Mathematical Methods for Systems Biology – Math/BMC/BMI/Biochem 609

 


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 lecture notes are here.

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

 

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

 

 

Undergraduate students: please consider working on a research project and participating in the UW Undergraduate Symposium.  

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* complex balanced? Explain your answer (and give an example if your answer is “yes”). 

                        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 Tuesday Feb 22,  just before class.)

 

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 Tuesday March 1,  just before class.)

 

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 3just 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). 

 

 

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

 

 1. Read chapter 6 from Martin Feinberg’s textbook “Foundations of Chemical Reaction Network Theory”. (You can skip the section about “Independent Subnetworks”.)

 

 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.

 

 

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Reading Assignment #6 (due Friday March 25,  just before class.)

 

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 Monday April 4)

 

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 dynamical system that has more than one steady state.    (Give an example of such system and such a Lyapunov function, and explain your answer.)

 

 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:  https://people.math.wisc.edu/~craciun/public/html/Math_BMC_Biochem_609_Spring_2022_SUGGESTIONS_FOR_PROJECTS.html

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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. (This is a follow-up of a problem from HW#7) Consider the standard Lotka-Volterra predator-prey system in 2D, for which all trajectories are periodic. Find a Lyapunov function for this system.   (Hint: note that it is not possible to find a strict Lyapunov function, but it is possible to find a Lyapunov function which is constant along each periodic trajectory of the Lotka-Volterra system. For more concrete suggestions look at the Wikipedia page for Lotka-Volterra equations.)

 

 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).

 

 

 

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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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