Mapping behavioral specifications to model parameters in synthetic biology
© Koeppl et al; licensee BioMed Central Ltd. 2013
Published: 12 August 2013
With recent improvements of protocols for the assembly of transcriptional parts, synthetic biological devices can now more reliably be assembled according to a given design. The standardization of parts open up the way for in silico design tools that improve the construct and optimize devices with respect to given formal design specifications. The simplest such optimization is the selection of kinetic parameters and protein abundances such that the specified design constraints are robustly satisfied. In this work we address the problem of determining parameter values that fulfill specifications expressed in terms of a functional on the trajectories of a dynamical model. We solve this inverse problem by linearizing the forward operator that maps parameter sets to specifications, and then inverting it locally. This approach has two advantages over brute-force random sampling. First, the linearization approach allows us to map back intervals instead of points and second, every obtained value in the parameter region is satisfying the specifications by construction. The method is general and can hence be incorporated in a pipeline for the rational forward design of arbitrary devices in synthetic biology.
Synthetic biology places emphasis on small, standardized molecular parts and devices, mostly operating at the transcriptional level [1, 2]. With standardization comes the need for rigorous quantitative characterization of such devices and for a compositional theory to reliably build larger systems from small canonical circuits. For now most synthetic circuits implemented in vivo were constructed from a small number of components with topology and parameter values found by trial-and-error. The development of larger synthetic systems necessitates the use of appropriate design methodologies. In silico analyses can provide significant insights into the construction of complex synthetic systems, but due to the poor quantification of experimental and micro-environmental conditions, the predictive capability of in silico models for in vivo implementations remains limited. Apart from experimental limitations, modeling attempts to date most often make simplifying assumptions about all the perturbations that a synthetic construct is facing in vivo. For instance, only a few studies account for the large extrinsic noise [3–5] and in particular the one introduced by variations of plasmid copy number . Incorporating those realistic in vivo constraints will make computational models more predictive, eventually enabling the upfront in silico optimization of transcriptional circuits. A first step toward this goal is to investigate the parameter dependency of certain behaviorial properties of a circuits. In systems biology attempts have already been made to address this problem, however, they either rely on purely local measures [7, 8] such as considered in classical sensitivity analysis [9, 10], or perform random parameter sampling  to determined parameter dependencies.
For a given circuit topology, kinetic parameters and other parameters that are involved in controlling the expression level of molecular species (e.g. promoter activity or number of ribosome binding sites) are important design parameters in synthetic biology. A major challenge is to find a set of parameters that satisfies the behavioral specification of a device . Computer science offers various languages to formally define the proper functioning of a piece of code or hardware. Such specification languages of formal verification are used to check important behavioral properties, such as liveness, safety or fairness . One convenient way to specify such properties is to use temporal logic, which is considered an extension of classical propositional reasoning, where propositional variables may change their truth values over time. A prominent such logic is the linear temporal logic (LTL), where the truth value of the propositions is interpreted over a linear timeline . Such techniques were already applied to investigate robustness of computational models in system biology .
with the stoichiometric matrix , the reaction flux vector and the parameter set.
The brute-force method of determining the parameter region that satisfies a certain behavioral specification usually proceeds by Monte Carlo sampling of parameter sets, generating corresponding trajectories according to (1), checking whether those satisfy S and finally retaining only those parameter sets that led to satisfied specification S. There are two immediate downsides of this approach. First, most draws will be unsuccessful for high dimensional parameter spaces, for tight specifications, or for both. Different approaches using an optimized sampling [11, 17] have been developed to mitigate this problem, but are not solving it as they require convergence of the sampling. Second, drawing parameter points in does not provide guarantees that those points belong to a connected domain of consistent parameter sets. Here we provide first attempts to tackle both problems.
Clearly, sampling a multivariate region with balls of same dimension allow for a complete coverage of the region - something that can only be extrapolated when using pointwise sampling . The question to efficiently sample a region with balls has been addressed in computational geometry and efficient randomized algorithms are available .
Nominal values and meaning of the kinetic parameters for the model of the synthetic sensor construct.
Basal transcription rate
Active-promoter transcription rate
mRNA degradation rate
Protein translation rate
3 (nMsec) −1
0.1 (nMsec) −1
Dimer dissociation rate
Inhibitor binding rate
Inhibitor unbinding rate
Dimer-promoter binding rate
Dimer-promoter unbinding rate
Protein degradation rate
We presented a novel method to determine the parameter region of a biochemical reaction network that is consistent with a certain dynamical, behavioral specification. We defined specifications in a novel and general way that requires only the specification map to be once differentiable with respect to the states of the underlying differential equations. We showed that by locally linearizing this map we can solve the desired inverse problem of finding a parameter region for a given specification. As regions, instead of points, are mapped back to parameter space the scheme is in principle able to cover (given some regularity conditions) the feature and parameter space - something that is not possible with point-wise sampling. We also discuss means for estimating the size of the local neighborhood in order to guarantee certain approximation errors. The computational framework allows a very flexible definition of biologically relevant behavorial features and efficient determination of the corresponding parameter region. Hence, the range of experimentally modifiable parameters, such as promoter binding strength can be determined upfront before the experimental synthesis of a synthetic construct.
Throughout this work we only considered models based on ordinary differential equations, but the outlined framework can be extended to include stochastic dynamical models through the use of moment closure methods, for instance. In general, the specification functional will then involve the expectation operator and Monte Carlo sampling may be required to approximate it. Methods from stochastic sensitivity analysis  can be applied in order to perform the local inversion.
Publication of this article was supported by the Swiss National Science Foundation (SNSF) grant number PP00P2_128503.
This article has been published as part of BMC Bioinformatics Volume 14 Supplement 10, 2013: Selected articles from the 10th International Workshop on Computational Systems Biology (WCSB) 2013: Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S10.
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