1. COMPUTER MODELS IN BIOLOGY Bernie Roitberg and Greg Baker

2. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions

3. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations

4. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes

5. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes Individual-based problems

6. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes Individual-based problems Stochastic problems

7. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions

8. THE EULER EXACT r EQUATION

9. HOW TO SOLVE THE EULER Start with lnR 0 /G ≈ r

10. HOW TO SOLVE THE EULER Start with lnR 0 /G ≈ r Insert ESTIMATE into the Euler equation. This will yield an underestimate or overestimate

11. HOW TO SOLVE THE EULER Start with lnR 0 /G ≈ r Inserted ESTIMATE into the Euler equation. This will yield an underestimate or overestimate Try successive values that approximate lnR 0 /G until exact value is discovered

12. SOME GUESSES

13. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations

14. THE CONCEPT For small changes in x (e.g. time) the difference quotient  y/  x approximates the derivative dy/dx i.e. dy/dx =  x  0  y/  x Thus, if dy/dx = f(y) then  y/  x≈ f(y) for small changes in x Therefore  y ≈ f(y)  x

15. THE GENERAL RULE For all numerical integration techniques: y(x +  x) = y x +  y

16. EULER SOLVES THE EXPONENTIAL dn/dt = rN  N/  t ≈ rN  N ≈ rN  t N (t+  t) = N t +  N Repeat until total time is reached.

17. NUMERICAL EXAMPLE N 0+  t = N 0 + (N 0 r  T) t = 0.1 N.1 = 100 + (100 * 1.099 * 0.1) = 110.99 N.2 = 110.99 + (110.99 * 1.099 * 0.1) =123.19 N.3 = 123.19 + (123.19 * 1.099 * 0.1) =136.73. …... N 1.0 = 283.69 Analytical solution = 300.11

18. COMPARE EULER AND ANALYTICAL SOLUTION

19. INSIGHTS The bigger the time step the greater is the error Errors are cumulative Reducing time step size to reduce error can be very expensive

20. RUNGE-KUTTA tt N

21. ∆y t = f(y t ) ∆ t y t+ ∆ t = y t + ∆ y t ∆ y t+ ∆ t = f(y t+ ∆ t ) y t+ ∆ t = y t + ((∆y t + ∆ y t+ ∆ t )/2)

22. COMPARE EULER AND RUNGE-KUTTA

23. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes

24. COMPLEX FITNESS LANDSCAPES Employing backwards induction to solve the optimal when state dependent Numerical solutions for even more complex surfaces –Random search –Constrained random search (GA’s)

25. TABLE OF SOLUTIONS Oxygen Energy 0.10.20.30.4 0.1 A A A R R 0.2 A R R R D 0.3 R R D D D 0.4 R R D D D 0.5 D D D D D

26. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes Individual-based problems

27. INDIVIDUAL BASED PROBLEMS Simulate a population of individuals that “know” the theory but may differ according to state

28. WHERE NUMERICAL SOLUTIONS ARE USEFUL Problems without direct solutions Complex differential equations Complex fitness landscapes Individual-based problems Stochastic problems

29. STOCHASTIC PROBLEMS Two issues: –Generating a probability distribution –Drawing from a distribution

30. FINAL PROBLEM What do you do with all those data?