Turing Patterns Are Common but Not Robust

bioRxiv preprint doi: https://doi.org/10.1101/352302; this version posted June 20, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Turing patterns are common but not robust

Natalie S. Scholes,*,1 David Schnoerr,*,1 Mark Isalan,1 Michael Stumpf,†,1,2

1Department of Life Sciences, Imperial College London, London SW7 2AZ, UK. 2School of BioScience and School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia.

Abstract they are capable of generating entirely self-organized, complex, repetitive patterns of gene expression (Fig- Turing patterns (TPs) underlie many fundamental de- ure 1A, B). velopmental processes, but they operate over narrow parameter ranges, raising the conundrum of how evo- TPs generally alter local concentrations of biochem- lution can ever discover them. Here we explore TP de- ical components, resulting in self-organized spatial sign space to address this question and to distill design patterns such as spots, stripes and labyrinths (Kondo rules. We exhaustively analyze 2- and 3-node biologi- and Miura, 2010). These patterns have unique and cal candidate Turing systems: crucially, network struc- useful biological properties: perturbing them results ture alone neither determines nor guarantees emer- in recovery and re-organization of the patterns ("heal- gent TPs. A surprisingly large fraction (>60%) of ing"), as an intrinsic property of the dynamical bio- network design space can produce TPs, but these are chemical interactions. This also implies that if there is sensitive to even subtle changes in parameters, net- variability in size across individuals, the TPs will au- work structure and regulatory mechanisms. This im- tomatically re-scale themselves, simply adding or sub- plies that TP networks are more common than pre- tracting pattern segments in response to different field viously thought, and evolution might regularly en- sizes. This is a valuable property to support changes in counter prototypic solutions. Importantly, we deduce size, both within existing populations and over evolu- compositional rules for TP systems that are almost tionary time. In addition, TP networks are extremely necessary and sufficient (96% of TP networks contain parsimonious, often employing just two or three bio- them, and 95% of networks implementing them pro- chemical species. This implies that they might be an duce TPs). This comprehensive network atlas provides economical solution for evolution to employ, wherever the blueprints for identifying natural TPs, and for en- repetitive self-organizing patterns are needed. gineering synthetic systems. Given these advantages, it is perhaps not surprising that TPs are regarded as the driving morphogenetic Introduction patterning mechanisms in many biological systems. These include bone and tooth formation, hair folli- Pattern formation is an essential aspect of develop- cle distribution and the patterns on the skins of ani- ment in biology and we have a wealth of examples how mals, such as fish and zebras (Raspopovic et al., 2014; complex, structured, multi-cellular organisms develop Sick et al., 2006; Jung et al., 1998; Nakamasu et al., from single fertilized cells. Many organisms develop 2009; Economou et al., 2012). However, despite sev- complex spatial features with exquisite precision and eral experimentally verified examples (Raspopovic et robustness, and this has been the subject of extensive al., 2014; Sick et al., 2006), the underlying complex- molecular and theoretical study (Green and Sharpe, ity in biological systems has often prevented identifi- 2015; Maini et al., 2012). cation of the precise molecular mechanisms governing Various mechanisms have been proposed to ex- the potentially large number of TPs in nature. A sec- plain developmental patterning processes, ranging ond key problem is that there is a paradox between from maternally inherited cues (Wolpert, 1969), to the apparent widespread distribution of natural TPs mechanical forces (Howard, Grill, and Bois, 2011) and the observation — from mathematical analyses and chemical reaction-diffusion networks or Turing (Gaffney, Yi, and Lee, 2016; Iron, Wei, and Winter, patterns (TPs) (Gierer and Meinhardt, 1972; Turing, 2004; Palmer, 2004; Meinhardt and Gierer, 2000; 1952). The latter were first proposed by Alan Turing Gierer and Meinhardt, 1972) — that kinetic param- in 1952 (Turing, 1952), and were later independently eters need to be finely tuned for TPs to arise. This described by Gierer and Meinhardt (Gierer and Mein- raises the questions of how evolution could ever dis- hardt, 1972). TPs are particularly intriguing because cover such tiny islands in parameter space and, even *Authors N. S. and D. S. contributed equally. if it could, how would the resulting developmental †Corresponding author: [email protected] mechanisms still occur robustly under noisy real con-

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ditions. alytical computations and allows the authors to ana- One approach to resolve these apparent contradic- lyze all 3-node and 4-node networks. However, these tions is to explore TP systems mathematically. While a mathematical simplifications lead to small number of rich mathematical literature on Turing patterns exists, networks being stable, and only a tiny fraction exhibit- the vast majority of studies analyze single, idealized ing Turing patterns, even compared to (Zheng, Shao, networks with fixed parameters (Gaffney, Yi, and Lee, and Ouyang, 2016). But shifting the stationary states 2016; Iron, Wei, and Winter, 2004; Liu et al., 2013). to zero has been shown to systematically misrepresent Although these studies have significantly increased the stability properties of real dynamical systems (Kirk our understanding of patterning mechanisms, they do et al., 2015; Maclean, Kirk, and Stumpf, 2015): as this not provide general guidelines for either the identifi- procedure destroys the dependencies between stabil- cation of naturally evolved Turing networks in biolog- ity of stationary states and the reaction rate param- ical systems, or for synthetic engineering of TP net- eters (Kirk et al., 2015) which results in high rates works (Scholes and Isalan, 2017; Borek, Hasty, and (frequently up to 90% or more) of misclassifying sta- Tsimring, 2016; Carvalho et al., 2014; Duran-Nebreda ble stationary states as unstable. and Solé, 2016; Boehm, Grant, and Haseloff, 2018; Therefore, an exhaustive analysis and comparison Cachat et al., 2016; Diambra et al., 2015; Cachat et of different, biologically relevant regulatory functions, al., 2016). A recent approach which has proved very as well as comprehensive sensitivity/robustness anal- successful in increasing our understanding of biolog- yses of Turing networks with respect to parameter ical design principles is the "network atlas" approach variations, still remain difficult to achieve. This is be- (Babtie, Kirk, and Stumpf, 2014; Ma et al., 2009). In cause there is a combinatorial explosion in the number this approach, biological networks that execute a par- of conditions to explore, and this is computationally ticular function are modeled exhaustively: for exam- nearly prohibitively expensive. ple, all 2 and 3-node inducible genetic networks that In this study, we develop a new computational achieve switch-like behavior were modeled by MA et pipeline capable of analyzing stable solutions for a al. (Ma et al., 2009). Similarly, all 3-node networks wide range of potential TP network structures. Briefly, that form a central stripe pattern in a developmental we use linear stability analysis to determine the emer- morphogen signaling gradient were modeled by the gence of stable TPs (see Methods) and the correspond- group of Sharpe (Cotterell and Sharpe, 2010; Schaerli ing wavelength of the pattern (Figure 1C). et al., 2014). Such approaches allow one to compare Thus, we generate an extensive analysis of 7757 network and parameter design space (Barnes et al., unique networks with up to three reacting species, 2011) with the resulting phenotypic map, resulting in testing them exhaustively for their ability to form TPs an atlas or guidebook for the design principles behind (we analyzed approximately 3 1011 different scenar- that function. A guidebook of potential mechanisms ios - amounting to 8 CPU years× computing time). As and design rules for discovering TP networks would summarized in Figure 1D, these are analyzed in terms help towards solving the problem of characterizing of (1) the network topology; (2) the regulatory func- molecular players in natural TP systems. Furthermore, tion; (3) the kinetic parameters; (4) the diffusion con- a comprehensive Atlas of Turing network space might stants of the different species. We consider both com- shed new light on the problem of how evolution could petitive and non-competitive regulatory mechanisms ever discover and stabilize systems which only ever (Figure 1E) and study their quantitative and qualita- function in tiny islands of parameter space. tive differences. In terms of progress towards making a TP net- In this way, we are able to systematically explore work atlas, two recent studies have begun to ana- what proportion of network topologies are capable of lyze larger sets of TP network topologies and param- generating TPs. Moreover, we rank these networks eters and have made important progress in our global with respect to their robustness to variations in net- understanding of Turing systems (Zheng, Shao, and work topology, kinetic parameters and diffusion rates, Ouyang, 2016; Marcon et al., 2016). Zheng et al. allowing us to determine which kinds of networks are analyze all 2- and 3-node networks for one particu- more robust. Using an unsupervised classifier, we thus lar choice of regulatory function (Zheng, Shao, and identify a set of irreducible or minimal networks from Ouyang, 2016). This study does thus not consider which all Turing networks can systematically be con- different regulatory mechanisms, and the number of structed. This in turn allows us to distill and test com- identified Turing networks is only a fraction of those positional rules for predicting whether a given net- identified here which is easily explained by the differ- work will support TPs. ences in how exhaustively the parameter-spaces are With these results in hand we can identify the net- explored. In a further study Marcon et al. use expo- works that are most suitable for downstream synthetic nential and sigmoidal regulatory functions and do nei- engineering under different physiological conditions. ther include a basal production rate nor a degradation There is growing interest in synthetic biology to en- term (Marcon et al., 2016). Moreover, they shift the gineer patterning systems from first principles (Basu regulatory functions such that the stable steady states et al., 2005; Schaerli et al., 2014; Borek, Hasty, and are always at zero concentrations. This allows for an- Tsimring, 2016; Carvalho et al., 2014; Duran-Nebreda

