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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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Turing patterns and morphogen degradation. Stripedpatterns obtainedby numerical integration of eqs 5 and 6, using the same parameters as Figure 3B (white dots), for three different values of μ: 0.45(A), 1.0 (B) and 1.5 (C). In all cases, the initial condition is asmall Gaussian perturbation of the unstable steady state at x = 25. In agreement with Figure 3B, the size of the pattern in the simulation increases with μ.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. For μ= 1.5 (C), the typical Turing pattern decays to the stable solutionafter its initial formation, at t = 75, as a consequenceof a field size effect.
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fig4: Turing patterns and morphogen degradation. Stripedpatterns obtainedby numerical integration of eqs 5 and 6, using the same parameters as Figure 3B (white dots), for three different values of μ: 0.45(A), 1.0 (B) and 1.5 (C). In all cases, the initial condition is asmall Gaussian perturbation of the unstable steady state at x = 25. In agreement with Figure 3B, the size of the pattern in the simulation increases with μ.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. For μ= 1.5 (C), the typical Turing pattern decays to the stable solutionafter its initial formation, at t = 75, as a consequenceof a field size effect.

Mentions: In Figure 3, we showtwo cross-sectionsof the volume displayed in Figure 2 (D = 0.01) with two different planes, where the heatmap isrelated to kmax scale. In panel A, theintersecting plane is rh = 5, and panel B corresponds to ra = 5. From this plot, we can conclude that the convex sideof the volume is related to higher values of kmax and consequently to patterns with smaller typical length.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. Alternatively,increasing the production of activator (or decreasing the productionof inhibitor) can also increase the pattern size. Figure 4 depicts the numerical solution of the eqs 4 and 5 for the parameter valuesindicated by the white dots in Figure 3B, i.e., D = 0.01, ra = 5, rh = 5, and threedifferent values of μ: 0.45, 1, and 1.5. We can see that thesize of the pattern increases with μ, as predicted.


Cooperativity to increase Turing pattern space for synthetic biology.

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

Turing patterns and morphogen degradation. Stripedpatterns obtainedby numerical integration of eqs 5 and 6, using the same parameters as Figure 3B (white dots), for three different values of μ: 0.45(A), 1.0 (B) and 1.5 (C). In all cases, the initial condition is asmall Gaussian perturbation of the unstable steady state at x = 25. In agreement with Figure 3B, the size of the pattern in the simulation increases with μ.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. For μ= 1.5 (C), the typical Turing pattern decays to the stable solutionafter its initial formation, at t = 75, as a consequenceof a field size effect.
© Copyright Policy
Related In: Results  -  Collection

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

fig4: Turing patterns and morphogen degradation. Stripedpatterns obtainedby numerical integration of eqs 5 and 6, using the same parameters as Figure 3B (white dots), for three different values of μ: 0.45(A), 1.0 (B) and 1.5 (C). In all cases, the initial condition is asmall Gaussian perturbation of the unstable steady state at x = 25. In agreement with Figure 3B, the size of the pattern in the simulation increases with μ.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. For μ= 1.5 (C), the typical Turing pattern decays to the stable solutionafter its initial formation, at t = 75, as a consequenceof a field size effect.
Mentions: In Figure 3, we showtwo cross-sectionsof the volume displayed in Figure 2 (D = 0.01) with two different planes, where the heatmap isrelated to kmax scale. In panel A, theintersecting plane is rh = 5, and panel B corresponds to ra = 5. From this plot, we can conclude that the convex sideof the volume is related to higher values of kmax and consequently to patterns with smaller typical length.Because μ is the ratio of the degradation rates of inhibitorrelative to activator, increasing inhibitor degradation or decreasingactivator degradation will increase the pattern size. Alternatively,increasing the production of activator (or decreasing the productionof inhibitor) can also increase the pattern size. Figure 4 depicts the numerical solution of the eqs 4 and 5 for the parameter valuesindicated by the white dots in Figure 3B, i.e., D = 0.01, ra = 5, rh = 5, and threedifferent values of μ: 0.45, 1, and 1.5. We can see that thesize of the pattern increases with μ, as predicted.

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
Related in: MedlinePlus