**Lecturer: **Gheorghe Craciun

E-mail: craciun *at* math *dot*
wisc *dot* edu

Classroom: B223 Van Vleck

Class time: Tue Thu
1:00 to 2:15pm

Office hours: Tuesdays 2:15pm,
via zoom.

**Course Content: **This course will provide a rigorous
foundation for mathematical modeling of biological systems. Mathematical
techniques include differential equations/dynamical systems. We emphasize
applications to biochemical pathways, chemical reaction systems, and population
dynamics (including mathematical models for understanding the spread of
infections through a population).

**Prerequisites:** Linear algebra (Math 340 or equivalent)
and Nonlinear Dynamical Systems (Math 415 or equivalent).

**Recommended
textbooks, lecture notes and papers**

**Textbooks:**

1. Martin Feinberg’s textbook “Foundations of Chemical Reaction Network Theory”. You can access a PDF of this book via the UW-Madison Library.

2. Steven Strogatz, *“Nonlinear Dynamics And
Chaos: With Applications To Physics, Biology, Chemistry, And Engineering”*.

3. Uri Alon, *“An
introduction to systems biology: design principles of biological circuits”,
second edition*.

**Lecture notes:****
**1. Martin Feinberg’s

2. Jeremy
Gunawardena’s *lecture notes* are here.**
**3. Eduardo Sontag’s

**Papers:**

1. You can find many
important examples of biochemical systems in the paper *“Sniffers, buzzers,
toggles and blinkers: dynamics of regulatory and signaling pathways in the
cell”*, by John Tyson, Katherine Chen, and Bela Novak here.

2. For the use of
the LaSalle invariance principle, see Eduardo Sontag’s paper *“Structure and
Stability of Certain Chemical Networks and Applications to the Kinetic
Proofreading Model of T-Cell Receptor Signal Transduction”* here.

**Grading:**

Midterm Exam: 30%

Final Exam: 30%

Homework and class
participation: 40%* *

*Additional credit for work
on research projects**. **SUGGESTIONS FOR PROJECT TOPICS*

*Other
opportunities for undergraduate research are described **here.*

*Consider also the **Math Dept Directed
Reading Program.*

**Software:** For simple numerical simulations you
can use the software "pplane". For
more complex models you can use MATLAB or XPP (you can download XPP, look at
examples, and read the XPP tutorial here.
Note that with XPP you can solve systems with any number of variables, and you
can even create animations).

**MATLAB Help**

Several *MATLAB
tutorials* are available here.

A free online textbook “Numerical Computing with MATLAB” by Cleve Moler is available here (see especially Chapter 7 “Ordinary Differential Equations”).

** **

** **

** **

**Homework
assignments:**

**Reading Assignment
#1** (due **Tuesday Feb 8, just** before class.)

1. Read the **Lecture
Notes** (see file “LECTURE_NOTES_WEEK_1.pdf” that you have received by **email**).
If you don’t have this file send me an email to
request it.

2. Watch the **youtube****
videos** at:

a. E-graphs vs. Chemical Reaction Networks — https://www.youtube.com/watch?v=nz2uuGCCzAc

b. General formula for polynomial dynamical systems generated by an E-graph G — https://www.youtube.com/watch?v=6pfFfeoLFow

c. Embedding reversible systems into Toric Differential Inclusions — https://www.youtube.com/watch?v=ps9iov8OtVE

** **

3. Prepare and
bring with you **2 questions** about things you have read in
these Lecture notes or have seen in these youtube
videos.

**Homework#1** (due **Thursday Feb 10, just** before class.)

** **

Solve and hand in
solutions to the following 3 questions (if you want, *you could also submit
your solution by email*):

1. Consider a (one-dimensional) dynamical
system on the real line. Can this system have periodic solutions? Explain your
answer.

*Hint:** you can use the “uniqueness” part of the Theorem on
existence and uniqueness of solutions of dynamical systems.** *

2. Give an example
of a two-dimensional dynamical system that has a limit cycle.

*Hint:** one way to find an example is to start in polar
coordinates and write differential equations in “r” and “theta” that give rise
to a limit cycle. Then you can get it into Cartesian coordinates by using a
change of variables.*

3. Give an example of a mass-action system that has one or more periodic solutions.

*Hint:** one way to find an example is to start with a
two-dimensional predator-prey model for which all trajectories are periodic and
then rewrite it as a mass-action system.*

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**Reading Assignment
#2** (due **Tuesday Feb 15, just** before class.)

1. Read the tutorial paper available at:

*https://people.math.wisc.edu/~craciun/PAPERS_NEW/Yu_Craciun_2018-Israel_Journal_of_Chemistry.pdf*

* *

2. Prepare and
bring with you **2 questions** about things you have read in
this paper, or about anything that we discussed during the previous weeks.

