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Design Principles of Cell Circuits with Paradoxical Components

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Design Principles of Cell Circuits with Paradoxical Components Design principles of cell circuits with paradoxical components Yuval Harta, Yaron E. Antebib, Avraham E. Mayoa, Nir Friedmanb, and Uri Alona,1 Departments of aMolecular Cell Biology and bImmunology, Weizmann Institute of Science, Rehovot 76100, Israel Edited by Daniel S. Fisher, Stanford University, Stanford, CA, and accepted by the Editorial Board March 23, 2012 (received for review November 5, 2011) Biological systems display complex networks of interactions both Results at the level of molecules inside the cell and at the level of inter- Fundamental Features of Cell Circuits. We consider models of cells actions between cells. Networks of interacting molecules, such as that interact using chemical signals, such as cytokines, which we will transcription networks, have been shown to be composed of denote generally as ligands. Cells can divide to form new cells of the recurring circuits called network motifs, each with specific dynam- same type, differentiate to new cell types, secrete ligands, and take ical functions. Much less is known about the possibility of such up ligands from the medium. Cells are removed by cell death, mi- circuit analysis in networks made of communicating cells. Here, we gration, or differentiation. We define cell circuits as collections of study models of circuits in which a few cell types interact by means interacting cells that perform specific dynamical functions. of signaling molecules. We consider circuits of cells with architec- To understand cell circuits, it is useful to study simple models tures that seem to recur in immunology. An intriguing feature of of their dynamics. In this section, we introduce notation and these circuits is their use of signaling molecules with a pleiotropic equations as a basis for the following sections. The dynamics of X fi or paradoxical role, such as cytokines that increase both cell the concentration of cells of type can be described using a rst- 1 growth and cell death. We find that pleiotropic signaling mole- order differential equation (Eq. ): cules can provide cell circuits with systems-level functions. These _ functions include for different circuits maintenance of homeostatic X ¼ βX − αX; [1] cell concentrations, robust regulation of differentiation processes, where α is the cell removal rate, and β is its proliferation rate. and robust pulses of cells or cytokines. The rate of new cell production, βX, is proportional to X, be- 1 β cause all cells arise from existing cells by cell division. Eq. T helper cells | IL-2 | Tregs | TGF- | IL-27 results in a sensitive situation: if β < α, X tends to zero, whereas if β > α, X tends to infinity (Fig. 1A). ene regulation networks are composed of a handful of re- Note that nonlinear terms can be added to Eq. 1 to prevent cell Gcurring circuit elements, called network motifs (1). Theory concentrations from diverging to infinity. For example, a nonlinear and experiments have shown that each network motif can carry proliferation rate that decreases as cells approach a limiting con- out specific dynamical functions in an autonomous way, such as centration [e.g., βX(1 − X/Xmax)] limits the population at the filtering noisy signals, generating output pulses, and speeding maximal carrying capacity Xmax,inwhichavailablevolumeorother responses (1). limiting factor runs out. In the present study, we seek mechanisms Here, we ask whether one can apply this approach to the level that can stabilize cell concentrations at values much smaller than of circuits made of interacting cells. For this purpose, we con- the carrying capacity of the body or organ in question. sider cells that communicate by means of secreted molecules. Eq. 1 stands in contrast to the typical equation of molecular These secreted molecules affect cell behaviors such as rate of circuits, where molecule m is produced at rate β and removed at α proliferation and cell death. Previous studies on such cell systems rate (1, 8, 9). The difference is that the production rate of β βm 2 attempted to include many cell types and interactions in a model molecules is usually zero order ( rather than ). Thus (Eq. ), involving numerous biochemical parameters and variables (2–5). m_ ¼ β − αm; [2] Other works focused on the effects of a single cell type responding, for example, to a ligand that it secretes itself; these which has a nonzero stable steady state mst = β/α without need works showed the interplay between cell to cell variability and for additional regulation (Fig. 1B). Specific molecular circuits positive feedback, leading to bistability (selection), formation of can show bistability (1, 8, 9), but these circuits require a special thresholds for immune response, and memory (6, 7). design, such