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and Solé, 2016; Boehm, Grant, and Haseloff, 2018; 1934 networks with two and three nodes, respectively. Cachat et al., 2016). Artificial TPs are expected to For 3-node systems, we further have to distinguish have eventual applications in nanotechnology, tissue between which nodes diffuse and which ones are as- engineering and regenerative medicine (Scholes and sumed to be stationary. For a system with two diffus- Isalan, 2017; Tan et al., 2018). Despite much effort in ing entities this results in 5802 networks (see Supple- this area, engineered TPs remain elusive. Our compre- mentary Document S1 for a complete list of network hensive Turing network atlas contains the "blueprints" graphs), where A and B denote diffusing nodes and required for identifying natural TPs, and guiding the C is assumed to be stationary. Additionally, 1934 net- engineering of synthetic systems. works exist for the case where all three nodes are allowed to diffuse. Having defined the network topologies, we need to Results specify the functions and regulatory mechanisms (Fig- Developing a tractable model to search for TPs ure 2B). In this work, we use Hill-type functions for the regulation as these have been found to fit well with To generate a network atlas for exploring the design experimental measurements of gene regulatory net- space of TP formation, we need to develop a model- works (Estrada et al., 2016; Becskei, Séraphin, and ing framework that copes with this computationally Serrano, 2001; Rosenfeld, Elowitz, and Alon, 2002; demanding task. Our goal is to study the dynamical Burrill and Silver, 2010; Gardner, Cantor, and Collins, behavior of spatially-distributed molecule concentra- 2000; Ferrell and Machleder, 1998; Ferrell, Tsai, and tions, and their capability to form stable spatial pat- Yang, 2011; Klumpp, Zhang, and Hwa, 2009). We terns, across the complete range of 2- and 3-node employ two different types of regulatory mechanisms: network topologies. Furthermore, we aim to employ fully non-competitive and fully competitive interac- different regulatory functions (competitive and non- tions (Figure 2B). For gene regulatory networks, the competitive Hill functions), over as wide a range of former corresponds to the situation where all tran- parameters as possible (e.g. diffusion, activation, and scription factors independently regulate the corre- repression, etc.) and necessary; we thus forgo mathe- sponding target gene (Figure 1E, left panel). By con- matical convenience and tackle biologically more real- trast, for the fully competitive case, all transcription istic models using state-of-the-art computationally in- factors compete for their shared or overlapping bind- tensive, but robust methods. ing sites (Figure 1E, right panel). In addition to the A network’s capability to form patterns is deter- regulatory interactions, we include a basal production mined by four factors which can be ordered hierarchi- rate and a linear degradation term for each species cally as shown in Figure 1D. First, a network’s topol- (see Methods for a detailed description and Figure 2B ogy needs to be defined, that is, the nodes and edges of for an example). the network. Next, the functional form of the interac- Our approach (Figure 2C) allows us to investigate tions fi need to be specified. Subsequently, the param- many different networks and conditions at the appro- eters of these functions need to be specified. Finally, priate resolution to determine whether they can gen- for a spatial model, we need to describe the molecular erate TPs or not. We sample parameters and initial diffusion processes. conditions sufficiently densely and are thus able to de- The first task is to define the network structure (Fig- termine with a high degree of certainty whether a sys- ure 2A). For this, we only consider connected net- tem can exhibit the hallmarks of a TP mechanism: (i) works which cannot be split into non-interacting sub- stability of the non-spatial dynamics; with (ii) simulta- networks, since these would comprise redundant in- neous instability of the corresponding spatial dynam- dependent sub-networks with smaller node numbers. ics. We further exclude networks with nodes that have no Accordingly, to find Turing instabilities, we first incoming edge, since such nodes do not experience need to identify the stable steady states of a given sys- any feedback from the other nodes and will hence tem, and subsequently study their dispersion relation always converge to a spatially homogeneous steady which is defined as the largest eigenvalue of the lin- state. The influence of such nodes on the rest of the earized system as a function of wavenumber q. (see network would thus not have any impact on spatial Methods for details). patterning. Moreover, we exclude networks that have If spatial diffusion is added, it is possible that devia- nodes with no outgoing edge as these would purely tions from the steady state of certain length scales (or, act as "read-out" modules; they do not feed back to alternatively, wavelengths) do not decay towards the the dynamics of the rest of the network. Finally, we homogeneous steady state, but instead become ampli- reduce the number of networks using symmetry ar- fied. This is called a diffusion-driven or Turing insta- guments (Ma et al., 2009; Babtie, Kirk, and Stumpf, bility. If only an intermediate range of length scales 2014), where simply relabeling nodes maps one onto experiences such an amplification, we speak of a Tur- the other. Only one network from such an equivalent ing I instability. In this case, a system typically forms group needs to be considered. a pattern of the wavelength for which the amplifica- This network pruning amounts a total of 21 and tion is maximal (see Figure 1C). Thus, we are able

3 Mammalian systems as an example for competitively regulated systems Mammalian systems as an example for competitively regulated systems In eukaryotic systems intercellular communication is highly developed and thus cell-to-cell signaling moleculesIn eukaryotic exist systems in abundance. intercellular External communication signals are is highlytypically developed conveyed and to thus the cell-to-cell cell by signaling signaling pathwaysmolecules downstream exist in abundance. of membrane-bound External receptors.signals are Binding typically to conveyed such receptors to the by cell activating by signaling or inhibitingpathways species downstream is often of of membrane-bound a competitive nature. receptors. Moreover, Binding truncated to such receptors versions by of activating activating or speciesinhibiting frequently species form is often inhibiting of a competitive counterparts. nature. We therefore Moreover, explore truncated mammalian versions systems of activating as a scenariospecies in frequently which competitive form inhibiting interactions counterparts. are dominating. We therefore We have explore identified mammalian the most systems robust as a competitivescenario in Turing which networks competitive (Figure interactions XX). are dominating. We have identified the most robust competitiveWe further Turing explored networks the generated (Figure network XX). atlas in the light of an existing network that was proposedWe further for engineering explored purposes: the generated a system network using atlas HGF in the (hepatocyte light of an growth existing factor) network and that NK4 was (aproposed truncated for version engineering of HGF) purposes: as activator a system and inhibitor, using HGF respectively (hepatocyte [?]. growth They both factor) mediate and NK4 a response(a truncated via the version c-Met of receptor HGF) as signaling activator pathway and inhibitor, that activates respectively or represses [?]. They a human both derived-mediate a MMP-1response promoter via the construct. c-Met receptor In a previous signaling study, pathway this that system activates was analyzed or represses as a a classical human 2-node derived- systemMMP-1 and promoter the authors construct. suggested In thata previous a single study, promoter thissystem construct was driving analyzed the as expression a classical of 2-nodeboth activatingsystem and and the inhibiting authors species suggested could that su affi singlece to promotercreate Turing construct Patterns driving [?]. the This expression system would, of both however,activating require and inhibiting differential species diffusion could (10 su fold)ffice andto create a co-operativity Turing Patterns factor [? of].>= This 2 system for feasible would, Turinghowever, pattern require generation. differential In reality, diffusion the (10 network fold) can and be a co-operativityseen as a 3-node factor network of >= in which 2 for c-Met feasible composesTuring pattern the central generation. node and In mediatesreality, the the network signals can between be seen both as a HGF 3-node and network NK4 (see in which Fig 7 c-Met C). Withcomposes this model, the central the network node and equals mediates the core the topology signals #28 between with both nodes HGF B and and C NK4diffusing. (see FigIndeed, 7 C). thisWith topology this model, is amongst the network the top equals 10 % of the the core most topology robust#28 topologies with nodes according B and to C our diff results.using. Indeed, Even so,this the topology classical is requirements amongst the found top 10 for % the of the 2-node most system robust persist topologies HGF according and NK4 to are our more results. likely Even to diffso,use the at classical quite similar requirements rates, making found for this the network 2-node only system implementable persist HGF if and HGF NK4 could are bemore modified likely to todi diffffuseuse at much quite more similar slowly. rates, To making decrease this the network necessity only of implementable differential diffusion, if HGF and could maintain be modified the originalto diffuse single much promoter more slowly. design, To the decrease network the atlas necessity reveals of that diff includingerential di aff positiveusion, and feedback maintain loop the onoriginal top of thesingle receptor promoter would design, suffice the to network make equal atlas di revealsffusivity that accessible including for a given positive network feedback (# loop63 BC).on top This, of however, the receptor though would improving suffice to robustness make equal with diff respectusivity accessibleto extracellular for given parameters network does (# 63 notBC). improve This, intracellular however, though robustness improving significantly robustness and with thus respect in total, to robustness extracellular is only parameters marginally does improvednot improve (1.3 fold). intracellular To significantly robustness improve significantly the design and (4.6thus fold), in total, an additional robustness direct is only interaction marginally betweenimproved HGF (1.3 and fold). NK4 To would significantly have to improve be engineered the design (NK4 (4.6 activating fold), an HGF). additional Overall, direct this interaction shows howbetween the network HGF and atlas NK4 can would help decipher have to engineering be engineered options. (NK4 activating HGF). Overall, this shows how the network atlas can help decipher engineering options. 2 formulas 2 formulas @A 1 1 = VA + bA µAA (1) @t@A · 1+(kAA1 )n · 1+( B1 )n = VA A kBA + bA µAA (1) @t · 1+(kAA )n · 1+( B )n A kBA @B 1 = VB + bB µBB (2) @t@B · 1+(kAB1 )n = VB A + bB µBB (2) @t kAB n · 1+( A ) bioRxiv preprintA doi:n https://doi.org/10.1101/352302; this version posted June 20, 2018. The copyright holder for this preprint (which was @A ( k ) notAAA certifiedn by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. @A= VA A ( ) B + bA µAA (3) @t · 1+( )nkAA+( )n = VA kAA kBA + bA µAA (3) @t · 1+( A )n +( B )n kAA kBA A n @B ( k ) ABA n @B= VB ( A ) + bB µBB (4) @t · 1+(kAB )n = VB kAB + bB µBB (4) @t · 1+( A )n 1 A kAB B 2 A B 3 A B 4 A B 5 A B 6 A B 7 A B A B “Slow activator” @fA D q2 @fA @A 1activating@B @fA 2 @fA Reaction & Pattern J = @A D1q @B (5) 8 @fA @f B 9 A B 10 A B 11 A B 12 A B 13 A B 14 A B J =2 B B 23 time @A @B D2q (5) 2 @fB @fB 23 inhibiting D2q 4 @A @B 5 4 2 5 @15A A A B 16 A B 17 A B 18 A B 19 A B 20 A B 21 A B = ↵ + 2 µ A (6) @t@A · BA A = ↵ + µAA Non(6)- @t · B time homogeneous formation @B 2 initial state Diffusion @B= A µBB (7) @t = · A2 µ B “Fast inhibitor” (7) time @t · B 2 2 C D Topology

