1. ODEs part 3

2. Network model example Model considers positive feedback loops involving signalling pathways and/or gene regulation. Multiple interlinked loops that combine fast and slow kinetics

3. Yeast cell polarization Cdc42: part of biochemical signalling pathway Stands out as playing a central role in establishing cell polarity in all eukaryotic cells, irrespective of the biological context. Budding yeast (the model organism Saccharomyces cerevisiae) reaches a critical size, then forms a bud.

4. Cell polarization requires Cdc42 activity In the absence of Cdc42p (yeast Cdc42), cell growth is no longer polarized but instead gives rise to large, round, unbudded cells. Cdc42 cycles between an inactive GTP-unbound state and an active GDP-bound state. Cdc42-GTP levels can be regulated either positively by guanine nucleotide exchange factors (GEFs) or negatively by GTPase- activating proteins (GAPs). In yeast, Cdc24p is the major GEF controlling Cdc42p activity and cell polarization

5. Spatial and temporal control of Cdc42p activity The local activation of Cdc42p at the presumptive growth site The spatial control of Cdc42p activity is also achieved by recruitment of Cdc42p itself to the site of polarized growth (feedback loop) Mathematical models suggest that the local concentration of Cdc42p is generated by a stochastic increase of activated Cdc42p on plasma membrane sites and is amplified by the positive feedback loop

6. Second fedback loop is through the actin cytoskeleton The cytoskeleton is a cellular "scaffolding" or "skeleton" contained within a cell's cytoplasm and is made out of protein. Here, actin filaments are shown in red, microtubules in green, and the nuclei are in blue.

7. Actin filaments are dynamic polymers whose ATP-driven assembly in the cell cytoplasm drives shape changes, cell locomotion and chemotactic migration. Actin filaments have polarity. The actin monomers all orient with their cleft toward the same end of the filament, called the minus end. Actin monomers spiral around the axis of the filament, with a structure resembling a double helix.

8. Fig. 2. Stereo pair images of the yeast actin cytoskeleton through the cell cycle. Amberg D C Mol. Biol. Cell 1998;9:3259-3262 Copyright © 1998 The American Society for Cell Biology

9. Cdc42p directly promotes the selforganization of septins, leading to the formation of a ring that demarcates the mother and the bud Actin structures accumulate at the bud site and, in budding yeast, these are responsible for most of the subsequent events associated with cell polarization Slow feedback loop is (hypothetically) actin-mediated transport of Cdc42. Evidence is that, in the presence of an actin-depolymerizing agent, there is rapid but unstable cell polarization Brandman hypothesis: Slow positive feedback loop is necessary for the stability of the polarized on state, whereas the fast loop is critical for the speed of the transition between the unpolarized off state and the polarized on state.

10. Results The single-slow-loop switch turned on and off slowly and filtered out noise. Adding a second slow loop produced a higher basal activity in the off state, a quicker switch from off to on, and a slower switch from on to off. [...] The single-fast-loop switch turned on and off rapidly [...] adding a second fast loop quickened the transition from off to on and delayed the transition from on to off. In the slow-fast switch, the output turned on rapidly, as a consequence of the kinetic properties of the fast loop, and turned off slowly as a consequence of the kinetics of the slow loop. This allows for independent tuning of the activation and deactivation times.

11. Methods

12. Intra-species competition model Model of intra-species competition combining uniform growth (r 1 ) with a saturating term. This is known as the logistic equation Note that (1-x/K 1 ) is positive while x<K 1, where K 1 is the carrying capacity of the system.

13. Inter-species competition model We consider an example that models the populations of two species that are competing for a common resource. In the absence of the other species, each species grows according to a logistic equation. However, the presence of one species lowers the per capita growth rate of the other species. One way to write the equations for this system is

14. Inter-species competition model Note that if y(0) = 0, then y(t) remains 0, and the equation for x(t) becomes the logistic equation Similarly, if x(0) = 0, then x(t) remains 0 and y(t) is governed by a logistic equation

15. Inter-species competition model Let's consider a specific example, in which r 1 = 1, K 1 = 1, β 1 = 1, r 2 = 3/4, K 2 = 3/4, and β 2 = 2/3. The differential equations are

16. To find the equilibria, we must solve The 1 st equation holds if x=0 or y=1-x. We consider each case separately in the 2 nd equation. 1. If x = 0, then the 2 nd equation implies y = 0 or y = 3/4. So two equilibria are (0,0) and (0,3/4). 2. If y=1-x, then the second equation implies So the other two equilibria are (1,0) and (1/2,1/2)

17. quiver function Syntax quiver(x,y,u,v) Description A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y). For example, the first vector is defined by components u(1),v(1) and is displayed at the point x(1),y(1).

18. using quiver vMin = -0.25; % bounds for plotting only vMax = 1; wMin = -0.01; wMax = 0.15; % build vector field x1list = linspace(vMin,vMax,20); x2list = linspace(wMin,wMax,30); [x1matrix,x2matrix] = meshgrid(x1list,x2list); dx1dt_matrix=zeros(size(x1matrix)); dx2dt_matrix=zeros(size(x2matrix)); for i=1:length(x2list) for j=1:length(x1list) F=dy_fhn([x1matrix(i,j);x2matrix(i,j)],0); dx1dt_matrix(i,j)=F(1); dx2dt_matrix(i,j)=F(2); end subplot(1,2,1) % prepare phase plane quiver(x1matrix,x2matrix,dx1dt_matrix,dx2dt_matrix,0.25) axis([vMin vMax wMin wMax])

19. Exercise Plot the x-y phase plane of the inter-species competition model with a quiver plot. Plot the nullclines (dx/dt=0 and dy/dt=0) Generate plots around each of the four fixed points and use the quiver plots to evaluate whether the fixed points are stable or unstable.