Non Linear Dynamics in Quantitative Biology
A.Y. 2024/2025
Learning objectives
The last years have witnessed a transition of the biological sciences from a qualitative to a quantitative (hard) science. However, this transition can only take place if biologists become familiar with the mathematical instruments that are the basis for studying the systems quantitatively. In this course, we will discuss several approaches to the modelling of biological systems, giving particular attention to regulatory circuits (transcriptional or sRNA-dependent and their integration) and metabolic systems with a discussion of how these models can be exploited to rationalize the process of metabolic engineering.
To achieve this task, we will present a few fundamental concepts of the field to then show how complicate dynamical behaviours can originate from relatively simple circuits thanks to the non-linearity and high interconnectedness that is intrinsic of biological systems.
We will also show how it is possible to provide a detailed characterisation of these behaviours through mathematics tools. The course will integrate theory and practical lessons, the latter using a software developed to provide help to modellers, called Copasi, in addition to the generic platform R.
During the course we will also introduce related concepts to show how the structure of the network onto which a process takes place can have huge effects on the dynamics of a system, a concept which is particularly useful when studying epidemic spreading that is particularly important in these days.
To achieve this task, we will present a few fundamental concepts of the field to then show how complicate dynamical behaviours can originate from relatively simple circuits thanks to the non-linearity and high interconnectedness that is intrinsic of biological systems.
We will also show how it is possible to provide a detailed characterisation of these behaviours through mathematics tools. The course will integrate theory and practical lessons, the latter using a software developed to provide help to modellers, called Copasi, in addition to the generic platform R.
During the course we will also introduce related concepts to show how the structure of the network onto which a process takes place can have huge effects on the dynamics of a system, a concept which is particularly useful when studying epidemic spreading that is particularly important in these days.
Expected learning outcomes
After this course, the student will be able to:
- Use biological information about a cellular system or population to build a workable mathematical model;
- Use experimental data of different kinds to estimate the parameters of the models under analysis;
- Reasoning critically about the possible assumptions at the basis of each model;
-Be able to communicate their results by using a rigorous terminology, which is at the basis of providing a fully understandable message to the audience.
- Use biological information about a cellular system or population to build a workable mathematical model;
- Use experimental data of different kinds to estimate the parameters of the models under analysis;
- Reasoning critically about the possible assumptions at the basis of each model;
-Be able to communicate their results by using a rigorous terminology, which is at the basis of providing a fully understandable message to the audience.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
The course will be roughly organized in the following topics (duration with respect to the duration of the course)
1/10: Recap on previously encountered models (population models, outbreak) and linear algebra concepts mainly using the first 3 chapters of Non-linear dynamics and chaos by S. Strogatz.
2/10: Competition models, from simple competition to the quasi-species concept and the mutation matrix. This is based on Evolutionary dynamics: exploring the equations of life, by M. Nowak.
3/10: What's Metabolic Control Theory? The summation theorems, flux and concentration control coefficients, elasticities. (frontal + practice). This large part will be based on many published papers by authors Kacser, Burns, Small, Reder, Smallbone and others.
1/10 Periodic systems: playing with the cell cycle. (Group practical activity and student-to-student teaching).
3/10: Practice on all topics throughout the course.
1/10: Recap on previously encountered models (population models, outbreak) and linear algebra concepts mainly using the first 3 chapters of Non-linear dynamics and chaos by S. Strogatz.
2/10: Competition models, from simple competition to the quasi-species concept and the mutation matrix. This is based on Evolutionary dynamics: exploring the equations of life, by M. Nowak.
3/10: What's Metabolic Control Theory? The summation theorems, flux and concentration control coefficients, elasticities. (frontal + practice). This large part will be based on many published papers by authors Kacser, Burns, Small, Reder, Smallbone and others.
1/10 Periodic systems: playing with the cell cycle. (Group practical activity and student-to-student teaching).
3/10: Practice on all topics throughout the course.
