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Multiscale Mathematical Biology

from individual cell behavior to biological growth and form

In the autumn semester of 2018, we will be teaching the sixth edition of the course "Multiscale Mathematical Biology" at the Mathematical Institute of Leiden University. This year, the course will run in the second half of the semester, starting from Wednesday October 17th, 2018. There will be lectures on Wednesdays, from 11:00-12:45 and again 13:30-15:15, and a computer lab on Fridays, 9:00-10:45.

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: Koen Schakenraad and Corine in 't Veld

English, unless all students speak Dutch

Location & time:
October 17th - December 14th, 2018, Wednesdays and Fridays, Leiden University, Snellius (Mathematics)
  • Lectures: Wednesdays 11:00-12:45 and 13:30-15:15 . University of Leiden, Snellius (Wiskunde), room 412. On November 14 exceptionally only 2 hours of lecture from 13:30-15:15.
  • Computer exercises and mini-projects: Fridays, 9:00-11:00, room 302-304. On 16/11 exceptionally in room 307-309.


Methodology: lectures, paper seminar, practical exercises, mini-project, exam.

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 and Biology, Minor Quantitative Biology, and other interested students from (Astro)Physics, Pharmaceutical Sciences, and so forth.

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 2018 and overview of topics

Exact content of each lecture session may change depending on progress along the way

Session 1  - October 17, 2018
  • Introduction: role of mathematical modeling in (systems) biology
  • Diffusion and transport phenomena, diffusion length, gradients
  • Turing patterns; Gierer-Meinhardt equations
  • Linear stability analysis


Computer lab 1 - October 19, 2018

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

Session 2 - October 24, 2018
  • 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...
  • Does biology actually not use Turing patterns? Gap genes in Drosophila
  • Introduction to Cellular automata
  • Why space matters: understanding growth of tumor spheroids


Computer lab session 2 / homework - October 26, 2018

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

Session 3 - October 31, 2018
  • Logistic growth; Gompertz law; Eden Growth
  • Cellular automata, classes of cellular automata
  • Probabilistic cellular automata
  • Margolus diffusion in cellular automata - example of Turing patterns using CAs
  • Introduction to agent-based and cell-based modeling
  • Introduction to the cellular Potts model
  • Applications: tumor growth, cell sorting, angiogenesis
Computer lab 3 / homework - November 2, 2018

Cellular Automata

Session 4 - November 7, 2018 - Enrico Colizzi

Prebiotic evolution:

  • The error threshold
  • Hypercycles
  • Emergent multi-level selection
  • Possibly: genotype-phenotype maps


Computer lab 4 / homework - November 9, 2018

Prebiotic evolution

Session 5 - November 14, 2018 - NOTE: 13:30-15:15

Guest lecture on cell mechanics by Koen Schakenraad, MI and LION
Computer lab 5:

Cellular Potts modeling

Session 6 - November 21, 2018
  • Other cell-based modeling frameworks:
    • Langevin equation
    • Vertex-based models
    • Dynamics, geometry, and topology of epithelial tissue
    • Subcellular-element models
    • Lattice-gas cellular automata
  • Multiscale Modelling
    • Development of Dictyostelium discoideum
    • Angiogenesis
    • Cancer modeling


Computer lab 6 - November 23, 2018


Mini-projects II


Session 7 - November 28, 2018


Computer lab 7: Mini-projects I (two guided lab sessions plus individual work)


Session 8 - December 5, 2018

Presentations on papers and mini-projects

Computer lab 8: Mini-projects I (two guided lab sessions plus individual work)


Session 9 - December 12, 2018

Presentations on papers and mini-projects

Computer lab 9: Mini-projects I (two guided lab sessions plus individual work)


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.



EXAM: 16 January 2019, 14:00-17:00, HL106-9
2nd Take: 15 March 2019, 10:00-13:00, 412


Paper Presentation Schedule


Course Material 2018