Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 2
Voltage Gated Ionic Currents
Christopher P. Fall and Joel E. Keizer
In Chapter 1 we introduced models of simple channel behavior but ignored the idea
that something might ow through such a channel. In this chapter we will learn how
to model curr
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 6
Intercellular Communication
John Rinzel
Orchestrating the activity of cell populations for physiological functioning of the brain,
organs, and musculature depends on transmission of signals, learning and memory devices, and feedback control syst
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 3
Transporters and Pumps
Eric S. Marland and Joel E. Keizer
Ionic channels are not the only mechanism that cells use to transport impermeant
species across membranes. Cells have developed a great variety of transport proteins
for moving both ions
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 7
Spatial Modeling
James P. Keener
All of the models considered in previous chapters have relied on the implicit assumption that chemical concentrations are uniform in space. This assumption is reasonable
when the region of space in which the reac
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 8
Modeling Intracellular Calcium
Waves and Sparks
Gregory D. Smith, John E. Pearson, and Joel E. Keizer
In this chapter we shall discuss a variety of intracellular Ca2+ wave phenomena, but
always from the perspective that the distance scales of in
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 9
Biochemical Oscillations
John J. Tyson
Biochemical and biophysical rhythms are ubiquitous characteristics of living organisms, from rapid oscillations of membrane potential in nerve cells to slow cycles of
ovulation in mammals. One of the rst bi
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 10
Cell Cycle Controls
John J. Tyson and Bela Novak
In recent years, molecular biologists have uncovered a wealth of information about
the proteins controlling cell growth and division in eukaryotes. The regulatory system
is so complex that it dee
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 12
Molecular Motors: Theory
Alex Mogilner, Timothy C. Elston, Hongyun Wang, and George Oster
Evolution has created a class of proteins that have the ability to convert chemical energy
into mechanical force. Some of these use the free energy of nuc
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 13
Molecular Motors: Examples
Alex Mogilner, Timothy C. Elston, Hongyun Wang, and George Oster
13.1
Switching in the Bacterial Flagellar Motor
As an example of the use of the numerical algorithm developed in Chapter 12, we rst
consider a model for
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
APPENDIX A
Qualitative Analysis of
Differential Equations
G. Bard Ermentrout and Joel E. Keizer
Nonlinear ordinary differential equations are notoriously difcult or impossible to solve
analytically. On the other hand, the solution to linear equations like
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 11
Modeling the Stochastic Gating
of Ion Channels
Gregory D. Smith
In previous chapters we have seen several kinetic diagrams representing various molecular states and transitions between these states due to conformational changes and the
binding
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 4
Fast and Slow Time Scales
James P. Keener and Joel E. Keizer
One of the hallmarks of cellular processes is their complexity. For example, in Chapter
3 we described a detailed model for the SERCA pump that might require 11 ODEs
and 22 kinetic con
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
Interdisciplinary Applied Mathematics
Volume 20
Editors
S.S. Antman J.E. Marsden
L. Sirovich S. Wiggins
Mathematical Biology
L. Glass, J.D. Murray
