Design principles of cell circuits with paradoxical components

Abstract

Biological systems display complex networks of interactions both at the level of molecules inside the cell and at the level of interactions between cells. Networks of interacting molecules, such as transcription networks, have been shown to be composed of recurring circuits called network motifs, each with specific dynamical functions. Much less is known about the possibility of such circuit analysis in networks made of communicating cells. Here, we study models of circuits in which a few cell types interact by means of signaling molecules. We consider circuits of cells with architectures that seem to recur in immunology. An intriguing feature of these circuits is their use of signaling molecules with a pleiotropic or paradoxical role, such as cytokines that increase both cell growth and cell death. We find that pleiotropic signaling molecules can provide cell circuits with systems-level functions. These functions include for different circuits maintenance of homeostatic cell concentrations, robust regulation of differentiation processes, and robust pulses of cells or cytokines.

Fig. 1.

Without special regulation, cell concentration dynamics are inherently unstable and sensitive to precursor cell levels, whereas dynamics of molecular circuits are stable. (A) Dynamics of cell concentration where cell proliferation rate is β and removal rate is α. (B) Dynamics of molecule concentration produced at rate β and removed at rate α reach a unique, stable steady state. (C) Dynamics of cells X₁ that differentiate from a precursor population X₀ at rate β and are removed at rate α with no divisions.

Fig. 2.

Cytokine control can provide a homeostatic concentration of cells and an OFF state at low initial cell levels. (A) Schematic of a circuit in which cytokine c is produced by cells X at rate β₂ and degraded at rate γ. The cytokine affects both the cell proliferation rate β(c) and the removal rate α(c). (B) An immune example of this circuit, where T_h cells produce IL-2, which increases both their proliferation rate and apoptotic rate. (C) Above a certain threshold in initial cell levels, cell concentration dynamics of circuit (A) converge to a unique steady state (the ON state) independent of initial amounts of X or c. At low initial cell levels, cell concentration dynamics decay to zero (the OFF state). Here, α(c) = c, , β₂ = 1, γ= 5, c(0) = 5, and X(0) = 1, 1.5, 2, 4, 16, and 32. (D–F) For X to be in steady state, its time derivative needs to be zero. There are two ways this can happen: either X = 0 or β(c) = α(c) (Eq. 4). In D–F, circles denote fixed points caused by crossing of β(c) and α(c), whereas squares denote fixed points caused by X = 0 without such a crossing (the OFF state). The latter occurs when c = 0. Full circles and squares mark stable fixed points; open circles and squares mark unstable ones. (D) When cell proliferation and removal rates both increase with c and cross each other with appropriate slopes, a stable steady-state solution is found, c* > 0. In addition, when c = 0, the cell population goes to a second stable fixed point (an OFF state) with no cells, X = 0. (E) When cytokine c inhibits proliferation rate and enhances removal rate, a single stable steady state occurs without a stable OFF state. (F) When cytokine c inhibits removal rate and enhances proliferation rate, the steady-state ON state solution is unstable.

Fig. 3.

Analysis of circuit topologies that yield homeostatic cell concentrations. (A) All 24 connected topologies with a single cell type and a ligand; 15 of 24 topologies can show steady-state cell levels that are independent on initial levels (highlighted in gray). Circuits marked with a star show both ON and OFF stable states. Circuits marked with p show a pulse in ligand levels. (B) A sample of the 280 connected topologies of a two-cell differentiation process controlled by a single ligand. The four topologies highlighted yield a steady-state differentiated cell level that is independent on the precursor cell level. (C) The four topologies highlighted in B and their corresponding model equations.

Fig. 4.

A circuit that produces a robust amount of differentiated cells by endocytosis of a cytokine by the precursor cells. (A) Schematic of a circuit in which cytokine c enhances the differentiation of X₁ cells from precursor cells X₀. The precursors also remove c by endocytosis. (B) An example of the circuit in differentiation of T-helper cells: CD4⁺ cells differentiation to T-helper cells (e.g., Th2 cells) is enhanced by IL-2. However, Treg cell level is roughly proportional to the CD4⁺ pool (5–10% of CD4⁺ levels; proportionality is marked as a dotted arrow). Tregs take up IL-2, thus reducing its levels. (C) Dynamics of X₁ cell concentration reach a homeostatic steady state that is independent of the levels of precursors X₀. Different lines represent different X₀ levels (contrast with Fig. 1C). Here, α = β₀ = 1, β₃ = 3, α₁ = 2, f(c) = c /c + 3, X₁(0) = 1, and c(0) = 10.

Fig. 5.

Cell circuits that produce a robust pulse of cells or cytokines. (A) An incoherent feed-forward loop of cells with a shared cytokine. (B) An example in which c = TGF_β enhances the differentiation of X₀ = CD4⁺ T cells to X₁ = T_h17 cells and Y = iTreg cells. iTreg cells inhibit the differentiation of CD4⁺ T cells to T_h17 by secreting IL-10. Note that iTregs also secrete TGF_β but at later times, after the dynamics considered here are over. (C) For a wide range of parameters, this cell circuit model can display fold-change detection and exact adaptation. The circuit generates a pulse of T_h17 cell levels in response to a change in TGF_β levels. The same fold change in TGF_β leads to the same pulse dynamics, regardless of absolute TGF-β, a feature known as fold-change detection. Here, α₁ = β₁ = β₂ = α₂ = 1, X₀ = 1, X₁(0) = 1, and Y(0) = 0.25. (D) A circuit that produces a pulse of output signal with shape and size that is independent on the triggering input signal strength. Precursor cells X₀ differentiate to X₁ cells when triggered by cytokine c₁ (the input). This differentiation is completed at an early time, and therefore, initial X₁ level [X₁(0)] for the second proliferative stage depends on the levels of precursors cell X₀, the differentiating cytokine c₁, τ (the duration of the first differentiation stage), and (other differentiation-activating cytokines in the environment). At later times, X₁ proliferates and secretes the output signal c₂, which enhances X₁ proliferation. The input cytokine c₁ inhibits production of c₂ (dynamic equations for X₁ and c₂ at the second stage are shown). (E) An example of the circuit: c₁ = IL-27 promotes X₁ = T_h1 differentiation but also inhibits T_h1 secretion of c₂ = IL-2. IL-2 is secreted by T_h1 and increases its proliferation rate. (F) The circuit dynamics show that, although (Right) X₁ levels depend on c₁ levels (different marking for each level), (Left) the shape and size of the dynamic pulse of c₂ levels are independent of c₁ or X₀ levels (all trajectories align). Here, α₁ = β₁ = β₂ = α₂ = 1, c₁ = 1, 2, 4, and 10, X₁(0) = c₁ [i.e., K(X₀, τ, ) = 1], and c₂(0) = 0. (G) Design principle for components with paradoxical pleiotropy in cellular and molecular circuits.