IGTC Graduate Summer School in Mathematical Biology

A PIMS/Accelerate BC sponsored Summer School in Mathematical Biology

University of British Columbia, Vancouver, May 11-June 11, 2008

Course Content and Material

For a detailed daily schedule click HERE.

For an overview of the month-long schedule click HERE.

For a summary of events by category and/or lecturer(s) click HERE.

For abstracts of talks click HERE.

Unless otherwise indicated, events will take place at the UBC campus of PIMS. Directions for PIMS can be found HERE.

Useful links

1. Information for IGTC computer labs is HERE

2. Leah's Worksheet/Chapter on Enzyme Kinetics in PDF and in Latex form. (May 13, 2008). Note: may contain typos.

3. Leah's Preliminary Worksheet/Chapter on Biochemical Dynamics in PDF and in Latex form. (May 14, 2008). Note: may contain typos.

4. Some literature papers that you may want to read and present as a project are given at the last pages of the worksheet on Biochemical Dynamics.

5. A simple guide on using XPP AUTO for a simple bistable equation is given PDF form. (May 14, 2008). (This example was done in tutorial on May 14).

6. A summary of what was discussed on the Tyson models for cell cycle appears here in PDF form. This includes a guide to the literature for possible projects/class presentations. A shortened version of the powerpoint slides appears here in PDF form. UPDATE: The chapter has been edited to reflect the revision that includes better AUTO settings and file to produce the stable limit cycle. The ode file that was contributed by Anmar Khadra is HERE.

7. A worksheet/chapter on simple polymerization can be found HERE. An XPP ode file to simulate polymer size distribution (asizedistr.ode) will be posted on the lab webpage.

8. Notes and a guided exercise to how scaling arguments can be used to reconstruct the mechanism of polymer self-assembly are HERE. An excel file containing data for the Flyvbjerg polymer self-assembly problem is HERE. Your challenge is to use this data together with scaling arguments to determine how this polymer self-assembles (i.e., what are the steps).

9. A few of you have been asking for This File, that for some reason is bouncing from email.

10. A short example and exercise for the Gillespie Method, from Bard's XPP book.

11. Some quick notes about Bard Ermentrout's XPP-AUTO lab, including the two .ode files (ml.ode and hhh.ode) and AUTO setting can be found HERE.

12. Here are three documents that Bard has assembled to help you read up about the material in his lectures: Bard's Chap7 , Bard's Chap7's Figures , and Bard's MBI Chapter .

13. Here is an example of how to get XPP to plot the solutions of reaction-diffusion pde's. The example is different from the one we did in class, but the XPP instructions are the same. (Figures in the Turing chapter were produced in the same way.)

14. Here is a selection of a few projects that Bard Ermentrout suggested in a workshop he gave at Utah. One or two of these may be suitable, but many are outside of the scope of what he talked about here.

15. Here is the summary of the XPP lab tutorial by Bard Ermentrout showing how to use AUTO for the tyson.ode problem and for hhh.ode. Notes were kindly written by Jiafen Gong and edited slightly (with graphs inserted) by Leah.

16. Another list (from Leah) with ideas and general instructions for projects that you may want to consider.. or feel free to suggest your own.

17. I have updated this chapter on the Turing reaction-Diffusion system. It now includes more of what I talked about in class.

18. Here are the slides from the lectures by Alex Mogilner in .pdf file format: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5.

19. Here are Leah's slides for the lecture about Rho proteins, cell polarization, and wave pinning from the final few days.

20. A link to a paper by Omer Dushek and others about how to combine diffusion of two chemicals (related to what was discussed in class).

21. Here is an updated summary of the course in case you need to present this to get credit from your university.

Lectures by Prof Karl Hadeler (May 12-23, 2008)

• (1) Modeling spread in space (May 12): The standard model for spread in space is the diffusion equation. We try to understand how this equation has been justified, for what range of phenomena it is a valid model and which mathematical properties it has. Then, guided by examples, we look for other, refined descriptions for movement in space like transport equations and we investigate how the different models are connected by limiting procedures.

• (2) Reaction and diffusion (May 13): We couple spread in space to reactions and get reaction diffusion equations as models for situations where spread and interaction act on the same time scale. We supply these models with boundary conditions, and we investigate the qualitative behavior of these systems, particular stationary states and stability. Specific biological mechanisms like chemotaxis will be discussed.

• (3) Traveling fronts (May 14): The typical asymptotic behavior of a reaction diffusion equation in the whole space is a traveling front. We exhibit the basic concepts like spread number and minimal speed of a traveling front and also some examples from ecology, infectious disease modeling, neurobiology, etc. Then we review the recent literature with respect to useful results and open problems.

• (4) Ecological models (May 15): Here we describe some of the classical models from ecology, genetics, adaptive dynamics and discuss concepts like carrying capacity, predator response, Allee effect, persistence. Using these models, we understand some basic concepts like bifurcations, exchange of stability, direction of bifurcation, Lyapunov function, invariant of motion, structural stability, and how these can be used in investigating model systems, e.g., for pattern formation.

