COMPUTER MODELS IN BIOLOGY

1. COMPUTER MODELS IN BIOLOGY Bernie Roitberg and Greg Baker

2. WHERE NUMERICAL SOLUTIONS ARE USEFUL • Problems without directsolutions

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 directsolutions

8. THE EULER EXACT r EQUATION

9. HOW TO SOLVE THE EULER • Start with lnR0/G ≈ r

10. HOW TO SOLVE THE EULER • Start with lnR0/G ≈ r • Insert ESTIMATE into the Euler equation. This will yield an underestimate or overestimate

11. HOW TO SOLVE THE EULER • Start with lnR0/G ≈ r • Inserted ESTIMATE into the Euler equation. This will yield an underestimate or overestimate • Try successive values that approximate lnR0/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 Dy/Dx approximates the derivative dy/dx i.e. dy/dx = Dx 0 Dy/Dx • Thus, if dy/dx = f(y) then Dy/Dx≈ f(y) for small changes in x • Therefore Dy ≈ f(y) Dx

15. THE GENERAL RULE • For all numerical integration techniques: y(x + Dx) = yx + D y

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

17. NUMERICAL EXAMPLE • N 0+Dt = N0 + (N0 r DT) 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. • …... • N1.0 = 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 N Dt

21. RUNGE-KUTTA • ∆yt = f(yt) ∆ t • yt+ ∆ t = yt + ∆ yt • ∆ y t+ ∆ t = f(yt+ ∆ t ) • y t+ ∆ t = yt + ((∆yt + ∆ 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

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?