Modelling Course in Population and Evolutionary Biology

1
Modelling Course in Population and Evolutionary
Biology
Introduction
Part I The Course Part II R Part III The
Modules (2 x 15 min break)

• 2 June 2009, Zürich

2
Part I The Course
3
People
Prof. Sebastian Bonhoeffer Course Director
Viktor Müller Course Instructor
4
People module developers

• Martin Ackermann
• Tobias Bergmiller
• Sebastian Bonhoeffer
• Lucy Crooks
• Florence Debarre
• David Fouchet
• Nicole Freed

• Roger Kouyos
• Dusan Misevic
• Viktor Müller
• Roland Regoes
• Olin Silander
• Orkun Soyer

5
Feedback

• This course takes place for the third time.
  Everything can be changed flexibly, and we
  appreciate your feedback!

6
Goals

• To get familiar with basic approaches in the
  modelling of biological processes
• To obtain a conceptual insight into interesting
  biological questions
• To learn to appreciate the excitement and utility
  of computational modelling in biology
• To learn team work

7
Focus how to make these transitions?

• Foreground modelling
• Background biology math

computer implementation
biological problem
math model/ algorithm
interpretation of model results
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Time table
Place CHN F 46
Daily schedule 9-13 Work on
modules 13-14 Lunch break 14-18
Work on modules Last day (15 June)
presentations in the afternoon

• NOTES
• You are free to take short breaks during the
  work sessions.
• You can run simulations in the weekends.

Breakdown 4 5 1 10 days total Introduction
½ day Module 1 3.5 days Module 2 4.5
days Preparing presentations 1 day Final
presentations ½ day

• Recommendation
• Complete Module 1 this week.
• Prepare slides on the fly.

flexible
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Organisation of the course

• Team and module choice
• Each team should have at least one member with
  some experience in programming
• Teams should choose two modules that use
  different methods
• The same module can be chosen by several teams
• Extensive development of a level 1 module may be
  accepted as level 2 at the instructors decision.
  In this case the team should choose another level
  1 module in addition.
• Work on modules
• 1 student/computer
• Instructors help as needed

10
Evaluation

• Marks will be based on
• performance during the course
• instructors monitor progress
• completion of modules
• model design
• implementation (functionality of R code)
• scientific results
• final presentation
• ppt or pdf slideshow on level 2 module results
• get the message across

Important note to enable individual evaluation,
each team member should be given responsibility
for particular tasks and participate in the final
presentation.
Students with no prior knowledge of R should also
be able to achieve the highest mark.
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Webpage
https//www.tb.ethz.ch/education/model

• modules
• R resources
• practical information

12
Part II R

• Note this section focuses on getting started
  with R and on some useful tricks. You should
  certainly read the designated chapters of
  Introduction to R and you are advised to have
  the R reference card at hand.

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What is R?

• R is an integrated suite of software facilities
  for data manipulation, calculation and graphical
  display.
• It is often used for statistics, but it can do
  much more.
• R is a free implementation of the S language.

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Downloading R
go to http//www.r-project.org/
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Downloading R
scroll down to ETH mirror
16
Downloading R
choose your operating system
17
Downloading R Windows
click on base
then click on the setup program
and save it to your hard drive.
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Installing R Windows

• Run the setup program from your hard drive.
• install to c\Program Files\R
• Set up a desktop icon for R
• right-click over Desktop
• New gt Shortcut
• browse to c\Program Files\R\R-2.9.0\bin\Rgui.exe
• right-click on desktop icon gt Properties gt Start
  from set to \course
• Double-click on the icon ready to go!

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Downloading and installing R Mac
CTRL click to download install disk image
Double-click on icon of the multi-package
"R.mpkg" contained in the R-2.4.1.dmg disk image.
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Installing packages in R
go to Packages gt Install packages

• select mirror in Zürich
• click on package odesolve

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Using R

• Type commands directly at the prompt (command
  line)
• separate commands by newline (ltENTERgt) or
  semicolon (ltgt)
• use braces to group several commands into an
  expression
• starting with a hashmark (), everything to the
  end of the line is a comment
• use vertical arrows to recall previous commands
• Load code from the file menu or with
  source(filename)
• Code is written as a plain text file.
• on Mac use Rs internal editor
• on Windows Tinn-R (http//www.sciviews.org/Tinn-R
  /) provides user-friendly editing of R code

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Getting help

• Type help(command) or ?command
• Or go to help menu.
• If these approaches fail to help
• call us.

