COMPUTER MODELS IN BIOLOGY

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COMPUTER MODELS IN BIOLOGY

• Bernie Roitberg and Greg Baker

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
• Stochastic problems

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions

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THE EULER EXACT r EQUATION
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HOW TO SOLVE THE EULER

• Start with lnR0/G r

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HOW TO SOLVE THE EULER

• Start with lnR0/G r
• Insert ESTIMATE into the Euler equation. This
  will yield an underestimate or overestimate

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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

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SOME GUESSES
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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations

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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

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THE GENERAL RULE

• For all numerical integration techniques
• y(x Dx) yx D y

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EULER SOLVES THE EXPONENTIAL

• dn/dt rN
• DN/Dt rN
• DN rN Dt
• N(tDt) Nt DN
• Repeat until total time is reached.

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NUMERICAL EXAMPLE

• N 0Dt 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

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COMPARE EULER AND ANALYTICAL SOLUTION
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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

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RUNGE-KUTTA
N
Dt
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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)

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COMPARE EULER AND RUNGE-KUTTA
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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes

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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 (GAs)

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TABLE OF SOLUTIONS
Oxygen Energy 0.1 0.2 0.3 0.4 0.4
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
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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems

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INDIVIDUAL BASED PROBLEMS

• Simulate a population of individuals that know
  the theory but may differ according to state

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WHERE NUMERICAL SOLUTIONS ARE USEFUL

• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
• Stochastic problems

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STOCHASTIC PROBLEMS

• Two issues
• Generating a probability distribution
• Drawing from a distribution

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FINAL PROBLEM

• What do you do with all those data?