• (5) Infectious disease modeling (May 16): Modeling the spread of infectious diseases plays an important role in present day Math Biology and in public health. Although the classical models are very simple mathematically, it is not so evident how one should model contacts, transmission, recovery, quarantine, vaccination, disease-related mortality or spread in space. In particular with respect to spread there are two competing types of models: contact distribution and diffusion.

• (6) Coupled systems and quiescence (May 20): Coupled systems is a hot topic which comes under the names of lattice differential equations, meta-populations, seasonal dependence etc. We will be very modest and couple only two scenarios like rainy season - dry season or active phase - quiescent (dormant) phase. We will elaborate the key features of such coupled systems. In particular, we will see how introducing a quiescent phase changes the dynamics.

• (7) Demography and structured populations (May 21): At present there are very general theories of populations structured by age and size which allow to write practically every model as a simple equation in some large space. On the contrary, our goal is to look at specific model systems with few parameters and establish the connection to the theory of differential delay equations which has been used frequently to model unknown reaction or transport steps. If time allows, we will consider the pair formation problem of demography.

• (8) Birth-death processes and diffusion models (May 22): Birth - death processes can be used to model macro-parasite (helminth) infections. We shall describe the modeling approach, its advantages and drawbacks, and routes towards simpler models. One of these simplifications is based on the idea to replace the (integer) parasite load by a continuous variable (level of infection).

• (9) Summary and Outlook (May 23): At this moment we do not specify topics for this section. During the preceding days it will turn out in what specific questions the students are interested and which of the previous topics should be discussed more profoundly. Also some emerging research areas can be outlined.

Lectures by Prof Bard Ermentrout (May 26-29, 2008)

• (1) The onset of rhythmicity (May 26)
  Abstract: In this lecture, I will start by describing the geometry underlying the onset of oscillations in smooth dynamical systems. Specifically, I will describe sub- and supercritical Hopf bifurcations, the saddle-node invariant circle bifurcation and the homoclinic bifurcation. In the second part of the lecture, I will discuss how to find the Hopf bifurcation in some ODEs and delay equations. I will use examples from neurobiology, chemistry, and cell biology. I will also explore the onset of oscillations in asymmetrically coupled networks.

• (2)What is phase? (May 27)
  Abstract: In this lecture, I will describe how perturbations afffect oscillators and in so doing introduce the concept of phase, phase resetting, and the adjoint. I will explain how the phase resetting curve (PRC) determines the behavior of a single oscillator when perturbed by periodic and noisy stimuli. I will also discuss how noise affects the PRC and its relationship to other measured quantites in a noisy environment. I will show that in some cases the PRC can be analytically determined near bifurcations and use the PRC to study pulse coupling between oscillators.

• (3) Weak coupling theory. (May 28)
  Abstract: In this lecture, I will consider very general coupled oscillator systems and focus on the method of weak coupling. Here I will discuss how the behavior of pairs of oscillators depends crucially on both the nature of the coupling and the intrisic dynamics of the coupling. Specifically, I will focus on how changes in the frequency and modulation by external factors can alter the ability of neurons to synchronize. I will also look at "stochastic" synchriny whereby common fluctuations are enough to synchronize uncoupled oscillators (the "Moran" effect in ecology).

• (4) Large networks of oscillators(May 29)
  Abstract: Here I review the behavior of chains of oscillators and then use this to study two-dimensional arrays. I illustrate and prove that in some topologies there can be many attracting solutions including synchrony, waves and spirals. I finally describe the Kuramoto model of all:all noisy connectivity and its bifurcation into clustered states.

Here is some basic reading material for lectures by Prof Bard Ermentrout:

• #131,132 on the Ermentrout publications page
• See also the online chapter on oscillators in Izhikevich's book

Lectures by Prof Alex Mogilner (June 4-10, 2008)

• (1) Introduction to mitosis (June 4)

• (2) Mitotic checkpoint (reaction-diffusion model) (June 5)

• (3) Chromosomal oscillations in mitosis (ODEs - numerical solutions, bifurcation analysis) (June 6)

• (4) Spindle assembly (PDEs, probability estimate, Monte Carlo simulations) (June 9)

• (5) Cutting edge modeling of mitotic spindle (agent-based simulations) (June 10).

• Mogilner A, Wollman R, Civelekoglu-Scholey G, Scholey J. Modeling mitosis. Trends Cell Biol. 2006 Feb;16(2):88-96.

• Wollman R, Cytrynbaum EN, Jones JT, Meyer T, Scholey JM, Mogilner A. Efficient chromosome capture requires a bias in the 'search-and-capture' process during mitotic-spindle assembly. Curr Biol. 2005 May 10;15(9):828-32.

