Transcript Nonlinear Dynamics and Chaos in Biological Systems ABE 591W, BME595U, IDE 495C Prof Jenna Rickus Dept.

Nonlinear Dynamics and Chaos in Biological Systems
ABE 591W, BME595U, IDE 495C
Prof Jenna Rickus
Dept. of Agricultural and Biological Engineering
Dept. of Biomedical Engineering
today
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goals of the class
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approach & recommendations
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syllabus
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what is special about and why study nonlinear systems
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why is nonlinear dynamics important for biological
systems?
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math as a framework for biology
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this weeks plan
what is the goal of this class?
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convince you that dynamics are important
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change your conceptual framework about how you think
about biological systems
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Arm you will tools for describing, analyzing, and
investigating dynamical models
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Arm you with tools for approaching complex behavior
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Demonstrate analogies between biological system and
other systems (learning from other disciplines)
What is this class about?
What is a dynamical system?
scope of the class
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What we will discuss is a subset of a broader field,
dynamics
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Dynamics is the study of systems that evolve with time.
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The same framework of dynamics can be applied to
biological, chemical, electrical, mechanical systems.
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We will focus on Biological Systems.
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All scales from gene expression to populations
Dynamical systems
-Typically expressed as differential equation(s)
(or discrete difference equation but we will focus on ODEs)
- systems that evolve with time and they have a “memory”
- state at time t depends upon the state at a slightly earlier time
where < t
- they are therefore inherently deterministic: state is determined by
the earlier state
Where x is going in future time
dx
 f (x)
dt
Is determined by where x is now
dynamic
approach and recommendations
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will use PowerPoint (concepts, images) and chalkboard (math)
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follow the text topics closely … READ!!!. I will supplement
with outside examples.
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will supplement the text for biological context
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take notes
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I will provide summary of notes when available, but do not
depend upon these
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do the homework!!
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Mathematica and Matlab will be required, you must use them.
go through syllabus
Why Do We Want to
Quantitatively Model Living
Systems?
Why do we care about dynamics for
biological systems?
What are some examples
of biological dynamic
systems?
Cardiac Rhythms
normal versus pathological
Ventricular fibrillation
Chaotic state of the voltage propagation in the heart
Ventricles pump in uncoordinated and irregular ways
Ventricle ejection fraction (blood volume that they pump) drops to
almost 0%
Leading cause of sudden cardiac death
Link to movie
http://www.pnas.org/cgi/content/full/090492697/DC1
Oscillations / Rhythms Occur in Nature
At all time scales
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Predator Prey Population Cycles (years)
Circadian Rhythms (24 hours)
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Biochemical Oscillations (1 – 20 min)
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sleep wake cycles
metabolites oscillate
Cardiac Rhythms (1 s)
Neuronal Oscillations (ms – s)
Hormonal Oscillations (10 min - 24 hour)
Communication in Animal and Cell Populations
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fireflies can synchronize their flashing
bacteria can synchronize in a population
Circadian rhythms
Are there biological clocks?
Biological Clocks
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Why do you think that you sleep at night and are awake
during the day?
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External or Internal Cues?
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What do you think would happen if you were in a cave, in
complete darkness?
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If you were in a cave .. Could you track the days you were
there by the number of times you fell asleep and woke up?
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i.e. 1 wake – sleep = 1 day = 24 hours?
Internal Clock
Humans w/o Input or External Cues
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diurnal (active at day)
constant darkness  ~25 hour clock (>24 hours)
wake up about 1 hour later each day
constant light shortens the period
Rodents
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nocturnal (active at night)
constant darkness ~23 hour clock period (<24 hours)
wake up a little earlier each day
constant light lengthens the period
time of day
Day
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Circadian Rhythms Everywhere!
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actually more rare for a biological factor to
not change through-out the 24 hour day
temperature
 cognition
 learning
 memory
 motor performance
 perception
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all cycle through-out the day
The Circadian Clock
Defined By:
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1. Period of ~24 hours
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2. synchronized by the environment
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3. temperature independent
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4. self-sustained (--- therefore inherent)
remind ourselves of definitions
variable: attributes of the equation or system that change
independent variable: a variable is active in changing the behavior
of the equation or system; typically not affected by changes in other
variables
(our independent variable will always be time in this class)
dependent variable: a variable that changes due to changes in
other variables
parameter: constants that determine the behavior and character of
the equation or system; impact how variables change
what is nonlinear?
mathematical definition
Linear Terms: one that is first degree in its dependent variables and
derivatives
x is 1st degree and therefore a linear term
xt is 1st degree in x and therefore a linear term
x2 is 2nd degree in x and there not a linear term
Nonlinear Terms: any term that contains higher powers, products and
transcendentals of the dependent variable is nonlinear
x2, ex, x(x+1)-1
all nonlinear terms
sin x
nonlinear term
other examples where x & y are the dependent variables
and time is the independent variable
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Linear
dy
dt
2
d y
2
dt
 dy 
sin t  
 dt 
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2nd order
not
2nd degree
Nonlinear
 dy 
 
 dt 
2
 dy 
sin x  
 dt 
xy
x
3
linear /nonlinear equations
linear equation: consists of a sum of linear terms
y=x+2
y(t) + x (t) = N
dy/dt = x + sin t
nonlinear equation: all other equations
y + x2 = 2
x(t) * y(t) = N
dy / dt = xy + sin x
most nonlinear differential equations
are impossible to solve analytically!