as positive feedback or autocatalysis such that β = Here, we study simple models of circuits made of a few β(m) is a function of m over some range. communicating cell types. Because many of the interactions in The intrinsic instability of Eq. 1 raises the need for regulation cell circuits are poorly characterized at present, we seek models of cell proliferation and removal rates. One mode of regulation in which the exact functional form of the interactions does not involves the response of cells to a ligand, with concentration that affect the conclusions; therefore, models have a degree of gen- is denoted by c. When ligand is produced by cells X and acts on erality. We also scan all possible topologies with a given set of proliferation and removal rates β and α in Eq. 1, feedback components to obtain the widest class of circuits that can per- mechanisms can arise. Below, we discuss simple cell circuits that form a given function. can carry out such feedback. We consider circuit designs that seem to recur in immunology. An additional fundamental process is differentiation, where cell An intriguing feature of these systems is the fact that many se- type X0 gives rise to a different cell type X1, and therefore (Eq. 3), creted signaling molecules (cytokines) are pleiotropic: they have multiple effects, sometimes antagonistic or paradoxical, such as increasing both proliferation and death of a certain cell type. We Author contributions: Y.H., Y.E.A., A.E.M., N.F., and U.A. designed research, performed find that pleiotropic signals, in the configurations suggested by research, analyzed data, and wrote the paper. immune circuits, can provide circuits with specific dynamical The authors declare no conflict of interest. functions. We discuss several such functions—homeostatic cell This article is a PNAS Direct Submission. D.S.F. is a guest editor invited by the Editorial concentrations, robust regulation of differentiation processes, Board. and robust pulses of cells or cytokines—and the corresponding 1To whom correspondence should be addressed. E-mail: [email protected]. immune circuit examples. The predicted functions can be readily This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tested experimentally. 1073/pnas.1117475109/-/DCSupplemental. 8346–8351 | PNAS | May 22, 2012 | vol. 109 | no. 21 www.pnas.org/cgi/doi/10.1073/pnas.1117475109 _ X 1 ¼ βX0 − αX1: [3] A B This simple situation results in dynamics of X1 that depend on the initial concentration of precursor cells X0 (Fig. 1C). One may expect that concentrations of precursors vary from individual to individual and in the same individual over time, raising the challenge of making the response of the system insensitive to such fluctuations (10–13). For the sake of simplicity, in the following analysis, we con- sider only the average cell dynamics, and therefore, no spatial component (7, 14) or stochastic components (6, 15) of the di- versity between cells are included. Such factors can be readily added in future studies. C Homeostatic Cell Concentration with an OFF State by Ligand Enhance- ment of both Proliferation and Removal Rates. As mentioned above, achieving a specified cell concentration requires control of pro- liferation and/or removal rates. We now discuss ways in which such regulation can be achieved. Consider a cell type X that produces ligand c that affects the cell’s proliferation rate β(c)anditsremovalrateα(c). The production– 4 removal equation is (Eq. ) Fig. 1. Without special regulation, cell concentration dynamics are in- herently unstable and sensitive to precursor cell levels, whereas dynamics of _ X ¼ðβðcÞ − αðcÞÞX; [4] molecular circuits are stable. (A) Dynamics of cell concentration where cell proliferation rate is β and removal rate is α.(B) Dynamics of molecule con- where c is produced by X (Eq. 5): centration 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 _ ¼ β X − γ c: [5] β α c 2 population X0 at rate and are removed at rate with no divisions. Eqs. 4 and 5 have a form known as integral feedback (16, 17). They allow a nonzero steady state only at a special ligand level explanations rely on negative feedback mechanisms, which can SYSTEMS BIOLOGY c*, at which proliferation rate equals removal rate β(c*) = α(c*). provide partial compensation for precursor levels (20, 21) (addi- The way in which β(c)andα(c) depend on c is important for the tional discussion in SI Appendix). behavior of the circuit. We consider here the situation found in + CD4 T-cell proliferation, where the cytokine c = IL-2 produced Analysis of All Circuit Topologies with One Cell Type and One Ligand by these cells is a pleiotropic signal for both their proliferation β α c Shows a Small Class of Mechanisms for Stable Cell Concentrations and death (18, 19). In this system, both and increase with (Fig. with an OFF State. To test the generality of the above mechanism, 2 A and B).
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