* *

**Homework#2** (due **Thursday Feb 17, just** before class.)

** **

Solve and hand in solutions
to the following 5 questions (if you want, *you could also submit your
solution by email*):

1. Is it possible that a *detailed
balanced* system is *not* *complex balanced*? Explain your answer (and
give an example if your answer is “yes”).

*Hint:** For this problem you want to argue that **any detailed balanced equilibrium is actually
also a complex balanced equilibrium **(and therefore any detailed balanced system must also
be a complex balanced system). [See also some examples of complex balanced
and detailed balanced systems on piazza.]*

2. Is it possible
that a *complex balanced* system is *not* *detailed balanced*?
Explain your answer (and give an example if your answer is “yes”).

*Hint:** For this problem you want to actually
find a complex balanced system that is **not** detailed balanced. I suggest you
could look for an example that uses a **weakly
reversible network which is *not* reversible **(because if it is not reversible then it cannot be
give rise to a detailed balanced system)*. * [See also some examples of complex balanced and
detailed balanced systems on piazza.]*

3. Is it possible
that a *complex balanced* system is *not* *weakly reversible*?
Explain your answer (and give an example if your answer is “yes”).* *

*Hint:** For this problem, have a look at the paragraph below
equation (7) on page 3 of the tutorial paper** **https://people.math.wisc.edu/~craciun/PAPERS_NEW/Yu_Craciun_2018-Israel_Journal_of_Chemistry.pdf*

4. Is it possible
that a *weakly reversible* system is *not* *complex balanced*?
Explain your answer (and give an example if your answer is “yes”).* *

*Hint:** For this problem you want to actually
find a weakly reversible system that is **not**
complex balanced. I suggest you could look for an example that uses a **weakly reversible network which has deficiency > 0.*

* *

5. Is it possible
that a *weakly reversible* ** linear** system is *not*

*Hint:** For this problem you want to show that any weakly
reversible linear system must have deficiency = 0, and then use the Deficiency
Zero Theorem (i.e., Theorem 2.8 in the tutorial paper).** *

* *

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** Reading Assignment #3** (due

1. Read the first 3 chapters from Martin Feinberg’s
textbook *“Foundations of Chemical Reaction Network Theory”**.
(You can skip sections 3.6, 3.7, and Appendix 3A for now.)*

* *

2. Prepare and **upload to piazza** **2 questions** about things you have read in
these chapters, or about anything that we discussed during the previous weeks.

* *

**Homework#3** (due **Thursday Feb 24, just** before class.)

1. Read all the
questions asked by other students this week on piazza, and
**write answers for at least 2 of them** (also on piazza). *Try to choose
questions that do not already have other complete answers, written by someone
else.** *

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** Reading Assignment #4** (due

1. Read chapters 4 and 5 from Martin Feinberg’s
textbook *“Foundations of Chemical Reaction Network Theory”**.
(Please pay especial attention to the **examples** in Chapter 5.)*

* *

2. Prepare and **upload to piazza** **2 questions** about things you have read in
these chapters, or about anything that we discussed during the previous weeks.

3. Have a look at
the new piazza post about **“Teams & suggested topics for projects and
presentations”. **As
we discussed in class, the team project and presentation replaces the midterm
exam.

**Homework#4** (due **Thursday ****March 3, just** before class.)

1. Read all the questions
asked by other students this week (or last week) on piazza,
and **write answers for at least 2 of them** (also on piazza). *Try
to choose questions that do not already have other complete answers, written by
someone else.** **Also, please make sure that you **do not erase the answer given by another student** when you post your own answer (for
example, you could make a copy of this other answer before you start writing
your answer).** *

* *

* **Reading Assignment and Homework #5** *(due **Friday ****March 11**)

* *

** **1. Read chapter 6 from Martin Feinberg’s textbook

2.
Prepare and **upload to piazza** **2 questions** about things you have read in this chapter, or about anything that we
discussed during the previous weeks.

3. Read all the questions asked by
other students this week (or last week) on piazza, and
**write answers for at least 2 of them** (also on piazza). *Try to choose
questions that do not already have other complete answers, written by someone
else.** **Also, please make sure that you **do not erase the answer given by another student** when you post your own answer.*

* *

* *

** Reading Assignment #6** (due

1. Read chapter 7 from Martin Feinberg’s textbook *“Foundations
of Chemical Reaction Network Theory”**. (Please pay especial attention
to all the **examples**
in this chapter.)*

*Homework #6** *(due **Friday ****March 25, just** before class.)

Solve and hand in
solutions to the following 3 questions (if you want, *you could also submit
your solution by email*):

** **1. Is it
possible that a weakly reversible reaction network has 3 nodes (i.e.,
complexes) and the dimension of its stoichiometric subspace is = 3? Explain your answer
(and give an example if your answer is “yes”).

2. Is it possible that a reaction network has a single linkage class and has deficiency = 2? Explain your answer (and give an example if your answer is “yes”).

