Self-assembly of molecular-scale devices [1] is one approach for fabricating molecular circuits. Often a variety of structures perform the same function (e.g., an AND gate). This variation allows selecting structures that not only achieve required functional behaviors when assembled exactly as designed, but are also robust, i.e., operate correctly even with some defects in components or their connectivity.

Robust circuits should function even with poorly characterized networks of component interactions. While particularly relevant for biological networks, this issue also arises in engineered systems when the statistical nature of self-assembly gives some variation in circuit properties.

To achieve this robustness, we propose a design principle of examining
the distribution of network structures giving the desired functionality and selecting those least sensitive to parameter variation. Applying this principle is particularly useful when performance distributions have an extended tail [2].

For circuits with few components, exhaustive enumeration of possible network structures can identify those particularly robust with respect to variation in component interactions. This computational task need not await the ability to fabricate the structures and so allows investigating possible designs prior to their construction.

For example, recent work on a nonlinear differential equation model of a small genetic circuit important for body patterning in Drosophila demonstrated a remarkable robustness with respect to the exact values of the 48 parameters in the model [3]. That is, many parameter sets, chosen unformly at random, enabled the model to reproduce the behavior of the real circuit. Apparently, the circuit's network topology was responsible for its remarkable robustness. The basic circuit occurs in many diverse organisms, so this kind of robustness may have been selected for.

Inspired by such examples, and the general design problem, we define an ensemble of small graphs with nodes that can represent molecular species (e.g. in the case of a biological circuit, transcription factors) and edges representing either activating or repressive interactions on other nodes. Such an ensemble is the set of non-isomorphic, directed, signed graphs. We exhaustively test this set (for up to 6 nodes) searching for networks most likely to have stable fixed points under random choices of edge weights (i.e., interaction parameters).

Our results can help identify circuit topologies most likely to be robust, shedding light on which circuits one might expect to observe in biological systems, as well showing which circuit designs may be most desirable for particular goals [4].

References

[1] G. M. Whitesides et al., "Molecular self-assembly and nanochemistry: A chemical strategy for the synthesis of nanostructures", Science 254:1312-1319 (1991).

[2] T. Hogg, "Robust self-assembly using highly designable structures", Nannotechnology 10:300-307 (1999).

[3] G. von Dassow et al., "The segment polarity network is a robust developmental module", Nature 406:188-92 (2000).

[4] J. Hasty et al., "Computational studies of gene regulatory networks: in numero molecular biology", Nature Reviews Genetics 2:268-279 (2001).