Table 1.
Summary of models for positive feedback driven switches and resulting behavior.
Figure 1.
Conceptual model of positive feedback.
(A) A simple 2-state model of positive feedback. Signaling molecules can either be in an active (red) or inactive (green) state. Molecules can transition between active and inactive states. Positive feedback occurs because active signaling molecules can recruit inactive molecules to change state. (B) Application of model to cell polarity. Here, active or inactive states correspond to signaling molecule localization on the membrane or cytosol, respectively. Signaling molecules may only be spontaneously activated (with rate ), or recruited (with rate
) if they are within the volumes
or
of the membrane, respectively. Active molecules can spontaneously transition to an inactive state (with rate
). (C) Signaling molecule flux between the membrane and the cytosol. The total number of molecules in the membrane and cytosol are denoted by
and
, respectively. The volume of the cell is denoted by
.
Table 2.
Examples of cluster formation in cell signaling systems with positive feedback.
Figure 2.
Repression, emergence, and loss of polarity for increasing concentrations of signaling molecules.
(A) Three regions of polarization behavior are shown: repression (blue bar); spontaneous emergence (cyan bar) and loss (red bar);. Black curve: averaged membrane fractions of molecules. Red curves: averaged probabilities of observing polarization; signaling molecules are considered clustered when more than 20 molecules are present on the membrane, and 50% of all molecules on the membrane are within a small region covering 15% (dotted curve), 20% (dashed curve), or 25% (solid curve) of the membrane. (*) indicates critical number of molecules . Results are averaged of 50 simulations, performed for each indicated value of
. Changing the minimum cluster size to 10 molecules from 20 does not affect results (not shown). (B) Kymographs of simulations for values of
chosen from the three regions shown in (A). (C) Positive feedback circuits give rise to switch-like behaviors in time and space. 0 min: the positive feedback circuit is initialized with
molecules, 10% of which are randomly distributed on the membrane, and polarity is repressed (
); 30 min (red triangle): 10% of the cytosolic molecules are reseeded to 10% of the membrane; 60 min: 200 particles are added to the cytosol, and polarity switches on (
); 90 min: 200 particles are removed from the cytosol and polarity switches off. Bottom panel: kymograph of simulation is as in (B); top panel: total number of molecules on membrane (gray curve and left axis) and total number of molecules in cell (red curve and right axis). Simulations were performed on a 1-D circular membrane (see Table 3 for model parameters).
Table 3.
Parameters used for simulations in Figure 2.
Figure 3.
Polarity is repressed below a critical total density of signaling molecules.
(A) Illustration shows stability of equilibrium values for cytosolic densities, and
, for varying
when
. Red: stable root; black unstable root. (B) Shown is equilibrium membrane fraction
of signaling molecules on the membrane
for various cell densities and different values of
. Cytosolic buffering occurs when
is nearly zero. (C) The probability,
, that the: cytosol contains exactly
molecules is shown for different total molecule numbers,
, scanned between 0 and
. Steady state probabilities of molecule numbers in the cytosol are computed from stochastic master equation (Protocol S1). Inset: zoom-in of transition region showing bimodality of probability distribution.
Figure 4.
Positive feedback can recurrently generate a single, polarized cluster of signaling molecules.
Figure 5.
Density-dependence of spatial clustering is observed for different spatial geometries.
(A) Modification of model shown in Figure 1B, removing the assumptions that the active and inactive molecules are spatially segregated into different spatial compartments, and that the inactive form is spatially homogeneous (infinite rate of diffusion). Here, both the active (red) and inactive (green) molecules can occupy the same compartment and diffuse at finite speeds given by rates and
, respectively. (B) Numerical implementation of the modified model shown in (A) for three different spatial geometries: (i) active molecules reside on the surface of sphere, while inactive molecules reside in the interior (polarity); (ii) both active and inactive molecules reside in a 3-D volume (cytosol); and (iii) both active and inactive molecules reside on a 2-D surface (membrane). For all geometries, we observe progression from buffered off state to localized clusters to homogeneous on state as the number of molecules is increased. (C) Phase plane diagram for the 2-D model as a function of molecule numbers and membrane area. Numerical simulations using the stochastic molecule simulator Smoldyn illustrate a density-dependent switch in clustering behavior. Inset: analytically computed phase plane diagram; “+” marks indicate locations of simulations; V in equations has dimensions of area (see labels at bottom (B)). All simulations in (B–C) were performed using the Smoldyn stochastic molecule simulator version 2.15 [51]; . Shown are results from running the stochastic simulation for 100 time units (see Protocol S1, Appendix for code and parameter values).