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Cooperativity to increase Turing pattern space for synthetic biology.

Diambra L, Senthivel VR, Menendez DB, Isalan M - ACS Synth Biol (2014)

Bottom Line: We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning.Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor.These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

View Article: PubMed Central - PubMed

Affiliation: Centro Regional de Estudios Geńomicos, Universidad Nacional de La Plata , Blvd. 120 No. 1461, 1900 La Plata, Argentine.

ABSTRACT
It is hard to bridge the gap between mathematical formulations and biological implementations of Turing patterns, yet this is necessary for both understanding and engineering these networks with synthetic biology approaches. Here, we model a reaction-diffusion system with two morphogens in a monostable regime, inspired by components that we recently described in a synthetic biology study in mammalian cells.1 The model employs a single promoter to express both the activator and inhibitor genes and produces Turing patterns over large regions of parameter space, using biologically interpretable Hill function reactions. We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning. We show how to control Turing pattern sizes and time evolution by manipulating the values for production and degradation relationships. More importantly, our analysis predicts that steep dose-response functions arising from cooperativity are mandatory for Turing patterns. Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor. These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

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Parameter region for Turing patterns. Region of the parameter spacewhere Turing instabilities develop (i.e., where kmax is positive) for nH =2 and D = 0.01 (A) or D = 0.1 (B).The region of pattern-forming parameter space increases when D decreases (i.e., a 100-fold difference in inhibitor diffusion,relative to slower activator diffusion, is more likely to yield Turingpatterns than is a 10-fold difference). Turing space also increaseswith the steepness of the Hill function nH (see Figures 7 and 8).
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fig2: Parameter region for Turing patterns. Region of the parameter spacewhere Turing instabilities develop (i.e., where kmax is positive) for nH =2 and D = 0.01 (A) or D = 0.1 (B).The region of pattern-forming parameter space increases when D decreases (i.e., a 100-fold difference in inhibitor diffusion,relative to slower activator diffusion, is more likely to yield Turingpatterns than is a 10-fold difference). Turing space also increaseswith the steepness of the Hill function nH (see Figures 7 and 8).

Mentions: The unstable state determined by eqs 12 and 13 allows Turing instabilities in a region of theparameter space spanned by D, μ, ra, and rh, which are terms related to combinations of diffusionrates, degradation rates, and binding affinities. Because we are dealingwith four-dimensional space, we can illustrate only some projectionsof this region on 3D space. In particular, we are interested in theprojections on the space spanned by parameters μ, ra, and rh for some values of D. In Figure 2, we depict the projection of the region of theparameter space where kmax > 0, i.e.,the region where Turing patterns can be developed, for two differentvalues of the ratio of diffusion of activator/inhibitor, D (0.01 and 0.1). Biologically, this corresponds to situations wherethe inhibitor diffuses 100- and 10-fold faster than that of the activator,respectively. As expected, the volume of this region decreases with D, meaning that it is easier to find Turing patterns whenthe inhibitor diffuses much faster than the activator (100-fold fasteris better than 10-fold faster; see also Table 1).


Cooperativity to increase Turing pattern space for synthetic biology.

Diambra L, Senthivel VR, Menendez DB, Isalan M - ACS Synth Biol (2014)

Parameter region for Turing patterns. Region of the parameter spacewhere Turing instabilities develop (i.e., where kmax is positive) for nH =2 and D = 0.01 (A) or D = 0.1 (B).The region of pattern-forming parameter space increases when D decreases (i.e., a 100-fold difference in inhibitor diffusion,relative to slower activator diffusion, is more likely to yield Turingpatterns than is a 10-fold difference). Turing space also increaseswith the steepness of the Hill function nH (see Figures 7 and 8).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4384830&req=5

fig2: Parameter region for Turing patterns. Region of the parameter spacewhere Turing instabilities develop (i.e., where kmax is positive) for nH =2 and D = 0.01 (A) or D = 0.1 (B).The region of pattern-forming parameter space increases when D decreases (i.e., a 100-fold difference in inhibitor diffusion,relative to slower activator diffusion, is more likely to yield Turingpatterns than is a 10-fold difference). Turing space also increaseswith the steepness of the Hill function nH (see Figures 7 and 8).
Mentions: The unstable state determined by eqs 12 and 13 allows Turing instabilities in a region of theparameter space spanned by D, μ, ra, and rh, which are terms related to combinations of diffusionrates, degradation rates, and binding affinities. Because we are dealingwith four-dimensional space, we can illustrate only some projectionsof this region on 3D space. In particular, we are interested in theprojections on the space spanned by parameters μ, ra, and rh for some values of D. In Figure 2, we depict the projection of the region of theparameter space where kmax > 0, i.e.,the region where Turing patterns can be developed, for two differentvalues of the ratio of diffusion of activator/inhibitor, D (0.01 and 0.1). Biologically, this corresponds to situations wherethe inhibitor diffuses 100- and 10-fold faster than that of the activator,respectively. As expected, the volume of this region decreases with D, meaning that it is easier to find Turing patterns whenthe inhibitor diffuses much faster than the activator (100-fold fasteris better than 10-fold faster; see also Table 1).

Bottom Line: We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning.Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor.These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

View Article: PubMed Central - PubMed

Affiliation: Centro Regional de Estudios Geńomicos, Universidad Nacional de La Plata , Blvd. 120 No. 1461, 1900 La Plata, Argentine.

ABSTRACT
It is hard to bridge the gap between mathematical formulations and biological implementations of Turing patterns, yet this is necessary for both understanding and engineering these networks with synthetic biology approaches. Here, we model a reaction-diffusion system with two morphogens in a monostable regime, inspired by components that we recently described in a synthetic biology study in mammalian cells.1 The model employs a single promoter to express both the activator and inhibitor genes and produces Turing patterns over large regions of parameter space, using biologically interpretable Hill function reactions. We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning. We show how to control Turing pattern sizes and time evolution by manipulating the values for production and degradation relationships. More importantly, our analysis predicts that steep dose-response functions arising from cooperativity are mandatory for Turing patterns. Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor. These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

Show MeSH