q max Regulatory function

2π /q max Function parameters

Diffusion constants

E Non-Competitive Competitive

Figure 1: Reaction networks and Turing instabilities (A) A network graph of the Gierer-Meinhardt model (Gierer and Meinhardt, 1972) as an example for a 2-node Turing network (top), with the corresponding ordinary differential equations below (bottom). Blue and red arrows indicate activating and inhibiting regulations, respectively. Species A activates both itself and species B, while species B inhibits species A. (B) The top left panel represents the diffusion profiles for species A (blue, slow activator) and the bottom panel for species B (red, fast inhibitor). Over time, small deviations in a noisy, non-homogeneous initial condition (Panel 2) can get amplified by the interplay of reactions and diffusion (Panel 3). For the given system this can lead to the formation of stable patterns (Panel 4). (C) An exemplary dispersion relation (real part of the largest eigenvalue of the linearized system as a function of wavenumber q.) of the system shown in (A). The wavenumber qmax for which the dispersion relation is maximal becomes amplified the strongest. This leads to the formation of a pattern with wavelength 2π/qmax as shown in the inset. (D) In this article we analyze four hierarchical factors determining a network’s pattern forming properties: the topology — the species and types of interactions between them; the regulatory function — the functional form of the interactions; kinetic parameters — parameters in the regulatory functions; and the diffusion constants of the different species. (E) Visualization of the two regulatory mechanisms analyzed in this study on a transcriptional level. Non-competitive regulation describes the case where transcription factors (TF) bind to independent TF sites and thus regulate the recruitment of RNA polymerase (RNAP) and transcription independently (left panel). In the competitive case, in contrast, TFs directly compete for the binding site.

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to screen networks and analyze the types of TPs they We perform a similar analysis here to identify min- form. Figure 2C summarizes the computational pro- imal TP motifs. To this end, we create a network at- cedure. las in which all networks that differ by a single edge are connected (see atlas for 2-node systems in Fig- Turing topologies are common but sensitive to ure 3B,C). Each connection between a network rep- regulatory mechanisms resents the addition/deletion of a single edge, or a change of sign of a single edge. Networks are subse- Analyzing all 2-node topologies (21 networks) and 3- quently sorted hierarchically according to their com- node topologies with two diffusors (5802 networks) plexity (here defined as network size). From all net- we find that more than 60% can exhibit Turing I In- works that can generate TPs, we identify two min- stabilities and thus we expect them to be capable of imal or "core topologies" for 2-node networks: #8 generating TPs (Figure 3). This large number of po- for the competitive case and #8 and #9 for the non- tential Turing networks is many fold higher than the competitive case. All other Turing networks can be number of networks identified in the literature to date constructed from these by the addition of one or more ( 700 topologies) (Zheng, Shao, and Ouyang, 2016; edges. Marcon≈ et al., 2016). Turing networks can show patterns in which the Importantly, we observe that subtle features beyond concentration maxima of the different molecular network structure influence a network’s pattern gen- species are either in phase or out of phase. "In phase" erating capability. The first difference appears in the refers to systems in which the maximal concentrations choice of regulatory mechanism. We find that there of both species coincide, whereas for out of phase are fewer competitive Turing topologies overall among mechanisms, the maximum of one species coincides the 2-node networks (Figure 3A). All five networks are with the minimum of the other (see Figure 4D). We detected for non-competitive mechanisms, of which analyzed the 2-node Turing topologies (given in Fig- only three are found for competitive interactions (Fig- ure 3B) with respect to their patterns’ phases, by nu- ure 3B). However, we find that the rarer competi- merically solving the corresponding PDEs. We find tive interactions are more robust to parameter vari- that all 2-node Turing networks with competitive reg- ations than non-competitive interactions, which re- ulations give rise to in-phase patterns. In the com- sults in more TP solutions within a given topology. petitive case, it appears that the networks #15 and Network #8 constitutes the classical Turing network, #20 inherit the patterning phase from the core net- which was analyzed by Alan Turing in 1952 (Turing, work #8, to which they can be reduced. 1952) (Figure 3B-D). The other four 2-node networks For non-competitive regulation, network #8 again have also been reported elsewhere (Zheng, Shao, and exhibits only in-phase patterning, whereas network Ouyang, 2016; Marcon et al., 2016). #9, which constitutes the second core network for For 3-node systems, we similarly find a group of non-competitive systems, shows out-of-phase pattern- Turing topologies that is shared by both regulatory ing. One might thus expect network #20 (which can mechanisms (2400 networks, Figure 3A). While a be reduced to either network #8 or network #9 by large fraction of topologies exhibit a Turing I insta- removal of one edge) to give rise to both in and out- bility, this is again mainly for non-competitive rather of-phase patterns. Our analysis shows that this is in- than competitive interactions. This result stands in deed the case. Interestingly, here the phase can be sharp contrast to the existing literature which mainly controlled by the diffusion constants: when A dif- highlights the network topology as the important fac- fuses faster than B, in-phase patterning is observed, tor for TP capability (Diego et al., 2017). By contrast, whereas if B diffuses faster than A, out-of-phase pat- our results suggest that network structure alone does terning is seen. In a sense, the choice of which node not suffice but that the choice of regulatory function contains the "fast" or "slow" species thus becomes a also critically determines a network’s Turing capabil- topology-related system parameter. Overall, key qual- ity. itative properties of TPs such as phasing appear to be mediated by the underlying core topologies. Minimal topologies define key properties such as pattern phasing Core topologies also specify phase properties for 3-node networks In order to determine key dynamic features of networks it has been shown to be advantageous to an- We next apply the complexity atlas reduction proce- alyze the minimal topologies necessary to achieve a dure to the 5802 different 3-node networks where particular behavior. For example, such an analysis two species can diffuse (3N2D). As for the 2-node revealed that the key motifs to achieve single stripe networks, we reduce 3-node networks to their core patterns mediated by external cues (French Flag pat- topologies. This leads to a hierarchical graph similar terns) are incoherent feed-forward loops (Ingolia and to the 2-node case shown in Figure 3B, but that can- Murray, 2004; Schaerli et al., 2014; Cotterell and not be depicted due to its large size (> 2400 nodes Sharpe, 2010). and > 104 dependencies). Within this atlas, we find

