The spatial dynamics of prion disease

Eco-evolutionary trade-offs in the dynamics of prion strain competition

Prion and prion-like molecules are a type of self-replicating aggregate protein that have been implicated in a variety of neurodegenerative diseases. Over recent decades, the molecular dynamics of prions have been characterized both empirically and through mathematical models, providing insights into the epidemiology of prion diseases and the impact of prions on the evolution of cellular processes. At the same time, a variety of evidence indicates that prions are themselves capable of a form of evolution, in which changes to their structure that impact their rate of growth or fragmentation are replicated, making such changes subject to natural selection. Here we study the role of such selection in shaping the characteristics of prions under the nucleated polymerization model (NPM). We show that fragmentation rates evolve to an evolutionary stable value which balances rapid reproduction of PrPSc aggregates with the need to produce stable polymers. We further show that this evolved fragmentation rate differs in general from the rate that optimizes transmission between cells. We find that under the NPM, prions that are both evolutionary stable and optimized for transmission have a characteristic length of three times the critical length below which they become unstable. Finally, we study the dynamics of inter-cellular competition between strains, and show that the eco-evolutionary trade-off between intra- and inter-cellular competition favours coexistence.

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.

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.

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

Protein misfolding refers to a process where proteins become structurally abnormal and lose their specific 3-dimensional spatial configuration. The histopathological presence of misfolded protein (MP) aggregates has been associated as the primary evidence of multiple neurological diseases, including Prion diseases, Alzheimer's disease, Parkinson's disease, and Creutzfeldt-Jacob disease. However, the exact mechanisms of MP aggregation and propagation, as well as their impact in the long-term patient's clinical condition are still not well understood. With this aim, a variety of mathematical models has been proposed for a better insight into the kinetic rate laws that govern the microscopic processes of protein aggregation. Complementary, another class of large-scale models rely on modern molecular imaging techniques for describing the phenomenological effects of MP propagation over the whole brain. Unfortunately, those neuroimaging-based studies do not take full advantage of the tremendous capabilities offered by the chemical kinetics modeling approach. Actually, it has been barely acknowledged that the vast majority of large-scale models have foundations on previous mathematical approaches that describe the chemical kinetics of protein replication and propagation. The purpose of the current manuscript is to present a historical review about the development of mathematical models for describing both microscopic processes that occur during the MP aggregation and large-scale events that characterize the progression of neurodegenerative MP-mediated diseases.

THE SPREAD OF PRION DISEASES IN THE BRAIN — MODELS OF REACTION AND TRANSPORT ON NETWORKS

In this paper we will present a modeling approach to describe the progression and the spread of prion diseases in the brain. Although there exist a number of mathematical models for the interaction of prions with their native counterpart, prion transport and spread is usually neglected. The concentration dynamics of prions, and thus the dynamics of the disease progression, however, are influenced by prion transport, especially in a medium as complex as the brain. Therefore, we focus here on the interaction between prion concentration dynamics and prion transport. The model is constructed by combining a model of prion-prion interaction with transport on networks. The approach leads to a system of reaction-diffusion equations, whereby the diffusion term is discrete. The equations are solved numerically on domains given as large networks. We show that the prion concentration grows faster on networks characterized by a higher degree heterogeneity. Furthermore, we introduce cell death as a consequence of increasing prion concentration, leading to network decomposition. We show that infectious diseases destroy networks similarly to targeted attacks, namely by affecting the nodes with the highest degree first. Relating the incubation period and disease progression to the process of network decomposition, we find that, interestingly, a long incubation time followed by sudden onset and fast progression of the disease does not need to be reflected in the overall concentration dynamics of the infective agent.

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.

Theoretical modeling of prion disease incubation

We apply a theoretical aggregation model to laboratory and epidemiological prion disease incubation time data. In our model, slow growth of misfolded protein aggregates from small initial seeds controls the latent or lag phase; aggregate fissioning and subsequent spreading leads to an exponential growth phase. Our model accounts for the striking reproducibility of incubation times for high dose inoculation of lab animals. In particular, low dose yields broad incubation time distributions, and increasing dose narrows distributions and yields sharply defined onset times. We also explore how incubation time statistics depend upon aggregate morphology. We apply our model to fit the experimental dose-incubation curves for distinct strains of scrapie, and explain logarithmic variation at high dose and deviations from logarithmic behavior at low dose. We use this to make testable predictions for infectivity time-course experiments.

