# Team:McGill/Modeling

(Difference between revisions)
 Revision as of 05:57, 21 October 2009 (view source)Fnaqib (Talk | contribs) (→Introduction)← Older edit Revision as of 06:14, 21 October 2009 (view source)Fnaqib (Talk | contribs) Newer edit → Line 71: Line 71: The above system was solved numerically using a forward Euler scheme in time and a centered difference scheme in space. Cyclical boundary conditions were assumed; meaning the spatial dimension formed a ring. This was chosen since simulating an approximate infinite line is computationally costly. However, this also allowed us to investigate two site geometries during one simulation (explained later). The ring was given a physical length of 50 and discretized into 500 intervals. For simplicity, separation distances will be reported in terms of numerical intervals rather than physical distance. The above system was solved numerically using a forward Euler scheme in time and a centered difference scheme in space. Cyclical boundary conditions were assumed; meaning the spatial dimension formed a ring. This was chosen since simulating an approximate infinite line is computationally costly. However, this also allowed us to investigate two site geometries during one simulation (explained later). The ring was given a physical length of 50 and discretized into 500 intervals. For simplicity, separation distances will be reported in terms of numerical intervals rather than physical distance. + =='''One Oscillator'''== + We first explored the different potential dynamics when the separation distance between an activating and inhibitory site was increased. + [[Image: Mcgill09OneOscillatorGeometry.png|frame|center|Figure 2 – One Oscillator – The red bar represents the inhibitory site, which remains fixed in position while the activating site, blue bar, is sequentially moved around the ring.]] + The following is an example of the dynamics observed when the two sites are at a distance where oscillations occur. + [[Image: Mcgill09One_activation_one_inhibitory_site_separated_by_11.png|frame|center|Figure 3 – One Oscillator – One activation and one inhibitory site separated by 11 intervals. The concentration of the activating molecule is measured at the activating site and the inhibitory molecule at the inhibitory site.]] =='''Appendix A - Parameters'''== =='''Appendix A - Parameters'''==

## Revision as of 06:14, 21 October 2009 ## Introduction

Many models examining intercellular signaling do not take into account the separation distances of the signaling bodies. We use a partial differential equation (PDE) based model to gain insight into spatially heterogeneous activation-inhibition intercellular signaling.

Two types of signaling molecules exist: activating and inhibiting. Each molecule is synthesized by a unique strain of cells and affects the synthesis rate of the other strain. Figure 1 – Activation-inhibition intercellular signaling – Activating molecule (A) synthesized and diffuses to increase synthesis of inhibiting molecule (B) in secondary strain. Inhibiting molecule also diffuses back to initial cell and decreases synthesis of activating molecule.

This is modeled using the following system of PDEs:

where Ψ1 and Ψ2 represent the concentrations of the activating and inhibiting molecules, respectively, γi the degradation constant, Di the diffusion constant, λi the maximal synthesis rate of molecule i, and δ the Dirac function. fi represents the Hill function describing the dependence on the opposing molecule:

where n, b, and θ are positive. To simplify the analysis, we chose equal parameters between the activating and inhibiting sites (Appendix A).

## Numerical Simulation

The above system was solved numerically using a forward Euler scheme in time and a centered difference scheme in space. Cyclical boundary conditions were assumed; meaning the spatial dimension formed a ring. This was chosen since simulating an approximate infinite line is computationally costly. However, this also allowed us to investigate two site geometries during one simulation (explained later). The ring was given a physical length of 50 and discretized into 500 intervals. For simplicity, separation distances will be reported in terms of numerical intervals rather than physical distance.

## One Oscillator

We first explored the different potential dynamics when the separation distance between an activating and inhibitory site was increased. Figure 2 – One Oscillator – The red bar represents the inhibitory site, which remains fixed in position while the activating site, blue bar, is sequentially moved around the ring.

The following is an example of the dynamics observed when the two sites are at a distance where oscillations occur. Figure 3 – One Oscillator – One activation and one inhibitory site separated by 11 intervals. The concentration of the activating molecule is measured at the activating site and the inhibitory molecule at the inhibitory site.

## Appendix A - Parameters

The standard set of parameters used to observe oscillations were taken directly from Shymko and Glass (1974). γ = 2 D = 2 λ = 54 θ = 1 b = 0 N = 8 Both strains were assumed to have identical parameters in order to simplify the model as well as explicitly observe the dependence of dynamics on separation distance.