5 Mammalian systems as an example for competitively regulated systems In eukaryotic systems intercellular communication is highly developed and thus cell-to-cell signaling molecules exist in abundance. External signals are typically conveyed to the cell by signaling Mammalian systems as an example for competitivelyMammalianpathways downstream regulated systems ofsystems as membrane-bound an example for receptors. competitively Binding regulated to such receptors systems by activating or inhibitingMammalian species systems is often as of an a example competitive for nature. competitively Moreover, regulated truncated systems versions of activating In eukaryotic systems intercellular communication isInspeciesIn highly eukaryotic eukaryotic frequently developed systems systems and form intercellular intercellular thus inhibiting cell-to-cell communication communicationcounterparts. signaling is We is highly highly therefore developed developed explore and and mammalian thus thus cell-to-cell cell-to-cell systems signaling signaling as a molecules exist in abundance. External signals aremoleculesscenariomolecules typically in exist exist which conveyed in in competitiveabundance. abundance. to the cell interactions External External by signaling signals signals are dominating. are are typically typically We conveyed conveyed have identified to to the the cell the cell by most by signaling signaling robust Mammalian systems as an example for competitively regulated systems pathways downstream of membrane-bound receptors.pathwayscompetitivepathways Binding downstream downstream Turing to such networks receptors of of membrane-bound membrane-bound (Figure by activating XX). receptors. receptors.or Binding Binding to to such such receptors receptors by by activating activating or or inhibitingIn eukaryotic species systems is often intercellular of a competitive communication nature.inhibitinginhibiting is highlyWe Moreover, further developedspecies species truncated explored is is and often often thethus versions of of generated cell-to-cell a a competitive competitive of activating network signaling nature. nature. atlas in Moreover, Moreover, the light truncated of truncated an existing versions versions network of of activating activatingthat was speciesmolecules frequently exist in form abundance. inhibiting External counterparts. signalsspecies Weproposedspecies are therefore typically frequently frequently for explore engineering conveyed form form mammalian inhibiting to inhibiting purposes: the cell systems counterparts. counterparts. by a system signaling as a using We We HGFtherefore therefore (hepatocyte explore explore mammalian growth mammalian factor) systems systems and NK4 as as a a scenariopathways in whichdownstream competitive of membrane-bound interactions are receptors. dominating.scenario(ascenario truncated Binding in in We which which version tohavesuch competitive competitive identified of receptors HGF) interactionsthe as interactions by activator most activating robust are and are or dominating. inhibitor, dominating. respectively We We have have identified [? identified]. They the both the most most mediate robust robust a competitiveinhibiting Turing species networks is often of (Figure a competitive XX). nature.competitiveresponsecompetitive Moreover, via Turing Turingthe truncated c-Met networks networks receptor versions (Figure (Figure signaling of XX). XX). activating pathway that activates or represses a human derived- speciesWe further frequently explored form the inhibiting generated counterparts. network atlasMMP-1 WeWe inWe therefore the further furtherpromoter light explored ofexplore explored an construct. existing mammalian the the generated generatednetwork In a previous systems thatnetwork network was study, as atlas a atlas this in in system the the light light was of of analyzed an an existing existing as networka network classical that that 2-node was was proposedscenario for in whichengineering competitive purposes: interactions a system are using dominating.proposedsystemproposed HGF and (hepatocyte for for the We engineering engineering authors have growth identified suggested purposes: purposes: factor) the that amost and a system a system single NK4robust using promoter using HGF HGF construct (hepatocyte (hepatocyte driving growth growth the expression factor) factor) and and ofNK4 both NK4 (a truncated version of HGF) as activator and inhibitor,(aactivating truncated respectively and version inhibiting [of?]. HGF) They species as both couldactivator mediate suffi andce a to inhibitor, create Turing respectively Patterns [?]. [?]. They This both system mediate would, a competitivebioRxiv preprint Turing doi: https://doi.org/10.1101/352302 networks (Figure XX). ; this (aversion truncated posted versionJune 20, of2018. HGF) The ascopyright activator holder and for inhibitor,this preprint respectively (which was [?]. 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[].?]. ThisC This diff systemusing. system Indeed, would, would, however,MMP-1 require promoter diff construct.erential diff Inusion a previous (10 fold) study, andhowever,thishowever, this a topology co-operativity system require require is was amongst di di analyzedff factorerentialfferential the ofas topdi >= diff ausionff 10 classicalusion 2 % for of(10 (10feasible the 2-nodefold) fold) most and and robust a a co-operativity co-operativity topologies according factor factor of toof >= our >= 2 results. 2 for for feasible feasible Even Turingsystem pattern and the generation. authors suggested In reality, that the a network single promoterTuring canso,Turing the be patternseen classical pattern construct as a generation. requirements 3-nodegeneration. driving network theIn In found reality, expression reality, in which for the the the networkc-Met of network 2-node both can system can be be seen persist seen as as a HGF a 3-node 3-node and network NK4network are inin more which which likely c-Met c-Met to composesactivating the and central inhibiting node andspecies mediates could sutheffice signals tocomposesdicomposes createff betweenuse at Turing the quite the both central central Patterns similar HGF node node and rates, [?]. and NK4 and This making mediates mediates(see system Figthis the 7 would,network the C). signals signals only between between implementable both both HGF HGF if and HGF and NK4 NK4 could (see (see be Fig modified Fig 7 7 C). 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Overall,A this shows B how the network atlas can help decipher engineering−1 options.0 hownot the improve network intracellular atlas6 can help robustness decipher significantly engineering7 how and options. thethus network in total, atlas robustness can@A help8 is decipher only marginally1 engineering1 options.9 10 B C = VA … + bA µAA (1) improved (1.3 fold). To significantly improve the design (4.6 fold), an additional@t direct· 1+( interactionkAA )n · 1+( B )n A kBA 2between formulas HGFA and NK4B would have to beA engineered22 formulasB (NK4 formulas activating HGF).A Overall,B this shows A B A B how the network11 atlas can help decipher engineering12 options. 13 14 15 @A 1 1 @A@A @B 11 1 11 = VA = VB + bB +µbBAB µAA (1)(2) = VA + bA µAA = VA kAA (1)n kAB n B n + bA µAA (1) B A TranslatingB networkskAA Ainton ODEBB equationsn A@t @t· 1+(B · k1+(AA) n· 1+() AB) n B A B @t · 1+( ) · 1+( ) @t · 1+(A ) ·A1+(kBA ) A kBA A kBA 2Non formulas-Competitive16 17 18 19 20 @A 1 1 A n @B 1 @A @B@B ( k 11) A B = VA k B + bA µAA = VB (1)AA + bB µBB (2) = VB AA n + bB nµBB = VA = VB (2)kAB n + bB+ bAµBBµAA (3)(2) @t · 1+( )kAB· 1+(n ) @t · A nkAB nB n @t · 1+(A ) kBA @t @t · 1+(·1+(1+() A+() ) ) 21 A kAA A kBA Competitive A @B A 1n @A ( ( AA)nn)n @A = VB ( ) + bB µBB @A @B (k(2)AAkAA) kAA kAB n = V kAB + b µ A = V@At · 1+( ) + bA µAA = AVA= V A(3)n B+ bn +µAbAB µAAA (3)(4)(3) A n A B n @t@t · ·1+(B A) nA+(n B) Bn B @t · 1+( ) +( ) @t 1+(· k1+(AAk ) +()kBAk ) kAA kBA AA kAB BA

A n @A ( k ) AA n n A AAn @B @fA ( ( 2) ) @fA @=B VA ( ) + bA µAA @B Dk(3)ABkABq kAAB n B n = V @A 1 + b@B µ B C Estimating @Turingt = V· BPattern1+( ) capability+(+ bB) µBB = BVB (4) AA n +BbB µBBB (4)(4) kAA A n kBA @Jt@t= · ·1+( ) n (5) @t · 1+( ) 1+(kABk ) kAB 2 @fB AB @fB D q23 Specify parameter boundaries Initial conditions: Stability analysis:@A @B 2 A n 4 5 @B ( k ) @f@AfA 2 2 @f@AfA @fA AB2 @fA DD1q q = VB D1q + bB µBB @A@A (4)1 2 @B@B @t @A · 1+( A )n @B JJ== @A A (5) Parameter Range J = kAB =(5)⇢ µ A (5) 22 @f@BfB @f@BfBA 233 (6) 2 @fB @fB 23 2 D q @t @A · B @B DD2q2q @A @B 2 @A @B V 0.1-100 4 5 4 @fA 2 @fA 5 4 5 D1q @A @B 2 b 0.1-100 J = 2 Without@@A@AA diffusionAA 2(D=0): System @A A = ⇢(5)A2 µ B 2 =@⇢fB µ@fBA 23 ==⇢⇢ µµBAAA (6)(7)(6) k 0.1-100 A D2q considered@@t@tt (6) stable···BB if real parts of @t @A· B @B 4 5 all eigenvalues < 0 µ 0.01-1 @A A2 2 n 2-4 @A @A@A 2 2 =ODE=⇢ ⇢A 2simulations:µ µBAA ==(7)⇢(6)⇢AA µµBBB (7)(7) @t@t · · B B @t@t · · Include diffusion (D ≠ 0): @A 2 Parameter 22 Range

2 … = ⇢ A µBB (7) @t · D1 1 D 10-3-103 2 2 Define parameter sampling grid:

If real part of any eigenvalue Identify steady states: > 0, classify instability type:

5 5

0 0

dimensionalgrid -5 -5 - RE (eigenvalue) RE (Eigenvalue) p Turing I Turing II -10 -10 0 5 10 0 5 10 Steady state 1 Steady state 2 q q

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Figure 2: Definition and analysis of networks (A) Definition of network structure. Left: 2- and 3-node networks containing all possible edges. The latter can be specified as either 0’s (no regulation), 1’s (activation) and -1’s (inhibition). From the set of all possible networks, we identify groups of networks that are equivalent to each other (redundant) and remove all but one network per group. Furthermore, we exclude networks containing any unconnected node(s). The resulting networks can be represented as a network graphs or by their adjacency matrices. The network shown here corresponds to network #8 (see Figure 3A). (B) Generation of ODEs equations. Each edge in a network corresponds to a Hill term in the ODE equations. The way these terms get combined depends on the regulatory mechanism. In the equations, V denotes the maximum level of expression and b the basal production rate. n is the Hill coefficient, indicating the "steepness" of the regulation. We include a linear degradation term with degradation constant µ for each species. (C) Workflow for estimating steady states and identifying Turing instabilities. Left panel: for each parameter, a range is specified and a logarithmic grid generated with three values per parameter, amounting to 531441 parameter combinations for fully-connected 3-node networks. Middle panel: for each parameter set, the corresponding ODEs are solved numerically until time t = 1000 for several different initial conditions. We use k-means clustering on the endpoints of the trajectories to find the steady states of the system. Right panel: the final step of the algorithm calculates the eigenvalues of the Jacobian for each steady state. For the Jacobian evaluated at zero diffusion, the real parts of all eigenvalues are required to be smaller than zero, corresponding to a stable steady state. Subsequently, diffusion is taken into account. A Turing instability exists if the real part of the largest eigenvalues becomes positive for some finite wavenumber q. Depending on the behavior of the eigenvalue as a function of the wavenumber q, we classify the instability into two types. In this article we only consider Type I instabilities as only these generate patterns with finite wavelengths.

