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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. - PowerPoint PPT Presentation
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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
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
• Stochastic problems
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
THE EULER EXACT r EQUATION
1= e- rx
x=0
3
! lxmx
HOW TO SOLVE THE EULER
• Start with lnR0/G ≈ r
HOW TO SOLVE THE EULER
• Start with lnR0/G ≈ r • Insert ESTIMATE into the Euler
equation. This will yield an underestimate or overestimate
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
SOME GUESSES
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 2
Guess r
Value
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 2
Guess r
Value0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 2
Guess r
Value
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
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
THE GENERAL RULE
• For all numerical integration techniques:
y(x + x) = yx + y
EULER SOLVES THE EXPONENTIAL
dn/dt = rNN/t ≈ rN
N ≈ rN t
N(t+t) = Nt + NRepeat until total time is reached.
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
COMPARE EULER AND ANALYTICAL SOLUTION
0
50
100
150
200
250
300
350
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
T
N
0
50
100
150
200
250
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
T
N
0
100
200
300
400
500
600
700
800
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
T
N
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
RUNGE-KUTTA
t
N
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)
COMPARE EULER AND RUNGE-KUTTA
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
t
N
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
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)
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
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
INDIVIDUAL BASED PROBLEMS
• Simulate a population of individuals that “know” the theory but may differ according to state
WHERE NUMERICAL SOLUTIONS ARE USEFUL
• Problems without direct solutions
• Complex differential equations
• Complex fitness landscapes
• Individual-based problems
• Stochastic problems
STOCHASTIC PROBLEMS
• Two issues:
– Generating a probability distribution
– Drawing from a distribution
FINAL PROBLEM
• What do you do with all those data?