23
Exiting R

• Type quit() or q()
• or close window.
• You can save all objects at quitting into .RData.
  Starting R from the same directory, the workspace
  is loaded and you can continue working where you
  stopped it.

Keep in mind if you do this, you may have
objects (variables, functions) defined that you
have long forgotten about.
Recommendation use this feature only for short
interruptions in your work, but not on a
day-to-day basis.
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A sample session
switch to R/Tinn-R Options/MainApplication/
/R/General/Path of your preferred
RGui Options/Shortcut customization
(main) R_Sending Line (edit ALTL) download
https//www.tb.ethz.ch/education/model/sample.r
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A sample session function definition
distance lt- function(p1,p2) xdiff p1"x"
- p2"x" ydiff p1"y" - p2"y" diff
sqrt(xdiff2 ydiff2) as.numeric(diff)
gt source(distance.r) gt point1 lt- c(x5,y3) gt
point2 lt- c(x1,y6) gt distance(point1,point2)
Modify the function to calculate distance in n
dimensions.
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Demo a simple reaction kinetics model
k1
k3
E S C E P
k2
EC constant etot
C etot - E
SCP constant stot
P stot S C stot etot S E
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Demo a simple reaction kinetics model
k1
k3
E S C E P
k2
28
Demo a simple reaction kinetics model
k1
k3
E S C E P
k2
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Demo a simple reaction kinetics model

• source and read reaction.r
• create pdf figure
• try
• simulationcopy
• simulation-copy
• comment pdf() and print on screen

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Miscellaneous notes

• Introduction to R read Sections 1-3 (pp1-15),
  5.1-5.4 (pp18-20), 5.7.1 (p22), 5.8-5.9 (p24), 6
  (pp26-28), 7.1-7.2 (pp30-31), 9 (pp40-41), 10.1
  (p42), 10.3 (p43), 12.1-12.2 (pp62-66), 12.4-12.6
  (pp67-75), 13 (p76). 44 pages total.
• Keep a working script for the solution of each
  exercise. When a module is completed, we will
  check the functionality of the scripts and
  discuss their answers to the questions.

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Part III The Modules
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The organization of modules

• Webpage brief description links for download
• Reader (PDF)
• biological and modelling background
• instructions to develop the model
• exercises (basic advanced/additional)
• Starting R script (not all modules)
• Glossary
• Literature Weblinks (optional reading)

• Unless otherwise stated in the reader, completion
  of a module requires solving all basic exercises.
• this rule is just as flexible as all the others.

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List of modules

• Level 1
• The logistic difference equation and the route to
  chaotic behaviour
• SIR models of epidemics
• Stochastic effects on the genetic structure of
  populations
• Within-host HIV dynamics estimation of
  parameters
• Within-host HIV dynamics the emergence of drug
  resistance
• Level 2
• Discrete vs. continuous time models of malaria
  infections
• Evolution of the sex ratio
• Rock-paper-scissors dynamics in space
• Spatial cooperation games
• Stability and complexity of model ecosystems
  Are large ecosystems more stable than small ones?
• Stochastic simulation of epidemics
• Unstable oscillations and spatial structure The
  Nicholson-Bailey model of host-parasitoid dynamics

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Level 1 modules
35
The logistic difference equation andthe route to
chaotic behaviour

• Basic problem
• many species have non-overlapping generations and
  may therefore be described better in discrete
  time
• logistic growth self-limitation
• discrete steps allow for overshooting ?
  oscillations, chaos
• General approach iterate difference equation
• Concepts
• chaos
• periodic behaviour
• bifurcations

36
The logistic difference equation and the route
tochaotic behaviour

• Methods
• time plots
• phase diagrams
• bifurcation diagrams
• Questions
• what types of behaviour are possible in the LDE?
• what defines chaotic behaviour?
• analyse bifurcation diagram
• introduce space

