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## - Publications list -

(* = joint authorship)#### Peer-Reviewed Publications

- Telek M,
**Feliu E**(2023) Topological descriptors of the parameter region of multistationarity: deciding upon connectivity.*Plos Computational Biology*. To appear. [Arxiv]*Abstract.*Switch-like responses arising from bistability have been linked to cell signaling processes and memory. Revealing the shape and properties of the set of parameters that lead to bistability is necessary to understand the underlying biological mechanisms, but is a complex mathematical problem. We present an efficient approach to determine a basic topological property of the parameter region of multistationary, namely whether it is connected or not. The connectivity of this region can be interpreted in terms of the biological mechanisms underlying bistability and the switch-like patterns that the system can create. We provide an algorithm to assert that the parameter region of multistationarity is connected, targeting reaction networks with mass-action kinetics. We show that this is the case for numerous relevant cell signaling motifs, previously described to exhibit bistability. However, we show that for a motif displaying a phosphorylation cycle with allosteric enzyme regulation, the region of multistationarity has two distinct connected components, corresponding to two different, but symmetric, biological mechanisms. The method relies on linear programming and bypasses the expensive computational cost of direct and generic approaches to study parametric polynomial systems. This characteristic makes it suitable for mass-screening of reaction networks. Hide - West R, Delattre H, Noor E,
**Feliu E***, Soyer OS* (2023) Co-substrate pools can constrain and regulate pathway fluxes in cell metabolism.*Elife.*To appear. [BioRxiv]*Abstract.*Cycling of co-substrates, whereby a metabolite is converted among alternate forms via different reactions, is ubiquitous in metabolism. Several cycled co-substrates are well known as energy and electron carriers (e.g. ATP and NAD(P)H), but there are also other metabolites that act as cycled co-substrates in different parts of central metabolism. Here, we develop a mathematical framework to analyse the effect of co-substrate cycling on metabolic flux. In the cases of a single reaction and linear pathways, we find that co-substrate cycling imposes an additional flux limit on a reaction, distinct to the limit imposed by the kinetics of the primary enzyme catalysing that reaction. Using analytical methods, we show that this additional limit is a function of the total pool size and turnover rate of the cycled co-substrate. Expanding from this insight and using simulations, we show that regulation of co-substrate pool size can allow regulation of flux dynamics in branched and coupled pathways. To support theses theoretical insights, we analysed existing flux measurements and enzyme levels from the central carbon metabolism and identified several reactions that could be limited by co-substrate cycling. We discuss how the limitations imposed by co-substrate cycling provide experimentally testable hypotheses on specific metabolic phenotypes. We conclude that measuring and controlling co-substrate pools is crucial for understanding and engineering the dynamics of metabolism. Hide **Feliu E***, Kaihnsa N*, de Wolff T*, Yürük O* (2023) Parameter region for multistationarity in n−site phosphorylation networks.*SIAM Journal on Applied Dynamical Systems.*To appear. [Arxiv]*Abstract.*This work addresses the problem of understanding the parameter region where a family of networks relevant in cell signalling admits at least two steady states, a property termed multistationarity. The problem is phrased in the context of real algebraic geometry and reduced to studying whether a polynomial, defined as the determinant of a parametric matrix of size three, attains negative values over the positive orthant. The networks are indexed by a natural number n and for all n, we give sufficient conditions for the polynomial to be positive and hence, preclude multistationarity. We also provide sufficient conditions for the polynomial to attain negative values and hence, enable multistationarity. These conditions are derived by exploiting the structure of the polynomial, its Newton polytope, and employing circuit polynomials. We also prove that the parameter regions that enable or preclude multistationarity are connected for all n. Hide**Feliu E***, Walcher S*, Wiuf C* (2022) Critical parameters for singular perturbation reductions of chemical reaction networks.*Journal of Nonlinear Science,*32:83. [Journal] [Arxiv]*Abstract.*We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path toward such reductions. In the present paper we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov-Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero), or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex balanced reaction networks (of deficiency zero) and first order reaction networks. Hide**Feliu E***Telek M L* (2022) On generalizing Descartes' rule of signs to hypersurfaces.*Advances in Mathematics.*408(A). [Journal] [arXiv]*Abstract.*We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations. Hide**Feliu E**, Sadeghimanesh A (2022) Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations.*Mathematics of Computation,*91, pp 2739-2769. [Journal][Arxiv]*Abstract.*Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration, we apply the formula to partition the parameter region according to the number of solutions or find a region in parameter space where the system has the maximal number of solutions. The motivation stems from the study of steady states of chemical reaction networks and gives new tools for the open problem of identifying the parameter region where the network has at least two positive steady states. We illustrate with numerous examples that our approach successfully handles a larger number of parameters than exact methods. Hide- Hayes C,
**Feliu E***, Soyer O. S.* (2022) Multi-site enzymes as a mechanism for bistability in reaction networks.*ACS Synthetic Biology.*Online first. [Biorxiv][Journal]*Abstract.*Here, we focus on a common class of enzymes that have multiple substrate-binding sites (multi-site enzymes), and analyse their capacity to generate bistable dynamics in the reaction systems that they are embedded in. Using mathematical techniques, we show that the inherent binding and catalysis reactions arising from multiple substrate-enzyme complexes creates a potential for bistable dynamics in a reaction system. We construct a generic model of an enzyme with n substrate binding sites and derive an analytical solution for the steady state concentration of all enzyme-substrate complexes. By studying these expressions, we obtain a mechanistic understanding for bistability and derive parameter combinations that guarantee bistability and show how changing specific enzyme kinetic parameters and enzyme levels can lead to bistability in reaction systems involving mjulti-site enzymes. Thus, the presented findings provide a biochemical and mathematical basis for predicting and engineering bistability in multi-site enzymes. Hide **Feliu E***, Lax C*, Walcher S*, Wiuf C* (2022) Quasi-steady state and singular perturbation reduction for reaction networks with non-interacting species.*SIAM Journal on Applied Dynamical Systems,*21:2. [Journal] [Arxiv]*Abstract.