MODELING OF BIOLOGICAL SYSTEMS-FEBRUARY 1998
General Modeling Section.
Silvina Ponce Dawson and John E. Pearson.
In this section of the course we will discuss basic issues regarding the
use of evolution equations for the modeling of natural systems, with emphasis
on differential equations. We intend to cover the topics described below,
although we will adjust this program depending on the students needs. We
will illustrate most of the ideas with numerical experiments that the students
will do. We will also use some illustrative laboratory experiments.
PROGRAM
General outline and features
The importance of dimensionless parameters will be emphasized. Concepts
from low dimensional dynamical systems theory will be introduced. Data
analysis and data-driven model building will also be discussed. The modeling
of systems with many degrees of freedom at various levels of description
will be described. The process of diffusion both by itself and when
coupled with chemical reactions will be discussed. Solutions to reaction-diffusion
systems will be described and simulated numerically. Different numerical
techniques will also be given. Finally, the particular case of the
dynamics of intracellular (and possibly intercellular) calcium will be
discussed applying many of the ideas introduced before. More specifically,
the following topics will be covered:
1. About modeling in general
General introduction on the different types of models we may build and
the different types of answers we may get from each type of model. Basically
some questions will be posed that we will try to answer during the course.
The difference between deterministic and non-deterministic models, between
microscopic and macroscopic descriptions, between qualitative and quantitative
types of agreement will be discussed.
2. Dynamical systems theory for low-dimensional deterministic systems.
Brief description of some concepts and tools of the theory. Phase space:
variables and parameters. Basic types of trajectories: fixed points, periodic
and chaotic orbits. Stable and unstable orbits. Attractors. Coexistence
of various attractors. Poincare maps and the Poincare surface of section.
The idea of topological equivalence. Bifurcations. Continuity and
bifurcations. The behavior of systems near bifurcations. Normal forms close
to bifurcations: the ability of knowing the dynamical equations in a set
of unknown variables. The existence of two timescales and the projection
onto a center manifold.
Numerical experiments: Using previously written programs the
students will get used to diverse visualization techniques and run a variety
of simulations to illustrate the different types of trajectories, the occurrence
of bifurcations and the reduction of the description to a small number
of variables.
3. Analysis of time-series data from determinisitc systems.
Fourier analysis of finitely sampled periodic time-series. Handling data
with gaps. Embeddings and phase space reconstruction. How many dimensions
we need to describe the measured data in a deterministic way. Model building
from time-series using a standard form and treating the system as a "black-box".
Numerical experiments: The students will be given a numerically
generated time series. They will have to determine the minimum number of
variables with which to reproduce the series and find an approximate vector
field using the techniques discussed. The values will then be compared
with the actual values of the system.
4. Systems with many degrees of freedom.
From the microscopic to the macroscopic description. The behavior of averages.
The effect of fluctuations. Behaviors not captured by averaged equations.
The effect of fluctuations as noise terms.
5. The diffusion equation
Derivation of the diffusion equation from a simple microscopic model. The
diffusion coefficient: kinetic theory. Solutions and properties of
the diffusion equation. Lattice gas simulation of the diffusion equation.
Numerical experiments: The students will run a simple lattice
gas simulation. This numerical experiment will also illustrate how to go
from a microscopic to a macroscopic description.
6. Reaction-diffusion systems
The focus in this section will be on two-component reaction-diffusion systems.
We will discuss the known behaviors that such systems posess and describe
the parameter regimes where these behaviors exist. Some of these behaviors
will be illustrated experimentally. We will emphasize the role that the
ratio of diffusion coefficients plays in determining the dynamics. We will
discuss the effect of immobile buffers on diffusion coefficients.
Numerical experiments:
To illustratete the rich class of behaviors possessed by two component
reaction-diffusion systems, we will numerically integrate the Gray-Scott
and Fitzhugh-Nagumo equations and visualize the
solutions.
7. Numerical techniques for reaction-diffusion systems.
We will discuss some of the standard pit-falls that occur in
the numerical integration of differential equations. These include stiffness
and/or numerical instability. Implicit and explicit integration schemes
will be touched upon with regard to numerical stability.
8. Calcium Dynamics
We will discuss in detail the de Young-Keizer model of calcium induced
calcium release (CICR). The analytic properties of this
important biochemical model will be discussed and compared to the
Gray-Scott and Fitzhugh-Nagumo systems. The model will be integrated numerically
and the solutions visualized. The break-down of the continuum approximation
for the distribution of release sites will be discussed.
Bibliography:
1. H.G. Solari, M.A. Natiello, and G.B. Mindlin, Nonlinear Dynamics.
A two way trip from physics to math. (Institute of Physics Publishing,
Bristol, 1996)
2. H.E. Nusse and J.A. Yorke, Dynamics: Numerical Explorations (Springer-Verlag,
New York, 1994)
3. A. Goldbeter, Biochemical Oscillations and Cellular Rythms. The
molecular bases of periodic and chaotic behaviour (Cambridge University
Press, Cambridge, 1996)
4. J.D. Murray, Mathematical Biology (Berlin ; New
York : Springer-Verlag, 1989.)
5. W.H. Press et al, Numerical Recipes: the art of scientifc computing,
(Cambridge ; New York, N.Y. : Cambridge University Press, 1989)
6. J. Crank, The mathematics of diffusion, (Oxford, [Eng] :
Clarendon Press, 1975).
7. J.C. Strikwerda, Finite Difference Schemes and Partial Differential
Equations, (Pacific Grove, Calif. : Wadsworth & Brooks/Cole Advanced
Books & Software, 1989).
8. G.W. de Young and J. Keizer, A single-pool inositol 1,4,5-trisphosphate-receptor-based
model for agonist-stimulated oscillations in Ca2+ concentration, Proc.
Nat. Acad. Sci. USA, 89:9895-9899,(1992).
9. Y.X.-Li and J. Rinzel, Equations for IP3 receptor-mediated Ca2+
oscillations derived from a detailed kinetic model: A Hodgkin-Huxley like
formalism. J. Theor. Biol. 166:461-473, (1994).
10. J.E. Pearson, Complex Patterns in a Simple System, Science,
261:189-192, (1993).