EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due May 1, 2013
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HOMEWORK
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due March 6, 2013
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EN580.223 BIOLOGICAL MODELS AND SIMULATION Midterm Exam
The Johns Hopkins University
Name (print 1: K?
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M'uitermam.
You have Minutes to complete
HW6 Due Wed 4/2
1D Autonomous:
Special cases:
a)
b)
stable
Stable: f ( x* ) 0
Unstable:
f ( x* ) 0
f ( x) f ( x* ) f ( x* )x f ( x* )x 2 O(x3 )
x x3
c)
unstable
x f ( x, y )
y g ( x, y )
det(J I ) 0
x Ax
x f (x) FP: f ( x* ) 0
f ( x* ) 0
x x3
2D Auton.:
M
Exam I Wed 4/16
Nullclines to find fixed points
Poincar-Bendixson Theorem to Prove Existence of Closed Orbit / Limit Cycle
- if you can create trapping region with no fixed points and all trajectories going in
- all trajectories must approach a closed orb
Morris-Lecar model of muscle cells:
Muscle contains primarily gated Ca and K activation gates
The Morris Lecar model involves a fast (instantaneous) Ca current, a more slowly evolving K
current, and a leak current (other ions slipping through):
dV
Cm
gCa
HW 7 due 4/9
Exam I Wed 4/16
Previously: Stability of fixed points determines local properties of trajectories
Today:
Methods to describe global properties of trajectories
Last time: Nullclines to find fixed points
Poincar-Bendixson Theorem to Prove Exist
Iterated maps:
Exam pickup in section
HW 9 due Wed Apr 30
Weve mostly been considering 2D systems, as higher dim systems are less amenable to phase
plane techniques. In 3D and higher the concept of a map is very useful.
Consider the system:
x x y x( x 2 y
HW 7 due 4/9
Exam I Wed 4/16
Kermack-McKendrick model of viral infection, now with limited (logistic) population growth
x = uninfected population
y = infected population
nonlinear infection rate kxy
x grows at rate r(1-ax)
growth rate ->0 at x=1/a
y dies
mean: 72
sigma: 16
Bifurcation and Chaos in Iterated Maps:
HW 8 due Wed Apr 30
A Poincare Map is an iterated map that describes successive trajectory crossings of a lower
dimensional surface: x k 1 P(x k )
x k 2 P(x k 1 ) P( P(x k )
Poincare Map of Rossle
Review for Midterm Wed 4/16:
graphing calculators allowed
nth order NLDE system of n 1st order NLDEs
existence & uniqueness of solutions to NLDEs, n DEs n constants
solutions of Linear DEs are sums of exponentials, sum of any two solns is also a solution
Mathematical Model of Mutual Repression
A
Transcription rate of A
ta A
dA
rA
k aA
dt
k aA A
B
k aB
B
tb B
dB
rB
k
dt
kbB B bB A
kbA
A
B=0
B=1
A
B
t
There are two final states, high A and low B, or low A and high B
Numerical solution is great, but it
Department of Biomedical Engineering
BME 580.223 Biological Models and Simulations, Part II
Instructor: Mike Beer, 573 Miller Research Bldg, [email protected]
TAs: Taeyoung Hwang [email protected], Robert Yaffe [email protected]
Models and Simulations
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 2, 2014
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HOMEWOR
Poiseuilles Law Flow in a circular vessel
p1
L
p2
Q
This is additional information for Problem 2 in HW5. Consider flow of a viscous fluid
of viscosity in a circular vessel of radius R and length L under the pressure
difference p=p1-p2. Then the volumetric
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due Friday March 7, 2014
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Biological Models & Simulations Midterm Exam
NAME:
EN580.223 BIOLOGICAL MODELS AND SIMULATIONS
The Johns Hopkins University
March 24, 2010
Name (print; also put your name on top-right of every page):
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Use pen
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EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 3, 2013
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EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 27, 2013
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HOM
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 20, 2013
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I affirm that the work I have turned in is my own and was not copied from any other source.
HOM
-The cover sheet should be attached, otherwise the total score will be reduced
10%.
- In question 1, students who do not use cm as the unit, or express dimensions
by powers of 10, will reduce 0.5% in each individual answer. Any answer
within 25% of the so
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 10, 2013
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I affirm that the work I have turned in is my own and was not copied from any other source.
HOMEWO
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 13, 2013
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I affirm that the work I have turned in is my own and was not copied from any other source.
HOM
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 17, 2013
Name (print):
Members of Study Group (or write, none):
Signature:
I affirm that the work I have turned in is my own and was not copied from any other source.
HOMEWO