*Quasi-steady state (QSS) reduction is a commonly used method to lower the dimension of a differential equation model of a chemical reaction network. From a mathematical perspective, QSS reduction is generally interpreted as a special type of singular perturbation reduction, but in many instances the correspondence is not worked out rigorously, and the QSS reduction may yield incorrect results. The present paper contains a thorough discussion of QSS reduction and its relation to singular perturbation reduction for the special, but important, case when the right hand side of the differential equation is linear in the variables to be eliminated. For this class we give necessary and sufficient conditions for a singular perturbation reduction (in the sense of Tikhonov and Fenichel) to exist, and to agree with QSS reduction. We then apply the general results to chemical reaction networks with non-interacting species, generalizing earlier results and methods for steady states to quasi-steady state scenarios. We provide easy-to-check graphical conditions to select parameter values yielding to singular perturbation reductions and additionally, we identify a choice of parameters for which the corresponding singular perturbation reduction agrees with the QSS reduction. Finally we consider a number of examples. Hide- Pascual-Escudero B,
**Feliu E**(2022) Local and global robustness in systems of polynomial equations.*Mathematical Methods in the Applied Sciences.*45:1, pp 359-382. [Arxiv][Journal]*Abstract.*In this work we consider systems of polynomial equations and study under what conditions the semi-algebraic set of positive real solutions is contained in a parallel translate of a coordinate hyperplane. To this end we make use of algebraic and geometric tools to relate the local and global structure of the set of positive points. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi=c, and provide a criterion to identify it. We consider the global property, namely that the whole positive part of the variety is contained in a hyperplane of the form xi=c, termed absolute concentration robustness (ACR). We show that zero sensitivity implies ACR, and identify when the two properties do not agree, via an intermediate property we term local ACR. The motivation of this work stems from the study of robustness in biochemical systems modelling the concentration of species in a reaction network, where the terms ACR and zero sensitivity are both used to this end. Here we have clarified and formalised the relation between the two approaches, and, as a consequence, we have obtained a practical criterion to decide upon (local) ACR under some mild assumptions. Hide **Feliu E***, Kaihnsa N*, de Wolff T*, Yürük O* (2022) The kinetic space of multistationarity in dual phosphorylation.*Journal of Dynamics and Differential Equations,*34, pp 825–852. [Arxiv][Journal]*Abstract.*Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. We model the dynamics of the concentrations of the proteins over time by means of a parametrized polynomial ordinary differential equation (ODE) system arising from the mass-action assumption. Since this system has three linear first integrals defined by the total amounts of the substrate and the two enzymes, we study for what parameter values the ODE system has at least two positive steady states after suitably choosing the total amounts. We employ a suite of techniques from (real) algebraic geometry, which in particular concern the study of the signs of a multivariate polynomial over the positive orthant and sums of nonnegative circuit polynomials. Hide- Torres A,
**Feliu E**(2021) Symbolic proof of bistability in reaction networks.*SIAM Journal on Applied Dynamical Systems*, 20:1, pp 1-37. [Arxiv] [Journal]*Abstract.*Deciding whether and where a system of parametrized ordinary differential equations displays bistability, that is, has at least two asymptotically stable steady states for some choice of parameters, is a hard problem. For systems modeling biochemical reaction networks, we introduce a procedure to determine, exclusively via symbolic computations, the stability of the steady states for unspecified parameter values. In particular, our approach fully determines the stability type of all steady states of a broad class of networks. To this end, we combine the Hurwitz criterion, reduction of the steady state equations to one univariate equation, and structural reductions of the reaction network. Using our method, we prove that bistability occurs in open regions in parameter space for many relevant motifs in cell signaling. Hide **Feliu E***, Rendall A D*, Wiuf C* (2020) A proof of unlimited multistability for phosphorylation cycles.*Nonlinearity*, 33:11. [Arxiv] [Journal]*Abstract.*The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n=1,2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result. Hide**Feliu E***, Kruff N*, Walcher S* (2020) Tikhonov-Fenichel reduction for parameterized critical manifolds with applications to chemical reaction networks.*Journal of Nonlinear Science.*30, pp. 1355–1380. [Arxiv] [Journal]*Abstract.*We derive a reduction formula for singularly perturbed ordinary differential equations (in the sense of Tikhonov and Fenichel) with a known parameterization of the critical manifold. No a priori assumptions concerning separation of slow and fast variables are made, or necessary. We apply the theoretical results to chemical reaction networks with mass action kinetics admitting slow and fast reactions. For some relevant classes of such systems, there exist canonical parameterizations of the variety of stationary points; hence, the theory is applicable in a natural manner. In particular, we obtain a closed form expression for the reduced system when the fast subsystem admits complex-balanced steady states. Hide- Cappelletti D*,
**Feliu E***, Wiuf C* (2020) Addition of flow reactions preserving multistationarity and bistability.*Mathematical Biosciences*, 320, 108295 [Arxiv][Journal]*Abstract.*We consider the question whether a chemical reaction network preserves the number and stability of its positive steady states upon inclusion of inflow and outflow reactions. Often a model of a reaction network is presented without inflows and outflows, while in fact some of the species might be degraded or leaked to the environment, or be synthesized or transported into the system. We provide a sufficient and easy-to-check criterion based on the stoichiometry of the reaction network alone and discuss examples from systems biology. Hide **Feliu E**(2019) On the role of algebra in models in molecular biology.*Journal of Mathematical Biology*, Perspective article. 80:4, pp 1159-1161. [Journal]- Conradi C*,
**Feliu E***, Mincheva M* (2019) On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle.*Mathematical Biosciences and Enginnering*, 17:1, pp 94–513. [Arxiv] [Journal]*Abstract.*Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study four simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parameterize the steady state and Yang's Theorem. Hide - Sáez M, Wiuf C,
**Feliu E**(2019) Nonnegative linear elimination for chemical reaction networks.*SIAM J Applied Mathematics*, 79:6, pp 2434–2455. [Arxiv][Journal]*Abstract.*We consider linear elimination of variables in steady state equations of a chemical reaction network. Particular subsets of variables corresponding to sets of so-called reactant-noninteracting species, are introduced. The steady state equations for the variables in such a set, taken together with potential linear conservation laws in the variables, define a linear system of equations. We give conditions that guarantee that the solution to this system is nonnegative, provided it is unique. The results are framed in terms of spanning forests of a particular multidigraph derived from the reaction network and thereby conditions for uniqueness and nonnegativity of a solution are derived by means of the multidigraph. Though our motivation comes from applications in systems biology, the results have general applicability in applied sciences. Hide **Feliu E**(2019) Sign-sensitivities for reaction networks: an algebraic approach.*Mathematical Biosciences and Engineering*, 16:6, pp 8195-8213. [Arxiv] [Journal]*Abstract.