Considerable progress has been made in identifying and characterizing the component parts of genetic oscillators, which play central roles in all organisms. Nonlinear interaction among components is sufficiently complex that mathematical models are required to elucidate their elusive integrated behavior. Although natural and synthetic oscillators exhibit common architectures, there are numerous differences that are poorly understood. Utilizing synthetic biology to uncover basic principles of simpler circuits is a way to advance understanding of natural circadian clocks and rhythms. Following this strategy, we address the following questions: What are the implications of different architectures and molecular modes of transcriptional control for the phenotypic repertoire of genetic oscillators? Are there designs that are more realizable or robust? We compare synthetic oscillators involving one of three architectures and various combinations of the two modes of transcriptional control using a methodology that provides three innovations: a rigorous definition of phenotype, a procedure for deconstructing complex systems into qualitatively distinct phenotypes, and a graphical representation for illuminating the relationship between genotype, environment, and the qualitatively distinct phenotypes of a system. These methods provide a global perspective on the behavioral repertoire, facilitate comparisons of alternatives, and assist the rational design of synthetic gene circuitry. In particular, the results of their application here reveal distinctive phenotypes for several designs that have been studied experimentally as well as a best design among the alternatives that has yet to be constructed and tested.

Mathematical models of biochemical systems provide a means to elucidate the link between the genotype, environment, and phenotype. A subclass of mathematical models, known as mechanistic models, quantitatively describe the complex non-linear mechanisms that capture the intricate interactions between biochemical components. However, the study of mechanistic models is challenging because most are analytically intractable and involve large numbers of system parameters. Conventional methods to analyze them rely on local analyses about a nominal parameter set and they do not reveal the vast majority of potential phenotypes possible for a given system design. We have recently developed a new modeling approach that does not require estimated values for the parameters initially and inverts the typical steps of the conventional modeling strategy. Instead, this approach relies on architectural features of the model to identify the phenotypic repertoire and then predict values for the parameters that yield specific instances of the system that realize desired phenotypic characteristics. Here, we present a collection of software tools, the Design Space Toolbox V2 based on the System Design Space method, that automates (1) enumeration of the repertoire of model phenotypes, (2) prediction of values for the parameters for any model phenotype, and (3) analysis of model phenotypes through analytical and numerical methods. The result is an enabling technology that facilitates this radically new, phenotype-centric, modeling approach. We illustrate the power of these new tools by applying them to a synthetic gene circuit that can exhibit multi-stability. We then predict values for the system parameters such that the design exhibits 2, 3, and 4 stable steady states. In one example, inspection of the basins of attraction reveals that the circuit can count between three stable states by transient stimulation through one of two input channels: a positive channel that increases the count, and a negative channel that decreases the count. This example shows the power of these new automated methods to rapidly identify behaviors of interest and efficiently predict parameter values for their realization. These tools may be applied to understand complex natural circuitry and to aid in the rational design of synthetic circuits.

An overarching goal in molecular biology is to gain an understanding of the mechanistic basis underlying biochemical systems. Success is critical if we are to predict effectively the outcome of drug treatments and the development of abnormal phenotypes. However, data from most experimental studies is typically noisy and sparse. This allows multiple potential mechanisms to account for experimental observations, and often devising experiments to test each is not feasible. Here, we introduce a novel strategy that discriminates among putative models based on their repertoire of qualitatively distinct phenotypes, without relying on knowledge of specific values for rate constants and binding constants. As an illustration, we apply this strategy to two synthetic gene circuits exhibiting anomalous behaviors. Our results show that the conventional models, based on their well-characterized components, cannot account for the experimental observations. We examine a total of 40 alternative hypotheses and show that only 5 have the potential to reproduce the experimental data, and one can do so with biologically relevant parameter values.

Background

The gap between genotype and phenotype is filled by complex biochemical systems most of which are poorly understood. Because these systems are complex, it is widely appreciated that quantitative understanding can only be achieved with the aid of mathematical models. However, formulating models and measuring or estimating their numerous rate constants and binding constants is daunting. Here we present a strategy for automating difficult aspects of the process.

Methods

The strategy, based on a system design space methodology, is applied to a class of 16 designs for a synthetic gene oscillator that includes seven designs previously formulated on the basis of experimentally measured and estimated parameters.

Results

Our strategy provides four important innovations by automating: (1) enumeration of the repertoire of qualitatively distinct phenotypes for a system; (2) generation of parameter values for any particular phenotype; (3) simultaneous realization of parameter values for several phenotypes to aid visualization of transitions from one phenotype to another, in critical cases from functional to dysfunctional; and (4) identification of ensembles of phenotypes whose expression can be phased to achieve a specific sequence of functions for rationally engineering synthetic constructs. Our strategy, applied to the 16 designs, reproduced previous results and identified two additional designs capable of sustained oscillations that were previously missed.

Conclusions

Starting with a system's relatively fixed aspects, its architectural features, our method enables automated analysis of nonlinear biochemical systems from a global perspective, without first specifying parameter values. The examples presented demonstrate the efficiency and power of this automated strategy.

In this report, we characterize the design principles of futile cycling in providing rapid adaptation by regulatory proteins that act as environmental sensors. In contrast to the energetically wasteful futile cycles that are avoided in metabolic pathways, here we describe a conditional futile cycle exploited for a regulatory benefit. The FNR (fumarate and nitrate reduction) cycle in Escherichia coli operates under two regimes - a strictly futile cycle in the presence of O2 and as a pathway under anoxic conditions. The computational results presented here use FNR as a model system and provide evidence that cycling of this transcription factor and its labile sensory cofactor between active and inactive states affords rapid signaling and adaptation. We modify a previously developed mechanistic model to examine a family of FNR models each with different cycling speeds but mathematically constrained to be otherwise equivalent, and we identify a trade-off between energy expenditure and response time that can be tuned by evolution to optimize cycling rate of the FNR system for a particular ecological context. Simulations mimicking experiments with proposed double mutant strains offer suggestions for experimentally testing our predictions and identifying potential fitness effects. Our approach provides a computational framework for analyzing other conditional futile cycles, which when placed in their larger biological context may be found to confer advantages to the organism.

Mechanistic models of biochemical systems provide a rigorous description of biological phenomena. They are indispensable for making predictions and elucidating biological design principles. To date, mathematical analysis and characterization of these models encounter a bottleneck consisting of large numbers of unknown parameter values. Here, we introduce the Design Space Toolbox v.3.0 (DST3), a software implementation of the Design Space formalism enabling mechanistic modeling without requiring previous knowledge of parameter values. This is achieved by using a phenotype-centric modeling approach, in which the system is first decomposed into a series of biochemical phenotypes. Parameter values realizing phenotypes of interest are subsequently predicted. DST3 represents the most generally applicable implementation of the Design Space formalism and offers unique advantages over earlier versions. By expanding the Design Space formalism and streamlining its distribution, DST3 represents a valuable tool for elucidating biological design principles and designing novel synthetic circuits.