12 and 20 core topologies for competitive and non- For the first core motif, the positive feedback loop, competitive regulation, respectively, all of which have three possible configurations exist: a direct positive four edges (Figure 4A). As in the 2-node case, the com- feedback (e.g. network #49) or an indirect positive petitive core topologies constitute a subset of the non- feedback consisting of either two positive or two nega- competitive systems. We also assess whether these tive edges to another node. The interaction can either core topologies give rise to in phase or out-of-phase be mediated via the other diffusing node (e.g. net- patterns, by solving the corresponding PDEs numer- work #65) or the non-diffusing node (e.g. network ically. In phase mechanisms (Figure 4A bottom) are #131). The second core motif consists of a negative observed less commonly than out of phase mecha- feedback loop on one of the diffusing nodes that is me- nisms (Figure 4A top). One might expect that in phase diated through the other diffusing node. This motif mechanisms are rarer with three nodes, since more can be of several different types and some examples species now have to be in phase. In contrast to the are depicted in Figure 4C. 2-node case, however, some competitive systems now also form out-of-phase patterns. In all core topologies, we solely observed either out of phase or in-phase patterns, which suggests that these minimal topologies exhibit unique behaviors. Having identified the two core motifs from studying Analyzing 3-node networks with three diffusing the 3-node core topologies, we re-analyzed all 2-node nodes, we find that all such 3N3D Turing networks are networks with respect to these motifs. We find that all also a 3N2D Turing network. We further find that the 2-node Turing networks do indeed possess both core core topologies for two and three diffusing molecules motifs, and all networks containing both motifs ex- coincide. This suggests that networks in which three hibit Type I instabilities in the non-competitive case molecules diffuse are mere expansions of the Turing (Figure 3A). For the 3-node networks, we find that parameter set, but that the instability driving compo- 96% of all networks containing both core motifs do ex- nent already exists in the two-diffuser systems. There- hibit Turing I instabilities. Since we can only sample a fore, in order to understand the core TP mechanisms finite number of parameters, we cannot categorically for 3-node systems, we have to focus on those with rule out that the missing 4% might also exhibit a Tur- two diffusing species. ing I instability for certain parameters. To reduce the probability of missing Turing I instabilities for these 5 Two core motifs account for more than 95% of networks, we ran additional simulations for 10 ran- Turing topology space domly sampled parameter sets per network. Analyz- ing all networks exhibiting Turing I instabilities, we Analyzing all 3-node core topologies with Turing I in- observe that 95% possess both core motifs. We con- stabilities, we identify two frequently occurring core firm by simulation that the remaining 5% topologies motifs: a positive feedback on at least one of the dif- do indeed give rise to TPs despite the absence of the fusing nodes, consisting of one or two edges (Figure core motifs. Therefore, the two identified core mo- 4B), and a diffusion-mediated negative feedback loop tifs together constitute an almost necessary and almost on both diffusing nodes (Figure 4C). sufficient criterion for Turing I instabilities.

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B activating inhibiting

A B A B A B A 1 B A 2 B A 3 B A 1 B A 2 B A 3 B 1 2 3 A B A B A B A B A 4 B A 5 B A 6 B A 7 B AA 4 B A 5 B A 6 B A 7 B A B A B A B A B 44 5 6 7 A 8 B A 9 B A 10 B A 11 B A B A B A B A B A 8 B A 9 B A 10 B A 11 B 8 9 10 11 8 9 10 11 complexity A B A B A B A B A B A B A B A B A B A B 12 13 14 15 16 A 12 B A 13 B A 14 B A 15 B A 16 B A B A B A B A B A B A A 12 B B A A 13 B B A A 14 B B A A 15 B B A A 16 B B A 12 B A 13 B A 14 B A 15 B A 16 B 1712 1813 1914 2015 2116 A 17 B A 18 B A 19 B A 20 B A 21 B A B A B A B A B A B 17 18 19 20 21 A 17 B A 18 B A 19 B A 20 B A 21 B 17 18 19 20 21

C D ID Non-Competitive Competitive

8 Level 1 1 2 3

9 No pattern

Level 2 4 5 7 6 8 9 10 11 15 complexity

16 No pattern

Level 3 12 13 17 14 15 16 20 18 19 21 20

A B A B

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Figure 3: Results for 2-node networks (A) Visualization of the sampled network topology space and fractions of Turing pattern generators (Turing topologies) for 2- and 3-node networks with 2 diffusing molecules. (B) Considered 2-node networks selected according to criteria described in Figure 2A. Blue and red edges indicate activation and inhibition, respectively. The networks are arranged according to their complexity (that is, the number of edges) with increasing complexity towards the bottom. Each network is given an ID number (1-21). Lilac boxes indicate the networks identified as Turing pattern generators. Dark lilac indicates non-competitive regulatory mechanisms, whereas the lighter shade indicates competitive ones. (C) Hierarchical graph of 2-node networks. Each node represents a network of a given ID number. Networks are connected by an edge whenever they can be transformed into each other by addition, deletion or modification (change of sign) of a single edge. For example network #1 can be transformed into network #4 by addition of a negative self-interaction on node A, and network #1 into network #2 by changing the sign of one edge from inhibition to activation (see networks in (B)). The nodes are colored according to the legend shown in (A), with different shades of lilac indicating for which regulatory mechanism the networks exhibit Turing I instabilities. (D) Exemplary 2-dimensional patterns for the identified 2-node Turing generating networks. The parameters for which the patterns were generated are given in Supplementary Document S3.

Differentiating Turing Instabilities - new instabil- for q , but remains positive instead. Since the ity types and their patterns dispersion→ ∞ relation does possess a global maximum in qmax and hence a finite wavelength that experiences The dispersion relation of a system is typically re- the strongest amplification, one might expect such sys- lated to the resulting pattern: the wavenumber qmax tems to lead to patterns anyway. Through numerical for which the dispersion relation assumes a global simulations we verify that this is indeed the case. This maximum (that is, the largest eigenvalue of the Jaco- result agrees with the conclusion of a recent preprint bian becomes maximal; see Methods) experiences the (Smith and Dalchau, 2018). An exemplary pattern largest amplification. We thus expect to see a pattern plot is visualized below the corresponding dispersion with wavelength 2π/qmax (see Figure 1C). relation in Figure 4D. Due to the extent of this analysis in terms of the Figure 4D panel 4 shows the Type IIb dispersion re- number of networks considered, we also observe some lation. Similarly to the Type IIa instability, the disper- qualitatively novel types of dispersion relations that sion relation becomes positive for some finite q1 and had not been reported previously in the literature. We assumes its maximum for q . However, in con- distinguish four different groups of dispersion rela- trast to the original Type IIa→ case, ∞ the dispersion re- tions. First, the "classical" Turing I instability fulfills lation becomes negative again for some intermediate three criteria: for q = 0, the system should be stable q2 > q1; but like the Type IIa case the maximum of (i.e. Re(λ) < 0); for a finite q we have Re(λ) > 0; and this dispersion relation occurs for q and thus we Re(λ) < 0 for q . This type of instability is the → ∞ → ∞ cannot get stable patterns since arbitrarily small wave- most commonly discussed mechanism underlying TPs. lengths experience the strongest amplification. To our We describe as a Turing II instability the case when the knowledge this type of Turing instability has not been steady state is stable for q = 0 (i.e. Re(λ) < 0), where reported in the literature before. ∗ there exists some threshold q such that Re(λ) > 0 for We would like to note that in the analyses described ∗ all q > q , and the dispersion relation becomes maxi- in the previous sections we merely distinguished sys- mal for q . Therefore, for q > q∗, the larger the → ∞ tems according to their patterning capability and re- mode q the stronger the amplification. Consequently, ferred to Ia and Ib jointly as "Type I", and IIa and IIb arbitrarily large modes and hence short wavelengths jointly as "Type II". Both types of instabilities are im- get amplified the most, which does not lead to a stable portant when considering the analysis of real-life sys- pattern with a well-defined wavelength. tems. Ib could be mistaken for a non-patterning sys- The previous cases are referred to as Type Ia and IIa tem as it does not fulfill a classical criterion for a Tur- instabilities. However, other types of dispersion rela- ing I instability, which may lead to underestimating tions exist for diffusion driven instabilities that do not the robustness of a system. Similarly, IIb can be mis- fall into either Type Ia or IIa. Figure 4D shows exam- taken for a Type I instability (if q is sampled over an ples for all four categories that we find. One of these insufficient scale) consequently leading to an overes- is similar to Type Ia and capable of producing stable timation of Turing robustness. patterns and we hence term it Type Ib. The other dispersion relations does not produce patterns and we de- Turing I instabilities are not a sufficient criteria note it by IIb. Type Ib instabilities fulfill both criteria for patterning of a negative real λ for q = 0 and possess a finite qmax for which the dispersion relation becomes positive and In development, cell-fate decisions are often governed assumes a global maximum. However, the dispersion by systems in which multiple stable steady states ex- relation does not fulfill the third criteria of Re(λ) < 0 ist (Harrington et al., 2014; Harrington et al., 2013;

