ODE/PDE α-synuclein Models for Parkinson’s Disease
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About this ebook
ODE/PDE Alpha-Synuclein Models for Parkinson’s Disease discusses a mechanism for the evolution of Parkinson’s Disease (PD) based on the dynamics of the protein a-synuclein, a monomer that has been implicated in this disease. Specifically, a-synuclein morphs and aggregates into a polymer that can interfere with functioning neurons and lead to neurodegenerative pathology. This book first demonstrates computer-based implementation of a prototype ODE/PDE model for the dynamics of the a-synuclein monomer and polymer, and then details the methodology for the numerical integration of ODE/PDE systems which can be applied to computer-based analyses of alternative models based on the reader's interest.
This book facilitates immediate computer use for research without the necessity to first learn the basic concepts of numerical analysis for ODE/PDEs and programming algorithms
- Includes PDE routines based on the method of lines (MOL) for computer-based implementation of ODE/PDE models
- Offers transportable computer source codes for readers, with line-by-line code descriptions relating to the mathematical model and algorithms
- Authored by a leading researcher and educator in ODE/PDE models
William E. Schiesser
Dr. William E. Schiesser is Emeritus McCann Professor of Chemical and Biomolecular Engineering, and Professor of Mathematics at Lehigh University. He holds a PhD from Princeton University and a ScD (hon) from the University of Mons, Belgium. His research is directed toward numerical methods and associated software for ordinary, differential-algebraic and partial differential equations (ODE/DAE/PDEs), and the development of mathematical models based on ODE/DAE/PDEs. He is the author or coauthor of more than 16 books, and his ODE/DAE/PDE computer routines have been accessed by some 5,000 colleges and universities, corporations and government agencies.
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ODE/PDE α-synuclein Models for Parkinson’s Disease - William E. Schiesser
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Preface
W.E. Schiesser Bethlehem, PA 18015, USA
This book discusses a mechanism for the evolution of Parkinson's disease (PD) based on the dynamics of the protein α-synuclein (α-syn), a monomer (mono) that has been implicated in PD. Specifically, mono α-syn misfolds and aggregates into a polymer (poly form), which can interfere with the functioning of neurons, that then leads to neurodegenerative pathology.
The transition of α-syn from the mono form to the poly form is modeled mathematically by a system of convection–diffusion–reaction (CDR) ordinary/partial differential equations (ODE/PDEs).¹ The computer-based integration (solution) of the CDR ODE/PDEs is implemented with a series of documented R routines² that are available via a download link so that the reader can duplicate the solutions reported in the book, then use the ODE/PDE model in computer-based experiments, for example, by varying the parameters and changing the form and number of ODE/PDEs.
Therefore, the book content is divided into two principal parts:
1. The computer-based implementation of a prototype ODE/PDE model for the dynamics of mono and poly α-syn. The R routines are discussed in detail, particularly linking sections of the code to the ODE/PDEs, plus initial conditions (ICs) and boundary conditions (BCs). Numerical and graphical output from the model is presented with a discussion of possible application to PD.
2. The detailed discussion of the methodology for the numerical, integration of ODE/PDE systems that can be applied to the computer-based analysis of alternative models of interest to the reader/analyst.
I hope this book is helpful in understanding the possible dynamics of PD, and more generally, neurodegenerative disease. I would be interested in knowing if this objective is realized, and would welcome comments and suggestions from the reader.
The R routines are available from http://www.lehigh.edu/~wes1/pd_download. Queries about the routines can be directed to wes1@lehigh.edu.
December 1, 2017
¹ "The physical system is a neuron which is a long, slender cell with two ends, the soma and the synapse. The ends are assumed to have spatially uniform concentrations of α-syn, mono and poly, and are therefore modeled by ODEs. The connecting axon has a significant length and therefore is modeled by PDEs. The solution of the ODE/PDE system gives the α-syn (mono and poly) concentrations as functions of space and time.
The intent of the solution is to demonstrate the space and time variation of the α-syn concentrations which could interfere with the signaling of the neuron that then leads to adverse neurodegenerative effects. The model might suggest a therapy for PD based on reducing the misfolding and aggregation of α-syn."
² R is an open-source scientific programming system that is easily downloadable from the Internet. It has utilities for linear algebra, numerical integration, and graphical output that facilitate the study of CDR ODE/PDE models.
Chapter 1
ODE Model Formulation
Abstract
This book discusses a mechanism for the evolution of Parkinson's disease (PD) based on the dynamics of the protein α-synuclein (α-syn), a monomer (mono) that has been implicated in PD. Specifically, as α-syn (mono) misfolds and aggregates into a polymer (poly), the poly form can interfere with the functioning of neurons that then leads to neurodegenerative pathology. A model for the α-synuclein dynamics and evolution based on ordinary and partial differential equations (ODE/PDEs) is implemented as a set of routines in R.
Keywords
Parkinson's disease; Protein α-synuclein; ODE/PDE model; Partial differential equations; R routines
Chapter Outline
Introduction, ODE Model Formulation
1.1 ODE Model
1.2 Main Program
1.3 ODE Routine
1.4 Subordinate Routine
1.5 Model Output
1.6 Routines for t Derivative
1.7 Routines for ODE Terms
1.8 Summary and Conclusions
Reference
Introduction, ODE Model Formulation
This book discusses a mechanism for the evolution of Parkinson's disease (PD) based on the dynamics of the protein α-synuclein (α-syn), a monomer (mono) that has been implicated in PD. Specifically, as α-syn (mono) misfolds and aggregates into a polymer (poly), the poly form can interfere with the functioning of neurons that then leads to neurodegenerative pathology.
The transition of α-syn from the mono form to the poly form is modeled mathematically in this chapter by a system of reaction ordinary differential equations (ODEs).¹ The computer-based integration (solution) of the ODEs is implemented with documented R routines² that are available via a download link so that the reader/analyst can duplicate the solutions discussed subsequently, then use the ODE model in computer-based experiments, for example, by varying the parameters and changing the form and number of ODEs.
1.1 ODE Model
Fig. 1.1 is a schematic representation of a neuron that illustrates the dependent and independent ODE (compartmental) model variables and associated ODE derivatives in t.
Figure 1.1 Schematic of neuron with the ODE derivatives in t .
Fig. 1.1 is based on the ODE model reported in [1]. The R routines that implement this model are discussed subsequently. A principal advantage of a computed-based model is that experimentation is easily carried out to vary the parameters and form of the model equations. The model that follows is therefore intended as only a prototype which can be used as the starting point for variations and extensions to investigate α-synuclein PD dynamics.
The following ODEs are statements of mass (conservation) for the compartmental soma and