EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due March 6, 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.
HOMEWOR
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
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
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
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
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
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, mbeer@jhu.edu
TAs: Taeyoung Hwang taeyoungh@gmail.com, Robert Yaffe robertbyaffe@gmail.com
Models and Simulations
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 2, 2014
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.
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
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.
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due April 3, 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.
HOMEWOR
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 27, 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.
HOM
EN580.223 BIOLOGICAL MODELS AND SIMULATION Midterm Exam
The Johns Hopkins University
Name (print 1: K?
Section:
Signature:
I affirm that the work I have turned in is my own and was not copied om any other source.
M'uitermam.
You have Minutes to complete
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 20, 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.
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
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
EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 13, 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.
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