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A Out-of-Phase Core Topologies

A A A A A A

B C B C B C B C B C B C 49 50 52 53 65 86

A A A A A A

B C B C B C B C B C B C 87 93 131 132 16 17

A A A A A A

B C B C B C B C B C B C 64 67 68 92 119 120

In-Phase Core Topologies

A A A A A A

B C B C B C B C B C B C 46 47 83 84 125 126

Non-Competitive Turing topologies Competitive Turing topologies activating inhibiting B C A A

A A B A B A B B C B C

L = 1 L = 2 L = 2 L = 2 L = 3 L = 4

diffusing node diffusing/non-diffusing node non-diffusing node L = path length

D Multistable Multistable Turing Ia Turing Ib Turing IIa Turing IIb Case 1 Case 2

No pattern

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Figure 4: Core networks and core motifs for 3-node networks (A) Top: 3-node core topologies that generate out-of-phase patterns (one species’ maximum concentration is shifted by half a period with respect to the other two). Bottom: 3-node core networks exhibiting in-phase patterns (all concentration maxima aligned). Colored frames indicate the regulatory mechanism (competitiveness) for which the networks exhibit Turing I instabilities. (B) Examples for the first identified core motif: positive feedback loop of length one or two on one of the diffusing nodes. The figure shows the different possibilities to achieve this. If the path length is two it can be mediated by either the other diffusing or the stationary node. (C) Examples for the second identified core motif: diffusion-mediated negative feedback on a diffusing node. One diffusing node has to have a negative feedback loop whose path includes the other diffusing node. This interaction can consist of two, three or four edges and the figure shows one example each. The core motifs in (B) and (C) are almost sufficient and almost necessary for TPs. (D) Examples of different Turing instabilities and resulting patterns. Note that for Turing IIa and Turing IIb instabilities no patterns are formed. Note also that in the multistable case 1, no pattern is generated despite the existence of a Turing I instability. Instead, starting from a perturbation around the stable steady state with Turing I instability (indicated as dashed lines) this moves to a second homogeneous steady state. This behavior is observed for 4% (non-competitive) and 14 % (competitive) of multistable network-parameter combinations exhibiting a Turing I instability. The parameters for which the dispersion relations and patterns were generated are given in Supplementary Document S3.

Moris, Pina, and Martinez Arias, 2016). In mam- To this end, we define four measures of robust- malian stem cells, for example, Nanog, Sox2 and Oct4 ness to parameter variation or uncertainty: robust- form a system possessing two stable steady states. ness to intracellular processes, robustness to extra- Here, each steady state corresponds to one specific cell cellular processes, topological robustness and total fate: either a cell remains pluripotent or it differenti- robustness. As intracellular parameters, we define ates. all kinetic parameters of the ODE equations (Figure We therefore must explore how multi-stability af- 2B) that describe the chemical interactions between fects TP formation. First, we find that systems with species within cells, and we define "intracellular ro- multiple stable steady states can exhibit Type I in- bustness" as the fraction of the analyzed parameter stabilities (Ia or Ib), either for a single or for several combinations that is capable of Turing pattern forma- steady states. The former has also been reported in a tion. recent preprint (Smith and Dalchau, 2018). Similarly, In addition to intracellular processes, the speed of when probing the patterning behavior of Type Ia, Ib, extracellular diffusion of molecules determines if a IIa and IIb instabilities, we simulate the spatial system network possesses a Turing instability. We accord- numerically for all systems and parameter combina- ingly define the "extracellular robustness" as the ro- tions for which either one or multiple Turing I insta- bustness of a Turing network to changes in the diffu- bilities are present. Even though most systems indeed sion constants, given that the intracellular parameters show the expected pattern formation (86% and 96 %, are fixed to values that can give rise to a Turing insta- for competitive and non-competitive systems, respec- bility (see Figure 5A). tively), some do not (14% for competitive and 4% for non-competitive). Rather than observing stable, re- Despite uncertainties in parameter values, it is fre- producible patterns, we find that these systems transi- quently the case in biological experiments that one tion from a perturbation around the steady state with cannot even be certain about the network topology a Turing I instability to one of the other stable steady of a given system (Babtie, Kirk, and Stumpf, 2014). states. Figure 4D, Panel 5, shows an example of such There might be additional, unknown regulatory in- behavior. We thus conclude that a Type I Turing insta- teractions between species, or assumed interactions bility is not a sufficient criterion for pattern formation might not be active. Accordingly we define "topolog- in multi-stable systems. Again this highlights the need ical robustness" as follows: for a given network, con- for a thorough analysis if we want to be able to predict sider all networks that can be generated by adding, and validate natural TP generating networks. removing or changing one edge (as exemplified in Fig- ure 3C for 2-node systems). Then the topological robustness is defined as the fraction of generated net- Defining quantifiable measures of robustness works that are capable of exhibiting Turing I instabil- In biological systems, parameter values are gener- ities (Figure 5A). ally known only with partial certainty, and sometimes Finally, we would like a measure for the overall ro- entire interactions are unknown (Kirk, Babtie, and bustness of a given network taking into account the in- Stumpf, 2015). It is therefore crucial not only to iden- fluence of all three different sources of uncertainties. tify networks that exhibit Turing instabilities, but also This "total robustness" is the product of the intracel- to assess their sensitivity with respect to these uncer- lular, extracellular and topological robustness (Figure tainties. 5).

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C D

E F

Figure 5: Quantifying a network’s Turing capability: defining four measures of robustness (A) Left: definition of intracellular (extracellular) robustness as the fraction of sampled kinetic (diffusion) parameter sets leading to Turing I instabilities. Right: definition of topological robustness as the fraction of neighboring networks (that is, networks into which a given network can be transformed by addition, deletion or modification of a single edge) that exhibit Turing I instabilities. The example shows that two out of six neighbors of network #8 are Turing networks, leading to a robustness 1/3. The total robustness is defined as the product of intracellular, extracellular and topological robustness. (B) Mean values for the different robustness measures for competitive (left) and non-competitive systems. 2N (2Nr) denotes 2-node systems for which V and b are varied (restricted to 100 and 0.1, respectively). 3N2D and 3N3D represent 3-node systems with two and three diffusing nodes, respectively. (C-F) Histogram plots of the four robustness measures comparing for both regulatory mechanisms. We find that competitive systems are on average more robust for intracellular processes, whereas non-competitive systems are topologically more robust. In terms of total robustness, a subset of 3N2D competitive networks constitute the best performing networks.

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A Total Complexity Robustness = number of edges 0.0038 9 A A A A A A

B C B C B C B C B C B C 5703 4491 4030 1372 475 126

8 A A A A A A

A A A A A A B C B C B C B C B C B C 5703 4491 4030 1372 475 126 B C B C B C B C B C B C 5651 4564 4147 1291 733 67

A A AA AA A A A A A 7

B C B CC BB CC BB CC B B C C B B C C B C 5703 44915651 40304564 13724147 4751291 126733 67

A A A AA AA A AA A

B C B B C C BB CC BB CC B CC BB CC B C 5703 44915651 6 40304564 13724147 1291475 733126 67

A A A A A A A A A A A A

B C B C B C B C B C B C B C B C B C B C B C B C 5703 4491 4030 1372 475 126 5651 4564 5 4147 1291 733 67 A A A A A A

A A A 4 A A A B C B C B C B C B C B C 0 5703 4491 4030 1372 475 126 B C B C B C B3 C B - C B C 5651 4564 4147 1291 733 67

A A A A A A B Top 10 Non-Competitive Networks: B C B C B C B C B C B C 5651 4564 4147 1291 733 67 A A A A A activating B C B C B C B C B C 4030 1372 1528 4491 3774

A A A A A inhibiting

B C B C B C B C B C 3775 4300 3881 1331 3772

C D

A C

B C

low high

concentration

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Figure 6: Most robust 3-node Turing networks with non-competitive regulation and two diﬀusing nodes (A) Robustness map. The ﬁgure indicates the total robustness of all analyzed 3-node networks arranged according to complexity. For each complexity class we show the most robust network, together with its ID number. (B) Ten most robust networks with numbers indicating the corresponding network ID. Total robustness decreases towards the right and bottom, i.e. #4030 is the most robust network. The absolute robustness values are given in Supplementary Document S2. (C) Local neighborhood atlas for the most robust non-competitive Turing network #4030 (center). Colored nodes indicate the total robustness value according to the colorscale shown in (A). (D) 2-dimensional pattern for the most robust non-competitive Turing topology #4030. Yellow and blue indicate high and low species concentrations, respectively, for nodes A, B and C. This is an out-of-phase pattern. The parameters for which the pattern was generated are given in Supplementary Document S3.

