Understanding Diffusion Models within environments

I. Introduction

A cell is the basic unit of out cellsim system. Our programming model uses a simple model of biological cells, in which cells communicate through two methods, cell emission and cell diffusion. Cell emission can only take place through cell walls whereas cell diffusion occurs through an environment.

To model cell diffusions, we need to model environments. Up to date, a global environment model that monitors cell diffusions in the system is already set up. The environment model not only enables cell diffusion to happen, but also helps us observe the program behaviors in different environments.

The following part of the document will clearly illustrate different diffusion models that had been adopted by the system.

II. Diffusion Models.

In the SystemModel, the chemical diffusion information including chemical concentration, chemical name and diffusion location etc. are stored in a vector List, which is a private data field of the Environment class. Environment has all the diffusioninformation and function findconcentration(). Systemmodel calles Environment to find out the concentration at a particular position whenever needed.

Figure 1. The figure is taken from Principles of Development (Wolpert, 2002), Kruppel gene expression in bicoid mutant, Fig. 5.18, pp.164. The figure has been modified to demonstrate the point.

As Fig. 1 shows, a high concentration gradient is diffused at left pole (anterior). The gradient is decreasing gradually as the distance between a particular point and the anterior is increasing. When distance traveled by gradient reaches certain threshold, concentration goes to zero and can thus be neglected for our purpose of calculation.

A: Fist Version-Linear Model

As Fig. 2 shows, to calculate the concentration at at a particular cell centered position:

concentration = chemical concentration (the orgiginal conc of diffusion) - distance (between the cell and the diffusion location).

Figure 2. Linear Model of Diffusion. Concentration is based on the distance traveled. Concentration=concentration-distance.

B: Second Version-a Model based on time steps:

This model has additional parameters including spreadrate, evaporrate, step, and forcegain. The new concentration is also affected by force which is a vector variable that increases or decreases the concentration value in a designated way. Before understanding the current model, first a few variable names are to be specified.

spreadrate stands for spreading rate of the chemical diffusion.

evaporrate stands for the evaporation rate of the chemical diffusion

timeS is the difference between the current simulation time and the time when diffusion actually occured, which is when the diffusion data are stored in the Environment class. In another word, timeS represents how long the diffusion has occured. Here, we use simulation step as a meter of time.

forcegain is a double that controls whether the diffusion is affected by the force vector.

spreadtime is the distance/spread rate

distance: the calculated distance between where the diffusion occurred and the position where the new concentration should be revolved.

As Fig. 3 shows, the new concentration is a function of time rather that distance if the step. If the timeS is less than the spread item, which is the pre-set time point at which the concentration should be at its highest value. The concentration is 0.0, Otherwise, timeS* evaporrate calculates how much of the diffusion chemical is evaporized at the particular time point and the new concentration = original chemical concentration - timeS * evaporrate, which is how much chemical is left un-vaporized.

Figure 3. Diffusion Model based on Time Steps.

Figure 4. Two biological circles have identical genes. A) (Above left) Biological circle with no internal forces at development step 15. b) (Above right) Biological circle with one force in X+ direction at development step 15. The force vector has value +1.0 in its X data field.

Usage of forces:

A notion of forces that affect how chemicals diffuse. This can affect the way cell programs execute. As Figure 4 below demonstrates, biological circles with identical genes at the same stage of cell development behave differently because of the force vector. Figure 4A shows a circle with no internal forces (force vectors are set to zero); figure 4B shows a circle with one internal force in X+ direction.

Idea of Segmentation:

One of the significant experiments done employing new diffusion model is segmentation, which is the most obvious feature of a fly. A fly has numerous body parts including mouth, eye, legs and so forth. Our goal is to construct a fly body and segment the body into a number of distinct regions. Cells in one region are unique so that the region represents a body part and has all the corresponding biological functions.

We have successfully segmented a model into two regions so far. As Figure 5 illustrates, identical chemicals are diffused at the same time from two polar cells. This results the segmentation of the biological object. Cells on the left represent one type of cells and cells on the right represent a different kind.

Figure 5. Segmentation of a biological object. The purple cell on the left is the left pole (anterior); the red cell from the opposite side is the right pole (posterior). Two identical chemicals are diffused at the same time from two poles. A distinctive boundary is formed in the middle (dark red cells). Yellow cells on the left and grey cells on the right have different genetic makeup.

C. Current version-Exponential Model & others:

1. Exponential model.

The current diffusion calculate gradient using an exponential model. The function is: Concentration = concentration* exp (-dist*factor), in which the smaller the factor, the higher the concentration.

Figure 6. Example of exponential diffusion model. concentration=20*exp(-distance*1).

One example of using exponential model is the bicoid protein illustrated in Fig. 8.

2. Bell-shape curve model.

The bell-shape curve model can be best illustrated as proteins giant and eve stripes2 in fig. 7. The function for this model is shown below:

Figure 7. The figure is taken from A First Course in Porbability (Ross, 2002, Figure 5.5, pp.199).

As fig 7. shows, the standard curve(with shrinkfactor =1) starts at nu-2*sigma, at where f(x) is zero. The center of the curve is at nu and the maximum height equals to .399/sigma. If the shrinkfactor is bigger than 1, then the bell-shape curve shrinks correspondingly.

3. Sudden-drop Model.

As fig. 8 shows, the gradient of hunchback is a constant if distance is less than nu. And the gradient forms a left-half bell shaped curve if distance is greater than nu.

4. Sudden-raise Model.

As fig. 8 shows, the gradient of kruppel is a constant if distance is greater than nu. And the gradient forms a right-half bell shaped curve if distance is less than nu.

Figure 8. Illustration is taken from Principles of Development (Wolpert, 2002), Fig. 5.22, pp.168. Bicoid employs exponential model; giant and evestripe2 employs bell-shape curve model; hunchback uses sudden-drop model and kruppel uses sudden-raise model.

We will create more interesting shapes employing various new diffusion models.