Preprint, 11th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems June 6-8, 2016. NTNU, Trondheim, Norway Exploring Design Principles of Gene Regulatory Networks via Pareto Optimality ? Irene Otero-Muras ∗ Julio R. Banga ∗ ∗ BioProcess Engineering Group, IIM-CSIC, Spanish Council for Scientific Research, Eduardo Cabello 6, 36208 Vigo, Spain (e-mail: [email protected], [email protected]). Abstract: One central problem in systems and synthetic biology is to characterize the biological functions of regulatory network motifs. Here we consider recent model-based exploration approaches used to identify motifs capable of performing a specific biological task. In this work, we propose an optimization based strategy where the motivation is twofold: on the one hand, to introduce efficiency and optimality in the search, by using global mixed integer nonlinear optimization methods. On the other hand, to incorporate multiple design objectives (Pareto optimality), in order to cope with realistic trade-offs observed in nature. The potential of this approach is illustrated through an example where we explore the design principles underlying stripe-forming motifs. Keywords: Global Optimization, Gene Regulatory N'etwork, Synthetic Biology, Systems Biology, Multiobjective Optimization 1. INTRODUCTION et al., 2009) or the formation of stripes (Munteanu et al., 2014; Rodrigo and Elena, 2011). Numerical approaches A gene regulatory (or transcriptional) network consists are usually based on the exhaustive exploration of the of a collection of DNA segments and their interactions topology spaces. Cotterell and Sharpe (2010) proposed to which together regulate biological functions by controlling link topologies together into a non-directed graph based the expression levels and temporal patterns in which gene on topological similarity, and then analyze the shape of a products appear (Karlebach and Shamir, 2008). Transcrip- resultant complexity atlas to determine the core topologies tional networks in living cells are complex, and one of the for a given function. challenges of systems biology is to uncover their structural In this work, we propose an optimization based approach design principles. to find patterns capable of specific biological tasks. In con- A gene regulatory network can be described by a graph, trast to exhaustive exploration, which is computationally where nodes correspond to genes, and edges indicate the expensive and becomes practically unfeasible for increas- transcriptional regulation of one gene by the protein prod- ing levels of complexity, our method aims to introduce uct of another gene. Milo et al. (2002) developed an algo- efficiency in the search, exploiting the potential of Global rithm to detect patterns of interconnections occurring in Mixed Integer Programming solvers (Otero-Muras and real networks more often than in randomized ones, identi- Banga, 2014). fying a first set of network motifs or basic building blocks. Moreover, our approach is multiobjective, allowing not To understand how specific functional outcomes or cellular only to find circuits with a specific functionality, but behaviours emerge from particular interactions of genes optimally performing with respect to a set of predefined and proteins, increasing effort is devoted to analyze the criteria. This is motivated by the fact that the levels of functionalities of these motifs and their interconnections complexity found in biological circuits cannot be explained (Ingram et al., 2006). by the accomplishment of a given function alone, sug- Different analytic and numeric approaches make use of dy- gesting multiple simultaneous (and potentially conflicting) namic models of biochemical networks to explore the map- goals. For example, even if a simple negative feedback is pings between the spaces of topologies and parameters and enough to generate oscillations, many oscillators found in the space of model behaviour (Otero-Muras et al., 2014). nature contain both negative and positive feedback loops. The goal is to find patterns, structural and/or parametric Tsai et al. (2008) demonstrated through a computational features associated to biological functions, like the capacity study a number of advantages conferred by the presence of for bistability and oscillations (Mincheva and Craciun, positive feedback in oscillators, namely period tunability, 2008; Otero-Muras et al., 2012), adaptive responses (Ma improved robustness and reliability. ? We acknowledge funding from the Spanish MINECO (and the Here we will also illustrate how the usefulness of multicri- European Regional Development Fund) project SYNBIOFACTORY teria optimization in synthetic biology design goes beyond (grant number DPI2014-55276-C5-2-R). obtaining a set of optimal trade-offs (Pareto front). We Copyright © 2016 IFAC 809 IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, Norway will show how the analysis of those Pareto solutions can As an example, we consider the three gene network in Fig. lead to a more systematic