Mechanics and Materials
R.V. Kohn
Systems and Control
S.S. Sastry, P.S. Krishnaprasad
Problems in engineerin
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
9 y U`i y Q C U yf E f FPfGcwYWtFreISgwdlWdecSlgD SrpwRe B ` B yf y ECE U jEC yC B j `B u`GtoqDTgFchIGV`chIpI!g dReFSdpe S Ut HC yC HCBE Tf Q t H tBE x yC t y U U y B x Qf EB j T y C X U gBE B X C X B a` d y H pPtGVFIFphFhwEhV`cGphrIYeVfhs`crIYprc#gPB#g
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
3qFVqmgCtX
H e kW W rG P a Y YW i acW kW z a HG P RG Pr Y kW EW q E [email protected]`1Ipq`SkpX'[email protected][email protected] ca a c gc n PGcW D wca HG rG caR kT RG Pr Y R YW i E PGT WcG ErT W R n H [email protected]`sdprptpdWkQdTbbplYsFEotXbap`IrbpSrgI3SE`lp1`SrbqpII
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
p U a UD 5 Xb E rGExEG B D X 5 xE X ` U Ex a @bE x x 5 UDb 5 i x a X a UD 5 Xb G i Ex rdWDfgW5F#r3rFHYdFfrfodgpWD3crHAyArwXYpAfha9crfgw5Y#rHprhXeYca XD Eb 5E sGE r 5 Xb i X 5 UDX 5 ` X X s @ DX 5 U Eb 5E sGE r 5 Xb E U D U X B D@ 8D x SX 8x u B s @ F3cdr
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
Chapter 8
Bounded Linear Operators on a Hilbert Space
In this chapter we describe some important classes of bounded linear operators on Hilbert spaces, including projections, unitary operators, and selfadjoint operators. We also prove the Riesz represent
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
h eeE r  z f T f hd`hf b h`f f X yhdf Tf f f z w X f Xd ry w X suheai1xugcxcheTIed x fusiuyiIcfw_ueegy egsasq n d` f Xd T ` yd T`dy T f w y h x yqy egWauar fEcadIugu x d` f Xd d` Xd T f t Tf Xd y` z ` fhf X x yqy exqaWaQY#iagsr de x vyguxy YAPqu gext
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
` T R h hf h h V R d p h f d d v R d T h ` d `X sqyy(YslogyWYqgSosyCtauuWUwgenygivWC R h V T r R X ` T h ` f X d h f d T R b d ` X X b CP nyyeiCPtWywneneWEYtao~nwq R h V T r X CP2ueUyYWC#2no d CPw# TdV qviT Vb ` eiguieYicnWqggig7Wx v ` T h ` ` R X T b
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
Errata Applied Analysis (Corrected in online les but not in Second Printing) p. 115: Replace statement of Theorem 5.53 with: A consistent approximation scheme is convergent if and only if it is stable. p.115: Replace last paragraph (Conversely.) of the pr
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
c 6 E83 e c a AAAwA g R 7U97D6 gydb` 3 1 Ft Uhh 4sq e VRrisr n X ( n X ( g c 6 E83 c a 4R 7U97D6 gydb` ccc y0( gv74p2Uqv9R96 gydb` X i 61 1 v8 6 d E 68 G c a ccc @ v@ G E@ 8 G q 1 E@ G 1 8 q 1 6 8 6 v c #7R4H17DsyYHGBV4DDy a p7@ UccD q RHGE h q c 5HD796
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
Index
backwards heat equation, 164
BakerCampbellHausdor formula, 123
ball
closed, 14
open, 14
unit, 4
Banach algebra, 39
Banach space, 8, 91
of linear operators, 110
BanachAlaoglu theorem, 120, 208
BanachSteinhaus theorem, 204
base of open sets, 84
ba
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
Measure Theory
V. Liskevich 1998
1
Introduction
We always denote by X our universe, i.e. all the sets we shall consider are subsets of X. Recall some standard notation. 2X everywhere denotes the set of all subsets of a given set X. If A B = then we often
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
CHAPTER 1
Dynamic Phenomena in Cells
Christopher P. Fall and Joel E. Keizer
Over the past several decades, progress in the measurement of rates and interactions of
molecular and cellular processes has initiated a revolution in our understanding of dynamic
Ordinary Differential Equations and Dynamical Systems
MATH 200

Spring 2014
ss
C8FuTgtcq208TmP3FcgoincgHb0meFd0gVle8jw1&1558swBB $PI9GEDiBXIEWVUTS2X$XIeDGf98H67 3 ' % ' e
sr bp b
k'b
@ u s
8e45fjvg's#T
g8gT37H4pR1TX3igT625pTHP8Aw1i150F8$YXIEWVUHSToBciWIp
l s sp 3 ' ' b
@
V'5qEywvvs! vT3Fsc9b'`eFdee
sHG7ThUb fBFY$h7I ' e zfT cIz