37
SIR models of epidemics

• Basic problem what factors govern the spread of
  infectious diseases?
• General approach
• numerical integration of ODE model
• Concepts
• basic reproductive ratio
• herd immunity
• Methods
• time plot
• phase portrait
• contour plot

38
SIR models of epidemics

• Questions
• what are the conditions for the outbreak of an
  epidemic?
• what fraction of a population is going to be
  infected?
• can partial vaccination be protective?
• model treatment, drug resistance and birth-death
  dynamics

39
Stochastic effects on the geneticstructure of
populations

• Basic problem
• genetic drift can destroy variation, counteract
  selection and build up associations between loci
• General approach
• simple population genetic models with mutation,
  selection, recombination and random sampling of
  offspring
• Concepts methods
• iteration of discrete time population genetics
  model
• interplay of selection and drift
• benefits of recombination
• sampling from binomial distribution
• Questions
• How does drift reduce the diversity that mutation
  builds up?
• How does drift affect the elimination of
  detrimental alleles through selection?
• How do bottlenecks affect the diversity at
  neutral and selected loci?
• What do effective population sizes tell about the
  magnitude of stochastic effects?

40
Within-host HIV dynamics 1estimation of
parameters

• Basic problem
• The apparent latency of HIV infection conceals a
  highly dynamic steady state. Perturbation by drug
  treatment reveals the dynamics.
• General approach
• Estimation of decay parameters by fitting a
  simple ODE model to real and simulated treatment
  data.

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Within-host HIV dynamics 1estimation of
parameters

• Concepts methods
• Model fitting Parameter estimation by
  non-linear minimization.
• Numerical simulation of ODEs.
• Questions
• What factors influence the quality of parameter
  estimation?
• How does random noise (measurement error) affect
  the estimation?
• What if treatment is not 100 effective?
• What is the effect of long-lived virus-producing
  cells?

42
Within-host HIV dynamics 2the emergence of
drug resistance

• Basic problem
• Mutations in the enzymes of HIV can render the
  virus resistant to drugs.
• General approach
• ODE models to simulate wild-type and mutant virus.

43
Within-host HIV dynamics 2the emergence of
drug resistance

• Concepts methods
• numerical simulation of ODEs
• mutation-selection equilibrium
• Questions
• What are the conditions for the emergence of drug
  resistance?
• How does the efficacy of the drugs affect the
  time to the emergence of resistance?
• Resistance mutations can exist in a
  mutation-selection equilibrium even before
  treatment how does this affect the emergence of
  resistance under therapy?
• What is the advantage of administering a
  combination of different drugs?
• Devise optimal treatment strategy

44
Level 2 modules
45
Unstable oscillations and spatial structure The
Nicholson-Bailey model of host-parasitoid dynamics

• Basic problem
• A discrete-time model of host-parasite
  interactions is unstable. Can the implementation
  of space stabilize the system?
• General approach
• Model host-parasite interactions and dispersal on
  a 2D lattice.

46
Unstable oscillations and spatial structure The
Nicholson-Bailey model of host-parasitoid dynamics

• Concepts methods
• Simulation of simple two-species difference
  equations
• Simulate spatial structure and observe emerging
  patterns
• Questions
• Why is the simple NB model unstable?
• What is the effect of spatial structure?
• What is the effect of lattice size and boundary
  conditions?
• Do initial conditions affect the outcome?
• Can parasitoids facilitate the coexistence of
  different host types?

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Spatial cooperation games

• Basic problem altruistic behaviour decreases the
  fitness of the actor. So how can it evolve and be
  maintained?
• General approach simulate iterated cooperation
  games in unstructured and spatially structured
  populations.
• Concepts
• Game theory Prisoners dilemma and snowdrift
  games.
• Spatial structure and the evolution of
  cooperation.

