1. The Chemotactic Paradox Amie Cubbon Denise Gutermuth

2. Chemotaxis –A response of motile cells in which the direction of movement is affected by the gradient of a diffusible substance. –The gradient alters probability of motion in one direction only

3. Dictyostelium Discoideum –Individual cells aggregate in response to a signal from ‘organizer’ cells –Chemoattractant is initially released by organizer –Chemical attaches to receptors on cell surface orienting the cell in the direction of movement –Signal is relayed by cell – causes an outward moving wave of signal

4. A Modeling Problem : Assume that the wave is moving in one spatial dimension Cell responds to increasing signal concentration moves forward (to the right) in direction of the wave wave back passes the cell cell senses a negative gradient and should change direction

5. Result: – net displacement to the left ie. negative value Problem: –In experiments, the cell continues to move to the right => net displacement should be to the right ie. positive value

6. Assumptions movement in one spatial dimension –ie. may move left or right on horizontal axis fixed and constant number of receptors on back and front of the cell wave of chemicals ‘retains its shape’ –ie. only displacement to the left speed of the wave is constant

7. The Model We assumed a gaussian (normal) distribution of chemical signal. s(t) describes the shape of the wave as a function of time. = Standard Deviation for the Gaussian distribution. We set this constant equal to one in our simulations. = 5 (Time for wave to reach the cell. This is also a constant value)

8. Front Receptors number of front receptors that are empty number of front receptors that are full are constants

9. number of back receptors that are empty number of back receptors that are full is a constant Since the filled front receptors inhibit the back receptors from picking up the signal, is now a function of There is a time delay b between the filling of the front and back receptors Back Receptors c are constants

10. Solutions Initial Conditions: Time scale: 0 to 20 minutes Let Assuming that there are a total number of 10 receptors on the back and front of the cell. Red = Green = Number of Filled Receptors over Time

11. Net Displacement Let Since the net area under the curve is positive (approx. 23 for this simulation)  over the time interval there are more front receptors that are filled than back  net displacement to the right Direction of Movement