Part IA Natural SciencesMathematical Biology

2010-2011

Please pick up one of each of the four different coloured handouts

on your way in

(We shall start at about 9.05am)

Welcome toMathematical Biology

Dr Nik CunniffeDepartment of Plant Sciences

njc1001@cam.ac.uk

Mathematical Biology – Lecture One

Topic : Introduction

Aim : 1) To introduce arrangements for lectures, practicals and supervisions 2) To introduce growth and rate curves

Objectives: 1) To summarise typical shapes of growth curves 2) To show some characteristic functions used for growth curves 3) To show how to compute rate curves from growth curves

Organisation: LecturesOrganisation: Lectures

• Tuesday, Thursday and Saturday (!) at 9am • Note after today we will start at 9 sharp, so can finish at 9.50 and give those of you studyingChemistry enough time to get to Lensfield Road

• For this part of the course, new material onlypresented on Tuesday and Thursday

• Saturday to consolidate and to aid the transitionfrom A Level study by worked examples etc.

• Other lecturers (and subjects) will differ

Practicals and Example ClassesPracticals and Example Classes

• Every Thursday during first two terms (including this afternoon)

• You will start at 2, 3.30 or 4.45 (you should have been told which group you are by now, I think)

• Require PWF user id and password; if you donot have one yet don’t worry too much, but do try to find out about your account by next week

• Alternate between practical and example class•Odd weeks: computer practical (MathCAD/R)•Even weeks: pen and paper exercises

Practicals arein the Titan Teaching Rooms

Practicals and Example ClassesPracticals and Example Classes

The demonstrators: a range of backgroundsThe demonstrators: a range of backgrounds

Erik/Rich: Plant Sciences

Jelena/Sabine: Genetics

Kristien: Zoology

SupervisionsSupervisions

• Extra support and help; for mathematics this can be invaluable

• Work will generally be based around example sheets distributed by the lecturers, although can also talk around the lecture notes etc.

•If your supervisions have not been arranged by end of next week or so then contact your DoS

• Also let DoS know if you have problems withyour supervision; part of their job is to ensureyou are happy

About meAbout me

• I apply mathematics to biological systems

• In particular diseases of crops and trees

• An accurate model allows prediction, and alsoassessment of the likely effect of any proposed control strategy (eg. chemicals, GM, biocontrol)

• I actually use the techniques I am going to teachyou; definitely not just maths for the sake of it

Enough about me…what about you?Enough about me…what about you?

• Studying exclusively, or at least mainly, biological options in Part IA Natural Sciences • You are here because you are interested in learning how mathematics can be applied to biological problems

• Have a minimum of AS level mathematics (or equivalent) basic requirement is that you have taken maths post-16 and know at least a bit about differentiation, integration and curve sketching

• No need to have studied any statistics before

BooksBooks

• No single good book for the entire course

• Some suggestions on page 2 of handout

• Should be no need to buy, but instead you should borrow books from College library as required

• However main source of information whenit comes to revision will be your lecture notes…

• …so try to make good ones!

Lecture notesLecture notes

• Big handout summarises the topics andshould be helpful for revision

• However after today I will be writing a loton the overhead projector, actually working the calculations you will eventually be tested on

• Try to balance listening, understanding and writing down enough to recreate what I’ve done

• Also tell me if I make a mistake!

Apart from lecture notes, today you also getApart from lecture notes, today you also get

And what will you learn?And what will you learn?

•This term•Modelling single populations•Compartmental modelling

•Next term•Analysing coupled differential equations•Probability and statistics

•Exam term•Matrix algebra and population models•Ecological modelling: coupled populations

By writing differential equations can derive equations governing dynamics of populations in terms of parameters (incorporating mechanisticassumptions on the processes governingpopulation growth)

Example: modelling single populationsExample: modelling single populations

Example: compartmental modellingExample: compartmental modelling

View physiological (and other systems) as systems of linked compartments, andderive equations for concentrations of substances…has applications in optimisation

Example: coupled differential equationsExample: coupled differential equations

Understanding systems like this allows us to understand models of biological populations interacting with each other…no species exists in isolation

Examples: statisticsExamples: statistics

As well as “standard” hypothesistesting (which may or may not be familiar from school), we concentrateon the assumptions underlying thetests and so when they are reliable

Example: matrix algebraExample: matrix algebra

In addition to relatively mundane tasks like solving equations, we willuse matrices to provide us with mathematical tools to understand certain ecologicalmodels e.g. host-parasitoid interactions, age-structured populations

