ODE/PDE Alpha-Synuclein Models for Parkinson’s Disease

220 Pages·2018·2.425 MB·English

by  W.E. Schiesser

ODE/PDE
α-SYNUCLEIN
MODELS FOR
PARKINSON’S
DISEASE

ODE/PDE
α-SYNUCLEIN
MODELS FOR
PARKINSON’S
DISEASE
W.E. Schiesser
PREFACE
This book discusses a mechanism for the evolution of Parkinson’s disease
(PD)basedonthedynamicsoftheproteinα-synuclein(α-syn),amonomer
(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 neurodegenerativepathology.
The transition of α-syn from the mono form to the poly form is mod-
eled mathematically by a system of convection–diffusion–reaction (CDR)
ordinary/partialdifferentialequations(ODE/PDEs).1 Thecomputer-based
integration (solution) of the CDR ODE/PDEs is implemented with a se-
ries 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
varyingtheparametersandchangingtheformandnumberofODE/PDEs.
Therefore, the book content is divided into two principal parts:
1. Thecomputer-basedimplementationofaprototypeODE/PDEmodel
forthedynamicsofmonoandpolyα-syn.TheRroutinesarediscussed
in detail, particularly linking sections of the code to the ODE/PDEs,
plusinitialconditions(ICs)andboundaryconditions(BCs).Numerical
and graphical output from the model is presented with a discussion of
possible application to PD.
2. Thedetaileddiscussionofthemethodologyforthenumerical,integra-
tion of ODE/PDE systems that can be applied to the computer-based
analysis of alternative models of interest to the reader/analyst.
1The physical system is a neuron which is a long, slender cell with two ends, the soma
andthesynapse.Theendsareassumedtohavespatiallyuniformconcentrationsofα-syn,
monoandpoly,andarethereforemodeledbyODEs.Theconnectingaxonhasasignif-
icant length and therefore is modeled by PDEs. The solution of the ODE/PDE system
givestheα-syn(monoandpoly)concentrationsasfunctionsofspaceandtime.
Theintentofthesolutionistodemonstratethespaceandtimevariationoftheα-syn
concentrationswhichcouldinterferewiththesignalingoftheneuronthatthenleadsto
adverse neurodegenerative effects. The model might suggest a therapy for PD based on
reducingthemisfoldingandaggregationofα-syn.
2Risanopen-sourcescientificprogrammingsystemthatiseasilydownloadablefromthe
Internet.Ithasutilitiesforlinearalgebra,numericalintegration,andgraphicaloutputthat
facilitatethestudyofCDRODE/PDEmodels.
vii
viii Preface
I hope this book is helpful in understanding the possible dynamics of
PD, and more generally, neurodegenerative disease. I would be interested
inknowingifthisobjectiveisrealized,andwouldwelcomecommentsand
suggestions from the reader.
The R routines are available from http://www.lehigh.edu/~wes1/pd_
[email protected].
W.E. Schiesser
Bethlehem, PA 18015, USA
December 1, 2017
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CHAPTER 1
ODE Model Formulation
Contents
Introduction,ODEModelFormulation 1
1.1 ODEModel 2
1.2 MainProgram 5
1.3 ODERoutine 10
1.4 SubordinateRoutine 12
1.5 ModelOutput 13
1.6 RoutinesfortDerivative 15
1.7 RoutinesforODETerms 23
1.8 SummaryandConclusions 32
Reference 32
INTRODUCTION,ODEMODELFORMULATION
This book discusses a mechanism for the evolution of Parkinson’s disease
(PD)basedonthedynamicsoftheproteinα-synuclein(α-syn),amonomer
(mono) that has been implicated in PD. Specifically, as α-syn (mono) mis-
foldsandaggregatesintoapolymer(poly),thepolyformcaninterferewith
the functioning of neurons that then leads to neurodegenerative pathol-
ogy.
The transition of α-syn from the mono form to the poly form is
modeledmathematicallyinthischapterbyasystemofreactionordinarydif-
ferentialequations (ODEs).1 The computer-basedintegration (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 ex-
periments, for example, by varying the parameters and changing the form
and number of ODEs.
1Thephysicalsystemisaneuronwhichisalong,slendercellwithtwoends,thesomaand
thesynpase.Theendsareassumedtohaveuniformconcentrations(nospatialvariations)
ofα-syn,monoandpoly,andarethereforemodeledbyODEs.Theconnectingaxonhas
asignificantlengthandthereforeismodeledsubsequentlybyPDEs.Forthisinitialdiscus-
sionthesolutionofthesystemofODEsgivestheα-syn(monoandpoly)concentrations
inthesomaandsynapseasafunctionoftime.
2R is an open-source scientific programming system that is easily downloaded from the
Internet.Ithasutilitiesforlinearalgebra,numericalintegrationandgraphicaloutputthat
