Raman and Wagner (2011) mapped the genotype space of the Saccharomyces cerevisiae target-of-rapamycin (TOR) signalling circuit by enumerating 69,120 alternative topologies, representing each by a single optimized parameter set, and clustering the resulting signalling dynamics into almost 300 phenotypes; they reported a highly skewed distribution of genotype-set sizes, genotype networks spanning genotype space, and a positive relationship between phenotype robustness and evolvability. Whether these findings reflect the circuit itself or the study's methodological choices - one parameter set per topology, optimization toward the canonical TOR behaviour, and one particular clustering of concentration trajectories - remains unresolved, as does the degree to which the enumerated topologies are realizable in vivo. The problem is to complete the map from circuit topology and kinetic parameters to dynamical phenotype for this circuit, and to test whether the reported genotype-space organization and robustness-evolvability relationship survive parameter variability, alternative optimization targets, and alternative phenotype definitions.

Background

The target-of-rapamycin (TOR) pathway is a nutrient-sensing signalling circuit of kinases, phosphatases and regulatory proteins, conserved from yeast to mammals and central to the control of cell growth. In systems-biology models such a circuit is written as coupled ordinary differential equations: a topology fixes which components activate or inhibit which others, while kinetic parameters (rate constants and binding affinities) fix the strengths of those interactions; the circuit's dynamical behaviour is the set of concentration trajectories of key signalling molecules after a stimulus.

A long-standing programme in evolutionary systems biology asks how biological systems can be simultaneously robust, meaning able to maintain their phenotype under mutational or environmental change, and evolvable, meaning able to produce novel heritable phenotypes. A central concept is the genotype-phenotype map, and in particular the genotype network: the set of genotypes that produce the same phenotype, connected by single small genetic changes. Computational studies of RNA secondary structure, proteins, gene-regulatory and metabolic networks have found recurring structural features: many genotypes per phenotype, a highly skewed distribution of genotype-set sizes, genotype networks extending far through genotype space, and phenotypes with the largest genotype networks being both the most robust and the most able to reach novel phenotypes (Ahnert 2017). Early work had shown comparable neutrality in simple feed-forward signalling circuits (Fernández and Solé 2007).

Raman and Wagner (2011) extended this programme to a biologically realistic, complex signalling circuit: the yeast TOR circuit. They enumerated 69,120 alternative biochemically plausible topologies, treated each topology as a genotype, defined the signalling phenotype through clustering of the concentration time-courses of key molecules (yielding almost 300 phenotype classes; Raman and Wagner 2011 report 286 clusters), and - because each topology involves tens to more than a hundred kinetic parameters - represented each topology by a single parameter set obtained by optimizing toward the canonical, wild-type-like TOR behaviour. They found that a minority of phenotypes accounts for the vast majority of genotypes, that phenotype genotype-sets form genotype networks spanning genotype space, and that phenotypes with larger genotype networks are simultaneously more robust and more evolvable: over a broad range, robustness facilitates phenotypic variability in this circuit. A sibling study applied the same framework to programmable logic hardware (Raman and Wagner 2011, Journal of the Royal Society Interface).

These findings, however, rest on methodological choices the authors themselves flagged as limiting: each topology sampled at a single optimized parameter set; an optimization target identical to the phenotype whose unusual dominance is being measured; phenotype classes defined by one particular clustering of trajectories; and an enumerated topology set whose correspondence to in-vivo biochemical diversity is untested. The unresolved question is whether the circuit's genotype-space organization - in particular the positive robustness-evolvability relationship and the prominence of the canonical phenotype - is intrinsic to the TOR circuit or an artifact of these choices, and what the complete map from topology and parameters to dynamical phenotype looks like.

Problem Statement

Determine whether the genotype-space organization reported by Raman and Wagner (2011) for the Saccharomyces cerevisiae TOR signalling circuit is intrinsic to the circuit or a consequence of their sampling and analysis choices, by completing the map from circuit topology and kinetic parameters to dynamical phenotype. Concretely: (1) estimate, for each enumerated topology, the set of distinct dynamical phenotypes reachable across its parameter space, stating the sampling or optimization effort and the coverage and representativeness criteria used, and assess whether assigning each topology a single phenotype from one optimized parameter set is representative of that map; (2) re-optimize parameters toward alternative target behaviours under matched search effort and acceptance criteria, and test whether the resulting genotype-sets show dominance and robustness comparable to that of the canonical TOR phenotype's genotype-set - that is, whether that phenotype's prominence is intrinsic rather than an optimization artifact; (3) quantify how the number, sizes, and connectivity of phenotype genotype-sets depend on the phenotype definition, including the choice of features, time points, normalization, and clustering of concentration trajectories; (4) assess the degree to which the enumerated biochemically plausible topologies correspond to circuits that are realizable or evolutionarily accessible in vivo, reporting the fraction of enumerated topologies whose interactions are each supported by in-vivo evidence (an interaction counts as supported when it is documented for S. cerevisiae in at least one primary experimental study) and the fraction reachable from the canonical topology through a chain of single-interaction changes that each preserve a viable signalling phenotype (a change preserves a viable phenotype when the modified topology still yields a bounded, stimulus-responsive concentration trajectory belonging to one of the phenotype classes of the same clustering; the reported fraction is the path-connected component, not the immediate single-edit neighbourhood). The circuit model, the enumeration of the biochemically plausible topologies (the experimentally motivated core pathway combined with the elementary topological variants), the canonical topology, the kinetic-parameter ranges, and the wild-type (canonical) target behaviour are those specified by Raman and Wagner (2011) and its supporting information, and are the fixed inputs for all four parts. The overarching objective is a validated statement of whether the positive robustness-evolvability relationship and the skewed, network-structured genotype-space organization of this circuit hold across parameter variability, alternative optimization targets, and alternative phenotype definitions, and whether the enumerated topologies are realizable or evolutionarily accessible in vivo - the common objective that links all four requested results. The four results (1)-(4) are jointly posed: together they establish the overarching validated statement, and each is needed for it.

References

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