Spatio-temporal dependence of the signaling response in immune-receptor trafficking networks regulated by cell density: a theoretical model

Abstract

Cell signaling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signaling pathways. In most experimental systems, ligand concentration and cell density vary within a wide range of values. Dependence of the signal response on cell density is related with the extracellular volume available per cell. This dependence has previously been studied using non-spatial models which assume that signaling components are well mixed and uniformly distributed in a single compartment. In this paper, a mathematical model that shows the influence exerted by cell density on the spatio-temporal evolution of ligands, cell surface receptors, and intracellular signaling molecules is developed. To this end, partial differential equations were used to model ligand and receptor trafficking dynamics through the different domains of the whole system. This enabled us to analyze several interesting features involved with these systems, namely: a) how the perturbation caused by the signaling response propagates through the system; b) receptor internalization dynamics and how cell density affects the robustness of dose-response curves upon variation of the binding affinity; and c) that enhanced correlations between ligand input and system response are obtained under conditions that result in larger perturbations of the equilibrium ligand + surface receptor [Please see text] ligand - receptor complex. Finally, the results are compared with those obtained by considering that the above components are well mixed in a single compartment.

Figure 1. Representation of the model.

Schematic representation of the production of ligand and receptors, receptor-ligand binding, internalization, recycling and degradation processes. Regions domains and components are, 1) extracellular medium: L = free ligand; 2) cell surface: LS = free ligand, RS = free receptor, and LRS = receptor-ligand complex; 3) within the cell: RI = internalized receptor, LRI = internalized complex, (*) = degraded products. Ligand is produced at rate by a source , and free receptors are synthesized by the cell at rate . The domains are not drawn to scale especially the cell surface.

Figure 2. Spatial domains.

Top: three-dimensional spatial domains: A) , , . The spherical surfaces and are so close that they appear overlapped; B) , other conditions as in A. Note the different scales in A) and B) although the volume of the cell is the same in both cases. Bottom: A) cross section representation of top domain A. The two circles (blue) and (red) can be now distinguished; B) cross section representation of top domain B.

Figure 3. Two-dimensional spatial distributions of ligand at four cell densities.

Spatial distribution of ligand computed at , nM, , , for four cell densities: A) high cell density, ; B) medium-high cell density, ; C) medium-low cell density, ; D) low cell density, . The rate constants used are given in Table 2.

Figure 4. Three-dimensional spatial distributions of ligand at high and low cell densities.

Spatial distribution of ligand computed at high (A, ) and low (B, ) cell densities. Note the difference of scale in both figures although the cell volume does not change. Other conditions as in Fig.3.

Figure 5. Concentration-distance profiles of ligand at high cell density.

Ligand profiles computed as a function of distance at high cell density () and nM for s (A) and s (B). These profiles are defined in the extracellular medium () and distance is measured in the radial direction from the cell surface, i.e. . The 2D plots (not shown) viewed from the cell surface appear blue-red in (A) and red-blue in (B).

Figure 6. Concentration-distance profiles of ligand at low cell density.

Ligand profiles computed as a function of distance at low cell density () for nM. The -values (s) are: a) 10; b) ; c) ; d) ; e) ; f) ; g) ; h) . These profiles are defined in the extracellular medium () and distance is defined as in Fig.5. The corresponding 2D plots (not shown) viewed from the cell surface appear blue-red.

Figure 7. Concentration-time profiles of ligand at medium cell density.

Concentration-time profiles of ligand computed at medium cell density () for nM. These profiles are defined in the extracellular medium () and distance () measured from the cell surface in a radial direction is: a) 0; b) 2.4; c) 7.4; d) 17.4; e) 42.4. The dashed line f) is the profile computed with the non-spatial model of ref. using the kinetic parameters given in this reference and in Table 2.

Figure 8. Spatio-temporal distributions of LRI and RI species.

Top:2D spatial distributions of internalized ligand-receptor complexes and free receptors (LRI and RI species) computed at medium-high cell density () and . The white space inside the larger circle is the extracellular medium. The concentration gradients of LRI and RI are established in opposite directions. Bottom: concentration-time profiles of LRI and RI at close proximity to the cell surface (, curves a) and at the center of the cell (, curves b). Other conditions as in Fig.3.

