Table Of ContentInterdisciplinary Applied Mathematics
Volume 32
Editors
S.S.Antman J.E.Marsden L.Sirovich
Geophysics and Planetary Sciences
Imaging,Vision,and Graphics
Mathematical Biology
L.Glass,J.D.Murray
Mechanics and Materials
R.V.Kohn
Systems and Control
S.S.Sastry,P.S.Krishnaprasad
Problems in engineering,computational science,and the physical and biological
sciences are using increasingly sophisticated mathematical techniques.Thus,the
bridge between the mathematical sciences and other disciplines is heavily
traveled.The correspondingly increased dialog between the disciplines has led to
the establishment of the series:Interdisciplinary Applied Mathematics.
The purpose of this series is to meet the current and future needs for the
interaction between various science and technology areas on the one hand and
mathematics on the other. This is done, firstly, by encouraging the ways that
mathematics may be applied in traditional areas, as well as point toward new
and innovative areas of applications; and, secondly, by encouraging other
scientific disciplines to engage in a dialog with mathematicians outlining their
problems to both access new methods and suggest innovative developments
within mathematics itself.
The series will consist of monographs and high-level texts from researchers
working on the interplay between mathematics and other fields of science and
technology.
Interdisciplinary Applied Mathematics
Volumes published are listed at the end of the book.
Dominik Wodarz
Killer Cell Dynamics
Mathematical and Computational
Approaches to Immunology
Dominik Wodarz
Department of Ecology and Evolution
University of California
321 Steinhaus Hall,Mail Code 2525
Irvine,CA 92697-2525
USA
[email protected]
Editors
S.S.Antman J.E.Marsden
Department of Mathematics and Control and Dynamical Systems
Institute for Physical Science and Mail Code 107-81
Technology California Institute of Technology
University of Maryland Padadena,CA 91125
College Park,MD 20742 USA
USA [email protected]
[email protected]
L.Sirovich
Division of Applied Mathematics
Brown University
Providence,RI 02912
USA
[email protected]
Cover illustration:The Killing,painting by Natalia Komarova
Mathematics Subject Classification (2000):37N25,46N60,62P10,92-xx,92Bxx,92B05
Library ofCongress Control Number:2006931001
ISBN-10:0-387-30893-8
ISBN-13:978-0-387-30893-7
Printed on acid-free paper.
© 2007 Springer Science+Business Media,LLC
All rights reserved.This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media,LLC,233 Spring Street,New York,
NY 10013,USA),except for brief excerpts in connection with reviews or scholarly analysis.Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication oftrade names,trademarks,service marks,and similar terms,even ifthey
are not identified as such,is not to be taken as an expression of opinion as to whether or not they
are subject to proprietary rights.
9 8 7 6 5 4 3 2 1
springer.com
To
my parents Klara and Hans-Walter, my brother Andi, Pinktche,
my wife Natasha, and our children Sophia and Paulina. Paulina
was born during the final stages of writing this book.
Preface
Systems biology and computational biology have recently become prominent
areas of research in the biomedical community, especially in the area of cell
biology.Giventhatmuchinformationongenesandtheirproteinproductshas
become available, the big question is how the individual components interact
and work together, and how this determines the functioning of cells, organs,
andorganisms.Longbeforethepopularityofsystemsbiologyinbiomedicine,
however, such approaches have been used successfully in a different area of
biology: population ecology. Research in the area of population dynamics in-
vestigated complex interactions between different populations of organisms,
such as the dynamics of competition and predation, food webs, community
structure, as well as the epidemiology of infectious diseases. In this field, the-
oretical biology and mathematical modeling have become an integral part
of research. Mathematical models allowed people to obtain interesting and
counter-intuitive insights into how complex interactions among different pop-
ulations can play out. Such mathematical studies not only gave rise to in-
teresting theoretical ideas, but also provided the basis for the design of new
experimental work and defined major questions and directions of research.
Around 1990, such population dynamic concepts, and the use of mathemati-
cal/computational approaches, started to be applied to the in vivo dynamics
between viruses and the immune system. These interactions have many sim-
ilarities to ecological, epidemiological, and evolutionary principles. Consider
theepidemiologicalspreadofapathogen(suchasthecommoncold)througha
population of hosts. Pathogens and parasites reproduce inside infected hosts,
are released from the hosts into the environment, and infect other susceptible
hosts. Similarly, viruses reproduce inside host cells in vivo. Infected host cells
produce new virus particles which are released into the extra-cellular envi-
ronment and which can infect further susceptible host cells. The population
dynamic principles are very similar in both cases.
The interactions between pathogens and the immune system is an ex-
tremelywideanddiversetopic.Thisbookconcentratesonaparticularbranch
oftheimmunesystem:killerTcells,orcytotoxicTlymphocytes(CTL).This
VIII Preface
reflects my own research interest and fascination. Killer T cells recognize in-
fectedcells,andattackthem.Therefore,thisbookfocusesonpathogenswhich
enterhostcells.Manypathogenscanreproducewithintheirhostwithouthav-
ing to infect cells. The main class of pathogens which do infect host cells are
viruses. In fact, viruses cannot reproduce without entering the host cells, be-
cause the cells’ metabolic machinery is required for replication. Hence, the
book considers killer T cell responses mostly in the context of viruses.