*This paper presents an algebraic framework to study*sign-sensitivities*for reaction networks modeled by means of systems of ordinary differential equations. Specifically, we study the sign of the derivative of the concentrations of the species in the network at steady state with respect to a small perturbation on the parameter vector. We provide a closed formula for the derivatives that accommodates common perturbations, and illustrate its form with numerous examples. We argue that, mathematically, the study of the response to the system with respect to changes in total amounts is not well posed, and that one should rather consider perturbations with respect to the initial conditions. We find a sign-based criterion to determine, without computing the sensitivities, whether the sign depends on the steady state and parameters of the system. This is based on earlier results of so-called injective networks. Finally, we address systems with multiple steady states and how to restrict the computations to stable steady states. Hide**Feliu E***, Müller S*, Regensburger G* (2019) Characterizing injectivity of classes of maps via classes of matrices.*Linear Algebra and its Applications*, 580, pp 236-261. [Journal][Arxiv]*Abstract.*We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including generalized monomial and monotonic (not necessarily continuous) maps. In fact, monotonic maps are special cases of component-wise affine maps. Further, we study compositions of maps with a matrix and other composed maps, in particular, rational functions. Our framework covers classical injectivity criteria based on mean value theorems for vector-valued maps and recent results obtained in the study of chemical reaction networks. Hide- Sadeghimanesh A,
**Feliu E**(2019) The multistationarity structure of networks with intermediates and a binomial core network.*Bulletin of Mathematical Biology,*81, pp 2428–2462. [Arxiv][Journal]*Abstract.*This work addresses whether a reaction network, taken with mass-action kinetics, is multistationary, that is, admits more than one positive steady state in some stoichiometric compatibility class. We build on previous work on the effect that removing or adding intermediates has on multistationarity, and also on methods to detect multistationarity for networks with a binomial steady state ideal. In particular, we provide a new determinant criterion to decide whether a network is multistationary, which applies when the network obtained by removing intermediates has a binomial steady state ideal. We apply this method to easily characterize which subsets of complexes are responsible for multistationarity; this is what we call the multistationarity structure of the network. We use our approach to compute the multistationarity structure of the n-site sequential distributive phosphorylation cycle for arbitrary n. Hide - Sadeghimanesh A,
**Feliu E**(2019) Groebner bases of reaction networks with intermediate species.*Advances in Applied Mathematics*, 107, pp 74-101. [Journal] [Arxiv]*Abstract.*In this work we consider the computation of Groebner bases of the steady state ideal of reaction networks equipped with mass-action kinetics. Specifically, we focus on the role of intermediate species and the relation between the extended network (with intermediate species) and the core network (without intermediate species). We show that a Groebner basis of the steady state ideal of the core network always lifts to a Groebner basis of the steady state ideal of the extended network by means of linear algebra, with a suitable choice of monomial order. As illustrated with examples, this contributes to a substantial reduction of the computation time, due mainly to the reduction in the number of variables and polynomials. We further show that if the steady state ideal of the core network is binomial, then so is the case for the extended network, as long as an extra condition is fulfilled. For standard networks, this extra condition can be visually explored from the network structure alone. Hide **Feliu E***, Helmer M* (2019) Multistationarity for Fewnomial Chemical Reaction Networks.*Bulletin of Mathematical Biology*, 81:4, pp 1089–1121. DOI: 10.1007/s11538-018-00555-z. [Arxiv][Journal]*Abstract.*We study chemical reaction networks with few chemical complexes. Under mass-action kinetics the steady states of these networks are described by fewnomial systems, that is polynomial systems defined by polynomials having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. We explore how the idea of Gale duality can be used to learn about the steady states of fewnomial networks. In particular, we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction. Hide- Sáez M,
**Feliu E**, Wiuf C (2019) Linear elimination in chemical reaction networks. In the book*Recent Advances in Differential Equations and Applications*. SEMA SIMAI Springer Series 18, pp 169-186. Eds. JL García Guirao, JA Murillo Hernández, F Periago Esparza. [Link] - Sáez M,
**Feliu E**, Wiuf C (2018) Graphical criteria for positive solutions to linear systems.*Linear Algebra and its Applications.*552, pp. 166-193. [Journal] [Arxiv]*Abstract.*We study linear systems of equations with coefficients in a generic partially ordered ring R and a unique solution, and seek conditions for the solution to be nonnegative, that is, every component of the solution is a quotient of two nonnegative elements in R. The requirement of a nonnegative solution arises typically in applications, such as in biology and ecology, where quantities of interest are concentrations and abundances. We provide novel conditions on a labeled multidigraph associated with the linear system that guarantee the solution to be nonnegative. Furthermore, we study a generalization of the first class of linear systems, where the coefficient matrix has a specific block form and provide analogous conditions for nonnegativity of the solution, similarly based on a labeled multidigraph. The latter scenario arises naturally in chemical reaction network theory, when studying full or partial parameterizations of the positive part of the steady state variety of a polynomial dynamical system in the concentrations of the molecular species. Hide **Feliu E**, Cappelletti D, Wiuf C (2018) Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States.*Mathematical Biosciences.*301, pp. 68-82. [Arxiv] [Journal]*Abstract.*We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady states, called node balanced steady states, is naturally associated with such a reaction graph. We show that complex and detailed balanced steady states are special cases of node balanced steady states by choosing appropriate reaction graphs. Further, we show that node balanced steady states have properties analogous to complex balanced steady states, such as uniqueness and asymptotical stability in each stoichiometric compatibility class. Moreover, we associate an integer, called the deficiency, to a reaction graph that gives the number of independent relations in the reaction rate constants that need to be satisfied for a positive node balanced steady state to exist. The set of reaction graphs (modulo isomorphism) is equipped with a partial order that has the complex balanced reaction graph as minimal element. We relate this order to the deficiency and to the set of reaction rate constants for which a positive node balanced steady state exists. Hide- Conradi C*,
**Feliu E***, Mincheva M*, Wiuf C* (2017) Identifying parameter regions for multistationarity.*PLOS Computational Biology.*13(10): e1005751. [Journal] [Arxiv]*Abstract.*Mathematical modeling has become an established tool for studying biological dynamics. Current applications range from building models that reproduce quantitative data to identifying models with predefined qualitative features, such as switching behavior, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce an algorithm to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The algorithm is based on a simple idea, the computation of the Brouwer degree, and creates a multivariate polynomial with parameter depending coefficients. Using algebraic techniques, the signs of the coefficients reveal parameter regions with and without multistationarity. We demonstrate the algorithm on models of gene transcription and cell signaling, and argue that the parameter constraints defining each region have biological meaningful interpretations. Hide - Marcondes de Freitas M, Wiuf C,
**Feliu E**(2017) Intermediates and Generic Convergence to Equilibria.