inference and understanding of 1 with genes A, B and C, where the net internal interaction the underlying design principles. This concept is somewhat matrix is given by: ! similar to the automated innovization approach recently 0 0 0 used in engineering design (Deb et al., 2014). Innovization Ω = !AB !BB !CB attempts to extract innovative design principles through !AC !BC !CC analysis of optimization results. and the gene A is induced by an external input I. The ODE As a proof of concept for our method we consider the system describing the dynamics of the network reads: problem of finding stripe forming motifs in 3-gene con- 1 A_ = − δA figurations, and compare our results with previously pub- 1 + exp(a − b(I)) lished studies based on exhaustive search (Munteanu et al., 1 2014). B_ = − δB 1 + exp (a − b(!ABA + !BBB + !CBC)) 1 2. METHODS C_ = − δC (2) 1 + exp (a − b(!AC A + !BC B + !CC C)) 2.1 Modeling framework Configurations with !AB!CB!BC < 0 give rise to inco- herent feedforward loops. In Fig. 1, the incoherent feed- Gene regulatory networks are represented as directed forward loop of type one IFF1 (! > 0;! > 0 and graphs, with nodes corresponding to genes, and edges AB AC ! < 0) and the incoherent feedforward loop IFF3 indicating their interactions. One arrow from gene A to BC (! < 0;! > 0 and ! > 0) are depicted. gene B indicates the transcriptional regulation of B by AB AC BC the transcription factor encoded by A. Starting from this model (Fig. 1) Munteanu et al. (2014) investigated in a recent work 3-gene configurations capable The dynamics of a gene regulatory network can be de- of translating a morphogen gradient into a single stripe scribed through a system of Ordinary Differential Equa- pattern. They considered a monotonically increasing input tions representing the mass balances of the species in- along a one-dimensional tissue of N isogenic cells, i.e., N volved. Detailed models might include promoters, RNA circuits with the same values of Y , W and the parameters polymerase, mRNA, proteins and complexes among species. a and b, and only varying the input I. Simpler models take into account time scale separation, lump transcription and translation into a single step (Ke- They found two incoherent feedforward motifs (IFF1 and pler and Elston, 2001) and consider only the levels of IFF3) as core topologies for stripe formation. the transcription factor proteins encoded by the network genes. In this way, for a n-gene network, the state vector of the ODE model, z(t) 2 Rn contains the levels of the n A A A proteins at time t. ωAC ω C In this work we use a connectionist model (Mjolsness AB ω C C CB ω et al., 1991) to describe gene regulation. This model is CC biologically-verified and extensively employed in the study ω B of developmental gene networks (Munteanu et al., 2014). B BC B Within this framework the regulation from gene Gi to IFF1 IFF3 gene Gj is characterised by two numbers: an integer yij 2 ωBB {−1; 0; 1g, coding for inhibition (−1), no action (0), and activation (1), and a strictly positive weight wij 2 R>0. Fig. 1. Graph of the connectionist 3-gene model with gene 2 Zn×n 2 Rn×n We can construct two matrices Y and W >0 A induced by an external input. The middle and right containing respectively the gene-gene interaction indices circuits correspond to incoherent feedforward loops of and the weights. type 1 and 3 respectively. The effective regulating input to a gene Gi is given by: 2.2 Mixed Integer Non Linear Programming formulation Xn χi = !jizj + αiI The search of a circuit performing a specific behaviour j=1 through the space of n-gene circuits can be formulated as where !ji = yjiwji, and the term αiI reflects the effect an optimization problem. of external inputs (in case the gene G is only affected i Within the modeling framework previously described, a by internal gene-gene interactions, the coefficient α = 0). i circuit (structure and parameters) can be characterized by The transcription rate is proportional to the sigmoidal- two vectors: w 2 Rr containing the weights (its elements filtering of the total contribution, such that the balance >0 are taken columwise from W ), and a vector of integer for the protein z encoded by G reads: i i variables y 2 Zr determining the interactions (its elements 1 z_i = − δzi (1) are taken columwise from Y ). Parameters that are fixed 1 + exp(a − b(χi)) in the ODE model describing the network dynamics are included in a vector k 2 Rk. where parameters a and b control the steepness and location of the threshold value of the regulation function, We can encode the specific desired performance of the and δ is the protein degradation rate constant. circuit by a suitable function J1(_z; z; w; y; k), such that 810 IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, Norway it reaches a minimum when the desired functionality is As a second optimization goal, we consider the protein achieved (importantly, the solution does not need to be production cost, which we assume proportional to the unique).