3. Is it possible that a weakly reversible reaction network has a single linkage class, has 4 nodes (i.e., complexes) that are all contained in a 2-dimensional plane, and has deficiency = 0? Explain your answer (and give an example if your answer is “yes”).

*Hint: for all
these problems you can focus on reaction networks with just 2 species.*

* *

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** Reading Assignment #7** (due

1. Continue reading chapter 7 from Martin
Feinberg’s textbook *“Foundations of Chemical Reaction Network Theory”**.
(Please pay especial attention to all the **examples** in this chapter.)*

Also, start reading Chapter 13, where some of the notions in chapter 7 are discussed in more detail.

*Homework #7** *(due **Monday ****April 4**)

Solve and hand in
solutions to the following questions (if you want, *you could also submit
your solution by email*):

** **1. Find
a Lyapunov function for a

2.
Find a Lyapunov function for a *dynamical system that has more than one
periodic orbit*. (Give an example of such system and such a
Lyapunov function, and explain your answer.)

3. Is it
possible that a complex balanced system has a Lyapunov function that is *not convex*? Explain your answer (and give an example if your answer is “yes”).

*Hint: for all
these problems you can focus on reaction networks with just 1 or 2 species.*

* *

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**FINAL EXAM CREDIT:** prepare and hand-in a ** class
project** as suggested here:

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*No homework is
due during the week of April 11-15.*__ __* Instead, please try to spend
some time this week to select a class project as described above (to be done
individually or in a small group, due May 6). **This class project can take the place of your final exam.** If you have any questions about
the class project please send me an email. *

*Homework #8** *(due **Thursday ****April 28**)

Solve and hand in
solutions to the following questions (if you want, *you could also submit
your solution by email*):

** **1.

2. *(This is
also a follow-up of a problem from HW#7)* Note that if *V(x)* is a Lyapunov function for
some dynamical system, and *F(x)* is a strictly monotone function, then *F(V(x))
*is also a Lyapunov function for that dynamical system. Use this remark to give an example of an 1D complex balanced system
has a Lyapunov function
that is *not convex*. *(Hint:
the 1D complex balanced system could be, for example, generated by the reaction
network X <-> 2X.)*

3. Find two different Lyapunov functions for a
dynamical system generated by the linear reaction network of the form *X1
-> X2 ->X3 -> X1*, for some choice of parameter values *k1, k2,
k3* (for example, you can take all of them to be 1).

**ALTERNATIVE FINAL
EXAM CREDIT:**
prepare and hand-solutions to 10 of the following 12 problems (due May 6; you don’t
have to do this assignment if you submit a class project for the final exam):

1. Explain why a reaction
network that has deficiency=0 *cannot* give rise
to periodic solutions. (Consider two cases: the case where the network is weakly
reversible, and the case where it is not weakly reversible.)

2. Give an example of
a reaction network that has deficiency=1 and can give rise to periodic
solutions for some choices of reaction rate parameters. Explain your answer and
include numerical simulation results that confirm your claim (obtained using “pplane” or any other software).

3. Give an example of
a reaction network that has conservation of mass and can give rise to periodic
solutions for some choices of reaction rate parameters. Explain your answer and
include numerical simulation results that confirm your claim (obtained using “pplane” or any other software).

4. Explain why a reaction
network that has deficiency=0 *cannot* give rise
to multiple positive equilibria that are stoichiometrically compatible. (Consider
two cases: the case where the network is weakly reversible, and the case where
it is not weakly reversible.)

5. Give an example of
a reaction network that has deficiency=1 and can give rise to multiple positive
equilibria that are stoichiometrically compatible. Explain your answer and
include numerical simulation results that confirm your claim (obtained using “pplane” or any other software).

6. Give an example of
a reaction network that has conservation of mass and can give rise to multiple
positive equilibria that are stoichiometrically compatible. Explain your answer
and include numerical simulation results that confirm your claim (obtained using
“pplane” or any other software).

7. Give an example of
a reaction system that is not complex balanced but has unique globally
attracting equilibrium.

8. Consider a 2D mass-action
system that has a limit cycle. Is it possible for this system to have *no*
positive equilibria? Explain your answer.

9. Consider a 2D mass-action
system that has a limit cycle. Is it possible for this system to have *two
different* positive equilibria? Explain your answer.

10. Construct a 2D
mass-action system that has a two limit cycles. Explain your answer and include
numerical simulation results that confirm your claim (obtained using “pplane” or any other software).

11. Construct a 2D
mass-action system that has a single positive equilibrium, and that equilibrium
is *repelling*. Explain your answer and include numerical simulation results
that confirm your claim (obtained using “pplane” or
any other software).

12. Construct a 2D
mass-action system that has a *repelling* limit cycle. Explain your answer
and include numerical simulation results that confirm your claim (obtained using
“pplane” or any other software).

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