*Bulletin of Mathematical Biology.*79:7, pp. 1662-1686. DOI: 10.1007/s11538-017-0303-4 [Arxiv] [Journal]*Abstract.*Known graphical conditions for the generic or global convergence to equilibria of the dynamical system arising from a reaction network are shown to be invariant under the so-called successive removal of intermediates, a systematic procedure to simplify the network, making the graphical conditions easier to check. Hide - Marcondes de Freitas M,
**Feliu E**, Wiuf C (2017) Intermediates, Catalysts, Persistence, and Boundary Steady States.*Journal of Mathematical Biology.*74:4, pp. 887–932. [Arxiv] [Journal]*Abstract.*For dynamical systems arising from chemical reaction networks, persistence is the property that each species concentration remains positively bounded away from zero, as long as species concentrations were all positive in the beginning. We describe two graphical procedures for simplifying reaction networks without breaking known necessary or sufficient conditions for persistence, by iteratively removing so-called intermediates and catalysts from the network. The procedures are easy to apply and, in many cases, lead to highly simplified network structures, such as monomolecular networks. For specific classes of reaction networks, we show that these conditions are equivalent to one another and, thus, necessary and sufficient for persistence. Furthermore, they can also be characterized by easily checkable strong connectivity properties of the underlying graph. In particular, this is the case for (conservative) monomolecular networks, as well as cascades of a large class of post-translational modification systems (of which the MAPK cascade and the n-site futile cycle are prominent examples). Since the aforementioned sufficient conditions for persistence preclude the existence of boundary steady states, our method also provides a graphical tool to check for that. Hide - Sáez M, Wiuf C,
**Feliu E**(2017) Graphical reduction of reaction networks by linear elimination of species.*Journal of Mathematical Biology.*74:1, pp. 195-237. [Arxiv] [Journal]*Abstract.*We give a graphically based procedure to reduce a reaction network to a smaller reaction network with fewer species after linear elimination of a set of noninteracting species. We give a description of the reduced reaction network, its kinetics and conservations laws, and explore properties of the network and its kinetics. We conclude by comparing our approach to an older similar approach by Temkin and co-workers. Finally, we apply the procedure to biological examples such as substrate mechanisms, post-translational modification systems and networks with intermediates (transient) steps. Hide - Song F, Sáez M, Wiuf C,
**Feliu E***, Soyer OS* (2016) Core signalling motif displaying multistability through multi-state enzymes.*Journal of the Royal Society Interface*, 13:121, 20160524. [Journal]*Abstract.*Bistability, and more generally multistability, is a key system dynamics feature enabling decision-making and memory in cells. Deciphering the molecular determinants of multistability is thus crucial for a better understanding of cellular pathways and their (re)engineering in synthetic biology. Here, we show that a key motif found predominantly in eukaryotic signalling systems, namely a futile signalling cycle, can display bistability when featuring a two-state kinase. We provide necessary and sufficient mathematical conditions on the kinetic parameters of this motif that guarantee the existence of multiple steady states. These conditions foster the intuition that bistability arises as a consequence of competition between the two states of the kinase. Extending from this result, we find that increasing the number of kinase states linearly translates into an increase in the number of steady states in the system. These findings reveal, to our knowledge, a new mechanism for the generation of bistability and multistability in cellular signalling systems. Further the futile cycle featuring a two-state kinase is among the smallest bistable signalling motifs. We show that multi-state kinases and the described competition-based motif are part of several natural signalling systems and thereby could enable them to implement complex information processing through multistability. These results indicate that multi-state kinases in signalling systems are readily exploited by natural evolution and could equally be used by synthetic approaches for the generation of multistable information processing systems at the cellular level. Hide **Feliu E**(2016) Sobre las soluciones positivas de sistemas de polinomios parametrizados en biología. (in spanish).*La Gaceta de la Real Sociedad Matemática Española*[Journal]*Abstract.*La evolución en el tiempo de las concentraciones de las especies químicas en una red de reacciones se modela comúnmente con un sistema de ecuaciones diferenciales polinomiales, bajo la denominada ley de acción de masas. Dichos sistemas dependen de numerosos parámetros cuyo valor es habitualmente desconocido. Los puntos de equilibrio relevantes del sistema son, así pues, las soluciones positivas de un sistema de polinomios que depende de parámetros desconocidos. Preguntas sobre el número de puntos de equilibrio positivos en función del valor de los parámetros han despertado gran interés en el análisis de redes de reacciones químicas. En este artículo introducimos el lector al formalismo matemático de la teoría de redes de reacciones y detallamos las preguntas relevantes que conciernen al número de puntos de equilibrio. Seguidamente damos algunos resultados recientes que responden parcialmente tales preguntas. El énfasis del artículo reside en presentar la estructura del sistema de polinomios bajo estudio, en resaltar las ventajas derivadas de su forma y en discutir las dificultades que nos encontramos al intentar responder las referidas preguntas. Hide- Müller S*,
**Feliu E***, Regensburger G*, Conradi C, Shiu A, Dickenstein A (2016) Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry.*Foundations of Computational Mathematics.*16:1, pp. 69-97. doi:10.1007/s10208-014-9239-3. [Arxiv] [Journal]*Abstract.*We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomials maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our results reveal the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients. Hide - Kothamanchu VB*,
**Feliu E***, Cardelli L, Soyer OS (2015) Unlimited multistability and Boolean logic in microbial signaling.*Journal of the Royal Society Interface*. 12:108, 20150234. [Journal]*Abstract.*The ability to map environmental signals onto distinct internal physiological states or programs is critical for single-celled microbes. A crucial systems dynamics feature underpinning such ability is multistability. While unlimited multistability is known to arise from multi-site phosphorylation seen in the signaling networks of eukaryotic cells, a similarly universal mechanism has not been identified in microbial signaling systems. These systems are generally known as two-component systems comprising of histidine kinase receptors and response regulator proteins engaging in phosphotransfer reactions. We develop a mathematical framework for analyzing microbial systems with multi-domain histidine kinase receptors We further prove that sharing of downstream components allows a system with n multi-domain hybrid histidine kinases to attain 3n steady states. We find that such systems, when sensing distinct signals can readily implement Boolean logic functions on these signals. Microbial cells are thus theoretically unbounded in mapping distinct environmental signals onto distinct physiological states and perform complex computations on them. These findings facilitate the understanding of natural two-component systems and allow their engineering through synthetic biology. Hide **Feliu E**, Wiuf C (2015) Finding the positive feedback loops underlying multi-stationarity.