Elisenda Feliu

Enzyme-sharing as a cause of multi-stationarity in signalling systems

Multi-stationarity in biological systems is a mechanism of cellular decision-making. In particular, signalling pathways regulated by protein phosphorylation display features that facilitate a variety of responses to different biological inputs. The features that lead to multi-stationarity 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 multi-stationarity in small motifs without feedback that repeatedly occur in signalling pathways. We derive an explicit mathematical relationship φ 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 φ 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 multi-stationarity, and extend the view that multi-stationarity in signalling pathways arises from multi-site 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.

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial 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 work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as 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.

An algebraic approach to signaling cascades with N layers.

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 that 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.

Simplifying biochemical models with intermediate species

Mathematical models are increasingly being used to understand complex biochemical systems, to analyse 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 modeller in choosing between models and in distinguishing their properties. Further, our work provides a formal way of comparing models that share a common skeleton.

Understanding Protein–Protein Interactions Using Local Structural Features

Protein-protein interactions (PPIs) play a relevant role among the different functions of a cell. Identifying the PPI 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 PPIs. 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 timescale. We use interacting and non-interacting protein pairs to classify the structural features that sustain the binding (or non-binding) behavior. Our study indicates that not only the interacting region but also the rest of the protein surface are important for the interaction fate. The interpretation of this classification suggests that the balance between favoring and disfavoring 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 PPIs. 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 unfavorable circumstances.

Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

Cellular Compartments Cause Multistability and Allow Cells to Process More Information

Many biological, physical, and social interactions have a particular dependence on where they take place; e.g., in living cells, protein movement between the nucleus and cytoplasm affects cellular responses (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 find conditions that ensure that multiple steady states cannot occur. Notably, we find that introducing species localization can alter the capacity for multistationarity, and we mathematically demonstrate that shuttling confers flexibility for and greater control of the emergence of an all-or-nothing response of a cell.

Variable Elimination in Chemical Reaction Networks with Mass-Action Kinetics

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 noninteracting species. The steady states are parameterized algebraically by the core variables, and a graphical condition is given that ensures that a steady state with positive core variables necessarily takes positive values for all variables. 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.

On the analysis of protein–protein interactions via knowledge‐based potentials for the prediction of protein–protein docking

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 physicochemical 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 with 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 physicochemical 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.

How different from random are docking predictions when ranked by scoring functions

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 Critical Assessment of PRedicted Interactions (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 reranking 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. Proteins 2010.