propagation of small amorphous assemblies called oligomers. We propose a model dedicated to studying those early stages, in the vicinity of a few neurons and after a polymer seed has been formed. We build a reaction-diffusion model, with a Becker-Döring-like system that includes fragmentation and size-dependent diffusion. We hereby establish the theoretical framework necessary for the proper use of this model, by proving the existence of solutions using a fixed point method.

On the Mechanism of Self-Assembly by a Hydrogel-Forming Peptide

Self-assembling peptide-based hydrogels are a class of tunable soft materials that have been shown to be highly useful for a number of biomedical applications. The dynamic formation of the supramolecular fibrils that compose these materials has heretofore remained poorly characterized. A better understanding of this process would provide important insights into the behavior of these systems and could aid in the rational design of new peptide hydrogels. Here, we report the determination of the microscopic steps that underpin the self-assembly of a hydrogel-forming peptide, SgI_37-49. Using theoretical models of linear polymerization to analyze the kinetic self-assembly data, we show that SgI_37-49 fibril formation is driven by fibril-catalyzed secondary nucleation and that all the microscopic processes involved in SgI_37-49 self-assembly display an enzyme-like saturation behavior. Moreover, this analysis allows us to quantify the rates of the underlying processes at different peptide concentrations and to calculate the time evolution of these reaction rates over the time course of self-assembly. We demonstrate here a new mechanistic approach for the study of self-assembling hydrogel-forming peptides, which is complementary to commonly used materials science characterization techniques.

Editorial: Network Communication in the Brain

Communication models describe the flow of signals among nodes of a network. In neural systems, communication models are increasingly applied to investigate network dynamics across the whole brain, with the ultimate aim to understand how signal flow gives rise to brain function. Communication models range from diffusion-like processes to those related to infectious disease transmission and those inspired by engineered communication systems like the internet. This Focus Feature brings together novel investigations of a diverse range of mechanisms and strategies that could shape communication in mammal whole-brain networks.

Protein-protein interactions in neurodegenerative diseases: A conspiracy theory

Neurodegenerative diseases such as Alzheimer's or Parkinson's are associated with the prion-like propagation and aggregation of toxic proteins. A long standing hypothesis that amyloid-beta drives Alzheimer's disease has proven the subject of contemporary controversy; leading to new research in both the role of tau protein and its interaction with amyloid-beta. Conversely, recent work in mathematical modeling has demonstrated the relevance of nonlinear reaction-diffusion type equations to capture essential features of the disease. Such approaches have been further simplified, to network-based models, and offer researchers a powerful set of computationally tractable tools with which to investigate neurodegenerative disease dynamics. Here, we propose a novel, coupled network-based model for a two-protein system that includes an enzymatic interaction term alongside a simple model of aggregate transneuronal damage. We apply this theoretical model to test the possible interactions between tau proteins and amyloid-beta and study the resulting coupled behavior between toxic protein clearance and proteopathic phenomenology. Our analysis reveals ways in which amyloid-beta and tau proteins may conspire with each other to enhance the nucleation and propagation of different diseases, thus shedding new light on the importance of protein clearance and protein interaction mechanisms in prion-like models of neurodegenerative disease.

A two-step biopolymer nucleation model shows a nonequilibrium critical point

Biopolymer self-assembly pathways are complicated by the ability of their monomeric subunits to adopt different conformational states. This means nucleation often involves a two-step mechanism where the monomers first condense to form a metastable intermediate, which then converts to a stable polymer by conformational rearrangement of constituent monomers. Nucleation intermediates play a causative role in amyloid diseases such as Alzheimer's and Parkinson's. While existing mathematical models neglect the conversion dynamics, experiments show that conversion events frequently occur on comparable timescales to the condensation of intermediates and growth of mature polymers and thus cannot be ignored. We present a model that explicitly accounts for simultaneous assembly and conversion. To describe conversion, we propose an experimentally motivated initiation-propagation mechanism in which the stable phase arises locally within the intermediate and then spreads by nearest-neighbor interactions, in a manner analogous to one-dimensional Glauber dynamics. Our analysis shows that the competing timescales of assembly and conversion result in a nonequilibrium critical point, separating a regime where intermediates are kinetically unstable from one where conformationally mixed intermediates accumulate. This strongly affects the accumulation rate of the stable biopolymer phase. Our model is uniquely able to explain experimental phenomena such as the formation of mixed intermediates and abrupt changes in the scaling exponent γ, which relates the total monomer concentration to the accumulation rate of the stable phase. This provides a first step toward a general model of two-step biopolymer nucleation, which can quantitatively predict the concentration and composition of biologically crucial intermediates.

How and why to build a mathematical model: A case study using prion aggregation

Biological systems are inherently complex, and the increasing level of detail with which we are able to experimentally probe such systems continually reveals new complexity. Fortunately, mathematical models are uniquely positioned to provide a tool suitable for rigorous analysis, hypothesis generation, and connecting results from isolated in vitro experiments with results from in vivo and whole-organism studies. However, developing useful mathematical models is challenging because of the often different domains of knowledge required in both math and biology. In this work, we endeavor to provide a useful guide for researchers interested in incorporating mathematical modeling into their scientific process. We advocate for the use of conceptual diagrams as a starting place to anchor researchers from both domains. These diagrams are useful for simplifying the biological process in question and distinguishing the essential components. Not only do they serve as the basis for developing a variety of mathematical models, but they ensure that any mathematical formulation of the biological system is led primarily by scientific questions. We provide a specific example of this process from our own work in studying prion aggregation to show the power of mathematical models to synergistically interact with experiments and push forward biological understanding. Choosing the most suitable model also depends on many different factors, and we consider how to make these choices based on different scales of biological organization and available data. We close by discussing the many opportunities that abound for both experimentalists and modelers to take advantage of collaborative work in this field.