*BMC Systems Biology*, 9:22. doi:10.1186/s12918-015-0164-0. [Journal]*Abstract.*Bistability is ubiquitous in biological systems. For example, bistability is found in many reaction networks that involve the control and execution of important biological functions, such as signalling processes. Positive feedback loops, composed of species and reactions, are necessary for bistability, and generally for multi-stationarity, to occur. These loops are therefore often used to illustrate and pinpoint the parts of a multi-stationary network that are relevant (`responsible') for the observed multi-stationarity. However positive feedback loops are generally abundant in reaction networks but not all of them are important for subsequent interpretation of the network's dynamics. We present an automated procedure to determine the relevant positive feedback loops of a multi-stationary reaction network. The procedure only reports the loops that are relevant for multi-stationarity (that is, when broken multi-stationarity disappears) and not all positive feedback loops of the network. We show that the relevant positive feedback loops must be understood in the context of the network (one loop might be relevant for one network, but cannot create multi-stationarity in another). Finally, we demonstrate the procedure by applying it to several examples of signaling processes, including a ubiquitination and an apoptosis network, and to models extracted from the Biomodels database. We have developed and implemented an automated procedure to find relevant positive feedback loops in reaction networks. The results of the procedure are useful for interpretation and summary of the network's dynamics. Hide- Jovanovic G, Sheng X, Ale A,
**Feliu E**, Harrington HA, Kirk P, Wiuf C, Buck M, Stumpf MPH (2015) Phosphorelay of non-orthodox two component systems functions through a bi-molecular mechanism in vivo: the case of ArcB.*Molecular Biosystems.*11:5, pp. 1348-59. [Journal]*Abstract.*Two-component systems play a central part in bacterial signal transduction. Phosphorelay mechanisms have been linked to more robust and ultra-sensitive signalling dynamics. The molecular machinery that facilitates such a signalling is, however, only understood in outline. In particular the functional relevance of the dimerization of a non-orthodox or hybrid histidine kinase along which the phosphorelay takes place has been a subject of debate. We use a combination of molecular and genetic approaches, coupled to mathematical and statistical modelling, to demonstrate that the different possible intra- and inter-molecular mechanisms of phosphotransfer are formally non-identifiable in Escherichia coli expressing the ArcB non-orthodox histidine kinase used in anoxic redox control. In order to resolve this issue we further analyse the mathematical model in order to identify discriminatory experiments, which are then performed to address cis- and trans-phosphorelay mechanisms. The results suggest that exclusive cis- and trans-mechanisms will not be operating, instead the functional phosphorelay is likely to build around a sequence of allosteric interactions among the domain pairs in the histidine kinase. This is the first detailed mechanistic analysis of the molecular processes involved in non-orthodox two-component signalling and our results suggest strongly that dimerization facilitates more discriminatory proof-reading of external signals, via these allosteric reactions, prior to them being further processed. Hide **Feliu E**(2015) Injectivity, multiple zeros, and multistationarity in reaction networks.*Proceedings of the Royal Society A*, 471:2173. DOI: 10.1098/rspa.2014.0530. [Arxiv] [Journal]*Abstract.*Polynomial dynamical systems are widely used to model and study real phenomena. In biochemistry, they are the preferred choice for modelling the concentration of chemical species in reaction networks with mass-action kinetics. These systems are typically parameterised by many (unknown) parameters. A goal is to understand how properties of the dynamical systems depend on the parameters. Qualitative properties relating to the behaviour of a dynamical system are locally inferred from the system at steady state. Here we focus on steady states that are the positive solutions to a parameterised system of generalised polynomial equations. In recent years, methods from computational algebra have been developed to understand these solutions, but our knowledge is limited: for example, we cannot efficiently decide how many positive solutions the system has as a function of the parameters. Even deciding whether there is one or more solutions is non-trivial. We present a new method, based on so-called injectivity, to preclude or assert that multiple positive solutions exist. The results apply to generalised polynomials and variables can be restricted to the linear, parameter-independent first integrals of the dynamical system. The method has been tested in a wide range of systems. Hide- Amin M, Kothamachu VB,
**Feliu E**, Scharf BE, Porter SL, Soyer OS (2014) Phosphate sink containing two-component signaling systems as tunable threshold devices.*PLOS Computational Biology*. 10:10, e1003890. [Journal]*Abstract.*Synthetic biology aims to design de novo biological systems and reengineer existing ones. These efforts have mostly focused on transcriptional circuits, with reengineering of signaling circuits hampered by limited understanding of their systems dynamics and experimental challenges. Bacterial two-component signaling systems offer a rich diversity of sensory systems that are built around a core phosphotransfer reaction between histidine kinases and their output response regulator proteins and thus are a good target for reengineering through synthetic biology. Here, we explore the signal- response relationship arising from a specific motif found in two-component signaling. In this motif a single histidine kinase (HK) phosphotransfers reversibly to two separate output response regulator (RR) proteins. We show that, under the experimentally observed parameters from bacteria and yeast, this motif not only allows rapid signal termination, whereby one of the RRs acts as a phosphate sink towards the other RR (i.e. the output RR) but also implements a sigmoidal signal-response relationship. We identify two mathematical conditions on system parameters that are necessary for sigmoidal signal-response relationships and define key parameters that control threshold levels and sensitivity of the signal-response curve. We confirm these findings experimentally, by in vitro reconstitution of the one HK-two RR motif found in the S. meliloti chemotaxis pathway and measuring the resulting signal-response curve. We find that the level of sigmoidality in this system can be experimentally controlled by the presence of the sink RR and also through an auxiliary protein that is shown to bind to the HK (yielding Hill coefficients of above 7). These findings show that the one HK-two RR motif allows bacteria and yeast to implement tunable switch-like signal processing and provides an ideal basis for developing threshold devices for synthetic biology applications. Hide - Kothamachu VB,
**Feliu E**, Wiuf C, Cardelli L, Soyer OS (2013) Phosphorelays provide tunable signal processing capabilities for the cell.*PLOS Computational Biology*, 9:11, e1003322. [Journal] [PDF]*Abstract.