facilitatethestudyofCDRPDEmodels.
ODE/PDEα-SynucleinModelsforParkinson’sDisease.
DOI:http://dx.doi.org/10.1016/B978-0-12-814614-9.00001-0 1
Copyright©2018ElsevierInc.Allrightsreserved.
2 ODE/PDEα-SynucleinModelsforParkinson’sDisease
Figure1.1 SchematicofneuronwiththeODEderivativesint.
1.1 ODEMODEL
Fig.1.1isaschematicrepresentationofaneuronthatillustratesthedepen-
dent and independent ODE (compartmental) model variables and associ-
ated ODE derivatives in t.
Fig.1.1isbasedontheODEmodelreportedin[1].TheRroutinesthat
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.
ThefollowingODEsarestatementsofmass(conservation)forthecom-
partmental soma and synapse.
Soma:
(cid:2) (cid:3)
d[A] [A]
V s =V Q −k [A] −k [A][B] − s −h [A], (1.1a)
s dt s A,s 1 s 2 s s T A s
A,1/2
d[B] [B]
s =Q h(t−t )h(t −t)+k [A] +k [A][B] − s ; (1.1b)
dt B,s 1 2 1 s 2 s s T
B,1/2
Synapse:
(cid:2) (cid:3)
d[A] [A]
V syn =V −k [A] −k [A] [B] − syn +h [A], (1.2a)
syn syn 1 syn 2 syn syn A s
dt T
A,1/2
ODEModelFormulation 3
Table1.1 VariablesinEqs.(1.1)to(1.8)
Symbol Genericsymbol Definition(units)
[A]s u1 monoα-synsomaconcentration(molm−3)
[B]s u2 polyα-synsomaconcentration(molm−3)
[A]syn u3 monoα-synsynapseconcentration(molm−3)
[B]syn u4 polyα-synsynapseconcentration(molm−3)
t t time(s)
d[B] [B]
syn =Q h(t−t )h(t −t)+k [A] +k [A] [B] − syn.
dt B,syn 1 2 1 syn 2 syn syn T
B,1/2
(1.2b)
Eqs. (1.1), (1.2) are stated with [A],[B],[A] ,[B] as the ODE depen-
s s syn syn
dent variables (defined in Table 1.1). These equations are first order in t
and thereforeeach requires an initial condition (IC).
A(t=0)=A ; B(t=0)=B , (1.3a,b)
s s,0 s s,0
A (t=0)=A ; B (t=0)=B . (1.4a,b)
syn syn,0 syn syn,0
Tofacilitatethesubsequentprogramming(coding)oftheseODEs,they
are restated in terms of dependent variables u , u , u , u .
1 2 3 4
Soma:
(cid:2) (cid:3)
du u
V 1 =V Q −k u −k u u − 1 −h u , (1.5a)
s dt s A,s 1 1 2 1 2 T A 1
A,1/2
du u
2 =Q h(t−t )h(t −t)+k u +k u u − 2 ; (1.5b)
dt B,s 1 2 1 1 2 1 2 T
B,1/2
Synapse:
(cid:2) (cid:3)
du u
V 3 =V −k u −k u u − 3 +h u , (1.6a)
syn syn 1 3 2 3 4 A 1
dt T
A,1/2
du u
4 =Q h(t−t )h(t −t)+k u +k u u − 4 , (1.6b)
dt B,syn 1 2 1 3 2 3 4 T
B,1/2
u (t=0)=u ; u (t=0)=u , (1.7a,b)
1 1,0 2 2,0
4 ODE/PDEα-SynucleinModelsforParkinson’sDisease
u (t=0)=u ; u (t=0)=u ; (1.8a,b)
3 3,0 4 4,0
k ,k ,Q ,Q ,Q ,Q ,T ,T ,V,V ,h,t ,t are parame-
1 2 A,s B,s A,syn B,syn A,1/2 B,1/2 s syn 1 2
ters (constants) that are given unit values for the ODE model formulation
in this chapter (so that the focus of the discussion is on the ODE model
formulation and computer implementation). These parameters are given
numerical values (with units) in Table 2.1 of Chapter 2 when the ODE
model is applied to PD.
An explanation of the LHS and RHS terms of Eqs. (1.1a) and (1.5a)
(for mono α-syn soma) follows.
d[A] du
• V s,V 1: accumulation (or depletion) of mono α-syn in the
s s
dt dt
soma. The units are (m3)(molm−3s−1) = mols−1 which should be the
units for the LHS terms (derivatives in t).
• VQ : Rate of external addition of mono α-syn to the soma. Units:
s A,s
(m3)(molm−3s−1) = mols−1.
• −Vk [A],−Vk u : First kinetic rate of conversion of mono to poly
s 1 s s 1 1
α-syn in the soma. Units: (m3)(s−1)(molm−3) = mols−1.
• −Vk [A][B],−Vk u u : Second kinetic rate of conversion of mono
s 2 s s s 2 1 2
to poly α-syn in the soma. Units: (m3)(mol−1 m3s−1) (molm−3)
(molm−3) = mols−1.
[A] u
• −V s ,−V 1 : Rate of decay of mono α-syn in the soma.
s s
T T
A,1/2 A,1/2
Units: (m3)(molm−3)(s−1) = mols−1.
• −h [A],−h u : Rate of transfer of mono α-syn from the soma to the
A s A 1
synapse (this term connects the soma and synpase in the model). Units:
(s−1 m3)(molm−3)= mols−1.
The terms in Eqs. (1.1b) and (1.5b) have analogous explanations and
units for poly α-syn in the soma. The one term that is different is
• Q h(t−t )h(t −t): h(t) is the Heaviside unit step that turns on this
B,syn 1 2
term at t=t and then turns it off at t=t to produce a pulse input of
1 2
poly α-syn to the soma. This pulse is clear in the numerical solutions
that follow.
Notethatthefirstkineticterm(linear,withk )isoppositeinsignformono
1
and poly α-syn) as well as the second kinetic term (nonlinear, with k ).
2
Eqs. (1.2) and (1.6) have similar interpretations, but for the synapse
rather than the soma.