### BIOMODELAT MATEMATIC (MATHEMATICAL BIOMODELING)

This course introduces the basic elements to address the problem of how to formulate very simple mathematical models capable of addressing a number of relevant problems related to cell and population growth, stability and instability and the logic of living systems. Most models are based on one-dimensional differential equations. These include cancer, cell growth and differentiation, epidemic spreading, population dynamics, gene and neural networks (at a very introductory level). The course is taught in catalan.

### CONTINGUTS / CONTENTS

1. Linear Models. Exponential growth. Estimating cell division time. Mortality and exponential decay. Constant growth + linear decay:
constitutive gene expression and other examples. Growth in cell cultures: early stages.

2. Linear diffusion between two compartments. Fick's law and relaxation to equilibrium. Chemical equilibrium.

3. Transitions between multiple states: chemical reaction chains and protein folding. Introduction to first-order linear
differential equations.

3. Bimolecular reactions: introduction to nonlinear models. DNA hybridisation and Cot curves.
Hybridisation curves and the structure of prokaryotic versus eukaryotic genomes. Receptor-ligand
kinetics and active transport. Imatinib and tyrosine-kinase interactions in cancer. Facilitated diffusion and active transport.

4. Population explosions and singularities. Second-order growth equations and critical times. Hyperbolic dynamics.

5. Logistic growth dynamics. Derivation of the logistic equation and its exact solution. Other models of growth under
constraints: Gompertz equation and embryonic development. The importance of spatial dynamics.

6. Linear stability and equilibrium (fixed) points. The concept of attractor. Bifurcations and bifurcation diagrams.
Potential function: definition, use and meaning.

7. Symmetry breaking and historical dynamics. Frozen accidents and multi stability. Path dependence and
evolution. Bottlenecks and biological diversity. Human diversity and cancer heterogeneity.

8. Epidemic spreading. Historical perspective and case studies. Evidence for transitions in epidemic thresholds.
The SIS model: assumptions and implications. Eradication thresholds and vaccination strategies.

9. Competition between populations. The constant population constraint (CPC) model. Competitive
exclusion. Applications to microbial populations and cancer dynamics.

10. Discrete dynamical systems. Boolean models and Boolean logic. Biological computation. McCulloch-Pitts logic neurons and
threshold neuron models. Computation in small neural systems. Genetic circuits and the lambda switch.

### VIDEOS/LECTURES

MIT lecture: Differential equations of growth

Transport across cell membranes and chemical equilibrium:

MIT Differential equations

Facilitated diffusion: the concept:

TED Talk. Global population growth by Hans Rosling:

Regulated/unregulated cancer cell growth

Stable/unstable equilibrium points

Population bottleneck and genetic drift

Pandemics and future epidemics TED Talks

Growth, competition and space in Physarum plasmodium: