COMPUTER MODELS IN BIOLOGY
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Transcript of 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
1= e- rx
x=0
3
! lxmx
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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
0
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Guess r
Value
0
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Value0
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Value
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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 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
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THE GENERAL RULE
• For all numerical integration techniques:
y(x + x) = yx + y
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EULER SOLVES THE EXPONENTIAL
dn/dt = rNN/t ≈ rN
N ≈ rN t
N(t+t) = Nt + NRepeat until total time is reached.
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NUMERICAL EXAMPLE
• N 0+t = N0 + (N0 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.• …...• N1.0 = 283.69• Analytical solution = 300.11
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COMPARE EULER AND ANALYTICAL SOLUTION
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T
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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
t
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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
0
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t
N
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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 (GA’s)
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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?