ODEs part 3

Network model example

Model considers positive feedback loops involving signalling pathways and/or gene regulation.

Multiple interlinked loops that combine fast and slow kinetics

Yeast cell polarization

Cdc42: part of biochemical signalling pathwayStands 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.

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

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

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.

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.

+

actin-ADP

actin-ATP

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

• 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.

Results

The single-slow-loop switch turned onand off slowly and filtered out noise.Adding a second slow loop produced a higherbasal activity in the off state, a quicker switchfrom off to on, and a slower switch from on tooff. [...] 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, andturned off slowly as a consequence of thekinetics of the slow loop. This allowsfor independent tuning of the activation anddeactivation times.

Methods

Intra-species competition model

Model of intra-species competition combining uniform growth (r1) with a saturating term. This is known as the logistic equation

Note that (1-x/K1) is positive while x<K1, where K1 is the carrying capacity of the system.

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

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

Inter-species competition model

• Let's consider a specific example, in which r1 = 1, K1 = 1, β1 = 1, r2 = 3/4, K2 = 3/4, and β2 = 2/3. The differential equations are

To find the equilibria, we must solve

The 1st equation holds if x=0 or y=1-x. We consider each case separately in the 2nd equation.

1. If x = 0, then the 2nd 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)

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).

using quiver

vMin = -0.25; %% bounds for plotting onlyvMax = 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); endend

subplot(1,2,1) %% prepare phase planequiver(x1matrix,x2matrix,dx1dt_matrix,dx2dt_matrix,0.25)axis([vMin vMax wMin wMax])

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.