*Achieving a complete understanding of cellular signal transduction requires deciphering the relation between structural and biochemical features of a signaling system and the shape of the signal-response relationship it embeds. Using explicit analytical expressions and numerical simulations, we present here this relation for four-layered phosphorelays, which are signaling systems that are ubiquitous in prokaryotes and also found in lower eukaryotes and plants. We derive an analytical expression that relates the shape of the signal-response relationship in a relay to the kinetic rates of forward, reverse phosphorylation and hydrolysis reactions. This reveals a set of mathematical conditions which, when satisfied, dictate the shape of the signal-response relationship. We find that a specific topology also observed in nature can satisfy these conditions in such a way to allow plasticity among hyperbolic and sigmoidal signal-response relationships. Particularly, the shape of the signal-response relationship of this relay topology can be tuned by altering kinetic rates and total protein levels at different parts of the relay. These findings provide an important step towards predicting response dynamics of phosphorelays, and the nature of subsequent physiological responses that they mediate, solely from topological features and few composite measurements; measuring the ratio of reverse and forward phosphorylation rate constants could be sufficient to determine the shape of the signal-response relationship the relay exhibits. Furthermore, they highlight the potential ways in which selective pressures on signal processing could have played a role in the evolution of the observed structural and biochemical characteristic in phosphorelays. Hide - Wiuf C,[PDF]
**Feliu E**(2013) Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species.*SIAM Journal on Applied Dynamical Systems*, 12 pp. 1685-1721. [Journal]*Abstract.*We present determinant criteria for the preclusion of non-degenerate multiple steady states in networks of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinetics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is derived from the determinant of the Jacobian of the species formation rate function. Using this characterization we further derive similar determinant criteria applicable to general sets of kinetics. The criteria are conceptually simple, computationally tractable and easily implemented. Our approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or non-catalytic kinetics, and also work based on graphical analysis of the interaction graph associated to the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph. Hide **Feliu E**, Wiuf C (2013) A computational method to preclude multistationarity in networks of interacting species.*Bioinformatics*, 29 pp. 2327-2334. [Journal]*Abstract.*Modeling and analysis of complex systems are important aspects of understanding systemic behavior. In the lack of detailed knowledge about a system, we often choose modeling equations out of convenience and search the (high-dimensional) parameter space randomly to learn about model properties. Qualitative modeling sidesteps the issue of choosing specific modeling equations and frees the inference from specific properties of the equations. We consider classes of ODE models arising from interactions of species/entities, such as (bio)chemical reaction networks or ecosystems. A class is defined by imposing mild assumptions on the interaction rates. In this framework, we investigate whether there can be multiple positive steady states in some ODE models in a given class. We have developed and implemented a method to decide whether any ODE model in a given class cannot have multiple steady states. The method runs efficiently on models of moderate size. We tested the method on a large set of models for gene silencing by sRNA interference and on two publicly available databases of biological models, KEGG and Biomodels. We recommend that this method is used as (a) a pre-screening step for selecting an appropriate model and (b) for investigating the robustness of non-existence of multiple steady state for a given ODE model with respect to variation in interaction rates.Hide**Feliu E**, Wiuf C (2013) Simplifying Biochemical Models With Intermediate Species.*Journal of the Royal Society Interface*, 10:87, 20130484. [Journal] [arXiv]*Abstract.*Mathematical models are increasingly being used to understand complex biochemical systems, to analyze experimental data and make predictions about unobserved quantities. However, we rarely know how robust our conclusions are with respect to the choice and uncertainties of the model. Using algebraic techniques we study systematically the effects of intermediate, or transient, species in biochemical systems and provide a simple, yet rigorous mathematical classification of all models obtained from a core model by including intermediates. Main examples include enzymatic and post-translational modification systems, where intermediates often are considered insignificant and neglected in a model, or they are not included because we are unaware of their existence. All possible models obtained from the core model are classified into a finite number of classes. Each class is defined by a mathematically simple canonical model that characterizes crucial dynamical properties, such as mono- and multistationarity and stability of steady states, of all models in the class. We show that if the core model does not have conservation laws, then the introduction of intermediates does not change the steady-state concentrations of the species in the core model, after suitable matching of parameters. Importantly, our results provide guidelines to the modeler in choosing between models and in distinguishing their properties. Further, our work provides a formal way of comparing models that share a common skeleton.Hide- Planas-Iglesias J, Bonet J, García-García J, Marín-López MA,
**Feliu E**, Oliva B (2013) Understanding protein-protein interactions using local structural features.*Journal of Molecular Biology*, 425 pp. 1210-1224. [Journal]*Abstract.*Protein-protein interactions play a relevant role among the different functions of a cell. Identifying the protein-protein interaction network of a given organism (interactome) is useful to shed light on the key molecular mechanisms within a biological system. In this work, we show the role of structural features (loops and domains) to comprehend the molecular mechanisms of protein-protein interactions. A paradox in protein-protein binding is to explain how the unbound proteins of a binary complex recognize each other among a large population within a cell and how they find their best docking interface in a short time-scale. We use interacting and non-interacting protein pairs to classify the structural features that sustain the binding (or non-binding) behaviour. Our study indicates that not only the interacting region but also the rest of the protein surface is important for the interaction fate. The interpretation of this classification suggests that the balance between favouring and disfavouring structural features determines if a pair of proteins interacts or not. Our results are in agreement with previous works and support the funnel-like intermolecular energy landscape theory that explains protein-protein interactions. We have used these features to score the likelihood of the interaction between two proteins and to develop a method for the prediction of PPIs. We have tested our method on several sets with unbalanced ratios of interactions and non-interactions to simulate real conditions, obtaining accuracies higher than 25% in the most unfavourable circumstances. Hide - Harrington H*,
**Feliu E***, Wiuf C, Stumpf MPH (2013) Cellular compartments cause multistability in biochemical reaction networks and allow cells to process more information.*Biophysical Journal*, 104:8 pp 1824-1831. [Journal] [arXiv]*Abstract.