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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?
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