PrP expression and replication by Schwann cells: implications in prion spreading

Prion infection relies on a continuous chain of PrP(c)-expressing tissues to spread from peripheral sites to the central nervous system (CNS). Direct neuroinvasion via peripheral nerves has long been considered likely. However, the speed of axonal flow is incompatible with the lengthy delay prior to the detection of PrP(Sc) in the brain. We hypothesized that Schwann cells could be the candidate implicated in this mechanism; for that, it has to express PrP(c) and to allow PrP(Sc) conversion. We investigated in vivo localization of PrP(c) in sciatic nerve samples from different strains of mice. We demonstrated that PrP(c) is mainly localized at the cell membrane of the Schwann cell. We also studied in vitro expression of PrP(c) in the Schwann cell line MSC-80 and demonstrated that it expresses PrP(c) at the same location. More specifically, we demonstrated that this glial cell line, when infected in vitro with the mouse Chandler prion strain, both produces the PrP(Sc) till after 18 passages and is able to transmit disease to mice, which then develop the typical signs of prion diseases. It is the first time that infection and replication of PrP(Sc) are shown in a peripheral glial cell line.

Mapping the parameters of prion-induced neuropathology

We present a theoretical framework that enables us to dissect out the parametric dependencies of the pathogenesis of prion diseases. We are able to determine the influence of both host-dependent factors (connectivity, cell density, protein synthesis rate, and cell death) and strain-dependent factors (cell tropism, virulence, and replication rate). We use a model based on a linked system of differential equations on a lattice to explore how the regional distribution of central nervous system pathology in Creutzfeldt-Jakob disease, Gerstmann-Sträussler-Scheinker syndrome, and fatal familial insomnia relates to each of these factors. The model then is used to make qualitative predictions about the pathology for two possible hypothetical triggers of neuronal loss in prion diseases. Pathological progression in overexpressing mouse models has been shown to depend on the site of initial infection. The model allows us to compare the pathologies resulting from different inoculation routes.

Scrapie infections initiated at varying doses: an analysis of 117 titration experiments

An analysis of 117 titration experiments in the murine scrapie model is presented. The experiments encompass 30 years' work and a wide range of experimental conditions. To check that the experimental designs were reasonably consistent over time, comparisons were made of size, duration, source of inoculum, etc., in each experiment. These comparisons revealed no systematic trends that would render invalid comparisons across experiments. For 114 of the experiments it was possible to calculate the dose at which half of the challenged animals were infected (the ID50). These 114 experiments were then combined on the basis of relative dose (i.e. tenfold dilution relative to the ID50). This created a data set in which over 4000 animals were challenged with doses of scrapie ranging from four orders of magnitude below to five orders of magnitude above the ID50. Analysis of this data reveals that mean incubation periods rise linearly with logarithmic decreases in dose. A one unit increase in relative dose (i.e. a tenfold increase in actual dose) will, on average, decrease the incubation period by 25 days. At ID50 the average incubation period in this data set is 300 days. Within a single dose, in a single experimental model, incubation periods have a distribution close to normal. Variability in incubation period also rises linearly as dose decreases. There is no age or sex effect upon the probability of infection, but female mice have incubation periods that are, on average, nine days shorter than their male counterparts and young mice have incubation periods that are longer by seven days. Although many of these patterns are apparent in the results of single titration curves, they can be more rigorously investigated by considering the outcome for thousands of mice.

Quantifying the kinetic parameters of prion replication

The mechanism of protein-only prion replication is controversial. A detailed mathematical model of prion replication by nucleated polymerisation is developed, and its parameters are estimated from published data. PrP-res decay is around two orders of magnitude slower than PrP-sen decay, a plausible ratio of two parameters estimated from very different experiments. By varying the polymer breakage rate, we reveal that systems of short polymers grow the fastest. Drugs which break polymers could therefore accelerate disease progression. Growth in PrP-res seems slower than growth in infectious titre. This can be explained either by a novel hypothesis concerning inoculum clearance from a newly infected brain, or by the faster growth of compartments containing smaller polymers. The existence of compartments can also explain why prion growth sometimes reaches a plateau. Published kinetic data are all compatible with our mathematical model, so the nucleated polymerisation hypothesis cannot be ruled out on dynamic grounds.