PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it seems to me it's either statistics or some ODE problems. Is there anything more PDE releated in biology?

Solution 1:

Mathematicians often look to other fields to gain inspiration for some of their research (the mother of all examples might be vector calculus and electromagnetism), so this is not a bad question. Lots of mathematical modelling in biology is very approximate and many models are entirely useless to the average experimentalist. An example of this are the PDEs e.g. "governing" neural transfer, where $u$ is the presynaptic firing rate and $v$ is a depolarization field :

$$\left(\frac{\partial}{\partial t} + (1-\Delta)\right)u(x,y,t) = C(x,y,t)v,$$ coupled to $$\frac{\partial^2}{\partial t^2} v + \frac{\partial}{\partial t}v + v = K(x,y,t)u.$$

In my view, more interesting PDEs arise in the study of biological membranes. For instance, for vesicles with a phospholipid bilayer, Helfrich found (based on his work in liquid crystals) that a possible energy for the membrane is $$E[u] = \kappa\int_\Sigma H^2 - K dS.$$ The absolute minimizer among surfaces of genus 1 for this energy is the Clifford torus. This problem provides interesting mathematical insight and matches biological findings very accurately.

One can also consider the energy for liquid crystals whose molecules follow a director field n:

$$\int |\nabla_s n|^2 = \int (div_s n)^2 + |n \times \nabla_s \times n|^2 + |n\cdot \nabla_s \times n|^2$$ where the $s$ always denotes ``surface'' and $\nabla_s \times$ is surface curl.

Solution 2:

The book referenced by @Henrik Finsberg is OK if you'd like to learn some basic facts about PDEs (and about their applications in biology). If you feel already confident about PDEs, try to get a hold of

• James D. Murray, Mathematical Biology, Vol. 2
• Benoît Perthame, Transport equations in biology
• Robert S. Cantrell, Chris Cosner, Spatial Ecology via Reaction-Diffusion Equations
• Akira Okubo, Simon A. Levin, Diffusion and Ecological Problems, Modern Perspectives

Solution 3:

There is a great book about this: Leah Edelstein-Keshet, Mathematical models in biology chapter 10.

Solution 4:

Just graduated in a masters course at ICL: "MSc Theoretical Systems Biology & Bioinformatics", I can say it was some of the most exciting stuff I had ever come across.

I think you would be more interested in the "Systems Biology" aspect - it covers dynamical systems theory (PDEs, ODEs, SDEs, Agent-based modelling, quantum molecular modelling, stability analysis etc, Gillespie simulations, chaos), control theory, graph theory (networks), biophysics, biotech, bioengineering and synthetic biology.

Useful Info:

• The more general realm of mathematics that goes beyond Biology is "complex systems" or "complexity science" (WIKI).

• Have a look at some of these online MIT lectures (HERE)

• Quote by Cohen(2004):

"Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better"