Figure 9. Concentration dependence of surface and internalized receptors and ligand-receptor complexes on cell density.

RS, RI, LRS, and LRI concentrations computed as a function of the radius of the extracellular medium per cell, . When is expressed in the cell density-values are given by cells/ml. Internalized species (receptors and ligand-receptor complexes) values were computed at the center of the cell (), while surface species were determined at . nM, min. Other conditions as in Fig.3.

Figure 10. LRI Concentration-time profiles at high and low cell densities.

Concentration-time profiles of LRI computed at high (, curves a) and low (, curves b) cell densities. LRI-values were obtained at the center of the cell (). The dashed lines (curves a' and b') are the profiles computed with the non-spatial model of ref. using the kinetic parameters given in this reference and in Table 2. Other conditions as in Fig.3.

Figure 11. Dose-response curves of LRI computed at different cell densities.

The LRI response was computed at the center of the cell () and expressed as the area under the concentration-time profiles for 10 hours. Integration was performed expressing the concentrations of in nM and times in min. The ligand concentration is given in nM. The values of () are: a) 100; b) 50; c) 30; d) 15.

Figure 12. Concentration-time profiles of LRS and LRI with and without endocytic downregulation.

Concentration-time profiles of LRS and LRI computed at two values of the endocytic downregulation ratio : curves (a,a') , curves (b,b') . X = LRS (solid lines), X = LRI (dashed lines). Panel A: curves obtained using the spatial model with . The spatial domain for LRS is and for LRI . The time profiles were computed at for LRS and at the center of the cell () for LRI. The parameters used for computation are those given in Table 2, except curves b and b' which were obtained with a 10-fold increase in the -value. Panel B: curves computed using the non-spatial model of ref. and the kinetic parameters given in this reference and in Table 2. Curves (a,a') , curves (b,b') .

Figure 13. Concentration-time profiles of LRS and LRI without recycling of internalized receptors.

Concentration-time profiles of LRS and LRI computed at two values of the endocytic downregulation ratio : curves (a,a') , curves (b,b') . X = LRS (solid lines), X = LRI (dashed lines). There is no recycling of empty receptors to cell surface (). Panel A: curves obtained using the spatial model. Panel B: curves obtained with the non-spatial model of ref.. Other conditions for panels A and B as in Fig.12.

Figure 14. Concentration-distance profiles of LRI as a function of .

Concentration-distance profiles of internalized ligand-receptor complexes computed for (medium cell density) and s as a function of diffusion coefficient. These profiles are defined inside the cell () and distance is measured in the radial direction from the center of the cell. The values of the diffusion coefficient ) are: a) 0.05; b) 0.1; c) 0.2; d) 0.5; e) 1. Other conditions as in Fig.3.

Figure 15. Extracellular, surface, and internalized response to step changes in ligand input rate.

Concentration responses of L, RS, LRS, and LRI to step changes in the ligand input rate. Two inputs of 10 s duration separated by a recovery phase of 40 s were considered. In both inputs ligand enters the system at a constant rate of 1 nM/s. , , . Panel A: ligand response computed at the outer region of the extracellular medium (, curve a) and at the interface with the cell surface (, curve b). Panel B: concentration-time profiles of empty (X = RS) and occupied (X = LRS) cell surface receptors obtained at . Panel C: concentration-time profiles of internalized ligand-receptor complexes computed at the surface of the cell (, curve a) and in the center of the cell (, curve b). Other conditions as in Fig.3.

Figure 16. LRS response to step changes in ligand input rate with and without induced endocytosis.

Concentration changes of L and LRS to step changes in the ligand input rate. Two inputs of 10 s duration separated by a recovery phase of 40 s were considered. In both inputs ligand enters the system at a constant rate of 1 nM/s. , . The ligand response was computed at the interface between cell surface and extracellular medium (). The surface ligand-receptor complex response was determined at . Panels A: . Panels B: . Other conditions as in Fig.3.