Prerequisites for admission
none
Teaching methods
The teacher will tentatively exploit the following types of lesson, which, will depend on the students he is working with. In the syllabus I propose a plausible plan, which is however to be considered as a draft and dynamical proposal.
Frontal: these are classical lessons with the aid of slides (rarely) or the blackboard (often) depending on the needs. Students are stimulated to intervene by questions / small problem solving / hypothesis formulation.
Practice with theory: these are lessons with theory concepts that I will try to highlight by means of workable examples at the computer; as a simple example, how the steady state condition Nv=0 looks like? Is it true (as Metabolic Control Theory states) that the control over a certain reaction is shared by all other enzymes, in different proportions?
Practice: a practical lesson at the computer, generally following a theory part to stress specific concepts and provide technical details on how to access and perform some calculation. It may also be more data-analysis oriented, for instance to integrate theory and available data.
Group practical activity: students are split in groups and let to work on different examples. The teacher is available for technical help and suggestions.
Student-to-student teaching: when working in groups, students will work on different aspects of a model or on the same aspect on slightly different models, such that the students might encounter varying difficulties that should be solved by collective effort. Next, each group will introduce the work done to the students from other groups. A small report might be asked on another group's work.
Open discussion: The lesson is anticipated by some bibliographical suggestions to the students and therefore requires some study time ahead. This is a discussion where the teacher briefly introduces an idea and then lets the students openly discuss about it.
Frontal: these are classical lessons with the aid of slides (rarely) or the blackboard (often) depending on the needs. Students are stimulated to intervene by questions / small problem solving / hypothesis formulation.
Practice with theory: these are lessons with theory concepts that I will try to highlight by means of workable examples at the computer; as a simple example, how the steady state condition Nv=0 looks like? Is it true (as Metabolic Control Theory states) that the control over a certain reaction is shared by all other enzymes, in different proportions?
Practice: a practical lesson at the computer, generally following a theory part to stress specific concepts and provide technical details on how to access and perform some calculation. It may also be more data-analysis oriented, for instance to integrate theory and available data.
Group practical activity: students are split in groups and let to work on different examples. The teacher is available for technical help and suggestions.
Student-to-student teaching: when working in groups, students will work on different aspects of a model or on the same aspect on slightly different models, such that the students might encounter varying difficulties that should be solved by collective effort. Next, each group will introduce the work done to the students from other groups. A small report might be asked on another group's work.
Open discussion: The lesson is anticipated by some bibliographical suggestions to the students and therefore requires some study time ahead. This is a discussion where the teacher briefly introduces an idea and then lets the students openly discuss about it.
Teaching Resources
S. Strogatz, Non linear dynamics and chaos
M. Nowak Evolutionary dynamics: exploring the equations of life
Pertinent bibliography from Kacser, Burns, Small will be provided ahead of lessons.
M. Nowak Evolutionary dynamics: exploring the equations of life
Pertinent bibliography from Kacser, Burns, Small will be provided ahead of lessons.
Assessment methods and Criteria
There will be no examination at the end of the course and students will get a grade based on their 'performances' throughout the course (continuous assessment). I will evaluate both your performances in terms of both technical and theoretical level (knowledge and technical skills), your ability to work in group (collaborative skill), to communicate your knowledge to others (communicative skill) and last but not least, your ability to provide constructive criticisms to the work of other students (reviewer skill). To be able to do this, I need students to participate to a great majority of lessons, therefore another important aspect in determining the final grade is the number of missed lessons.
BIO/11 - MOLECULAR BIOLOGY - University credits: 1
BIO/18 - GENETICS - University credits: 1
BIO/19 - MICROBIOLOGY - University credits: 4
BIO/18 - GENETICS - University credits: 1
BIO/19 - MICROBIOLOGY - University credits: 4
Practicals: 32 hours
Lessons: 32 hours
Lessons: 32 hours
Professor:
Brilli Matteo
Professor(s)