1. Part IA Natural Sciences Mathematical 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)

2. Welcome to Mathematical Biology Dr Nik Cunniffe Department of Plant Sciences njc1001@cam.ac.uk

3. 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

4. Organisation: 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 studying Chemistry enough time to get to Lensfield Road For this part of the course, new material only presented on Tuesday and Thursday Saturday to consolidate and to aid the transition from A Level study by worked examples etc. Other lecturers (and subjects) will differ

5. Practicals 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 do not 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

6. Practicals are in the Titan Teaching Rooms Practicals and Example Classes

7. The demonstrators: a range of backgrounds Erik/Rich: Plant Sciences Jelena/Sabine: Genetics Kristien: Zoology

8. Supervisions 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 with your supervision; part of their job is to ensure you are happy

9. About me I apply mathematics to biological systems In particular diseases of crops and trees An accurate model allows prediction, and also assessment of the likely effect of any proposed control strategy (eg. chemicals, GM, biocontrol) I actually use the techniques I am going to teach you; definitely not just maths for the sake of it

10. 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

11. Books 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 when it comes to revision will be your lecture notes… …so try to make good ones!

12. Lecture notes Big handout summarises the topics and should be helpful for revision However after today I will be writing a lot on 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!

13. Apart from lecture notes, today you also get

14. 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

15. By writing differential equations can derive equations governing dynamics of populations in terms of parameters (incorporating mechanistic assumptions on the processes governing population growth) Example: modelling single populations

16. Example: compartmental modelling View physiological (and other systems) as systems of linked compartments, and derive equations for concentrations of substances…has applications in optimisation

17. Example: coupled differential equations Understanding systems like this allows us to understand models of biological populations interacting with each other…no species exists in isolation

18. Examples: statistics As well as “standard” hypothesis testing (which may or may not be familiar from school), we concentrate on the assumptions underlying the tests and so when they are reliable

19. Example: matrix algebra In addition to relatively mundane tasks like solving equations, we will use matrices to provide us with mathematical tools to understand certain ecological models e.g. host-parasitoid interactions, age-structured populations

20. Example: ecological modelling We consider (for example) predator-prey, competition, epidemics

21. 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 epidemics Me (again): Plant epidemics

22. Time for a break I will try to have a short break near the middle of each lecture if possible, but whether or not this happens will depend on my timing If I manage this, it allows all of us to recover from the first half hour… Suggest introduce yourself to a neighbour if you don’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…

23. 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

24. Empirical Growth Curves i.e. results of experiments Lots of examples in Fig 1.2; I will concentrate on 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?

25. Population of the USA Population x10 6 Year

26. Nutrient uptake by Escherichia coli Nutrient uptake (  mol g -1 ) Minutes

27. Population dynamics of yeast Quantity of yeast Hours

28. Density of pathogenic fungus Population (g -1 soil) Days

29. Fitted Growth Curves Examples in Fig 1.3 in handout Attempt to summarise the data by drawing a curve 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]

30. Population x10 6 Year Population of the USA

31. Nutrient uptake (  mol g -1 ) Minutes Nutrient uptake by Escherichia coli

32. Quantity of yeast Hours Population dynamics of yeast

33. Population (g -1 soil) Days Density of pathogenic fungus

34. Rate 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

35. Population x10 6 Year Rate Population of the USA

36. Nutrient uptake (  mol g -1 ) Minutes Rate Nutrient uptake by Escherichia coli

37. Quantity of yeast Hours Rate Hours Population dynamics of yeast

38. Population (g -1 soil) Days Rate Days Density of pathogenic fungus

39. Summary 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 responses using mathematical techniques (differential equations, really)

40. Some final words, for today at least Remember, practical this afternoon in Titan rooms A possible route to Chemistry Dept. (in case you need it)

41. 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

42. …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 Infection risk by 2010

43. Even helps me with less “academic” pursuits 10 th July 2008…one of history’s darkest days…

44. Even helps me with less “academic” pursuits