Competitive 3-node systems are the most robust Robustness maps of 3N2D topology space to re- Turing networks veal the most robust networks and their neigh- borhoods

Due to the large number of 3-node Turing topologies we visualize their total robustness in a "robust- We compute the intracellular, extracellular, topologi- ness map" shown in Figures 6A (non-competitive) cal and total robustness for all 2-node and 3-node net- and 7A (competitive). We group networks accord- works (Figure 5). We find that Turing networks with ing to their complexity; for each complexity class we competitive interactions are more robust than non- additionally depict the most robust network (right competitive ones, in particular with respect to intra- panel). However, these networks only constitute a cellular parameters. This is consistent for 2-node sys- fairly small fraction of the top 10 most robust networks tems ( 1.7-fold), as well as 3-node systems with two (three and two for competitive and non-competitive, ∼ ( 2.3-fold) and three ( 2.1-fold) diffusing species respectively; Figures 6B and 7B). Overall, networks ∼ ∼ (see Figure 5B). As the intracellular robustness varies with complexity of 5-6 (competitive) and 6-7 (non- over about two orders of magnitude more than the competitive) are the most robust topology groups. We extracellular and topological robustness, competitive find no significant correlation between robustness val- systems are also in total more robust ( 1.5 2-fold). ues and topology complexity. This further suggests ∼ − Our results demonstrate that the choice of regulatory that increasing complexity does not generally lead to a interactions can have significant influence on a net- larger robustness. All robustness measures (intracel- work’s Turing capability: non-competitive topologies lular, extracellular, topological and total robustness) are more likely to be able to generate TPs whereas which we derived from this analysis are provided in competitive systems that do generate TPs are more ro- Supplementary Document S2; network identifiers cor- bust. respond to those in the network graphs provided in Supplementary Document S1. Due to the large number of topologies, we are not Due to computational cost, the V and b parameters able to illustrate the full network atlas to show how were restricted for the analysis of 3-node systems (see these networks are related. Instead, we provide a lo- Methods). Consequently, to compare like-for-like, we cal neighborhood atlas for the single most robust non- calculated the robustness for 2-node systems under competitive and competitive networks (Figure 6C and the condition that V and b were fixed to the same 7C) and a corresponding 2D PDE solution (Figure 6D values. With this restriction on parameter space, 2- and 7D). The local neighborhood atlas contains all the node systems are on average more robust to intracel- networks that can be generated from the central net- lular variations than 3-node systems ( 2.7-fold for work by adding, deleting or modifying one edge. Most competitive and 3-fold for non-competitive∼ regula- edge changes lead to a pronounced drop in Turing ro- tions). On the other∼ hand, 3-node systems are on av- bustness, although it is still possible for evolution to erage significantly more robust to extracellular ( 2.4 "walk" from one topology to another while still main- (competitive) and 1.5-fold (non-competitive)∼ and taining a TP. topological ( 2-fold)∼ variations than 2-node systems. Even though∼ in total the average robustness of 2 com- Overall, one of the most important conclusions from pared to 3-nodes is not significantly different, we do this study is that Turing network topologies and mech- find that amongst the top (most robust) networks for anisms are more common within random networks either case, 3-node systems are more than 4-fold more than previously thought. Therefore, it is possible that robust in total than the top 2-node systems. It is thus evolution can move through these less robust — but likely that TP networks in nature will be composed of relatively common — prototypic solutions, towards at least three interacting species. one of the more robust networks.

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A A A A A A A

Total Complexity TopB networkC B C B C B C B C B C 5703 4491 4030 1372 475 126 Robustness = number of edges per class

0.0124 9 A AA A A AA AA A

B C BB CC B C B C BB CC BB CC B C 5703 44915651 40304564 41471372 1291475 733126 67 8 A A A A A A A A A A A A

B C B C B C B C B C B C B C B C B C B C B C B C 5703 4491 4030 1372 475 126 5651 4564 4147 1291 733 67

A 7 A A A A A A A A A A A B C B C B C B C B C B C 5651 4564 4147 1291 733 67 B C B C B C B C B C B C 5703 4491 4030 1372 475 126

A A A A A A

A A A A A A B C B C B C B C B C B C 5703 4491 4030 6 1372 475 126 B C B C B C B C B C B C A A A A A A 5651 4564 4147 1291 733 67

A A A A A A B C B C B C B C B C B C 5703 4491 4030 1372 475 126 B C B C B C 5B C B C B C 5651 4564 4147 1291 733 67

A A A A A A 4 B C B C B C B C B C B C 0 5651 4564 4147 1291 733 67 3 -

B Top 10 Competitive Networks:

A A A A A activating B C B C B C B C B C 1291 733 725 1807 4147

A A A A A inhibiting

B C B C B C B C B C 1261 1423 726 1474 2924

C D

A C

B C

low high

concentration

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Figure 7: Most robust 3-node Turing networks with competitive regulation and two diﬀusing nodes (A) Robustness map. The ﬁgure indicates the total robustness of all analyzed 3-node networks arranged according to complexity. For each complexity class we show the most robust network, together with its ID number. (B) Ten most robust networks with numbers indicating the corresponding network ID. Total robustness decreases towards the right and bottom, i.e. #1291 is the most robust network. The absolute robustness values are given in Supplementary Document S2. (C) Local neighborhood atlas for the most robust competitive Turing network #1291 (center). Colored nodes indicate the total robustness value according to the colorscale shown in (A). (D) 2-dimensional pattern for the the most robust competitive Turing topology #1291. Yellow and blue indicate high and low species concentrations, respectively, for nodes A, B and C. This is an in-phase pattern. The parameters for which the pattern was generated are given in Supplementary Document S3.

Discussion ble’ across a Turing network (more than 60% of networks considered here can produce TPs), even though Turing’s pattern generating mechanism, later inde- for most architectures the existence of a TP depends pendently rediscovered by Gierer and Meinhardt, is an crucially on parameters. Being common but not very elegant way in which purely biochemical mechanisms robust to parametric and structural changes could sug- can give rise to reproducible and self-organizing spa- gest that many different architectures are used in na- tial patterns. Despite initial unease (and sometimes ture to generate TPs (as was already hinted at in outright hostility) over them being relevant and ro- some of Meinhardt’s work (Meinhardt, 2013)). Once bust mechanisms of patterning, TPs are now widely a structure is in place with suitable regulatory interac- accepted and have become an important cornerstone tions and reaction rate parameters, and the resulting of modern developmental biology. TP confers an evolutionary advantage, natural selec- Given their provenance, it is perhaps not surprising tion is likely to maintain this mechanism. that TPs have also received close attention by mathe- While network structure by itself neither guaran- matical modelers interested in biological pattern for- tees nor implies the existence of a TP, the overwhelm- mation. But these have typically focused on single ing majority of TP generating mechanisms embed the models, exploring them in great detail. The large- hallmarks encapsulated by two core motifs: a positive scale in silico surveying of potential TP models is a feedback on at least one of the diffusing nodes and much more recent phenomenon. There are two po- a diffusion-mediated negative feedback loop on both tential pitfalls in such analyses: (i) computational cost diffusing nodes. It is tempting to speculate that larger may require simplified models or prohibit exhaustive systems will also reflect these compositional rules and analysis; (ii) automating any mathematical analysis, have these core motifs embedded; a basic TP gener- but in particular something as subtle as pattern for- ating motif could thus, for example, be regulated in a mation is non-trivial unless we have very precise crite- more nuanced manner. ria by which stability, robustness and patterns can be Finally, our exhaustive analysis reveals a spectrum scored. Our approach addressed these issues explic- of Turing-like instabilities, and subtle dependencies itly and from the outset, and the algorithm employed between these and the eventual pattern formation. here is capable of analyzing a wide range of different These are naturally easily missed when analyzing in- network structures and is independent of functional dividual Turing systems, or applying simplifying mea- choices for regulatory mechanisms and rate functions. sures in large-scale surveys. The rewards of a thorough and comprehensive anal- The analysis presented here provides us with a set ysis of 2-node and 3-node candidate Turing network of blueprints for TP generating mechanisms that can models are also considerable. This analysis clearly guide the search for naturally evolved Turing systems, demonstrates that the structure of the network alone as well as the de novo engineering of biosynthetic sys- cannot determine whether a TP exists; this is in line tems. For the latter, in particular, competitive net- with a large body of work on network dynamics (In- works should be a safer bet, due to their increased gram, Stumpf, and Stark, 2006), but seems to directly (intracellular) parametric robustness. This tension be- contradict results from other studies on TP mecha- tween commonality and robustness is perhaps one of nisms. These, however, (i) fail to fully assess the de- the most fascinating (or vexing) features of Turing pendence of TP on regulatory mechanisms and reac- mechanisms. tion rate parameters (largely because a mathematically more convenient model structure was imposed); Methods and (ii) their models are special cases of the more general and more comprehensive treatment here. A Semi-Formal Summary of our TP search Perhaps the most surprising result of this analysis This first section provides an overview of the computational is how common networks capable of producing TPs approach taken here; more technical detail is provided in the are. Evolution has had many opportunities to ‘stum- following sections; with this information in hand it should be