• Methods
• Spatially explicit simulation of population
  interactions on a lattice
• Cellular automaton

48
Spatial cooperation games

• Questions
• How does spatial structure affect the evolution
  of cooperation?
• What is the effect of the payoff parameters
  (cost, benefit)?
• Investigate the effects of
• neighbourhood size (3,4,6)
• updating scheme (synchronous vs. asynchronous
  pair-wise vs. multiple competitions)
• population size (500, 1000, 2000)
• heterogeneous environment
• on the evolution of cooperation and the
  significance of spatial structure.

49
Rock-paper-scissors dynamics in space

• Basic problem can intransitive fitness
  interactions facilitate the maintenance of
  diversity?
• General approach model local competition in a
  cellular automaton
• Concepts
• intransitive interaction AltB, BltC, CltA
• density dependent selection

lt
lt
lt
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Rock-paper-scissors dynamics in space

• Questions
• How does the maintenance of diversity depend on
• the type and strength of fitness interactions
• initial population size and species frequencies
• The distance over which organisms
  interact/disperse?
• What factors affect the magnitude of population
  fluctuations?
• How do the dynamics of the system change when
  there are greater numbers of species interacting?
• What is the effect of disturbance (e.g. local
  fires) on the maintenance of diversity?

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Stability and complexity in model ecosystems

• Basic problem Does complexity help stability?
• General approach study stability of randomly
  generated multi-species Lotka-Volterra systems.
• Concepts methods
• Connectivity, diversity and stability of an
  ecosystem/network
• Numerical simulation of (large) systems of
  ordinary differential equations
• Questions
• How does ecosystem stability depend on the size
  (i.e. number of species) and connectivity of the
  ecosystem?
• What are useful measures of ecosystem stability?
• Does the coexistence of a set of species depend
  on the order in which they were introduced into
  an ecosystem?
• How does the ecosystem respond to the removal or
  invasion of a species?
• How does stability change if some interactions
  are predatory?

52
Discrete versus continuous-time modelsof malaria
infections

• Basic problem
• Malaria parasites reproduce in discrete
  generations. What is the effect of simplifying
  this to continuous-time models?

53
Discrete versus continuous-time modelsof malaria
infections

• General approach
• Compare discrete and continuous-time models of
  malaria.
• Concepts methods
• Numerical simulation of ODEs and difference
  equations
• Trade-offs and evolutionary optimum
• Questions
• How to parameterize the models to achieve maximal
  equivalence?
• Can you obtain identical behaviour?
• What level of gametocyte investment maximises
  transmission?
• Model an immune function/compartments/variable
  investment

54
Evolution of the sex ratio

• Basic problem why is the typical sex ratio 11?
• General approach
• simulate a population of males and females
• sex ratio of offspring determined by a diploid
  locus in the mother
• introduce sex ratio mutants and run until
  evolutionary equilibrium
• Concepts methods
• evolutionary optimization
• individual-based modelling
• stochastic simulation
• Questions
• optimal sex ratio for various inheritance schemes
  of the SR gene
• what happens if the sexes have different survival
  or cost?
• what if the SR gene is located on a sex
  chromosome?

55
Stochastic simulation of epidemics

• Basic problem
• Introduce stochasticity and discrete populations
  into the SIR model
• General approach
• Stochastic modelling with the Gillespie algorithm
• Concepts methods
• Comparison of deterministic and stochastic models
• Simulation and analysis of stochastic models
• Questions
• What is the extinction probability of the
  infection for different R0's?
• Does the average dynamics of the stochastic model
  differ from the deterministic SIR model?
• Are population sizes across runs normally
  distributed?

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Note

• Completion of a module is defined flexibly you
  are free to deviate from the pre-defined
  exercises and work on your own ideas. Consult the
  instructor.
• Keep a working version of your script for each
  exercise.

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Background information

• Ecology and Evolution II Populations
• 701-0273-00l WS (Bachelor BIOL 5th term/ Master
  UWIS)
• by Sebastian Bonhoeffer

58
Reminder team and module choice

• Each team should have at least one member with
  some experience in programming
• Teams should choose two modules that use
  different methods
• The same module can be chosen by several teams
• Extensive development of a level 1 module may be
  accepted as level 2 at the instructors decision.
  In this case the team should choose another level
  1 module in addition.

59
Contact

• viktor.mueller_at_env.ethz.ch