Mary Kloc '06, with Mentor Sheryl Hemkin, studies
Calcium Oscillations using Knot Theory
Summer Science Proposal, supported by Kenyon College HHMI 2000
Biological Calcium Oscillations Modeled Using Knot Theory
Calcium oscillations have numerous functions in a wide variety of cells and are integral to the regulation of many biological processes. Their known functions range from the regulation of heart beats, circadian rhythms, and hormone secretion to roles in the fertilization and wound healing processes. They are also present in other cells, such as astrocytes, in which their function remains unknown. Astrocytes are the most abundant cell type in the central nervous system. They are situated next to neurons and make up the blood brain barrier. Considering the importance of Ca2+ oscillations in such a diverse array of cells throughout the body, their presence alone in astrocytes suggests that they must have a function there as well and it has been inferred that they may be aiding in the entrainment of cells during an epileptic seizure.1 However, before the question of function can be answered, we must first understand how they arise and behave in the cell.
This study will utilize the astrocytic calcium system that I have previously explored. The model2 (Figure 1) consists of a system of three differential equations, which describe [Ca2+] oscillations in the cytosol, mitochondria, and endoplasmic reticulum of astrocytes, which are glial cells in the central nervous system.
Numerical integration, using the Gear algorithm, is used to produce concentration vs. time data that can be compared to experimental data. The Gear algorithm is a predictor-corrector numerical integration scheme that uses the stiff method, which has a variable stepsize. The variable stepsize is more efficient for stiff equations when high precision is needed. [Ca2+]cyt vs. [Ca2+]ER vs. [Ca2+]mit can be plotted to produce three dimensional phase portraits, such as those shown in Figures 2 and 3. Comparison of these plots reveals the large change from chaotic to simple periodic behavior when only a single parameter (in this case, the fractional external stimulation) is changed; and this is consistent with experiment.3 In general, to improve this model or other models like it, intuition and extrapolation is used. I want to try to lend structure to this process. Since these biological systems have mathematical rules that govern them, I want to use a more structured mathematical approach to improving the model. It is known that dynamical systems in a chaotic regime, such as [Ca2+] oscillations in astrocytes, can tie themselves in knots in 3-D space. I plan to probe for knots in chaotic phase planes similar to and including the one shown in Figure 2. and use the principles of knot theory to tie the governing mathematics to the biological behavior.
The field of knot theory first emerged in 1867 when Lord Kelvin presented a paper to the Royal Society that hypothesized that atoms were composed of knots in ether. Although this application of knots was shown to be incorrect shortly after it was proposed, knot theory has remained an integral part of the field of chemistry. In organic and materials chemistry, researchers are attempting to build molecules that contain different types of knots; the simplest knotted molecule created so far (shown in Figure 4) was synthesized in 2000.7 Knot theory also has its place in biochemistry and molecular biology, in which knotted proteins and strands of DNA are currently being studied. These knotted biological molecules are shown to have properties that are vastly different than their unknotted topological isomers.8 I plan to incorporate knot theory into chemistry in yet another way: through the analysis of computational and experimental data for [Ca2+] oscillations in astrocytes.
Figure 4: 4
A phase diagram containing a suspected three-dimensional knot, such as one of those generated during my previous research, must be projected onto a plane (called a knot projection) for mathematical analysis. The plot is then distorted by means of a series of Reidemeister moves to create a plot in which only topologically important crossings or links (those that involve the knot) are present. Reidemeister moves are mathematical transformations that change only the projection of a link or crossing, but not the link itself. Knot invariants, which are usually polynomials such as the Jones or Alexander knot polynomials, are then used to distinguish different types of knots from one another; in this project, knot invariants will be used to distinguish between the “unknot” (which lacks the traditional knot) from any other kind of knot.9 Examples of the unknot and other knots are shown in Figure 5.
The most important concept from the field of knot theory to be applied in this project is that the fundamental topological properties of a knot must be invariant over continuous transformations: one type of knot cannot be turned into another type of knot by way of any continuous transformation. Most relevantly to this case, the unknot cannot be continuously transformed into any other kind of knot.
The mathematics of knot theory may prove to be very important when applied to dynamic biological systems. From a biochemical point of view, the statement above implies that a periodic (unknotted) state cannot move to a chaotic (knotted) state through a continuous transformation; I want to show that certain types of behavior are invariant under this set of topological principles. I will demonstrate that periodic behavior of any period is topologically equivalent to any other periodic behavior, but is not topologically equivalent to chaotic behavior. I plan to characterize aspects of this biological system according to the mathematical operations and transformations that are permitted by the rules of knot theory; find biological parameters for which the system is invariant to a change in value; and gain further insight into how system moves from a chaotic to a periodic regime when a continuous transformation of this type is not possible.
1. Speelman B., Larter R., Worth RM “A Coupled ODE Lattice Model for the Simulation of Epileptic Seizures” Chaos 9: 795-804 (1999)
2. Robyn Harshaw, Mary Kloc, Sheryl Hemkin. “Mitochondrial Influences in Astrocytic Ca2+ Oscillations” (in preparation)
3. Cornell-Bell A., Finkbeiner SM, Cooper MS, Smith SJ “Glutamate induces calcium waves in cultured astrocytes: long-range glial signaling” Science 247: 470-473 (1990).
Adams, Colin. “Why knot:
knots, molecules, and stick numbers”. http://plus.maths.org/issue15/features/knots/
Flapan, Erica. When Topology
Meets Chemistry: A Topological
Look at Molecular Chirality.
6. Houart G, Dupont G, Goldbeter A. Bursting, chaos, and birhythmicity originating from self-modulation of the inositol 1,4,5-trisphosphate signal in a model for intracellular Ca2+ oscillations B Math Biol 61 (3): 507-530 MAY 1999
7. Marhl, M., S. Schuster, M. Brumen, and R. Heinrich. 1998. “Modeling oscillations of calcium and endoplasmic reticulum transmembrane potential role of signaling and buffering protiens and of the size of the Ca2+ sequestering ER subcompartments.” Bioelec. Bioener. 46: 79-90
Murasugi, Kunio. Knot Theory
and Its Applications.
Roberts, Justin. http://math.ucsd.edu/~justin/190.html