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straightforward to implement and repeat our analysis (soft- homogeneous steady state, but are instead amplified. This ware is made available, below, too). is then the diffusion-driven or Turing instability. If only an To identify potential Turing instabilities, we first con- intermediate range of length scales experiences such an am- sider the stability of the non-spatial system. Suppose we plification, we speak of a Turing I instability. In this case, a have a system of n interacting molecular species with time- system typically forms a pattern of the wavelength for which dependent concentrations xi(t), i = 1, . . . , n. We model the the amplification is maximal (see Figure 1C). dynamical behavior of such concentrations in terms of a sys- Mathematically, the stability is determined by the depen- tem of ordinary differential equations (ODEs), dence of the real part of the largest eigenvalue of the Jaco- bian matrix of the system on the wavenumber q (see below for more details). We also call this dependence the d xi(t)= fi(x(t)), i = 1, . . . , n, (1) dispersion relation. For zero wavenumber, a spatially con- dt stant system, the real part of the largest eigenvalue is neg- where x(t)=(x1(t),...,xn(t)), xi(t) ∈ R is the concen- ative, if evaluated at a stable steady state of the non-spatial tration of the ith species at time t. Equations of the form in system. If, however, the dispersion relation becomes posi- (1) cannot typically be solved exactly for nonlinear functions tive for some wavenumber, these become amplified and the fi. However, efficient numerical algorithms exist to solve steady state unstable. For cases where the dispersion rela- such equations approximately, leading to time trajectories of tion has a maximum for a finite value of q, a Turing I insta- the system, that is, to solutions xi(t) (see Figure 2C for ex- bility is present. If the dispersion relation remains positive amples). f(x(t)) = (f1(x(t)),...,fN (x(t))) in equation (1) and becomes maximal for large wavenumbers, we speak of encodes the interactions between the different species. If the a Turing II instability. In this case deviations on arbitrarily xi denote protein concentrations, for example, f(x(t)) may small length scales become amplified, and no stable pattern encode the regulatory mechanisms between the proteins. is formed. We thus only search for Turing I instabilities in Figure 1A shows a network representation of the Gierer- the following. See Figure 1A for a visualization of this phe- Meinhardt model, and its governing ODEs (Gierer and Mein- nomenon and the sections below for mathematical details. hardt, 1972). The nodes indicate the interacting species and Therefore, to find Turing instabilities we first need to arrows the direction of interaction. We distinguish between identify the stable steady states of a given system, and sub- two possible types of interactions: activating (blue arrows) sequently study their dispersion relation. Figure 2C summa- and inhibiting (red arrows), which we encode in the corre- rizes the computational procedure. sponding components of f(x(t)), whose functional form is For each candidate network we perform this procedure not specified by the graph (we employ Hill functions in the for a wide range of kinetic and diffusion parameters. We reaction equations). therefore have to specify the intracellular parameters deter- We next include the spatial diffusion of molecules and mining the regulatory functions fi, as well as the diffusion extend the model in Equation (1) to a spatial setting. In constants. The former consist of the parameters k (dissoci- this case the molecule concentrations xi(t) become space- ation rates), V (scaling factors), b (basal production rates) dependent concentration fields xi(r, t), where r denotes the and the degradation rate µ (see Equations in Figure 2B), spatial location. These fields satisfy the set of coupled partial which we vary across biologically relevant values between differential equations (PDEs) 0.1 and 100 (0.01-1 for µ). The extracellular/diffusion parameters are varied over a range between 10−3 and 103. Due to computational cost, we vary the V and b parameters ∂ x (r, t) = D ∇2x (r, t) + f (x(r, t)), i = 1,...,N, only for 2-node networks, but fix them to 100 and 0.1, re- ∂t i i i i (2) spectively, for 3-node networks. In this way we were still able to screen 3 × 1011 network-parameter combinations for which is obtained from Equation (1) by adding the diffu- TP formation, which we believe is the largest study of its 2 sion term Di∇ xi(r, t), where Di is the diffusion constant of kind to date. the ith species and ∇ the gradient with respect to position r, and concentration is now a function of space and time, Definition of networks and ODEs xi(r, t). The next step is to screen for the formation of stable Networks patterns through diffusion-driven instabilities (Maini et al., To generate all possible n-node networks we first compute 2012). In this case, for any small spatial fluctuations one all possible (n × n)−matrices with elements 0, 1 and −1, may expect the concentrations xi(r, t) to become spatially where a 0 (1/ − 1) represents the absence (presence) of an constant again for large times. For many systems, this is activating or inhibiting interaction/edge. The number of indeed the case, but for some systems the interplay of diffu- matrices is reduced by considering only connected networks sion and reactions can lead to the molecules’ concentrations and accounting for symmetries. We also remove matrices forming spatial patterns with certain wavelengths that are that correspond to networks including nodes without any stable and reproducible in time. incoming or outgoing edges. Each remaining matrix M In the Turing pattern framework we start with, x∗, which serves as an adjacency matrix for a network, where the is the stable steady state concentration of the non-spatial sys- element Mij being 1 (−1) represents a positive (negative) ∗ tem in (1), i.e. fi(x ) = 0, i = 1,...,n in Equation (1): if edge from node i to node j. the system is in state x∗, it remains there for all times, and if the system is close to x∗ it will converge towards x∗. If System of ODEs spatial diffusion of molecules is included into the model, as For each adjacency matrix we then construct the correspond- described Equation (2), it is possible that deviations from the ing set of ODEs. Each non-zero entry in the adjacency ma- steady state of certain length scales do not decay towards the trix corresponds to a Hill-type term which are combined in

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+ − ∗ either a non-competitive or competitive case. If Si (Si ) de- perturbation around x notes the set of positive (negative) edges ending in node i, ∗ then in the non-competitive case, the different regulators act x(t)= x + δx˜(t), (5) (and saturate) independently of each other, and we have, with a small constant δ R, and then linearise Equa-   ∈ Y 1 tion (1), to obtain fi(x1, ..., xN ) =Vi ·  nij  (3) kij j∈S+ 1 + x ∂ i j x˜(t)= Jx˜(t)+ O(δ). (6)   ∂t Y 1 ×  nij  + bi − µixi. xj The Jacobian, J, is defined as j∈S− 1 + k i ij ∂fi(x1, . . . , xn) Ji,j = , i, j = 1,...,N. (7) Here, Vi is the maximal induced production rate, bi the ∂xj basal production rate, µi the degradation rate, kij the concentration value at which the regulation of the ith Equation (6) constitutes a linear dynamical system species by the jth species is half its maximal value, and with steady state x˜ = 0. This steady state is asymptot- nij is the corresponding Hill coefficient determining ically stable if and only if the real parts of all eigenval- the steepness of the response. ues of the matrix J are negative. This in turn means ∗ In the competitive case the different regulators com- that x is a locally asymptotically stable steady state, pete for the binding site which leads to an additive in the sense that there exists a neighborhood around ∗ combination of terms: x such that any solution of the ODEs starting from this neighborhood asymptotically converges to x∗. Accordingly, it is sufficient to compute the eigenval-

fi(x1,..., xN ) = bi µixi (4) ues of the Jacobian defined in Equation (7) to assess − the local stability of a steady state of Equation (1). x nj j To assess if such a stable steady state can exhibit a k + ij diffusion-driven instability, we need to analyze Equa- j X∈ S + Vi nj nj . · xj xj tion (2). Similarly to the case without diffusion, we 1 + + perturb the system around x∗ to perform a linear sta- + kij − kij j X∈ S j X∈ S bility analysis of this steady state, but this time with a harmonic wave with wave-length q:

∗ iqr Numerical analysis x(r, t) = x + δx˜(t)e , (8) with a small constant δ R. Inserting this into Equa- Steady state estimation ∈ We generated a customized Matlab (R2016a) script to tion (2) and expanding to first order in δ, one obtains find steady states numerically. For a given system the a linear dynamical system similar to the one in (6), ˜ parameters are chosen from a logarithmic grid and for but with a modified Jacobian J given by: each set of parameters the system of ODEs is solved J˜ = J q2D, (9) numerically until time t = 1000 using the Matlab ODE − solver ode15s. Whenever the algorithm encounters where D = diag(D1,...,Dn) is a diagonal matrix numerical problems, the (slower but more robust) al- with the diffusion constants Di on the diagonal. gorithm ode23s is invoked. For differential diffusion of the molecular species, The resulting trajectory is checked to have con- the Jacobian in Equation (8) can have eigenvalues verged to a steady state. Next, we solve the system with positive real part for finite wave vector q; when n again for 3 initial conditions (Geest, 2016), where this occurs the stable steady state of the system in n is the number of species. The endpoints of each Equation (1) becomes unstable with diffusion, and resulting trajectory are clustered using k-means we speak of an diffusion-driven Turing instability. clustering assuming that the steady states are given Depending on the behavior of the largest eigenvalue by the centroids of the resulting clusters. Using the of J˜ for large q we distinguish different types of cluster centroids as an initial condition, the system Turing instabilities, see Figure 4D. of ODEs is solved again to verify that the system has converged sufficiently. For steady states that are ap- Workflow proached via damped oscillations, we do not solve the We implement the estimation of stable steady states ODEs numerically but instead search for a fixed point of the non-spatial system and identification of Turing of the ODEs directly using the Matlab function fsolve. instabilities into an automated workflow. Overall, the computational analysis consists of the steps: Stability Analysis The stability of a steady state x∗ of Equation (1) is i) Define a network and its governing ODE equa- assessed by a linear stability analysis. We add a small tions.

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