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  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

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)

What is this class about? What is a dynamical system?

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

Dynamical systems - 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 -Typically expressed as differential equation(s) (or discrete difference equation but we will focus on ODEs) Where x is going in future time Is determined by where x is now dynamic

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.

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

Oscillations / Rhythms Occur in Nature At 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

Circadian rhythms Are there biological clocks?

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?

 Humans w/o Input or External Cues 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 nocturnal (active at night) constant darkness  ~23 hour clock period (<24 hours) wake up a little earlier each day constant light lengthens the period Internal Clock Day time of day

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

The Circadian Clock  1. Period of ~24 hours  2. synchronized by the environment  3. temperature independent  4. self-sustained (--- therefore inherent) Defined By:

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

Linear Terms: one that is first degree in its dependent variables and derivatives x is 1 st degree and therefore a linear term xt is 1 st degree in x and therefore a linear term x 2 is 2 nd 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 x 2, e x, x(x+1) -1 all nonlinear terms sin x nonlinear term what is nonlinear? mathematical definition

other examples where x & y are the dependent variables and time is the independent variable  Linear  Nonlinear 2 nd order not 2 nd degree

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 + x 2 = 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 linear system: system of linear equations nonlinear system: system of equations containing at least 1 nonlinear term system 1 system 2 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?  nonlinear implies interactions!! the impact of x 1 is always the same the impact of x 1 on y depends on the value of x 2 there is an interaction between x 1 and x 2

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.

interactions! genes do not act independently! organisms eat other organisms cells send signals to their neighbors

Interactions! Biological interactions are at all scales : from molecules to the planet 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.

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!!

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

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 --> infinity time ---->

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

biological dynamics can be complex  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 time ---> can even have aperiodic behavior that goes on forever but never repeats! we have necessary oscillations in our dodies that we WANT to be stable

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

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?

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

Major Concepts and Implications  Complicated dynamic behavior can arise from simple equations and therefore simple models and interactions.

dimensionality  one dimensional systems stable constant ss, unstable constant ss, or blow up to infinity  two dimensional systems can also oscillate  higher dimensions - chaotic systems in this course we will try to consider a new framework for approaching biological systems Systematically step through increasing complex systems: * we will better define dimensionality in next lecture