*Many biological, physical, and social interactions have a particular dependence on where they take place. In living cells, protein movement between the nucleus and cytoplasm affects cellular response (i.e., proteins must be present in the nucleus to regulate their target genes). Here we use recent developments from dynamical systems and chemical reaction network theory to identify and characterize the key-role of the spatial organization of eukaryotic cells in cellular information processing. In particular the existence of distinct compartments plays a pivotal role in whether a system is capable of multistationarity (multiple response states), and is thus directly linked to the amount of information that the signaling molecules can represent in the nucleus. Multistationarity provides a mechanism for switching between different response states in cell signaling systems and enables multiple outcomes for cellular-decision making. We combine different mathematical techniques to provide a heuristic procedure to determine if a system has the capacity for multiple steady states and to find conditions that ensure that multiple steady states cannot occur. Notably, we find that introducing species localization can alter the capacity for multistationarity and mathematically demonstrate that shuttling confers flexibility for and greater control of the emergence of an all-or-none response. Hide **Feliu E**, Wiuf C (2013) Variable elimination in post-translational modification reaction networks with mass-action kinetics.*Journal of Mathematical Biology*, 66:1 pp 281-310. [ Journal] [arXiv]*Abstract.*We define a subclass of Chemical Reaction Networks called Post-Translational Modification (PTM) systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such as system are solutions to a system of polynomial equations with as many variables as equations. Even for small systems this task is daunting. We develop a mathematical framework based on the notion of acut , which provides a procedure to reduce the number of variables in the system. We show that a cut corresponds to a connected component in the species graph and is a conservation law. Further, we provide a criterion for when all conservation laws can be derived from cuts. Hide**Feliu E**, Wiuf C (2012) Preclusion of switch behavior in reaction networks with mass-action kinetics.*Applied Mathematics and Computation*. 219:4, pp 1449-67. [Journal] [arXiv]*Abstract.*In this work we extend the characterization of injectivity via the Jacobian criterion first developed by Craciun and Feinberg for chemical reaction networks with outflow reactions to arbitrary chemical reaction networks taken with mass action kinetics. Injective chemical reaction networks do not have the capacity to admit multiple positive steady states for any rate constants and within each stoichiometric class. It is shown that a network is injective if and only if the determinant of the Jacobian of the system of ordinary differential equations associated to the network never vanishes. The determinant is a polynomial on the species concentrations and the rate constants, and its coefficients are fully determined. Previous works apply to chemical reaction networks whose stoichiometric space has maximal dimension. Here we present a direct route, independent of the dimension of the stoichiometric space which precludes at the same time the existence of degenerate steady states. Hide**Feliu E**, Wiuf C (2012) Variable elimination in chemical reaction networks with mass action kinetics.*SIAM Journal on Applied Mathematics*. 72:4 pp 959–981. [Journal] [arXiv]*Abstract.*We consider chemical reaction networks taken with mass action kinetics. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop an algebraic framework and procedure for linear elimination of variables. The procedure reduces the variables in the system to a set of "core" variables by eliminating variables corresponding to a set of non-interacting species. The steady states are parameterized algebraically by the core variables, and a graphical condition is given for when a steady state with positive core variables necessarily have all variables positive. Further, we characterize graphically the sets of eliminated variables that are constrained by a conservation law and show that this conservation law takes a specific form. Hide**Feliu E**, Wiuf C (2012) Enzyme sharing as a cause of multistationarity in signaling systems.*Journal of the Royal Society Interface*, 9:71 pp 1224-32. [Journal] [arXiv]*Abstract.*Multistationarity in biological systems is a mechanism of cellular decision making. In particular, signaling pathways regulated by protein phosphorylation display features that facilitate a variety of responses to different biological inputs. The features that lead to multistationarity are of particular interest to determine as well as the stability properties of the steady states. In this paper we determine conditions for the emergence of multistationarity in small motifs without feedback that repeatedly occur in signaling pathways. We derive an explicit mathematical relationship between the concentration of a chemical species at steady state and a conserved quantity of the system such as the total amount of substrate available. We show that the function determines the number of steady states and provides a necessary condition for a steady state to be stable, that is, to be biologically attainable. Further, we identify characteristics of the motifs that lead to multistationarity, and extend the view that multistationarity in signaling pathways arises from multisite phosphorylation. Our approach relies on mass-action kinetics and the conclusions are drawn in full generality without resorting to simulations or random generation of parameters. The approach is extensible to other systems. Hide- Knudsen M,
**Feliu E**, Wiuf C (2012) Exact Analysis of Intrinsic Qualitative Features of Phosphorelays using Mathematical Models.*Journal of Theoretical Biology*, 300 pp 7-18. [Journal] [arXiv]*Abstract.*Phosphorelays are a class of signaling mechanisms used by cells to respond to changes in their environment. Phosphorelays (of which two-component systems constitute a special case) are particularly abundant in prokaryotes and have been shown to be involved in many fundamental processes such as stress response, osmotic regulation, virulence, and chemotaxis. We develop a general model of phosphorelays extending existing models of phosphorelays and two-component systems. We analyze the model analytically under the assumption of mass-action kinetics and prove that a phosphorelay has a unique stable steady-state. Furthermore, we derive explicit functions relating stimulus to the response in any layer of a phosphorelay and show that a limited degree of ultrasensitivity (the ability to respond to changes in stimulus in a switch-like manner) in the bottom layer of a phosphorelay is an intrinsic feature which does not depend on any reaction rates or substrate amounts. On the other hand, we show how adjusting reaction rates and substrate amounts may lead to higher degrees of ultrasensitivity in intermediate layers. The explicit formulas also enable us to prove how the response changes with alterations in stimulus, kinetic parameters, and substrate amounts. Aside from providing biological insight, the formulas may also be used to avoid time-consuming simulations in numerical analyses and simulations. Hide **Feliu E**, Knudsen M, Wiuf C (2012) Signaling cascades: consequences of varying substrate and phosphatase levels.*Advances in Systems Biology*in*Advances in Experimental Medical Biology*series. Vol 736, Part 1, pp 81-94. Eds. II Goryanin, AB Goryachev. Springer. [Journal]*Abstract.*We study signaling cascades with an arbitrary number of layers of one-site phosphorylation cycles. Such cascades are abundant in nature and integrated parts of many pathways. Based on the Michaelis-Menten model of enzyme kinetics and the law of mass-action, we derive explicit analytic expressions for how the steady state concentrations and the total amounts of substrates, kinase and phosphatates depend on each other. In particular, we use these to study how the responses (the activated substrates) vary as a function of the available amounts of substrates, kinase and phosphatases. Our results provide insight into how crosstalk and external regulation of cascades effect the cascade response. Hide**Feliu E**, Knudsen M, Andersen LN, Wiuf C (2012) An Algebraic Approach to Signaling Cascades with n Layers.*Bulletin of Mathematical Biology*, 74:1, pp 45-72. [Journal][arXiv]*Abstract.