• Doncic A, Ben-Jacob E, Barkai N. Evaluating putative mechanisms of the mitotic spindle checkpoint. Proc Natl Acad Sci U S A. 2005 May 3;102(18):6332-7.

• Grill SW, Kruse K, Ju"licher F. Theory of mitotic spindle oscillations. Phys Rev Lett. 2005 Mar 18;94(10):108104.

• Maly IV. Diffusion approximation of the stochastic process of microtubule assembly. Bull Math Biol. 2002 Mar;64(2):213-38.

• Ne'de'lec F. Computer simulations reveal motor properties generating stable antiparallel microtubule interactions. J Cell Biol. 2002 Sep 16;158(6):1005-15.

Course Sylabus

• Here is a Course Syllabus. This is provided for students who wish to take this course for credit under the Western Deans' Agreement (Western Canada).

Lectures by Leah Keshet (May 11-June 11, 2008)

• Introduction to biochemical modeling (May 13) This lecture will include a rapid survey of how biochemical and enzyme kinetic models are formulated and analyzed, starting from simple qualitative methods. Scaling and dimensional analysis, time scales, quasi-steady states, and Michaelian and sigmoidal kinetics will be considered. I will briefly describe XPP AUTO and how it can be used to study bifurcations in a simple example.

• Biochemical models, bistability, bifurcations (May 14 AM) I will discuss competitive and noncompetitive inhibitors, dimerization, and kinetics leading to bistability in chemistry and ecological setting.

• (Optional afternoon tutorial): dimensional analysis, qualitative dynamics (May 14) In the afternoon session of the same day, I will discuss such topics in more detail for students who have not had exposure to these methods.

• Applications to cell division, cell signaling kinetics (May 15) I will here survey models of cell division cycle (research of John Tyson), and possibly other intracellular signaling.

• Polymerization kinetics and length distributions (May 16) This lecture will introduce biopolymers such as actin and microtubules. I will show how classic stage-structured models can be used to describe polymer length dstributions.

• Actin and microtubule polymerization in a lamelipod (May 20) Here I will describe classic work by Dogterom and Leibler on microtubule dynamics and on the spatial distribution of actin filaments in a 1D model of a cell.

• Microtubule kinetics and growth (May 21-22) This lecture will be devoted to understanding how dimensional anlaysis can be used to understand the mechanism underlying polymer assembly from data for polymer mass over time (work by Flyvbjerg).

• Tutorial: Bifurcations (May 21) A brief survey of simple bifurcations.

• Microtubule kinetics and growth (May 22) See above.

• Stochastic approaches to molecular motion (May 23) Here we will consider thermal fluctuations, random walks, and stochastic approaches.

• Growing polymers and ratchets (May 23) This will continue the previous lecture. We will study the Langevin and Smoluchowski equation, and discuss the thermal polymerization ratchet.

• How cells move - filopod and lamelipodial growth (May 26) Here we survey some recent PDE models for the distribution of actin filaments in a growing cell. The models will illustrate time scale separation, and physical considerations.

• modeling the shape of the cell (May 27) We discuss work by Mogilner et al on the shape of a moving cell, the graded radial extension model, and insights from the analysis into the biology underlying the mechanism of cell motility.

• Mechanisms for pattern formation (May 28) This lecture will consist of a survey of mechanisms for pattern formation, including both local (Turing) and nonlocal (integro-PDE) examples. A tutorial/lab on the same day will allow students to explore such mechanisms using simulations.

• Orientational order and angular patterns (June 2) We extend the ideas in the previous lecture to alignment and angular pattern formation, and describe a model for turning (in ants) and alignment (in filaments) due to angular interactions. We show how previous ideas on stability can be generalized to include spatio-angular patterns.

• Cell polarization models (June 3) Before a motile cell decides in which direction to move, it has to polarize chemically and select a 'front' and a 'back'. In this lecture we survey some recent models in the literature for how cells perform such chemical computations. We show parallels with lateral inhibition and Turing-based pattern formation.

• Wave phenomena and application to polarization (June 4) Here I will describe some results on waves in reaction-diffusion systems and show how a wave-basd mechanism for cell polarization differs from some of the previous examples described in my last lecture.

• Eulerian swarming models (June 5) The survey on waves will lead us to a side route whereing we explore models for social aggregations. I will describe some of the literature on this topic and include an example of locus swarming and models related thereby.

• Individual-based models for swarming (June 6) In constrast to the above, I then discuss individual-based (Lagrangian) models, and describe several examples of recent work on this fascinating subject.

• Models of diabetes (June 9) This lecture will be devoted to a quick survey of some biomedical modeling, including models for Type 1 (autoimmune) diabetes. One of the aims will be to show how methods that we found useful in other systems will reappear and contribute to this new area.

• Summary and perspectives (June 10) This is left open for several possible directions that we will determine based on the progress made up to this point.

Other details about the content of the course will be posted here from time to time.