Alternative Hydrophobic Core in Proteins—The Effect of Specific Synergy

Proteins with a high degree of sequence similarity representing different structures provide a key to understand how protein sequence codes for 3D structure. An analysis using the fuzzy oil drop model was carried out on two pairs of proteins with different secondary structures and with high sequence identities. It has been shown that distributions of hydrophobicity for these proteins are approximated well using single 3D Gaussian function. In other words, the similar sequences fold into different 3D structures, however, alternative structures also have symmetric and monocentric hydrophobic cores. It should be noted that a significant change in the helical to beta-structured form in the N-terminal section takes places in the fragment much preceding the location of the mutated regions. It can be concluded that the final structure is the result of a complicated synergy effect in which the whole chain participates simultaneously.

Protein misfolding and aggregation in neurodegenerative diseases: a review of pathogeneses, novel detection strategies, and potential therapeutics

Protein folding is a complex, multisystem process characterized by heavy molecular and cellular footprints. Chaperone machinery enables proper protein folding and stable conformation. Other pathways concomitant with the protein folding process include transcription, translation, post-translational modifications, degradation through the ubiquitin-proteasome system, and autophagy. As such, the folding process can go awry in several different ways. The pathogenic basis behind most neurodegenerative diseases is that the disruption of protein homeostasis (i.e. proteostasis) at any level will eventually lead to protein misfolding. Misfolded proteins often aggregate and accumulate to trigger neurotoxicity through cellular stress pathways and consequently cause neurodegenerative diseases. The manifestation of a disease is usually dependent on the specific brain region that the neurotoxicity affects. Neurodegenerative diseases are age-associated, and their incidence is expected to rise as humans continue to live longer and pursue a greater life expectancy. We presently review the sequelae of protein misfolding and aggregation, as well as the role of these phenomena in several neurodegenerative diseases including Alzheimer's disease, Huntington's disease, amyotrophic lateral sclerosis, Parkinson's disease, transmissible spongiform encephalopathies, and spinocerebellar ataxia. Strategies for treatment and therapy are also conferred with respect to impairing, inhibiting, or reversing protein misfolding.

The Role of Exosomes/Extracellular Vesicles in Neural Signal Transduction

Exosomes, in a broad sense extracellular vesicles (EVs), are secreted from several cells and also exist in cerebrospinal fluid (CSF); they contribute to signal transduction not only between neural cells but also among hematopoietic cells. In addition to the peripheral nervous system, the association of regeneration and EVs has also been reported in the central nervous system, for example, following a spinal cord injury. Furthermore, it has become clear that major causative factors of neurodegenerative diseases are transmitted by EVs; thus, EVs are involved in the pathogenesis of neurodegenerative diseases. In particular, we would like to outline the relationship between neurophysiology and neurological disorders centered on EV-mediated communication between neural and glial cells.

Models of Network Spread and Network Degeneration in Brain Disorders

Network analysis can provide insight into key organizational principles of brain structure and help identify structural changes associated with brain disease. Though static differences between diseased and healthy networks are well characterized, the study of network dynamics, or how brain networks change over time, is increasingly central to understanding ongoing brain changes throughout disease. Accordingly, we present a short review of network models of spread, network dynamics, and network degeneration. Borrowing from recent suggestions, we divide this review into two processes by which brain networks can change: dynamics on networks, which are functional and pathological consequences taking place atop a static structural brain network; and dynamics of networks, which constitutes a changing structural brain network. We focus on diffusion magnetic resonance imaging-based structural or anatomic connectivity graphs. We address psychiatric disorders like schizophrenia; developmental disorders like epilepsy; stroke; and Alzheimer's disease and other neurodegenerative diseases.

Comparison of the structure of Aβ(1-40) amyloid with the one in complex with polyphenol ε-viniferin glucoside (EVG)

The analysis of amyloid structures is much easier recently due to the availability of the solid-state nuclear magnetic resonance technique, which allows the determination of the 3D structure of amyloid forms. The amyloidogenic polypeptide Aβ(1-40) (PDB ID 2M9R, 2M9S) in its soluble form is the object of analysis in this paper. The solubility of this polypeptide is reached due to the presence of a complexed ligand: polyphenol ε-viniferin glucoside. Two forms of complexes available in the PDB were taken for analysis with respect to the presence of a hydrophobic core in the 3D structure of these complexes. The idealized hydrophobic core structure assumed to be accordant with the 3D Gauss function distribution was taken as the pattern. The aim of this analysis is the possible further comparison to the structures of the hydrophobic core present in amyloids. It is shown that the discordant (versus the 3D Gauss function) fragments present in amyloids appear accordant in the discussed complexes.