*Posttranslational modification of proteins is key in transmission of signals in cells. Many signaling pathways contain several layers of modification cycles that mediate and change the signal through the pathway. Here, we study a simple signaling cascade consisting of n layers of modification cycles, such that the modified protein of one layer acts as modifier in the next layer. Assuming mass-action kinetics and taking the formation of intermediate complexes into account, we show that the steady states are solutions to a polynomial in one variable, and in fact that there is exactly one steady state for any given total amounts of substrates and enzymes. We demonstrate that many steady state concentrations are related through rational functions, which can be found recursively. For example, stimulus-response curves arise as inverse functions to explicit rational functions. We show that the stimulus-response curves of the modified substrates are shifted to the left as we move down the cascade. Further, our approach allows us to study enzyme competition, sequestration and how the steady state changes in response to changes in the total amount of substrates. Our approach is essentially algebraic and follows recent trends in the study of posttranslational modification systems. Hide**Feliu E**, Aloy P, Oliva B (2011) On the analysis of protein-protein interactions via knowledge-based potentials for the prediction of protein-protein docking.*Protein Science*, 20:3, pp 529-541. [Journal]*Abstract.*Development of effective methods to screen binary interactions obtained by rigid-body protein-protein docking is key for structure prediction of complexes and for elucidating physico-chemical principles of protein-protein binding. We have derived empirical knowledge-based potential functions for selecting rigid-body docking poses. These potentials include the energetic component that provides the residues with a particular secondary structure and surface accessibility. These scoring functions have been tested on a state-of-art benchmark dataset and on a decoy dataset of permanent interactions. Our results were compared to a residue-pair potential scoring function (RPScore) and an atomic-detailed scoring function (Zrank). We have combined knowledge-based potentials to score protein-protein poses of decoys of complexes classified either as transient or as permanent protein-protein interactions. Being defined from residue-pair statistical potentials and not requiring of an atomic level description, our method surpassed Zrank for scoring rigid-docking decoys where the unbound partners of an interaction have to endure conformational changes upon binding. However, when only moderate conformational changes are required (in rigid docking) or when the right conformational changes are ensured (in flexible docking), Zrank is the most successful scoring function. Finally, our study suggests that the physico-chemical properties necessary for the binding are allocated on the proteins previous to its binding and with independence of the partner. This information is encoded at the residue level and could be easily incorporated in the initial grid scoring for Fast Fourier Transform rigid-body docking methods. Hide**Feliu E**, Oliva B (2010) How different from random are docking predictions when ranked by scoring functions?*Proteins*, 78:16, pp 3376-85. [Journal]*Abstract.*Docking algorithms predict the structure of protein–protein interactions. They sample the orientation of two unbound proteins to produce various predictions about their interactions, followed by a scoring step to rank the predictions. We present a statistical assessment of scoring functions used to rank near-native orientations, applying our statistical analysis to a benchmark dataset of decoys of protein–protein complexes and assessing the statistical significance of the outcome in the CAPRI scoring experiment. A P-value was assigned that depended on the number of near-native structures in the sampling. We studied the effect of filtering out redundant structures and tested the use of pair-potentials derived using ZDock and ZRank. Our results show that for many targets, it is not possible to determine when a successful re-ranking performed by scoring functions results merely from random choice. This analysis reveals that changes should be made in the design of the CAPRI scoring experiment. We propose including the statistical assessment in this experiment either at the preprocessing or the evaluation step. Hide- Burgos Gil JI*,
**Feliu E***(2012) Higher arithmetic Chow groups.*Commentarii Mathematici Helvetici*, 87:3 pp 521-587. [Journal] [arXiv]*Abstract.*We give a new construction of*higher arithmetic Chow groups*suitable for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analog, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow group of Burgos. Our new construction is shown to be functorial and is endowed with a product structure, which is commutative and associative. Hide **Feliu E**(2011) On uniqueness of characteristic classes.*Journal of Pure and Applied Algebra*, 215:6, pp 1223-42. [Journal] [arXiv]*Abstract.*We give an axiomatic characterization of maps from algebraic K-theory. The results apply to a class of maps from algebraic K-theory to any suitable cohomology theory or to algebraic K-theory, which includes all group morphisms. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Lambda and Adams operations on higher algebraic K-theory. We show that the Adams operations defined by Grayson agree with the ones defined by Gillet and Soule. Hide- Burgos Gil JI*,
**Feliu E***, Takeda Y (2011) On Goncharov's regulator and higher arithmetic Chow groups.*International Mathematics Research Notices*, 2011:1, pp 40-73. [Journal][arXiv]*Abstract.*In this paper we show that the regulator defined by Goncharov from higher algebraic Chow groups to Deligne-Beilinson cohomology agrees with Beilinson's regulator. We give a direct comparison of Goncharov's regulator to the construction given by Burgos and Feliu. As a consequence, we show that the higher arithmetic Chow groups defined by Goncharov agree, for all projective arithmetic varieties over an arithmetic field, with the ones defined by Burgos and Feliu. Hide **Feliu E**(2010) A chain morphism for Adams operations on rational algebraic K-theory.*Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology,*5:2, pp 349-402. [Journal][arXiv]*Abstract.*For any regular noetherian scheme X and every k>0, we define a chain morphism between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by Q. These morphisms induce in homology the Adams operations defined by Gillet and Soulé or the ones defined by Grayson.Hide**Feliu E**(2010) Adams operations on higher arithmetic K-theory.*Publications of the Research Institute for Mathematical Sciences*, 46:1, pp. 115-169. [Journal][arXiv]*Abstract.*We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The deﬁnition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soulé, by means of the homotopy groups of the homotopy ﬁber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The deﬁnition relies on the chain morphism representing Adams operations in higher algebraic K-theory given previously by the author. In this paper it is shown that this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang.Hide- Planas-Iglesias J, Bonet J, Marín-López MA,
**Feliu E**, Gursoy A, Oliva B (2012) Structural Bioinformatics of Proteins: predicting the tertiary and quaternary structure of proteins from sequence. Chapter 10 in*Protein-Protein Interactions, Computational and Experimental Tools*, Intech. Edited by Weibo Cai. [Online access] **Feliu E**(2004) Function theory of higher logarithms. Arbeitsgemeinschaft mit Aktuellem Thema: Polylogarithms.*Oberwolfach Report*, 1:4, pp 2543-45. [Journal]**On higher arithmetic intersection theory,**2007.

#### Book chapters

#### Other scientific contributions

#### PhD Thesis

Advisor: José Ignacio Burgos Gil.[Fulltext][Introduction]

With a qualification of
Excellent Cum Laude, with European mention.

Awarded with the **Josep
Teixidor Prize** from the *Institut d'Estudis Catalans*, Sant Jordi Awards, 2010.