# Distilling Robust Design Principles of Biocircuits Using Mixed Integer Dynamic Optimization

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## Abstract

**:**

## 1. Introduction

- (i)
- Robustness is taken into account using a multi-criteria optimal design loop without introducing significant extra computational effort, thus ensuring good scalability.
- (ii)
- It allows for simultaneous searching through parameter and topology spaces, and it can handle arbitrary topologies with different types of interactions (activation, repression).
- (iii)
- It generates optimal trade-offs between robustness and user-defined performance metrics, i.e., the so-called Pareto-set of best solutions.
- (iv)
- Robust design principles can be distilled by post-processing the Pareto set.

## 2. Methods

- ${y}_{i}=1$ if device i is actively regulating the dynamics; and
- ${y}_{i}=0$ otherwise.

- ${y}_{ij}=-1$ if species i negatively regulates species j;
- ${y}_{ij}=+1$ if species i positively regulates species j; and
- ${y}_{ij}=0$ otherwise (species i does not regulate species j).

- (i)
- The dynamics of the biochemical network:$$f(\dot{z},z,w,y,k)=0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}z\left({t}_{0}\right)={z}_{0}$$
- (ii)
- Biophysical and/or design requirements in form of additional equality and/or inequality constraints:$$h(z,w,y,k)=0$$$$g(z,w,y,k)\le 0$$
- (iii)
- Upper and lower bounds for the decision variables:$${w}_{L}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}w\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}{w}_{U}$$$${y}_{L}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}y\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}{y}_{U}$$

**eSS**[17] and

**MISQP**[18] has shown to work well in our previous studies of automated biocircuit design [13,14]. When evaluating performance of biochemical circuits, we need to define a threshold $\u03f5$ such that a circuit is considered to be functional if:

**eSS**and

**MISQP**provides not only the optimal solution but also a number of near-optimal intermediate samples. To enrich the sampling of the near-optimal parameter space, we use multiple runs (i.e., a multi-start strategy) for the meta-heuristic.

- 1.
- Encode the dynamics of the biochemical regulation network in a Mixed Integer framework (Equation (1)).
- 2.
- Define the cost function J scoring circuit performance, and the threshold $\u03f5$ determining whether a circuit is functional.
- 3.

- 4.

- 5.
- Trim the sampling (final and intermediate solutions corresponding to functional circuits) keeping the designs with a performance cost below the threshold $\u03f5$ (see Figure 1a).
- 6.
- Classify and analyze the solutions attending to their topology.
- 7.
- For each topology $\theta $, collect all the functional circuits with the same structure and compute $PerP\left(\theta \right)$ and $RobP\left(\theta \right)$ (see Figure 1b,c).
- 8.
- From the set of functional structures, prune the Pareto-optimal topologies in terms of $PerP\left(\theta \right)$ and $RobP\left(\theta \right)$ to obtain a Pareto front (see Figure 1d).

## 3. Results

- ${y}_{kj}=-1$ if enzyme k negatively regulates enzyme j;
- ${y}_{kj}=+1$ if enzyme k positively regulates enzyme j; and
- ${y}_{kj}=0$ otherwise (enzyme k does not regulate enzyme j).

**eSS-MISQP**, reaching optimum in less than 4 min of computation time (using Matlab 2015b (Mathworks, Natick, MA, USA) on a PC with Intel 2.8 GHz Xeon processor). To enrich the sampling, multiple runs were executed (around 2000), storing all the solutions achieving the target performance ($\u03f5\le 3.0$). This step resulted in 23,273 solutions.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MIDO | Mixed Integer Dynamic Optimization |