So what do we do???
linear and nonlinear systems
system 1
dy1
 2 y1  y2
dt
dy2
 y1  3 y2
dt
linear system: system of
linear equations
system 2
dy1
 2 y1 y2  y2
dt
dy2
 y1 y2  3 y2
dt
nonlinear system: system of
equations containing at least 1
nonlinear term
we can use tools such as Laplace transformations to assist in
solving linear systems of differential equations
can’t use for nonlinear systems!!
what is nonlinear conceptually?
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nonlinear implies interactions!!
y  2x1  x2
the impact of x1 is always the
same
y  2x1 x2  x2
the impact of x1 on y depends on
the value of x2
there is an interaction between x1 and x2
biology is nonlinear … why?
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Why are biological systems almost always nonlinear?
INTERACTIONS!
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The entities of biological systems (organisms, cells,
proteins) NEVER exist or function in isolation.
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Life By Definition is Interactive!!
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Reproduction is the key to life and by definition requires or
results in 2.
interactions!
genes do not act independently!
organisms eat other organisms
cells send signals
to their neighbors
Interactions!
Global perspective:
actions of people impact the earth which in turn
impact the health of people
molecular interactions … are the key to life
A single protein is not alive. But a collection
of interacting proteins (and other molecules)
make up a cell that is alive.
Biological interactions are at all scales :
from molecules to the planet
biochemical reactions
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biochemical reactions are
rarely spontaneous single
molecule reactions
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many are facilitated by
enzymes
k
A 
B
dB
 kA
dt
k2
A  E 
 AE  B
k1
k 1
dB
 k2 * AE ;
dt
dAE
 k1 * A * E  k 1 * AE
dt
usually 2 or more molecules coming together to form complexes
INTERACTIONS!!
biological dynamics
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Why are dynamics important to biological systems?
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Temporal behavior of proteins, cells, organisms
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metabolism, cell growth, development, protein production,
aging, death, species evolution … all are time dependant
processes
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Inherent complexity in biological systems both in time and
space
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temporal patterns are related to structural ones --- we will
look at the inherent “structure” of the models and
equations
traditional biological framework
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think in terms of equilibrium processes
time ---->
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time ---->
We often (without realizing it) assume a biological
system has a stable and constant steady state as
time --> infinity
this framework is not explicit .. It is engrained in
our thinking based on how we are taught
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DANGEROUS!
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it is dangerous to have a subconscious framework!
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influences thinking without anyone realizing it
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We making assumptions about the system and its dynamics
without explicitly stating them
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can lead to faulty interpretation of data
biological dynamics can be complex
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as time goes to infinity … response doesn’t have to go to a
single constant value
could for example have oscillations that
would go on forever unless perturbed
we have necessary oscillations in our
dodies that we WANT to be stable
time --->
can even have aperiodic
behavior that goes on forever
but never repeats!
If we cannot “solve” them …what can we do??
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Many nonlinear equations are impossible to solve
analytically …. but they are still deterministic and we can
know their behavior through other methods
Poincaré and The three body problem: predicting the motion of the
sun, earth and moon
Turns out to be impossible to solve
Analytically … cannot write down
The equations for their trajectory
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
BUT you can can answer
questions and make limited
predictions about the system
What can we ask?
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Does a system have a threshold? How can we predict that
threshold? Can we change it? What does it depend upon?
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Is the system stable? If I perturb the system what will happen to
it? Will it return to the same steady state or go to a new one?
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Can my system oscillate? Under what conditions?
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Given an initial state, what will happen to the trajectory of my
system? What about for all possible initial conditions?
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Are their regions of qualitatively different behavior of my system?
And specifically how do my parameter values impact the transition
from one state to another?
Major concepts and implications?
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Deterministic does not mean practically predictable!
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Some systems are highly sensitive to initial conditions
Small errors in measurement of the system state at time t can quickly
amplify into large errors in the predicted state at t + t
We can never measure the current state with infinite precision
Led to the popular concept and
coining of the phrase of
“The Butterfly Effect”
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF(Uncompres sed) dec ompressor
are needed to see thi s pic ture.
Lorenz 1963
2004
Major Concepts and Implications
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Complicated dynamic behavior can arise from simple
equations and therefore simple models and interactions.
dimensionality
in this course we will try to consider a new framework for
approaching biological systems
Systematically step through increasing complex systems:
one dimensional systems
stable constant ss, unstable constant ss, or blow up to infinity
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two dimensional systems
can also oscillate
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higher dimensions - chaotic systems
* we will better define dimensionality in next lecture