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COMPUTER MODELS IN BIOLOGY

COMPUTER MODELS IN BIOLOGY. Bernie Roitberg and Greg Baker. WHERE NUMERICAL SOLUTIONS ARE USEFUL. Problems without direct solutions. WHERE NUMERICAL SOLUTIONS ARE USEFUL. Problems without direct solutions Complex differential equations. WHERE NUMERICAL SOLUTIONS ARE USEFUL.

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COMPUTER MODELS IN BIOLOGY

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  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?

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