Example: ecological modellingExample: ecological modelling

We consider (for example) predator-prey, competition, epidemics

Who is going to teach you…Who is going to teach you…

Cerian Webb: Animal epidemics

John Trapp: Mathematical education

Andrea Manica: Population biology and evolution

Rufus Johnstone: Behavioural ecology

Colin Russell: Human epidemicsMe (again): Plant epidemics

Time for a breakTime for a break

• I will try to have a short break near the middle of each lecture if possible, but whether or not thishappens will depend on my timing

• If I manage this, it allows all of us to recoverfrom the first half hour…

• Suggest introduce yourself to a neighbour if youdon’t know them already; Freshers’ “week” is one of the very few times in your life when talking to total strangers will not get you odd looks…

Mathematical Biology…a gentle start!

Topic : Introduction

Aim : 1) To introduce arrangements for lectures, practicals and supervisions 2) To introduce growth and rate curves

Objectives: 1) To summarise shapes of growth curves 2) To show some characteristic functions for growth curves 3) To show how to compute rate curves from growth curves

Empirical Growth CurvesEmpirical Growth Curves

•i.e. results of experiments•Lots of examples in Fig 1.2; I will concentrateon a subset (can discuss others, which will all be covered eventually, with supervisors if you wish)•Note that our definition of a “population” is pretty broad, includes concentrations, sizes, etc.• Think about

•How fast is initial growth?•Does growth saturate?•At what population size?•And at what time?•Is there a decline after initial growth?

Population of the USAPopulation of the USA

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Nutrient uptake by Escherichia coliNutrient uptake by Escherichia coli

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Population dynamics of yeastPopulation dynamics of yeast

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Density of pathogenic fungusDensity of pathogenic fungus

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Fitted Growth CurvesFitted Growth Curves•Examples in Fig 1.3 in handout

•Attempt to summarise the data by drawing acurve of form y = f(t) through the points

•Emphasises main qualitative features

•Allows us to compare populations in terms of relatively few numbers; the “parameters”

•Sometimes difficult to decide on “best” function …we shall attempt this by “modelling”rather than by just guessing [as is often done]

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Population of the USAPopulation of the USA

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Nutrient uptake by Escherichia coliNutrient uptake by Escherichia coli

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0 3 6 9 12 15 18

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Population dynamics of yeastPopulation dynamics of yeast

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Pop

ulat

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(g-1 s

oil)

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Density of pathogenic fungusDensity of pathogenic fungus

Rate CurvesRate Curves

•Fitted growth curves are entirely descriptive

•No insight into underlying processes

•Much more informative can be the rate of change, as this actually shapes the growth

•Summarise using the “Rate Curve” (ie. dy/dt vs t)

•Later lectures show how to go “backwards”, from biological assumptions -> rates -> function of population size at each time

0

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1750 1800 1850 1900 1950

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1750 1800 1850 1900 1950

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Population of the USAPopulation of the USA

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0 10 20 30 40

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r ien

t upt

ake

(m

ol g

-1)

Minutes

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Nutrient uptake by Escherichia coliNutrient uptake by Escherichia coli

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Population dynamics of yeastPopulation dynamics of yeast

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(g-1 s

oil)

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Rat

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-2

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Density of pathogenic fungusDensity of pathogenic fungus

SummarySummary

•Introduced the course, and explained logistics

•Considered some simple population dynamics

•Interpreted in terms of rates of change, corresponds to processes shaping the growth

•This part of the course uses mathematical modelling to link assumptions about these

processes with population responsesusing mathematical techniques (differentialequations, really)

Some final words, for today at leastSome final words, for today at least

Remember, practical this afternoon in Titan rooms

A possible route toChemistry Dept.(in case you need it)

Maths helps me at the very small scale…Maths helps me at the very small scale…

•Pathozone for fungal pathogen Rhizoctonia solani•Probability of infection of a plant as a function of initial distance of inoculation and time

…and at the very large scale…and at the very large scale

•Californian epidemic of Phytophthora ramorum which has already killed up to a million trees in US

•Water mould from same genus as potato late blight

•Also known as “Sudden Oak Death”

•Increasingly appearing in the UK

Disease IntensityHigh : 127

Low : -128

Host Vegetation

Infection risk by 2010

Even helps me with less “academic” pursuitsEven helps me with less “academic” pursuits

10th July 2008…one of history’s darkest days…

Even helps me with less “academic” pursuitsEven helps me with less “academic” pursuits