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Multiscale Mathematical Biology

from individual cell behavior to biological growth and form

In the autumn semester of 2017, we will be teaching the fifth edition of the course "Multiscale Mathematical Biology" at the Mathematical Institute of Leiden University. The course introduces students to the mathematical and computational biology of multicellular phenomena, covering a range of biological examples, including development of animals and plants, blood vessel growth, bacterial pattern formation and diversification, tumor growth and evolution. The course is also part of the Minor "Quantitative Biology"; students therefore come from a mix of scientific backgrounds, ranging from biology to mathematics and physics. 

For more information, see the program below, or
write Roeland an email. The course consists of a series of lectures,  practical assignments using biological modeling environments, and a final project.



Lecturer: Roeland Merks + guest lecturers

Assistants: Claudiu Antonovici and Koen Schakenraad

English, unless all students speak Dutch

Location & time: University of Leiden, Snellius (Wiskunde), room 405, Wednesday afternoons, from September 6th-December 13th, 2017; Lectures, 13:45-15:30; computer exercises and mini-projects, 15:45-17:30, every other afternoon (more or less; see schedule for exact dates)

Methodology: lectures, paper seminar, practical exercises, mini-projects.

Required knowledge: Basic background in biology is useful (in particular cell biology or developmental biology) but not required. Some programming skills and familiarity with numerical algorithms and differential equations are useful, but not required.

Evaluation: 1) Practicum assignments, 30%; 2) Final project + literature presentation 30%; 3) Written exam, 40%

For students of: 3rd Bachelor and 1st Master’s Mathematics, Minor Quantitative Biology, and other interested students

Course description:

Biological systems are so complex, that biologists often need to call in the help from mathematicians and computational scientists. These questions constitute a rich source of applied mathematical problems, for which often a range of mathematical and computational techniques need to be combined with one another. Mathematical insight into dynamical systems, pattern formation, complex networks, multiscale dynamics and parallel processing turn out to be a tremendous help while trying to ‘make sense of life’.

This course will in particular introduce you to the mathematical modeling of healthy and diseased multicellular organisms, like ourselves. A key question is how cells cooperate to create biological structure, and how this biological structure feeds back on gene expression. The focus will be on how to sharpen one’s intuition on the emergence of biological systems and patterns by using and further developing a variety of continuous and discrete mathematical models of biological systems. Mathematical techniques include ordinary-differential equations, partial-differential equations, cellular automata, Hamiltonian systems, and in many cases combinations of those. This course will cover a range of multicellular phenomena, including development of animals and plants, blood vessel networks, bacterial pattern formation and diversification, tumor growth and evolution.

At the end of course students will have an overview of and some hands-on experience with a range of mathematical and computational techniques that computational biologist use in the study of collective cell behavior and biological pattern formation. They are familiar with recent literature on multiscale biological modeling and they have some experience with constructing basic computational models and hypotheses of phenomena described in the biological literature.

Course material:

Slides will be handed out after the lectures. Reading material: papers. Suggested background reading: books (will come up with a list later). There are no suitable textbooks on this topic.

Schedule 2017

Session 1  - September 6, 2017, 13:30-14:15 and 14:30-15:15
  • Introduction: role of mathematical modeling in (systems) biology
  • Diffusion and transport phenomena, diffusion length, gradients

Session 2 - September 13, 2017, 13:30-14:15, 14:30-15:15 + computer lab (15:30-17:15)
  • Turing patterns; Gierer-Meinhardt equations
  • Linear stability analysis
  • Biological example (trichome patterns on leaves)
  • More biological examples: hair follicles, skin patterns of zebrafish. So, biology seems to use Turing patterns, with some extra feedback loops here and there. But does it always involve chemical signaling...


Computer lab session 1 / homework - September 13, 2017, 15:30-17:15

Paper & pencil work and numerical simulations - Derive conditions for Turing instability analytically and develop intuition using numerical simulations

Session 3 - September 20, 2017, 13:30-14:15 and 14:30-15:15
  • Does biology actually not use Turing patterns? Gap genes in Drosophila
  • Introduction to Cellular automata
  • Why space matters: understanding growth of tumor spheroids
  • Logistic growth; Gompertz law; Eden Growth


Session 4 - September 27, 2017, 13:30-14:15 and 14:30-15:15
  • Cellular automata, classes of cellular automata
  • Probabilistic cellular automata
  • Margolus diffusion in cellular automata - example of Turing patterns using CAs
Computer lab session 2 / homework - September 27, 2017, 15:30-17:15

Mutual exclusion of gene expression: modeling the gap gene network in Drosophila melanogaster using the Sharp & Reinitz (1995) model.

October 4th. No lecture due to week of "Leidens Ontzet"

Session 5 - October 11, 2017, 13:30-14:15, 14:30-15:15 - Lecture by Lisanne Rens
  • Introduction to agent-based and cell-based modeling
  • Introduction to the cellular Potts model
  • Applications: tumor growth, cell sorting, angiogenesis


Computer lab session 3 / homework - October 11, 2017, 15:30-17:15

Cellular Automata

Session 6 - October 18, 2017, 13:30-14:15, 14:30-15:15
  • Branching morphogenesis
  • Diffusion-limited aggregation - fractal dimensions
  • Coral growth
  • Kidney branching

Session 7 - October 25, 2017, 13:30-14:15, 14:30-15:15
  • Multiscale Modelling
  • Development of Dictyostelium discoideum
  • Tumor modeling


Computer lab session 4 / homework - October  25, 2017, 15:30-17:15

Experiment with cellular Potts models using CompuCell3D. Cell sorting models; angiogenesis, vasculogenesis

Session 8 - November 1, 2017, 13:30-14:15, 14:30-15:15
  • lattice-free cell-based models
  • Vertex-based models
  • Dynamics, geometry, and topology of epithelial tissue
  • Subcellular-element models

November 8, 2017 - No lecture or rescheduled lecture

Session 9 - November 15, 2017, 13:30-14:15, 14:30-15:15
  • Plant modeling


Mini-projects (two guided afternoons plus individual work)

The mini-projects will involve a creative modeling exercise: to try model (and explain) an existing (“easy”) phenomenon, and build a model with existing tools: cellular Potts, cellular automata, or whichever method the student deems best for his/her project. Several project suggestions will be made, and students can propose their own.

Sessions 10-12 -  November 22, 2017, November 29, 2017, December 6, 2017, 13:30-14:15, 14:30-15:15 - Seminars
I will prepare sets of three to four recent papers on a specific topic (plant modeling, biomechanics, gastrulation, cellular Potts modeling, etc.). Students will each select a topic and present/lead a critical discussion of the papers in a seminar.

Depending on the number of students and/or the progress of the lectures, we might use part of this time to work on the mini-projects or present them, or I may add a regular lecture on of the topics.


Computer lab session 5-7 - November 15, 2017, November 22, 2017, November 29, 2017, December 6, 2017 15:30-17:15

  • Working on mini-projects
Session 13 - T.B.D.

EXAM: January 17 2018, 14:00-17:00, room B1
2ND EXAM: February 14 2018, 14:00-17:00, room 405


Mini-project Presentation Schedule (November 29th)

Course Material 2017