Nonlinear Dynamics and Chaos in Biological Systems

1
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
2
today

• goals of the class
• approach recommendations
• syllabus
• what is special about and why study nonlinear
  systems
• why is nonlinear dynamics important for
  biological systems?
• math as a framework for biology
• this weeks plan

3
what is the goal of this class?

• convince you that dynamics are important
• change your conceptual framework about how you
  think about biological systems
• Arm you will tools for describing, analyzing, and
  investigating dynamical models
• Arm you with tools for approaching complex
  behavior
• Demonstrate analogies between biological system
  and other systems (learning from other
  disciplines)

4
What is this class about?What is a dynamical
system?
5
scope of the class

• What we will discuss is a subset of a broader
  field, dynamics
• Dynamics is the study of systems that evolve with
  time.
• The same framework of dynamics can be applied to
  biological, chemical, electrical, mechanical
  systems.
• We will focus on Biological Systems.
• All scales from gene expression to populations

6
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 ? lt t
• - they are therefore inherently deterministic
  state is determined by the earlier state

Where x is going in future time
Is determined by where x is now
dynamic
7
approach and recommendations

• will use PowerPoint (concepts, images) and
  chalkboard (math)
• follow the text topics closely READ!!!. I
  will supplement with outside examples.
• will supplement the text for biological context
• take notes
• I will provide summary of notes when available,
  but do not depend upon these
• do the homework!!
• Mathematica and Matlab will be required, you must
  use them.

8
go through syllabus
9
Why Do We Want to Quantitatively Model Living
Systems?
Why do we care about dynamics for biological
systems?
10
What are some examples of biological dynamic
systems?
11
Cardiac Rhythms

• normal versus pathological

12
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

13
Oscillations / Rhythms Occur in NatureAt all
time scales

• Predator Prey Population Cycles (years)
• Circadian Rhythms (24 hours)
• sleep wake cycles
• Biochemical Oscillations (1 20 min)
• metabolites oscillate
• Cardiac Rhythms (1 s)
• Neuronal Oscillations (ms s)
• Hormonal Oscillations (10 min - 24 hour)
• Communication in Animal and Cell Populations
• fireflies can synchronize their flashing
• bacteria can synchronize in a population

14
Circadian rhythms
Are there biological clocks?
15
Biological Clocks

• Why do you think that you sleep at night and are
  awake during the day?
• External or Internal Cues?
• What do you think would happen if you were in a
  cave, in complete darkness?
• 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?
• i.e. 1 wake sleep 1 day 24 hours?

16
Internal Clock

• Humans w/o Input or External Cues
• diurnal (active at day)
• constant darkness ? 25 hour clock (gt24 hours)
• wake up about 1 hour later each day
• constant light shortens the period
• Rodents
• nocturnal (active at night)
• constant darkness ?23 hour clock period (lt24
  hours)
• wake up a little earlier each day
• constant light lengthens the period

time of day
Day
17
Circadian Rhythms Everywhere!

• actually more rare for a biological factor to not
  change through-out the 24 hour day
• temperature
• cognition
• learning
• memory
• motor performance
• perception

all cycle through-out the day
18
The Circadian Clock
Defined By

• 1. Period of 24 hours
• 2. synchronized by the environment
• 3. temperature independent
• 4. self-sustained (--- therefore inherent)

19
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
20
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(x1)-1 all nonlinear terms sin x nonlinear
term
21
other examples where x y are the dependent
variables and time is the independent variable

• Linear

• Nonlinear

2nd order not 2nd degree
22
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???
23
linear and nonlinear systems
system 2
system 1
linear system system of linear equations
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 cant use for nonlinear
systems!!
24
what is nonlinear conceptually?

• nonlinear implies interactions!!

the impact of x1 is always the same
the impact of x1 on y depends on the value of x2
there is an interaction between x1 and x2
25
biology is nonlinear why?

• Why are biological systems almost always
  nonlinear?
• INTERACTIONS!
• The entities of biological systems (organisms,
  cells, proteins) NEVER exist or function in
  isolation.
• Life By Definition is Interactive!!
• Reproduction is the key to life and by definition
  requires or results in 2.

26
interactions!
genes do not act independently!
organisms eat other organisms
cells send signals to their neighbors
27
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
28
biochemical reactions

• biochemical reactions are rarely spontaneous
  single molecule reactions
• many are facilitated by enzymes

usually 2 or more molecules coming together to
form complexes INTERACTIONS!!
29
biological dynamics

• Why are dynamics important to biological systems?
• Temporal behavior of proteins, cells, organisms
• metabolism, cell growth, development, protein
  production, aging, death, species evolution all
  are time dependant processes
• Inherent complexity in biological systems both in
  time and space
• temporal patterns are related to structural ones
  --- we will look at the inherent structure of
  the models and equations

30
traditional biological framework

• think in terms of equilibrium processes
• We often (without realizing it) assume a
  biological system has a stable and constant
  steady state as time --gt infinity

time ----gt
time ----gt
31
this framework is not explicit .. It is engrained
in our thinking based on how we are taught

• DANGEROUS!
• it is dangerous to have a subconscious framework!
  
• influences thinking without anyone realizing it
• We making assumptions about the system and its
  dynamics without explicitly stating them
• can lead to faulty interpretation of data

32
biological dynamics can be complex

• as time goes to infinity response doesnt 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 ---gt
can even have aperiodic behavior that goes on
forever but never repeats!
33
If we cannot solve them what can we do??

• 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
BUT you can can answer questions and make limited
predictions about the system
34
What can we ask?

• Does a system have a threshold? How can we
  predict that threshold? Can we change it? What
  does it depend upon?
• 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?
• Can my system oscillate? Under what conditions?
• Given an initial state, what will happen to the
  trajectory of my system? What about for all
  possible initial conditions?
• 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?

35
Major concepts and implications?

• Deterministic does not mean practically
  predictable!
• 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
Lorenz 1963
2004
36
Major Concepts and Implications

• Complicated dynamic behavior can arise from
  simple equations and therefore simple models and
  interactions.

37
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
• two dimensional systems
• can also oscillate
• higher dimensions - chaotic systems

we will better define dimensionality in next
lecture