EMBL‐CRG Systems Biology Research Unit, Centre for Genomic Regulation, The Barcelona Institute of Science and Technology, Barcelona, SpainUniversitat Pompeu Fabra (UPF), Barcelona, SpainInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
A. In order to achieve multiple functions it has been proposed that circuits can be structurally modular, i.e they allocate distinct highly interconnected and non‐overlapping sets of genes to each individual function (Di Ferdinando et al, 2001; Solé & Valverde, 2008; Clune et al, 2013; Ellefsen et al, 2015). In this scenario modules do not overlap.
B, C Partial module overlap (Panovska‐Griffiths et al, 2013; Sorrells et al, 2015). The AC/DC circuit is able to alternate between distinct behaviors upon a change in the strength of its gene interactions. This circuit is formed by the superimposition of two distinct modules, a mutual inhibition motif and a repressilator motif, that combine under the same topology. As the strength of specific repressive interactions is adjusted, the AC/DC circuit switches between two distinct dynamical behaviors, that is, a bistable switch or oscillatory behavior, using the mutual inhibition and repressilator circuits, respectively.
D. Hypothetical scenario describing a complete module overlap: The same collection of interacting genes is essential to both functions.
We explore multi‐functional circuits capable of two qualitatively distinct multi‐cellular patterns: lateral induction and lateral inhibition. Analogous to biological processes such as the progression of the morphogenetic furrow in Drosophila (Sato et al, 2013), lateral induction leads to the propagation in time and space of a given gene expression state. In contrast, lateral inhibition describes processes such as neurogenesis, where a fine‐grained pattern of alternating cell fates is formed (Daudet & Lewis, 2005; Petrovic et al, 2014).
While genes (represented by black and yellow nodes) interact identically in both tissues/contexts, an external input signal termed the context signal C allows the circuit to switch between functions. The context signal (pink arrow) affects the basal expression level of one of the genes in every cell of the tissue. A circuit achieves lateral induction when it causes a progressive spread of expression from trigger T (thick black and white arrow) which is received by the central cell of the tissue. A circuit achieves lateral inhibition when it causes consecutive cells to be in alternating gene expression states. In subsequent figures, we use a simplified 2‐cell representation where, for simplicity, the inter‐cellular circuit is only shown in one direction (from the first cell to the second).
Figure 3.Dynamical mechanisms of lateral induction and lateral inhibition
A. Color‐coded complexity atlas that contains all two‐gene circuits able to achieve lateral induction or lateral inhibition. Nodes are circuit topologies and edges link those with a single‐topological change (addition or removal of a gene interaction). Topologies (nodes) are colored according to their function: Blue and red exclusively hold circuits capable of induction or inhibition, respectively. Green topologies are capable of both induction and inhibition (see Fig 4). The atlas layout, where topologies are ordered according to their number of regulatory links, reveals the core motifs at the tip of stalactites.
B, C Minimal core circuits to achieve induction (D0–D5) or inhibition (H0–H5) are classified into three distinct mechanisms for each function. Alternative mechanisms correspond to distinct spatiotemporal courses of gene expression to achieve a given function. The dynamical strategy of each mechanism is captured in the unique final profile. While simulations take places on a one‐dimensional row of 33 cells, for increased clarity, most graphic representations show 15 cells.
Figure 4.Parameter spaces for mono‐functional and bi‐functional circuits
A–D Unlike mono‐functional topologies (A, B), green topologies (C, D) (see Fig 3A) are capable of induction and inhibition depending on the values of their gene interactions. (C, D) Each function occupies a distinct region in parameter space.
Subregion of the complete atlas (Appendix Fig S5A) where strong multi‐functional circuits are shown in black. Of the 13 bi‐functional core circuits (Appendix Fig S5A and B), four are shown here.
Compatible combinations of two core induction and inhibition circuits are candidates to multi‐functionality, labeled A to G.
Multi‐functional motifs show distinct modular properties. Hybrids are composed of two separable modules, or sub‐circuits, while emergent circuits cannot be decomposed into distinct sub‐circuits. As such, hybrid circuits visually appear as the sum of two induction and inhibition circuits—the union of two mono‐functional stalactites—while emergent circuits “emerge” at higher levels of complexity within a stalactite.
Figure 6.Function‐switching mechanism and decomposability of hybrid circuits
How is hybrid C capable of performing both functions upon a change in the context signal? We use the simplified 2‐cell model of Box 2 with parameters (wA = 0.41, wB = 5.49, wC = −0.30, αA = 6.93, αD = 12.79). The context signal changes the position and number of steady states through a pitchfork bifurcation (Strogatz, 2014) (Bifurcation 2). This bifurcation drives the trajectory to access different attractors found in regions of the phase portrait corresponding to induction (θ2) and inhibition (θ4) patterns, respectively.
Phase portraits of the mono‐functional modules (induction and inhibition) that build hybrid C.
The nullclines of hybrid C can be decomposed into sub‐parts which correspond to the induction and inhibition modules.
A–D For circuit with parameters (wA = −0.05, wB = −7.98, wC = 6.47, wD = 9.61, αA = 6.40, αD = 6.81), we follow how concentrations of the four species (Dc1, Ac1, Dc2, Ac2) evolve in time as emergent circuit Activation‐Inhibition AI1 achieves (A, B) inhibition or (C, D) induction. In (B), we see how the phase portrait of AI1 (gray box) is equivalent to that of the mono‐functional inhibition circuits (orange box). (D) The lateral induction pattern results from a pursuit behavior (Verd et al, 2014, 2017) where the horizontal then vertical movement of the attractor θ5 deviates the trajectory which exhibits a sudden change in direction. This dynamic (gray boxes) is not equivalent to the dynamics seen in the minimal lateral induction circuits (orange boxes).
E. Structural view: The lateral inhibition function can be reduced to a sub‐circuit which is indeed the minimal circuit H1, but the lateral induction function cannot—it requires the full circuit.
Figure 8.Pattern transitions to model real biological systems
Real biological systems are found where lateral induction precedes lateral inhibition in the same tissue. In the Drosophila eye, an initial wave of differentiation (morphogenetic furrow) progresses through the tissue to later give rise to a fine‐grained pattern of R8 photoreceptor cells (Sato et al, 2013). In the chick's inner ear, a continuous domain of precursor cells, that is, patch of prosensory cells, gives rise to a mosaic of hair cells and supporting cells (Daudet & Lewis, 2005; Petrovic et al, 2014).
We chose to model hybrid G' circuit with parameters (wA = −4.93, wB = 3.22, wC = 0.43, wD = 0.14, αA = 15.86, αD = 8.62). The context signal is treated as a time‐dependent cue to model the transient nature of tissue environment. We show the final pattern and phase portraits for different values of C from 0 to 1 (Box 2). The context C behaves as a bifurcation parameter (subcritical pitchfork type): Two unstable states coalesce into a new fixed point that changes its stability: from stable to unstable. This type of bifurcations leads to a discontinuity in pattern transition.