The purpose of this book is to review how mathematical and computa-
tional approaches can be useful to help us understand how killer T cell re-
sponsesworktofightviralinfections.Further,theaimistodemonstratethat
such mathematical and computational approaches are most valuable when
coupled with experimental work through collaborations. The writing style
also reflects this interdisciplinary spirit. While the topic of the book is math-
ematical modeling, the text is written in a way such that experimental im-
munologists and virologists should be able to understand the arguments, and
to see the biological implications of theory. Experimental readers are encour-
aged to skip the equations, and to focus on the biological interpretations and
discussions.
The work which is reviewed here builds on earlier mathematical research
on virus dynamics which considered in detail the relationship between the
evolutionary dynamics of viruses in vivo and the development of disease in
pathogenic human infections such as HIV. Some of this research is summa-
rizedintheintroductorychapter,andathoroughreviewisgiveninaprevious
book by Martin Nowak and Robert May in the year 2000 (Virus dynamics,
Oxford University Press). While the current book aims to cover many im-
portant aspects of killer T cell dynamics, it is not intended to address all
important issues relevant to the understanding of killer T cell responses. The
bookconcentratesonthosetopicswhichhavebeenrelativelywellworkedout,
and where theory has been coupled with experimental work in a meaningful
way,orwherepublishedexperimentalworkprovidesconcreteapplicationsand
case studies. It provides a personal view of the subject, guided by my long
standingcollaborationswithvariousexperimentallaboratorieswhichworkon
thebiologyofkillerTcellresponses,andwhichhaveguidedmeovertheyears.
My work, and this book, would not have been possible without the en-
thusiastic input and guidance from my experimental collaborators. During
myPhDyears,PaulKlenermanintroducedmetoLCMV(lymphocyticchori-
omeningitis virus) infection, and subsequently to Hepatitis C virus (HCV)
and cytomegalovirus (CMV) infection. Our interactions continue to inspire
my theoretical work. Charles Bangham worked with me on the dynamics of
immune responses to human T cell leukemia virus (HTLV-1), and I am ex-
tremely grateful to him for his continuous guidance throughout my career.
Many of the theoretical concepts in the context of HIV infection came about
throughmyinteractionswithJeffLifson,whosemonkeysbroughtnewdimen-
sions to my work. I am grateful to Peter Doherty who shared many valuable
immunological insights in the context of murine influenza virus and gamma
Preface IX
herpes virus infections. Last but not least, much of my work would not be
possible without my collaboration with Allan Thomsen on CTL dynamics
in murine viral infections, especially LCMV. His interest in theory, and his
openness to apply theoretical thinking to experimental design has not only
benefited the quality of the mathematical research, but has also been and
continuous to be a tremendous amount of fun.
While my interactions with experimental laboratories has certainly been
vital for the development of my work, all this would not have been possible
without the guidance of my PhD advisor Martin Nowak, who introduced me
tothisfieldandtosomeofmyexperimentalcollaborators,andwhoopenedup
manyvaluabledoorsandopportunities.Similarly,interactionswiththeoretical
biologistsandevolutionarybiologistsinspiredmeandinfluencedmythinking.
TheseincludeFranciscoAyala,SteveFrank,DavidKrakauer,AlunLloyd,and
Alan Perelson. Finally, the Department of Ecology and Evolutionary Biology
at the University of California Irvine provided a stimulation and productive
environment for writing this book.
Irvine, California, USA Dominik Wodarz
September 2005
Contents
1 Viruses and Immune Responses: A Dynamical View ....... 1
1.1 Viruses................................................. 3
1.2 Basic Immunological Background ......................... 5
1.3 Experimental Mouse Models of CTL Dynamics.............. 13
1.4 Human Pathogenic Infections ............................. 15
1.5 Virus Dynamics and Mathematical Modeling................ 18
1.6 Immune Response Dynamics: Structure of the Book ......... 23
2 Models of CTL Responses and Correlates of Virus Control 25
2.1 Virus Dynamics ......................................... 26
2.2 Simplest Model for CTL Dynamics ........................ 28
2.3 Saturated CTL Expansion................................ 31
2.4 Precursor Versus Effector CTL ............................ 32
2.5 Programmed CTL Proliferation ........................... 34
2.6 Summary............................................... 40
3 CTL Memory.............................................. 41
3.1 The Generation of CTL Memory: Biological Background ..... 43
3.2 Mathematical Models of CTL Memory ..................... 43
3.3 Resolution of Primary Infection: Mathematical Predictions ... 45
3.4 Resolution of Primary Infection: Experimental Studies ....... 47
3.5 Protection Against Rechallenge: Mathematical Predictions ... 49
3.6 Protection Against Rechallenge: Experimental Studies ....... 51
3.7 Summary............................................... 53
4 CD4 T Cell Help .......................................... 55
4.1 Comparison of the Two Helper Pathways ................... 63
4.2 How Does Help Work? ................................... 63
4.3 Infection Dynamics in Helper Deficient Hosts ............... 66
4.4 Summary............................................... 70