MINLP | Mixed Integer Nonlinear Programming |

ODE | Ordinary Differential Equation |

eSS | Enhanced Scatter Search |

MISQP | Mixed Integer Sequential Quadratic Programming |

NFBLB | negative feedback loop with a buffering node |

IFFLP | incoherent feedforward loop with a proportioner node |

## References

- Marchisio, M.A.; Stelling, J. Computational design tools for synthetic biology. Curr. Opin. Biotechnol.
**2009**, 20, 479–485. [Google Scholar] [CrossRef] [PubMed] - Marchisio, M.A. Introduction in Synthetic Biology: About Modeling, Computation, and Circuit Design; Springer: Singapore, 2018. [Google Scholar]
- Boeing, P.; Leon, M.; Nesbeth, D.N.; Finkelstein, A.; Barnes, C.P. Towards an Aspect-Oriented Design and Modelling Framework for Synthetic Biology. Processes
**2018**, 6, 167. [Google Scholar] [CrossRef] [PubMed] - Lormeau, C.; Rybinski, M.; Stelling, J. Multi-objective design of synthetic biological circuits. IFAC PapersOnLine
**2017**, 50, 9871–9876. [Google Scholar] [CrossRef] - Mayne, D.Q.; Polak, E.; Sangiovanni-Vincentelli, A. Computer-aided design via optimization: A review. Automatica
**1982**, 18, 147–154. [Google Scholar] [CrossRef] - Asmus, J.; Müller, C.L.; Sbalzarini, I.F. Lp-Adaptation: Simultaneous Design Centering and Robustness Estimation of Electronic and Biological Systems. Sci. Rep.
**2017**, 7, 6660. [Google Scholar] [CrossRef] [PubMed] - Barnes, C.P.; Silk, D.; Sheng, X.; Stumpf, M.P.H. Bayesian design of synthetic biological systems. Proc. Natl. Acad. Sci. USA
**2011**, 108, 15190–15195. [Google Scholar] [CrossRef] [PubMed][Green Version] - Woods, M.L.; Leon, M.; Perez-Carrasco, R.; Barnes, C.P. A statistical approach reveals designs for the most robust stochastic gene oscillators. ACS Synth. Biol.
**2016**, 5, 459–470. [Google Scholar] [CrossRef] [PubMed] - Sunnåker, M.; Zamora-Sillero, E.; Dechant, R.; Ludwig, C.; Busetto, A.G.; Wagner, A.; Stelling, J. Automatic generation of predictive dynamic models reveals nuclear phosphorylation as the key Msn2 control mechanism. Sci. Signal.
**2013**, 6, ra41. [Google Scholar] [CrossRef] [PubMed] - Zamora-Sillero, E.; Hafner, M.; Ibig, A.; Stelling, J.; Wagner, A. Efficient characterization o high-dimensional parameter spaces for systems biology. BMC Syst. Biol.
**2011**, 5, 142. [Google Scholar] [CrossRef] [PubMed] - Jen, E. Robust Design: A Repertoire of Biological, Ecological, and Engineering Case Studies; Oxford University Press: New York, NY, USA, 2005. [Google Scholar]
- Ma, W.; Trusina, A.; El-Samad, H.; Lim, W.; Tang, C. Defining network topologies that can achieve biochemical adaptation. Cell
**2009**, 138, 760–773. [Google Scholar] [CrossRef] [PubMed] - Otero-Muras, I.; Banga, J. Automated design framework for synthetic biology exploiting Pareto optimality. ACS Synth. Biol.
**2017**, 6, 1180–1193. [Google Scholar] [CrossRef] [PubMed] - Otero-Muras, I.; Banga, J. Multicriteria global optimization for biocircuit design. BMC Syst. Biol.
**2014**, 8, 113. [Google Scholar] [CrossRef] [PubMed] - Otero-Muras, I.; Henriques, D.; Banga, J. SYNBADm: A Tool for optimization-based automated design of synthetic gene circuits. Bioinformatics
**2016**, 32, 3360–3362. [Google Scholar] [CrossRef] [PubMed] - Otero-Muras, I.; Banga, J. Design principles of biological oscillators through optimization: Forward and reverse analysis. PLoS ONE
**2016b**, 11, e0166867. [Google Scholar] [CrossRef] [PubMed] - Egea, J.A.; Marti, R.; Banga, J.R. An evolutionary method for complex-process optimization. Comput. Oper. Res.
**2010**, 37, 315–324. [Google Scholar] [CrossRef][Green Version] - Exler, O.; Schittkowski, K. A trust region SQP algorithm for mixedinteger nonlinear programming. Optim. Lett.
**2007**, 1, 269–280. [Google Scholar] [CrossRef] - Müller, C.L. Stochastic Methods for Single Objective Global Optimization. In Computational Intelligence in Aerospace Sciences; American Institute of Aeronautics and Astronautics, Inc.: Reston, VA, USA, 2014; pp. 63–112. [Google Scholar]

**Figure 1.**Details of the method. (

**a**) Convergence curve of a single run of the optimization algorithm; solutions with performance values under the threshold $\u03f5$ are collected for further analysis. (

**b**) Global performance of structure $\theta $: frequency distribution of performance value J (median highlighted in red). (

**c**) Robustness of topology $\theta $: interquartile range (IQRs) for each of the active parameters. (

**d**) Pareto front of topologies with optimal trade-off of global performance and robustness.

**Figure 2.**Circuit topologies and adaptive response: (

**a**) design space; (

**b**) topology including an incoherent feedforward loop with a proportioner node (IFFLP) motif; (

**c**) topology including a negative feedback loop with a buffering node (NFBLB) motif; and (

**d**) biochemical adaptation typical response as defined in [12]

**Figure 5.**Frequency distributions of the cost (performance J) for each of the nine most frequent structures. Medians are indicated in red.

**Figure 8.**Pareto solutions with best compromise between robustness and global performance: (

**a**) dynamics of one particular circuit with topology $P3$; (

**b**) performance and (

**c**) robustness for topology $P3$; (

**d**) dynamics one particular circuit with topology $P4$; and (

**e**) performance and (

**f**) robustness for topology $P4$.

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**MDPI and ACS Style**

Otero-Muras, I.; Banga, J.R.
Distilling Robust Design Principles of Biocircuits Using Mixed Integer Dynamic Optimization. *Processes* **2019**, *7*, 92.
https://doi.org/10.3390/pr7020092

**AMA Style**

Otero-Muras I, Banga JR.
Distilling Robust Design Principles of Biocircuits Using Mixed Integer Dynamic Optimization. *Processes*. 2019; 7(2):92.
https://doi.org/10.3390/pr7020092

**Chicago/Turabian Style**

Otero-Muras, Irene, and Julio R. Banga.
2019. "Distilling Robust Design Principles of Biocircuits Using Mixed Integer Dynamic Optimization" *Processes* 7, no. 2: 92.
https://doi.org/10.3390/pr7020092